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A with multiplication defined by (A ® a)(fj, ® b) = \n 0 (Xb + pa + ab) ) = w (a) > { eW = W / We = U > (a)) by Proposition 2.1.8. Hence, T a is included in J > (0) (in a natural sense). Again the desired equation is an immediate consequence of either side of equation (4) of Corollary 3.3.4. Hence, (e) holds under hypothesis ( 1 ). Next we will establish (e) under the alternative hypotheses that either /(0) = 3 or 0 ^ S p (a ) holds. A t the same time we will establish the first sentence o f (d). For a £ A and any / € T a, let T be a suitable contour for (a, f ) that has winding number 0 around zero if zero is not in Sp(a). For any A G T we can write ( A - a ) " 1 = A " 1 - A " 1 (A - 1 a)". (7) Thus we obtain ))). □ y / y v and >p: X / X v —> y / y v induced by ip always ( have a closed graph / are continuous ). (c) Let Z be a ( normed linear space / Banach space), and let ip-.y —> Z be a linear map. Then Z^,0{p is the closure of ip{yv ). Furthermore, ip o ip ( has a closed graph / is continuous ) if and only ifip{yv ) is {0 }. Ifipoipis continuous, then there is a constant M independent of Z and ip satisfying ) without altering the inclusion. We postpone the opposite inclusion until after we have proved the other results. If ip o ip has a closed graph, then Z ^0ip = {0 } implies ip(yv ) = {0 } by the inclusion we just noted. Suppose, conversely, that ip{yv ) vanishes. Then ip oip can be factored through the continuous map (p of conclusion (b): ip o ip = ip o ip where xp(y + y^) = ip(y) for all y € y. Thus ip o (a)|| for all a 6 A. Hence (a) holds. (b): O f course B m is the Brown-McCoy or strong radical of B. Let M be a maximal modular ideal of B. Then, B / M has an identity 1 which is in the center and satisfies p b /m ( I) = 1- Theorem 2.2.14 shows that B /M is a spectral semi-normed algebra. Consider the map is an ideal by Proposition 6.1.8(d), a' is a norm on B/M. Let b € B^, and e > 0 be arbitrary. Then there is an a € A satisfying ||a|| + ||(6 - y?(a)) -I- M\\ < ||c‘! 1|6—<^(a)|| < e. Since e > 0 was arbitrary, we conclude ( / ( b + M ) = 0 and hence b G M . Thus, Bv is included in M . Since M was an arbitrary maximal modular ideal, we conclude Bv C Bm ;c): If B is strongly semisimple, then Bv = {0 } by (b). But this is exact ly the assertion that ip has a closed graph. □ Cc-'M itions on the Target Algebra The surjectivity of the homomorphism in Theorem 6.1.3 severely limits the application of that theorem. There are several cases where the hypoth esis of surjectivity can be relaxed. We give two. B from another Ba nach algebra A is continuous if and only if no element b 6 B satisfies r r = i»"s = ), S 2n(a i,a 2, . . . ,a 2n) is in A j . Therefore A / A j satisfies the polyno mial identity S2n. (c) => (b): Obvious. (b) => (a): Theorem 7.1.16, particularly (10), (11) and the definition of S, establishes this. O Here is a very special case. Theorem 1.1.20 describes the case in which a Banach algebra B is a full matrix algebra over some subalgebra A which is it3e!f a unital algebra. If A is commutative we easily show that B has a H'.’ Micient family of Ti-dimcnsional representations. In fact we only need to assume that B is a unital spectral subalgebra of M n(A ). Let tp\B —► Mn(A ) be an isomorphism onto a spectral subalgebra. Let $ = {<^7 : 7 € 1'a } where e e'aments 709 single vUued extension property 296 singly generated algebra 318 singularity set o f ip 576 socle e^l ^ : f rlgh 671 soft she if 648 spectra! mapping theorem 199 rcrm 212 ’ ius p fo) 196, 208 ret olution o f S 278 se ni-norm 212, 226, 252, 687 examples 188 stability o f 219 el.kxe 244 su ^algebra 245 examples 190 ay ithesis 434, 640 spec*T’’ * • Sp(a) o f a 196 be andary o f 253 continuity o f 222, 362 spectrum o f W C L °°(G ) 434 spectrum o f an algebra 304 sphere, unit 14 square roots in Banach algebras 360 spectral theory o f compact operators 289 SR-algebra 210 stalk 6?6 *tandarc ideal 372
V A, fj, G C; o, b 6 A .
(16)
Clearly the map a >—> 0 © a embeds A isomorphically into this algebra. Furthermore 1 © 0 is an identity element. W e can partially alleviate the
1- I . i l
Norms and Semi-norms on Algebras
19
awkwardness o f this construction by the following device, which was already used by Segal [1941], 1.1.11 D e fin itio n Let A be an algebra. If A is not unital, denote the algebra described above by .A1. If A is unital, denote the algebra A itself by A 1. We will call A 1 the unitization of A . The advantages of this convention on the meaning o f A 1 will begin to become apparent when we discuss the spectrum. From now on we will always consider A iis h subset of A 1 under the (imbedding a m 0 $ a. Wc will also suppress the direct sum notation ® and will usually denote the multiplicative identity of A 1 by 1 whether or not A is unital. Furthermore, for any A G C, the element A1 of A 1 will be denoted simply by A. Even when calculating in an algebra A , which may not be unital, it is frequently convenient to use notation which must be interpreted in A 1. For instance, the expression a(X + b )(p + c) represents an element in A . We use it since it is easier to understand than Aiia + Aac. + fj.ab + abc. Similarly we will routinely denote the smallest ideal o f an algebra A containing an element a o f A by A l a A l . We remark that if X is an ideal o f A , then it is also an ideal of A 1 since A 11 A 1 C 1 holds. If (A ,cr) is a nonunital semi-normed algebra, there are several ways of extending the semi-norm cr to a semi-norm cr1 on A 1. Since different extensions are useful under different circumstances, we make no permanent choice. One method of extending cr which always works, but is frequently unnatural, is to define cr^A + o) = )A| + cr(o)
V A + o g A 1.
(17)
W<» restrict attention to algebra norms in the rest of this discussion. 1.1.12 Definition Let || ■|| be an algebra norm on an algebra A. (a) If || • || satisfies ||a|| = sup{||o6||, ||6a|| : b G A i } for all o G A , then it is called a regular norm. (!:;) If A is nonunital, define the unitization ||■ ||i o f || • |j on A 1 by ||A + a ||1 = |A| + ||a||
V A + a e . 4 1.
(c) If A is nonunital and || ■ || is regular, define the regular unitization j| ■||« of || ■|| on ^t1 by ||A + fl||ft ~ sup{ ||A6 -j- ab||, ||A6 -i- 6a|| : fe G A i }
V A + a 6 A *.
A number of authors have defined an analogue of || ■||fi by ||A + a||'ft = su p {\\Xb + ah| : b G A i } . However, if there is a right identity j in A (but no left identity), then the equation ||— 1 + j\\'R = 0 shows that ||• ||'ft is not a norm on A 1. Our definition avoids this problem.
20
1: Normed Algebras and Examples
1.1.13
1.1.13 Proposition Let || • || be an algebra norm on an algebra A. (a ) I f A is unital, then || • || is regular if and only i f it satisfies ||1 || = 1 . (b) I f A is nonunital, ||• |h is a regular norm on A 1 which extends || • || on A. and any rex/ular norm |||• ||| on A 1 which extends ||■|| on A satisfies: |||A -f- ajj| < ||A 4- a||i
V A 4- a G A^.
(c) I f A is nonunital and ||■|| is regular, ||• ||« is a regular norm on A 1 which extends ||• || on A , and any regular norm ||| • ||| on A 1 which extends || ■|| on A satisfies: ||A+a||;j < jjjA + aj|| < ||A 4- a||i < (1 4- 2e)|| A 4- a||#
V A 4- a G A 1.
Proof (a ) and (b): Obvious. (c): Any A + o G A 1 and b e A i satisfy ||A6 4- o 6|| = |||A6 4- ob||| < 111A 4- o||| |||b||| < |||A 4- a|11 and the similar inequality with b on the left. This gives the first inequality and the second has already been noted. In the final inequality, e is the base of the natural logarithms. Since the proof is easy with the numerical range techniques of Section 2.6, we postpone it until just after Theorem 2.6.4. □ Locally Convex Spaces Besides semi-normed linear spaces, there is another class of semi-topolog ical linear spaces that we will frequently meet. These are the locally convex semi-topological linear spaces. W e will review their theory briefly and in formally. For proofs and more details, the reader should consult a book on functional analysis or one of the many books on topological linear spaces, e.g., Kelley and Namioka et al. [1963], Robertson and Robertson [1964], Kothe [1971] and Schaefer [1971]. 1.1.14 D e fin itio n A locally convex semi-topological linear space is a semitopological linear space X that satisfies one o f the following two equivalent properties: (a) There is a neighborhood base U. at zero consisting o f balanced convex subsets. (b) There is a collection P of semi-norms on X such that the topology of X is the weakest topology in which each of these semi-norms is continuous. Given condition (a), condition (b) follows by constructing the Minkowski functional ou of each V contained in U , where cry is defined by <Ju(x) = inf{< G R + : x G tU },
V lG ^ .
Then P may be chosen as {a y : U G U ). Conversely, if the family P of semi-norms is given, one can take U to be the set of all subsets U F,n = {x G X :
)
< n l,
a G F },
Norms and Semi-norms on Algebras
1.1.15
21
where F ranges over finite subsets of P and n ranges over N. These state ments require some checking, which we omit. A standard form o f the Hahn - 3anach theorem applied to the second definition shows that there are one ugh continuous linear functionals on X to separate the cosets o f the subspsce which is the closure of zero. This fact is more familiar in the case of locally convex topological linear spaces (i.e., locally convex semitopological linear spaces satisfying the Hausdorff separation axiom), where it asserts that there are enough continuous linear functionals to separate points. Finally, we mention that a locally convex topological linear space is metrizable if and only if the set P in condition (b) can be chosen as a countable set. In this case, the metric may be chosen as OO d (x ,y ) =
2~n m in {l,
V *, y € X ,
(18)
n —1
where {
22
1: Normed Algebras and Examples
1.1.16
We denote ( X ’ ) ' by A*** and call it the double dual of X . The natural map k or k,v of X into its double dual is defined by evaluation: k (x )(w ) = w (x )
V x 6 A ;u > G A * .
The weakest topology on X relative to which each element of A * is continuous will be called the weak topology or the A * -topology. The weakest topology on X * relative to which each element of n (X ) is continuous will be called the weak*-topology or the A -topology. Note that the Banach space dual A * is always a Banach space, whether or not A is complete. The natural map k is always an isometric linear injection. It is also easy to see that the weak and weak* topologies are locally convex. The Algebra o f Linear Maps The linear space (under pointwise linear operations) of all linear maps of one linear space A into another y (normed or not) will be denoted by C (X , 3>). If y equals A , we simplify this to £ ( A ). Then £ ( A ) is an algebra when multiplication is composition of maps. Unless the contrary is specif ically stated, all subalgebras of C (X ) will be given these same algebraic operations. We state the next definition formally because o f its importance in what follows. 1.1.16 D e fin itio n For any normed linear spaces A and y the subspace of £ (A , y ) consisting of continuous linear maps will be denoted by B (A , 30 and considered as a normed linear space under the operator norm ||T|| = sup{||Tx|| : x € X i}
V T G B ( A ’).
(19)
Whenever the linear spaces are obvious from context, we will simply call elements o f B ( X , y ) operators. Topological statements about B (X , y ) always refer to the norm topol ogy unless another topology is specifically mentioned. The strong operator topology on B (X , y ) is the topology o f pointwise convergence with the norm topology on y . The weak operator topology on B ( X , y ) is the topology of pointwise convergence when y carries its weak topology. When y equals A , we write B(X) for B ( X , y ) and consider it as a normed algebra under composition as multiplication. Many readers will already know that B( A ) is a Banach algebra if A is a Banach space, and that B(X) is a topological algebra under both the strong operator topology and the weak operator topology. In any case, these statements are easily verified. Observe that B( A , y) is complete if y is complete (whether or not X is complete).
Norms and Semi-norms 011 Algebras
1.1.17
23
The Ideal o f Finite-rank Operators 1.1.17 Definition Let W , X , y and Z be normed linear spaces. For elements y & y and u> G X *, we define an operator y ® u> € B (X , y ) by (y ® w )(z ) = u>(z)y
'i z G X .
( 20 )
We denote the linear span of all such operators by B f (X , ^ ) and call oper ators in this space finite-rank operators. We denote B jr(X , X ) by B p (X ). The tensor product notation (® ) will be explained in Section 1.10. For arbitrary S G B (W , * ) , T G B (y , Z ) , y G y , u> G * * , z G Z and r G y it is easy to check the formulas (z ® r)(j/g> u>)
=
r(y )z ® u >
T(yg>u>)
=
T(y)g>u>
(y ® u ))S
=
yg> S *(w ).
(21)
where z ® r belongs to B f W , Z ) , z ® u> and T (y ) ® w belong to B f ( X , Z ) and y ® 5 ” (ta;) belongs to B f (W , > ’)• Also the dual map o f y® w G B f ( X , y ) is given by (y ® u ) m= u ® /c(y) e BF ( y * , x * ) . Now suppose IV, and Z fill equal X in the above formulas. Then these formulas show that B f ( X ) is an ideal of B (X ). We prove more. 1.1.18 Proposition Let X be a normed linear space. The set o f operators in B (X ) with finite-dimensional range is the set n B f (X ) -
: n G N; i j G X\ u)j G X * }
(22)
j= i of finite-rank operators. It is an ideal of B {X ) which is included in every non-zero ideal. Proof Clearly every operator in B f ( X ) has finite-dimensional range. Con versely, if T G B (X ) has finite-dimensional range, let { x i , x 2, ■. ■,x n} be a basis for its range. Use the Hahn Banach theorem to extend linear func . ■. ,wn} o f X * satisfying tionals on this range space to elements U j(x k) = <5jfc. Then
n T =
Xj ® T * (u>j) 3= 1
is the desired expansion. Suppose J is a non-zero ideal of B (X ). Then there is a non-zero operator T e l and a vector z G X satisfying T z ^ 0. By the Hahn-Banach theorem
24
1 : Normed Algebras and Examples
1.2.1
there is a continuous linear functional r 6 X ' satisfying t ( T z ) = 1. Hence for any x G X and any u> £ X * we can write x ® u> as x ® u> — (x ® t ) ( T z ® u>) = (x ® t ) T ( z ® w) which proves that x ® w belongs to the ideal I .
□
This proposition explains why we call B f ( X ) the ideal o f finite-rank operators. Note that the equations in (21) show that 8 f ( X , y ) is a type o f generalized ideal in the complex of spaces B ( X , y ) as X and y vary. The ideal B x ( X ) of compact operators will be defined and discussed in §1.7.7.
1.2
D o u ble Centralizers and Extensions
A double centralizer (sometimes called a double multiplier) of an algebra A is a pair of maps of A into itself. The set of double centralizers of A forms a unital algebra T>(A) under natural operations. We will begin by introducing the regular representation which defines a homomorphism of the original algebra onto an ideal in T>(A). The mapping is surjective if and only if A is unital and, in this case, it is actually an isomorphism. In the situations o f most interest, this homomorphism is injective and one often thinks of the original algebra A as a subalgebra of the double centralizer algebra 'D (A ). In many circumstances this embedding o f A into T>(A) is a more natural way o f adding an identity to a nonunital algebra. (A fter Proposition 1.2.6, we will mention three classes o f examples which are given in more detail later in this chapter.) Moreover, T>(A) is an example o f an extension of the algebra A : that is, there is a short exact sequence { 0} -* A -
V (A ) -
V (A )/ A -
{ 0 }.
It turns out that this short exact sequence is universal for the algebra A in the sense that ai] short exact sequences {0 } -
A -
B -
C-
{0 }
can be constructed from this one and a naturally defined homomorphism o f C into T>(A)/A- When we think o f B in such a short exact sequence as being constructed from the “simpler” algebras A and C, we call it an extension. Thus, this is a satisfying way of constructing and classifying extensions o f A . We explore all these ideas in this section starting with the regular representation. The Regular Representations 1.2.1 Definition Let A be an algebra. For each o g A , let L a and R a be the linear maps in C (A ) defined by L 0( 6) = ab
and
Ra{b) = ba
V b 6 A.
( 1)
1.2.1
Double Centralizers and Extensions
25
W e call the map L (i.e., a ►-» L a) the left regular representation o f A and the map R (i.e., a h-> R a) the right regular representation o f A . The kernel A l a = {a € A : ab = 0 V b € -4} of L is called the left annihilator o f A and the kernel A h a = {a G A : ba = 0 V 6 G A } of R is called the right annihilator o f A . The intersection A a o f the left and right annihilators o f A is called the annihilator o f A ■ The extended left and right regular representations are the maps L 1 and R 1 of A into L ( A 1) defined by L\(Xb) v R^(X + b)
=
Ao + ab
=
Xa + ba
. V a e A , X + b e A 1.
( 2)
Note that the left and right regular and extended regular representations are bounded if the original algebra is normed. In that case, we give them their operator norms, using the norm ||A + a|| = |A| + ||a||
V A G C; a e A
(3)
for A 1 if A is nonunital. Then it is easy to see H I = ll^ ll = P i l l
V a G A.
(4)
In particular, L 1 and R 1 axe injective. When A is noncommutative, the right regular and extended right regular representations are anti-representar tions rather than representations in the general terminology introduced later, but the present usage is well established. It is of some interest that the extended left regular representation was used by Charles Saunders Peirce [1881] to prove that every finite-dimensional algebra could be represented by matrices. See Thomas Hawkins [1972] for remarks on the context in which this proof was given. Note that the annihilator ideals satisfy ( A l a )2 = ( A r a )2 = ( A a )2 = { 0 }
(5)
and hence the annihilator ideals themselves are { 0 } if A is semiprime (Defi nition 4.4.1 below) and o fo rtio ri if A is semisimple (Definition 4.3.1 below). O f course, they vanish if A is unital or even approximately unital (Defi nition 5.1.1 below). Following Wang [1961], Ronald Larson [1971] says a Banach algebra A is without order if A l a — {0 } or A r a = {0 }. This ungainly terminology is more common in commutative algebras. Cent'alizers When the annihilator ideal is zero, there is another more natural way of embedding A into a unital algebra. The following construction was first explicitly studied by Gerhard P. Hochschild [1947]. It was introduced into analysis by Barry E. Johnson [1964a], [1964b], [1966], who was apparently
26
1: Normed Algebras and Examples
1.2.2
unaware o f Hochschild’s work and abstracted the idea from the work of James G. Wendel [1952] and Jeffery D. Weston [I960) (c/. our remarks below on Ju-kwei Wang’s [1961] study of multipliers). See also Sigurdur Helgason [1956], John Dauns [1969] and Dauns and Karl H. Hofmann [1968], [1969].
1 . 2 .2 D e fin itio n Let A be an algebra. A ( left centralizer / right centralizer ) o f A is an element ( L / R ) G C (A ) satisfying ( L(a b ) = L (a )b / R (ab) = aR(b) )
V a ,6 e A
(6)
A double centralizer of A is a pair (L , R ) where L is a left centralizer, R is a right centralizer and together they satisfy aL(b ) — R (a)b
V o, b G A .
(7)
The algebra T>(A) o f double centralizers of A is the set o f double centralizers with pointwise linear operations and with multiplication defined by ( L u R l ) ( L 2 , R 2) = ( L l L 2 , R 2R l )
( 8)
for all ( L \ , R i) and ( L 2, R 2) in V (A ). If .4 is a normed algebra, then the set V b (A ) — { ( L , R ) £ T>(A) : L , R G B(^4)} is called the algebra o f bounded double centralizers and is given the norm *
||(L, *)|| = max{||Z||, p H }
V (L , R ) g V B (A ),
(9 )
w here ||I>|| and ||i?|| are the operator norms of L and R , respectively. A ( left / rig h t) centralizer is simply a map in the commutant ( (R a ) ' / ( L a ) ' ) of the { right / l e f t ) regular representation in C (A ). As mentioned above, this definition of double centralizers is due to Hochschild [1947]. Johnson [1964a] considered any pair (L, R ) of maps in C (A ) which satisfies equation (7). W e will show in Theorem 1.2.4 below that such a pair is a double centralizer in our terminology if both the one-sided annihilator ideals are zero. Therefore, our definition agrees with Johnson’s definition in this case, which is the most frequently considered case anyway. (Note that the identities ( A l a )2 = ( A r a )2 = {0 } imply that any semisimple or even any semiprime algebra (as defined in Chapter 4 below) satisfies this condition.) W lien A l a and A r a are not both zero (particularly if their intersection A a is zero), the definition given here is more convenient. It is easy to check that V ( A ) is an algebra under the stated operations and that T>b (A ) is a subalgebra when it is defined. Clearly (/, I ) is an iden tity element for V ( A ), where I is the identity map in C (A ). Moreover the map a *-* ( L Q, Ra) is a homomorphism (called the regular homomorphism)
Double Centralizers and Extensions
:.2 ,3
27
of A into T>(A) which maps into T>b(A ) when A is normed. The kernel of the regular homomorphism is obviously the annihilator ideal A l a O A r a = A a of A . I f A is unital, then for all o G A and all ( L , R ) G T>(A) we have L (a ) = 1L (a ) = R ( l ) a = L r ( i ) ( o ) and R (a ) = R (a )l = a L ( 1) = R m ^ (a ). Note also that L ( 1 ) = 1Z/(1) = /£(1) 1 = R ( 1). Thus the regular homomor phism is surjective if and only if A is unital. If A is a normed algebra, it is simple to check that T>B (A ) is a normed algebra under the norm given in the definition. O f course the identity element o f T>b (A ) satisfies ||1 || = ||(/, / )|| = 1 . For a ll 0 , b G A and ( L , R ) G T>(A), we have L aL (b )
= = = =
R R a(b) L L a(b) R aR{b)
aL(b) R(ba) L(ab) R (b)a
= = = =
R (a )b bR (a) L (a )b bL(a )
= = = =
LR(a)b, RR(a)b, LL(a)b, R[j(a)b.
Therefore the image of the regular homomorphism is an ideal in V ( A ) and we have ( L , R ) ( L a, R a)
=
(^L(a)i R l ( u))
(^fl> R a )(L , R )
=
(^H(a)) RR(a))
V (L ,R )e V (A ); a e A .
(11)
If the annihilator ideal is zero and A is considered as embedded in T>(A) ( via the regular homomorphism), then these relations can be written as (L , R )a = L (a ), a (L , R ) = R (a )
V (L , R ) g V ( A ); a e A .
(12)
We will now state some of these results formally and add a few more. 1.2.3 P roposition Let A be an algebra. (a) T>(A) is a unital algebra. (b) The regular homomorphism o f A into T>(A) has the annihilator ideal A a as kernel and its range is an ideal. It is surjective if and only i f A is unital, in which case it is an isomorphism. (c) I f either A 2 = A or the annihilator ideal A a zero, then the fo l lowing conditions on (L , R ) G V { A ) are equivalent: (c i) (L , R ) belongs to the center o f V ( A ) . (c2) L = R, (C3 ) L and R are both left and right centralizers. (d ) I f the annihilator ideal is zero, then A is commutative if and only if T>(A) is commutative. Proof It only remains to show (c) and (d). Under the hypothesis of (d) the regular homomorphism is injective, so A is commutative if D ( A ) is conur„utative. Th e implication (C 3 ) => (c j) shows the converse. Hence it remains only to prove the equivalence o f the conditions in (c).
28
1: Normed Algebras and Examples
1.2.3
First we will show, without any restriction on A , that either (c i) or (c2) implies (C 3 ) . The second of these implications is immediate from the definitions. If (c i) holds, then any o, b G A satisfy L(ab) — L L a(b) = L aL (b ) = a L(b ) and R (ab) = R R b(a) = R b R (a ) — R(a)b. Hence ( c j implies (C3 ). Next we will show that (C3 ) implies (c i) and (c2) under either o f the hypotheses of (c). Suppose (C 3 ) holds. Then any o, b G A and any left centralizer V satisfy L L '(a b ) — L (L '(a )b ) — L '(a )L (b ) — L '(a L (b )) = L 'L (a b ). A similar calculation for right centralizers shows that (C 3 ) implies (c j) when A satisfies A 2 = A . Also any o, b G A and any double centralizer (L ', R !) satisfy the two relations b L L ‘ (a ) = L (b L '(a )) = L(R ?(b)a) = R , (b )L (a ) = b L 'L (a ) L L '(a )b = L '(a )L (b ) = L '(a L (b )) = L '(L (a )b ) = L 'L {a )b . A similar calculation for R >R shows that (c3) implies ( c i ) when A satisfies A a = {0 }. I f A a = { 0 } and (C3 ) are both true, then the identities L (a )b = L(ab) = aL (b) = R (a )b and bL(a) — R (b )a = R(ba) = b R (a) for all o, b G A imply (C2 ). I f A 2 = A and (C3 ) both hold, then the identity L (ab) = aL (b ) = i2(d)6 = R (ab) for all a, b G A implies (c2). O A linear operator T = L = R G C (A 1) satisfying condition (c) in the above theorem is an example o f a multiplier. For any algebra A , we define T € C (A ) to be a multiplier if it satisfies T (a )b = aT{b)
V a, b G A .
I f the ( left / right ) annihilator ideal of A is zero, then the calculation ( cT(ab) = T (c)a b = cT (a )b / T (a b )c = ab T (c) = a T (b )c ) for all c G A shows that T is a ( left / right ) centralizer. Hence Proposition 1.2.3(c) shows that when the annihilator ideal is zero, the set o f multipliers is simply the center of the double centralizer algebra. Multipliers were defined by Helgason [1956] and first systematically studied on commutative Banach algebras (with annihilator ideal { 0 } ) by Ju-kwei Wang in his thesis [1961]. They are still most often considered in this context. The standard reference for their theory is Ronald Larsen [1971]. However, we are aware o f a number of papers which purport to use multipliers on noncommutative algebras but apply properties which hold only in the commutative case. Hence we shall not use the term “multiplier” again. The second statement in the next theorem is our first example o f au tomatic continuity, in that a purely algebraic condition on a map implies that it is continuous. Both results are due to Johnson [1964a). A related result is given in Proposition 5.2.6.
1.2.4
Double Centralizers and Extensions
29
1.2.4 T h eorem Let A be an algebra satisfying A l a — A r a = {0 }. (a) I f L and R are arbitrary (not necessarily linear) maps o f A into A satisfying the double centralizer condition (7) aL(b) = R (a )b
V o, b G A ,
then (L , R ) is a double centralizer. (b) I f A is a Banach algebra and (L , R ) is a double centralizer, then L and R are bounded linear maps, so T>b {A ) equals T>(A). In this case, 2?(.4) = V b ( A ) is a Banach algebra under its norm and the regular homo morphism is contractive.
Proof (a): For any o, b, c G A and any A, n € C, we get c L (\ a + fib) = R (c )(X a + fib) — X R (c)a + fiR (c)b = c(AL (a ) + i*L (b )), and cL(ab) = R (c)ab — cL(a)b. Since c G A is arbitrary and the right annihilator of A is zero, we conclude that L is a left centralizer. Similarly, R is a right centralizer. (b): Suppose A is a Banach algebra, {a n} „ eN is a sequence in A converg ing to zero and the sequence { L ( o „ ) } ngn also converges to some element b in A . Then any c G A satisfies cb =
lim cL (a n) =
n —*oo
'
'
lim R (c )a n = 0.
n —*oo
Hence, A r a = { 0 } implies 6 = 0. Thus L is closed and, by the closed graph thecrem, it is continuous. Similarly R is continuous. It is easy to check that T>B( A ) is a Banach algebra since the limit in B (A ) of a sequence o f left or right centralizers has the same form. Clearly the regular homomorphism is contractive. □ If A is an algebra satisfying A a = { 0 } , then V (A ) is a unital algebra which contains an ideal isomorphic to A under the regular homomorphism. If A is a Banach algebra satisfying A l a — A r a — { 0 }, the last theo rem shows that V (A ) — 1 > b (A ) is a unital Banach algebra which contains an ideal continuously isomorphic to A under the regular homomorphism. Hence, under suitable restrictions on the annihilator ideals, the construc tion of V ( A ) provides another way to embed an algebra in a unital algebra and a Banach algebra in a unital Banach algebra. (Examples are mentioned after Proposition 1.2.6.) Note that A equals T>(A) if and only if A is uni tal. For technical reasons, it is sometimes desirable to replace D { A ) by the poss:bly smaller unital subalgebra V ( A ) z + A . For a discussion of this, see Dauns [1969] and Dauns and Hofmann [1968], [1969]. Left and Right Idealizers 1.2.5 Definition Let A be a subalgebra of an algebra B. The left idealizer of A in B is the set {6 G B : bA C A ) . The right idealizer is defined similarly. The idealizer Ba of A in B is the intersection o f the left and right idealizer.
30
1: Normed Algebras and Examples
1 .2.6
Clearly the ( left / right ) idealizer is the largest subalgebra of B in which .4 is a ( left / right ) ideal. Similarly the idealizer is the largest subalgebra in which A is an ideal. If b belongs to the left idealizer of A in B , then L b maps A into A and its restriction to A is a left centralizer. The same remarks hold for the right idealizer and right regular representation. Hfnre if 6 belongs to the idealissor of A ih B, then (Lb\A, /£b|»4) is a double centralizer. This is the most natural way in which double centralizers arise. See the proof of the following result. 1.2.6 P r o p o s itio n Let A be a subalgebra o f an algebra B and let B a be the idealizer o f A in B. Then there is a natural homomorphism 0 o f B a into T>(A) defined by: 0{b) = ( L b\A, Rb\A)
V b e B A.
(13)
The homomorphism 0 extends the regular homomorphism o f A into V ( A ) . It is the only such homomorphism i f the annihilator o f A is {0 }. The kernel o f 0 is {6 € B : bA = Ab = { 0 } } . I f A is an ideal in B, then 0 maps B into V ( A ). I f B is a normed algebra, then 6 is a contractive map into V b (A ). P roof The description o f the kernel o f 0 and the last sentence are obvious. Thus we only need to check the uniqueness assertion. Let 0: Ba - * 1)(A) be any homomorphism which extends the regular homomorphism. Let a € A and b € B ji be arbitrary and denote 0(b) by ( L , R ). Then formula (10) gives (Lab,Rab) = 0(ab) = 0(a)0(b) = ( L a, R a) ( L , R ) = ( L R{a), R R(a))Since A satisfies A a — {0 }, this implies R(a) = ab. A similar argument gives L(a) = ba. Hence we conclude that 0(b) = ( L , R ) = (Lb\A, Rb\A) (where we use the left and right regular representations of B ), proving the uniqueness. □ Examples o f Double Centralizer Algebras Let fl be a locally compact topological space. In §1.5.1 we will more for mally introduce the following two important commutative Banach algebras of bounded continuous complex-valued functions on fl. Both are Banach algebras under pointwise multiplication and the supremum norm: Il/lloc = sup{|/(w)| : w e n } . Let C (f i) be the algebra o f all bounded continuous functions and let C'o(fJ) be the algebra o f continuoiis functions vanishing at infinity. Clearly Cq(£1) is a closed ideal in C (Q ). Hence by Proposition 1.2.6 there is a natural homomorphism of C (Q ) into Z?(C'o(n)). In §1.5.1 we show that this is an isometric algebra isomorphism onto 'D (C0(Q )). Similarly, in §1.7.1 we will show that the double centralizer algebra of the ideal B k ( X ) of compact operators on any Banach space X can be
2.7
Double Centralizers and Extensions
31
identified with the algebra B (X ) o f all bounded linear operators on X . In fact, we show that this result extends to a general class of ideals in B (X ). Finally in §1.9.13 we will show that for any locally compact group G the double centralizer algebra of L 1 (G ) can be identified with the measure algebra M (G ). We have already noted that the use of this identification by Wendel [1952] provided an important stimulus to the study of double centralizer algebras in analysis. Automorphisms The proof of the next result is straightforward. When derivations axe introduced in Chapter 6 , a similar result will be proved there. 1.2.7 P r o p o s itio n Let A be an algebra in which the annihilator ideal is zero and let a be an automorphism o f A ■ Then we may define an automor phism a of T>(A) by a (L , R ) = (a o L o a - 1 , a o R o a T 1) which extends a when A is viewed as a subalgebra o f V ( A ) . Then a —►a de fines an isomorphism o f A u t(A ) onto the subgroup o f A ut(T > (A )) consisting o f those automorphisms fo r which A is invariant as a subset o f V ( A ) ■ I f A is normed, we may replace the word “automorphisms”, by “homeomorphic automorphisms”, everywhere. I f a is an isometry, so is a. Topological and Geometric Categories o f Normed Algebras There are two categories of normed algebras (or normed linear spaces) in common use. Since confusion between them sometimes occurs, it is worthwhile to point out the distinction clearly here. We will only describe the case of normed algebras, since the case of normed linear spaces is exactly similar. In the more common and important category, the morphisms are con tinuous algebra homornorphisms. That is,
32
1 : Normed Algebras and Examples
1.2.7
The second category has contractive algebra homornorphisms as its morphisms (i.e., algebra homornorphisms < p :A —* B satisfying ||i^(a)|| < ||a|| for all a € A ). Hence, an isomorphism is an isometric algebra isomorphism. In this category an algebra with two unequal but equivalent algebra norms corresponds to two non-isomorphic objects. Thus, in particular, a finite dimensional algebra, which is a single object in the topological category, is the underlying space of infinitely many non-isomorphic objects in this sec ond category. We will call this the geometric category o f normed algebras. If we restrict the objects in these two categories to be Banach algebras, we get the topological and geometric categories o f Banach algebras. The terms “dominate” and “majorize” for comparison o f two norms which we introduced above, correspond to these two categories. By simply deleting all reference to multiplicative structure, we get the topological and geometric categories o f normed linear spaces and the topo logical and geometric categories o f Banach sjtatxs. Many arguments and constructions in this work may be carried out in either category. We will seldom point this out explicitly. Instead we will carry out the construction in whichever category seems best. FVom time to time, the differences in the properties of these two cate gories o f Banach algebras do become significant. The following is a good example. Example 4.8.11, which is due to Robert C. Busby [1971a], shows that the short five lemma o f category theory fails in the geometric category o f Banach algebras. Hence the theory of extensions is more complicated in that category. (In the geometric category, the bijection o f Theorem 1.2.11 has a (well identified) subset of Ext (C ,A ) as its range. For details, see the original paper by Busby [1971a]. In the terminology o f that article, our construction o f the double centralizer algebra shows that a Banach algebra with zero right and left annihilator ideals is faithful, and the various argu ments using the open mapping theorem establish the algebraic properties o f the topological category o f Banach algebras.) Classification o f Extensions We now turn to the use of the double centralizer algebra in the classi fication o f extensions. W ith any class o f mathematical objects, one wishes to analyze the complicated examples in terms o f the simple examples. To do this, one needs to know how to build up complicated examples from simple ones. There are many such constructions for topological algebras— direct sums, tensor products, direct and inverse limits, etc. However, we will investigate here the fundamental case of extensions. One is given two algebras A and C and asks how to classify the algebras B in which A can be embedded as an ideal in such a way that B /A is isomorphic to C. For algebras, the theory was partially worked out by Hochschild [1947]. The Banach algebra case turns out to be no harder. We follow the treatment
Double Centralizers and Extensions
1 2.8
33
of Busby [1971a] which was originally limited to C*-algebras [1967], [1968]. See the first reference for further details. (The reader should be aware that in some contexts B is called an extension of A whenever A is a subalgebra o f B. See Example 3.4.17 for information on this.) An algebra is simple (in the technical sense) if it has no ideals except { 0 } and itself. A topological algebra is topologically simple if it has no dosed ideals except { 0 } and itself. Many algebras with which we will deal can be built up in a finite number o f steps by successive extensions from topologically simple algebras. We need some technical definitions. 1.2.8 D e fin itio n A short exact sequence of Banach algebras is a triple of Banach algebras A , B and C together with a pair o f continuous homomorphisms t p : A —* B and ip\ B —►C such that ip is injective, its image
— ► A
B
C
— ► 0
(14)
The terminal arrow { 0 —* A / C —> 0 ) encodes the assumption that (
— ► A
-¥-+
B
C
— ► 0
(15)
0
(16)
of Banach algebras. Tw o extensions
0
— ,
i
i
B
1 ,
C
-h
0
-
4
^
S'
^
C
— ► 0
of A by C are said to be equivalent if there is a homeomorphic isomorphism 6 :3 - * B ' such that the diagram
34
1: Normed Algebras and Examples
1.2.10
commutes, where I_a. and Jc are the identity maps on A and C, respec tively. The set o f equivalence classes of extensions o f A by C is denoted by Ext(C ,A ). The extension (15) is called a semidirect product if there is a continuous homomorphism —* B with ip ° \ the identity map on C. A short exact sequence is said to be split if it represents a semidirect product. A semidirect product is called a direct product if x (C ) is the kernel o f a continuous surjective homomorphism n onto AA simple but interesting example o f a split extension is given in 5.3. When the maps
V a € A; c e C
which makes the isomorphism into a homeomorphic isomorphism. The images o f A and C in B are both closed ideals, and another way to describe a direct product is to say that it is the sum of two closed ideals with intersection { 0 }. In a semidirect product, each element 6 is the sum of 6 — \ ° ip(b) and X°ip(b), where the first summand is in the image o f
Let (18)
1.2.11
Double Centralizers and Extensions
35
be an extension o f A byC and let t :C ' —* C be a continuous homomorphism. Then there is an extension
0
— ► A
B'
C'
— ► 0
(19)
of A byC' and a continuous homomorphism 6:B ' —* B such that the diagram
0
_
A
T*
T" 0
.4
c
B
i
B'
—►0
T' V-',
C'
—►0
is commutative. This extension of A by C' is unique up to equivalence. P roo f Let B x C' be the Banach algebra with B x C as its underlying set, with pointwise algebraic operations and with norm given by ||(6, c7)J| = max{||6||, He'll}. Let B' be the closed subalgebra { ( 6, d ) e B x C : ip(b) = r ( c ') } . Define the maps
— ► A
11A 0
— ► A
B
C
j>
jr
B"
C
— > 0 (2 1) — ► 0
also commutes. Define 0'-.B" —►B' by 0 '(b ") — (0 "(b "), ip "{b ")), so that it is a continuous homomorphism which makes the analogue o f diagram (17) commute. The argument preceding this theorem shows that 6' is a homeomorphic isomorphism, so that the two extensions are equivalent. O Recall that after Proposition 1.1.3 we defined CHom(^4, B ) to be the set o f continuous homornorphisms from a normed algebra A to another B. 1.2.11 Theorem Let A and C be Banach algebras with the first satisfy ing A r a — A l a — {0 }. Let 0— * A -^ > 'D (A )-—->'D (A )/A — >0 be the short exact sequence in which p is the regular homomorphism and a is the nat ural map. The construction o f Theorem 1.2.10 applied to this short exact sequence gives a bijection o f CH om (C,T>(A)/A ) onto E x t(C ,A ). Further more, the extension corresponding to t 6 C H o m (C ,V (A )/A ) is a semidi rect product if and only if there exists a map to € CHom(C, 'D (A )) satisfying
1: Normed Algebras and Examples
36
1.2.11
r = a o u> and this extension is a direct product i f and only i f t is the zero map. P roo f The remarks following Theorem 1.2.4 show that T>(A) = V g ( A ) is a Banach algebra. For any r e CHom(C, V (A )/ A ), the construction of Theorem 1.2.10 gives the following commutative diagram. 0
— *
A
-*->
V (A )
-Z-*
V (A )/ A
—♦
0 ( 22 )
]r
0
—
—
c
0
where the lower line is an extension of A by C, which is unique up to equivalence. This establishes a map o f CHom(C, V (A )/ A ) into Ext(C, A ). We must show that the map is both injective and surjective. We will prove injectivity first. Suppose that 0— >A-^-+B— *C— >0 and 0 — ►A B' C — ►0 are two extensions o f A by C constructed from r:C -> V ( A )/ A and r ': C - * V (A )/ A , respectively. Let 6:B - * T>(A) and O': B ' —►V { A ) be the maps associated with this construction, which satisfy Oot p — p — 9 'o t p '. I f these extensions are equivalent, there is a homeomorphic isomorphism 8" : B —* B’ which makes the following diagram commute. A JL* & c — > 0
|7A
|e" JfU
B'
(
|'c C
— ► 0
We have 0 o ip — p = &' o ip' = (9' a ff" ) o ip. Therefore, the uniqueness assertion o f Proposition 1.2.6 implies 6 = 6' o 6" . Since any element of C can be written as ip(b) for some b e B, the calculation r ( t />(£>))
=
a o 6 (b) = a o 9' o &"(b)
=
t ' 04 , ' o0"(b) = r'(iP(b))
VbeB
proves r = r'. Hence the map is injective. Now suppose an extension 0
—*
A
B
C
— .
0
of A by C is given. Proposition 1.2.6 shows that there is a continuous homo morphism ff:B —> T>(A) satisfying p — 8 cup. Define r € CH om (C, T>(A)/A) by r(\p(b)) = 0 0 6 (b) for all ip(b) € C. First we check that r is well de fined. If V'(b') = ip(b), then there is some a e A satisfying b' = b 4-
1 AO
Sums, Products and Limits
37
Definition 1.2.8, that C is homeomorphically isomorphic to B/ip(A) with its quotient norm, shows that r is continuous. Now the diagram in (22) satisfies all the conditions of diagram (20) and thrs, by uniqueness, 0 — >A-^-+B-^-*C — >0 is the extension defined by r up to equivalence. Hence the map is surjective. Finally we prove the results about semidirect and direct products. First, suppose t can be written a s r = i r o w f o r u 6 CHom(C, V ( A ) ) . Then it is easy to check that \\C —* B C V ( A ) x C defined by x ( c) — (w (c),c) shows that the extension splits (i.e., the extension is semidirect). Conversely, if the splitting map x: C —» B is given, we may define ui to be 6 a The statement about when the extension is a direct product is even easier to check. □ We have discussed extensions o f Banach algebras because this is the case in which we are most interested. However the reader can easily strip away all the norm and continuity assumptions from the above discussion and obtain an improvement of Hochschild’s original theory [1947] for extensions of algebras. Hochschild also considers the case in which the annihilator ideal is not necessarily zero.
1.3
Sums, Produ cts and Limits
In this section we introduce a number of ways in which new algebras or Banach algebras can be constructed from a collection o f algebras. No deep results are proved. The last section mentioned the direct product of two algebras as a special case o f an extension. When more than a finite number o f algebras are involved, direct products and direct sums are two very different generalizations of this simple case. Direct Products Let A and C be Banach algebras and give the Cartesian product A x C pointwise algebra operations. Define a norm ||• || by ||(a,c)|| =max{||a||,||c||}
V (a ,c) € A x C.
This direct product satisfies a universal mapping property. If {.4 “ : a € A } is a family of objects in a category, then an object A together with maps (called projections) na: A —> A a for each a € A is called a product of {,A “ : a e A } if, whenever another object A ' and another collection o f maps 7^a- J ! —►A a for each a e A is given, there exists a unique map ( p : A '—> A such that Tr'a = na oip for all a e A This definition determines the product up to isomorphism. It is easy to see that the direct product introduced abov : with projection (a, c) = a and ^ ( a , c) = c satisfies this definition in both the topological and the geometric category of Banach algebras. More
38
1: Normed Algebras and Examples
1.3.1
generally, if {.4 1* : a t >1} is any finite set of Banach algebras, the Cartesian product x ag_4^4 a can be made into a Banach algebra by coordinatewise algebraic operations and the P -n o rm : ll(aQ)Qe 4 ll = max{||a„|| : a e A }
V (aa ) aeA € x aeAA a .
It is again clear that this construction with the obvious definition o f pro jections gives a product in either category of Banach algebras. In the topological category of Banach algebras this construction satis fies another universal mapping property, so long as {.4 ° : a € A } is finite. Let { A u : a € A ) be a collection of objects in a category. An object A together with maps (called injections) r]a'.A a —> A is called a coproduct of { A a : Ot G A } if, whenever another object A ! and another collection of maps T]'u : A a —* A ! for each a € A is given, there exists a unique map i p : A —> A ' satisfying q'a = ip ° »/Q for each a e A. This definition deter mines a coproduct up to isomorphism. It is again easy to check that the construction described above, together with the obvious injection maps, defines a coproduct in the topological category o f Banach algebras. To get a coproduct in the geometric category, we merely need to change the norm, so long as { A a : a € A } is finite. The norm we need in the -norm: li (aa)a€.4 II l = ^ l!aa|| a£A
^ ( aa)a£A £
•
When we come to infinite families { A a : a G A } of algebras, normed algebras or Banach algebras, the construction of the product and coprod uct diverges more decisively, and becomes dependent on whether algebras, normed algebras or Banach algebras are involved. Throughout this work we always interpret the Cartesian product x aeJ4«Aa of a family o f sets as the collection of all functions a: A —►Uae,4.4 a satis fying a (a ) e A a for each a € A. An element o f the Cartesian product is called a cross section when considered as a function. The direct product fIQg,4.4 a o f a family { A a : a € A } of algebras is sim ply the Cartesian product * a e .4A a made into an algebra by coordinatewise algebraic operations. 1.3.1 D e fin itio n Let {.4 ° : a 6 A ) be a family of normed algebras or Banach algebras. The t°°-direct product H^eAA a of { A a : a € A } is the subset o f the Cartesian product x n f AA a consisting o f all cross sections a such that l l ai L = sup{||a(a)|| : a € A } (1) is finite. The subset is made into a normed algebra or Banach algebra by coordinatewise algebraic operations and the f°°-norm defined by equation (I)-
Sums, Products and Limits
i.3.2
39
It is easy to see that the above definitions subsume those given earlier and define products in all o f the relevant categories when the projections are defined in the obvious way. Direct Sums ( Let { A a : a € A ) be a collection of algebras or normed algebras. The direct sum ® Qg 4 ^4a o f { A a : a € A ) is the subset o f the Cartesian product x o 6 i4^ ° consisting of all cross sections which are zero except at a finite nu’Tiber of elements of A. (Th e finite set of a with o (a ) non-zero varies with a e ©c>e,4-4 “ .) This set is made into an algebra by coordinatewise algebraic operations. I f the summands A a are normed algebras, then the direct sum is a normed algebra with the t l -norm defined by N i t = £ ||a(a)||. aeA The direct sum just described is frequently called the algebraic direct sum. In order to define the direct sum in the categories o f Banach algebras, arid for many other purposes in this work, we need the notion o f an un ordered sum o f numbers. Let A be an arbitrary index set and for each a e A , let aa be a complex number. The unordered sum fla of this arbitrary set of numbers is defined as follows. Let T be the collection o f all finite subsets o f A. When T is ordered by inclusion, it is a directed set. For each F € T define ap — Y I c z f a«- Then {ap-Jp-gjr is a net. If this net has a limit, then this limit is the unordered sum. If the net has no limit, then ilie sum does not exist. If it should happen that each number aa (a € A ) is non-negative, then it is easy to see that s u p {a f : F g F ] is the limit o f the net (and hence the unordered sum) if and only if it is finite. Hence for non-negative series we define s u p {a f : F e !F} to be the unordered sum, whether or not it is finite. 1.3.2
D e fin itio n
Let {.4 “ : a € .4} be a collection o f Banach algebras.
The Banach algebra direct sum A a o f {.4 Q : a e A } is the subset of the Cartesian product x aeAA a consisting o f all the cross sections a such that the unordered sum
INK = £
M a )ll
(2)
is finite. This set is made into a Banach algebra by coordinatewise linear operations and the ^'-norm defined by equation ( 2 ). The definitions o f direct sums o f algebras, normed algebras, and Banach algebras just given subsume the previous definitions and give coproducts in each of the relevant categories with respect to the obvious injections. Suppose {.4 ° : a e A } is a collection of Banach algebras. Then its Banach
40
1: Normed Algebras and Examples
1.3.2
algebra direct sum ® ^ g/4.4Q is just the completion of its normed algebraic direct sum © Qe,4-4 a . Internal Direct Sums The direct sums we have just discussed are sometimes called external direct sums. There are corresponding notions o f internal direct sums. Let A be an algebra, let A be an index set, and let { l a : a € A } be a family of ideals o f A . Then the sum J^aeA this family is the set o f finite sums o f elements from Uae/( I n. The algebra A is calk'd the internal direct sum o f the family { I Q : a e A } if A is the sum o f the family and satisfies
1/5 = { 0 }
r n
V a € A.
aeA\{a] When these conditions hold, we write A = ® QgA%a- It is clear that (external) algebraic direct sum ® a 6 ,4.4 “ is the internal direct sum of ideals = {a € (Bae A A a : a(/3) = 0 for all 0 ^ a }. Conversely, if A is internal direct sum o f { ! “ : a € A }, then A is naturally isomorphic to external direct sum ® ae^ Z a . I f A is a nonned algebra, we say it is the internal direct sum o f family { ! “ : a € A } if each 1 “ is a closed ideal, A = A^a ^ d r n
^
= {0 }
V a € A,
the the the the the
(3 )
/3€A\{a} where the bar denotes closure. Again it is easy to check that the external direct sum © a e^^4Q of a family {^4“ : a € A } of normed algebras is the internal direct sum o f {Z “ : a € A } , where each T a is defined by I a = {a € ® a eA-A0 : a ((i) = 0 for all 0 ^ a }. Suppose A is the internal direct sum o f { I Q : q e A }. The definition of a coproduct guarantees that there is a map (in either category) of the external direct sum onto A . The map is an algebra isomorphism but not necessarily an isomorphism in either category o f normed algebras. In either category of Banach algebras we say that A is the internal direct sum o f ( J “ : a € A ] if each T a is a closed ideal, is dense in A and Xa fl $^/3e 4 \{a} ^ 's zero for each a 6 A. Clearly an external direct sum is an internal direct sum as before. If A is the internal direct sum o f ( I “ : a e A }, then the definition of a coproduct shows that there is a map (in either category) of the external direct sum ©ag^X 0 onto a dense subset o f A (which includes Y I c ^ a J Q), but this map need not be a homeomorphism nor onto A . Subdirect Products Finally, we discuss what are usually called subdirect sums o f algebras and nonned algebras, but what are more correctly called subdirect prod
:.3.3
Sums, Products and Limits
41
ucts. These arise naturally in many contexts as will be shown in Chapter 4 end elsewhere. It should be noted at the outset that representation o f an al gebra as a subdirect product furnishes relatively little information about its structure unless additional information, such as that discussed in Chapter 6 , is available. Subdirect products were first considered by H. Priifer [1925] for commutative rings and by Gottfried Kothe [1930] for noncommutative rngs. For further information and history, consult Neal H. McCoy [1947]. The normed case is an obvious extension of the non-normed case. Sub direct products are simply algebras, normed algebras or Banach algebras together with a suitable embedding as a subalgebra, or normed subalgebra o f a direct product. Direct sums are examples o f subdirect products when provided with the obvious embedding into the direct product. 1.3.3 D e fin itio n Let A be an index set and let : a € .4} be a family of algebras. A subdirect product o f { A a : a € .4} is a subalgebra A of Ha€AA a such that for each a € A, na: A —* A a is surjective. The subdirect product in either category o f normed algebras or Banach algebras is simply a normed or complete normed subalgebra A o f I I Q6 ^^4a defined in the appropriate category with the same restriction that each tt0 :.4 —►A a should be surjective. Let A be a subdirect product in any o f these categories and, for each fi € A, define X/} to be {a € A Q x aeA A a : a(f3) = 0 }. Clearly, Xa is an iteal which is closed in the nonned cases. Furthermore, each na can be identified with the natural map A —►A/Xa — A a. Finally, n ae^ I a = { 0 } holds. Conversely, if {Xa : a € A } is any collection o f ideals (closed in the nonned case) with n a6AI a — { 0 }, then A is isomorphic to a subdirect product o f {A/Xa : a € >1}. This is the way in which subdirect products most commonly arise. A ll o f the foregoing discussion o f direct products, direct sums, internal d rect sums and subdirect products applies to linear spaces, normed linear spaces and Banach spaces. Simply dropping all references to multiplicative structure, (and thus replacing ideals by linear subspaces) makes this clear. Later we will need to deal with direct sums in the category of Hilbert spaces. The discussion o f that case is postponed until needed. Inductive or Direct Limits We begin by defining inductive limits which can be considered as a massive extension o f direct sums. W e will then give one example which is important in the definition o f K-theory. The example is often called the a jebra of matrices of arbitrary size. 1 3.4
D e fin itio n
Let (j4, < ) be a directed set, and for each a € A let
/ a be an algebra. For each a, (3 £ A with a < /?, let
42
1. Normed A lg eb ra arid Examples
1.3.4
a < /? < 7 € /I, let th ese h o m o m o rph ism s s a tis fy :
*p~ra — ^ 7,1 °
(4)
The above set of data will be called an inductive (or direct) system o f alge bras. If, in addition, each Aa is a normed algebra and each ipa is continuous, the inductive system will be called an inductive system o f normed algebras. Finally if all these conditions hold and, for each a € A , the limit superior lim sup H ^ q II
(5)
peA
is finite, then the inductive system will be called a normed inductive system o f normed algebras. (When all the A a are Banach algebras, the uniform boundedness principle shows that this last condition is satisfied if and only if limsup ^gj4 ||v?/3a(
1.3.5
Sums, Products and Limits
43
general inductive systems, the inductive limit is essentially an increasing union o f the <pa {A a )1.3.5 T h e A lg e b r a M 00(A ) o f M a tric e s o f A r b it r a r ily L a rg e S ize o v e r an A lg e b r a A This is an important general example o f a normed inductive limit of algebras. (In this case and others where the connecting homornorphisms are isometries, the normed inductive limit and algebraic inductive limit coincide.) Let A be an algebra or a normed algebra and let M n( A ) be the algebra o f n x n matrices over A described in Definition 1.6.9. I f A is a normed algebra or a Banach algebra, then M n( A ) is a normed algebra or a Banach algebra under various natural norms as noted before Proposition 1.6.10. Construct an inductive system with the positive integers N as index set, the algebra M n(A ) corresponding to n e N and with the connecting maps defined by sending each smaller matrix into a larger one by putting it in the upper left-hand corner and completing the matrix with zeroes. Clearly this is tin inductive limit, and each homomorphism is an isometry under any of the norms mentioned above if A is a nonned algebra or a Banach algebra. The limit is essentially the union o f matrix algebras o f arbitrary size over A , so that any algebraic operation with a finite number of elements can always be carried out in a specific size matrix algebra over A , but the size may increase as needed in any series o f computations. Operations in this algebra bring out some o f the more subtle structure o f the original algebra A . It is used in most definitions o f K-theory. O f course we could complete M oc(A ) or any other normed algebra. It is worth noting that if A is a C*-algebra, then the completion of M X (A ) has an interesting interpretation. First, M o o (A ) itself is isometrically iso morphic to a dense subalgebra o f the tensor product o f A with the Banach algebra o f all compact operators on separable Hilbert space. This tensor product has a natural norm. The appropriate norm for the inductive limit (relative to which this isomorphism is an isometry) comes from applying a certain operator norm to the individual matrix algebras, which extends the C*-norm of the original algebra. Then the completions of the algebra of Kre'irarily large matrices and o f the tensor product are naturally isometrxaily isomorphic. The importance of this tensor product was recognized before the natural inductive limit construction of M o o (A ) was appreciated. (See Section 1.10 and Volume II.) Projective or Inverse Limits Jh again begin with a definition o f this concept which has about the same relationship to direct products as the inductive limit has to direct s u i t s . In some sense it is the dual construction to that o f inductive limits. In the case o f inductive limits, we were interested in limits in the category of n armed algebras, but here we are primarily interested in the category of
44
1: Normed Algebras and Examples
1.3.6
topological algebras. We will again give one example. I
1.3.6 D efinition Let [A , < ) be a directed set, and for each a € A let A „ be an algebra. For each a, ft € A with a < /3, let
= ‘fia defined as follows. Let
n Aa : ppaiap) = aa
for all (3 > a } ,
(7)
a€ A
and endow this collection o f coherent cross sections with the obvious pointwise operations. For each a, the map <pa is just the projection map onto the a factor. In the case o f a projective system of nonned algebras, this same algebraic projective limit can be given the weakest topology making all of the projection maps 1pa continuous. This topology, which is just the relativized topology from the direct product, is seldom a norm topology. It easy to check that these projective limits satisfy the universal prop erty that if B is another algebra (or topological algebra) with a family of (continuous) homornorphisms # Q: B —> A {, satisfying 0 't'# = ^ a, then there is a unique map 'P: B —►A satisfying tpa o for all a e A. 1.3.7 A P rojective Lim it A lg e b ra This example should help make the concept more comprehensible. Consider the directed set of all com pact subsets of the real line ordered by inclusion. For each compact subset K , let A k be the Banach algebra C ( K ) o f all continuous complex-valued functions on K with the supremum norm ||• I U Then
1.3.7
Sums, Products and Limits
45
Uitraproducts and Ultrapowers These concepts have proved valuable in Banach space theory, but are
01 iy beginning to be used in Banach algebra theory. We will provide a brief introduction. Details can be found in Brailey Sims [1982) and Stefan Heinrich [1980]. Let I be an infinite index set. A filter o f subsets o f I is a nonempty family F o f nonempty subsets o f I satisfying: (1) I f A , B € F , then i f l B e f , and (2) I f A e F and A C B C I , then B € F . (Some authors allow a filter o f subsets to contain the empty set 0. Then ths collection o f all subsets o f I is the only filter o f subsets containing 0. W ith this convention, our filters are simply called proper filters.) Probably tfe2 most familiar filter is the family o f all neighborhoods o f a point in a topological space. Given any nonempty subset S o f /, the family o f all subsets of I which include S is a filter denoted by F s - Still another filter on I is the Frechet filter o f all the subsets o f I which have finite complement in I. A filter o f subsets is called an ultrafilter if it is maximal (under inclusion) among filters. Every filter is included in some ultrafilter. It is easy to see that a filter o f subsets is an ultrafilter if and only if for each subset A of I exactly one o f A and I \ A belongs to the filter. Another easy argument shows that this is equivalent to: If A U B belongs to F , then either A € F or B € F holds. Hence for any ultrafilter F , Pl(^4 : € F } 's either a singleton set or empty. Note that T s introduced above is an ultrafilter if and only if 5 = { p } is a singleton set. In this case we write F v instead of T s - The only ultrafilters wi h p|{,4 : A £ F } nonempty sure those o f the form T p. We call these fixed ( or trivial) ultrafilters and all others free ultrafilters. Hence an ultrafilter is fixed if and only if it contains a finite subset, and it is free if and only if P itA ■■ A € F ) = 0. Every filter o f subsets is included in an ultrafilter. We will require that our ultrafilters be countably incomplete. This means that the ultrafilter is not closed under countable intersections. It is equivalent to requiring that the intersection of the sets belonging to some countable subset o f the ultrafilter be the empty set. It certainly implies that the ultrafilter is free. From now on F will be a particular countably incomplete ultrafilter on I. For each i 6 /, let A ' be a ( Banach algebra / Banach space ). Our construction begins with the f^ -d irect product n^g/^4‘ o f { A ‘ : i € 1} of Definition 1.3.1. The case in which all the A ’ equal a fixed A is important. Fo:- any a € H f^ jA ', we can define ||a|| to be the unique non-negative minbersuch that for every e > Otheset { i € I : ||a||-e < ||a(i)|| < ||a||+e} is in F . W e will take the existence, uniqueness and other properties o f ||■||
■
46
1: Normed Algebras and Examples
1.3.8
for granted. Let B be the quotient of 11^, A 1 modulo the subset of elements with norm 0 . Then B has the structure of a { Banach algebra / Banach space ). 1.3.8 Definition The ( Banach algebra / Banach space ) B defined above is called the ultraproduct with respect to T o f the family {-4'}<e/. If all A * = A , then it is called the ultrapower o f A with respect to T . It is not easy to give an illuminating example, but the reader may look at the theory of ultraprime ideals introduced in Definition 4.1.11. The paper by Michael Cowling and Gero Fendler [1984] has interesting applications to representations on Banach spaces ( cf. Chapter 4 below). See also the proceedings o f a conference edited by Nigel J. Cutland [1988].
1.4
A ren s M ultiplication
Introduction Richard Arens defined two products on the double dual A ** o f any nonned algebra A [1951a], [1951b]. The first systematic study o f these products was carried out by Paul Civin and Bertram Yood [1961]. A useful compilation and simplification o f results up to that date was published by John Duncan and Seyed Ali Reza Hosseiniun [1979]. (See also Harald Hanche-Olsen [1980] for a different approach to a related problem.) Each product makes A * ’ into a Banach algebra and the canonical in jection k: A —» A ** is a homomorphism for both products. (This canonical injection is just the evaluation map: k(a)(u>) = w (a) for any o 6 A and u> € A ‘ .) We shall denote the two products by juxtaposition and by a dot, respectively, (i.e., we use f g and f • g for the two products o f / and g in In most cases the two products do not agree and the double dual algebra with either product is too large, too complicated and too badly behaved to be useful without some simplification. However, there are special cases (notably the C*-algebras which will be considered in the second volume o f this work) in which this algebra is easy to comprehend and exploit. In other cases, quotients of the algebra modulo various natural ideals are easier to exploit. In all cases that have been studied in detail, the structure is intimately connected with that o f the original algebra and has interesting, if sometimes complicated, relationships to significant mathematical objects. In this section we explore identities and one-sided identities in the Arens products, connections with double centralizers, the case in which the two products agree and a few simple examples. Tw o representations which also stem from Arens’ very general construction are defined. Although the two Arens products are usually called the first and the
1.4.1
Arens Multiplication
47
setfohd Arens product, they stand on a completely equal footing since their definition simply depends on choosing left or right first. The first product is coa:inuous in the weak* topology in its first (left) variable for any choice of the second (right) variable in A '* ; it is also continuous in the same topology in its second variable for any choice of the first variable in the canonical image of A in A " ■ (That is, f } g is continuous in / e .4** for any g e ' A ” , and g h-> k ( a)g is continuous in g for any a e A .) The second product enjoys the reflected continuity properties. (That is, g >—* / • g is continuous in g e A ** for any / € A **, and / > - * / • /c(a) is continuous in / for any a € >1.) Basic properties o f the weak* topology guarantee that any element / € A ** is the limit of a net in k ( A ) within the ball o f radius ||/|| (cf. Dunford and Schwartz [1958] V.4.2). These facts allow the calculation o f both products, but the reader should be warned that this is a subject on which an unusual number of false results have been published, and several of the errors stem from attempting to use continuity results in proofs. The entirely algebraic original construction is less subject to erroneous interpretation. We now give that definition, which guarantees the existence o f products with the continuity properties just described. Definitions and Basic Properties 1.4.1 Definition Let A be a normed algebra, let k : A —> A ** be the nat ural injection and let L and R be the left and right regular representations of A. For any a e A and w € A *, define elements uia and 0w o f A * by
811(1
Wa = ( £ a ) * M
aU = (R a )*(w ).
(1)
For any ui 6 A ' and / € A **, define elements fui and uij o f A * by fu>(a) - /(w0)
and
w/(a) = / (aw)
( 2)
for all a € A . Finally for any f , g € A * ' and w € A *, define elements f g and / •g o f A ** by fg (u >) = / ( 9w)
and
/ • g(w) - g (u f ).
(3)
These two products in A ** are called the first and second Arens product, respectively. When the two products coincide, the algebra A is said to be Arens regular. We gather some elementary properties o f these two products. 1.4.2 Theorem
The elements uia and auj belong to A " and satisfy: u>a{b) = u)(ob)
l k . l l < IM I ||a||
au{b) = u>(ba)
(4)
||„w|| < IM I ||a||
(5)
48
1: Normed Algebras and Examples (u}a) b = u }ab
( au ) b = a(ub)
a(bV) ~ abU
1.4.2 (6)
fo r all ui e A ' and a,b 6 A - The elements fU) and uif belong to A * and satisfy H H I < |M|||/|| IK|| < M\\f\\ (7) (/w)o = f ( u a) k(o)W —
o(w/) = (o w)/
( 8)
wii(a) = wa
(9)
/or all a € A , u) € A * and f e .4*’ . Finally the two products satisfy
l\fg\\ < l l / l l W faw
=
11/ - d l < l l / l l IM I
/ (s w )
* (a )/ = « ( a ) • / = (£ « )* * (/ )
w / 9
=
( a ’/ ) s
/ « (a ) = / • « ( a ) = (/**)**(/ )
/c(a)/c(6) = /c(a) • /t(6) = k ( ab)
do) ( H
)
( 12 ) (13)
fo r all a,b € A , u € A * , f , g € A " . Each product makes A '* into a Banach algebra, and k is an injective homomorphism from A to A " with respect to either Arens product. The two products agree whenever one o f the factors is in k (A ). Furthermore, the map ( L . A '* - B(-4*) / R - A " - B ( A ') ) defined by ( Z , (w ) = fui j Rf{u>) = u>f ) is a { homomorphism / anti-homomorphism ) with respect to the ( first / second ) Arens product. The following maps are continuous in the weak* topology (I.e., the A * topoLogy) on A ' * : f
f9
g>~>n(a)g
9 - + f -9
( 14)
f >-> f ■n(a )
(15)
fo r all a € A and f , g € A ‘'*. I f ip. A —> B is a continuous ( homomorphism / anti-homomorphism ), then ifi, , :A " * —►B ** is a continuous { homomorphism / anti-homomorph ism ) with respect to ( either Arens product / the opposite /Ireus products ) on A ** and B *m. Proo f A ll these results are easy to obtain from step by step calculations starting with the three step definition. □ We will show in Proposition 2.5.3(e) that n (A ) is a spectral subalgebra o f A * mwith respect to either Arens product and in 4.1.15 that various kinds o f ideals in A and A " are closely related. In Proposition 8.3.7 and Theorem 8.7.14 we will give more details on the relationship between minimal ideals in A and A " . Fereidoun Ghahramani, Anthony To-Ming Lau [1988] and Ghahramani, Lau and Viktor Losert [1990] show that, for locally compact groups G and
1-4,2
Arens Multiplication
49
H , any isometric isomorphism
or equivalently
,
.
for d l a € A and u> e A *. A ( left / rig h t) identity for the { first / second ) Arens product must also be a { left / right ) identity for the { second / firs t) Arens product and is less common. This implication follows from the fact that an element e € .4** is a { left / right ) identity for the { first / second ) Arens product if and only if it satisfies
{ e(/W) = f ( w ) / e ( v f ) = / M )
V / s r . 6 / ,
(17)
and these conditions imply (16) (in reverse order) by equation (9). For reasons to be considered in Chapter 5 below, it is fairly common, and quite interesting, for A ** to have a right identity for the first Arens product which is also a left identity for the second. W e call such an element a m-'xed identity for A **. Let us denote one by e € .4**. This condition is equivalent to assuming eui = u> = we or, equivalently, that e(u>a) — u>(a) = e (Qw) for all cu € A * and a € A - There is no reason to expect t h a a mixed identity is unique. In fact, any e! € A ** such that e and
50
1: Normed Algebras and Examples
1.4 3
e' agree: on ( L ^ ) ' { A ' ) + is also a mixed identity. Anticipating the terminology and results of Chapter 5, we note that A " has a mixed identity if and only if A has a bounded two-sided approximate identity. In this situation we say that A is approximately unital. Similar results involving one-sided approximate identities hold for a right identity for the first Arens product or a left identity for the second product individually even if they are not mixed identities. If A " has a ( left / right ) identity for either product, then equations (16) and (4) show that the annihilator ideal ( A b a / A l a ) ) is zero. The situation on uniqueness of mixed identities changes if A " has a (two-sided) identity element for either one of the Arens products. In that cam: any mixed identity must equal this unique (two-sided) identity since ( e — eej — e' / e ~ v!c e‘ ) if a is an identity and e' is a ( right / left ) identity. When A ** is unital or when it just has a mixed identity, there is another way to consider the Arens product which is helpful in some situations. In slightly less generality, it was first published by Duncan and Hosseiniun [1979]. Recall that Theorem 1.4.2 already showed the existence o f a ( ho momorphism L : A ** —* B ( A ') / anti-homomorphism R: A " —> B (A * ) ) defined by ( Lf(u>) = fui / Rf(u>) = ujf ). In fact we have already derived in equations ( 1 ) and ( 8 ), but not yet stated explicitly, that the range of ( L / R ) is included in the commutant { ( ( L a ) * ) ' / ( ( R a ) 'Y )• When A ’ * has a one- or two-sided identity we get the following.
1.4.3 Proposition Let A be a nonned algebra. Then ( I / R ) has the whole commutant ( ( ( L a ) * ) ' / ( ( R a ) * ) ' ) o* its range i f and only i f A ** has a ( right / left ) identity fo r the ( first / second ) Arens product. Further more, ( L / R ) is a homeomorphic { isomorphism / anti-isomorphism ) of A " with the ( first / second ) Arens product onto ( ( ( L a ) ' ) ' / ( ( R a ) * ) ' ) i f and only i f there is a two-sided identity fo r the ( first / second ) Arens product. Finally, ( L / R ) is an isometry i f and only i f the two-sided identity fo r the ( first / second ) Arens product has norm one. P roo f We work only with L since the case o f R is similar. Suppose L has all o f the commutant as its range. Then there is some e G A** satisfying L e = I € B (A ’ ). This means eui = oj, from which it follows that e is a right identity for the first Arens product. If L is an isomorphism, then e must be a two-sided identity and if L is an isometry, ||e|| must be one. Conversely, suppose e g A " is a right identity for the first product. Let S be arbitrary in the commutant ( ( L a )* )'- Define f s G ^4** by /s(w ) = e(S(u>) ) for all u) € A * ■ Then equation (16) shows fs(<*>o)_= e(S(uia) ) = e (S o (L a)^(uj)) = e ((L a)* o S (u i)) = e(S (u})a) = S (u i)(a ), so L maps f s onto S. Thus L is surjective. Now suppose e is a two-sided identity for the first product. Since /uj is just L f ( u ) , equation (17) shows that L is injective.
1A 4
Arens Multiplication
51
Finally if e is a two-sided identity element of norm one, then the map we have just constructed from the commutant to A " is contractive. Since L is obviously always contractive, we conclude that it is an isometry. □ This result shows that when A * ’ is unital for one o f the Arens products its Arens multiplication may be defined in terms of the multiplication in ( ( L a Y ) ' or ({R-a Y Y 38 a subset of B (A * ) induced by the appropriate map L or R. The existence o f an identity in >1** can be described in terms of the existence of a particularly strong kind of approximate identity in A , but we will not discuss this here since approximate identities are not introduced until Chapter 5. It is of considerable interest that if either / is a right idealizer of k (A ) in A ** or if is left idealizer of k(-4) in .4**, then L j and R g commute in B (A * ). It is easy to see that these two operators commute exactly when the o ix ed associative law g • (h f ) = (g ■h ) f holds for all h € A **. Hence they commute for all /, g € A "* if A is Arens regular or k(.A) is a one-sided ideal in A ** with respect to one, and hence both, Arens products. Relationship to the Double Centralizer Algebra It should not be surprising that there is an intimate connection between the Arens products on A ** and the double centralizer algebra D (A ). We will define various maps between subalgebras o f >1**, T>(A) and V ( A * r ) under minimal hypotheses on A beginning with the following two easy propositions. Some o f these ideas are related to Laszlo Mate [1967] and Pak Xen Wong [1985]. 1.4X- Proposition Let ( L , R ) be a double centralizer o f a Banach algebra A in which the annihilator ( ideal A a is / ideals A l a and A b a are ) { 0 }. Then (18) is an isometric injective isomorphism o f { T>b { A ) / T>(A) = T>b { A ) ) into V b ( A " ) when A * ' is endowed with either Arens multiplication. Note that L ** and R** leave k ( A ) invariant as a subset o f A * * . P roo f Under the second alternative hypothesis, all double centralizers are bounded by Theorem 1.2.4. (In that case the same theorem shows that it is enough to consider the third and sixth lines below.) One easily checks the following results for sill a € A , ui G A " , and g € A **: L'{u>)a R*(u>)a £ *K )
9J?*( w) i?*( 9u,)
-
wL(o)
= = = = =
Ufi(a) L W fi“ (s)W L "(g )U
aL *(u j) aR * ( w) L(a)W
= = = =
L'(aU>) fl(a)W R*(aU>)
R (tJ)» £ * (w» )
=
R'(u>g)
=
52
1: Normed Algebras and Examples
1.4.5
These equations show that ( L ” / R " / (L **, R " ) ) is a bounded ( left / right / double ) centralizer whenever ( L / R / (L , R ) ) has the same property, no matter which of the Arens products we use. Since ( S o T )* * equals and ||S|| equals ||S**||, the maps L >—> L ** and R >—> R ** are homornorphisms and isometries. (This proof could also be based on looking at L " and R " as extensions by continuity of L and R , respectively.) □ 1.4.5 C o r o lla r y I f A " has a mixed identity e and ( L , R ) is a double centralizer o f A , then they satisfy the following identities: R “ ( f ) = f R » ( e ) = / L **(e ) and L ** (/ ) = L * * (e )- / = i?*‘ ( e ) - /
{
fo r all f € A " . Thus the map (L , R ) h-. ( L m\ R “ ) o f V { A ) into V ( A ” ) has range in (the image o f ) A ** fo r one of the Arens products i f and only i f there is a two-sided identity e € A** with respect to this product. In that case, ( L , R ) L ° '( e ) = R mm(e ) is a homeomorphic isomorphism o f T>(A) = T>b ( A ) into A * * , which is an isometry i f and only i f the identity has norm one. Proof As noted already, the existence of a mixed identity ensures that A l a and A r a axe zero so the stronger form of the last proposition is available. Equations (19) now follow by applying the basic properties of double centralizers to ( L " , R " ) and the equations f e ~ f or e • / = /, respectively. In particular, (/ **,/ **) is a norm one, two-sided identity for V ( A **) in whichever Arens product is being used. It can be identified with e in A " if and only if e is an identity for A ** in that Arens product. (Recall that, if any algebra like A " has an identity o f norm strictly greater than one, then the embedding o f A ** into T>(A**) is an isomorphism but not an isometry.) □ When A ' m is not unital, the situation becomes more complicated, and in particular it may be necessary to consider quotient algebras. We begin to study this situation by noting that the idealizer A ^ of k(.A) in A ** does not depend on which Arens product we use, since the two products agree when one factor is in k (A ): A'X = { } e >1** : M A ) = f ■k {A ) c k (A ) and n ( A ) f = k(.A) • / C k(.A )}. Proposition 1.2.3 describes the obvious homomorphism of this set into V ( k ( A ) ) , and we may consider this as a map into T>(A). Technically we can define this as / _► 9 ( f ) = ( L f . R f ) where (21) L/ = K~l o L f O K
and
R f = k ~* o R j o k
Arens Multiplication
1A3
53
with ( L f / R f ) being the ( left / right ) representation of / in A '* with either of its Arens products. Now it is easy to check
( L f y = R f e ((R Ay y
and
( R f y = If e((LA yy
( 22 )
in t e r m s of the notation used in Theorem 1.4.2 and Proposition 1.4.3. In the next proposition, we examine the second conjugates o f the actions L f and R f. For any subset S of A *, we use S x to denote the subset of A ’ * cons;sting of those functionals which vanish on 5. 1.4.8 P r o p o s itio n The subset ( ( L a ) * ( A * ) x / (R A y ( A * ) x ) is an ideal in A ** with, respect to the ( first / second) Arens product which is the kernel o f the maps { L and R / R and L ) . In fact, any element f from this ideal satisfies ( A * * f = 0 / / ■A ” = 0 ). (23) The subset A ^ = (R A y ( A * ) x n ( L A y ( A * ) x = {( i ? ^ ) * M * ) + ( ^ ) * M * ) } x is an ideal in A™ and is the kernel o f the homomorphism 8 : A % —> 'D (A ), defined above as 0 (f) = (L f , R f )
V / G ^ *.
(24)
The vwo Arens products agree on A*2 /^"n an(^ the map induced by 6 is an •isomorphism onto V ( A ) . In particular, A ^ is zero i f A ** has an identity fo r e ther Arens product. Wnen the map 8: A*£ —> T>(A) is followed by the embedding (L , R ) (L * * ,R * * ), the image o f f G A a is a double centralizer in either Arens product which agrees as a pair o f maps with ( L j \ R ^ ) where the super script numerals indicate the regular representation with respect to the second and first Arens multiplication, respectively. Proof We consider only the second case (for variety). For / G (R a )* ( A * ) 'L, we htve f ( auj) = 0 for all a G A and w G A *; this implies w/ = 0 from whicl the identification as kernels and equation (23) follow directly. (Note that L / R ) can be defined on the appropriate one-sided idealizer set which includes the annihilator set { (R a ) * ( A * ) x / ( L a ) * ( A * ) ± ).) Since R is an anti-homomorphism with respect to the second Arens product defined on all o f A , the set is a (two-sided) ideal even though (23) only shows that it is a right ideal. We can show directly that g ■/ is in this ideal for any € ^ **. We see g ■f { au>) = / ( ( „ « ) , ) = / ( „ K ) ) = 0 if / G ( R a W ^ . Suppose / € A **, g G A “ and gic(a) = n(b). Then f 9M
= / (6w) = p «(a )(w / ) = / •g (auj).
The case o f is similar so the two products do agree on the quotient. Now consider the last statement of the proposition. We again prove the
54
1: Normed Algebras and Examples
1.4.6
second case. For / € A * * , g € A*^ and ui € A *, we get ( R g ) * '(f ) (u j ) = f ( ( R t r ( u > ) ) = f ( g w ) = /ry(w ) = / # > ( / ) M .
Note that this proof has established more than stated in the proposition in several cases. □
( L ,R ) V {A )
V (A ") ( 4 n), 4 n))
(25)
The last three propositions can best be visualized in terms o f the com mutative diagram above, which represents the case in which the nth Arens product has an identity e either for n = 1 or 2. (A ll the maps are defined and interesting without this restriction, but the diagram is no longer com mutative in general.) Each map is a homeomorphic isomorphism onto its range and is isometric if e has norm one. Thus we see that if at least one o f the Arens products has an identity, then a substantial part of A " behaves very well indeed: just like the double centralizer algebra. Consider approximately unital algebras for a moment. If such an algebra is Arens regular, then it has only one mixed identity which is in fact a (two-sided) identity for both Arens products. Hence, in the context o f approximately unital normed algebras, A ** being unital for one of the Arens products is a weak version of A being Arens regular. Arena Semxregularxty A still weaker version of Arens regularity has been introduced by Michael Grosser [1984] and studied there and subsequently by the same author [1987] and by others. An approximately unital Banach algebra A is said to be semiregular if it satisfies R '* (e ) = L " { e ) for all mixed identities e. (The idea behind the definition is that an algebra is semiregular if and only if the two embeddings o f double centralizers, based on using their left or their right centralizers, agree for any mixed identity.) Corollary 1.4.5 shows this
1 . 4.7
Arens Multiplication
55
is true whenever A ** has an identity for at least one o f the Arens products. The above remarks show that Arens regular algebras are semiregular. The latter result and the easy fact that commutative approximately unital Bantxl algebras are semiregular were established in the first cited paper. The case of commutative algebras shows that semiregularity is strictly weaker than the possession of an identity for at least one Arens product. :Losert and Harald Rindler [1984] showed that, for a locally compact group G, the algebra L l (G ) is semiregular if and only if it is either discrete or commutative. This contrasts with the case of regularity where L l ( G ) is regular if and only if G is finite, a result proved by Young [1973a] after Civin and Yood [1961] handled the abelian case. Similarly, the algebra B a ( X ) oi approximable operators (i.e., those uniformly approximable by finiterank operators (Definition 1.1.17 above)) on a Banach space X is regular if and only if X is reflexive (Young [1976]). Much weaker conditions are knovm for semiregularity (Grosser [1987]). We shall treat these results in a preliminary way later in this section and in more detail in Section 1.7 after introducing more o f the relevant concepts. However, two special cases completely solved in the last reference are particularly interesting. Consider the algebra B a (C (Q .)) o f operators on the Banach space o f sill continuous complex-valued functions on a compact Hausdorff space Jl. This algebra o f operators is semiregular if and only if $1 is scattered, where scattered signifies that every nonempty subset contains an isolated point. Let /i be a finite measure and let L l (fi) be the usual space of (equivalence classes) of absolutely /i-integrable functions. Then B a {L 1(h )) is semiregular only when L l (fi) is finite-dimensional (in which case, of course, the algebra is Arens regular). Before discussing Arens regularity, we introduce three simple examples. Examples of Arens Multiplication fo r Commutative Algebras For a commutative algebra A , Definition 1.4.1 immediately implies aui = iva and fui = ujf, hence f g ~ - g - f (note reversal of factors) for all a & A , u) € A ' and j , g £ A **. Prom this it easily follows th^t a commutative algebra is Arens regular if and only if one, hence both, Arens products are commutative. In the following examples, this observation will be useful. For a fuller discussion of this and many other results on the Arens products for general commutative Banach algebras, see Theorem 3.1.12. 1.4.7 A re n s M u ltip lic a tio n in (co)** ^ £°° This is perhaps the simplest example in a nonreflexive space and shows that the Arens multiplication is often the natural multiplication on the double dual. (O f course, in a Banach algebra which is reflexive as a Banach space so that A and A ** are canonically isomorphic under k , both the Arens products agree with the original multiplication.) As usual, let co be the collection o f all sequences converging to zero with elementwise algebra operations and the supremum
56
1.4.7
1: Normed Algebras and Examples
norm:
I M L = sup{|an |
:n£ N}.
We identify t 1 (the space o f all sequences for which the norm, X.'
IN , = X > n l ’ n~l
is finite, with elementwise linear operations) with the dual o f cq as follows: OO
u i:ix —►(cq)*
is defined by
Wb{a) = ^
anbn
n=l
for all a 6 co and i 6 f Then we can identify i° ° (the space of all bounded sequences with the same supremum norm as cq and elementwise linear operations again) with the dual of t l ~ (co)* by: OO
f : t ° ° -> (£ *)' where f„ (b ) = £
anbn
n= 1
for all a G i ° ° and b G t x. W ith these identifications (which the reader can easily check are isometric linear isomorphisms), the canonical injection of cq into its double dual identified as i° ° is just the usual inclusion as spaces o f sequences. Thus we will suppress the symbol for this injection. Now the following results axe immediate:
0 (w(,) = (w (,)0 = uJab
where (ab)„ = a„bn
V d G c o ; 6 g < *;
/(w(,) = (w6)/ = uifb
where (/&)„ = /„&„
\/f £ e°°; b £
( f 9 ) n = V - g ) n = fn 9 n
V / ,flG / “ .
Thus, in this case, the two products agree and just give the elementwise multiplication with which we would certainly have endowed t°° in the first place. This example is unusual. Most algebras axe not Arens regular and gen erally the double dual is quite complicated. The fact is that Co is an example, o f a commutative C*-algebra. Commutative C*-algebras will be introduced in Chapter 3, particularly Theorem 3.2.12. Not necessarily commutative C*-algebras axe briefly introduced in §1.7.17 and will be studied at more length in the second volume of this work. There we will show that any C*-algebra is Arens regular and that its double dual has a very natural interpretation as a universal W*-algebra (or von Neumann algebra) “com pletion” , o f the original algebra. Theorem 3.2.12 shows that every unital commutative C*-algebra has the form C (ft ) for some compact space ft and vice versa. Hence it is obvious
1 . 4.8
Arens Multiplication
57
that i ' ~ £°° is C ( f l ) for what turns out to be a rather complicated uxtre aely disconnected connected space. For generalizations, see Samuel Kaplf n [1985], 1 ■ S A re n s M u ltip lic a tio n on w h en £l has P o in tw is e M u l tip lic a tio n There are two possible fairly natural multiplications on l l : point'visc and convolution multiplication. We will examine the Arens multiplic t.ion in both cases. Although both multiplications are commutative on £' only the first is Arens regular. This is particularly interesting since, as we will note below, certain Banach space properties force Arens regular ity. This example shows that no Banach space property can characterize Arent regularity. Tt make these two examples more interesting, we wish to exhibit a reasonahiy concrete interpretation of the double dual space o f £l . It turns out I lat there are two concrete representations, each o f which has some interest. But before we even describe these concrete representations of ~ (co)***, we note a special property o f any triple dual Banach space X***-. it has a canonical contractive projection 7r onto X * . Any linear sub space which is the range o f a bounded projection 7r is complemented by the closed linear space (1 —1 In this case this complementary subspace is precisely « ( A ’) X, as the reader may easily check. The construction o f this projection will be given in Section 1.7. The first concrete realization o f (£°°)* ~ ( t 1)** — (co)*** we mention is the; given by Dunford and Schwartz ([1958] IV.8.16) as the space 6a (N ) of all complex, bounded, finitely additive set functions where the duality is implemented by simply integrating the sequence in £°° by the set function in 6g (N ). In this case the two complementary subspaces defined in the last paragraph are simply: 7r (ia (N )) is the space of countably additive measures on N which can obviously be identified with i 1 by the Radon-Nikodym theorem; and ( I — 7r )( 6a (N )) is the space of finitely additive measures which vanish on all finite sets. The second representation of the dual of £°° comes from noting that f.°° is a unital, commutative C*-algebra. Hence, by what is essentially a corollary o f Theorem 3.2.12 described in §3.2.15, £°° can be identified with the Banach space C'(/3N) of all continuous complex-valued functions on a compact Hausdorff space 0N, called the Stone-Cech compactification of N. There is a continuous injection of N onto a dense subspace o f /3N, and the function in C(/3N) corresponding to / 6 t°° may be considered as just the extension of / by continuity to all of /3N. Then the Riesz representation theorem identifies ~ (C(/3N))* with the Banach space M (8 W ) of all complex regular Borel measures on /3N. This time the pair of complementary spaces consists of the subspace of measures supported
58
1: Norrned Algebras and Examples
1.4.9
on N C /?N (naturally identified with £l , again by the Radon-Nikodym theorem) and the space o f measures supported on the complementary space 3N \ N. We now consider the Arens products arising from elementwise multipli cation on t 1 following Civin and Yood [1961], Theorem 1.4.2. In describing the situation, we use several concepts which will not be introduced until later in this work, and some of the ideas come from our discussion in Chap ter 3 of the Arens multiplication in commutative algebras 3.1.12. To begin with, the situation is similar to that of the last example. We identify £°° with the dual of i 1 ^ (co)* by: uj:£°°
-> ( t l y
where
OO u>a(b) = £ anbn n= I
for all a e t°° and b e t 1. We will use { «o / ) for the natural map of { Co / t 1 ) into its double dual space. We immediately see fc(w0) = (wa)fc = uiab
where
(ab)n = anbn
for all a £ t°° and 6 € £l . However, this shows that these linear functionals are all in «o(co), from which it follows that all Arens products with at least one factor from the complementary subspace to k i ( ^ ) in the triple dual space decomposition must be zero. Call this complementary subspace 71 = Ko(co)x - Thus the direct sum decomposition (£1) '* — « i ( ^ ) © TZ is an algebra direct sum, multiplication on (£*)** is commutative for either Arens product and thus the two products agree Moreover, it is easy to see that any non-zero homomorphism of ( l (and hence of (^* ) ’ * under either interpretation o f this as a space of measures) into C is given by evaluation at a positive integer n. A ll these evaluations belong to the dual space of ( l since they are obviously all contractions. Their closed linear span is just the image of co in £°° under the embedding «o-'Cq —» £°°. Thus the subspace of elements of (^’ )** which vanish on all these homornorphisms is just TL. Hence 1Z is the Gelfand radical and the Jacobson radical of (£ ')**, and £l (or equivalently k ^ 1)) is Jacobson semisimple. Thus, with pointwise multiplication on £1, the Arens multiplication on its double dual is extremely simple but badly behaved in that it has an enormous annihilator ideal.
1.4.9 Arens Multiplication on — M(/3N) when £l has Convo lution Multiplication In this example, it will be tidier if we consider the entries in the sequences in £x to be indexed starting with zero. Obviously this makes no difference to the Banach space structure. The convolution product is then defined by n { b * d )n = Y s bn- kdk k=0
V n £ N°; b,d £ £l
Arens Multiplication
59
which is easily seen to be commutative. Let Sn € £l be the sequence which is zero except at n where it is one (so (Sn)rn = Snrn, the Kronecker delta). Then So is a multiplicative identity and in general 6n * 6m — 6n+m so that £l with this product may be interpreted as the semigroup algebra of N° with addition for its semigroup operation. This time an easy calculation shows OO b (^ a )
(^ a )b
— ^aob
w h ere
(fl O 6 )n
fln + m 6m
m=0 for ell n G N°, a G £°° and b € £l . However, the formula for the next step in the definition would not help us to see that this algebra is not Arens regular. Instead we look at a very concrete construction using property (f) o f Theorem 1.4.11 below. Consider two sequences {& „} and { d „ } in £l where bo = 6q and, for n > 1, bn = and dn = <52an. These are chosen so that simple inequalities show that all bn * dm are distinct if their indices are different. Hence the characteristic function of {2 2n_1 + 22m : m < n } in t°° gives 0 for one o f the limits in equation (28) o f Theorem 1.4.11(f) and 1 for the other limit. Therefore this algebra is not Arens regular. The idea for this example comes from Young [1973b] where he proves the following surprising and comprehensive theorem. Semigroup algebras are defined in §1.9.15 below. 1.4.SO Theorem The follotuing are equivalent fo r any locally compact topological semigroup S in which multiplication is at least singly continuous: (a ) The measure algebra M ( S ) (with the topology o f S ) is Arens regular. (’:>) The algebra £l ( S ) is Arens regular. (c) There do not exist sequences {u n} and {u m} in S such that the sets {u nvm : m > n ) and {u nvm : m < n ) are disjoint. (d) The semigroup operation can be extended to the Stone -Gech com pactification (3S o f S as a discrete space. • When these conditions hold, £l (S )** with its Arens product can be natu rally identified with M(/3S) equipped with the convolution product based on the extended semigroup operation. V/e refer the reader to the original paper for a proof, although one can be based on our next result and is suggested by the last example. Arens Regularity Recall that an algebra is called Arens regular if the two Arens products agree on its double dual. This topic has attracted most of the attention devoted to Arens multiplication, at least until recently. As we will note, Ar ens regularity is uncommon. Duncan and Hosseiniun [1979] first formu lated the following comprehensive theorem this way, using ideas crystallized by Young [1976] from Pym [1965], and J. Hennefeld [1968].
60
1.4.11
1: Normed Algebras and Examples For any
uj
e A ' define
(K , - . A ^ A ' / P ^ .A -* A ' )
by
( A w( a ) = w a / P w(a) = 0w )
(26)
for all a 6 A . These maps satisfy ( K ( f ) = f u > / K ( f ) = ujf )
“ d
( A :*(/)(
(27) )
for all f , g & A **. The linear functional w on an algebra A is called weakly almost periodic if either (and hence both) and P w are weakly compact operators. This means explicitly that the weak closure o f thc^ets {w „ : a € A i } and {„ w : a € ,4 i} have compact closure in the weak topology (or A ** topology) on A " . A well known result (cf. Section 1.7 below or Dunford and Schwartz [1958] V I.4.2) states that a linear map is weakly compact if and only if its double dual is included in the canonical image of the original target space o f the linear map. We will use this criterion below.
1.4.11 T h e o r e m (a) (b) (c) (d) (e) (f )
The following are equivalent fo r a Banach algebra A : A is Arens regular. For each f 6 A ** the map g •—* fg is weak* continuous. For each g 6 A ** the map f t - * f ■g is weak* continuous. For each u> € A ’ the map Aw is weakly compact. For each uj € A * the map P w is weakly compact. Any two bounded sequences { a n} and { 6n} in A and uj € A * satisfy lim lim w (a „ 6m) = lim lim w (an6m). n
m
m
n
(28)
Prxwf The equivalence of (a), (b ) and (c) follow from the weak* continuity properties o f the Arens products contained in Theorem 1.4.2 and the density o f k(.A) in A " . The second result in equation (27) shows that if A is Arens regular, then it satisfies (d) and (e). Conversely, if either o f the maps in (d) and (e) is weakly compact, then its dual is continuous from the weak* topology on A ** to the weak topology (op. cit. V I.4.7) on A * which, by the first part of equation (27), implies (b) or (c). Finally, equation (28) is Alexander Grothendieck’s criterion [1954] for weak compactness o f either of the sets {w a : a G .A i} or { aw : a € A \ ). □ The following corollary can be based on criterion (f) of the last theorem, but the results were first noted by Civin and Yood [1961] where they were derived from the last sentence o f Theorem 1.4.2 above.
1.4.12 C o r o lla r y
A subalgebra or quotient algebra o f an Arens regular
algebra is Arens regular.
Arens Multiplication
61
Arens [1951b] gave the first example o f a non Arens regular commutative algebra. Civin and Yood [1961] proved many interesting results chiefly r e la i’ig to group algebras. In particular, they showed that for a locally compact abelian group G, L 1 (G ) is never regular unless G is finite. This was extended to arbitrary locally compact groups by Young [1973a]. Ian G. Craw and Young [1974] show that no locally compact group G has a weight function with Arens regular Beurling algebra (cf. 1.9.15 below) unless G is discrete and countable. Despite these negative results, Sheila A. McKilligan and t Ian J. W hite [1972] proved interesting properties o f quotient algebras for gioup algebras as well as more general Banach algebras. See also Civin [1961', [1962a], [1962b], Oshobi and Pym [1981], [1984], [1987] and Oshobi [1988 (see review). ¥. 3 will postpone until Volume II the interesting proof that for a Hilbert space f i , B (H ) and therefore all its closed subalgebras are Arens regular. Since the Banach algebra C o (fi) o f continuous complex-valued functions vanishing at infinity on any locally compact space Q is a closed subalgebra of so: :>e B ( f i ) (cf. Section 1.5, §1.7.17 and Theorem 3.2.12), any uniform algebra (3.1.8) or quotient o f a uniform algebra is Arens regular. The norm closure B a ( X ) is Arens regular for a Banach space X if and only if X is reflexive, but little positive is known about the Arens regularity o f B (X ). We mentioned additional results on Arens regularity in our discussion of semiregularity above. Our examples have shown that two different multiplications on the same Banach space can differ with respect to Arens regularity. Nevertheless, sufficiently strong conditions on the Banach space structure, short of reflexiv’ty, do imply regularity. (See Gilles Godefroy and Iochiun [1988].) Similar results have been obtained by Ulger [1987] and [1990a] and Angel Rodriguez-Palacios [1987], who consider a kind o f super regularity for Bar uach spaces. The Banach space of any C*-algebra satisfies this condition. P ;m and Ulger [1989] consider the Arens regularity o f inductive limit algeb as and Ulger [1988a] deals with projective tensor product algebras. Ulgc: [1990c] shows that the Banach algebra C (fl, A ) o f all continuous functions from a compact Hausdorff space fl to a Banach algebra A (with supre num norm) is Arens regular if and only if A is Arens regular. D. ncan and Hosseiniun [1979], following McKilligan and W hite [1972], establish a variant of Theorem 1.4.11 in which the maps A*, and P „ are required to be compact. This is equivalent to either Arens product being jointly continuous in each variable in the bounded weak* topology described in Dunford and Schwartz [1958] V.5.3. Bt fore the last theorem, weakly almost periodic elements o f the dual space A * o f an algebra A were defined. A linear functional w on an algebra A is called almost periodic if both A w and P w are compact operators. This rsear
explicitly that the the sets {w 0 ; a € A i } and { au : a £ A i } have
62
1 : Normed Algebras and Examples
1.4.13
compact closure in the norm topology on A *. If A is the usual group algebra L l (G ) of a locally compact group G, then A * can be identified with L ° ° (G ). This example motivates the name assigned to these functionals. When G is the real line under addition, the concept agrees with classical terminology. The sets of weakly almost periodic and almost periodic functionals on A will be denoted by A ’w AP and A'a p , respectively. The book by Robert B. Burckel [1970] gives a useful compendium up to the time o f its publication. See also Andrew Tonge [1980] and Jesus Gil de Lamadrid and Loren N. Argabright [1990]. k (A
) as an Ideal in A **
The ideas used in Theorem 1.4.11 allow us to say when k (A ) is a one- or two-sided ideal in A ' * . So many papers have been published on this inter esting question that the following simple straightforward characterization due to Seiji Watanabe [1974] is particularly satisfying. See also Michael Grosser [1979b], Ghahramani [1984]. 1.4.13 P r o p o s itio n The following are equivalent fo r a normed algebra A : (a ) k (A ) is a ( left / right ) ideal in >1 **. (b ) ( lia / L a ) is weakly compact fo r each a € A . (c) ( (R a )' / ( L a)* ) is weakly compact fo r each a £ A Hence, k ( A ) is a two-sided ideal i f and only i f both R a and L a are weakly compact f o r all a € A . P roo f Gantmacher’s theorem (cf. Dunford and Schwartz [1958] VI.4.8) shows the equivalence of the last two conditions. Condition (b) is equiva lent to (R * * ,L * * ) having range in k (A ) (op. cit. VI.4.2). However, as we have noted earlier, (R a) * * ( f ) = / « ( « ) and ( L a) " ( f ) = n ( a ) f , proving the equivalence to (a). □ Arens Representations In his original papers, Arens’ construction was extremely general. In particular, the situation he described applies not only to algebras but to certain representations as well. Representations will be studied systemat ically in Chapter 4, but here we will define a continuous ( representation / anti-representation ) T o f a Banach algebra A on a Banach space X to be a continuous ( homomorphism / anti-homomorphism ) T : A —* B (X ) with the image o f an element a € A under T denoted by T„. The Arens representations arising from a representation T are two representations T 1 , T 2: A ** —> B ( X " ) where T 71 is a representation with respect to the nth Arens product for n = 1,2. These representations extend T in the sense that the following diagram is commutative where ** denotes the map which assigns to any operator S its double dual S " . (The map on top is a homomorphism (or representation) with respect to the nth Arens product,
1-4.14
Arens Multiplication
63
as noted above.) ,4**
B (X * * )
J-
j ”
A
(29)
B (X )
Although the construction we are about to give is derived from Arens’ original papers, it was hardly utilized until quite recently. The present treatment is based on Palmer [1985] (cf. Grosser [1987]). The basic defini tion is similar to that of the Arens products. 1.4.14 D e fin itio n Let T : A —> B (X ) be a continuous representation of a ncmied algebra A on a normed linear space X . For any a G A , x G X and t (z X *, define elements ra G X * and x t G A * by r a = (Ta) * ( r )
and
xr ( a ) = r ( T a( x ) ) - Ta(x ).
(30)
For any a £ A , x £ X , T € X * , f € A ** and F G X **, define elements f t G A * and r / G X * by f t ( o)
= F ( r a)
and
r f {x ) = } { x t ) .
(31)
Finally, for any / G A **, t € X * and F G X **, define elements T j and T? of B ( X " ) by T } (F )(t ) = } ( f t )
and
T f ( F ) ( r ) = F ( r f ).
(32)
The ;wo maps T 1 and T 2 of >1** into B (X * * ) are called the first and second Arens representations, respectively. V/e again gather some elementary properties of these two representa tions. 1.4.15 T h e o re m Let T : A —> B (X ) be a continuous representation. The elements r a and xt belong to X * and A * , respectively, and satisfy: IM
< imi ||r|| 1 1ci11 ( r a) b = Tab
( 33)
||,r|| < ||r|| ||r|| ||*|| b(xT)(a)
( 34)
= Tab(x)
fo r all a,b G A , x € X and t € X * . The elements pT and Tf belong to A * and X * , respectively, and satisfy: ||F r|| < ||T|| ||r|| ||F|| ( f ' 7" ) * * —
f ( 7 ’o )
k (x
)T ~
xT
x{^f)
||T/|| < ||T|| ||r|| ||/|| {xT)f
T K{a) ~
Ta
Tf g “
(35) iT f ) g
( ^
)
64
1: Normed Algebras and Examples
1.5.0
fo r all a £ A , x € X , r € X ' , f , g € A ** and F € ft”*’ . Finally, the two maps X 1 and T 2 o f A " m to U ( X **) are continuous representations with respect to the first and second Arens product, respectively. Also T l and T 2 have bound ||T|| and each makes diagram (29) commutative. Proof All these results are again easy to obtain from step by step cal culations starting with the three step definition. The only intermediate equation we have not displayed above is t '{F )T — g (F t ) for all r 6 X * , g £ A " and F e X " . ’ □ Similar constructions are possible for anti-representations. Section 1.7 gives an explicit construction for the first Arens representation o f the alge bra o f approximable operators. Since the construction involves ideas and notation which will not be introduced until that section, we postpone it.
1.5
Algebras of Functions
The next six sections, beginning with this one, are simply intended to give the reader a stock of examples to which the theory presented in this work can be applied. The discussion in all these sections shows how natu rally Banach algebra concepts arise in examples. A t least at the beginning of each section, the exposition has been kept elementary. The present section is extremely brief because the topic belongs more naturally to the subject matter o f Chapter 3, which studies commutative algebras. In that chapter, we will show that a commutative Banach algebra, and more generally a commutative spectral algebra (cf. Section 2.4 for the definition), can be represented as an algebra o f bounded continuous functions with pointwise multiplication. In this section we briefly introduce such algebras. Let fi be a set and let C be a collection o f complex-valued functions on fi. I f this collection o f functions is closed under addition, multiplication and scalar multiplication, then it is an algebra under pointwise operations. Let us agree that, when we call any such collection o f functions an algebra, we mean that the algebraic operations are pointwise unless we clearly specify some other operations. If the functions in C are sill bounded, then the supremum norm or uniform norm defined by ll/lloo = s u p {| / M | :u ,e fi}
fe e
is an algebra norm on C. Convergence with respect to this norm is uniform convergence. Thus if C should be closed under uniform convergence, then it would be complete under the supremum norm. Hence C would be a Banach algebra. Let w be a point in fi. Then evaluation at ui defines a homomorphism 7 u,-C —►C by 7 „ ( / ) = f(u i) for all / e C. Unless all the functions in C
iA l
Algebras o f Functions
65
vai.ish at ui, 7 ^ will have all of C as its range. Hence ker(7 ^ ) will be an ideal of C which is maximal among proper ideals. (To see this, note that if 1 were an ideal properly between ker(7 w) and C, then 7 W(J ) would be a non-zero proper ideal of C.) Results in the next two chapters (particularly Corollary 2.2.3, Proposition 2.4.12 and the first portion o f Section 3.1) will show that in many circumstances, every ideal which is maximal among proper ideals of C will arise in this way. In particular, Theorem 3.2.12 and §3.2.15 show that the algebras of continuous functions considered below have this property. Consider the algebra B ( f l ) o f all bounded functions on fi. This is ob viously uniformly closed so the remarks above show that it is a Banach alg ibra under the supremum norm. There are various uniformly closed subalgebras o f B ( f i ) which we will consider in this work including certain sutaigebras o f measurable functions. However, the most important exam ples arise when fl is a topological space and we consider certain classes of bounded continuous functions on fi. l.li l A lg e b ra s o f C on tin u ou s Fu n ction s Let fi be a topological spa ’.e. We will define two algebras o f bounded continuous complex-valued functions on fi, each o f which is a commutative Banach algebra under the supremum norm. Let C (f i) be the algebra o f all bounded continuous complex-valued functions on fi, and let C 0(fi) be the algebra of all contin uous complex-valued functions on f i which vanish at infinity. (A function / 0 1 fl is said to vanish at infinity if for every constant £ > 0 there is a compact subset K of fi satisfying |/(w)| < e for all w G f l \ K . ) It is easy to che ’k that each of these algebras is a Banach algebra under the supremum norn. We usually assume that the space fi is completely regular and Hausdorff. (Some authors include Hausdorff as part of the definition o f completely regular, e:g., Gillman and Jerison [I960].) This condition can be defined by requiring that singletons are closed sets (i.e., fi is T x) and that, for any closed subset K C fl and any point ui G f i \ K , there is a continuous function /: fi —►[0 , 1 ] satisfying f ( K ) = {0},
f(uj) = 1.
It Jf easy to see that the concept o f complete regularity is not changed if we merely assume that / belongs to C ( f l ) , instead o f requiring that it have ranr e in [0,1]. In §3.2.15, we show that for any topological space fi there is a completely regular Hausdorff topological quotient space fi such that the natural map o f C (f i) onto C ( f i ) is an isometric isomorphism. Thus the assumption that fi is completely regular is no loss o f generality from the viewpoint o f Banach algebra theory. 'lecall that Urysohn’s lemma shows that a normal space (which we ass’-me Hausdorff) is completely regular. It is easy to show that a compact
(Ui
1: Normed Algebras and Examples
1.5.1
space is normal, that a subspace o f completely regular space is completely regular and that a locally compact space (which is a subspace o f its onepoint compactification) is therefore completely regular. In §3.2.15 we also show that C o (fi) is naturally isometrically isomorphic to C’o(ft), where fi is a locally compact quotient o f fi. Hence we will usua&ly assume that fi is locally compact when discussing Co (ft). We remark that the algebra Coo(fi) o f continuous complex-valued functions on fi with compact support is a dense ideal o f Co(ft) when fi is locally compact. The Stone-Weierstrass theorem gives a useful criterion for a subalgebra of C 0(ft) to be dense. In particular, it establishes the last remark. Since this theorem is frequently stated in less generality than we will want, we take this opportunity to state the result in the form we will need. If 0 is locally compact, then a subalgebra o f Co(ft) is dense if it is closed under complex conjugation, separates the points of 0 and does not vanish identically at any point o f fi. The last two requirements simply mean that, for any two distinct points ui and ui' o f fi, there are functions / and g in the subalgebra satisfying / (w ) / /(u/) and g(ui) / 0. See Robert B. Burckel [1972], [1984], Thomas J. Ransford [1984b], Jaroslav Zemanek [1978] for recent extensions and proofs. The commutative Banach algebras C (ft ) and Co(ft) are characterized as Banach algebras by Theorem 3.2.12. Note that C (ft ) is always unital and that C o (fi) is unital if and only if fi is compact, in which case the two algebras coincide. Furthermore C o (fi) is an ideal in C (ft). We will now show that extension o f the regular homomorphism described in Proposition 1.2.6 naturally identifies C ( fi) with the double centralizer algebra of Co(ft). T h e o r e m Let ft be a locally compact Hausdorff space. Then the natural homomorphism of C'(ft) into the double centralizer algebra 23(6'o(ft)) of C o(Q ) is an isometric isomophism onto V (C o (fl)). Proof To simplify notation, we write C, V and A for C (ft), X>(Co(ft)) and C o (fi), respectively. Proposition 1.2.3(c) and (d) show that T> is commuta
tive and that any double centralizer ( L , R ) € T> satisfies L = R so that L and R are both left and right centralizers. Hence we may identify T> with the Banach algebra o f left centralizers on A under composition as product and the operator norm. W ith this identification the natural map 6 : C —►T> has the form < H f)(g) = f g V/eC;,?eA We will now construct an inverse for this map. Choose L G T>. Let g and h be two functions in A which are non-zero at w € ft. Then the equation L (g )h = g L (h ) shows that L (g ){u i)/g(io) is independent of g so long as g(u>) ^ 0. Hence we can define /z,: ft —> C by / l(w ) = L (g )(u i)/g(ui) for any g which is non-zero at w. Since the same g may be used in some neighborhood of ui, we see that f l is continuous. It is bounded by ||L||, so
1.5.1
Algebras o f Functions
67
f i belongs to C. Clearly the map L h f L is an inverse for 0. Both maps are contractive, so they are isometric isomorphisms between C and T>. □ For later use we also need the following result on C'o(fi). T h e o re m Let fi be a locally compact Hausdorff space and let 1 be an ideal of C o (fi). Either there is some w G fi on which every g e l vanishes o r l includes Coo(fi), so 1 is dense in C o(fi). I f I is the kernel o f a non-zero homomorphism 7 : C o (fi) —> C, then the form er case holds and 1 = {/ G C o (fi) : f(u i) = 0 } so 7 (/ ) = /(w ) fo r all
f
e
C0(Sl)
P ro o f Suppose there is no u S f i on which every g e l vanishes. Choose a non-zero h e Coo(fi) For each w G supp(A), choose gu G J such that gu (w) = 1. If g* is defined by ff* (r ) = pw(r )* for all r G fi, then SwSuj = | G J satisfies 1 < g (u ) for w G supp(/i). Next choose k e Coo(fi) so that k(uj) = 1 for all w G supp(fc) and k(uj) = 0 if g(ui) < 1/2. Then kg~' is defined and belongs to Coo(fi), so h = (k g ~ 1)gh belongs to 1. Suppose 1 — ker(7 ). The former case holds since 7 is non-zero. Hence there is some w G fi on which every g e ker(7 ) vanishes. The range of a homomorphism is a subalgebra, so 7 maps onto C. Choose h G C o (fi) so that h(u>) = 1. Then for all / G C o (fi), 7 vanishes on (/ — 7 (/ ))ft, so (/ M - 7 (/ ))l= 0 . □ If i i - is a subset of fi, then it is clopen ( = closed and open) if and only if its characteristic function belongs to C (fi). Furthermore, an element / of C ( fi) is an idempotent (i.e., f 2 = f ) if and only if it is a characteristic function. Hence, fi is disconnected if and only if C ( f i) contains idempotents different from 0 and 1. Similarly, C o (fi) contains idempotents different from 0 if and only if fi has at least one nonempty compact clopen subset. A deep gereralization of these results, due to Silov [1953], is given in Corollary £.4.13 below. A simple application o f the Stone-Weierstrass theorem to the remarks of the last paragraph shows that, for a compact Hausdorff space fi, C ( fi) is generated as a Banach algebra by its set of idempotents if and only if fi is totally disconnected. Many other connections are known between the algebra structures of C (f i) and C o (fi) and the topological structure o f fi. Sane will be given in Chapter 3. The Gelfand theory presented in Chapter 3 shows that for any com mutative Banach algebra C there is a canonical locally compact space Tc and a canonical homomorphism 0:C —> C o(Tc). This space Tc is compact if and only if C is unital in which case C q(T c ) = C (F c ). In many inter
68
1: Normed Algebras and Examples
1.5.2
esting cases, the canonical homomorphism 0 is injective. Theorem 3.2.12 characterizes those algebras for which the canonical homomorphism is an isometric isomorphism. 1.5.2 T h e D isc A lg e b r a The above remarks justify the claim made earlier that C o (fi) and C (Q ) are universal commutative algebras in some sense. However, they are far from typical. We now present a simple but somewhat more general example. Let D be the unit disc, O = {A e C : |A| < 1}, and denote its interior by D° = {A G C : |A| < 1} as usual. The disc algebra j4 (0 ) is the algebra of continuous complex-valued functions on D which are analytic (=holomorphic) on its interior. These functions are bounded since D is compact. The algebra is closed in C (O ) and therefore complete in the supremum norm since the uniform limit o f analytic functions is analytic. (To see this, one may use Morera’s theorem: a continuous function on 0 ° is analytic if its integral around any circle in 0 ° vanishes.) Hence yt(0 ) is a commutative united Banach algebra. The maximum modulus principle shows that the map which restricts a function in .4(0 ) to the boundary T of D is actually an isometric isomorphism onto its range. This range is a proper subset of C '(T ) since the continuous function which takes each ( £ T to its complex conjugate certainly does not belong to it. These facts will be proved to gether with many other interesting properties of the disc algebra in §3.2.13. 1.5.3 T h e A lg e b r a C 1 (T ) as th e Q u o tien t o f a U n ifo r m A lg e b r a The algebra C '( T ) is just the linear space of functions on T which are continuously differentiable In the sense o f differentiation along T. (A n easy definition o f this differentiation is given at the beginning o f Section 1.8 if the reader wishes to see it.) This is a Banach algebra under pointwise multiplication and the norm
ll/llc * = ll/lloo + l l / ' I L
Vf e
c ‘ (T).
(T o check completeness, write / as an indefinite integral o f / '.) The following construction is due to Sten Kaijser [1976]. Consider the collection of all continuous functions / :T x D —►C. This is a Banach algebra under the supremum norm and pointwise product. W e will consider a closed subalgebra A defined by two conditions. First, for each £ € T the function A •—* /(£, A) should belong to the disc algebra. Any such function clearly has a series expansion /(£, A) = /n(C)^n in terms o f continuous functions /„ on T. Note that the first two terms in the expansion of the product f g have coefficient functions (f g )o = fo9o and (/
f M j ’ W 'd s . Jo
Matrix Algebras
’ .3.1
69
T iis condition obviously defines a subspace closed in the supremum norm. It. can be restated as requiring that fo be differentiable and satisfy /q = f x,
in which form we see that it defines a subalgebra. Hence .A is a Banach algebra o f functions under pointwise multiplication and the supremum (or uriform) norm. Such algebras are called uniform algebras. Note that the pr ivious two examples discussed uniform algebras also. We wish to write C J(T ) as a quotient o f A. The ideal J involved is de ined to be the collection o f functions f in A with expansions satisfying fo = f i = 0. This also gives a nontrivial example o f a split extension in the terminology o f Definition 1.2.9 since we may embed C 1 (T ) into A by an isomorphism x : X ( / ) ( C , A ) = / ( C ) + / '( C ) A .
T iis embedding is isometric by the maximum modulus principle. It is obvious that the quotient A/1 can be identified with C X(T ) and that x is a right inverse for the natural quotient map.
1.6
M atrix Algebras
For each n e N, the set M „ o f all n x n (complex) matrices is an algebra un ier the usual linear operations and matrix multiplication. The expression ( \ . )n x n stands for the matrix ( An A21
Aj 2
...
A i„ \
A22
■• •
A2n
V A„i
An2
* * ■ Ann /
The product o f ( K j ) nxn and (t* ij)nXn is the matrix
^ 2 AikHkj j ' nxn
Tb. se algebras form the most familiar reasonably representative family of algebras in which calculation is easy. It is surprising how often confusion or £ji ill-founded conjecture can be cleared up by checking the 2 x 2 matrix ah jra. We urge the beginning reader, who is probably already acquainted wit 1 the basic facts, to become facile with such calculations. O f course, A/„ hes dimension n2, has an identity element (which we denote by I ) and is no :ommutative unless n equals 1 . 1 ,: 1 M a t r ix U n its The usual choice o f a linear space basis for M „ is the E kt : 1 < k ,t < n } o f m atrix units where E kl has 1 in the ^-position ana zero elsewhere (i.e., the ij- e ntry is given by ( E k%
= SikSji,
70
1: Normed Algebras and Examples
1.6.2
where ,* is the Kronecker delta). The matrix units satisfy the product law: E 'j ■E kl = 6jk E lt V i, j, k,£. This relation provides an abstract description o f a matrix algebra: An algebra is isomorphic to M „ if and only if it contains a linear space basis ( E u : 1 < k,£ < n } satisfying the above product law.
1.6.2 The Trace We may define a linear map Tr: M „ —►C by describing its value on the above basis by Tr(.E‘-’ ) = <*>,_, for all i , j . This map is, of course, just the usual trace, defined explicitly by n 1=1 From the product law for matrix units we see T r ( £ IJ E kt) = 6jk6u = T r (E ktE ii ) . Hence we have the trace identity T r (A B ) = Tr(JBA)
V A, B e M „.
By using the product law for matrix units, it is easy to show that any linear function satisfying the trace identity is a multiple of the trace as defined here. For some purposes, the normalized trace which equals ^TV on M n is useful. 1.6.3 M a t r ix R ep rese n ta tio n s o f L in e a r O p e ra to rs We briefly review the connection which exists between linear operators on a finite-dimensional linear space with dimension n and n x n matrices. Let V be a linear space o f dimension n. Let V and W be two ordered bases for V with basis vectors r 1, t'2, . . . , t>" and u>', w2, . . . , wn , respectively. (A n ordered basis is simply a basis listed in a fixed linear order.) For each T € £ (V ), define n2 complex numbers {A i:, : 1 < t, j < n } by n i= l Denote the matrix (AtJ)nxn by M V (T ). For any other ordered basis such as W, M w { T ) is defined similarly. The main fact about this construction is summarized in the following theorem.
Theorem.
Let V be an ordered basis fo r the n-dimensional vector space V. Then M y (as defined above) is an algebra isomorphism o f C (V ) onto M „ .
1.6.4 Change of Bases The matrix corresponding to a given operator depends on the choice of ordered basis. For example, let V be the twodimensional linear space consisting of all differentiable functions /: R —> C
Matrix Algebras
1.5.5
71
such that f " = f . Consider the two ordered bases V and W given by the functions (ex p (-), e x p (- (- ))) and (cosh(-), sinh(-)), respectively. Differen tiation defines a linear operator D in £ (V ) and we have «.-< * > > -( J
;
J ).
We wish to recall the effect of changing bases. We continue the notation established above. Given the two ordered bases V and W , we can define an element C vw of £ (V ) by n
n
n
»=1
»=1
v
53 V *= 1
eV .
Note that C yw is invertible with inverse C w v • we obtain n
T C v w vJ — Tup = 5 3
Denoting M ^ ( T ) by
n
= C vw 5 3 My1'*-
»=1
i= l
Multiplying both sides by C y ^ and recalling the definition of M y (T ) , we gei: M W ( T ) = M v (C y lw T C v w ) = M v { C v w r xM v { T ) M v (C v w ). This shows that M w (T ) and M \r{T) are similar matrices. By reversing the above argument, we also see that any matrix similar to M y (T ) will be M w { T ) relative to a suitable ordered basis W . The trace identity shows that similar matrices have the same trace. Thus the foregoing discussion shows that, for any finite-dimensional linear space V and any linear operator T 6 £ (V ), we may define the trace of T by T r (T ) = T t( M v ( T ) ) where the choice of the ordered basis V is irrelevant. 1.6.5 Determinants It is possible to define the trace and the deterinmant of a linear operator on a finite-dimensional space directly without introducing bases (cf. e.g., Nickerson, Spencer and Steenrod [1959]), but th's is sufficiently difficult that it is seldom done. We will not do it here. Hence we will assume that the reader is familiar with a definition of the determinant of an n x n matrix and the following three basic facts. We use D e t(M ) to denote the determinant of an n x n matrix M . (1) D et(M iV ) = D et(M )D et(AT) V M , N € M n. ( 2 ) D e t(M ) = 0 if and only if M is not invertible.
72
1: Normed Algebras and Examples
1 .6.6
(3) D et(/) = 1. Properties ( 1 ) and (3) and the similarity of M y (T ) and M w { T ) show that D e t(M j/ (T )) depends only on T and not on the choice of ordered basis V . Hence, from now on, we often consider Det to be a multiplicative map from £(V ) into C (defined by D et(T') = D e t(M y (T ')) where the choice of V is irrelevant). Consideration o f the extended left regular representation shows that ev ery finite-dimensional algebra is isomorphic to a subalgebra o f some matrix algebra. W e will show later that every finite-dimensional prime algebra is isomorphic to M n for some value o f n. See Chapter 4 for terminology and Theorem 8.1.1 for a proof. Each M n is a simple unital algebra. This means that there are no ideals except { 0 } and M n. Proposition 1.6.10 establishes a more general result. Every finite-dimensional semiprime algebra is uniquely an internal di rect surn o f finitely many algebras, each of which is isomorphic to M n for some n, and every such direct sum is strongly semisimple. Finally ev ery finite-dimensional algebra is uniquely decomposable as the linear space internal direct sum of a (finite-dimensional) nilpotent ideal and a (finite dimensional) semiprime subalgebra.
1.6.6 Historical Remarks These last results were proved by Elie Car tan [1898] but are called the Wedderburn or W edderbum-Artin structure theorems because o f important generalizations due to Wedderburn [1907] and Emil Artin [1926], They are proved at the beginning o f Chapter 8 . Matrix algebras have an interesting history. They were invented and named by Arthur Cayley [1855], [1858]. However, quite independently, and from very different motives, they were re-invented by both Benjamin Peirce [1870] and his son Charles Saunders Peirce [1873], James Joseph Sylvester stated [1884] that at Johns Hopkins University in 1881, he gave the first course on matrix theory ever given anywhere. For additional historical comments, particularly on the interesting separate viewpoints of Benjamin Peirce and C. S. Peirce, see Thomas Hawkins [1972] and Karen Hunger Parsha11 [1985]. 1.6.7 N o rm s o n M n The algebra o f n x n matrices can be made into a normed algebra in many ways. Since any norm on a finite-dimensional space is complete, M n together with any of its algebra norms is a Banach algebra. Any two norms on a finite-dimensional space are equivalent, so that only when one is examining geometric rather than topological properties does it matter which algebra norm is used. The simplest norm for most
Matrix Algebras
73
calculations is the Hilbert-Schmidt norm defined by n
n
1/2
for all {d ij)nxn € M n. The submultiplicative property is verified by an application of the Cauchy-Schwarz inequality. Let C " be the n-fold Cartesian product o f C made into a linear space by :he usual pointwise linear operations. When the elements of C ” are wrr ten as column vectors, matrix multiplication defines an isomorphism bet’veen M „ and the algebra £ ( C " ) o f linear operators on C ". I f C " is provided with a norm so that it is a Banach space, then (because o f finitedip jnsionality) the Banach algebra B (C n) o f bounded linear operators is just £ (C n) provided with the corresponding operator norm. When these operator norms are carried back to M n through the isomorphism mentioned abo e, we get a large family o f algebra norms on M n. W e will give the operator norms on M 2 corresponding to the three most common norms on C 2. The operator norms for
corresponding to the ^-norm ( j|(«5, e)|| = |<5| + |e|), the ^-norm (||(i5, e)|| = (|<5|2 + |e|2) 1/2), and the £°°-norm (||(5,e)|| = max{|<5|, |e|}) on C 2 are, respec lively, llal|/i
=
max{|a| + J7 J, |/3| + |5j}
l|a||/a
=
2 - 1/2{|a|2 + |/3|2 + |7|2 + |^|2 + [(| a |2 + |/3|2 + |7 |2 + \6\2) 2 - 4|a6 - /37 |2]l/2} l/2
lla ll/°°
=
max{ja| + \P\, |7l + |<5|}.
The first and last of the above norms are easily checked. The second norm can >e most easily calculated from the properties of self-adjoint elements in C -algebras discussed in Volume II.
1 .6 .J T en so r P r o d u c ts o f M a tric e s We wish to recall an old-fashioned concrete construction of the tensor product o f two matrices. Let a = {ai j ). , Xn b = (b*j)mxm two matrices. Their tensor product a ® 6 is the r m x nm matrix ( c y )nmxnm defined as follows:
74
1: Normed Algebras and Examples
1.6.9
Example:
Another way to describe this product is to note E
® E k t = E { kn~ n + ' ) ( t n - n + j )
y 1 < i, j < n; 1 < k , t < m
where E * * is the ij-matrix unit in M„ and E kt is the A:£-matrix unit in M m. It is easy to check that this operation satisfies the following identities: (Aa + fic) ® 6
=
A (a ® b) + n (c ® b)
a ® (Ab + fid)
=
A(a ® b) + n (a ® d)
(a ® 6 )(c ® d )
=
a c ® bd
for all a, c € M n, 6, d € M m and A, /i € C. The formula for tensor products o f matrix units shows that the set o f all finite sums o f tensor products a ® b for a 6 M n and 6 € M m (which we would normally denote by M n ® M m) is all of M nm. Hence M m 0 M n is isomorphic to M n ® M m. The maps
a •—» a ® /
a € Mn
b>-> I ® 6
6 € Mm
are unital algebra isomorphisms of M n and M m, respectively, into M nm ~ M n ® M m. Note that the images under these isomorphisms of M n and M m in M nm are commuting unital subalgebras. Each is exactly the commutant of the other (Definition 1.1.4). Theorem 1.10.6 gives a general framework for this construction and show that the algebra M nm ~ M n ® M m together with these embeddings is the solution to a universal mapping problem. The Algebra M n( A ) o f n x n Matrices over an Algebra A 1.6.9 D e fin itio n Let A be an algebra. For each i and j satisfying 1 5 i , j < n let ai3 be an element o f A . Then ( a ^ )nxn represents the matrix / an ai2 ■• • flin > <*21 <*22 ■• • «2n
\®nl
®n2
•••
Ann'
We use M n(A ) to denote the set of all n x n matrices with entries from A The sum o f two matrices (a ,j)nxn and (b ,j)nxn in M „ ( A ) is (otj + 6 y )nxn, and the product of these matrices is
I
aik^ki )
Matrix Algebras
75
It is easy to chock that M n(A ) is an algebra. In Section 1.10, M n(A ) v/i!l be identified as the algebra tensor product of M n and A . The map which sends any element a of A to the diagonal matrix which has a at each position on the main diagonal (i.e., at each position with J — j ) and 0’s elsewhere is an isomorphism onto the subalgebra o f diago nal matrices. Another way to embed A isomorphically into M „ (A ) is to send each a € A onto the matrix with this element in its upper left corner (i.e. its 1 ,1-position) and 0’s elsewhere. Clearly there are many other sim ilar isomorphic embeddings. If A is a normed algebra or a Banach algebra, then M n( A ) is a normed dgabra or a Banach algebra under various natural norms. For instance, the matrix (a ij)nxn acts on the column vector ( 6 1 , 62, . . . , 6„ ) with entries 'ro'n A 1 by matrix multiplication a i2
•
f b i \
* £ £ = 1 aikbk >
021
<*22
•
<*2n
62
£ f c = l a 2 kbk
Vanl
«n2
•
flnn )
U n /
V £ f c = l a n k bk /
/a n
•
a in \
Clearly this action defines a bounded linear operator when the space (.41)" carries any one o f several natural norms such as ||(&i, 62, . . . , 6„)||oo = sup{||6i||, H&2 II,...., ||6„||}. The matrix (d ij)nXn can now be given the operator norm for this bounded linear operator. I f A is normed, M n (A ) may also be given the HilbertSchmidt norm: ||(oy)n>
M n (1 ) for some ideal T of A .
, Let A be a unital algebra and let B = M n(A ). Let e € B be the matrix E 11 and let ui € B be the matrix with ones on the diagonal immediately
76
1: Normed Algebras and Examples
1.6.11
below the main diagonal and in the upper right corner (the In-position) and zeroes elsewhere. It is easy to check that e and w satisfy the conditions of the next theorem. 1.6.11 T h e o r e m Let B be a unital Banach algebra. Then B is isomorphic to M n(A ) fo r some unital Banach algebra A and some n € N i f and only i f there are elements e and w in B satisfying n
wn — 1 = ^ 2 ,v P ~ l ewl ~i
and
ettri ~ l e = S^e
V j = 1 ,2 ,..., n. (1)
In this case, A is isomorphic to the subalgebra eBe and B is isomorphic to M n(eBe) under the isomorphism tp defined by
V 6 € B; j , k = 1 ,2 ,... ,n.
(2)
Furthermore, the map 0: eBe —» B defined by n
0(ebe) = y e tis ’ ~1ebew1~J
VbeB
(3)
is an isomorphism ont^ its range 6(eBe) C B. Each element o f 0(eBe) commutes with e and w. Finally, the center o f B is the image 0 ((e B e )z ) o f the center o f eBe under 0. P roo f The discussion preceding the statement o f the theorem shows that if B is isomorphic to M n( A ) then B contains elements satisfying (1). In this case eBe is obviously isomorphic to A . Suppose, conversely, that elements e and w satisfying (1) can be found in B. For any a, b e B and j , k = 1 ,2 ,..., n we get
(tp(a)tp(b))jk
=
=
n £ ¥>(<1 ) ^ ( 6) ^ i- 1 n 5 3 eU}1 ~3a (w '~ 1e2w1~%)bw1~ke = tp(ab)jk i=\
which shows that tp is a homomorphism. To see that it is surjective, note V b 6 B.
n
b = 5 3 w*~l ewl ~ib j =l fc=i
n
n
i= i fc=i
1.6.11
M atrix Algebras
77
For any a, b € B and j , k = 1 ,2 ,..., n the calculation n
w '~ 1ebew1~ '')w k~ 1 = Sjkebe «=i
shows that
( : ? ) - ( f d \ -c
-b \ ( a a ) \ c
b\ _ f d a - be d ) - \ - ca + ac
-a b + ba' —cb + da d b -b d \ —cb + a d ) ‘
I f a, b, c and d belong to a commutative algebra, the expressions on the right are just the determinant times the identity matrix, but if these elements do not commute, the right sides may be quite complicated. There are also four different ways to calculate the determinant. For specific examples, consider the algebra M 2 (M 2 ) o f 2 x 2 matrices over M 2 . I f we choose
-(;
» ) - ( “
i)
;)■
then the determinant calculated in any o f the four possible orders is invert ible but the matrix A
=
(
°
c
d
<4>
)
is not. In the other direction, if
•-(!! ?)■*-(!! J ) - ( ?
2)
S).
tie n the element A defined in (4) satisfies A 2 — I so that it is invertible even though its determinant is not invertible when calculated in any o f the four orders. Criteria for invertibility and one-sided invertibility in terms cf the determinant have been developed (cf. S. S. Kesler, and Naum Ya. Krupnik [1967] or Krupnik [1987] Theorems 1.2 and 1.3).
78
1: Normed Algebras and Examples
1.7.0
The following criterion is occasionally useful for matrices in M ^ iA ). If A has the form (4) and a is invertible, then A is invertible if and only if d — ca~l b if> invertible since A satisfies / I Vca->
0\ f a lyVO
0 W l d — ca~1bJ \0
f l - 'i A 1 ) '
C f the remarks following the proof of Theorem 2.2.14 and Danrun Huang [1992].
1.7
Operator Algebras
Algebras of operators on linear spaces are among the most important algebras to which the theory of this work applies. The extended left regular representation shows that every Banach algebra is isometrically isomorphic to a (necessarily closed) subalgebra of B (X ) for some Banach space X . The last section deals with the case of finite-dimensional vector spaces, so in this section we will usually assume that the vector spaces lack a finite basis and are therefore infinite-dimensional. Interesting results on infinite-dimensional linear spaces require some topology on the space and, in keeping with the general emphasis of this book, we will require a norm. In fact, the most common situation will be that in which the operators act on a Banach space. A t this point we do not have enough terminology for a detailed discus sion o f algebras of operators. Therefore, we will concentrate on introducing a few important ideals that occur in the algebras B (X ). The theory of these ideab becomes somewhat messy unless the Banach space is fairly well behaved. Prom another viewpoint, we are merely acknowledging that a perfectly general Banach space may be wildly poorly behaved. Fortu nately, most Banach spaces which appear in applications (other than as counterexamples) are relatively tame. Hilbert spaces are the best behaved of all infinite-dimensional Banach spaces. However, the study o f operator theory on Hilbert space is domi nated by the extra structure provided by the Hilbert space adjoint which makes many interesting algebras of operators into *-algebras. Since the second volume of this work is devoted to the study of *-algebras, we pre fer to postpone most of the study o f operators on Hilbert space until that volume. (A short summary is included in 1.7.17 below.) Therefore we will state results for operators on Banach spaces which satisfy an array of de sirable properties. There are many excellent books for further reading of which we mention a few. Dunford and Schwartz [1958] remains a basic ref erence but for newer developments Joseph Diestel and John Jerry Uhl, Jr. [1977], [1983] and Joram Lindenstrauss and Lior Tzafriri [1977], [1979] are particularly useful. See also Mahlon M. Day [1973], Diestel [1984], Graham
Operator Algebras
79
J. O. Jameson [1987], Vitali D. Milman and Gideon Schechtman [1986], Gilles Pisier [1986], Ivan Singer [1970], [1981], Nicole Tomczak-Jaegermann [1989], Dick van Dulst [1978] and Bernard Beauzamy [1982], 1.7.1 R e fle x iv e B an ach Spaces The most important distinction in Banach space theory is between reflexive and nonreflexive spaces. If A" is a normed linear space, then its double dual space X ** is just the collection of continuous linear functionals on its dual space X * . Thus evaluation provides a natural linear map k : X —* X ** defined by k ( x ) ( uj)
= uj( x )
V i G i t 1; u> G X * .
It ;s easy to see that this is an isometric linear isomorphism of X into X * * . Thus it is often useful to think o f X as a subspace of X " . D e fin itio n A Banach space is called reflexive if the map « , defined above, is surjective. It is important to note that the definition requires that k be the map which establishes a linear isometry between X and X * * . Robert C. James [1951] provided an example o f a (nonrefiexive) Banach space J (called the James space) which is isometrically isomorphic to J ** but in which k ( J ) is a codimension one subspace of J ' * . (Alfred Andrew and William L. Green [1930] show that J is a Banach algebra under pointwise multiplication and that the Arens product on J ** is the unitization o f J . They determine the automorphism group of this algebra in detail. See also Shashi Kiran and A jii Iqbal Singh [1988].) T h eo rem The following are equivalent fo r a Banach space X : (a ) X is reflexive. (b) X * is reflexive. (c) The weak topology (i.e., X**-topology) on X * equals the weak* topology (i.e., X-topology). (d) The unit ball o f X is compact in its weak topology. (e) The unit ball o f X * is compact in its weak topology. P roof (a)=>(c): This is obvious. (c)=>(e): Alaoglu’s theorem (Dunford and Schwartz [1958], V.4.2) shows this. (e)=>(b): Goldstine’s theorem (Dunford and Schwartz [1958], V.4.5) asserts that the image under k o f the unit ball of any Banach space ^ is dense in the unit ball y * * of its double dual space in the weak* topology c f ohat space. When this topology is transported back to 3A by /e, it is just weak topology. Hence if X j is weakly compact, its image under k must fcs all o f X * '" . Thus k is surjective. (b)=>(a): Let k 0: X —* X ,m and K\: X * —* X *** be the natural maps. Condition (b ) shows that any n in X *** has the form fl = Ki(a>) for some
80
1: Normed Algebras and Examples
1.7.1
ui 6 X " . Therefore fi(/co(x)) = k i ( ui) ( k,o( x ) ) — Kq ( x ) ( uj) = w (x) for all x £ X shows that Q is zero if it vanishes on /co(A’). The Hahn-Banach the orem then shows that kq( X ) is norm dense in X * * . Since «o is an isometry, its range is complete and thus norm closed. Therefore kq is surjective. Now the equivalence o f (d ) and (e) follows from the equivalence of (a) and (b). □ Relatively few Banach spaces are reflexive. The best known examples are I ? and t? for 1 < p < oo. Hilbert spaces are reflexive. In fact if 'H is a Hilbert space, then it is isometrically real-linearly isomorphic to H * but the isomorphism is conjugate linear rather than complex linear. In a nonrefiexive space, the tower o f dual space X , X * , X *’ , ... usually grows very rapidly (Th e space J , mentioned above, is a weird exception.) In fact, nonreflexive Banach spaces X with X * separable are comparatively rare and concrete representations for X *** are even rarer among nonreflex ive spaces. The theorem implies that every closed subspace and every quotient space o f a reflexive Banach space is reflexive. We mention two additional interesting properties which characterize reflexive Banach spaces. The weak compactness o f the unit ball shows that any continuous linear functional on a reflexive Banach space actually attains its norm at some point in the unit ball. (I.e. for each u> € X * , there is some x € X\ satisfying u j ( x ) = ||w||.) This property characterizes reflexive Banach spaces (James [1972]). I f a reflexive Banach space is given an equivalent norm, it is again reflexive by condition (c) of the last theorem. Thus, in particular, it is a dual Banach space. Again this property is characteristic. That is, any nonreflexive Ba nach space X has an equivalent norm in which it is not isometrically linearly isomorphic to a dual Banach space. (See van Dulst and Singer [1976] and van Dulst [1978], 4.8 for a slightly stronger result.) The following result is due to Nicholas J. Young [1976]. T h e o r e m I f X is a reflexive Banach space, then every continuous linear functional on B k ( X ) is weakly almost periodic. Conversely, i f X is a non reflexive Banach space, then no non-zero continuous linear functional on B f - ( X ) (with its operator norm ) is weakly almost periodic. Recall (from Theorem 1.4.11) that a normed algebra A is Arens regular if and only if every continuous linear functional on A is weakly almost periodic. Also, subalgebras and quotient algebras of Arens regular algebras are Arens regular. For a normed algebra A (such as B p ( X ) ) which is not complete, Arens regularity and the weakly almost periodic character of linear functionals do not change when the algebra is replaced by its completion. There are two easy and interesting corollaries.
.7.2
Operator Algebras
C o ro lla ry Let X be a Banach space. and only i f X is reflexive.
81
Then B k { X ) is Arens regular i f
C o ro lla ry Let ( X / y ) be a ( nonreflexive / reflexive ) Banach space. Then there is no non-zero continuous homomorphism o f B p ( X ) into B ( y ) . Young shows that reflexive Banach spaces X exist for which B ( X ) is not Arens regular. In fact, he exhibits a reflexive Banach space X for which B k ( X ) is Arens regular and B k ( X ) ” with Arens multiplication is homeomorphically isomorphic to B ( X ) (cf. §1.7.14 below), but B ( X ) is not even Arens regular. Hence an algebra A may be Arens regular while its double dual A** (with Arens multiplication) is not Arens regular. Pym [r 383] gives an easier example. Ulger [1988b] proves a stronger form of Arens regularity for B k ( X ) when X is reflexive. 1.7.2 T o p o lo g ic a l and G e o m e tr ic C a te g o rie s o f B an ach Spaces In Section 1.2 we mentioned the existence of two categories of Banach algebras. We have the same phenomenon in the present case also. W e consider the topological category, in which the morphisms are continuous linear maps, to be more fundamental than the geometric category, in which the morphisms a-e contractive linear maps. In the first o f these categories, objects are isomorphic if there is a homeomorphic linear isomorphism between them, ar.d thus the objects are really topological algebras in which the topology is derived from a complete norm. In the geometric category, two objects are isomorphic if there exists an isometric linear isomorphism between them, and thus the precise value of the norm itself is part of the isomorphic struc ture of an object. Traditionally, two Banach spaces are called isomorphic if they are isomorphic in the topological category and isometric if they are isomorphic in the second category. We observe this distinction. We will stress properties of Banach spaces which are isomorphism invariants. We have already noted that reflexivity o f a Banach space is an isomorphism invariant. We wish to introduce four additional properties related to refiexivity. In two cases it requires care to define them in an isomorphism invariant way. Ws begin by considering a more restrictive condition. 1.7.3 S u p e r-re fle x iv e B an ach Spaces Let us briefly discuss Hilbert s ' stce as a special case of a Banach space. We noted earlier that Hilbert space is usually defined in terms o f an inner product and this gives rise tc the Hilbert space adjoint o f a bounded linear operator and thus to the *- algebra structure of algebras o f operators. A Hilbert space inner product (cf. 1.7.17) can easily be introduced into any Banach space for which the norm satisfies the parallelogram law: x + y f + \ \ x - y f = 2 \ \ x f + 2\\yf
82
1: Normed Algebras and Examples
1.7.3
in such a wav that the inner product defines the given norm. (A proof is included in Volume II.) This condition may be considered as asserting the perfect roundness o f the unit ball, and it turns out that reflexivity is related to the absence of infinite-dimensional flatness on the surface of the unit ball. James A. Clarkson [1936] introduced a condition which he called uniform convexity. D. P. Milman [1938] showed that it implied reflexivity although it is not enjoyed by all reflexive Banach spaces. (See Kosaku Yosida [1965] V.2.2 for a proof.) It is a sort of weak form of the parallelogram law and, like it, is certainly not an isomorphism invariant. Uniform convexifiability, super-reflexivity and related properties were studied extensively by James, but the final characterization is due to Per Enflo [1972] (cf. van Dulst [1978]). D e fin itio n A Banach space X is called uniformly convex if the following implication holds for any two sequences {x „}n e N and {t/n}neN o f elements in the unit ball: (ll^ ^ l!- l)^ (H Z n - y n | | - 0 ). A Banach space X is called uniformly smooth if the norm is uniformly differentiable in the sense that the limit: lun t-o
~ ll^ll t
exists uniformly for x, y G X \, x ^ 0. A Banach space y is said to be finitely representable in a Banach space X if, for every finite-dimensional subspace T o f ^ and every e > 0, there is a linear injection 6 of T into X satisfying
(l-e)llwll
Vj/SJF.
Finally, a Banach space X is said to be super-reflexive if any Banach space y which is finitely representable in it is reflexive. Finite representability is not a demanding condition. For instance, for every Banach space, X ** is finitely representable in X . More surpris ingly, there is a reflexive Banach space U in which every Banach space is finitely representable. The space U is just the ^-direct sum of the spaces C n for all n 6 N where the individual pieces carry the sup norm: IKA i, Aa,. . . , A„)[|oc, = inax{]Ai [, |A2|,. . . , |An|}. Obviously U is not superreflexive. Moreover, I 2 is finitely representable in any infinite-dimensional Banach space. T h e o r e m A Banach space is uniformly convex i f and only if its dual space is uniformly smooth. A uniformly convex or uniformly smooth Banach space is reflexive.
1 .7.4
Operator Algebras
83
'The following conditions are equivalent fo r a Banach space X : (a) X is super-reflexive. (b) X * is super-reflexive. (c ) X is uniformly convex in an equivalent norm (i.e., it is isomorphic to a, uniformly convex space). (d) X is uniformly smooth in an equivalent norm (i.e., it is isomorphic to a uniformly smooth space). (e) X has an equivalent norm m which it is both uniformly convex and uniformly smooth. We prove only that a uniformly convex space is reflexive. B y considering the converse of the definition, it is easy to see that a space is uniformly convex if and only if for every e > 0, the number
6(e) = in f{1 - | | ^ | | : x ,y e
\\x - y|| > e)
is positive. (A deeper and surprising result states that, if 6(e) is positive for any single value of e with 0 < e < 2, then the space is uniformly convex (van Dulst [1978] p. 188).) We have already noted Goldstine’s result that the'unit ball of a Banach space is dense in the unit ball of its double dual with respect to the weak* topology on the latter space. Thus any x ** € X ** of norm 1 is the weak* limit of a net { x a }aeA Q X\. The net { ( x a + xp)/2}^a /3^ AyA also converges to x**. Hence the weak* lower semicontinuity of the norm implies that this net satisfies the condition in the uniform convexity criterion and thus lim(Q ^j ||xQ — xp\\ = 0. Therefore the original net is a Cauchy net in the norm. Hence it must converge to an element of X which n sends onto x**. See van Dulst [1978] for a full proof. We will mention other conditions below which are equivalent to superreflexivity. However we will omit the large number o f equivalent, more geometrical properties based on the growth of trees in the space. Those who wish to learn more should consult the very readable book by van Dulst [1978].
1.7.4 W e a k S eq u en tia l C o m p leten e ss The weak topology on a Banach space is seldom a metric topology (except in the comparatively rare case when its dual space is separable). Hence there is no reason to expect that, for example, a weak limit point of a sequence will be the weak limit of a subsequence. Nevertheless, since the very beginning of the subject, numerous results have been found showing that sequences are unusually useful in the weak topology. Probably the best known result is the EberleinSm'jlian theorem which asserts that weak compactness of a subset 1C of a Lanach space is equivalent to weak countable compactness (i.e., every sequence has a weak limit point in AC) and to weak sequential completeness (i.e. every sequence has a subsequence converging weakly to an element in .2). ?or an easy proof see Robert W hitley [1967].
1: Normed Algebras and Examples
1.7.5
We now introduce an important sequential property which is enjoyed only by certain Banach spaces. D e fin itio n A sequence { i „ } n6N in a Banach space X is said to be a weak Cauchy sequence if for each to & X * the sequence o f numbers {w (x n) } ngN is a Cauchy sequence. The Banach space X is said to be weakly sequentially complete if every weak Cauchy sequence has a weak limit in X. P r o p o s itio n
A reflexive space is weakly sequentially complete.
P roof A corollary of the uniform boundedness principle shows that a yeak Cauchy sequence is bounded. Since the weak and weak* topologies agree on a reflexive space, the sequence must have a weak limit by the Alaoglu and Eberlein Smulian theorems. (This result can be established without the last theorem.) □ In addition to reflexive spaces, most Banach spaces of measures and L 1and ^'-spaces are weakly sequentially complete. See Dunford and Schwartz [1958], pp. 374-379. 1.7.5 D u a l Spaces Obviously a reflexive Banach space is always the dual of some other space, namely its own dual. W e have already noted that reflexive spaces are the only Banach spaces which, when given any equivalent norm, are still isometric to a dual space. However, a space X which is a dual space in some norm has interesting properties. This is particularly true if it satisfies some other property such as separability or possession o f a basis. P’or any dual space, there is a canonical projection o f X " * onto X " . However, this property does not characterize dual spaces-Banach spaces with this property are known which are not dual spaces in any equivalent norm (e.g. Z^1([0 ,1 ])). We now give the interesting construction of this projection for a dual space. Let X be a Banach space and let /to: X —* X ** and k \ : X " —► X *** be the canonical injections. Then for the dual « (*: X *** —> X * of /to, the map k S o k ,:* * -
X*
is the identity by an easy calculation. Hence k y
- . *.
V»***
o Kq, a
•»>***
— ►t K
is a norm 1 projection onto the subspace n \ ( X ' ) . We will denote this projec tion by 7r. It is easy to see that the kernel of n is the closed complementary subspace ( I — n ) ( X " * ) = kq(X )-1-. 1.7.6 T h e R a d o n -N ik o d y m an d K re T n -M ilm a n P r o p e r tie s The Radon-Nikodym property has great importance in many areas but it arose
1.7.1
Operator Algebras
85
in the study o f vector measures. The excellent book by Diestel and Uhl [1977] presents many equivalent conditions, summarized on page 217 (cf. [1983] also). Because space does not permit a discussion of vector measures here, we will use one o f the easy geometric conditions in our definition. Our chief interest lies in another consequence o f the Radon-Nikodym property which will be introduced after certain ideals o f operators have been defined. D e fin itio n Let Af be a Banach space. (a) Let K be a convex subset of X . A point x € K is said to be an extreme point of K. if it is not the midpoint of a positive length line segment with end points in K. Similarly, x € AC is a strongly exposed point of K if there is a linear functional u 6 X * with the following two properties: (1) uj( x ) is strictly greater than u>(y) for any other y 6 K and (2) if any sequence { x n}.,gN in K satisfies lim w (xn) = u>(x), then the sequence converges to x. (b) X is said to satisfy the ( K rein -M ilm an / Radon-Nikodym ) property if evwry nonempty norm closed bounded convex subset o f X is the norm closed convex hull o f its ( extreme / strongly exposed ) points. It is not known whether the two properties introduced in (b ) are actually equivalent. The classical Krein-Milman theorem asserts that, in any locally convex topological linear space, every nonempty compact convex subset is the closed convex hull o f its extreme points. Hence, in a dual Banach space, a nonempty norm closed bounded convex subset is the weak* closed convex hull if its extreme points. If the Banach space were reflexive, so that the weak* and weak topologies agreed, this would make the set the weak closure and hence the norm closure of its convex hull. Obviously a strongly exposed point is an extreme point. These two remarks establish part of the next theorem. See Diestel and Uhl [1977] for a full proof. T h e o re m A reflexive space or a separable dual space has the RadonNikodym property. Any space with the Radon-Nikodym property has the K rein-M ilm an property and the two properties agree on dual spaces. Finally, a Banach space is super-reflexive i f and only i f every space finitely representable in it has either the Radon-Nikodym or the KrexnM tin an property. It every Banach space finitely representable in a given space X has a certain property, then we say that X has the super-property. Thus the above theorem shows that super-reflexive is equivalent to super-RadonNikodym and to super-KreTn-Milman. It turns out that there are other properties for which the super version is equivalent to super-reflexive. We introduce some briefly. e rst let us consider the girth of the unit ball in a Banach space. To unde stand this concept, consider first two-dimensional real space where our geometrical insight is good. We will introduce three possible unit balls
86
1: Normed Algebras and Examples
1.7.6
and then measure the distance around them in the norm which they define on the space. A circular disc defines the ordinary Euclidean norm and its girth is 2n. A hexagon or parallelogram centered around the origin give girths of 6 and 8, respectively, and it turns out that these are the extreme values. In a higher dimensional space, we wish to measure the minimal distance of a closed curve on the surface of the unit sphere which contains the antipode —x o f any point x in the curve. It is easier to think of finding the shortest path from any point to its antipode and doubling the length of this path. The rather astounding fact is that, for some infinite-dimensional Banach spaces, the distance on the surface of the unit sphere from a point to its antipodal point can be as short as 2 (i.e., the same as the straight Une distance through the origin.) A Banach space in which this can happen is aptly called flat and a space where it does not is called nonflat. We will show that X = C ([0 ,1]) is flat. Consider the path x: [0,2] —►X defined by x( s) = f , where if if
0 < t < s/2, a/2 < t < 1.
(1)
It is easy to check that this path satisfies ||/,|| = 1 and \\ft — f r || = |a — r| for all s and r in [0,2] and /2 = —fo-
Theorem
I f a Banach space X is flat, then so is its dual X * . In this case, X is not reflexive and its dual X * is not separable.
We can embed C ([0 ,1]) isometrically into £°° by simply enumerating the rational numbers in [0,1] and mapping a function to the sequence o f its values. Hence t°° = cj* is flat, but Co is not since its dual i 1 is separable. It turns out that l l is not flat either. We consider two more properties. A Banach space is said to have the Banach-Saks property if each of its sequences { x „ } n g N has a subsequence { x nt for which the sequence { n -1 Y12=\ x m. }neN o f Cesaro means con verges. Uniformly convex spaces have the Banach-Saks property and spaces with the property are reflexive, but the converses o f both these implica tions fail. In particular, the Banach space U , introduced in the discussion o f super-reflexive spaces, has the Banach-Saks property. Finally, a Banach space X is said to be ergodic if for every linear isometry T € B ( X ) the sequence { ( n + 1) ' 0T k} n f ^ of Cesaro means converges to zero in the strong operator topology. Reflexive spaces are known to be ergodic but, again, the converse fails. Both of these concepts were introduced in order to state the following theorem.
Theorem
The following six properties o f a Banach space are equiva lent: super-reflexive, super-Radon-Nikodym, super-Krein-Milman, super nonflat, super- Banach-Saks and super-ergodic.
1.7.7
Operator Algebras
87
All these results are proved in Diestel and Uhl [1977] and van Dulst [19781 Before we discuss the final two properties of Banach spaces which we want to survey, the existence of bases and the approximation property, we will introduce several ideals of operators.
1.7.7 The Ideals of Approximable and Compact Operators The ideal B f ( X ) of finite-rank operators on a normed linear space X was in troduced in Definition 1.1.17. Proposition 1.1.18 shows that B f ( X ) is an ideal of B ( X ) which is included in every non-zero ideal of B ( X ) . We recall the formulas. Let W , X, y and Z be normed linear spaces. For elements y € y and w € X * , define an operator y ® u € B ( X , y ) by (y
VzSzX.
(2 )
We denote the linear span o f all such operators by Bp (X , iV) and call oper ators in this space finite-rank operators. We simplify B f ( X , X ) to B p ( X ) . For arbitrary S 6 B { W , X ) , T € B ( y , Z ) , y € y , u € X * , z 6 Z and r € y * these operators satisfy (z 0 r ) ( y
—
T(y)z®u) T (t/ )® w
( y<8>uj)S =
y ®S * ( u i )
(x® u >)*
u
=
(3)
0 k ( x ).
We introduced tensor product notation in the last section in a prelimi nary way. The concept will be studied more systematically in Section 1.10. When this is done, it will be seen that Bj ?( X) is isomorphic as a linear space to the algebraic tensor product X ® X *, and that the notation given for the operators x ® ui is consistent with the general notation. Furthermore, a matrix representation for the operators x ® ui can be constructed explic itly as a generalization o f the explicit construction o f the tensor product M n ® M m given in the last section. To make things simpler, we will assume that X is finite-dimensional and that { x i , i 2 , . . . , i n} is a basis for X . Let U 2 , ... ,wn} be the dual basis for X * used in the proof o f Proposition 1.1.18. Then the matrix for x & u is nxn'
where x = X)"= i^ * x t and u = Y^j=\ When x is considered as an n x 1 matrix (a column vector) and ui is considered as a 1 x n matrix (a row vector) so that the evaluation ui(x) is just matrix multiplication, then the above formula for the matrix o f x ® and the explicit formula for the matrix a ® b (for a € M „ and b € M m) given in the last section agree.
88
1: Normed Algebras and Examples
1.7.7
R e m a rk and C o n ven tio n We are about to introduce a number of ideals of B ( X ) . However, all these ideals may be extended to be subspaces of the normed linear space B ( X , of bounded linear operators from X to y. In this setting they still form a sort o f generalized ideal in the sense indicated by equations (3) above. Prom now on, we will freely use our terminology for operators in B ( X , when this is convenient even when the definition is given only for operators in B ( X ) . In every case definitions and proofs are obvious generalizations. In a Banach algebra, the closed ideals are usually more important than the nonclosed ideals. It is easy to see that B f ( X ) is never closed unless X is finite-dimensional, in which case it equals B ( X ) . D e fin itio n The closure of B f ( X ) is called the ideal of approximable operators and denoted by B a ( X ) . Obviously this ideal is included in each non-zero closed ideal o f B ( X ) . For some common Banach spaces X , it is t he only proper closed ideal of B(X). D e fin itio n An operator T in B ( X ) is said to be compact if the norm closure o f T ( X \ ) is compact in the norm topology o f X . The set of compact operators in B ( X ) is denoted by B k ( X ) . P r o p o s itio n B k { X ) is a non-zero closed ideal in B ( X ) which includes the ideal o f approximable operators. In an infinite-dimensional Banach space it is always proper (i.e., not equal to B ( X ) ) . P roo f To see that B k ( % ) is closed, we use the fact that a subset o f X has compact closure if and only if it is totally bounded. The sum of two compact operators is compact because the sum of two compact sets is compact. If K is a compact operator end T an arbitrary bounded operator, then T K ( X \ ) is the continuous image of a compact set and hence compact. Similarly K T ( X i) is included in ||T||A'(A’1) and hence is compact. Thus the compact operators form an ideal. Any finite-rank operator is compact since the unit ball o f a finite di mensional normed space is always compact. Hence B k ( X ) is non-zero and includes the ideal o f approximable operators. If X is infinite-dimensional, we may choose a strictly increasing sequence o f closed subspaces X (1) C X (2) Q X ^ C • •• C X . Hence by the Riesz lemma, stated and proved below, we may choose a sequence of unit vectors Ifn € X (n) satisfying ||t/m — t/„|| > 1 — £ for m n. This shows that X\ is not compact. Hence the identity operator I is not compact. □ The following result was proved by Frigyes Riesz [1918],
1,7.7
Operator Algebras
89
R ie s z L e m m a I f y is a proper closed subspace o f a normed linear space X , then fo r every e > 0 there is a vector x € X \ y satisfying ||x|| - 1 and ||x 3>|| = inf{||x + y\\ : y 6 3>} > 1 - £■ P roo f We start with an arbitrary z € and note that ||z+,y|| is positive sine 3 ^ is closed. Hence there is a (non-zero) vector y € y satisfying II-z + y|| < ||2 + 3^1(1 - £) ~ l - An easy calculation shows that the vector x = ||z + t/||_1(z + y) has the desired property. □ The fact that B ( X ) has a nontrivial (i.e., a non-zero, proper) closed ideal when X is infinite-dimensional contrasts with B ( X ) ~ M n for a finite dimensional normed linear space X of dimension n. In the last section we noted that M n has no ideals except { 0 } and M n. For all classical separable Banach spaces, the ideals o f approximable and compact operators are equal. It was a long standing open question as to whether this was true for all separable spaces. Enflo [1973] settled the question by giving a counterexample. Schauder’s theorem (Dunford and Schwartz [1958] VI.5.2) states that an operator is compact if and only if its adjoint is compact. We prove the next proposition both for historical reasons and because we will use it later in this section. P r o p o s itio n For any Banach space X , a compact operator sends weakly com argent sequences into norm convergent sequences. P roo f Suppose T is a compact operator and { x n} ners C X is a sequence converging to x in the weak topology. For any ui G X * , the numbers ujTxn - u)Tx = uj( T x „ - T x ) = T * (w ) ( xn — x) form a sequence converging to zero. Hence { r i n} neN converges weakly to T x. Suppose { T x n} n6N does not converge to T x in norm. Then there is a constant 6 > 0 and a subsequence which we again denote by {a ; „ }„ eN such that { r a ; „ } n6N remains outside the 6-ball around T x . Since {z „ }n e N converges weakly, the uniform boundedness principle shows that it is a bounded sequence. Since T is compact, this shows that there is a subsequence o f { T x n} n(EN which converges in norm to some element y outside the 6-ball around T x . However this would imply that the subsequence converges weakly to y which is impossible since it converges weakly to T x. This contradiction proves that fT x n} n6N converges to T x in norm. □ C o ro lla ry F or any Banach space X , a norm continuous operator is also wmk'.y continuous (i.e. any operator in B ( X ) is continuous as a function from X with its weak topology to X with its weak topology). Proof Use nets instead of sequences in the first step of the last proof.
□
90
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1.7.8
If A" is reflexive, the property of compact operators proved in the above proposition characterizes them. This is an immediate consequence of the Eberlein-Smulian theorem and the results on reflexive spaces given above. David Hilbert [1906] originally defined compact operators just on Hilbert space in terms of this characterization. The definition o f compact operators that we have used is due to Frigyes Riesz [1918]. Calkin [1941], in one o f the earliest papers dealing with the ideal theory of a Banach algebra, showed that when X = H is a separable Hilbert space then every ideal of B ( H ) lies between B f ( H ) and B k ( H ) . The result is known to hold for some additional Banach spaces (I. A. Feldman, Israel Tzutikovic Gohberg and A. S. Markus [I960]) and to fail for others (Dunford and Schwartz [1958] Chapter V I, and Seymour Goldberg [1966]). Calkin also classified all the ideals o f B ( H ) . We discuss several other ideals related to Bk (<¥) next. See Section 2.8 below for the spectral properties o f compact operators.
1.7.8 Completely Continuous, Weakly Compact and Strictly Sin gular Operators An operator in B ( X ) is called completely continuous if it maps weakly convergent sequences into norm-convergent sequences. It is easy to see that the set B c c ( X ) of completely continuous operators is a closed ideal of B { X ) . We have shown that this ideal includes B k { X ) and that it equals B k ( X ) if X is reflexive. The term “completely continuous” with this meaning was used by Hilbert [1906] for operators on Hilbert space, where it has the same meaning as “compact” , before the term “compact” , was invented. An operator T in B ( X ) is called weakly compact if the weak closure o f T X i is compact in the weak topology o f X . Because bounded linear operators are weakly continuous, it is easy to see that the set B w k { X ) of weakly compact operators is an ideal of B ( X ) containing the ideal of compact operators. Since a Banach space Z is reflexive if and only if its unit ball is weakly compact, it is also easy to see that Z is reflexive if and only if for all Banach spaces ( X / y ) , all operators ( R G B ( X , Z ) / Q e B ( Z , }> )) are weakly compact. The fact that B w k ( X ) is a closed ideal follows from the next theorem formally stated below. Gantmacher’s theorem asserts that an operator is weakly compact if and only if its adjoint is. For a proof of this and other assertions given here, see Dunford and Schwartz [1958] VI-4. The following result, proved there, is often useful.
Theorem
An operator T G B ( X , ^ ) is weakly compact i f and only i f it
satisfies: T ” ( X **) C K( y ) . This theorem leads to an interesting exact sequence for any nonreflexive
Operator Algebras
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Banach space X 0 - » BW K ( X ) -> B ( X ) -> B ( X ” /k ( X ) ) where any T £ B ( X ) is sent to T defined by T ( F + k ( X ) ) = T * * ( F ) + k ( X ) for F 6 A'**. This may be regarded as a faithful representation (cf. Defi nition 4.1.1 below) of the weak generalized Calkin algebra B ( X ) / B w k ( X ) . If X = J is the James space, this provides a homomorphism o f B ( J ) onto C with kernel B w k ( J ) K. Garth Dales, Richard J. Loy and George A. Willis [1992] construct aii interesting Banach space related to the above considerations. For each ? > 2, they first construct an analogue J p of the James space based on t p, so J 2 is the ordinary James space. For q > p, they show B (J q, J p) = Bi ( ( J q , J p ) , and for q < p every operator in T £ B ( J q, J p) can be written uniquely in the form A + W for A € C and W € Bw h:(Jq, Jp)- Let W be the t l direct sum of the spaces J Pk where p\ = 2 and {pjt}fceN is a strictly increasing sequence. The operators on W have a matrix representation with matrix entries restricted as noted above: compact above the main diagonal and the sum o f a scalar and a weakly compact operator on and below that diagonal. This allows the Banach algebra B( W) to be analyzed in detail. For instance, the operators for which the matrix entries on the main diagonal are weakly compact (i.e., zero scalar part) form a closed ideal I which satisfies B ( W) = I + V where V ~ i°° is the closed subalgebra of diagonal operators with purely scalar diagonal entries from i° ° . Also S (W ) — K + T where K. is the closed ideal of operators with the scalar part of each matrix entry zero, and T is the closed subalgebra of operators with purely scalar matrix entries. We refer the reader to the original paper for substantial further detail. The first part of the next theorem is an important result of William J. Davis, Tadeusz Figiel, William B. Johnson and Aleksander Pelczynski [1974]. The last paragraph o f the theorem comes from Kaijser [1976].
Theorem
A n operator P 6 B ( X , y ) is weakly compact i f and only i f there is a reflexive Banach space Z and linear operators R 6 B ( X , Z ) and Q 6 B ( Z , ^ ) satisfying P = Q R . Moreover, Q may be chosen injective and contractive, and S may be chosen to have the same norm and kernel as P and dense range in Z . Suppose T : A —> B ( X ) and S: A —> B ( y ) are continuous representations o f a normed algebra A satisfying P T a( x ) = Sa( P x ) fo r alia € A and x 6 X . I f ? is weakly compact and Z, Q and R are as described above, then there is a continuous representation U : A —> B ( Z ) satisfying R T a(x ) = Ua( R x ) and QUa( z) = Sa( Qz) fo r all a £ A , x £ X and z 6 Z .
Partial P roo f For each n £ N define a new norm on ^ by IMIn = inf{||.P|| 2_n/"2 ||x|| + ||y — Px\\ : x £ X }
V y £ y.
(4 )
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1 : Normed Algebras and Examples
1.7.8
Taking x = 0 Ln (4) gives
llvlln < IM I
V v e y- n e N
(5 )
and ||y|| < ||Px|| + ||y - Px\\ < ||P|| ||x|| + 2n/2||y - P x || implies llyll < 2n/2||y||„
Vy e
n e N.
(6)
V x e X.
(7 )
Hence all these norms are equivalent. Note also
l\Px\\n < 2 -"/ 2||P|| ||z|| D efin e Z by Z — {y e y :
||j/||2 < oo}, and define a norm on Z by
(8)
111*111= ( X > l l n \n=l
From (7) we see i/i
IliP z lll < ( f > - " / 2 |!P|| ||x ||)2
Vx e X
\\P\\
(9 )
\n=l so P X C Z. Hence we may define R e B(X, Z ) and Q e B(Z, }>) by Rx = P x
V x € X;
Qz - z
V z 6 Z.
(10)
Inequality (9) shows ||.R|| < |(P|| and / oo
\ V2
IIQ 2|| = / X J ( 2 'n/2P I I ) 2J
/ 00
\ ^ 2
< ( X > H » )
= 111*111
V z e 2
(11 ) shows HQII < 1. Hence P = Q R implies \\R\\ = ||P|| and ||Q|| = 1. If R X is not dense in Z, replace Z by the closure o f RX. This proves all the results except that Z is reflexive and the statements about representations. For the former fact we refer the reader to either Diestel and Uhl [1977], VIII.4.8 or Albrecht Pietsch [1980], 2.4. We now turn to the proof o f the last paragraph. The final equation in the theorem and our definition of Q show that we must define U by Ua( z) = Sa( z) for all a e A and z € Z. Hence we need to check that Sa(z) is in Z for all a e A and z G Z. Note l|50»l|n
X)
<
inf{||P||2-n/2 ||T0x|| + ||5ay - P T 0x||
<
inf{||T.|| ||P|| 2 -"/ 2 H IU H + ||S.|| ||y - P x || : x e X }
<
max{||T||, 1)5)|} ||a|| |jy||n
V a e A; y e y
Operator Algebras
:.T 8
93
where we used the intertwining equation P T a = SaP. This shows that Sa senc'.s elements satisfying condition (7) into other elements satisfying that equation. I f it is necessary to replace this original choice of Z by the closure o f P X , the intertwining equation shows again that Saz will still belong to Z if z does. Hence we are free to define U : A —> B ( Z ) by Ua( z) = Sa( z) for all c, € A and 2 6 2 . We also conclude that the norm of the representation U is at most max{||T||, ||S||}, □ . Banach space X is said to have the Dunford Pettis property if every wea.dy compact operator from X into any Banach space y is completely continuous. Obviously no infinite-dimensional reflexive space has this prop erty, but most classical nonreflexive spaces do (Dunford and Schwartz [1958] p. 511), In fact, if X * has the Dunford-Pettis property so does X, but the converse fails. The composition o f a weakly compact operator followed by a completely continuous operator is compact by the Eberlein-Smulian theorem, since the image o f a bounded sequence will have a weakly convergent subsequence. Hence for any space with the Dunford-Pettis property, the square o f the ideal Bw K ( X ) of weakly compact operators is the ideal o f compact opera tors. (This is the first example we have given of a natural ideal for which the square is different and also a natural ideal. The example is rather un usual in that the square is closed in the norm in which the original ideal is complete.) If fi is a compact Hausdorff topological space and X = C ( f i) is the Banach space of continuous functions on fi under the supremum norm, then the ideals of weakly compact and o f completely continuous operators actually coincide and they are proper and properly include the ideal o f com pact operators unless X is finite-dimensional (or equivalently, fl is finite). Weakly compact operators were introduced by Kosaku Yosida [1938] and Shizuo Kakutani [1938]. ,4n operator in B ( X ) is called strictly singular if it does not have a bounded inverse on any infinite-dimensional subspace. The set B s s ( X ) of strictly singular operators in B ( X ) is a closed ideal which includes Bcc(X)C\ B w k ( X ) . This class of operators was introduced by Tosio Kato [1958]. We conclude this discussion by noting the inclusions which hold between these classes o f operators from one Banach space to another. I f X and y are arbitrary, they satisfy
BK( x , y )
c Bc c ( x , y ) n B WK( x , y ) c Bss( x, y) .
Whe:: y is reflexive, these inclusions can be improved:
BK( x , y )
c Bc c ( x , y ) c Bss( x , y ) c BWK( x , y ) = B( x, y) .
Finally, if X is reflexive instead, we get even more:
sK( x , y ) = Bc c { x , y )
c Bss( X , y ) c BWK( x , y ) = B( x, y) .
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1.7.9
For further results on these classes of ideals, see Goldberg [1966] and Gustafson [1968]. 1.7.9 Bases It turns out that there is a fairly natural idea of a basis in a Banach space, but not all separable spaces have bases. A space with a basis is easily seen to be separable. D e fin itio n A sequence { x rl} neiN in a or Schauder basis for X if for every x {^n}n€N of scalars so that ^nXn a basis {xjt}fc6N and a positive integer
Banach space X is called a basis in X there is a unique sequence converges to x in the norm. For n, define the nth basis projection
P n: X -> s p {x 1,x 2 l. . by P n(52kLl = £ it = i x* ■ For a basis { x n}n6N) the sequence of linear functionals {z * } neN defined by OO
is called the biorthogonal system defined by the basis. A basis { x n} n€N is said to be shrinking if for each u j £ X ' the sequence {(/ — P n )*(‘*’)}neN converges to zero in norm, and boundedly complete if for every sequence of scalars {A n} neis for which the supremum o f the set {II : n G N }, is finite, the series A „ x „ converges. For very simple examples, consider the obvious bases for Co and l l (de fined by the Kronecker delta {<5n}neN) which are, respectively, shrinking and boundedly complete. Since all classical separable Banach spaces have bases, Banach asked whether this was true of all separable Banach spaces. The question was not settled until Enfio [1973] constructed a counterex ample. Banach also noted that the uniform boundedness principle shows that the sequence of basis projections is norm bounded. Note that the basis projections have the expansion P n = Y l'j—i x i ^ x j *n terms ° f the biorthogonal system. The next theorem gives some interesting properties o f the concepts just introduced. Proofs can be found in Lindenstrauss and Tzafriri [1977], l.b. T h e o re m I f a Banach space X has a shrinking basis, then its biorthogonal system is a boundedly complete basis fo r X *. A space with a boundedly complete basis is isomorphic to a dual space. I f a dual space X * has a basis, then X has a shrinking basis so X * has a boundedly complete basis. A Banach space X with a basis is reflexive i f and only i f it satisfies one and hence all o f the following equivalent conditions. (a) The given basis is both shrinking and boundedly complete. (b) Every basis fo r X is shrinking. (c) Every basts fo r X xs boundedly complete.
1.7.10
Operator Algebras
95
1.7.10 T h e A p p r o x im a tio n P r o p e r t y In a space with a basis, the se quence { P n}neN o f finite-rank projections converges to the identity operator in some sense, which we will now make explicit. Alexander Grothendieck [1S55], [1956] discovered the right way to present the property involved. D e fin itio n A Banach space is said to have the approximation property if its identity operator I can be uniformly approximated on every compact subset K. by finite-rank operators (i.e., for every e > 0 there is a finiterank operator T (depending on K. and e) satisfying \\Tx — x|| < e for all x (z 1C.) It is said to have the bounded approximation property if there is some finite constant B so that the finite-rank operator T can be chosen to satisfy ||T|| < B independent of K. and e. I f B = 1, then the Banach space is said to have the m etric approximation property. A Banach space is said to have the compact approximation property if its identity operator can be uniformly approximated on every compact subset by compact operators. If the approximating compact operators can be chosen from a bounded set, the space is said to have the bounded compact approximation property. A.s we just note, the bounded sequence of basis projections { P n} neN approximates the identity operator in this fashion for a space with a basis. Thus all Banach spaces which have a basis also enjoy the bounded approx imation property. Enflo’s counterexamples [1973], which we have already mentioned twice, were actually spaces which failed to have the approximar tion property. His examples were simplified and supplemented by Alexander M. Davie [1973b] and Andrzej Szankowski [1978] so that now subspaces of Co and l p for 1 < p < 2 and 2 < p < oo are known which lack the approxi mation property. In fact they do not even have the compact approximation property. Thus reflexive spaces may fail to have the compact approximar tion property. Spaces are known (even with separable duals) which have the approximation property but not the bounded approximation property and with the compact approximation property but not the bounded com pact approximation property. Willis [1992d] has recently given an example cf a separable reflexive Banach space with the bounded compact approx imation property which does not have the approximation property. It is no; ’ :nown whether all separable spaces with the bounded approximation proparty have a basis (cf. Lindenstrauss and Tzafriri [1977] and [1979]). The only naturally occurring Banach space known to lack the approximar tioc property is B( H) for a Hilbert space H, Szankowski [1981]. In §5.1.12 we show the connection between these properties of X and the existence of approximate identities in various ideals o f operators in B(X).
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1.7.10
Basis
B A P - ^ (A P a B C A P )
AP
BCAP
CAP The diagram above summarizes the easy implications between these con cepts (omitting the metric versions). We are about to note that the approxi mation property implies that the finite-rank operators are norm-dense in the compact operators. This establishes the equivalence BAP<=>(APABCAP). A ll the other implications are obvious. The preceding remarks show that no implication arrow can be added to this diagram unless it turns out that B A P implies the existence o f a basis (for separable spaces). Here is an elementary result which applies only to the bounded case. T h e o re m Let X be a Banach space and let B be a finite constant. Then X has the ( bounded / bounded com pact) approximation property with bound B i f and only i f fo r every e > 0 and every finite subset X\, x 2, .. ■, x n o f X there is an operator T £ ( B f ( X ) / B k ( X ) ) satisfying ||T|| < B and IIT x j -Xj\\ < £ fo r j = 1 ,2 ,... ,n. P roo f Let AC be an arbitrary compact subset of X . Cover AC with open e/( B + 1) balls centered at x i , x 2, , x n. If T is the operator guaranteed in the condition and x € AC, then they satisfy \\Tx — z|| < || T (*-*,)| | + ||Txj — Xj\\ + \\x3 — x\\ < 2e for some j . The converse is immediate. □ Before stating the next theorem, which is due to Grothendieck [1955], we mention two more ideas important in the proof. Compact subsets of a Banach space can be characterized as those included in the convex hull of some sequence converging to zero. Our definition of the approximation property mentions uniform approximation on compact sets. This corre sponds to a locally convex topology on B ( X , y ) , the topology o f uniform convergence on compact sets, which is given by the collection of semi-norms: {o n : AC is a compact subset of X } , where <J/c(T) = sup{||Tx|| : x £ AC}. Prom now on, we will call this the Grothendieck topology on B( X, y) . The proof of the next theorem depends on showing that the linear functionals
Operator Algebras
1.7.10
97
on B( X, }>) which are continuous in the Grothendieck topology are exactly those given by: OO
OO
T >-* y 'u > n (T x n) for { i „ } „ 6N C X\ { w „ } „ 6N c y * ; 5 3 llw«llllx '*ll < 00• n —l
n —l
T h e o re m The following assertions are equivalent fo r any Banach space X. (a) X has the approximation property. (b) For every Banach space y, B p ( X , y ) is dense in B ( X , y ) in the Grothendieck topology. (c ) For every Banach space y, Bp( y, X ) is dense in B ( y , X ) in the Grothendieck topology. (d) Every choice o / { x n } n6N C X and {w n } n6rs C X * such that OO
OO
5 3 1 k . h i m < 00 and 5 3 v n{ x) xn = 0 fo r all x G X n=1
n —l
sc isfi.es , w „ ( x „ ) = 0. (c) For every Banach space y, Bp(y, X ) is dense in B i ( ( y , X ) in the no Tn topology. Moreover, X satisfies the ( m etric / bounded ) approximation property i f and only i f there is some finite constant B , so that in (b) o r (c) finiterank operators in the { unit ball / B-ball ) approximate operators in the unit ball and in (d) the last condition is replaced by | tJn(T’xn)| < 1 for all T e B p ( X ) i and the conclusion is replaced by ( | wn(®n)| S 1 / i£ ~ l"n (* „ )| < £ > . i f BK( y, X ) and B p ( y , X ) in condition (e) is replaced by B k (X, y ) and Bp' X, y) , we get a characterization of when X* has the approximation property. Thus we see that the ideal o f compact operators equals the ideal of i pproximable operators in B ( X ) if either X or X* has the approximation prcperty. Attractive proofs of all these results can be found in Lindenstrauss and Tzafriri [1977], I.e. W ith the ideas given in the last paragraph, they are all easy. In particular, note that (d) is just the statement that any linear functioned, continuous in the Grothendieck topology, which vanishes on JI rank one operators must vanish on the identity. Hence it is obviously equivalent to (a). Despite the next theorem, there is a space with a basis (ai :i hence the approximation property) and a separable dual for which the dutJ space fails to have the approximation property. T h e o re m A Banach space has the approximation property i f its dual space does. Hence a reflexive space has the approximation property i f and only i f its dual space does. A reflexive space or a separable dud space with the approximation property has the m etric approximation property.
1. 7.11
1: Normed Algebras and Examples
98
1.7.11 N u c le a r O p e ra to rs The closed ideals o f operators considered above are all (possibly) larger than the ideal of compact operators. The ideal considered here lies between the compact operators and the finite-rank operators. Both the ideal o f nuclear operators and the ideal o f integral op erators (considered next) are complete in their own norms which dominate the operator norm. We shall continue to make definitions for operators in B ( X ) (thus getting ordinary ideals, which is our main interest) but to use the fact that these are ideals in the extended sense when considered for operators from one space to another. The book by Pietsch [1980] provides an interesting view of these and other operator ideals. D e fin itio n An operator T G B ( X ) is said to be nuclear if there exist sequences { x n} n€N C X and {w n}n6N Q X ' satisfying
OO
oo ||u>„||||xn|| < oo and T (x ) =
n=1
u n( x ) x n
V x e X.
(12)
n —l
The set of all nuclear operators is denoted by B n ( X ) and is provided with the nuclear norm
OO
OO
i m u = i n f { £ I K ||IK|| : T is represented by n=l
x n ®u>„}. n=l
O f course, the representation mentioned in the definition o f the norm is that described in equation (12). Obviously the adjoint o f a nuclear operator is nuclear, but the converse fails (Pietsch [1980] 8.2.6) unless the dual space has the approximation property. The norm inequality in the next theorem is characteristic of ideals defined by norms. T h eo rem The collection B n ( X ) o f nuclear operators is an ideal in B ( X ) . It is a Banach algebra in its own norm which is greater than or equal to the operator norm and satisfies the inequality: ||/ZT5||iv < ll-ft||||r||jv||S|| fo r R, S e B ( X ) and T G BN ( X ) . Moreover, \\x ® w||jv = |kl|IM| fo r all x € X and u> G X ' and || • || is the largest norm on B p ( X ) satisfying this condition. A ll the above results are easy to prove. Nevertheless, the nuclear opera tors exhibit some mysterious behavior in perfectly arbitrary Banach spaces which was first identified by Grothendieck [1955], [1956]. This arises from considering their relationship to tensor products. Since we will put off a general discussion of tensor products until Section 1.10, we could avoid these complications here. However, we wish to introduce the duality the ory involved in the two classes o f ideals introduced in this and the following section, and that deeply involves their relationship with tensor products. We begin by considering the ideal B j r ( X ) of finite-rank operators as the algebraic tensor product X <%>X*. We have already considered the operator
I
’ .7.11
Operator Algebras
99
norm on B f ( X ) and its completion in this norm B a { X) . In the language o f iensor products, this norm is called the injective tensor norm and the
icnpletion B a ( X ) corresponds to the injective tensor product, denoted by X ® X * , which carries this norm. There is another natural norm for T 6 B f ( X ) , the projective tensor norm: n
n
ll^llp = inf{^||xfc||||w fc|| :T = ^ x * ® w fc} k—1 k=l
(13)
where the infimum extends over all n e N; x k € X ; u>k e X *. I f we complete X ® X " ~ B f { X ) in this norm instead o f the operator norm, we g e t the projective tensor product denoted by X ® X * . It is not hard to see that this norm is greater than or equal to the operator norm so that there is a natural contractive linear map rj: X ® X * —►BA( X) . Also, every element of the projective tensor product can be represented in the form xn ® where { x „ } n6N C X and { w „ } n6N C X * satisfy Y.™=i llw n||lkn|| < oo. The infimum of the last sum over all such representations is the projective tensor norm. Thus it is obvious that the norm decreasing map r) sends X ® X * onto Bn( X) . One would certainly hope that this map was an isometry, but in fact it may have a kernel. The remarks we have already made show that Bn ( X ) is isometrically isomorphic to X ® X * / ker(?7), but for a finite-rank operator the two inequalities n
n
n
II Y ] x k ® W fc|| < || Y \ x k ®Wfc||7V < || V 'x f c ® Wfcllp fc=l
fc=l
fc=l
are both strict in general. (The first norm is the operator norm which equals the injective tensor norm, the second is the nuclear norm which is calculated as the infimum over countable representations o f the operator and the third is the projective norm which involves only finite representations of the finiterank operator.) The approximation property for X is precisely the condition needed to ensure that tj is injective and hence that the last two norms agree for every (nuclear) operator. This is the content o f condition (d) o f the .main theorem on the approximation property which shows that a countable sum which vanishes as an operator is already zero in the projective tensor product. Zn fact condition (d) has another interpretation. It asserts that the map OO
OO
y ^ U n( x n), n=l
n=l
which is certainly well defined and therefore contractive in the projective tensor product, is also well defined (and therefore contractive) on the ideal of nuclear operators. Since this map is defined by the classical formula for
100
1: Normed Algebras and Examples
1.7.12
a trace, we will call it the trace. Hence, when X has the approximation property and T e B n ( X ) , we define OC
OO
T r (T ) = ^ a J n( i n)
where
T —
71— 1
x n <8>ujn, n:- 1
{ i n}neN C X , {u>n} „ eN C X * and ||i„||||u>„|| < oo. The definition makes the following trace identity and inequality clear for any T £ B n ( X ) and S e B ( X ) , T t ( T S ) = T r (S T ) \Tt(TS)\ < IITIUHSII. Hence nuclear operators induce continuous linear functionals on B ( X ) . More is true as we will show in the next theorem. When we examine the projective tensor product more closely, we will see that, for any Banach space, B ( X m) is naturally isometrically isomorphic to the dual o f X ® X * under the inap which sends an operator T onto the linear functional T defined by OO
f(r ) = £ n —1
OO
T(u>n) ( x n)
where
r = ^
x n ® u>„.
n —1
When X has the approximation property and we apply this construction to the ideal of nuclear operators instead o f the projective tensor product, we get the second statement of the following result. T h e o re m I f X has the approximation property, t h e n B A ( X) equals B k ( X ) . AUo, the map T <—> T establishes an isometric linear isomorphism between B ( X *) and the Banach space dual o f B y ( X ) . P roo f The first statement follows from (e) of the main theorem on the approximation property, nd we have already outlined the proof of the second. D If the Banach space Af is a Hilbert space H , then it has the approxi mation property since every separable subspace has an orthonormal basis which is a Schauder basis. Furthermore, the nuclear operators can be iden tified with the trace class operators, and H ' can be identified with H (by a conjugate linear isometry). Hence the second result states that the dual of the ideal of trace class operators is the algebra o f all bounded linear operators. This special case is better known and has proved more useful (so far) than the more general result. It will be discussed in Volume II. 1.7.12 In te g r a l O p e ra to rs Next we identify the dual space of B a ( X ) . This will turn out to give a new ideal of operators on A* in a natural way.
7,12
Operator Algebras
101
Any operator T in B ( X ) defines a linear functional on B f ( X ) = X
102
1: Normed Algebras and Examples
1.7.13
nuclear operators. Theorem V III.3.8 of the reference cited in the last proof gives the easy result that X has the metric approximation property if and only if the nuclear and integral norms are equal for all operators in B ^ ( X ) . We combine these results to get the representation we need o f the dual and bidual of the compact operators. T h e o re m I f X * has the m etric approximation property and finite-rank operators are dense in the integral operators on X * , then B ^ i ( X ' ) is iso metrically linearly isomorphic to the Banach space dual o f B k ( X ) under the action T (S ) = Tr(TS*)
V T G B n (X*)-, S G BK ( X ) .
Hence B k ( X ) " is isometrically linearly isomorphic to B ( X * * ) under a map 0 which satisfies 0(/c( K ) ) = K * * f o r all K G B k ( X ) .
Corollary I f X is reflexive and has the approximation property, then B k ( X ) * is isometrically linearly isomorphic to B ^ ( X * ) as in the theorem and B k ( X )** is isometrically linearly isomorphic to B ( X ) under a map 0 which satisfies 0 ( k ( K ) ) = K ** f o r all K e B k ( X ) . We close by quoting two related results o f Grothendieck [1955] on in tegral operators from Diestel and Uhl [1977]. Theorem V III.2.11 shows the nontrivial fart that an operator is integral if and only if its adjoint is. Theorem V III.4.6 shows that if X * has the approximation property, then it has the Radon Nikodym property if and only if B i ( X , y ) and B ^ ( X , y ) and their norms are equal for all Banach spaces 1.7.13 T h e A re n s P r o d u c ts on B a { X )** We now wish to calculate the first Arens representation ( c f Section 14) for the double dual of the algebra B k ( X ) o f compact operators. This was first done in the case of Hilbert space. We will do it here for a wide class of Banach spaces, but. not for arbitrary ones, since the exact nature of the Arens products on the double dual o f B k ( X ) or B a ( X ) is not known for perfectly general Banach spaces. The results are due to Palmer [1985]. Let us assume that thr Banach space X * has the metric approximation property and that the finite-rank operators are norm dense in the integral operators on it, so that X satisfies the last theorem. We will calculate the first Arens representation for A = B k ( X ) and the representation T which is just the given action of the operators in B k ( X ) on X. This Arens representation is an algebra homomorphism o f B k ( X )** with its first Arens product into B ( X mt). It is satisfying that this homomorphism is the same map as the isometric linear bijection 0 just developed from purely Banach space considerations. Before stating a comprehensive theorem, we give the meaning of the various steps in the construction of the first Arens representation and Arens product using the notation introduced above.
1.7.13
Operator Algebras
103
An arbitrary linear functioned on { A / A * ) has the form ( S / T ) -3i { 5 G B f j ( X ' ) / T e B ( X * * ) ). Using the above notation, for r G X ' , K G B k ( X ) and F G X**, Definition 1.4.14 gives tk
= K*(r)
pr = (r ® F )v
T l(F ) = T(F).
Hence the first Arens representation Tj, o f T £ .4** is just the natural action of T on X**. Since this is essentially an independent calculation, it allows us to give a converse to the last theorem o f the previous section. We conclude by stating the main result from the cited paper, which shows that if B a ( X )** is even a little like B ( X " ) , it is isometrically alge braically isomorphic to it when the former carries its first Arens product. See the original paper for additional details, including a formula for the second Arens product. T h e o re m The following are equivalent fo r a Banach space X : (a) The dual X * o f X has the metric approximation property and B p ( X * ) is dense in B j ( X * ) . (b) There is a left identity element o f norm one fo r B a { X ) * * with its first Arens product. (c) The first Arens representation o f B A ( X )** is an isometry onto B ( X * * ) . (d) A ll the following are natural isometric linear isomorphisms which are also algebra isomorphisms when both spaces have natural algebra structure: Ba ( X ) = Bk ( X ) ~ X & X * BA{ x y = BK { x y ~ B i ( x * ) = b n ( x *) ~ B a ( X ) * * a B( X* * ) . Many common Banach spaces satisfy these conditions. See Lindenstrauss and Tzafriri [1977] and Diestel and Uhl [1977] for examples. In contrast, B a ( X ) is Arens regular only for reflexive Banach spaces. For reflexive Banach spaces and Hilbert spaces, we have more. C o ro lla ry The following are equivalent fo r a reflexive Banach space X : (a) X has the approximation property. (b) The two Arens products on B k ( X ) " agree and the natural injection k o f B k ( X ) into B k ( X ) * * can be extended to an isometric algebra iso morphism o f B ( X ) onto B k ( X ) * * which is compatible with the inclusion o f B k ( X ) i n B ( X ) . Moreover B k ( X ) * is isometrically isomorphic to B n ( X * ) . C o ro lla ry I f H is a Hilbert space, then (b) o f the last corollary holds ( with X — H ). Moreover the ideal Bm( H) o f nuclear operators is precisely the ideal B t t CH) ° f trace class operators and is included in B k ( H ) . The isometric linear isomorphisms Bk { H Y ~ BTr { H )
and
S r r (H )* ss B ( H )
104
1: Normed Algebras and Examples
1.7.14
are both implemented by the pairing
(T, S) = TV( TS)
V T e B(H)\ S e B iv (W ).
1.7.14 Double Centralizers of Ideals of Operators In the last example, we calculated the Arens product on the double dual of B k ( X ) for special Banach spaces and identified it with B ( X ) in the reflexive case. In this example, we will show that B( X) can also be identified with the double centralizer edgebra of B k ( X ) for any Banach space X. Actually our results hold for a general class of ideals in B ( X ) Let X be a Banach space, and let A be a non-zero ideal in B( X) which is complete in its own norm ||| • ||| (which need not be the restriction o f the operator norm on B( X) ) . Proposition 1.1.18 shows that A must include Bf ( X) . The ideals B( X) , BK ( X) , BCC( X) , BWK( X) , BSS( X) , BN ( X ) and B j ( X ) (artiong others) satisfy all the requirements which we will subse quently make for A. Proposition 1.2.6 gives a homomorphism 0 of B( X) into ' D( A) defined by 0(T) = ( Lt\A, Rt\A) for all T e B( X) . The inclu sion B f ( X ) C A shows that 6 is injective. Theorem 1.2.4 shows that the operators Lt\A and Rt\A are continuous. We will now investigate when 6 is surjective. This w ill« ccur if and only if every left centralizer has the form Lt \A. Prom now on, we will f-ssume that the norm ||| • ||| on A is a reasonable norm when restricted to I f ( X ) C A (i.e., it lies between the injective and projective tensor norms cf. Section 1.10) and satisfies max{|||T5|||,\\\ST\\\} < |||S||| • ||T||
V S ^ T e
B( X) .
To simplify notation, we will replace Lt\A and Rt\A by T # and T 1 *. Thus ( * ) is a contractive injective homomorphism of B ( X ) into the subalgebra £ C B( A) of continuous left centralizers on A, and ( ^ is a contractive injective anti-homomorphism of B( X) into the subalgebra TZ C B( A) of continuous right centralizers on A. We claim that ( # ) is an isometric isomorphism of B ( X ) onto C. To prove this, let L £ C, x, y G X and u>, r £ X * satisfying u>(y) = 1 be arbitrary, and define z € X by z = L ( x g> ui)y. Then we get L (x ® r ) = w ( y ) L ( x ® r ) = L ( ( x <S>uj)(y <S>r ) ) = L ( x ® uj)y ® r = z ® r. Clearly z depends linearly and continuously on x and is independent of r. Hence the map x i —* 2 belongs to B(X)\ if we denote it by T then the restriction of I to BF ( X ) satisfies L\Bf ( X ) = L T \Bp(X). Since ||| • ||| is a reasonable norm, we get ||T|| < ||L||. Hence for any S e A, x € X and r e A ” we have L ( S ) x & t = L ( S x ® r ) = L ( ( S x ) © r ) = ( T S x ) ® r = T * ( S ) x ® r. This implies L = T * . Therefore * is an isometric isomorphism as claimed. (Note the curious fact that the restriction of L to B y ( X ) determines L
1 7.15
Operator Algebras
105
even though B p ( X ) need not be dense in A (e.g., A = B ( X ) ) . ) For any ( L , R ) e V ( A ) , L — T * implies R = 7* so that T ( T * ,1*) is an isometric isomorphism of B ( X ) onto T>(A). When we apply similar calculations to the homomorphism ( b), the ansv srs are surprisingly different. Before doing so, we note that the Banach s : -xe adjoint (* ) is an anti-homomorphism o f B ( X) into B( X* ) which is surjs. „ive if and only if X is reflexive (Dunford and Schwartz [1958], V I.9.13). Let R e TZ be arbitrary Choose to e X ' and y € X satisfying u>(y) = 1. Then, for any r € X * , define a € X * by a = ( R ( y ® t ) ) * ( ui ). For any x = X , the calculation R ( x ® r ) = u ( y ) R ( x ® r ) = R ( ( x ® u ) ( y ® r ) ) = x£J)u>R(y®r) = x ® a shows that a is independent o f the choice of u) and y. Cl ;arly it depends linearly and continuously on r. Hence the map r a be’ongs to B ( X * ) . If we call this map R, then we have R ( x ® t ) = x <8>R(t ) for all x 6 X and r 6 X * from which we conclude ||fl|| < ||.ft|| since ||| • ||| is . reasonable norm. Hence for any S 6 A , x e X and r 6 X * we have x ® ( R ( S ) ) * t = x ® t R ( S ) — R ( x ® t S) = R ( x ® S *r) = x ® R S *t . Therefore R ( S ) * equals RS* for any S 6 A. If A" is reflexive, R may be written as T * for some T e B ( X ) so we have R = T b. Thus the map T •-> T* is an isometric anti-isomorphism o f B ( X ) onto TZ when X is reflexive. We conclude by briefly considering the case in which X is not reflexive asd A is either B k ( X ) or B w k ( X) . Then well known results which can be found in Dunford and Schwartz [1958] VI.4.2, V I.4.8, and V I.5.2 show that, for any T € B ( X **) and any S € A , the restriction S**T\x belongs to A C B( X) . This, together with the results obtained above, shows that the map T ►-+ R T . defined by rt
. ( s ) = 5 **r*u
V T e B(X*y, S e A
is tn isometric isomorphism of B ( X *) onto 1Z. The above example is due to Jo.' nson [1964a], but the methods used here (and the generality obtained) ay: ear to be novel.
l A A S N o rm s and Isom orp h ism s o f B( X) In one of the earliest papers on Banach algebra theory, Meier Eidelheit [1940a] proved several remark able theorems about the algebra B ( X ) of all bounded linear maps of a Ba iach space X into itself. He showed that an algebraic isomorphism be tween B ( X ) and B ( y ) must come from a homeomorphic linear isomorphism between X and y . He also showed that every automorphism of B( X) is an r automorphism and that any Banach algebra norm on B( X) must be equivalent to the operator norm. (W e will enormously generalize this last resr.lt in Chapter 6.) Here we will prove slightly stronger statements using his simple but powerful ideas. We also give a related result based on an argument of Bertram Yood [1958].
1: Normed Algebras and Examples
106
Th eorem
Let { X /
1.7.15
y ) be a Banach space and let ( A /
B ) be a
norm closed subalgebra o f ( B ( X ) / B ( y ) ) including ( B p (X ) / B p (y ) ). I f $ : A —>B is an algebra isomorphism, then there is a homeomorphic linear isomorphism P : X —> y satisfying $ (S ) = P S P - 1
V S £ A.
P roo f Choose a non-zero element z £ X and a linear functioned r £ X * satisfying t ( z ) = 1. Then 2 0 r is an idempotent operator of rank one. Hence $ (2 ® r ) is also idempotent. Several easy arguments show that $ (2 ® r ) also has rank one. For instance, note that (2 ® r ) A ( z ® r ) is the one-dimensional space C 2 ® r. Hence $ (2 0 t ) B $ ( z 0 r ) must also be onedimensional. Since the image includes $ (2 &) r ) B p ( y ) $ ( z 0 r ), this proves that 4>(2 0 t ) is a rank 1 idempotent. Hence we may write $ (2 ® r ) = w ® a for some w £ y and o £ y * satisfying cr(w) = 1. We can now define the operator P . For any x £ X , $>(x ® r ) = $ (x ® t ) $ ( 2 ® r ) = $ ( x ® t ) ( w ® a ) = y ® <x for some y £ y which depends linearly on x. Define P to be the operator which sends x to y. We can write P as P ( x ) = y ® cr(w) = $ ( z ® r)(u ;)
Vx £ X
and its inverse as P ~ l (y ) - x ® r ( z ) — $ _1(y ® < r )( 2 )
Vy£
For any S £ A and a; €
Eid elh eit’s Lem m a Let A be a closed subalgebra o f B ( X ) which includes B jr (X ). Then a subset S is bounded i f and only i f fo r every T £ B p (X ) there exists an e > 0 such that I — X T S is invertible f o r all A 6 eC\ and all S£S. P roo f The identity operator is the only operator which could serve as an identity element in A . If it is not in A , put it in. Suppose S is bounded by M . Choose e = (M||T|| + l ) -1 . Then |A| < e implies ||ATS|| < 1. Hence the series ^^°=0(A T 5 )n converges in norm, so the calculation O
O
O
O
O
O
(/ - X T S ) £ ( A T S ) n = ^ ( A T S ) n - ^ ( A T S ) n+1 = I n=Q
n =0
n=0
is justified. It shows that / — X T S has a right inverse. Similarly, it has a left inverse and hence is invertible.
L 7.!5
Operator Algebras
107
Conversely, suppose that S is not bounded. By a corollary of the uniform boundedness principle and the Hahn-Banach theorem (Dunford and Schwartz [1958], II.3.21), there exist x e X and u> 6 X * such that {o>(5x) : S G 5 } is not bounded. Let T be the operator x ®u>. Then for any e > 0, there is an S £ S satisfying |w(5x)| > e ~ l . Hence we see that the condition fails by taking A = (u>(Sx))_1 and noting (7 — AT S )x = 0 .0 C o ro lla ry 1 I f X and y are Banach spaces, any algebraic isomorphism from B( X) to B ( y ) (o r even between closed subalgebras which include B f ( X ) and BF ( y) , respectively) is continuous. C o ro lla ry 2 phism.
Every algebra automorphism o f B (X ) is an inner automor
C o ro lla ry 3 Let A be a norm closed subalgebra o f B (X ) which includes B p (X ). Then any two complete algebra norms on A are equivalent. P roo f This follows from the lemma. Suppose a sequence { T „ } neis converges to zero in another complete algebra norm. Then we can find a sequence {
Pro6f If A is an ideal, it includes B p (X ) by Proposition 1.1.18. The proof is by contradiction. Let the algebra norm on A be ||| • |||. Suppose it does not dominate the operator norm. Then we may find a sequence {T n} ners satisfying |||Tn||| = 1 fo r a ll n 6 N and ||Tn|| —►oo. Now let x 6 X and u> G X * be arbitrary. Then |w(T„(x))| |||a: ® u>||| = |||x ® ujTn x ® a>||| < |||x ® w|||2. Hence, in particular, {|a>(Tn(x))| : n G N } is bounded for each x 6 X and ui e X *. The uniform boundedness principle thus implies that {|j7 ’n || : n € N } is bounded, contradicting our assumption. □
108
1: Normed Algebras and Examples
1.7.16
1.7.16 G e n e ra liz e d C a lk in A lg e b ra s and F red h o lm O p e ra to rs In the earliest American paper on Banach algebras with an abstract “ modern” viewpoint, Calkin [1941] introduced an algebra which has come to be known as the Calkin algebra. It is the quotient algebra B (H )/ B k ('H ), where H is a separable Hilbert space. This algebra is simple (i.e., it has no nontrivial ideals) and unital. In Volume II we will further discuss this algebra and Calkin’s representation of it. Here,we generalize to the case in which X is a Banach space by defining Q (X ) = B (X )/ B k ( X ) to be its generalized Calkin algebra. We have seen Banach spaces in which Bw k ( X ) lies strictly between B k ( X ) and B (X ) so that the generalized Calkin algebra is not always simple. Furthermore, in some of these spaces (e.g., those with the Dunford-Pettis property, in cluding most classical nonreflexive spaces) the square o f the ideal of weakly compact operators is the ideal o f compact operators. In that case, the gen eralized Calkin algebra has a closed non-zero ideal with square zero. This is a type o f bad behavior studied in more detail in Chapter 4 below. See also Selwyn R. Caradus, W illiam E. Pfaflenberger and Bertram Yood [1974], John J. Buoni and Albert J. Klein [1979] and Michael J. Meyer [1992c]. An operator in B (X ) is called Fredholm if its image in Q (X ) is invert ible, Riesz if its image in Q (X ) is topologically nilpotent, and inessen tial if its image in Q (X ) belongs to the Jacobson radical Q ( X ) j o f Q (X ). The terms “topologically nilpotent” and “Jacobson-radical" are defined in the next chapter. The literature on FVedholm operators is enormous (cf. Caradus, Pfaflenberger and Yood [1974], Kato [1966], Martin Schechter [1971] and Section 8.5 below). Riesz operators are discussed briefly in Sec tion 2.8 below, where the name is explained, and in great detail in Bruce A. Barnes, Gerard J. Murphy, M. R. F. Smyth, and Trevor T . West [1982]. For the other notions introduced above, see Anthony F. Ruston [1954], David C.Kleinecke [1963], West [1966], and Pfaflenberger [1970]. 1.7.17 H ilb e r t Spaces an d C *-a lg e b ra s .Although intensive study of these topics is postponed until Volume II, a brief overview will be useful here. A ll details can be found in Volume II. Let H be a linear space. An inner product on H is a map ( V ):H x H —> C which is linear in the first factor, conjugate linear in the second and satisfies (x ,x) > o
V i e W \ {0 }.
An inner product defines a norm on 7i: ||x|| = ( x , x ) 1/2
V ieW .
A Hilbert space is a linear space with a fixed inner product such that the corresponding norm is complete. The most important property of Hilbert
Operator Algebras
1.7.17
spaces is that for any continuous linear functioned there in a unique ('lenient y e H satisfying u>(x) = (x ,y )
109 uj
on a Hilbert space H
V x e H.
Furthermore, y also satisfies ||j/|| = ||w||. Hilbert spaces are reflexive and even super-reflexive. They have or thonormal bases which are always both shrinking and boundedly complete bases in the terminology of §1.7.9 above. Hence Hilbert spaces have the baunded approximation property with bound 1. Moreover if H is a Hilbert space, the idead of compact operators is the only nontrivial closed ideal in B (H ) and includes all proper ideals. There is an interesting continuum of ideals C p for all 0 < p < oo in B (H ) deeply connected with the P ’ spaces. Each ideal is a Banach algebra in its own norm ||•||p. and is the completion of the ideal o f finite-rank operators in this norm. Here C °° is just B k (H ) and C p is continuously embedded in C q if p < q. Furthermore C l is the ideal of nuclear operators which is also the ideal of trace class operators (with a different, but equivalent, definition), and C 2 is the ideal B h s CH) of ffilbert-Schmidt operators. This last set is itself a Hilbert space in its own inner product and norm. It can be defined either abstractly or in terms of vtry concrete integral operators. The inner product on H gives rise to the Hilbert space adjoint in B (H ): for any T e B( H) , there is a unique operator T * in B ( H) satisfying (T * x , y) = (x, T y )
V x, y e n .
The map T i-+ T * from B (H ) to B (H ) satisfies the next definition. D e fin itio n
An involution on an algebra .A is a map ( * ): A —* A satisfying
(a + b)* (A a )' (aby (a *)*
= = = =
a* + b * A* a* b*a* a
(additive) (conjugate homogeneous) (anti-multiplicative) (involutive)
for all a,b e A and A € C. An algebra with a fixed involution is called a -algebra, and a Banach algebra with an involution is called a Banach *-c.lgebra. The following condition is called the C*-condition: ||a*a|| = ||a||2
Va e A
(C*-condition).
(15)
A Banach algebra in which the norm and involution are related by the C* condition is called a C*-algebra. The simple definition o f a C*-algebra does not suggest the profound cfiect the C*-condition has on the structure of an algebra. In Chapter 3
110
1: Normed Algebras and Examples
1.7.17
we will show that commutative unital C*-algebras can be characterized as these algebras isometrically isomorphic to the algebra C ( f l) of all contin uous complex-valued functions on some compact Hausdorff space fl [cf. §§1.5.1, 3.2.12 and 3.4.17). The remarks preceding the above definition show that B (H ) is a C*-algebra when H is a Hilbert space. Hence any subalgebra of B (H ) which is closed under the involution and also norm-closed is a C*-algebra. The fundamental Gelfand Naimark theorem (Corollary 9.5.5 in Volume II) asserts the converse: any C*-algebra is isometrically isomorphic to a norm-closed and involution-closed subalgebra o f B (H ) for some Hilbert space H. A ll the ideals o f B (T i) mentioned above sure closed under the involution, but only the ideal of compact operators is closed in the operator norm. A number o f formulas are known to calculate the norm of a C*-algebra from its *-algebra structure. Hence, in particular, every homomorphism tp between C*-algebras which respects the *-algebra structure (i.e., satis fies tp(am) = (tp (a ))m for all a) is continuous (even contractive), and every isomorphism which respects the involution is an isometry. These results show that, in the category o f ^-algebras, the norm o f a C*-algebra is a consequence of the other structure. Similarly in the geomet ric category o f Banach algebras (where morphisms are contractive algebra homomorphisms and isomorphisms sure isometric), the involution in a C*algebra is a consequence o f the shape o f the unit ball. See Section 2.6 and especially Proposition 2.6.8 for this viewpoint. Even in the topological category o f Banach algebras (where morphisms are continuous algebra ho momorphisms and hence the norm is only determined up to equivalence), the involution can be derived from consideration o f bounded subgroups. A ll this will be discussed further in Volume II. Propositions 2,3.17 and 2.3.18 show that every C*-algebra is semisimple (cf. Definition 2.3.2 and Section 4.3). Even the Banach space structure of a C*-algebra is strongly restricted by the definition. Any Banach algebra with the underlying Banach space o f a C*-algebra is Arens regular. Also if a C*-algebra is linearly isometric to the dual o f some Banach space, that Banach space is uniquely defined in a strong sense. In Chapter 6 we need to know that the double centralizer o f any C *algebra is again a C*-algebra and that the regular homomorphism is an isometric isomorphism which preserves the involution. We will give the easy proof here. Let A be a C*-algebra. Recall that the norm of a bounded double centralizer is given by \\(L, fl)|| = max(||L||, ||rt||)
V (L , R ) € V (A ).
The C*-condition shows directly that the hypotheses o f Theorem 1.2.4 sure satisfied (so every double centralizer is bounded) and that the regular ho-
1.8.0
Group Algebras on T
111
momorphism is an isometric isomorphism |M|2 = Hflfl’ H = ||L0(a*)|| < ||La||||a|| < ||a||2 and similarly for the right regular representation R. T>(A) is given by ( L , R Y = ( R ', L * )
V
Va € A The involution on
(L ,R )e V (A )
where R * and L* are defined by T '( a ) = (T (a * )Y
V
T e C (A ); a e A .
T: is easy to check that this does define an involution. Theorem 1.2.4 shows that T>(A) is a Banach algebra under its norm. In the present case each (L, R) € 'D (A ) satisfies ||L||
||(L,rt)*||
| | (i,R ) * { L , R)||
=
sup{||L(a)|| : a € A \ ) = sup{||L(a)6|| : a, b € A i }
=
sup{ ||aR(b) || : a, b e A i } = sup{||i?(6)|| : b € A i }
=
W l
=
||(i?*,L*)|| = ||fl’ ||-sup{||JR * (a )| | :a 6 > l1}
=
sup{||(i?(a*))*|| : a € A i } = ||i?||
=
\\(L,R)\\
=
||JPL|| = sup{||R*L(a)b|| : a , b e A i }
=
sup{||L(a)L*(6)|| : a, b 6 A i }
>
sup{||£(a)L(a)*|| : a e A i ] = sup{||L(a)||2 : a e A i }
=
||L||2 = ||(i,R)||2 > \\(L,R)*(L,R)\\.
and
Hence T>(A) is a C*-algebra.
1.8
Group Algebras on T
This section provides elementary concrete realizations o f many phenom ena that we will discuss more abstractly and in much more generality later. See particulaurly Sections 1.9 and 3.6. Here we will look in detail at the various spaces of functions on the circle group which sure algebras under convolution multiplication or sure in some other way related to harmonic analysis in this setting.
112
1.8.1
1: Normed Algebra.-, and Examples
1.8.1 Basic Facts and the H aar In tegral The symbol T will always represent the set of complex numbers of absolute value one, considered as a group under multiplication. (T stands for torus since this group is sometimes called the one-dimensional torus.) This group will always carry the relative topology it inherits as a subset o f the complex plane. Locally this is also the topology of the line as we will explain in greater detail below. The group operations (i.e. the multiplication map (A,/j) i—> Xji from T x T to T and the inversion map At-* A-1 from T to T ) sure obviously continuous so that T is an abelian, compact, connected topological group. The map h: R —►T defined by h (t) = ei( is a group homomorphism o f the additive group o f R onto T which is also a local homeomorphism. Thus h identifies T with the quotient of R modulo the subgroup 2nZ. Prom another viewpoint this makes it clear that T is an abelian, real, compact, connected, Lie group which has h: R —►T as its simply connected covering group with fiber it\ (T ) « Z. When /: T —►C is any function, it will be convenient to denote f o h by adding a tilde: /. This identifies complex-valued functions on T with 2 periodic functions on R. Another useful viewpoint is to consider the restriction o f / to the closed interval [—tt, n}. This identifies complex-valued functions on T with functions on [—w, 7r] which have the same value at —7r and n. We use the map h to define integration and differentiation of complexvalued functions on T. For any suitable function /: T —> C, we define 5T
J
f(X )d \
IT h (t)d t
—■K
IT f ( t )d t
= ~ j
—-K
f ( e u )dt.
—-K
( N . B. The integral defined here is not a contour integral although on oth’e'r occasions we will use the notation f r f(X )d X to denote a contour integral on a contour T in C .) If f is Riemann integrable, this integral may be interpreted as a Riemann integral, but usually we consider it as a Lebesgue integral. It has the following remarkable properties J f(n \ )d X = j r f(X )d X
V n e T,
J f ( X - l )dX = J f ( X ) d X .
(1)
(2)
Translation invariance, as given in equation (1), is the characteristic prop erty o f a Haar integral to be defined in Section 1.9 below. Equation (2) is a property of unimodular Haar integrals.
1.8.2
Group Algebras on T
113
Whenever / :T —►C is a function such that / has at least an almost everywhere defined derivative, we will define /' to be the function satisfy ing ( f ')~ = f • We will transport other properties o f functions on [—7r, -rrj to functions on T in a similar fashion. Frequently we prefer to work with . instead o f / because o f the more elementary appearance o f the computa tions. 1.8.2 S om e B an ach Fu n ction Spaces We wish to define a number of Banach algebras o f complex-valued functions on T. Let us begin by discussing the Banach spaces involved. Their linear operations sure always Y- Mintwise. Readers not already familiar with these Banach spaces can work c it their properties as exercises after consulting Dunford and Schwartz [ 958], Edwin Hewitt and Kenneth A . Ross [1963], [1970], or Walter Rudin [i 962], We will make some clarifying remarks after listing the Banach spaces o ' complex-valued functions on T and their norms. Eanach S pace N o rm L 1 = {absolutely integrable functions} |j/||x = /T |/(A)|dA L 2 = {square integrable functions} ||/||2 = ( f T |/(A)|2dA)*^2 L °° = {essentially bounded functions} jj/lj^ = ess sup{|/(A)| : A 6 T } C = {continuous functions}
ll/lloo = SUP{I/(A)| : A 6 T }
C B V = {continuous functions o f
II/IIcbv ~
bounded variation}
ll/lloo
||/||v = s u p {£
A C = {absolutely continuous functions} C 1 = {continuously differentiable functions}
+ WfWv where |/ (*,•) - / (t ,_ i)| }
||/IUo = ll/lloo + ll/'lli ||/||c l = H/H^ + ||f'W ^
The supremum in the definition o f ||/||v is extended over all n € N and all sets {<0,
E i —i
n
~
< *
K/ p “
and
f W j ) ~ ( f P ~ f ) ( t j - 1)| < e.)
j =i
A function /: T —►C is defined to be absolutely continuous if / is abso lutely continuous on [—7r, 7r], Hence, / belongs to A C if / is differentiable almost everywhere and its derivative belongs to L 1 and satisfies t
f ( t ) = / (-*■ ) +
J
f '( r ) d r
VteR .
114
1: Norimxl Algebras and Examples
1. 8.3
This observation will be used several times. In particular, it gives the completeness o f A C . ( I f {/ n} „ eN is a Cauchy sequence in A C , then there is a g £ L 1 such that {/ ^ }neN converges to g in L l . Hence if we define f ( —tr) bv Um /n( —tt) and define / 6 A C by f ( t ) = / ( —n ) 4- f l „ g ( r ) d r , then {/ n } n£N converges to / in A C .) Alternatively, it is not particularly hard to see (e.g., H ewitt and Kauri R. Stromberg [1965], 18.1) that the norms II ‘ l i e b v ^ II ’ H a c 8ire actually equal on A C and that A C is a closed subspace o f C B V , so that it is complete. The proof of the completeness of C 1 is similar to the (first) proof of the completeness o f A C . The inclusions C 1C A C C C B V C C c r c L 2 a '
(3)
sure obvious from the definitions. (O f course the inclusion C C L °° needs some interpretation, since the elements of L °° are equivalence classes of almost everywhere equal functions. If one of these equivalence classes con tains a continuous function /, it contains only one and the two interpre tations o f ll/U^ agree. This gives meaning to the inclusion C C L °° and shows that C is closed in L 00.) The fact that T has measure one, and other elementary considerations, give the inequalities ll/ll,
<
ll/ ll2 <
ll/ I L
<
WfWcBv
=
Wfhc
<
ll/ llc-
(4)
Therefore each inclusion in (3 ) expresses a continuous embedding of the smaller Banach space into the larger. The embeddings C —> L °° and A C —* C B V sure actually isometries.
1.8.3 Trigonom etric Polynom ials We will next discuss the linear space o f trigonometric polynomials. It is dense in all o f the above Banach spaces except L °° and C B V . For each n € Z, define hn: T —►C by hn(X ) = An so that h „ (t) is just e,nt. The set T of trigonom etric polynomials is the set o f finite linear combinations o f functions in H = {h n : n € Z }. If p = ctkhk is a trigonometric polynomial, then we call the smallest possible n in this expression the degree of p. Clearly T is an algebra under pointwise multiplication, which is closed under complex conjugation, separates the points o f the compact space T and contains the constant functions. Since these are the hypotheses o f the complex form of the Stone-Weierstrass theorem (cf. 6.1), we conclude that T is dense in C (i.e., dense in the uniform norm || • H^). From the regularity o f the measure on T, one can conclude that C is dense in both L 1 and in L 2 in their respective norms. This fact and the inequalities || • ||1 < || • Hqq and I! ' H2 — II ’ Hoc sh°w that T is dense in L l and L 2 (e.g., for any s > 0 and / € L 2, find g G C and h e T satisfying ||/ - »||2 < § and ||g - hW^ < §, and thus conclude 11/ - h\\2 < ||/ - s||2 + ||s - h\\2 < | + ||p - Alloc < £0
1.8.4
Group Algebras on T
115
Clearly T is not dense in L °° since it is included in C which is closed in L ° ° . Similarly T is not dense in C B V since it is included in the closed subspace A Z . The density o f the trigonometric polynomials in A C can be established as follows. For any / € A C and any e > 0, choose g € T satisfying lls — f ' l Ii < §• Next define h € T by h (t) = / ( —n ) + ^ g (r)d r. Then we have ||/ - h\\AC = s u p | ^ f
C f '(r ) - g { r )) d r \ + \\g - / %
J
< ^
\ f '( r ) - g ( r ) \ d r + ~ < £ .
An entirely similar argument proves that the trigonometric polynomials sure dense in C 1. 1.8.4 Banach A lge b ra Structure So far we have discussed these seven spaces as Banach spaces. It is easy to check that the five Banach spaces L °°, C, C B V , A C and C 1 are all Banach algebras under pointwise multiplication. This is not the Banach algebra structure in which we are primarily interested here. We wish to consider all seven Banach spaces as Banach algebras under convolution multiplication, which we sure about to define. To distinguish these two Banach algebra structures, we will use the symbols L l , L 2, C { * ), C B V (* ), A C ( *) and C 1(* ) for the Banach algebras with convolution multiplication and use L °°, C, C B V , A C and C 1 to denote either the Banach spaces or the Banach algebras with pointwise multiplication. 1.8.5 Convolution Prod u cts For any two functions f , g € L l we denote their convolution product by / * g and define it by f* 9 (n )
J
=
f(X )g (X ~ 1fi)dX
V M€ T
T
n 7 T ,W
±
=
J
f(t)g (s -t)d t
V s € R.
(5)
—■K
I * j (§1.9.1 below gives an elementary explanation o f the origin of these formu las.) Application of equations (1) and (2) allows us to rewrite this as f*g((j)
=
f
f( f j , X) g( X 1) dX
l f ( X ~ l )g(X ii)d X
=
j
f{ n X ~ l )g(X)dX V M€ T
(6)
7/bid. implies f * 9 = 9 * f-
(7)
116
1: Normed Algebras and Examples
1.8.6
An application of Fubini’s theorem and equation (1) shows 2 w
w
(sf) J\Jhms-t)dt\d,
=
(8)
—ir
k
ir
<
So the norm on L 1 is submultiplicative. (Hereafter we omit the proof that various functions are measurable and other measure theoretic sub tleties. Many o f these results are checked explicitly in Hewitt and Ross [1963] or Hewitt and Stromberg [1965].) We could now check the other properties necessary to show that L 1 is an algebra by arguments similar to those already given. (Only associativity is nontrivial.) Actually we prefer to postpone this until we have introduced the Fourier transform at which point these results become obvious. Anticipating this discussion, we will regard L l as a commutative Banach algebra and prove that the other con volution algebras are subalgebras or ideals in L 1. We will also derive the submultiplicative property for each norm.
1.8.6 L 2 is an Ideal in L 1 For / € L l and g € L 2, the Cauchy-Schwarz inequality gives
-*)la)1/a
= (^F / ir
ir
ir
< ± j |/(t)l Id t ± J \m\ \g(s~t)\2dt = \\f\\1i- J |/(0| ls(s—0l2dt. —k
—-k
—n
This result together with Fubini’s theorem gives is
11/* flll22= ^ J
A
\(f
* flK*)|2^ <
-K
(
*
J Ij -X
2 *
< ll/ll,
*
\
2
I/(t)fl(s - 0M* j ds
V-W
/
*
J J 1/(01 \9(s - t)\2dsdt = ll/ll^yij2. —-K—-K
Hence we have established
ll/* s lla < ll/llills ll3
V / e L ' u e L 2,
(9)
1.8 "
Group Algebras on T
117
so that L 2 is an ideal in L l . Inequalities (4) and (9) show that the norm on L 2 is an algebra norm.
1.8/ L ° ° (* ) is an id ea l in L 1 It is even easier to show that L ° ° (* ) is also in ideal o f L 2. For any / 6 L 1, g g L °° and a g [—n, 7r], we have "K
"K
17 Z g ( ‘ )\ < ~ J \ f ( t ) g ( s - t ) \ d t < ± - j \ f ( t ) \ M ^ d t = H/IMIsIL—ir
—n
This establishes
H / * s lL < ll/ llills lL
(10)
V f € L 1-, g € L °°.
so th it L ° ° (* ) is an ideal in L 1 and in L 2. This inequality and (4) show ll/ * »IL < ll/ IU IM L
So L '° ( * ) is a Banach algebra.
1.8.8 T h e Convolution P rod u ct o f Functions from L 1 and L°° is Continuous To establish this and several related results we need a lemma. For any /: T —> C and any s 6 R we will define /»: T —►C by /s>A) = /(Ae*“ )
V A6 T
or
V < 6 R.
/»(<) = /(< + « )
(11)
In all of the Banach spaces under consideration, note that the norm satisfies ll/ ll =
II/*II
f° r
the space and all a € R.
L em m a Let X be one o f the Banach spaces L 1, L 2,C ,A C , or C 1. Then fo r each f € X , the map 1 1—> f t from R to X is continuous. P roo f For any s, r € R, any / € X and any e > 0, we choose a p € T satisfying ||/ — p|| < |, where || • || is the norm in X . Then we have II/ , - f r ||
<
||/. -
=
IK / - P ) . l l +
Po ll
+ Up, IIP . -
Prll
+
P rll +
||Pr
~ /r||
| | ( / - p ) r || <
-
+
Up . -
P r ||.
Hence it is enough to show that any given p € T and r € R satisfy lims_,P ||pa —pr || = 0 in the norm o f each o f the Banach spaces under discus sion. \fter choosing an n such that p can be written as p = afc/ifc for suitable coefficients a *, one can easily establish these continuity properties. n T'\is lemma easily establishes the inclusion L 1 * L °° C C.
(12)
1: Normed Algebras and Examples
118
1.8.9
This implies that I/oc(* ) and C (* ) are both ideals in L l and hence in L 2. For / € L 1, g 6 L °° and r, s € R we have
\(f * 9) (* ) ~ ( f * 9) (r)\
=
-^1
J (j(s
+ t ) g ( - t ) - f ( r + t)g (-t)jd t\
—■K n
- bJ ~ —n
<
ll/ .- M IJ M L .
Since the first factor on the right can be made arbitrarily small, f * g is continuous. Note that equation (12) also implies that the ideal (L ° ° ( * ) ) 2 in L ° ° (* ) is included in C. This ideal clearly contains T (each h„ is idempotent under convolution multiplication so T 2 = T ) and hence we have ( ( L 00( * ) ) 2) - = C ' / L 00(*).
1.8.9 C B V ( * ) is an Ideal in L °°(* ) This follows from the inequality 11/ * aWcBv - II/ IU Is IIc b v which we now prove.
V / € L°°; g € C B V
(13)
Given / and g as above, we have ||/ * p||oo <
ll/IIJIsll, < ll/ILllslloo bX Previous results. For any partition t0 = - n < ti < • • • < tn = n we have: n X ^ l ( /
*a){tj)
~
( f * a ) { t j - i)l
<
" i t 5 Z ^~ J l/W I
<
ll/lloo^: J
3= 1
l
}
^(tj
n
- t )-
g(tj-1 - t)\dt
- a-t(tj-x)\dt
IT <
l l / l l o o ^ / l l s l l v *
=
l l / I U I s l l v
—n Hence (13) follows from I I / *
s
II c
b v
=
l l / * s l L + l l / * s l l v
^
l l / I L ( N L + l l s l l v )
=
I I / I I o o IIs II c
b v
Combining again with (4) shows that the norm on C B V is submultiplica tive.
1.3.10
Group Algebras on T
119
1.E.10 A C (* ) is an Id e a l in L 1 To complete the discussion of the relation of these convolution algebras we will show (/ *< ? )' = /*
V f e L 'i g e A C
(14)
which implies / * g € A C and H / * fllU c < ll/ lliN U c
V f e L l -,g e A C .
(15)
Thus A C ( * ) is an ideal in L l and hence in the other convolution algebras which contain it. Furthermore (14) and (10) show that C 1( * ) is an ideal in L l . As in the earlier cases, we conclude that the norms on A C (* ) and C'1(* ) sure submultiplicative. To prove (14), assume / € L 1 and g € A C . Then the equation g (s ) = g ( —r ) 4g '(r )d r for all s € K implies K
J
f * g ( s ) - f * g (-*■ ) =
f ( t ) ( g ( s - t) - g ( - 7T - t))d t
—7T n
= i
8 —t
^ /
J
/(<)
—n
—
&
n
J J f { t ) g '( u - t ) d u d t
g '(r ) d r d t = - ^
n —t
s
— k --k
*
a
= -^: / / f(t)9'(u - t)dtdu = J ( f * (g')J(u)du. —7r —it
—n
Differentiating this equation gives (14). The equation also shows directly thai f * g is an indefinite integral and therefore belongs to A C . 1.3.11 L 2 as a H ilb e r t space In order to introduce the Fourier transform " natural way, we will discuss the Hilbert space L 2 in more detail. Its iiir.ar product is *
‘K ( f , 9 ) = / fW g (X y d \ = i
I
f (t )g (t)* d t.
T
The functions hn, introduced above, satisfy
0 Eence H = { hn : n € Z } is an orthonormal set in L 2 since this is just the defining property o f such a set. Recall that an orthonormal set H in a
120
1: Normed Algebras and Examples
1.8.12
Hilbert space H is called an orthonormal basis if it satisfies one (and hence all) o f the following equivalent conditions: (a ) H is a maximal orthonormal set. (b) If x € 'H satisfies (as, h) = 0 for all h € H , then x = 0. (c) H is the closed linear span o f H. (d) For all x 6 H , the unordered series x = £
(x ,h )h
he H converges to x in the norm o f H. (e) For all x, y € 7i, the unordered series (x , v ) = ^ 2 (x ,h )(h ,y ) he H converges to (x, y). (f ) For all x € H , the unordered series iixii2 = x ; i ( x , / o i 2 he H converges to ||x||2. The series in (d) is called the Fourier expansion of x, and each o f the last two equations is sometimes called Parseval’s identity. (The easy proof o f the equivalence o f these properties can be found in any book discussing Hilbert spaces.) Since the set T = span(H ) of trigonometric polynomials is dense in L 2, the orthonormal set H = {h n : n € Z } is an orthonormal basis. 1.8.12 F ou rier-P la m ch erel T ran sfo rm s For each / € L 2, let us define a function /: Z —►C by IT f ( n ) = (f , h n ) = J f ( \ ) h n(\ y d \
f ( t ) e ~ inidt.
(16)
Condition (d) for an orthonormal basis shows (17) nez
where convergence is in the norm o f L 2. Condition ( f ) shows ll/lb2 = £ n€Z
l/ (n)|2
V / € L 2.
(18)
121
Group Algebras on T
.3
Hence the map / <-* / is an isometric linear map o f L? into the Hilbert space £2 = £2( Z ) = {square summable, doubly-infinite sequences}. Since E n ( Z fl"^n converges to an element / in L 2 for any a € £2, we see that the map f >-* f is surjective and hence is a unitary isomorphism o f L 2 onto £2. We vill call this map the Fourier-Plancherel transform. 1.8.13 Id e a l S tru ctu re o f th e B anach A lg e b r a L 2 Next we wish to ■:? amine the algebraic structure o f L 2 and o f the Fourier-Plancherel transform. A trivial calculation gives n 7^ m
f. 0, h „*h m = { [
^
V n, m e Z.
:
( 19)
Hence the orthonormal basis H consists of orthogonal idempotents so that L2 =
®Chn r»€I
(20)
is bo;h an orthogonal direct sum decomposition o f the Hilbert space L 2 and a Baaach algebra internal direct sum decomposition o f the algebra. The formula f * h n = f(n )h n V / € L 2; n € Z (21) provides an easy proof that T is an ideal in L 2 (and in all our other con volution algebras when the function / i—* / has been extended to L 1) and that the closed ideals o f L 2 are exactly the subspaces o f the form 5 x = { / € L 2 :(/,/ i„ ) = 0
V n € 5}
whero 5 is an arbitrary subset o f Z. (The ideal T is the socle o f all o f these algebras in terminology that will be introduced in Chapter 8.) F tially we come to the most interesting result. For any / and g in L 2 and t n y n e Z, Fubini’s theorem and translation invariance, equation (1), give
J j
/*
f(t )g (s -t )d te
—n —x 2 it =
(^ ) J —X
f ( t ) e - ' nt
,nBds
ir
J
g ( s - t ) e - in< -V d s d t
—-K
IT
=
h
J
f ^ e~ 'ni9 (n ) dt = f (n )g (n ).
(22)
122
1: Normed Algebras and Examples
1.8.14
The Cauchy-Schwarz inequality shows that the pointwise product o f two functions in t 2 is in t l — t l ( Z ) = {doubly-infinite summable sequences} C I 2. Hence we may, and will, consider i 2 to be an algebra under pointwise mul tiplication. Thus, we have shown that the Fourier-Plancherel transform is not only a unitary isomorphism but also an algebra isomorphism of L 2 (with the convolution product) onto £2 (with the pointwise product). 1.8.14 F o u rier 'IYan sform o n L 1 Some important parts o f the above discussion can be generalized. Except for the inner product, each of the expressions for f ( n ) given in (16) makes sense for any / € L 1. Hence we use them to define a linear map / •-* / from L 1 into the linear space of complex sequences. This map is called the Fourier transform. It is immediate that all n € Z satisfy |/(n)| < H/Hp so that / is always a bounded sequence. In fact a stronger result is easy for us to prove now, since we know that T is dense in L 1.
Riem ann Lebesgue Lem m a
The Fourier transform f o f any function
f € L l belongs to Co = {doubly-infinite sequence converging to zero at oo and — oo}. P roo f For any f € L l and any e > 0, we can find a trigonometric polynomial p satisfying \\f —p||i < e. Then for n greater than the degree o f p, we have |/(n)| = |(/ - p)(n)| < ||/ - p||! < £■ Hence / belongs to the space Co-
□
The space Co is always regarded as a Banach algebra under pointwise multiplication. The calculation (22) remains valid for f , g € L l and shows that the Fourier transform is a contractive homomorphism of L 1 into Co. Our next task is to show that the Fourier transform is injective and hence an isomorphism onto its range. The analogous result for the FourierPlancherel transform was easy since we could reconstruct any / 6 L 2 from its Fourier-Plancherel transform via the series / = £ n e z f ( n )h n which converges in L 2. The formal series /(n)/in is called the Fourier series o f /, whether or not it converges in any sense. 1.8.15 T h e F e jer K e r n e l Unfortunately the Fourier series for / € L 1 seldom converges in the norm o f L l . However the series is summable by various standard summability methods. We will investigate only one— Cesaro summability. For a (doubly-infinite) series we w'^ use to denote the nth symmetric partial sum Sn = 23fc=-n a * ’ we use
Group Algebras on T
123
converges), then the sequence {e r „ }„ eN converges, but that the converse faiis in many interesting cases. Since the limit o f the latter always agrees with the limit o f the former when that limit exists, it is a value that can be assigned in place o f a sum for some nonconvergent series. Suppose / belongs to L 1. Formula (21) shows that the Cesaro mean
k=0 j= -k of the series J^n€Z f ( n ) h „ is given by * „ (/ ) = * » * / ,
(23)
where K „ is the trigonometric polynomial
*
- - ±
* = —n
( ' - &
(24)
) *
which is called the Fejer kernel. Clearly K n satisfies 7T ±
j —
k n{t)d t = 1.
(25)
7T
It &Iso satisfies Kn > 0 and.
Vn € N
(26)
2i r - 6 lim J - [ n—.oo 2n J
K n(t)d t = 0
V<5> 0.
(27)
6
B o ti o f these properties sure immediate consequences o f the identity
n + 1 V sin(|)
)
.t
which in turn follows from the calculation 4(n + 1) (sin
) 2 K n{t) = ( - e « + 2 - e " « ) £
(n + 1 - |fc|)eifc‘ —n
= _ C<(»+1)‘ + 2 - e-<("+D ‘ = 4 ^sin
•
We will now show that if / belongs to one o f the Banach spaces L 1, C, /./J or C 1, the sequence {<7n(/ )}n eN converges to / in the norm o f that
124
1: Normed Algebras and Examples
1.8.15
space. O f course this Cesaro series also converges in L 2 since the Fourier series itself actually converges there.) The only properties o f a„ ( / ) we will use are those given by (23), (25), (26) and (27). Denote L l ,C ,A C or C l by X and the norm of X by ||• ||. We will use only the following properties o f these Banach spaces: for any / € X , the function « » - * / , is 27r-periodic and continuous from R to X and satisfies ||/„|| = ||/|| for all s € R. In the proof o f the next theorem we will use Riemann integrals o f certain continuous ^-valued functions. The definition o f these integrals, the proof that they exist, and the proof of the other basic properties which we use can be copied from any careful calculus textbook discussion using Riemann sums. F e je r’s T h e o re m Let (X , || • ||) be a Banach space such that fo r each f € X and s € R an element f a € X is defined satisfying: ll/«ll = ll/ll,
s>~* f , is continuous from R to (X , ||• ||)
and f , is -periodic in s. Let {i
= i-
j k n( t ) ( f -
f t )dt +
±J
-6
k „ ( t ) ( f - h )d t.
S
For the first integral, (26) and (25) give s 11^
J
K n( t ) ( f - f t )dt\\ < - ^
j
ATn(<)d<m ax{||/-/t ||} = m ax{| | / -/ t ||}
-i
which approaches zero with 6, by our assumption on the continuity o f 1 1—> /(. For the second integral, after the choice o f 6, the assumption ||/t|| = ||/|| for all t G R gives 2-K-6
IIi J 6
k n{ t ) ( f - f t )dt\\<2\\f\\^
2*-6
j 1 5
k n(t)d t
Group Algebras on T
S.16
125
w’lich also approaches zero with n, this time by (27).
□
This result shows that the Fourier transform is injective on L 1 since / = 0 implies an( f ) = 0 for all n. We summarize our results on transforms: C o ro lla ry The Fourier-Plancherel transform is a unitary map and an algebra isomorphism o f L? onto £?. The Fourier transform is a contractive injective homomorphism o f L 1 into cq. 1.3.16 T h e In verse F o u rier T ra n sfo rm o n £l We will now reverse oi r point of view briefly. Consider the Banach space £l = £l (Z ) of doublyininite absolutely summable complex sequences (i.e., functions a from Z to C satisfying X )n€Z|a„| < oo) with norm a i-+ Hal^ = ]C n6zla nl- We cs i make f x into a Banach algebra by defining a convolution product
(a ♦ <9)(m) = y : atnPm-n
V a t f P e l 1.
nez T f see that a *f3 belongs to P again and satisfies (Jot * /3||t < Hallill/Jllj, we us3 the sequential form of Fubini’s theorem: lla * @\\l —
\a n 0 m - n \ = m €Z n€Z
la n| n€Z
\Pm -n\ =
lla llill/ J | li'
m €Z
The rest o f the axioms for a Banach algebra are now most easily checked by reference to (29) below. We will follow the common practice o f denoting this Banach algebra by £l even though £l could be considered a Banach algebra under pointwise multiplication. For any sequence a € £l , we define a: T —* C by 6 (A ) = £ a „ A "
VAST,
or
6 = £ a n/i„
n€Z
(28)
n€Z
wl ere the first series converges absolutely and uniformly, and therefore the s:: >nd series converges in the norm of C to an element o f C. The map « - i d i s called the inverse Fourier transform. Since uniformly convergent 3e: ies sure integrable term-by-term, we find (a)T = a
V a € E1.
1.1.17 T h e F o u rier A lg e b r a In fact, it is obvious that a function / in C has an absolutely convergent Fourier series if and only if / belongs to £1, in which case it has a uniformly convergent Fourier series. Let us denote th ; space o f such functions by A = A (T ) and make A into a Banach space h j transporting the norm of £x via ( ') (or ( “)), i.e.,
II/IL = £ | / ( " ) I «€ Z
v
/€A.
126
1: Normed Algebras and Examples
1.8.18
In fact, A is a Banach algebra under pointwise multiplication. Its closure under pointwise multiplication is a consequence o f the identity fa = f * 9
v /> 3 e A
(29)
which, o f course, can be rewritten as a/3 = (a * (3J for any a, (3 € £l . Since the proof o f this is essentially the same as (22) above, it is omitted. This algebra A is called the Fourier algebra of T. The restriction of the Fourier transform to A is again called the Fourier transform. We have proved:
P rop osition
The Fourier transform is an isometric algebra isomorphism
o f A onto i 1. (O f course the isometry part o f this proposition is a bit o f a fraud since the norm o f A was defined to make it true.)
1.8.18 O th er A lgebras on T under the Pointw ise Prod u ct We have already remarked that C, C B V , A C and C 1 sure, like A, Banach algebras under pointwise operations. We will close this section by showing that C 1 is included in A and some related results. This will depend on the following simple identity: (/ 7 (n ) = i n f { n )
V / G AC-, n € Z
(30)
which is established by integration by parts: * ( / 7 ( n )
h
=
/
/ ' ( <) e _ m t *
—■K 7T
~hJ f(t)(~in)e~intdt
=
+
= *"/(«)•
—1r
If / belongs to C l , then / ' belongs to C C L 2. Hence the Cauchy-Schwarz inequality gives
E
!/(")!
=
E
e(/')i«)i
n € Z \ {0 }
n € Z \ {0 }
<
( E \ n € Z \ {0 }
( ^ " E ^ 'H n ) ) 2] n€Z
= -y/'||2
J
Hence / belongs to A as claimed. W e have already shown that the convolution product o f a function in L 1 and one in L °° gives a continuous function. Thus convolution is a
1.8.19
Group Algebras on T
127
smoothing operation. This is important in the more classical applications. Here is another amusing indication o f the same thing. Consider / € A C and g € A. Results (14) and (15) show f * g € A C and (/ * g )' = f * g. Hence ((/ * g ) ') A = ( f 'J g is the pointwise product o f a sequence in cq er.d a sequence in t l and hence belongs to l l . Therefore (/ * g )' belongs tc A and hence to C. Thus we have proved the inclusion A C ♦ A C C 1. Moreover, the convolution product o f two functions without everywhere defined derivatives can have a continuous derivative. 1.3.19 H is to ric a l R em a rk s Th e study o f Fourier series was one of the main sources of functional analysis. Daniel Bernoulli (1700-1782) and Leer surd Euler (1707-1783) first used what we would now call Fourier series. They did not develop their ideas, and not much progress was made on this ep^roach until Jean Baptiste Joseph Fourier (1768-1830) used trigonometric series in a problem on heat. His methods were severely criticized by some of liis contemporaries, but this motivated him to further develop his ideas for a decade until he wrote his famous book Theorie analytique de la chaleur [1822]. In this book he claimed that “any function” could be expanded in a cc:’ 73rgent Fourier series. A t that time, mathematicians were often not very precise about convergence and were extremely vague about the meaning o f tli3 word “function” . Fourier’s claim spurred a closer examination o f the corrupt o f a function. Peter Lejeune Dirichlet (1805-1859) stated essentially our modern view o f functions and in [1829] he introduced his kernel and ^sod'it to prove the first rigorous convergence theorem for Fourier series. Since his theorem and method o f proof sure still important we will review them briefly. A function /: R —> C is said to be piecewise smooth on an interval [a, 6] if there is some partition a = to < C be piecewise smooth on [—w, n\ and 2%-periodic. Then the Fourier series converges pointwise to f ( x ) at every point o f continuity and it converges to f (x + ) + f(x ~ )
2 o.t cr, y point o f discontinuity. I f f is also continuous, then the series con venes to f absolutely and uniformly on R. Dirichlet’s proof was based on the following function known as D irich-
128
1: Normed Algebras and Examples
1.9.1
le t’s kernel
The first expression in (31) shows that the convolution product with any function in L 1 is the nth partial sum of the Fourier series: / * D n(t ) =
n £
f ( k ) e ikt.
k——n
The middle expression in (31) shows the same thing for Fourier series writ ten in the standard real form. The last expression in (31) allows an ele mentary evaluation o f the limit of { / * D n( t ) } nea as n increases. Around the beginning of this century, functional analysis began to de velop around problems dealing with integral equations, many of which had originally arisen in the context of Fourier series (e.g., Eric Ivar Fredholm [1900], David Hilbert [1904], [1906], Erhard Schmidt [1907], [1908]). Just before the Second World War increasingly algebraic methods, which arose naturally in the study o f Fourier series and its lineal descendent, harmonic analysis, were a main stimulus to the definition o f Banach algebras (e.g., Norbert Wiener [1932], Arne Beurling [1938]).
1.9
Group Algebras
We will introduce some o f the most important group algebras here and give a few o f their basic properties. Since we sure postponing the discussion of involutions to Volume II, and involutions play a prominent role in the theory of group algebras, we will not go into the theory o f group algebras very deeply here. However, additional information on group algebras will be developed later in the present volume as new topics are explored. See particularly Section 3.6 (Commutative Group Algebras) and Example 5.1.9 (approximate identities in group algebras). We will denote the identity element in a group by e unless some other notation is established. For more details we recommend Hewitt and Ross [1963] and Hans Reiter [1968], Jean-Paul Pier [1990] is a fascinating historical survey. 1.9.1 T h e A lg e b ra ic G ro u p A lg e b r a Fundamentally, most group algebras are devices for introducing linear operations into a group. We will call the simplest possible group algebra o f a group G the algebraic group algebra. This algebra, which we will denote by £q(G ), consists o f all complex-valued functions on G each o f which is non-zero at only finitely many elements of G. Each of these functions / may be thought of as a
.9.1
Group Algebras
129
formal sum / ~ 5 Z f ( u) u u€G
o ' finitely many complex multiples of elements o f G. I f two elements / e.nd g of the algebraic group algebra sure interpreted in this way, then the fc miliar rules o f computation suggest the following definitions of addition, sc alar multiplication and multiplication: £ / («)«+ fl( u) u uGG uGG
=
f ( u) u
=
A u6G
^ 2 f (u )u ^ 2 g { v )v ueo weG
] C ( / ( U) + S (u ))u uGG
xf ( u) u u6G
=
^ 2 f{u )g (v )u v . ueG veG
Thus we define the linear operations pointwise. I f we collect terms in the expression for the product, we get
weG u6G Hence we sure led to the definition o f the convolution product f * 9 (u ) = '^ 2 f ( v ) g ( v ~ 1u)
VueG .
(1)
u€C
It s obvious that these operations make the algebraic group algebra into an algebra. (Note that the above discussion would not need to be changed if another field (or even ring) were substituted for the complex field. Algebraic group algebras with respect to other rings and fields play an important role in the theory of finite groups. See Charles W . Curtis and Irving Reiner [1981, 1987]. However, we will consider only complex scalars.) The algebraic group algebra has a number o f disadvantages. It ignores the topology of G when G has a nontrivial topology, as we will often suppo: e. However even for finite groups (which are necessarily discrete in this discussion, since we will insist that topological groups have a Hausdorff toj ology) the algebraic group algebra is not a complete set o f invariants. As a matter o f fact the dihedral group o f order eight (with two generators u snd v satisfying u4 = v2 = e and uv = vu3) and the quarternion group ( w ± two generators u and t> satisfying u4 = e, u2 = v2 and uv = vu3) ha’ s isomorphic group algebras. (A n explicit isomorphism is obtained by usi.ig the bases (e, u, u2, u3,v + u2v, uv + u3v, w, uw) where in the first case w : 3 v — u2v and in the second case w is i(u v — u3v ).)
130
1: Normed Algebras and Examples
1.9.2
As we will show below, the normed algebra (£q(G ), || • ||,) obtained by endowing the algebraic group algebra with its natural f'-norm ll/lli = £
l/ (« )l
V / e^ (G )
(2)
ug G
is a complete set of invariants for the group G. (This norm is submultiplica tive as shown in (4) below, thus verifying that (£q(G ), || • ||j) is a normed algebra.) When thinking about topological groups we will call groups without topology, discrete groups. For discrete groups, this normed algebra is a fairly satisfactory group algebra. It is complete, and hence a Banach alge bra, if and only if G is finite. In order to be able to make use of the theory of Banach algebras when G is not finite, it is useful to replace (£q(G ), || • ||j) by its completion. This completion is the most natural and useful group algebra for discrete groups. We will now introduce it formally. 1.9.2 T h e ^ - G r o u p A lg e b r a D e fin itio n satisfying
We begin with a formal definition.
Let G be a group. Let ( l (G ) be the set of functions f : G —* C 11/11!= £ | / ( « * ) l < 0 0 . ueG
(3)
Let t* {G ) have the structure of a normed algebra with pointwise linear operations, the norm || ■||j defined above and the convolution product given in (1) above. Then £*(G ) is called the £l -group algebra o f G. W ith the above definition, £l (G ) is a Banach algebra. It is easy to check that ||• ||j is a complete norm. To show that the product is well-defined, we check that the infinite series (1) converges absolutely. This calculation also shows that the norm is submultiplicative. We have II/* slli
1 5 Z / ( v ) s ( v _ lu )l
= u6G
(4)
vZG
- Y1Y1 i/(w)s(i,-lu ) i = 5Z t*GG v £ G
=
v£
£ l / M l £ l f f ( « O I = ll/llillslli w
i- f^ i is(w_lu)i
G u£G
v /, g e £*(G)
£G
where we have used the fact that G = { v ~ 1u : u G G } holds for any fixed v € G. Now it is straightforward to check that £ '(G ) is an algebra (and hence a Banach algebra) under this multiplication. For each u € G , define Su to be the function on G which takes the value one at u and zero elsewhere. Then it is easy to see that 6f is a multiplicative
Group Algebras
J.3
131
identity element for t 1( G ) and that each 6U is invertible and satisfies (6U) *= ^ ,,-1
6U6V = 6UV
V u ,v e G .
(5)
” ;r>ce the map u i—» 6U is a group isomorphism of G into the multiplica tive group o f invertible elements of i l (G ). Each element o f i l (G ) may be expressed uniquely as an absolutely norm-convergent series o f complex "i'ltip le s of these elements: / = £ / ( “ >*» ueG
V / e^ (G ).
(6)
This is a rigorous interpretation of the formal sums introduced above in the discussion of algebraic group algebras. Since u i—> 6U is an isomorphism o f G into the multiplicative group of . A 1(G ), we see that £l (G ) is commutative only when G is abelian. Con versely, the absolute convergence of the expansion (6) for / € £l (G ) gives an easy proof that £l (G ) is commutative when G is abelian. Thus £l (G ) is commutative i f and only i f G i s abelian. If / belongs to the center of £l (G ), then it satisfies 6U * f * 6U-1 = / for all u € G. In fact, the expansion above shows that this is a necessary and sufficient condition for / to belong to the center of G. Hence / € i 1(G ) belongs to the center of ^ ( G ) if and only if it is constant on conjugacy classes. (A conjugacy class in a group G is a set of the form C\, — {uvu~* : • € G } for some v g G .) An £l function can have a constant non-zero . value only on finite sets. Hence £ *(G ) has a nontrivial center (i.e., different from CSe which is always included in the center) if and only if it has some 5 r :ie conjugacy classes besides the trivial class { e } . Groups which have only infinite conjugacy classes (except for { e } ) are called Infinite Conjugacy Class (abbreviated IC C ) groups. They provide important examples of von Neamann algebra factors as we shall show subsequently. i <• 1.9.3 R e c o v e rin g G fro m £l (G ) W e now show how a discrete group G may be recovered from the Banach algebra £l (G ). Our previous remarks show that the norm must play an essential role in the process. Note that tbe extreme points of the unit ball ^ ( G ) i o f ^ ( G ) have the form (6 U for §ome ( € T and some u € G. This follows immediately from the expansion (6). Thus the set E of extreme points forms a group. Let h: G —►T be any homomorphism. Then the set H h — {h (u )6 u : u e G } is ® subgroup of E which has the property
11/ +
g\\ =
ll/ll + IMI
v / lS6 fl,
(7)
130
1: Normed Algebras and Examples
1.9.2
As we will show below, the normed algebra (£q(G), || • j|,) obtained by endowing the algebraic group algebra with its natural f'-norm ll/lli = £
l/(**)l
v / e *o( G)
(2)
«eG
is a complete set of invariants for the group G. (This norm is submultiplica tive as shown in (4) below, thus verifying that (£q(G ), || • ||j) is a normed algebra.) When thinking about topological groups we will call groups without topology, discrete groups. For discrete groups, this normed algebra is a fairly satisfactory group algebra. It is complete, and hence a Banach alge bra, if and only if G is finite. In order to be able to make use of the theory of Banach algebras when G is not finite, it is useful to replace (£}t(G ), || • ||j) by its completion. This completion is the most natural and useful group algebra for discrete groups. We will now introduce it formally. 1.9.2 T h e ^ - G r o u p A lg e b r a D e fin itio n satisfying
We begin with a formal definition.
Let G be a group. Let f 1( G ) be the set o f functions /: G - * C
ll/ll, = £ |/(«*)| < oo.
(3)
ueG
Let f ] (G ) have the structure of a normed algebra with pointwise linear operations, the norm ||• ||1 defined above and the convolution product given in (1) above. Then £x(G ) is called the £l -group algebra o f G. W ith the above definition, £*(G ) is a Banach algebra. It is easy to check that || • ||j is a complete norm. To show that the product is well-defined, we check that the infinite series (1) converges absolutely. This calculation also shows that the norm is submultiplicative. We have
II/* slli = £ 1 £ / (v)s,(trlu)i u6G
-
£
u£G
£
ueC u cG
=
i/(w)ff(v_1“ )i =
£
£
i/(v )i
iff(v~lu)i
ueG ueG
D / ( ” )l £ I s M I = ll/llillslli w(zG
V /, ® € £ '(G )
where we have used the fact that G = {v ~ l u : u £ G ) holds for any fixed v € G. Now it is straightforward to check that £l (G ) is an algebra (and hence a Banach algebra) under this multiplication. For each u e G , define 6U to be the function on G which takes the value one at u and zero elsewhere. Then it is easy to see that b, is a multiplicative
J.3
Group Algebras
131
identity element for t l ( G ) and that each 6U is invertible and satisfies (6U) 1 = 6u-i
6U6V = 6 UV
V a , t € G.
(5)
” ~,r>ce the map u >—> <5U is a group isomorphism of G into the multiplica tive group of invertible elements of £l (G ). Each element o f ^1(G ) may bs expressed uniquely as an absolutely norm-convergent series of complex multiples of these elements: / = £ / («)* „ ueG
V f £ £ 1(G ).
(6)
This is a rigorous interpretation o f the formal sums introduced above in the discussion of algebraic group algebras. Since u i-* 6U is an isomorphism o f G into the multiplicative group of . ■£1(G ), we see that I 1(G ) is commutative only when G is abelian. Con versely, the absolute convergence of the expansion (6) for / € £l (G ) gives an easy proof that £l (G ) is commutative when G is abelian. Thus £l (G ) is commutative i f and only i f G is abelian. If / belongs to the center of i l (G ), then it satisfies 6U * f * 6u- i = f for all u £ G. In fact, the expansion above shows that this is a necessary and sufficient condition for / to belong to the center of G. Hence / £ t x(G ) belongs to the center of ^1(G ) if and only if it is constant on conjugacy classes. (A conjugacy class in a group G is a set of the form C\, = {uuu-1 : ' ** e G } for some v £ G .) An t 1 function can have a constant non-zero . value only on finite sets. Hence l l {G ) has a nontrivial center (i.e., different from CSe which is always included in the center) if and only if it has some 2r.::e conjugacy classes besides the trivial class { e } . Groups which have only infinite conjugacy classes (except for { e } ) are called Infinite Conjugacy Clcss (abbreviated IC C ) groups. They provide important examples o f von Neumann algebra factors as we shall show subsequently. J
11
1.9.3 R e c o v e rin g G fro m l l (G ) W e now show how a discrete group G may be recovered from the Banach algebra £l (G ). Our previous remarks show that the norm must play an essential role in the process. Note that the extreme points o f the unit ball l l (G )i of £l (G ) have the form for §cme ( 6 T and some u £ G. This follows immediately from the expansion (6). Thus the set E o f extreme points forms a group. Let h: G —* T be any homomorphism. Then the set H h = {h (u )6 u : u £ G } is = subgroup o f E which has the property 11/ + g\\ = ll/ll + y i
V /, p e Hh-
(7)
132
1: Normed Algebras and Examples
1.9.4
Furthermore t * (G ) is the closed linear span o f Hh. Conversely, suppose H is a subgroup of E satisfying (7). Then H contains at most one element of the form (6 U for each fixed element u e G. Hence H equals {h (u )6 u : u € H } , where H is some subset of G and h: H —* T is some function. Since H is a subgroup, H must be a subgroup of G and h must be a homomorphism of H into T. If in addition f ( G ) is the closed linear span o f H , then H must be G. Hence H is isomorphic to G. Therefore we can recover G, up to isomorphism, as any subgroup H o f E which satisfies (7) and which has ( l (G ) as its closed linear span. In fact, in Volume II we will use the assignment G t-* (G ) to construct an isomorphism between the category o f groups and group homomorphisms and a certain restricted category of Banach algebras. The following result is immediate from what we have proved. T h e o re m Two groups G and H are isomorphic as groups i f and only if t l (G ) and £l ( H ) are isometrically isomorphic as Banach algebras. 1.9.4 In tro d u c tio n t o R ep rese n ta tio n s Representations of algebras will be described systematically in Chapter 4. W e introduce some o f the ideas informally now in conjunction with representations of groups. A rep-* resentation of a group G on a linear space V is a group homomorphism V: G —r C (V ) g of G into the group of invertible linear operators on V. If V is a representation of G on V, then it is clear that V can be “extended” to a representation (i.e., an algebra homomorphism) T v of the algebraic group algebra (1 (G ) of G on V. We simply write T / («) = £ / ( « ) u£G
v/e/S(G )
where there is no problem o f convergence since only finitely many terms of the sum are non-zero. It is clear that T v is an algebra homomorphism and hence a representation of (1 (G ) on V. A normed representation of a group G on a Banach space A' is a group homomorphism V :G - > B ( X ) G of G into the group o f invertible bounded linear operators on X . A normed representation V is said to be bounded if there is a common bound B for all the operators Vu (i.e., ||14|| < B for all u € G ). Given a bounded representation of a group G on a Banach space X , we can “extend” V to a continuous representation T v of ( l (G ) (i.e., a norm-continuous algebra homomorphism T v -.t.l (G ) —> B ( X ) ) simply by writing r / = £
uec
/(u )V u
v/ e^ G ).
( 8)
1.9 .5
Group Algebras
133
Clearly the sum converges absolutely and T v is a continuous homomor phism of £l (G ) satisfying |jT,v' || < B if B is a bound for V . Note that the continuity of T v has nothing to do with any continuity property of V. If G is a group and T is a unital representation o f £\(G) or ^1(G ), then w< may define a representation V T o f G by V j = TSu
VueG.
(9)
If T is a continuous representation o f £ *(G ) on a Banach space, then V T is bounded. Clearly T i—►V T and V i—►T v are inverse constructions. Hence we can identify the class of all representations o f a group G with the cl e s s o f representations of £\(G ), and we can identify the class of bounded representations of G on Banach spaces with the class of all continuous representations o f £l (G ) on Banach spaces. These identifications preserve all the important properties o f the representations. For instance, if V is a representation of G on V, then a linear subspace W o f V is invariant with respect to V (i.e., Vux e W for all u e G and all a: € W ) if and only if it is invariant with respect to T v (i.e., T j x g W for all / g £q(G) and all x € W ). Similarly if V is a bounded representation o f G on a Banach space X , then a closed linear subspace W is invariant with respect to V if and only if it is invariant with respect to T v . Here is a natural and important example of these ideas. (Others will be discussed in Volume II.) Let G be a group and, for any u € G, let Vu G B(£1(G ) ) be defined by Vu( f ) = « » * / = . /
V / e^ G ).
It clear that this is a bounded representation o f G with bound 1. It is also clear that T v is the left regular representation o f £l (G ) on itself. ” In tro d u c tio n t o T o p o lo g ic a l G ro u p s We now extend the fore go, g considerations to topological groups. First we will define topological groips and give a few basic facts about them. The important work by Ed 'in Hewitt and Kenneth A. Ross [1963, 1970] is the best reference for adcrtional facts. Unfortunately, our notation does not always agree with the.rs. 4. topological group is a group which is also a Hausdorff topological spa ;e in which the multiplication map from G x G to G and the inversion map from G to G, defined by (u, v) i—> uv and u i—> u” 1, respectively, continuous. Many authors do not require that a topological group be . iausdorff but we shall. Since inversion is its own inverse, inversion is actually a homeomorphism. These conditions ensure that, for any fixed v e G, the maps u i—> uv and u t—> vu are homeomorphisms with inverse maps u i—> uv-1 and u i—> v _1u, respectively. Each o f these maps will carry a neighborhood system at the identity e into a neighborhood system
134
1: Normed Algebras and Examples
1.9.6
at v. The inverse maps have the inverse effect. Thus the topology of a topological group is completely determined by the neighborhoods of e and can be described by giving a neighborhood base at e. An abstract description of such a neighborhood base is given in Hewitt and Ross [1963], Theorem 4.5. Many properties o f topological groups can be proved in terms of these bases. A subset U in a group G is called symmetric if it satisfies U = U ~ l , where U ~ l is just {u -1 : u £ U ). It is useful to know that any neighborhood V o f e contains a symmetric neighborhood e.g., V fl V ~ 1. Hence a base o f neighborhoods at e may always be chosen to consist of symmetric neighborhoods. Space forbids extensively developing the fundamentals of the theory o f topological groups. We shall assume the most basic properties tacitly since they are almost self-evident. Complete proofs o f all these facts, and some less basic ones, will be found in Hewitt and Ross [1963], Sections 4 through 8. For less obvious facts, we will give references. In particular, we assume the following results which are more or less obvious: the closure o f a subgroup is a subgroup which is abelian or normal if the original subgroup has the corresponding property; the coset space G / N = {uJV : u e G } of a closed normal subgroup AT of a topological group G can be given the structure o f a topological group in such a way that the natural map G —* G / N is a continuous open homomorphism; the kernel of a continuous homomorphism is a closed normal subgroup; the product U S = {uv : u e OO
U , v £ 5 } of an open subset U with any subset S is open; |J U n is an n=l open subgroup if U is a symmetric neighborhood of e. Here are some slightly less evident basic facts with references to Hewitt and Ross [1963]. I f K is a compact subset and C is a closed subset, then K C is closed (Theorem 4.4); if V is a neighborhood o f e, then there is a neighborhood U of e with U C V (Corollary 4.7) and if K is a compact subset, then there is a neighborhood U of e satisfying u U u ~ l C V for all u € K (Theorem 4.9); a subgroup is open if and only if it has nonvoid interior, and open subgroups are closed (Theorem 5.5). If tp:G —* H is a continuous homomorphism with the closed subgroup N as kernel, then there is a continuous isomorphism of G / N onto H . This need not be a homeomorphism. For example, consider the identity map o f R<j (the addi tive group of real numbers with the discrete topology) onto R (the same group with its usual topology). 1 .9 .6 L o c a lly C o m p a c t G ro u p s A locally compact group is a topolog ical group which is locally compact as a topological space. Such groups have very special properties which we will briefly discuss later. However, to begin with, we only need the definition. Note that any group considered discrete topological group is a locally compact group. The additive group R and the multiplicative group T = € C : |£[ = 1} are nondiscrete
as a
■.9.7
Group Algebras
135
locally compact groups under their usual topologies. Also, any closed mul tiplicative group G o f n x n real or complex matrices is a locally compact group under the relative topology it inherits as a subset o f R " or C n . For a nondiscrete locally compact group G, we can of course construct ^1(G ), but it ignores the topology. (In fact in this case we will consider £l (G ) to be the group algebra of G j — the discrete group which is isomorphic to G as a group.) In order to find a natural group algebra which involves the topology o f G , we first note that P ( G ) is isometrically isomorphic to the dual space o f the space Co(G') (the space of all complex-valued functions on G vanishing at infinity, i.e., those functions / ;G - * C such that for every e > 0 there is a finite set off which the values of / are less than e in absolute value). The duality is implemented by 9 ^ ( f ^ £ / ( u)ff(u) ) u
'1 g e P ( G ) \ f e c o ( G ) .
(10)
The following general considerations allow us to define a convolution prod uct in a much wider context. (For historical comments on the development given here, see Hewitt and Ross [1963], §19 Notes].) 1.9.7 C o n v o lu tio n o n D u a l Spaces Let G be a group and let / : G - * C be a function. For any element u G G, we may define two new functions „/ : G —> C and /„: G - C by ' ’
Uf ( v ) = f ( u ~ 1v)
and
/ «(v ) = / (w ti'‘ )
V v G G.
(11)
W e also define a function /: G —> C by f(u ) = / (« " * )
V u G G.
(12)
It is easy to check that these definitions satisfy «(»/ ) (u / )'
=
« «/ ,(/ .)» = / »»,/ = /
=
(/ ) u-i
and
(13)
(/ „ )" = „ - i ( / )
for all u, v G G. The function ( uf / fu ) is called the ( left / r ig h t) trans late o f f by u, and the function / is called the reverse o f f . (Unfortunately is no standard universally accepted notation for these important concspts. Although our main reference for locally compact groups is Hewitt and Ross [1963], we have reluctantly chosen notation which is at odds with il’:.3 notation used there.) For the given function /: G —+ C we also define a function /: G —» C by f (u ) = / («)*
VbgG
(14)
where / (« )* is simply the complex conjugate of f ( u ) . (The notation f * TC'ild be more natural here, and we use this notation in some other con■jsxis for the function defined by conjugating the values. However, it will
136
1: Normed Algebras and Examples
1.9.7
be convenient to reserve /* for another concept when we deal with most convolution algebras of functions on groups.) Let G be a group and let X be a linear space of complex-valued functions on G. Let A be a linear space o f linear functionals on X . We will define a very general notion o f convolution on A when A and X satisfy certain conditions. First we assume that X is closed under left translation: V f e x - , u eG.
uf€X
(15)
For any ui € A and / e X , we define a function “'/: G —* C by " / ( « ) = w („ - ./ )
VueG .
(16)
V f e X ; we A.
(17)
Second we assume wf e X
Note that " / (e ) = u j(f ) shows that w •—» (/ >—►“'/ ) is a linear injection of A into C (X ). Given any two linear functionals u/, r e A , we define their convolution product to be the function uj * t : X —* C given by u j * r ( f ) = u j ( Tf )
V f eX.
(18)
Clearly w * t is a linear functional, and our third assumption is: uj
*r e A
V
ui,
r e A.
(19)
e A; f e X.
(20)
A routine calculation now gives «.T J =
V w,
t
This implies that A is an algebra under convolution multiplication and that *—* (/ •—» " / ) is an injective algebra homomorphism of A into C (X ). Retaining our previous assumptions, let us further assume that X is & normed linear space under the supremum norm: ll/lloo = sup{|/(u)| : u e G } and that .4 is included in the space A * o f continuous linear functionals on X . It follows immediately that any f e X , u e G and uj e A satisfy: ||„/|| = ll/ll
and
r/ | | < | M | ||/||.
From this we conclude l|w *r|| < ||w|| ||r||
Vuj,reA
so that the norm is submultiplicative and A in ti normed algebra,
(21)
I >.8
Group Algebras
137
If we apply the above theory to X = Co(G) and A = co(G )* we obtain a Beaach algebra structure on co(G )* which is isometrically isomorphic to the convolution algebra £1(G ) defined in §1.9.2. The isomorphism is given by t*' isometric linear isomorphism o f l x{G ) onto c o (G )' given in (10) above. 1.9.8 T h e G ro u p M ea su re A lg e b r a M ( G ) If G is an arbitrary locally compact group, then the Banach space C o(G ) of all continuous functions from G to C which vanish at infinity is an obvious candidate to serve in the role of X . This choice o f X with the choice A = X * satisfies all of the assumptions made above. This is easily checked when it is noted that for each function / in the dense set o f continuous functions which vanish off a (variable) compact set the map u i—►uf of G into C q(G ) is continuous. Hence C o(G )* is a Banach algebra under convolution. For the rest of this example we will assume that G is locally compact and that C0( G ) ' has the Baaach algebra structure just defined. It is customary to use the Riesz representation theorem (Rudin [1974] Theorem 6.19 or Hewitt and Ross [1963], Chapter 3) to identify C q{G )* with the Banach space M ( G ) o f all complex regular Borel measures on G, ant then to call this convolution algebra the measure algebra o f G. O f course, a measure /x € M ( G ) corresponds to the linear functional
( 22) W i en this identification is made, the convolution product takes on the attractive form
whore n and v are any two measures in M (G ). The norm on M (G ) , which agrees with the norm on C q(G )* under the above identification, is just ll/xll = H ( G )
V n e M (G )
wh:re |/x| is the tottil variation positive measure associated with the complex roe? sure fi. Note that for a discrete group G, the Radon-Nikodym theorem identifies M ( G ) with ^1(G ), in such a way as to make our present discussion a e re c t generalization of that given in §1.9.2. Since the present discussion uses no measure theory, we will often con tinue to denote our convolution algebra by C o(G )* and think o f its elements as continuous linear functionals on C o(G ). We believe this approach is concep;ually simpler. However, in other sections of the book we will usually adopt common practice and write M ( G ) for this algebra. W e note that m u A o f any proof o f the Riesz representation theorem is algebraic in nature
138
1: Normed Algebras and Examples
1.9.8
and makes sense in a wider context than just that o f algebras o f continuous functions. This will be explored further in Volume II o f this work. Next we wish to prove that G i—►C o(G )* is a covariant functor. In order to do this, we must construct a continuous algebra homomorphism i?-.C0{ G ) '
C0{H Y
for each continuous group homomorphism
v / e C 0(/0.
However, unfortunately, the map tp just defined does not usually map C q(H ) into C q(G ). (Consider tp to be the exponential map t i—> elt of R onto T = {A € C : |A| = 1 }.) A t first sight, this appears to be a fated flaw. Fortunately it is not. The map
V u € Co (G y-, f 6 C 0( H )
where uj is u> considered as a strictly continuous linear functional on C (G ). Fortunately, all o f this agrees with the identification o f C 0( G ) ' with A f(G ): the action of a measure fi € M ( G ) on a function / e G (G ), given by formula (22), is just the strictly continuous extension o f its action on C q(G ). Then formula (23) shows that the map Tp, which we have just defined, is a homomorphism. A ll this will be established in a wider setting in Volume II. For now, we have proved: T h e o re m Let G and H be locally compact groups. I f
h
f d & p ) = / f o ipdfi J
V / € C b (ff); /i 6 M (G ).
g
Equivalently,
2.9
Group Algebras
139
1.9.9 R e c o v e rin g G fro m M (G ) We now show that the abstract Banach eJgebra C o(G )* ~ M (G ) determines G up to homeomorphic isomorphism. Thus the functor G *-♦ C o{G )* provides a complete set o f invariants for locally compact groups. For each u € G, let 6U be the corresponding evaluation linear functioned in C o (G )* defined by M / ) = / («)
V / 6 C o(G).
(24)
Then it is easy to see that the map u i—►6U
Vu € G
is; an injective group homomorphism of G into the group of invertible eleirents o f Cq( G) * under convolution multiplication. It is injective since C\)(G) separates the points o f a locally compact space G. The convolution algebra l l (G ) is obviously isometrically isomorphic to the closed linear span o f {<5U : u e G } in C o(G )* under the map / -»£ / («)*» ueG
V / e t l (G ).
(i-iote that this gives a more concrete interpretation of formula ( 6) above.) Since G can be embedded in C o (G )*, it is obvious that G is abelian (i.e., commutative) if C q( G ) ' is commutative. The reverse implication holds as we will soon see, or the reader may consult Hewitt and Ross [1963], Theorem 19.6, for a proof based on Fubini’s Theorem. Now suppose we are presented with A — C q(G )* ^ M ( G ) as an abstract Banach algebra and wish to recover the topological group G from it. The extreme points of the unit ball are just the elements o f the form (S u for ( 6 T and u g G. This is easy to see when considering M (G ). Hence the set of extreme points of A i form a group which is isomorphic to T x G. When starting from an abstract Banach algebra, in order to get a specific isomorphism, and thus identify G as a subgroup, we must find an element ty in the Gelfand space of A . In the case of M (G ), this can be thought of as the map 7 (/x) = f G dp. In terms of 7 , we define G A to be the subgroup of
140
1: Normed Algebras and Examples
1.9.10
G. This was done by Roger Rigelhof in his thesis [1969b], but it seems likely that further improvements may be possible. Proofs or references can be found there for remarks in the previous paragraph which need proof. In [1965] and [1966] Marc A. Rieffel characterized the group algebras of commutative and finite groups, respectively. For commutative groups the characterization makes heavy use of the order structure on the group al gebra. Frederick P. Greenleaf [1965] characterized the group algebras of compact groups. This characterization also relies on the special nature of the group in an essential way and involves the existence of minimal ideals (see Chapter 8). The Banach algebra G'o(G)* is so large and complicated that it is poorly understood. Hence another smaller Banach algebra, which is less natural than C o(G )* in a number o; respects, is usually taken as the group algebra o f a nondiscrete locally compact group G. This is the L 1-group algebra. In order to define it we need to introduce Haar measure. 1.9.10 T h e H a a r F u n ction a l Let G be a locally compact group and let C oo(G ) be the linear space (under pointwise operations) of all continuous functions / from G to C which have compact support, where the support o f / is defined by: supp(/) = {u e G : f ( u ) ^ 0 } - . The cone o f non-negative real-valued functions in C oo(G ) will be denoted by C o o(G )+ . If h is a non-zero function in G'oo(G)+ , it is natural to think o f its translates uh for u 6 G as all having the same size as h. Hence the size o f a linear combination n
tk ukh k= 1
tk £ R+;ufc £ G
with non-negative coefficients should be times the size of h. Using this idea, the order properties of Coo(G )+ and an elaborate limiting proce dure which allows the comparison function h to be taken with smaller and smaller support, it is possible to find a linear functioned on Coo(G) which precisely measures the notion of size crudely described above. We refer the reader to Hewitt and Ross [1963], §15 Notes, for the long and interesting history of these ideas. (Attractive proofs of the theorem are also given in: Glen E. Bredon [1963], Erik M. Alfsen [1963] and Leopoldo Nachbin [1965].) Since the decisive step in generalization was taken by Alfred Haar [1933], the resulting linear functional is called the left Haar functional or left Haar integral. We state its properties formally. T h eo rem Let G be a locally compact group. There exist maps A: Coo(G) —> C and A :G —* R* satisfying:
Group Algebras
3.9.10
141
(a) A is linear. (b) A (/ ) > 0 fo r all non-zero f e C o o(G )+ . (c ) A ( „ / ) = A ( / ) fo r all f € C q o (G ); u e G. (d) I f A': Cqo(G) —►C satisfies (a), (c) and (b ') A (/ ) > 0 fo r all f € C qo(G )+ , then there is some c e R+ satisfying A ' = cA. (e) A is a continuous group homomorphism into the multiplicative group o f positive real numbers. (f) A (/ ) = A (u )A (/ „) = A (/ A ) fo r all f e Cbo(G); u e G. Note that (d) asserts that the Haar functional is unique up to multi plication by a positive real number. The homomorphism A : G —*■ R* is called the modular function o f G. Its existence follows directly from the uniqueness result in (d). Its continuity depends on the uniform continuity of translation in the supremum norm ll/lloo =
8UP { I / ( U )I
u e
G }
on Coo(G). If G is a discrete group, then (d ) shows that A (/ ) = £ / ( « )
V / g Coo(G) = £q{G )
u6G
is he Haar functional, so that A is the constant function with value 1. T r s modular function is also the constant function with value 1 if G is ab ilian (use (f), (c) and the identity /„ = u/) or compact (since A (G ) is a compact multiplicative subgroup of R* ). Locally compact groups for which the modular function is identically equal to 1 are said to be unimodular. The uniqueness result (d) shows that if one can guess a functional A sr.t sfying (a), (b), and (c), then this is the Haar integral. Thus we easily ii"-.'. OO
J
A (/ )=
f(t )d t
V / G Coo(R)
— OO
for the additive group R and 2tt A (/) = ^ J
f ( e i l ) dt
V / G Coo(T) = C (T )
o for fche multiplicative group T. O f course any positive multiple o f these linear functionals could also serve as the Haar functional, but in the case of compact groups such as T, it is customary to normalize the Haar integral so t j<‘ , it satisfies A ( l ) = 1. Hewitt and Ross [1963] have many more examples in f p 5 , 16.
142
1: Normed Algebras and Examples
1.9.11
Property (b) of the theorem shows that A is a positive linear functional on Coo(G). A standard construction in measure theory, which is carried out m sufficient generality in Hewitt and Ross [1963] §11, associates a unique positive regular Borel measure on G with each positive linear functional on Coo(G). We will denote the completion of the measure on G associated with A by A and call it the (left) Haar measure o f G. We will often continue to emphasize the functional A rather than the measure A, but we record here the basic properties of left Haar measure. Paul R. Halmos [1950] gives a converse (essentially due to Weil [1940]) to this theorem. T h e o re m Let G be a locally compact group. There is a complete, extended real-valued, positive measure A defined on a o-algebra B\ o f subsets o f G and satisfying: (a) By includes the o-algebra B o f Borel subsets o f G. (b) A( U ) > 0 fo r all nonempty open subsets U o f G. (c) A( K ) < oo fo r all compact subsets K o f G. (d) A(u S ) = A(S ) fo r all S € B\;u e G where uS = {uv : v € S }. (e) A((7) = sup{A(AT) : K C U and K compact} fo r all open subsets U o f G. (f) A(S) = M { X ( U ) : S C U and U open} fo r all S e B. (g) J/A' is any other measure satisfying (a), (b), (c), (d), (e), (f), then there is a positive constant c satisfying A'( S ) = cA (S )
V S e B x.
(h) A (/ ) = / fd X fo r all f G Coo(G) is a Haar functional. 1.9.11 T h e G ro u p A lg e b r a L 1(G ) For each locally compact group which we consider, we make a choice once and for all o f a particular Haar functional A and related Haar measure A from among all possible positive multiples. We denote by L 1(G ) the usual Banach space L 1( A) o f (equiv alence classes o f) functions absolutely integrable with respect to A with norm
G
It becomes a Banach algebra when convolution multiplication is defined by g * h (u ) =
I g (v )h (v l u )d v
V g ,h € L 1(G ); u G G.
(26)
To prove that the integrand is BA-m easurable and that g * h belongs to L l (G ) requires a careful argument and Fubini’s theorem. We may interpret L X(G ) as a subalgebra, indeed an ideal, in M (G ) by considering the Radon-Nikodym theorem. Left Haar measure on G is finite, and hence an element of M ( G ), if and only if G is compact, but we may
1.9.11
Group Algebras
143
always consider the subspace M a{G ) o f M { G ) consisting o f those measures absolutely continuous with respect to Haar measure. The Radon-Nikodym ; ::3orem provides a bijective isometric linear isomorphism between this sub3 jt.ce and L 1(G ). Calculations then shew that this is an algebra isomor phism and the subspace is an ideal. From now on we will regard L 1(G ) as an ideal in M (G ) under this embedding, whenever that is convenient. See Hewitt and Ross [1963] for proofs. Table 1, on the next page, records all the standard facts about L 1(G ) s:zd M (G ) with the conventions and notation we use in this work. We have already noted that the map u t-» „ / is continuous from G into C q(G ) for any locally compact group G and any / € C q(G ). It is 'raquently useful to know that u h J is continuous from G into L l (G ) for ar'y / 6 L 1(G ). The proof is similar to the one outlined in §1.9.7 for the continuous case but relies on the regularity of Haar measure. A converse wes noted by Walter Rudin [1959] and rediscovered by Irving Glicksberg [1973]. This allows M a(G ) ~ L ( G ) to be defined independent o f Haar measure. P r o p o s itio n Let G be a locally compact group. A measure fi in M ( G ) belongs to M a( G ) i f and only i f v >-* is continuous from G into C ) for any Borel set E . P ro o f Suppose /x satisfies the criterion and E is a Borel set with Haar measure zero. Let g be the characteristic function o f a Borel set A with Saiie measure so that g, and hence g * n, belong to L 1(G ). Then f /i(v-1 E )d v = g * n (E ) = [ Ja
g * fid \ = Q
Je
shows that fi(v ~ 1E ) is zero almost everywhere with respect to Haar mea sure. Continuity assures ft(E) = 0. □ Elsewhere (Palmer, [1973]), we have fully developed another simple con struction of the L l -group algebra which involves no measure theory. Each g 6 Cqo{G ) determines an element A g in C q(G )* by the formula As( f ) = A ( g f )
V / g C 0(G ).
To see that A g is continuous, we note |AS(/)| < A ( M ) < A(||/||J fl|) < AdsDH/H^ for any / G C o(G ). By considering A g(g ), it is easy to see that this is an injection of C’oo(G) into C o(G )*. A little more work (involving g/(\g\ + £ ) ) shows that g i—> A g is an isometry with respect to the norm
144
1: Normed Algebras and Examples
1.9.1l
T able 1: Form ulas for a Locally Com pact G ro u p G For any function f : G —* C and any u, v £ G we define: /(«*) = / ( « * - * ) ;
/(«*) = / (« )* ;
u f{v ) = / (u _1v);
f u{v) = f ( v u ~ l ).
Then this notation satisfies: u(/) =
u(vf)
=
Uu/ )
{ f)u — U~X/> u (/ v)
(u / )vi
(/ u )v — / u v
Denote left Haar measure by dA, du or dv and the modular function by A . For u e G and fi, v e M (G ), define 6U, fi, Ji and fi * u in M ( G ) by 6JE) = | J
J
u \ E. U
M E ) = ,i(E ~ l )
h d (j.* v =
j
J1(E) = /*(£?)*
h (u v )d n (u )d v (v )
Then any f , g € L l (G ), u, v e G , J u fd X =
V h G C q(G ).
M (G ) and h € Cf^G ) satisfy:
J fdX\ J f udX = A (u ) j
J fA d X = J fdX;
V Borel sets £ ;
fd X ;
A = A -1
J fd X = J fA d X (when defined).
So right Haar measure is given by A dA. * f ” uf \
f *
=
=: ^u—l )
^111) =
n * f{u )
=
J f ( v ~ l u )d fi(v );
fi* f
=
J Ufdn{u)\
f* n (u )
=
J A ( v ~ 1) f ( u v ~ 1)d fi(v );
f * (J,
=
J A f udfi(u);
/ * ff(u)
= = =
/ f ( v )g ( v ~ 1u)dv S f (^ v )g (v ~ 1)dv / A ( v ~ 1) f ( u v ~ 1)g (v )d v
f *g
=
/ ^ ( v _ 1 ) / ( v _ 1 )ff(v u )d w ;
= = =
f f ( u ) »gdu f f u- i g{u)du f A (u )/ U g(u)du
=
f
A (u )/ (i i ) u-ig d u .
For /i € C'o(G), ifc 6 L°°(C J, /i, ^ 6 M (G ) and /, 9 g L ‘ (G ) define ^ (z * ) =
J hdn
and
ujk( f ) =
J kfdX.
Then this notation satisfies:
u/fc(/i * «/) = W„.h(/i) = W h .A ^ ):
Wfc(/ * ff) = UJg.k(f) = Wfc.A/(»)•
1.9.12
Group Algebras
145
Then L 1(G ) is just the completion o f Coo(G) with respect to this norm. It may also be identified with the closure o f ^Cuu(G) in C o(G )*. To see the way in which this embedding is related to the multiplication or: C q(G )* we need a technical result on A. Let i f be a compact subset of G and let h e C o o(G )+ be a function satisfying h (u ) = 1 for all u € K . •- y / e C qq(G ) with support in K satisfies |A(/)| < A(|/|) < Adl/H^/i) < A(ft)||/||oo, w lere A (h ) depends only on K and not on the particular choice of /. Let ui call a linear functional A ' on C oo(G ) tame if for each compact K there is a constant B k satisfying |A'(/)| < BkII/IIoq
V/eCoo(G)
with
supp(/) C K .
In addition to A, the restriction to C oo(G ) of each w € C q (G )* is obviously ta lie. Tame linear functionals satisfy a weak form o f Fubini’s theorem which is strong enough for our present purposes. The simple proof is based on using the Stone-Weierstrass theorem to show that any function / in Coo(G x G ) can be uniformly approximated by expressions o f the form n £ / j ( u)flj(w)
for
and
fj,9 j€ C o o (U )
j =i where U is an open set with compact closure related to the support o f /. A r med with this result, we can now define and calculate all the properties of L l (G ) as an ideal in C q(G )*. A large number of results are known for G or L 1(G ), both when G is abelian and when it is compact. W e have already noted that G is unimodulrr in both cases. In §3.2.6 we note the easy fact that G is a [M AP] group wl: 3n it is either compact or abelian. L 1(G ) is strongly semisimple (Section 4.5) whenever G belongs to [M AP]. The Proposition in §5.1.9 shows that G xas small invariant neighborhoods (or equivalently, L 1(G ) has a central approximate identity) in both these cases. In §7.4.10 we will note that G belongs to [F IA ]— and hence L l (G ) is completely regular under either hypothesis. Finally, in Volume II we will show that G is amenable if it is either abelian or compact. The relations between these and many other similar classes of locally compact group are studied in detail in Palmer [19?8]. Some of these results are also given in Volume II. i . . 12 F u n ctoria l P r o p e r tie s o f L l ( G ) The functorial properties of L ' \G) are rather meager. Let G and H be locally compact groups and let >p: ' —* i f be a continuous homomorphism. In §1.9.8 above, we constructed a contractive algebra homomorphism ip: M ( G ) —» M ( H ) and this, of course, ."H’-'ies L l (G ) into M ( H ) . An easy computation gives the meaning of 'ip(f)
146
1.9.12
1: Normed Algebras and Examples
for / G L 1( G ) when
V u N € G/N
(27)
where the subscript v on A N indicates that A is acting on f{ u v ) considered as a function o f v G N only. The left invariance of A N shows that tp’N ( f ) is well defined on the topological space G / N = { u N : u G G }. Let tp^\G —+ G / N be the natural (continuous and open) quotient map. Then the support V l i i f ) is included in the compact set <^/v(supp(/)) and an easy argument shows that u i-+ A„ / (u v) is a continuous map on G so that tp*N ( f ) is continuous on G / N , i.e.,
V f
G
C oo(G )
satisfies the hypotheses of (d) in the first theorem in §1.9.10 so that it is a multiple o f Haar functional on C oo(G ). By a suitable normalization, we have W eil’s formula [1940], 2nd ed., pp. 42-45: Aa ( f ) = AG JNN A ? f { u v ) f f(u )d u =
Jg
f
V /
G
f f{ u v )d v d (u N )
or
C o o (G ) V /
G
(28)
Coo(G) .
J g /n J n
We can now state the main result for case (b). T h e o r e m Let G be a locally compact group and let N be a closed normal subgroup. Let the Haar functionals o f G, N and G / N be normalized so that W eil’s formula holds. Let tpN :G —> G / N be the natural map and let Crx)(G) —* C oo(G / N ) be as defined in (27). Then tp'N induces a surjec tive algebra homomorphism ip'N o f L 1(G ) onto L 1(G / N ) which establishes an isometric isomorphism between L l ( G )/ ker(y>Jv ) and L l (G / N ).
i.o.r.3
Group Algebras
147
To prove this, we state a lemma from which the theorem follows by the most general abstract considerations. L err-n a Under the hypotheses o f the theorem, tp*N maps C o o ( G) onto CW {G / N ) and satisfies l l ^ ( / ) l l i = “ W M l i :
-
«
/ («)= / \ 0,
ifu e^fsu p pf*)) if u
Then / is obviously continuous on ipfj ( K ) and is zero on the open comple ment of this set. Its support is included in supp(/i). Hence / belongs to C oo(G ) and satisfies
f& fo) =
/ / o
V / e C 0( H ) ; n e M ( G ) .
JG
I f K is another locally compact group and ip :H - * K is another con tinuous open group homomorphism, then o ip: L 1(G ) —+ L l ( K ) satisfies ip o ip = ip o Tp. Jor a related result see Antoine Derighetti [1978], 1.9.’ ? W e n d e l’ s T h e o re m Next we give the result of James G. Wendel [’ 952] which represents M (G ) as the double centralizer algebra V ( L 1(G )) of L \G) whenever G is a locally compact group. For each fi 6 M (G ) ~ CS(C )'\ define and l \ in B ( L l (G )) by L ltf = H * f
R p f = f * ft
V / eL 'fG )
148
1: Normed Algebras and Examples
1.9.13
so that L and R are merely the restrictions of the left and right regular representations of M (G ) to the ideal L l {G ). Clearly we have ||I I < M l and ll-fyJI < ||/i|| for all /i G M ( G ) and we will soon see that equality actually holds in these inequalities. Theorem 1.2.4 shows that for any double centralizer ( L , R ) o f L l ( G ) , L and R are bounded operators if the annihilator ideals L 1 ( G ) l a and L 1 ( G ) r a are both zero. We will show more than this: any /, g G L 1 ( G ) sat isfy ll/ll = sup{||/*si|| : g G L ' { G ) i } and ||g|| = sup{||/*si|| : / G L ' ( G ) i } Hence when P ( L 1(G )) is given its natural norm ||( L , f l)|| =
m a x {||L ||,
||A ||}
(L ,R )
V
G
V ( L '( G ) ) ,
the embedding of L 1(G ) into V ( L 1(G ) ) is isometric. Obviously, the map 0 :M ( G ) —> V ( L l (G )) defined by 0 (n ) = ( Z ^ , ^ ) is a contractive homo morphism, but even more is true. T h e o r e m I f G is a locally compact group, the map 6: M (G ) —►V ( L l (G )) is an isometric isomorphism of M ( G ) onto V ( L l (G )). Proof It is enough to show that each ( L , R ) G V ( L l (G ) ) can be expressed as ( L ^ R p ) for some /i G M ( G ) satisfying ||/i|| < ||L|| < ||(L, /2)||. Let V be the family of all compact neighborhoods of e in G. Order V by reverse inclusion (i.e., for V, W e V V < W means W C V ). This makes V into a directed set. For each V G V, let e y be A (V )-1 times the characteristic function of V . Example 5.1.9 below shows that the net { e y } y €y satisfies: lim e v * p r o v in g
b o th
||/||
/
=
=
/;
s u p { ||/
lim f * e v = f * g\\ : g
G
V /
e
L 1(G ), and
a ls o
(29) ||g ||
=
* g \ \ : / G L H G ) i } f o r a l l /, g G L l {G ). Let ( L , R ) G V ( L l (G ) ) and / G L 1(G ) be given. Here is another conse quence o f (29) which uses the continuity of L also: s u p { ||/
L f = L lim ey * f = lim L (e y * f ) = lim hy * f , where hy = L (e y ). vgv vev v tv Clearly { h y } v e v is a net in the closed ball ||L||M(G)i. Also ||L||M (G)i = ||L||Co(G)J is compact in the weak*- or Co(G)-topology by Alaoglu’s the orem. Therefore some subnet {h y } v e w converges to some ji G ||L||M(G)j in the C0(G )-topology. Let / and g be arbitrary elements of Coo(G) so that f *g also belongs to Coo(G). Given any e > 0, there is some Vo G W such that V > Vo implies * g (u ) h y (u ) du < £.
1.9.13
Group Algebras
149
By ' ubini’s theorem we have
J f ( v) J 9 (v
i
1u)dfi(u)dv —
J f ( v ) J g (v
< £. l u)h\r(u)dudv
Taking the limit of V > Vo in W gives j y f ( v )[L n (9 )(v ) - L{g)(v)\dv
=
— lim
J g (u ~ l v )h v (u)du]dv
< e.
Since / G C oo(G ) and e > 0 were arbitrary, we see that L ^ (g ) = L (g ) holds for each g G Coo(G). Since both
and L are continuous and C'oq(G ) is
densj in L l (G ), we conclude L = L^ where ||/i|| = ||Z>||. Furthermore, for any f , g & L 1(G ) we have R ( f ) * g = f * L (g ) = / * /i * g = R ^ i f ) * 9Hence we have shown that ( L , R ) equals for some G ||L||M (G)j as we wished to do. O Note that the fact that the left centralizer L came from a pair (L , R ) G D ( L ‘ (G )) was only invoked to justify the use of Theorem 1.2.4 to prove that L was continuous. Under the present circumstances, Proposition 5.2.6 below can be used in place of Theorem 1.2.4 to prove the continuity of L. In fact, we have shown that any left centralizer is a bounded linear map of the ! >rm for some /i G M (G ). Wendel [1952] used the above result to prove: C o ro lla ry A map : L l ( G ) -> L 1( G ) is an isometric left centralizer i f and only if it has the form L (f)= C u f
V / G L\G )
fo r s m e u G G and £ G T. From this he concluded: T h e jrem Let G and H be locally compact groups and let T : L l (G ) —* L 1( H ) be a contractive algebra isomorphism. Then there is a ,wmeomorphic group isomorphism
V ueH
where c is the constant value of the ratio \ m p (E )j\ q ( E ) f or each mea surable set E with finite non-zero measure. In particular G and H are isom >rphic as topological groups and T is actually an isometry.
150
1: Normed Algebras and Examples
1.9.14
The proof is based on the obvious fact that T L T ~ l is a left centralizer o f L l ( H ) when L is a left centralizer of L l (G ). I f L is isometric, then it can be shown that T L T ~ l is also. Thus, with a little work, the previous result gives a map
x (u)jy«
v ti €
h
where and L 'u are defined by L ^ (u>/ = *,(„)/ and L 'ug = ug for all u e H , f e L l {G ) and g £ L l (H ). For details, see the original reference. (Contractivity can be relaxed; see Nigel J. Kalton and Geoffrey V . Wood [1976] and W ood [1983], [1991]. Also see Roger P. Rigelhof [1969a], Adegoke Olubummo [1979].) The major consequence of this result is that the Banach algebra L 1(G ) (with its norm), considered as an abstract Banach algebra, is a complete set of invariants for G. In [1966] Greenleaf described a procedure for recovering the topological group from the abstract Banach algebra A = L 1(G ). First construct the double centralizer algebra T>(A). (W e have just seen that V ( A ) is isometrically isomorphic to M (G ) . ) Let G (A ) be the subgroup of the multiplicative group o f V ( A ) consisting of those pairs (L , R ) in which both L and R are surjective isoinetries and let 70 be some multiplicative linear functional on V (A ). Under the identification of T>(A) with M (G ), we have seen that (^ (L ^ G )) is the subgroup {£<$„ : ( € T, u E G } and that there is always a multiplicative linear functional. Let G (A ) be the subgroup
G (A ) = {« € 6 ( A ) : t d M = Hull} of T>(A) and endow G (A ) with the relativized strict topology (Definition 5.1.5) of T>(A) (induced by its ideal A ). Greenleaf [1966] shows that G (L l (G )) is homeomorphically isomorphic to G under the map of G into M (G ) given by u i-» 7 o( 6u)* 6„. This construction is then used to characterize those Banach algebras iso metrically isomorphic to L 1(G ) for some compact group G. Jyunji Inoue [1971] introduces an interesting group algebra intermediate between L l (G ) and M (G ). It is the closure in M (G ) of the sum of L * ( G T) where r varies over all locally compact group topologies stronger than the given topology and U (G T) is just the corresponding group algebra. This algebra plays an important role in Joseph L. Taylor’s [1973a] structure theory of M ( G ) for G a locally compact abelian group. See 3.6.17 for related results. 1.9.14 R ep rese n ta tio n s o f T o p o lo g ic a l G ro u p s A representation V of a group G on a linear space X is a group homomorphism o f G into the group C ( X ) g of invertibles in C (X ). I f X is normed and the representation
£.14
Group Algebras
151
is into B ( X ) g , it is called a normed representation. I f G is a topological group, a normed representation V o f G on a Banach space X is said to be ( norm / weakly / strongly ) continuous if the map V : G —►B (X ) is cc'T.inuous with respect to the given topology of G and the { norm / weak operator / strong operator ) topology on B (X ). Note that boundedness and continuity o f representations are completely independent, with neither implying the other. Obviously norm continuity implies strong continuity which implies weak continuity. It turns out that norm continuity is too strong a concept to be o f much interest. (Except o f course when X is finite dimensional so that the various topologies on B (X ) agree.) As we will show, weak continuity frequently implies strong continuity. Note that the reconstruction o f G from L l (G ) carried out above de pended on the representation V :G —►B (L 1(G )) given by Vu( f ) = 6 u * f = « f
V u e G ; f e L 'C G ) .
This representation is strongly continuous. (To prove this, note that C oo(G ) is dense in L l (G ) and that u h-> /„ is continuous with respect to the supremum norm on Coo(G).) The following results will frequently be useful. Let G be a locally com pact group and let V : G —►B (X ) be a representation o f G on a Banach space X such that u h-> Vuz is bounded and weakly measurable for any fixed z e X . (For formal definitions of these almost self explanatory terms, ses the appendix to this section where the Bochner integral and its basic properties are also defined and discussed.) For any / € L l (G ), we may use the Bochner integral to define T jz =
J f (u )V uz du
V z € X
as an element of X . (The integrand is almost separably valued since supp(/) is cr-compact.) Clearly the map T j (i.e., z h-> T j z ) is linear. We now show that it belongs to B (X ) and that the resulting map T v : L l (G ) —* B (X ) is a weakly continuous representation o f L 1(G ) on X . The uniform boundedness 7,r;:~/uple shows that { Vu : u £ G } is a bounded set o f operators in B (X ). Let 3 be a bound for V and let z G X and w G X * be arbitrary. We get \u{Tjz)\
=
\ J t t u M Vu(z ))d u \ < j
<
ll/IU M IB IM I.
m u)\ |u;(K.(z))| du
Weak continuity follows by replacing / by / — g in the above expression. Fubini’s Theorem gives
152
1: Normed Algebras ajid Examples w(T/CgW )
1,9.14
~
J J /(v)fl(v_ 1w)dvw(V'„(z))du J J f{ v )g { v ~ l u )u (V u{z))d u d v
=
J J f(v)g(u )u > (V vu(z))d u d v
=
J J f ( v )u (V vVu(z ))d v g (u )d u
=
=
J g (u )o j{T jV u(z))d u J g (u )(T f y (u )(V »(z ))d u
=
( r / ) * ( « ) ( r / ( 2)) = u {t J t V { z )).
=
For later use, we need to note also that if V is weakly continuous at e e G (i.e., if V is a continuous function at e e G into B (X ) with its weak operator topology), then each z € X satisfies € { T jz : / G L l (G )} ~ . If this were to fail, then the Hahn-Banadi tiuurem would give an element w €
V f
G
L l {G )
which in turn implies that u (V u( z )) is zero almost everywhere and hence is zero at e since it is continuous there. This gives the contradiction 0 ^ a>{z) = ui(Ve( z )) = 0, which establishes z e { T j z : / e L 1( G ) } “ . A weak consequence is that { T j z : f e L 1(G ), z € X } is dense in X . (Actually Theorem 5.2.2 and Example 5.1.9 below show that { T j z . f € L \ G ) \ z <EX} equals X and that each z
G
X may be written as T j y for some /
G
L l (G )
and y € { T j z : f G L l ( G )} \ In fact in the latter expression y may be chosen as close to z as desired and / may be chosen with H/Hj < 1.) The definition of the Bochner integral makes it clear that a closed V mvariant subspace o f X is T v -invariant. In order to prove that a closed r l -mvariant subspace is V-inv&riant, it seems necessary to make some continuity assumption on V . In the terminology of Definition 4.2.1, these two results will show that V is topologically irreducible if and only if T v is. FYom now on, in addition to the other assumptions made above, we assume that V is continuous with respect to the weak operator topology
1r
4
Group Algebras
153
on B (X ). Then we may use the Bochner integral to define a continuous _y
representation T
of M (G ) on X by
T^z
—J Vuz d n (u )
/i€ M (G );
V 2 6 X.
The "act that this defines a continuous representation of M ( G ) is proved by calculations similar to those above using results from the appendix as before.
Note the identities T j
=
(where A is Haar measure and
fX { B ) = f B fdX for each Borel set B ) and Vu = T%u for all / _y
G
L l (G )
and ' ( e G. Th e latter identity shows immediately that any T -invariant subspace is ^-invariant. Suppose y is a closed T v invariant subspace. We showed above that any z € X can be written as lim T Y z for a suitable n —*oo
sequence { / » } nCN in L } {G ). Hence for any K Z = K
G
Jn
M (G ) , we have
,im T f z = lim T ^ Jnz. _y
Hence if z belongs to any T''"-invariant subspace, then T^ z does also. We have shown: T h e o re m I f V is a weakly continuous, bounded representation o f a lo cally compact group G on a Banach space X such that u Vuz is almost separubly valued fo r each z G X then V is strongly continuous. Moreover, i f T i' : L 1(G ) —* B ( X ) a n d T V : M ( G ) —* B (X ) are defined by the Bochner integrals
j } {u )V uz dx T ^ z = J Vuz dfi(u)
T jz —
V f V n
G
G
L \ G ); z
G
X
M (G ); z € X
then the following are equivalent fo r a closed subspace y o f X : (f ) y is V-invariant; (b ) y is T v -invariant; —V (e) y is T -invariant. It only remains to show that V is strongly continuous. It is enough to show continuity at c since V is a group homomorphism. Suppose z e X and e > 0 are given. Choose / e L l (G ) satisfying \\TYz — z\\ < e/(2 + 2B ) (where, as before, B is a bound for { ||141| : u 6 G }). By the continuity of tb3 map u h-> „/ (Hewitt and Ross [1963], Theorem 20.4), there is a neigi horhood N of e such that u G N implies ||/ — u/|| < e/2B. Then, for any j; G JV, we have l\Vuz - z||
<
\\Vuz - V uT]fz\\ + \\VuT ]fz -T y z \ \ + \\Tyz-z\\
<
B\\z - T/z|| + ||ru/_ /z|| + IIT j z - z \ \ < £.
154
1: Normed Algebras and Examples
1.9.15
Hence V is strongly continuous. Note that this conclusion is automatic if V' is a unitary representation on Hilbert space since the weak operator and strong operator topologies agree on the set of unitary elements as we will show in Volume II. C o ro lla ry In the last theorem, the hypothesis that u • > Vuz is almost separably valued is unnecessary if V is assumed strongly continuous or if X ts separable or reflexive. Proof We only used the hypothesis to ensure that the Bochner integral was defined. Hence this hypot hesis may be dropped if the Banach space X is reflexive, for then the simpler integral defined in the appendix may be employed. Furthermore, if V is strongly continuous, then the map u t—> Vuz will be strongly measurable for each 2 X with respect to measures in M (C ). Thus iu this case also the above argument is valid. Finally if X is separable, the hypot hesis mentioned is automatically satisfied. □ 1.9.15 F o u rier A lg e b ra s , B e u rlin g A lg e b ra s a n d S em ig ro u p A l gebras We wish to briefly introduce some variations on group algebras. Fourier algebras and Fourier Stieltjes algebras will be studied in Volume II. Here we merely mention that they are commutative algebras of complex valued functions with dual spaces which are convolution algebras. The functions can be interpreted as belonging to certain representation spaces. Pierre Eyrnard [1964] gave a systematic theory in the noncommutative case. The whole theory can be extended to Kac algebras introduced by L. I. Vainerman and Grigoril Isaakovic Kac [1974], Michel Enock and JeanMarie Schwartz [1979], John Ernest [1967] and Masamichi Takesaki [1972]. See Jean De Canniere [1979], De Canniere, Enock and Schwartz [1979], De Canniere and Ronny J. E. Rousseau [1984]. See also Martin E. Walter [1986], [1989]. D e fin itio n A weight function u on a locally compact group G is a lo cally bounded, measurable, positive-valued function w: G —►R+ satisfying uj(e) = 1 and u (u v ) < u>(u)ui(v) for all u, v G G. If w is a weight function, then the Beurling algebra on G defined by u> is the Banach space L l (G , u ) = { / : ( ? —►C : ll/Hu, = / |/(u)|w(u )d u < o o }, Ja provided with convolution multiplication. It is easy to check that L l (G ,u i) is the subalgebra of L l (G ) consisting of functions / satisfying fui e L * (G ). It is a Banach algebra in its own norm || • ||^. If we replace ai by ui{u) = infx;{sup{u;(ttv) : v G N } : N G fC} (where K is the family of compact neighborhoods of e in G), then u> is an upper semi-continuous weight function with L l (G ,u i) equal to L ' ( G , w ) as
Group Algebras
155
a subset of L l {G ) and having an equivalent norm. Hence we may further restrict weight functions to be upper semi-continuous without essential loss o f generality. Bcurling algebras share many properties with Z^-group algebras, but their .weight functions can be chosen to introduce subtle differences. For instance, Beurling algebras are approximately unital (see Chapter 5) and unital if the group is discrete. If N is a normal subgroup, the quotient algebra induced by the homomorphism G —* G / N is a Beurling algebra on G / N . Arne Beurling essentially introduced these algebras into classical harmonic analysis in [1938]. Yngve Domar [1956] obtained many results for general locally compact abelian groups. Beurling algebras have also been studied on noncommutative locally compact groups. See also Bruce A. Barnes [1971b], Ian G. Craw and Nicholas J. Young [1974], John Liukkonen and Richard Mosak [1977], B. A. Rogozin and M. S. Sgibnev [1980], Niels Gr 0",baek [1983], W ilfried Hauenschild, Eberhard Kaniuth and Ajay Kumar [1983], William G. Bade, Philip C. Curtis, Jr. and H. Garth Dales [1987]. For other group algebras and related notions, see Satoru Igari and Yuichi Kanjin [1979], John G. Romo [1980a], [1980b], Klaus Hartmann [1981], S. A . Antonyan [1983] and David Gurarie [1984]. If 5 is a semigroup, we can introduce convolution multiplication on t \ S ) by / **(«)=
£
f(v )g (w )
V f , g e t '( S ) .
{v,w£S : u=t>u>} This makes £l (S ) into a Banach algebra which we will call the semigroup al-gijra o f S. This algebra can have quite different properties depending on the structure of S. If S is a unital semigroup (i.e., if it has an identity element c for its multiplication), then 6e is an identity element for i l {S ). Example 4.8.5 below deals with the semigroup algebra o f a free semigroup. I f u>: S —►R+ is a weight function satisfying the inequality of the last definiCion, then the Banach algebra £l (S,u>) can be defined. The very special semigroups which have an analogue of Haar measure have been studied, and for them analogues o f L l (G ) and L 1( G , w) can be defined. For further inforEi&jlcn on semigroup algebras see Young [1973b], John F. Berglund, Hugo D. Junghenn and Paul Milnes [1978], [1989], Ghahramani [1980], [1982], [1983], [1989], Lau [1983], Ghahramani and McClure [1990], Bade [1983] G.-fznbaek [1988], [1989a], Roderick G. McLean and Hans Kummer [1988] John Walter Baker and A li Rejali [1989] and John Duncan and Alan L. T. Paterson [1990].
i.S.T O A ppen d ix: Integration o f Banach Space Valued Functions In this appendix we will define and state the major results on: (1) a
156
1: Normed Algebras and Examples
1.9.16
relatively elementary Lebesgue integral for reflexive Banach space valued functions, (2) the Bochner integral, and (3) other closely related matters. Proofs will not be given. They can be found in Hille and Phillips [1957],. 3.73.9, Yosida [1965], Chapter V, 4 and 5 or Diestel and Uhl [1977], Chapter II. The first and third references contain much more material while the second gives the major results more concisely. Dunford and Schwartz [1958], Chapter 3, and other parts of Diestel and Uhl’s book give closely related theories. Throughout this appendix, let (S, B, n ) be a measure space, where /j. is either a non-negative extended real-valued, or a complex-valued, countably additive set function with the a-field B of subsets of S as domain. Let X be a Banach space and let /: S' —►X be a function. Then / is said to be weakly B-measurable if the numerical valued function w o / is measurable for each w € X ‘ . Let / be a weakly B-measurable function, and assume moreover that w o f belongs to for each w 6 X " . We can always define an A”*'*-valued integral of / with respect to /i by the following procedure. Consider the map W \ X ’ - L '{ n ) defined by W (w ) = w o f . This is clearly linear and closed. Hence it is bounded by the closed graph theorem. Now consider the map w i-»
J w (f (s )) dn(s) = J W (w )(s )d n (s )
V w eX *.
This is also a linear map which must be bounded since W is. Therefore it is an element o f X ** which could be denoted by / f(s )d )i(s ). This integral (which was defined independently by Gelfand [1938] and Dunford [1938]) has most of the usual properties of an integral. Its great defect is that the integral of / belongs to a larger space than the range of /. To remedy this defect, we will only use this integral for reflexive spaces X where we naturally interpret the integral of / as an element of X again. The easily obtained elementary properties of this integral are given, for instance, in Hille and Phillips [1957], 3.7. A function f : S —> X (where as before (5, B,/j.) is a measure space and X is a Banach space) is said to be n-simple if it has only a finite number of values and assumes each non-zero value on a B-measurable set of finite /i-measure. A function /: S —* X is said to be strongly fi-measurable if in the norm of X it is the pointwise /i-almost everywhere limit of /i-simple functions. A function f : S —* X is said to be fj.-almost separably valued if there is a |i-null set E in B such that f ( S \ E ) is a separable set in the relativized norm topology of X . Clearly strong /^-measurability would be difficult to check directly. Hence the following theorem of Billy J. Pettis [1938] is important. It is proved in Yosida [1965], V.4.
1.9.16
Group Algebras
157
T h e o re m A function f : S —* X is strongly fi-measurable i f and only i f it is weakly B-measumble and almost separably valued. Next we turn to the Bochner integral which is defined in non-reflexive spaces, but is less general than our previous integral when that integral is defined. When both integrals are defined they agree. A function f :S —> X is said to be Bochner ^-integrable if there exists a sequence { / „ } „ g n of /i-simple functions from S to X which converges in the norm of X pointwise /i-almost everywhere to / in such a way that it satisfies
The Bochner integral of a /i-simple function / = XEhXk (where E i, E i, . . . , E n is a collection of disjoint sets in B with finite /z-measure, XEk ’
The Bochner integral with respect to fi of a Bochner /i-integrable function f - . S - ^ X is
where { / n}ngN*s any sequence of /i-simple functions satisfying the defining property of Bochner integrability. O f course one must check that this inte gral is well defined since it apparently depends on the choice of a sequence. This js checked in both references given above. The usefulness of this integra,' depends largely on the following criterion for Bochner integrability first obtained by Salomon Bochner [1933]. T h e o re m A function f : S —> X is Bochner integrable i f and only i f it is strongly B-measurable and satisfies J ||/(s)||d|/i|(s) < oo. By combining the last two theorems we can obtain criteria for Bochner integrability which are easily checked in many situations. T e Bochner integral is easily seen to be a linear function o f the inte grand which satisfies
Furthermore if ^ is another Banach space, T : X —> y is a bounded lin ear function and f : S —> X is Bochner integrable, then T o / is Bochnerintegrable and satisfies
158
1: Normed Algebras and Examples
1.10.0
These results and a number of others are proved in our references. Finally we make a convention. When G is a locally compact group, we will use all of the above terms referring to the measure space (G, B, A), where B is the a-field of Borel sets and A is left Haar measure without specifying B and A. Thus we speak of weakly measurable functions or Bochner integrable functions rather than weakly fi-measurable and Bochner A-integrable functions.
1.10
Tensor Products
A number o f the previous examples have involved tensor products of one sort or another. Perhaps the most interesting example is the set Bj ?( X, y) of bounded finite-rank operators from a normed linear space X to another normed linear space ^ as presented in Definition 1.1.17. This can be iden tified with the algebraic tensor product X $4 y ’ defined below. In §1.6.8 we also noted that the matrix algebra M nm o f nm x nm-complex matrices could be identified with M n ® M m even as an algebra. Here we shall give a more systematic, but still brief, development of this important topic. Other useful discussions can be found in Robert Schatten [1950], [1960], Grothendieck [1954], [1955], Joseph Diestel and John Jerry Uhl, Jr. [1977] Johann Cigler, Viktor Losert and Peter Michor [1979] and Alexandr Yakovlevic Helemskii [1989b], [1993]. The latter three references concentrate on tensor products of Banach modules, a subject which we postpone until we treat cohomology theory. Schatten [1943a], [1943b], [1946] and Schatten and John von Neumann [1946] and [1948] began the study of tensor products o f Banach spaces but Grothendieck’s work, cited above, profoundly advanced the subject. A number of useful subsequent papers have presented in detail ideas and results which were implicit but far from explicit in his work. In this section we give basic facts about tensor products, with some em phasis on tensor products o f Banach algebras. We usually state our results for n-fold tensor products. Later in this work we will present additional results on tensor products o f Banach algebras which involve concepts not yet introduced. See §§3.2.18, 4.2.25, 4.2.26, 5.1.4 and 7.1.19. Tensor products are a device for reducing multilinear phenomena to linear ones. We are most interested in those tensor products of Banach spaces which are again Banach spaces. The two most important products of this kind are the projective tensor product and the injective tensor product. All other complete tensor products are intermediate between these two. We consider the purely algebraic case first. This is an area in which an unusual number of errors have been published. This probably stems from the fact that many proofs are utterly trivial, but in the Banach space theory some results which seem “obvious” are actually false.
1.10.1
Tensor Products
159
Algebraic Tensor Products 1.10.1 D e fin itio n Let X ' , X 2, . . . , X n and >> be linear spaces. A map / of P '1 x X 2 x ■■■ x X n into y is called multilinear if it satisfies j i .1 I } . . . , X k —\ , \ x k -j- f i x k , X k i , . . . , x n )
— A/ (X\, . . . ,X k—i , x'k, Xk + l, . . . , x n)+ / i/ (x 1, . . . , Xk—1, Xf., Xk+ 1, . . . , x n) "cr f 11 k e { 1, 2, . . . ,n} , \ ,n G C, x'k, x k £ A’* and x j e X i for j ^ k. Denote the set of all multilinear maps of X 1 x X 2 x • • • x X n into y by C = An algebraic tensor product of X 1, X 2, . . . , X n is a pair ( Z , p ) con sisting of a linear space Z and a map p £ C ( X l , X 2, . . . , X n\ Z ) called the ’ tensor map such that, if y is any linear space and / is any map in , X 2, . . . , X n; y ) , then there is a unique linear map h : Z —> y satisfy ing / = hop. As usual, we will denote p(x\, x 2, ■■- , x n) by x i ® x 2 ®- • rad call such elements elementary tensors. t The universal mapping property o f this definition and the uniqueness property in the next theorem can be illustrated by commutative diagrams.
X 1 x X 2 x ■• ■ x X n
X 1 x X 2 x ■■• x X n
1.13,2 T h e o re m I f X l , X 2, . . . , X n are linear spaces, then an algebraic tensor product ( Z , p ) exists. Each t £ Z can be written in the form m t = ^ x Xj ® X2,j ® j= i
® x nj
where
x kj £ X k
(1)
fo r s o m e m £ N. Both Z a n d (Z ,p ) are unique in the sense that i f ( 2 ' ,p') is another pair satisfying the definition, then Z and Z ' are linearly isomorphic under an inverse pair o f unique linear maps h : Z —* Z ' and h': Z 1 —> Z satisfying p' = h o p and p = h' o p '.
(2)
Proof We construct a linear space Z and a map p of X 1 x X 2 x • • ■ x X n into 2 such that the range of p generates 2 (giving uniqueness) and such that any element o f C ( X ' , X 2, . . . , X n\y ) factors through p uniquely. Let
160
1: Normed Algebras and Examples
1.10.2
Z ' be the huge linear space of formal linear combinations of the elements of the Cartesian product X 1 x X 2 x ■■• x X n. Let Z " be the linear subspace generated by the set of elements of the form
1j
{
"I"
Xk±\, ■• * , %n) ~ (^ 1) 2*2>■■• j ^n)
— ( x j , X 2 , . . . , X k —l , X k , X f c + 1 , . . . , X n ) ,
( 3)
( x , , x 2, ■■■, Xk„ ,, \ xk, x k+u . . . , x n) - X (x i , x 2, . . . , Xn) : V fc G {1 ,2 ,. . . , n }; x k,x'k G X k- A G C }. Let Z be the quotient Z ' j Z " and let p: X 1 x X 2 x ■■■ x X n —> Z be defined by p ( x i , X 2, ... , x n) = ( x j , x 2, ... ,x „ ) + Z " . Denote this element by X] ® x 2®- ■- ®xn. Hence each element in Z can be written in the form (1) for suitable m € N and x kj € X k. It is easy to see that this construction does provide an algebraic tensor product: given / € C ( X l , X 2, . . . , X n; }>) {ts in the definition, we define h by h (x i <8 x 2 ® ■• ' ® £ n ) = f ( x i , X 2 , ■■• , x n) which is well defined by multilinearity. In order to see that an algebraic tensor product is unique up to the natural notion o f isomorphism, let ( Z , p ) and ( Z ', p ') both be algebraic tensor products of X 1, X 2, . . . , X n. Then by Definition 1.10.1 there are unique linear maps satisfying condition (2). The uniqueness of the identity maps i: Z —<■ Z and Z ' —> Z ' show i = h! o h and %' = h o h!. Thus Z and Z ’ are isomorphic by a pair of inverse isomorphisms h and h! which intertwine with the tensor maps in the sense that they satisfy (2). This is the proper definition of an isomorphism between the pairs ( Z , p ) and (Z V ). □ From now on we will speak of “the algebraic tensor product” rather than “an algebraic tensor product” and denote Z by X 1 ® X 2 ® ■■• ® X n. Note that the definition of the algebraic tensor product establishes a linear isomorphism
r]: C { X 1, X 2, . . . , X n-,y) - > C ( X l ® X 2® - - - ® X n, y )
(4)
where r}( f ) = h, in the notation above. Clearly, r] is the inverse of the linear isomorphism g >-> gop of C { X * ® X 2®- ■ onto C ( X l , X 2, . . . , X n; y) . If m < n, it is easy to construct a linear isomorphism X 1 ® X 2 ® ■••® A"*
(5)
~ ( * l ® X 2 ® • ■■® X m) ® ( X m+l ® X m+2 ® ■■• ® X n). Hence any algebraic tensor product can be built up as a succession of alge braic tensor products between pairs of linear spaces: X 1 ® A”2 ® - - ® X n ~ ((• ■• ((A "1 ® X 2) ® A”3) ® • • •) ® A”1-1) ® X n. (6)
1.10.3
Tensor Products
161
Because of this construction, it is usually enough to deal with the tensor product of two spaces. 'Ihe definition of bilinear maps (i.e., multilinear maps defined on the prod act of two spaces) gives obvious canonical linear isomorphisms C ( x \ x 2-,y)
c ( x \ c ( x 2, y ) ) ~ c ( x 2, c ( x \ y ) ) .
(7)
If m < n, this generalizes to the dual expression of (5) , c ( x \ x 2, . . . , x n-,y) ~ c ( x \ x 2, . . . , x m- , c ( x m+ l , .
y))
but the special case (7) is more important. 7 e need the following simple results. 1.10 3 L e m m a Every non-zero element t in X 1 0 X 2 can be written as m
(8 )
t= j= l where {xfc.i, £*,2, • • ■, k ~ r.
<* linearly independent set fo r both k = 1 and
P roof We may assume the expression (1) (with n = 2) has been chosen with m mfniinal. In order to show that this expression satisfies the conditions given for (8), by symmetry, it is enough to show that ± i,m is not equal to a linear combination: X ijtn = H this were true, we would have m—1 t ~ XI j-1
0 (X2>^ + ^3x ^m)
which contradicts the minimality of m.
□
As we have noted in previous examples, there is a linear map of X ® 3^ into -Z(y, X ) given by x ® u>(z) = w (z)x
V x € X; w G 3^; z € 3*.
(9)
Its n nge is the set C p ( y , X ) of all (not necessarily bounded) finite-rank linea maps. The lemma implies that this map is injective as we will show in the proof of Proposition 1.10.4. There is also a linear map of X ® 3^ into , 'l ( X , y y given by x ® w (T ) = w (T x )
V x e X; w e y * ■, T e C ( X , y ) .
We wish to highlight a special case of (9).
(10)
162
1: Normed Algebras and Examples
1.10.4
1.10.4 P rop osition Let X be a linear space and let y be a subspace of X * . Then X ® y is an algebra under the multiplication (x ® y) ( z ® w) = y( z) x
V x, z, € X; y,w e y.
Equation (S ) above establishes an algebra isomorphism of X ® y onto a sub algebra of the ideal C f - ( X ) of not necessarily bounded finite-rank operators on X . If y = X ^, the isomorphism is onto C f ( X ) . Proof Injectivity of the map defined in (9) is the only property which may not be clear. We may suppose that any non-zero t = xj ® y j € X ® y satisfies the lemma. Since yj is not in tin- linear span of y2, 2/3 , ■■•, Un, there is some z € X satisfying yi ( z) ^ 0 and yk(z) — 0 for k = 2 ,3 ,..,, n. Thus t(z) = xi ^ 0. The proof of Proposition 1.1.18 above shows that C p ( X ) is the range of the isomorphism when y = XK □ There is a linear map of 6: X 1 ® X 2 ® given on elementary tensors by
® X n —►C ( X 1^, . . . , X n C )
n
0(x-l (6x2<6- ■-<&xn) ( w U u)2, . . . ,uJn) = ]^ [ Ulk( x k)
V xk e X k-, U!k e X ki.
fc=l
( 11) The existence of each of these three maps follows immediately from the definition of the algebraic tensor product. We will presently show that they are all injective. Hence they could be used to construct the algebraic tensor product. We can also write (11) in a dual fashion. If uik G X kt for k — 1,2,... ,n, we can define (0 ui2 to ■• ■00 u)„ 6 C ( X * , X 2, ..., X n-,C) by n U>1 ® u>2 ® ' • ■ & u Jn ( x u x 2, . . . , X „ ) =
J J U k(x k) k—
V xk e
X k\ u k G X k1.
1
( 12 )
This notation is consistent with a variant of equation (4) above.
1.10.5 P r o p o s itio n Let X 1, X 2, ... , X n be linear spaces. For each k = 1 ,2 ,... ,n , let y k be a linear subspace of X k^ which agrees with Z * when restricted to any finite-dimensional subspace Z of X k. The map o -.x 1 ® x 2 ?> ■■■® x n
c ( y \ y 2, . . . , y n-,c>
defined on elementary tensors by (11) is a canonical well defined injective linear map. Hence, for each non-zero tensor t e X 1® X 2
.'0.6
Tensor Products
163
Furthermore, if B k is a Hamel basis for X k for k = 1 ,2 ,..., n then ® x 2 ® ■• ■ ® i n : xk € B k; k = 1 ,2 ,..., n } is a Hamel basis for
Proof It is obvious from multilinearity that 6 is well defined and its injec t iv it y will follow from the next argument.
Since any algebraic tensor product may be built up from successive sigcbraic tensor products of pairs, we may use the lemma for a proof by induction on n. The case n = 1 is obvious (taking the algebraic tensor product of one linear space to be the space itself). Suppose the result has been established for n — 1 and write X 1 ® X 2 ® ■• • ® X n as y ® X n where ^ = X 1 ® X 2 ® ■■■® X n~l . By the lemma, we may write t = , tj ® xnj where { t u t2,.. - , t m} C >> and { x „ , i , x ni2). ■• ,x n,m} Q X n are linearly independent. The induction hypothesis gives us u k 6 y k for k — 1 ,2 ,... ,n — 1 with 0(
Ti ® r 2 ® ■■■® Tn: x l ® * 2 ® ■••® x n - y 1® y 2® ■■•® y 1
( 13)
defined on elementary tensors by I i
T } ® T 2®- • - ® r n(x i ® x 2®- ■-® x n) = T i ( x i ) ® T 2( x2) ® ■■■
164
1: Normed Algebras and Examples
1.10.6
product o f elementary tensors is defined by
(oi ® o 2 8 • • ■® an)(bi ® 62 ® • • ■® bn) = a ^ i ® 02^2 ® ■■• ® on6n
(15)
/or ai/ a*, 6* £ // each A ^ is unital, then the map tpk- A^ —» .4 defined by ipk(a) = l ® l ® - - - ® l ® a ® l ® . . . ® l (with a G . 4 ^ in the kth position) is an injective unital algebra homomorphism for each k = 1 ,2 ,..., n. These homomorphisms satisfy ‘Pk(a)'Pt(b) = w(&)v>*(a)
V M
= 1 ,2 ,..., n;
a
e .4(,£>; 6 € A {t). (16)
Finally, the (n + 1)-tuple (A , tp\, <^2 , ■■■, <£n) m uniquely determined up to the natural notion of isomorphism by the following universal mapping property: For any (n + 1)-tuple (B,tjji,ijj2, ■■• ,4>n), consisting of a unital algebra B and n unital algebra homomorphisms tpk: A ^ —> B satisfying the analogue of (16), there is a unique unital algebra homomorphism 0: A —* B which satisfies 0 o tpk = ipk for each k = 1 ,2 ,...,n . For n = 2, the proposition is illustrated by the following commutative diagrams. The second one shows the isomorphism result.
Proof The easiest way to see that the multiplication (15) is well defined'is to note that iri the explicit construction (in the proof of Theorem 1.10.2) of th<- algebraic tensor product as a quotient Z '/Z " , Z " is an ideal when Z ' is given the obvious multiplication which induces this product. Hence the algebraic tensor product is an algebra under this definition. (The universal
mapping property of algebraic tensor products can also be used to give a simple proof that this product is well defined, cf. Kjeld B. Laursen [1969a], p. 470.) Since 1 & 1 ® ■• ■® 1 is an identity element, A is unital. The maps
Tensor Products
l . ’ O.e
165
are obviously injective unital homomorphisms satisfying (16) so it only remains to establish that the map h :A —►B of Definition 1.10.1 is a unital aig bra homomorphism 6 in the present setting. By induction on n (using (6) ,, we may restrict attention to the case n = 2 where we find 6{(a ® b)(c ® d))
=
h(ac ® bd) = ,ipi(ac)'ip2{bd) =
=
tin (a)V’i (c)xp2(&)V>2(d) = V’i ( 0 ) ^ 2(<#1 {c)ip2 (d)
=
8(a ® b)8(c ® d)
V a,c € A ^ ; b,d € A ^ ■ □
When the algebraic tensor product is considered as an algebra in this v/" . we will call it the algebra tensor product. (Distinguish this from al gebraic tensor product and tensor algebra which have different meanings.) The universal mapping property above indicates that within the category of unital commutative algebras and unital algebra homomorphisms, the algebra tensor product defines a coproduct. Note also that M n ® M m, in troduced in §1.6.8, is an algebra tensor product and the notation associated with it is consistent with the present notations. If A is an arbitrary algebra, then the algebra M n( A ) of all n x n matrices with elements of A as entries can be identified with M n ® A as defined above. If A is unital, the algebra homomorphisms
implies
ati 1 = ae,2 = ■■•,
= 0.
'"his condition is satisfied by the tensor product since for each i with 1 < i < m and each k ^ t we can find w* G satisfying Wfc(afcj) = 0 for ') ^ i and u)k(dk,i) — 1- If wt G A ^ is arbitrary, this gives m 0 — w i ® u>2 ® •■■£5>wn I ^
\>=1 which implies a;,,- = 0 since u>i €
al(j ® a2 j ® ■•■<8> a„tJ J = w ^ a ^ ,)
/ was arbitrary. Also, i was arbitrary.
166
1: Normed Algebras and Examples
1.10.7
The present condition implies the universal mapping property since it ensures that the obvious definition of 6 on products can be extended to sums of products. The Projective Tensor Product of Banach Spaces For finite-dimensional linear spaces and algebras, the above theory is quite complete and satisfying. However for infinite-dimensional spaces we naturally wish to introduce topological restrictions. To begin with these restrictions cause no problem, but if we wish to have the tensor product of Banach spaces be a Banach space, more complications arise. We can form the algebraic tensor product of Banach spaces, and some times this construction is useful. For instance, B p ( X , y ) was defined that way. These algebraic tensor products can be provided with natural norms in several ways, but they will essentially never be complete (unless one factor is finite-dimensional). We usually wish to have the tensor product of Ba nach spaces be a Banach space. There are three main ways to accomplish this as we will show in more detail here. First we may apply Definition 1 .1 0 .1 to the category of Banach spaces, producing the projective tensor product. We may apply the construction noted after Proposition 1 . 1 0 . 5 to get the injective tensor product. Finally w<- may consider various norms (satisfying mild and natural restrictions) on the algebraic tensor product and then complete it. It turns out that the projective tensor product and injective product are special cases of the third approach using the largest and smallest suitable norms, respectively. 1.10.7 D e fin itio n Let X 1, X 2, . . . , A "1,and ^ be normed linear spaces. A multilinear function /: X 1 x X 2 x • • ■x X n —> is said to be bounded if there is a number B satisfying ll/i-Ti, X-)........a-„)t| < R|k,|| ||a-2||-"!!-r1,!l
V x k € X k.
The set o f bounded multilinear maps of X 1 x X 2 x • • • x X n into is denoted by B ( X ' , X 2...... X n- y ) . For each / B ( X l , X 2, X n; y ) let ll/ll be the infiminn of the numbers IJ satisfying the above inequality. The projective tensor product of X } , X 2, . . . , X n is a pair ( Z , p ) where Z ~ X ! ^ X 2r/j ' /jXn is a Banach spare and the tensor map p belongs to B ( X 1, X 2, . . . , X n; Z )i and satisfies the following universal property: If is any Banach space and / is any map in B ( X l , X 2, . . . , X n\y ) , then there is a unique continuous linear map h: Z —* y satisfy ing / = h o p and \\h\\ < ||/||. It is easy to see that / h-» ||/|| is a norm on B ( X 1, X 2, . . . , X n\y) which is complete if y is complete. Note that the projective tensor product of
(possibly incomplete) normed linear spaces is a Banach space. It is natu rally equivalent to the projective tensor product of their completions. We
!0.8
Tensor Products
167
need an analogue of Theorem 1.10.2. The proof of uniqueness is no different, but to establish existence we will show how the projective tensor product can be constructed as the completion o f the algebraic tensor product rel ative to a norm which we will call the projective tensor norm. The proof fir'; this is a norm rather than just a semi-norm is particularly interesting. a.10.8 Th eorem If X 1, X 2, . . . , X n are normed linear spaces, then their projective tensor product exists and is unique in the natural sense. The pro jective tensor product X 1® X 2® ■■■ " may be realized as the completion of the algebraic tensor product Z = X l ® X 2® ■■•® X n with respect to the following expression which is a norm on Z
{
m
n
|
m
II
: 1~
j= l* = l
® 3=1
(
v te i
J
(17) where the infimum is extended over all of the representations for t with m € N and xkj G X k. For any t 6 X l® X 2® ■■■ and any e > 0, we can find sequences C X k converging to zero for each k = 1 ,2 ,..., n and satisfying t= x i j ® x2j ® • • ■® xn<J and oo
n
n P < Y , n \ M ^ m P^ j = l k-^l
( 18)
Proof As noted above, the uniqueness of the projective tensor product is defined and proved in a way analogous to that given in Theorem 1.10.2. The expression (17) is easily seen to be a semi-norm on Z . Let 6\Z — X 1 ® X 2 ®
® X n -> B ( X lm, X 2* ,... , X nt; C )
be the canonical linear map given by the analogue o f equation (11). Propo sition 1.10.5 applies to the present situation directly so 0 is injective. For M i v t G Z, the inequality | | 0 (t)ll < ||<||p is obvious and shows that | |■ ||p is t\ norm not merely a semi-norm. Hence we may define Z to be the com pletion of Z in this norm, leaving the tensor map p unchanged. Clearly ■p C B ( X * , X 2, . . . , X n\Z ) is bounded with norm 1 when Z is defined tliio way. Furthermore, if ^ and / satisfy the conditions in the defini tion, then the definition of the algebraic tensor product gives a map h of 71* ® X 1®- ■ into y satisfying / = hop. Any elementary tensor satis fies \}h{x1® x 2® - - - ® x n)W = \\f(xi,x2, . . - , x n)\\ < ll/ll III!II ||x2||---||a:„||. F»?'ce h satisfies ||/i|| < ||/|| relative to the projective tensor norm. Thus can be extended by continuity to be a linear map with the same norm defined on all of the projective tensor product and having the desired prop erties. Hence ( Z , p ) satisfies the universal mapping property.
168
1: Normed Algebras and Examples
1.10.9.
In order to prove (18), choose a sequence { t m} mgN in the algebraic tensor product satisfying ||< - <m||p < e/2m+l for each m. Then ||
1.10.9 P r o p o s itio n For any normed linear spaces X 1, X 2, ... , X n, the Banach space dual of X l® X 2<8>■■■® X n is naturally isometrically isomor phic to B i X 1, X 2, ... , X n;C) . In particular, for any normed linear spaces X and y , (<-f®y)* is natu rally isometrically linearly isomorphic to B ( X , y * ) . Proof The definition gives a contractive map (/ •-+ h) o f the second space into the first which has the contractive map / i n / i o p a s inverse. For ( w e ( X ® y y / T e B ( X , y ) ) define ( Tu e B ( X , y * ) / urr e ( X ® y ) * ) by ( Tw( x ) ( y ) = ui(x ® y) / w r (x ® y) = T ( x ) ( y ) )
V
x 6 X\ y € y .
These yield inverse contractive linear isomorphisms.
□
The above construction o f a natural predual Banach space for any Ba nach space of bounded linear operators into a dual space is fundamental. It is the precursor of Shoichiro Sakai’s characterization [1956] o f von Neumann algebras as C*-algebras which have preduals. See Volume II. We can now explain the use of the name “projective” .
1.10.10 P r o p o s itio n Let X k and y k be Banach spaces and let Tk € B ( X k, y k) be a bounded linear map for k = 1 ,2 ,..., n. Then there is a well defined continuous linear map Tx ® Ti ® • • • ® Tn: X l® X 2K ■■■® X n - »
• ••® y *
(19)
1 10.11
Tensor Products
169
defined on elementary tensors by T\ ® T 2{8>- • - ® T n( x i ® x 2®- • -<8ia:n) = T i ( x 1) ® T 2(x 2)(8i- • - ® T n( xn) (20) vji'-h norm ||Ti ® T2 ® • • • ® T„||p = n * = i II1* IIFor k = 1 ,2 ,..., n, let Tk be the natural projection of X k onto X k/ Z k where Z k is a closed linear subspace of X k. Then T\ ® T2 <8> ■■■ <8> Tn is a projection of X 1® X 2® ■■■® X n onto the quotient Banach space X l ® X 2® ■■■® X n/ ker(Ti ® T2 ® ® Tn). Proof In (13) we have already remarked that T\ <8>T2 ® ® Tn is a well defined linear map between the algebraic tensor products. Hence Theorem 1.10.8 and the following calculation, where t = YIJL y xi ,j< S>x2j<S>- ■-QXrij, show that it is well defined with the asserted norm as a continuous linear map (with the asserted norm) between the projective tensor products: m
n
n
m
n
e n ii^ M ii < n ^ i i e i i M -
j = i *=1
*=1
j = i *=1
We simplify notation in the proof o f the second claim by setting n = 2. It s enough to consider the case in which only the first subspace is non zero, because the product o f quotient maps is a quotient map. W e write X 1 — X and Z 1 = Z and X 2 = y . Let e > 0 be arbitrary. Apply (11) and the definition of the quotient norm to t 6 ( X / Z ) ® y to find {in }m 6 N C X and {j/m} m6N c y satisfying t - £ £ U (® j + Z ) ® y, and OO
II^IU S
\\xi llllj/jll ^ ||t|| + e. This shows that T\ ®T% is a projection i=i onto a quotient space with its quotient norm. Q The following is an important case o f the projective tensor product for oui purposes. For a suitable product measure theory see Hewitt and Ross [1933], pp. 150-157. 1.10.11 P r o p o s itio n Fork = 1 ,2 ,..., n let ( M k,fik) be a pair consisting of a locally compact topological space M k provided with a positive regular Borel measure fik. Then fi ® f 2 ® • • ■® / „(m i,m 2, . . . , m „) = f i ( m i ) f 2( m2) • •• / „(m „) dcj
V f k € L l (/**); mk G M k
nes an isometric linear isomorphism of L 1( f i i ) ® L 1(n2) ® ■■■® L l (/j,n)
o m o L 1(/ii x pi2 x ■■ ■ x fin ).
Let X be a Banach space. Then f <8>x (m ) = f ( m ) x
V / € L 1(fii)\ x € X
170
1: Normed Algebras and Examples
1.10.11
defines an isometric linear isomorphism, of L l ( p i ) ® X onto the space L x( n\ , X) of Bochner fii-mtegmble X-valued functions on A f1. Proof Consider the second case first and replace ( M l , fii ) by (M , fi). The map (/, x) f x (where f x ( m ) = f ( m ) x for all m fc A f) is a bilinear map o f L 1( n) x X into L x( ii, X), so there is a contractive linear map r from the projective tensor product to L}( pt,X). It is not hard to show that the space o f continuous functions o f compact support from M to X is dense in L l ( n , X ) and hence the image o f r is dense. The proof will be complete when we prove that the map r is an isometry. We are reduced to showing that each t = fj ® Xj G L*(/i) ® X satisfies ||r(<)|| = |jf||p. Theorem 1.10.9 shows that we may identify ( L x(/j) ® X )* with B ( L 1( fi),X*) . Since the f} are supported on a cr-finite portion of (M , fi) and the x} are included in a finite-dimensional subspace o f X, we may apply the Dunford-Pettis theorem (Dunford and Schwartz [1958],V I . 8.6) to show that any element u> (z ( L H n ) '/j X ) ' haw the form w ( / ® i ) = J f( m) x*( rn) ( x) d/i ( m) for some function x ' : M —> X * satisfying ess supp{||x*||} = |M|. Since this can be rewritten as ui(t) =
j
x*(m )(T(t)(m ))dfi(m ),
we find ||f||p < sup{|w(t)| : W G ( L \ f i ) ® X ) * } < ||r(f)|| as we wished to show. Now the first result with n = 2 follows by setting X = L 1(/x2)- As usual, the general case follows by taking successive tensor products as in (6). O We will briefly introduce the trace on certain projective tensor prod ucts. Consider the projective tensor product X ® X * . Any element can be expressed as a series t = xn ® It is easy to see that the expression
satisfying £3^11 ||*n|| ||^n|| < oo.
OO
7 > (0 = ] T ^ n ( z « ) n= 1
(21)
(which we will call the trace) is well defined and satisfies |7>(t)| < ||i||p.
(22)
When X is finite-dimensional, the matrix construction in §1.7.7 shows that this trace agrees with the familiar trace on square matrices. In §1.7.11 above and §1.10.25 below, we will note that it also agrees with other notions of traces of certain operators on infinite-dimensional Banach spaces.
1.10.12
Tensor Products
171
Projective Tensor Products o f Banach Algebras
We will now extend the results o f Theorem 1.10.6 to Banach algebras. These ideas were first explored by Bernard R. Gelbaum [1959] and Jun Tomiyama [I960]. The projective tensor product is the only one o f the Ba'lach space tensor products which always preserves Banach algebra struc ture, but we will discuss some other special cases below. Other results on tensor products o f general Banach algebras are contained in T . Keith Came [1981]; Gelbaum [1959], [1961], [1962],[1965], [1970]; Jesus Gil de Lamadrid [1833], [1965a], [1965b], [1967]; Kjeld B. Laursen [1969b], [1970]; Arnold Lebow [1968]; Richard J. Loy [1970b]; Anna Maria Mantero and Andrew Tonge [1979] and [1980]; R. D. Mehta and Mahavirendra Vasavada [1985]; Jen Okniriski [1982]; U. B. Tewari, M. Dutta, and Shobha Madan [1982]; Tomiyama [1960], [1972]; A li Ulger [1988a]. This list omits all o f the many papers relating to tensor products o f C*-algebras and W*-algebras. 1.10.12 P r o p o s itio n Let A^l\ A^2\ . . . , A ^ be Banach algebras. Then the projective tensor product A 1 - 1) (SiA^ (?) • • • is a Banach algebra un der the natural product and the projective tensor norm. Proof The obvious calculation shows that the projective tensor norm is submultiplicative with respect to the multiplication on the algebraic tensor product. Hence the multiplication introduced in Theorem 1.10.6 can be extended to the projective tensor product by continuity. □ When the projective tensor product o f Banach algebras is regarded as a Banach algebra under this multiplication, we will call it the projective Banach algebra tensor product We will note below that the other Banach tensor products we introduce later do not necessarily satisfy the above proposition. Combining Theorems 1.10.6 and 1.10.8 and previous argu ments, we get the following result. 1.10.13 C o ro lla ry Let A^2\ . . . , A ^ be unital Banach algebras with identity elements of norm 1 and let A be their projective Banach algebra tensor product. For k = 1 , 2 let tfk' - A^ A be the natural con tractive injective unital algebra homomorphisms defined in Theorem 1.10.6. Than the ordered (n \)-tuple (.A, tpi, tp2 , • • •,
172
1: Normed Algebras and Examples
1.10.14
The following important result is a corollary of Proposition 1.10.11 and Corollary 1.10.13. 1.10.14 C o r o lla r y Let G 1,G 2, ... ,G n be locally compact groups. Then L 1( G l ) ® L 1( G 2) ® ■■■® L l (G n) is isometrically algebra isomorphic to L l ( G 1 x G 2 x ■■■ x G n) under the map described in Proposition 1.10.11. Proof It only remains to check that the products agree. We may reduce to the case o f two locally compact groups G and H . Then f , g € L 1( G ) and h, k e L l ( H ) satisfy
J J f ® h(w, z)g ® k(vu~1u, z~1v)dw dz = J f(vu)g(w~1u)dw J h( z) k( z~1v)dz = f * g ® h*k(u,v).
(/ 55 h) *( g ® k)(u, it) =
Hence the product in the algebra tensor product agrees with convolution in L l { G x H ) . Q Somewhat similar results are given by Jesus Gil de Lamadrid [1965b] for the projective tensor product o f the Banach space A f(O ) o f finite reg ular Borel measures on a locally compact space fl with smother Banach space X. Then M ( f l ) ® X is given as the space o f certain A'-valued mea sures on il at least when 0 satisfies a fairly general condition. In partic ular, M ( G l ) ® M { G 2) ® ■■■® M ( G n) is isometrically algebra isomorphic to M ( G l x G 2 x ■■■x G n). Tewari, Dutta and Madan [1982] prove interesting partial converses to these results. They show that if A ® B is isometrically isomorphic to L l ( G ) or M ( G ) for some locally compact abelian group G, then both A and B have the same form. The proof depends on Marc Rieffel’s characterization [1965] o f these group algebras for locally compact abelian groups. We remark on smother consequence of our definition of projective tensor products. Note that a Banach algebra consists of a Banach space A together with a multiplication map M : A x A —> A defined by M( a , b) = ab. Clearly M belongs to B( A, A; A ) and has norm at most one. Hence we may regard it as a contraction in B ( A ® A , A) . The associativity o f multiplication is then expressed by the equation M o (M ® / ) = A / o (/ ® M ), where /: -4. —*■^4. is the identity map. Clearly any contractive linear map M : A ® A —> A satisfying this condition determines a Banach algebra structure on A. Hence all of Banach algebra theory can be stated in these terms. We will seldom be tempted to do this. However when confronted with a Banach algebra A and a Banach algebra structure on A " , the above procedure may be used to give A ’ the structure of a Hopf algebra. See Theorem 4.2.26 for an interesting related result concerning injective alytbras in which multiplication is well defined and continuous from the injective tensor product.
i . 10.15
Tensor Products
173
T i e Varopoulos Algebra The projective tensor product of C ( f l i ) and C (n 2) may be embedded in C (O i x ^ 2) but it is not the whole algebra unless or is finite. W e will prove this only in the following interesting case first considered by Nicholas Th. Varopoulos [1965], [1967]. The elegant presentation is due in part to the review of [1965] by Carl S. Herz. See also Jean-Pierre Kahane [1970], These introductory comments will be generalized in Section 3.6. Let G be a compact abelian group with normalized Haar measure (i.e., A(<7) = 1) and let G be the set o f all continuous group homomorphisms fro n G into T. We will regard G as a discrete group under pointwise mu!tiplication. The Fourier algebra A ( G ) on G is the subalgebra o f C ( G ) consisting of all those functions which can bo expressed in the form f(u) = ^
g(h)h(v.)
VueG
for some j e ^ ( G ) .
(23)
heG
In Jection 3.6 we will give the easy proof that if / G A ( G ) satisfies (23), tbe i g is uniquely determined and satisfies g( h) = f ( h ) = JG f ( u ) h ( u ) ' d u for »ach h e G . Thus we may give A ( G ) the norm jj/||,4 = ||/||i. This maces A ( G ) into a Banach algebra since any two functions f , g e A ( G ) satisfy f g - f * g where f * g ( h ) = J2keG f ( k) a ( k~ lh) and hence \\fg\\A <
il/IUIM Uj e t V ( G x G ) be the projective tensor product C (G )® C (G ) considered as f, subalgebra o f C ( G x G ) by setting (/ ® g) (u, v) = f ( u ) g ( v ) for any u,v e G as usual. Any F € A ( G x G ) belongs to V ( G x G) , since it can be written in the form F ( u ’ v) ~ Sh.fcgG P ( h ® k) h(u)k(v) where we havj simply noted that each element o f G x G has the form ( h ® k)(u, v) = h( u)k(v) for some h , k e G . Hence we have continuous inclusions A { G x G ) -> V ( G x G ) - » C ( G x G).
1.1 C. 15 V a ro p o u lo s ’ T h e o r e m n r - iclized Haar measure. Define M : C ( G ) -» C ( G x G) M f(u,v)
=
PF(u)
=
f(uv)
f
Let G be a compact abelian group with
and
P: C ( G x G ) -> C ( G )
V / e C ( G ) ; u, v e G
F ( v , v ~ lu)dv
by
and
V F £ C ( G x G); u e G .
JG
The i P o M is the identity map on C ( G ) and the restriction of ( M / P ) dfiji'i es an isometric linear isomorphism, of ( the Fourier algebra A ( G ) / the
1: Normed Algebras and Examples
176
1.10.19
The reader can find many similar results. For instance there is a nat ural isometric linear embedding of X ® y onto a closed linear subspace of Next we note the functorial property of this tensor product and explain its name.
1.10.10 P r o p o s itio n Let X k mid y k be, Banach spaces and let Tk e B ( X k, y k) be a bounded linear map for fc — 1 , 2 , . . . , n. Then there is a well defined continuous linear map
n ® t 2 ®■••® Tn: x l® x 2® •••®*r* - yl®y2®■■■®yn defined on elementary tensors by T i ® T 2 ® - - ® T n(xi ® x2 ® ■■■® xn) = T i ( x i ) ® T 2( x2) ® ■■- ® T n( xn) with norm. ||7i ® T2 ® ■■■® TnH^, = For each k = 1 , 2 letTk be the natural injection of a closed linear subspace X k into y k. Then T\ ® T 2 ® ■■■® T n is an isometric injection of
x 1® * 2® •••g )* " mto y 1® y 2® ■■■® y n. Proof We have already noted (twice) that T\ ® T 2
||T, ® T2 ® ■■■® T„(t)|U = sup{| Y , m
= sup{| E
n
II
M T k M ) \ : u k G ( y k) 1}
n
n
T;(u;k) ( x ki])\ :ujk e ( > * ) ! }
J = lfc = l
n
<
m
n
n
n nTfciisup{ix:n^(^)i:^ e(**>;>=n n^n
k= l
j=lfc=l
fc=l
In order to see the injectivity result, consider the obvious map 6 o f the al gebraic tensor product Z = X l ® X 2®- ■ ® X n into B ( y u , y 2* , . . . , C). Because of the Hahn-Banach theorem, the norm on the image of Z is no different from the norm on 6 ( Z ) in B ( X 1' , X 2' , . . . , X n*\C ). But the for mer norm is that of Z considered hh a mibspace of y 't h y 2® ■■■® y n and the latter is its norm as e. subspace of X 1® X 2® ■■■® X n. Since these are the respective injective tensor norms, T\ ® T2 ® ■■■® Tn is injective and isometric. D There are Banach spares for which the projective tensor product does not satisfy the second paragraph of Proposition 1.10.19 and others for which
Tensor Products
1.10.20
177
cbt injective tensor product does not satisfy the last paragraph of Proposit 3n 1.10.10. It is seldom true that the injective and projective tensor prc ducts of two Banach spaces agree, but Gilles Pisier [1981], [1983] contrt licted a conjecture of Grothendieck by giving an example of a Banach sj;s:cc X satisfying X<&X = X
V (w ,t ) £ fi.
Definition 1.10.17 shows that this injection into C (f i) is an isometry. 1.10.20 T h e o r e m Let X andy be Banach spaces, and let !! = A'1* x y 1 * be the compact set defined by the weak* topology. Then A f( f i ) is isometrically isomorphic to the dual space (A'gO^)* of the injective tensor product under the map n >—* fi defined by rn
m
> rx(z 2 xi ® w ) = £
j i
-
/ UJ(xi ) T (y j')d^ (UJ' T ')
V 0 e M (Q );X j € X ;g j e y .
j i Jn
Proof It is immediate that fj, is a continuous linear functional with norm bounded above by ||^||. Suppose, on the other hand, that a continuous linear functional on X ® y is g'ven. The Hahn-Banach theorem shows that it can be extended from the isometric image of X<%>y in C ( f l ) to all o f C (f i) without increasing its norm. Then the Riesz representation theorem shows that it has the form fi fo~ some fi € A f(f i) with ||^|j no bigger than the norm of the original linear functional. □ Injective Tensor Products of Algebras We will give two related examples of injective tensor products of alge bras. Let f i i , 0,2,. . . , f i „ be sets and, for each k, let A( llk) be a linear space of complex valued functions on fi*. For each k, let fk £ A(flk) be given and define fl ® /2 ®
® /n (w 1,U12, . . . , U n) = fl(uii)f2(ul2) ■■■/n(w„).
(25)
Then f\ ® f 2 ® • • •<8>/„ is a function on fii x x • • • x fi„ . Since the map ( f i , j 2, . . . , f n) l~* fi <81/2®• • '® / n is clearly multilinear, this defines a linear
178
1: Normed Algebras and Examples
1.10.21
map of .4(^1) 155-4 ( ^ 2)® • • -8.4(1},,) into the set T — T(i\\ x f i 2 x • • • x ftn) of all complex functions on J2i x 0 2 x • • • x S2n, If each A(ilk) is an algebra under pointwise operations, then this map i.s a homomorphism when T is also considered as an algebra under point wise operations. We state the most interesting case formally. Let fi be a locally compact Hausdorff space. Recall the Banach algebra C o(O ) defined in §1.5.1. 1.10.21 P r o p o s itio n // S21, 0 2, . . . , Cln are locally compact Hausdorff spaces, then the map indicated above establishes an isometric algebra iso morphism of C o t f i O & C o ^ ) ® • ■■®
=
sup { l E n ( / i , j > 2(/2,.,)|: Tk e c 'o(nfc)J } j= i
=
sup{||
m
T i(/i,j)/ 2 ,j||o° : n e Coini)*} i =i m
=
sup{| y v i (/1j ) / 2,j(ufr)| : Tl e C (0 i)J ;w 2 £ 0 2} i =i
—
s M W T . h A ^ h j W - : ^ 2 e o 2} j= i
=
su p {| 5 2 / 2>j(w i)/ 2 j(w 2)| : Wfc G ilk) = 11/Hoc. j=i
m
Hence the map is an isometry. Thus it can be extended to an isometry of C q(Q% )
□ If n is a locally compact Hausdorff space and A is a Banach algebra, then the set (?o(n ..4) of continuous functions from 0 to A which vanish at infinity is a Banach algebra under pointwise operations and the supremum norm. The elements of C o (n )® ,4 can be interpreted as elements o f Co(n, ,4) through the definition
Tensor Products
1.10.22
179
V / G C o (n ); a G A\ w G 0 .
(/ ® a ) ( w ) = / (w )a
1.10.22 P r o p o s itio n Lei Cl be a locally compact space and let A be a Banach algebra. Then the map described above induces an isometric algebra isomorphism of C o(ft)(d A onto Co(n, ^4). Proof The proof is similar to that o f the last proposition exccpt for showing that the image of the tensor product is dense. Clearly the set Coo(ft, A ) of c6ntinuous functions from 0 to A which have compact support is dense. Heri«e we may assume that 0 is compact. Let / e C( il, A ) and e > 0 be given. Choose f(uii), f ( w 2) , . . . , /(wm) in f(Cl) so that for each f(ui) with w G Cl there is some j satisfying ||/(w) — /(wj)|| < £■ For each j , define Vj C Q, by Vj = {w G 0 : ||/(u>) - f ( u j )|| < £} so that Vi,V2, ... ,Vm is an opei cover for ft. There is a partition of unity e1,e 2, ... ,e m subordinate to tnis cover. By this we mean that each ej is a continuous function of 0 into [0,1] satisfying ej(u>) = 0 for w ^ Vj and '%2’j Li ej ( UJ) — 1 for all w 6 O. (For the existence o f such a partition o f unity, see Bourbaki [1966], IX.4.3.) Thus we have m ll/ M -
m ® / ( wj ) ( w)II
=
j=1 ^
H
- /(wj)|| < e
Vwen.
{ j ’. u & V j }
This proves that the image of the tensor product is dense as desired.
□
Other Complete Tensor Products There are many similarities between projective and injective tensor products. Their most remarkable property is that each is defined inde pendent of the particular Banach spaces occurring in the tensor product. Both Schatten [1950] and Grothendieck [1954], [1955] investigated this phe nomenon and the latter found 14 equivalence classes o f such norms. Their study is based on constructing norms on the tensor products of finite dimensional subspaces, but we will give no further details. Both the projective and injective tensor products are the completion of the algebraic tensor product with respect to a suitable norm. Both norms .satisfy the following definition. The term “ reasonable” and the next theo rem are due to Grothendieck, while the term “uniform” was used already be Schatten. We will not use his term “cross norm” . We are giving the next definition in the geometric category o f Banach spaces, as is most common and often suitable. However the reader should be av/are that modifications suitable for the topological category o f Banach algebras have been studied. Gil de Lamadrid [1967] called a norm on the algebraic tensor product admissible if it dominates the injective tensor norm
180
1: Normed Algebras and Examples
1.10.23
and is dominated by the projective tensor norm (cf. Theorem 1.10.24(d) below). This terminology has been adopted by others, but we are not aware that his name “modular” for a uniform norm up to topological equivalence has seen recent use. In the cited paper he offers a more general setting in which modular norms can be defined. 1.10.23 D e fin itio n Let X l , X 2, . . . , X n be normed linear spaces. Let II • ||a be a norm on the algebraic tensor product Z = X 1<% >X 2 <%>■■■ <$ X n. The norm induced on Z = X 1* ® X 2* ® ■■■® X nm by its embedding into the dual space of ( Z, || • |[Q) defined on elementary tensors by n VlEfc € X k\ UJk € ( X ky k-^1 is called the dual norm and denoted by || • |ja. . A norm on Z is called a reasonable norm if it satisfies ||ii ® x 2 8 - - - ® x „ | | a
<
||xi|| ||j-2||• • • ll^nll
||u»i 0 U>2 ® ■■• ® W„||a*
<
||wi 11|\hl211' ' ' I
Vxke X k
and
11 V Ulk £ ( X k)*
(26) (27)
The completion of Z with respect to a reasonable norm || • ||a on Z ,is denoted by ,V1®.Va® • ■■®.V". A reasonable norm on Z is called uniform if all choices of Tk G B ( X k) satisfy
||7i ® r 2 ® •■• ® T„|| = ||Ti|| ||T2||•••||T„|| whore the norm on the left is the operator norm in B ( Z ) . A reasonable norm Q Ot Ot || • ||a on Z is called nuclear if the natural projection of X 1
E f = i I l L i u k,j(xk,i)-
1.10.24 T h e o re m Let X l , X 2, . . . , X n be normed linear spaces and denote X ' ® X 2 ® • • • ® X n and X 1* ® X 2* ® • • • ® X n* by Z and Z , respectively. (a) For reasonable norms, the inequalities (26) and (27) are actually equalities. (b) The dual of a reasonable norm is a reasonable norm. (c) Both the projective tensor norm and the injective tensor norm on Z are uniform reasonable norms. When either is applied to Z it is the dual of the other on Z. (d ) A norm || • ||a on Z is reasonable if and only if it satisfies
lltllu. < ll*lla < 11‘ llp
vtez.
(28)
Tensor Products
M 0,34
181
Thus the injective and projective tensor norms are, respectively, the smallest and largest reasonable norms. Prooj (a): Given any x* £ X k, choose w* G [ X k)\ satisfying wt(xfc) = ||x*j| for ef ch k. Then (27) shows ||wi ® w2 ® • • • ® w„||a. < 1 which implies ||xi 8 x2 ® • • • ® x„||a > n * = i ll*fc||i 88 we wished to show. T ie proof for (27) is the dual of the one just given except that this time .ve must consider sequences {x fc „ }n6M C X k satisfying ||x*n|| = 1 ar>d lirnw/;(xfc„) = ||wfc||. (b): Let || • ||a be a reasonable norm. Since (26) for j| • ||Q« is just (27) for || • ||a , we need only establish (27) for || • ||Q-. By induction, it is enough to consider the product o f two normed linear spaces X and Let ( F e X " / G G >'** ) be arbitrary. Goldstine’s theorem (Dunford and Schwrxtz [1958], V.4.5) allows us to choose a sequence ( { x 7} 76r / {jA yK er ) satisfying { x 7 - » F / y7 - » G ) and { ||x7|) < ||F|| / ||jj7|| < ||G|| ). An arbitrary tensor u — YfjLi ® Tj >n X * ® y * satisfies m
m
| (F ® G )(u )|
=
5 3 f k )G ( tj ) = lim j - 1
<
J= l
limsup llx^H Ily^H ||u||Q. < ||F|| ||G|| ||u||Q.,
establishing ||F®G||Q.. < ||F||||G||. (c , (d): The first paragraph of Proposition ( 1.10.10 / 1.10.19 ) shows that the ( projective / injective ) tensor norm is uniform. First we will prove inequality (28) for any reasonable norm || • ||Q. Any Uk G (-Vfc)I and any t G 2 satisfy
|u>i ®
® •* * ® ^n(0l ~ 11^1 ® ^2 ® * ** ® WnjjcK* |j^||a ™ ||^||a*
Since the weak tensor norm of t is defined to be the supremum over all the expressions on the left hand side, we conclude ||t||u, < ||£||a Any t = jy jL i x iJ ® X2J ®
® %nj € 2 satisfies
m \\t\la ^
m n l l ^ l j ® %2,j ®
® ^njlla “
j= 1
^ ll^fc.jll j = l k —l
Since !|tjjp is defined to be the infimum over fill the expressions on the right hand side, we get ||<||Q < ||*||p, completing the proof o f (28). F xt we show that || • ||„, and || • ||p are reasonable. Equation (26) is clear or both norms. The definition of the weak tensor norm shows |u>i ® u;2 ® ■• • ® w„(£)| < ||wi|| ||w2||• • • ||wn||j |£j |u>
V tJk G ( X h)*\ t e Z
which implies (27) for || • ||„,. These results imply
IM U < ll«l|p* < IM L* < IMIp
v # e i
182
1: Normed Algebras and Examples
1.10.24
which gives (27) for || • ||p when u is an elementary tensor. Let us calculate the dual of the projective tensor norm for any u G Z:
IMIp-
=
sup{|u(t)| : t e £; ||t||p < 1}
=
sup{
T. u ( x l , ]
x 2,j
}=1 x k,3 e X k-
Ill'll-1 } j = 1 1
m <
suP { £
X2J
| «(x i,j ®
® ■• •
0 x n ,j)|
:
.7= 1 m
n
xk,j G X k; 53 f l H^oll < U J= 1
1 n
=
SUP{|U(X! ® X2 ®
—
IMI«> — IM Ip' •
® In)| : x k G X k\ J J ||Xfc|| < 1}
k= 1
Finally we calculate the dual o f the injective tensor norm for any u _ _ JZ, ) Wntj G Z:
m
-
sup{|u(i)| : f e 2 ;
< 1}
m
<
su p {]T ) lwi J ® W2,J ® • • • ® w„,j(<)| : t e z - , ||<||w < 1 } >=i ^ s u p f l w i , , ®U2, j ® ••• ® w „ j ( t ) | : t G 2 ; ||£||w < 1}
j= i m
=
n
e n j ~ i k=i
ii^-ii-
The projective tensor norm o f u is just the infimum over all the expressions occurring on the last line, giving the desired inequality and equality. Finally suppose ]| • ||a is a norm on Z satisfying (28). When t is an ele mentary tensor, (28) becomes (26) with an equality sign. By the definition of the dual norm, (28) implies
IM L = IMIp- < IMU* < IMU* = IMIp
v
u e z.
When u ii an elementary tensor, this gives (27) with an equality sign.
□
Because o f inequality (28), it is obvious that there are contractive linear maps between the completions o f X 1<8>X 2 ® ® X n in the corresponding
1.10j25
Tensor Products
183
norms: X l® X 2® ■■■® X n - * X ' g i X 2® ■■• ® x n - * x l® x 2® • • •® x n. C a u tion : Even though these maps are extensions by continuity o f the iden tity map on the algebraic tensor product, they may have nontrivial kernels. In particular, a formula which makes sense on the space B a ( X , y ) ~ y ® X * of approximable operators is not well defined on the projective tensor prod uct, 1.10.25
G e n e ra l T en so r P r o d u c ts o f B an ach A lg e b ra s
It is natu
ral to ask when a reasonable norm || • ||a on the algebraic tensor product (considered as an algebra) A = - 4 ^ ® A ^ ® • • •® - 4 ^ o f Banach algebras _4 (i),_<4 (2) ) ^4 (") defines a Banach algebra structure on the completed tensor product A " — A ^ (giA^ <8 ■• • The obvious answer, that this happens exactly when the norm || • ||a is submultiplicative, is useful, since submultiplicativity may be as easy to check directly as some o f the prop erties we mention below. This question was first considered by Gelbaum [1982] and Gil de Lamadrid [1963], [1965a]. Subsequently, T . Keith Came [1978] provided another type of answer, although he seems to have been un aware o f the previous work. W e will summarize some o f this work without giving full details. If || • ||a is a uniform norm, then B = B (A ^ ) <8>B ( A ^ ) ® ■• • ® B (A t'n^) clearly acts on (A , || • ||Q) in a bounded way: m
IMOIU
— ^ ] 11^1,j ®
® *‘ •® ^n,j(^)Ha
j= 1 m
<
n
E I I I I t mIIII
V « e B ; i e A
j = 1 k=l
where u = YljLi ® ^ 2,j ® • ■• ® Tnj Hence we may define the operator norm ||• ||opa on B relative to ( A, || ■||Q) and this is bounded above by the projective tensor norm. The dual action o f u* = on A* also satisfies ||u*||opo < ||u*||p, showing that || • ||opc( is reasonable. This operator norm is obviously submultiplicative. Gil de Lamadrid’s main result essentially follows from the above remarks. P r o p o s itio n Let A = A 1,1'1® A ^ ® • • • ® .A^") be Banach algebras such that the left regular representation L of each is an isometry. Let ||• ||a be a uniform reasonable norm on A = A ^ <8>A ^ ® • • •® . A ^ . For each t e A, define ||<||^ to be ||L( ||opQ where || • ||opa is defined as indicated above on B = B ( ^ 1) ) ® B ( ^ 2) ) ® - - . ® B ( ^ " ) ) . Then\\-\\0 is an algebra norm on A , and || • ||a is submultiplicative if and only i/|| • ||Q = || • ||^.
184
1: Normed Algebras and Examples
1.10.26
*= • -Sxin.j G A note that L t is simply Y ^ = i ® La*, &■ • j € B. Now it is easy to see that if ||-||a is submultiplicative, then the two norms aie equal. The converse is immediate. Note, however that the very ease of this proof shows that proving || • ||Q = || • \\p is likely to involve checking submultiplicativity directly. As Gil de Lamadrid noted, the main force o f this theorem is to prove the existence of a reasonable number of algebra norms on tensor product algebras such as A Came [1978] considers the natural tensor norms in Grothendieck’s sense mentioned above. He gives several necessary and sufficient conditions for such a norm to be submultiplicative. These conditions depend on a sort o f mixed associativity o f the metric tensor products. Using these criteria, he is able to show that representatives o f three o f the fourteen equivalence classes o f natural tensor norms which Grothendieck defined and studied are submultiplicative. It is obvious that the tensor product algebra o f two united algebras A 1) and A i s united, even when the algebraic tensor product A 1) ® A 1) is completed under some norm. Richard J. Loy [1970b] proves a very general Ol converse for Banach algebras: if A = A ^ d ^ A 1' is the completion o f A 1) ® A :) under some reasonable (or even just admissible) algebra norm, the A is united if and only if A 1' and A 2) are. Gelbaum [1962] contains a similar but more restricted result. For an extension o f these results to approximate identities see Proposition 5.1.4. A number o f the papers cited before Theorem 1.10.12 above deal with the ideal theory o f algebra tensor products. In particular, Gelbaum [1959], [1961], [1962], [1970], Tomiyama [1962], Lebow [1968] and Laursen [1970] fall in this category with the two 1970 papers correcting errors in the first pair. See also Came [1981], Some will be discussed in subsequent chapters. 1.10.26 R ea so n a b le H ilb e r t S pace N o rm s a n d C * - N o r m s §1.7.17 introduced Hilbert spaces and C*-norms on *-algebras. They will be studied much more intensively in Volume II. However, they present an important natural situation in which reasonable norms arise which differ from the projective and injective tensor norms. If W 1, 7i2, . . . , 7in are Hilbert spaces, it is natural that we should wish to complete their algebraic tensor product to be a Hilbert space again. Since the norm on a Hilbert space is most often defined in terms of the inner product, we give Z = Ti1 ® H 2 ® ® H n an inner product and then use that to define a norm. For elementary tensors we define (Xi ® x2 ® • • ■<8>X „ , 2/1 ® y2 ®
n ® yn) = JI(z/t,yfc) fc=l
V xk,yk e H k.
Clearly this expression can be extended by bilinearity to ■■ ®Hn and satisfies the properties of an inner product. The only question which
[.■
.26
Tensor Products
185
might arise is whether the inner product is definite, i.e., whether ( 6 2 and (t, u) = 0 for all u G Z implies t — 0. However, the semi-norm given by this inner product is certainly a reasonable norm, so it is larger than the injective tensor norm and really is a norm rather than just a semi-norm. Explicitly the norm is 1/2
m
i'o w let A ^ \ A ^2\ ... be C*-algebras and denote their algebraic t e r . >r product (considered as an algebra) by A. W e may introduce an inv ution in A by setting a* equal to ® a%j ® ■■■® a* j when 2^Li aiJ ® a2,j ® •• • ® an,j- It is easy to see that this is well defined ant’ satisfies the four properties o f an involution given in §1.7.17. Thus we I w . made A into a *-algebra. In Volume I I we will consider a number o f s o f *-algebras in which there is a largest C*-norm, called the GelfandNaii.iark norm, for each algebra. It turns out that A has this property and that the Gelfand-Nalmark norm 11•117 is a reasonable norm on it. We denote ____
7
7
7
the completion o f A with respect to || • ||7 by A — The unitization of any C*-algebra has an algebra norm satisfying the C*-cindition. Thus it is again a C*-algebra. Example 5.1.11 below shows tha C*-algebras are approximately united. This can be used to show that any 0*-norm on A can be extended to the tensor product o f the unitiza tion® o f the A Hence we may suppose all these algebras were already unital. In the category of united C*-algebras (in which morphisms are *homomorphisms which are necessarily contractive), A 7 satisfies exactly the same universal mapping property as the projective tensor product o f united Banach algebras does in the category o f united Banach edgebras. V ?hen we introduced C*-algebras, we noted that they were characterized by t 3 property that each one has at least one isometric isomorphism into B { 'H (for a suitable Hilbert space Ti) which sends the involution in the C^-M gebra onto the Hilbert space adjoint map in B ( H ) . Suppose that for each k = 1,2, . . . , n , ipk’- A ^ —> B ( H k) is such a map. Then for each elementary tensor ai ® a2 ® ® a „ G A, equation (15) shows how to interpret B(?i). It turns out, though it is far from obvious, tha lie operator norm on ip(a) is independent o f the choice o f the
186
1: Normed Algebras and Examples
1.10.26
The largest and smallest C ‘ -norms do not always agree on the tensor product o f C*-algebras. A C*-algebra B is called nuclear if they do agree on any tensor product o f B with any other C*-algebra. It is also known that the smallest C*-norm on the tensor product of two C*-algebras is strictly larger than the injective tensor norm unless at least one o f the factors is commu tative. Since Co(Jl) is the typical example of a commutative C*-algebra, this shows that Propositions 1.10.21 and 1.10.22 cannot be extended. Sc<> Kadison and Ringrose [1986] for further details.
2
The Spectrum
Introduction The spectrum is undoubtedly the most important concept in the theory of algebras. In fact, it is primarily the spectrum which differentiates the theory o f algebras from the general theory of rings. For an element a in an algebra A, the spectrum is a collection o f complex numbers, denoted by Sp(a), which is a sort o f shadow o f the element. (This is the origin of the name.) In finite-dimensional algebras, it is always nonempty and finite (cf. Section 2.7). However, in perfectly general algebras it may be empty or unbounded and thus too intractable to be useful. In this introduction to the chapter we will only define the spectrum o f an element o in a united algebra A - In this case Sp(a) = {A € C : A I — o has no inverse in 4 } . Thus the spectrum o f a matrix in the algebra M n o f all n x n-matrices is the set o f eigenvalues, and the spectrum o f a function / in the algebra o f all continuous functions on the unit interval [0,1] is the range o f the function: {/ (< ) : i 6 [0,1]}. In general, the spectrum offers a way to study the phenomenon of invertibility. Note that it is an isomorphism and automorphism invariant; under an isomorphism between algebras the spectrum of an element and its image are the same. After this introduction, we will begin the detailed portion o f the chap ter by extending this definition to nonunited algebras. This will involve the concepts o f quasi-multiplication and the quasi-inverse. In perfectly genered algebras the spectrum has severed useful properties including the spectral mapping theorem (Theorem 2.1.10). W e also explore its possible patholo gies with examples and a few simple results. Zero plays a special role in the spectrum and must be treated separately in the statement o f many theorems. The problem is that the spectrum does not behave well with respect to homomorphisms or subalgebras unless they are unital. It be haves much better in the category o f unital algebras where all algebras and homomorphisms are unital. The spectral radius p(a) of an element a G A is defined by p(a) = sup{|A| : A G Sp(a)}.
187
188
2: The Spectrum
2.0
We prove (Theorem 2.2.2) that any element in an algebra A with a non trivial algebra semi-norm a satisfies the inequality: lim a(an) lln < p(a) n—»OO
V a € A-
This shows, in particular, that the spectrum of such an element is not empty (since the spectral radius would be —oo in that case). The Gelfand-Mazur theorem (Corollary 2.2.3), which asserts that the complex field is the only normed division algebra, is an immediate consequence of this result. Next we introduce the concept o f spectral semi-norms. Many properties characterize these semi-norms. In particular they are the algebra seminorms a for which there is some constant C (in fact we may always choose ( ' -- 1) satisfying p(a) < Ca( a) V a e A. Alternatively, they are the algebra semi-norms satisfying Gelfand’s spectral radius formula: p(a) = lim « r (a " )1/n V a e A. n —»oo
This is the limit denoted by cr°° in Theorem 1.1.10. Israel Moiseevic Gelfand [1941a] proved this formula for complete algebra norms, which are there fore the best known examples o f spectral semi-norms. Furthermore, spec tral semi-norms are exactly those algebra semi-norms relative to which the multiplicative group o f invertible elements is open. Additional character izations are contained in Theorems 2.2.5, 2.2.14, and 2.5.7, Propositions 2.2.7 and 2.5.15 and Corollary 2.4.8. Many other properties o f spectral semi-norms are developed. The following list (with specific references to results in this work) shows that they arise frequently in analysis. They will play a prominent role in the rest of the work. Examples of Spectral Semi-norms (a) Any complete algebra norm (Gelfand [1941a]; Corollary 2.2.8). (b ) Any algebra norm (not necessarily complete) on a completely regular semisimple commutative Banach algebra (Rickart [1953]; Theorem 3.2.10). (c) Any algebra norm (not necessarily complete) on a C*-algebra (Rickart [1953], cf. Cleveland [1963]; Theorem 6.1.16). (d ) Any algebra norm (not necessarily complete) on any two-sided ideal o f B ( X ) or on any closed subalgebra of B ( X ) which includes all finite-rank operators, where B ( X ) is the algebra of all bounded linear operators on a Banach space X (Yood, [1958]; Theorem 2.5.17). (e) Any algebra norm (not necessarily complete) on a modular annihi lator algebra (Yood [1958] cf. Bonsall [1954a]; Theorem 7.4.12). (f) The operator norm on a full Hilbert algebra (Rieffel [1969b]; Volume II).
2.0
Introduction
189
(: i The Gelfand-Nalmark semi-norm on a hermitian Banach *-algebra (P t , or [1972], Ralkov [1946] and Pt&k [1970], [1972]; Theorem 10.4.11). (•') Any algebra semi-norm on a Jacobson-radical algebra (Corollary 2.3.4;. The , acobson Radical and Fundamental Theorem N iwburgh’s theorem (Theorem 2.2.15) shows that the spectrum and spcct al radius are upper semi-continuous functions with respect to any spectral semi-norm on an algebra. In order to formulate one o f the most fundamental results on spectral semi-iorms, we introduce in a preliminary fashion the Jacobson radical, which will be fully discussed in Chapter 4. It has a special role in connec tion with the spectrum and spectral semi-norms. The Jacobson radical can be described as the largest ideal on which the spectral radius is identically 0 or, alternatively, as the largest ideal such that the spectrum o f any element in the algebra agrees with the spectrum o f the image o f this element modulo the ideal. However in order to prove that any algebra has such a largest ideal, and for other reasons, we introduce algebraically irreducible represen tations and define the Jacobson radical of an algebra as the intersection o f the kernels o f all such representations. After a brief introduction of these concepts, we are able to prove the fundamental theorem of spectral semi-norms (Theorem 2.3.6) which states that the limits o f any sequence with respect to different spectral seminorrr.i differ by an element in the Jacobson radical. The proof uses a recent lemma due to Thomas J. Ransford [1989]. This is a rather spectacular result since spectral semi-norms are not at all unique. It has a number of zor'z'l jies, among which Barry E. Johnson’s [1967a] uniqueness of norm theon m (Corollary 2.3.10) is the best known. This theorem asserts that any two complete norms on a semisimple algebra must be equivalent. Commutativity and Modular Ideals Theorem 1.1.10 and the Gelfand spectral radius formula show that, on a commutative algebra with a spectral semi-norm, the spectral radius is both subadditive and submultiplicative. It turns out that it is enough to assume that the algebra is commutative modulo the Jacobson radical. Commutative algebras exist where the spectral radius is always finite-valued but is neither subadditive nor submultiplicative (Example 2.1.7). We will show Theorem 2.4.11) that if the spectral radius has either property, then it has both, and that then the algebra must be commutative modulo its Jacob ;on radical. (This is a full converse o f the first statement above, since the sc ectral radius is a spectral semi-norm under these hypotheses.) \ ■' briefly introduce modular ideals. They will be further discussed in Chap' jr 4. Here they are used to characterize spectral semi-norms as those
190
2: The Spectrum
2.0
algeb ra semi-norms in which all maximal modular ideals are closed (Corol lary 2.4.8) and to describe the kernels o f multiplicative linear functionals on an algebra, i.e., algebra homomorphisms into the complex number held (Theorem 2.4.12). Spectral Algebras and Spectral Subalgebras A spectral algebra is an algebra on which some spectral semi-norm can be defined. We emphasize that it is not an algebra with a particular choice o f spectral semi-norm, but merely an algebra on which it is possible to define at least one spectral semi-norm. Most spectral algebras have many spectral semi-norms, and no particular one o f them is preferred above the others. Proposition 2.4.2 gives a purely algebraic description of spectral algebras. These algebras play a fundamental role throughout this work. Note that any Banach algebra is a spectral algebra, and Theorem 2.5.18 characterizes spectral algebras in terms of the existence o f homomorphisms into Banach algebras. The Gleason-Kahane-Zelazko theorem (Theorem 2.4.13) shows that a linear functional ip: A —►C on a spectral algebra is a homomorphism if and only if it satisfies ip(a) € Sp(a) V a € A. ln the next chapter we will show that a commutative algebra satisfies the Gelfand theory if and only if it is a spectral algebra. Chapter 4 con tains the proof that any irreducible representation o f a spectral algebra is strictly dense, generalizing results o f Nathan Jacobson [1945b] and Charles E. Rickart [1950], The chapter concludes with a thorough discussion o f spectral subalgebras. These are the subalgebras such that an element o f the subalgebra has the same non-zero spectrum whether calculated in the subalgebra or in the larger algebra. N. Bourbaki [1967] calls unital subalgebras full (pleine) if they are spectral subalgebras, but many interesting cases of spectral subalgebras are not unital subalgebras, as the following list o f spectral subalgebras shows. (I f not otherwise indicated, the statements have easy proofs which are gathered in Proposition 2.5.3.) Examples of Spectral Subalgebras (a) Any one- or two-sided ideal. (b) The coinniutarit of any subset. Hence the center of an algebra and any maximal commutative subalgebra. (c) The “corner” subalgebra eAe, for any idempotent e G A. (d) Any finite-dimensional subalgebra or any subalgebra in which every element satisfies some polynomial equation. (<:) Any modular annihilator subalgebra of a normed algebra (Corollary
Introduction
191
(f) Any closed subalgebra o f a Banach algebra in which each element or ilvi subalgebra has nowhere dense spectrum relative to the subalgebra (Corollary 2.5.11). (g) Any closed *-subalgebra o f a hermitian Banach *-algebra (Theorem 10.4.19). (h) The range o f the left or right regular or extended regular represen tation. (i) The image k ( A ) of A in A " with either Arens product. (j) Any intersection o f spectral subalgebras or any spectral subalgebra o f a spectral subalgebra. As one would expect, a spectral subalgebra of a spectral algebra is again a spectral algebra in its own right. (However, a spectral subalgebra of a nonspectral algebra need not be a spectral algebra. Despite this, we believe the name “spectral subalgebra” , is appropriate, since spectral subalgebras art* the appropriate kind of subobject in the category of spectral algebras. The morphisms in this category are just algebra homomorphisms.) This gives a long list o f spectral algebras. For instance, any (not necessarily CIOS': i ) ideal in a Banach algebra is a spectral algebra. The theory o f topological divisors of zero extends to spectral semi-norms, and hence its consequences Eire available for spectral algebras, even though they do not carry a particular spectral semi-norm. Many results on spectral subalgebras arise in this way. Tw o propositions (Propositions 2.5.12 and 2.5.13) give conditions under which an algebra without topological quasidivissrs o f zero is isomorphic to the complex field. The last result (Theorem 2.5.18) characterizes spectral algebras as those algebras A for which A / A j can be embedded in a Banach algebra. This characterization remains valid even if the subalgebra is required to be dense and the Banach algebra is required to be semisimple. It gives insight into the reason for the similarity between the spectral theory of spectral algebras and Banach algebras. Historical Notes W e conclude this introduction with a few historical remarks on the ori gin of the concept of spectrum. David Hilbert can be credited with the introduction o f the words “spectrum” , and “spectral theory” , and with the beginning of the subject [1904, 1906; cf. 1912]. It arose out o f his work on integral equations. These were first used by Niels Henrik Abel, but their systematic study was begun by Eric Ivar Fredholm [19001. Fredholm’s work inspired Hilbert to spend most of the first decade of the twentieth century working on integral equations, and this work was the origin o f many of the fundamental ideas of functional analysis. In particular, Hilbert ex ploited a numerical parameter in the integral equations lie wished to solve to understand the behavior in terms of the various values of this parame
192
2: The Spectrum
2.1.0
ter. Although his formulation does not agree with the modern placement o f the parameter, his spectrum is closely related to our modern concept. Hilbert even gave a fairly sophisticated spectral theorem which expresses the integral operator as a generalized integral over its spectrum. From our modern viewpoint, Hilbert, and later his student Erhard Schmidt, dealt mainly with the case of compact operators, where the spectrum resembles the spectrum of a matrix much more closely. Indeed, Fred holm, Hilbert and Schmidt used limits of matrix approximations as one of their main tools. However, Hilbert did begin with a more general spectral theorem involving a treatment of continuous spectrum. It is a great irony o f history that the word “spectrum” , was first used in twentieth century science to describe the discrete lines in the light emitted by heated gases when that light was bent through a prism. Hilbert was well aware o f this use of the word. These lines represent the energy of jumps between quantized states of the atoms involved. After considerable development of quantum mechanics, in which Hilbert played a role, it finally became clear that the physical spectrum o f an atom could best be described mathematically in terms o f the spectrum o f an operator on Hilbert space. Hilbert is on record as saying that he had no hint of this connection when he introduced the concept o f the spectrum in analysis. The connection received an early definitive treatment in John von Neumann’s influential book [1927]. For more historical remarks see Lynn A. Steen [1973] and Constance Reid [1970]. We will not discuss continued fractions in Banach algebras. The reader will find a bibliography in Hans Denk and Max Riederle [1982].
2.1
Definition of the Spectrum
Quasi-multiphcation and Quasi-inverses We now begin the detailed portion of this chapter by extending the concept of spectrum to nonunital algebras. Recall that for any element a in a unital algebra A we define its spectrum and spectral radius by: Sp(a)
=
{A G C : A l — a has no inverse in A} ,
p(a)
=
sup{|A| : A G S p (a )},
and (1)
respectively. If the spectrum is empty or unbounded, then the spectral radius will have the value —oo or oo, respectively. When no element has empty or unbounded spectrum, the spectral radius is a non-negative real valued function on the algebra— a beginning for analysis and geometry. We can extend this definition to a not necessarily unital algebra A by saying that the spectrum of an element in A is the spectrum (as just defined) of its image in the unitization A 1 of A. According to this definition, 0 will
2.1.1
Definition of the Spectrum
193
belcng to the spectrum o f an element a unless A is already unital and a is inve rtible in A. This is the accepted definition and should always be kept in mind. However, for both technical and theoretical reasons it is desirable to give a logically equivalent definition of spectrum entirely in terms of A rather than its unitization A 1. In order to do this we must introduce a concept to take the place o f invertibility in nonunital algebras. O f course such algebras have no invertible elements, but by recalling that we really wish to describe the conditions under which an element in A has an inverse in A 1, we are led to the correct concept. It was first used by Sam Perlis [1942] (cf., Jacobson [1945a] and Irving Kaplansky [1947a]). The particular form of the quasi-product which is now standard was introduced in Einar Hilie’s influential book [1948]. 2.1.1 D e fin itio n The qtiasi-product of any two elements a and 6 in an algebra is denoted by a o b and defined by a o b = a + 6 — ab.
(2)
We shall also call this binary operation quasi-multiplication. (Note that 0 acts as an identity element under quasi-multiplication.) I f a o b = 0 holds, then a is said to be a left quasi-inverse for b, and b is said to be a right qua -inverse for a. An element which is both a left and a right quasi-inverse for c is called a quasi-inverse for o, and is denoted by a9. The terms right quasi-invertible, left quasi-invertible, and quasi-invertible are used in the usr.: way. An element which does not have a (two-sided) quasi-inverse is said to be quasi-singular. The set o f quasi-invertible elements in an algebra A is called the quasigre*\> of A and is denoted by A qa ■ Similarly, in a unital algebra the set of invertible elements is called the group of A and is denoted by A g The first result in the next proposition explains the quasi-product. In particular, it implies that a quasi-inverse is unique if it exists, that an element which is both left and right quasi-invertible is quasi-invertible and that the quasi-inverse of a is in the double commutant of a: aq G { a }" . I f A is unital, it shows that the simple map a i—> 1 — a
V a € .4
(3)
(which is its own inverse) is a semigroup isomorphism of the ( multiplicative semigroup of A / quasi-multiplicative semigroup of A ) onto the ( quasimultiplicative semigroup of A / multiplicative semigroup of A )■ Hence it is aka a group isomorphism o f the { group A g / quasi-group AqG ) onto the ( quasi-group A^ g / group A g )• If A is nonunital, the situation is more complicated: the above map establishes group isomorphisms between the ncr: nl subgroup *4^0) ~ {A + a G A q : A = 1} of the group A q of A 1 and
194
2: The Spectrum
2.1.2
the quasi-group A^a of A. The group A q is homeomorphically isomorphic to the direct product of the multiplicative group C* of non-zero complex numbers and A^a under the isomorphism A - a H-* (A, A-1a). Obviously, the inverse o f this group isomorphism sends the second factor onto the normal subgroup o f A q just described. Warning: Although quasi-multiplication is associative (and commuta tive if and only if A is commutative), it does not behave well with respect to the linear structure. Instead o f the distributive law we get: (a + 6 ) ° c = a o c + 6 o c — c
and
c o ( a + 6) = c o a + c o 6 —c
(4)
a o A6 = A(o o 6) + (1 — A)o
(5)
and scalar multiplication gives: Ao o b = A(o o b) + (1 — A)6
and
as short calculations will show.
2.1.2 Proposition
Let A b e an algebra. (a ) The following are equivalent for any a, 6, c € A : ao6 = c
and
(1 — a ) ( l — 6) = 1 — c.
(6)
(b) Quasi-multiplication is associative and the quasi-group is a group under quasi-multiplication, with 0 as its identity element. (c ) If A is embedded as a one- or two-sided ideal in some larger unital algebra B (with identity element 1), then the following are equivalent for any non-zero A e C and a € A : \ ~ la e A qa
and
(A I - a) e Ba-
(d) If A is not unital, then the quasi-invertible elements in A 1 are given by: = {1 — A + Aa : A € C \ {0 }; a € A^ g } and the quasi-inverse for such an element of A 1 is given by: (1 - A + Ao)9 = 1 - A " 1 + A " 1o«. Proof The first two claims are immediate. If A-1 a has 6 as a quasi-inverse, then ( A 11 - A-1 6) is an inverse for (A l - a) in B. Conversely, if c is an inverse for (A l - a) in B, then —ac = —ca is a quasi-inverse for A_1a in A. The last claim is also routine. □ The next proposition shows that the quasi-product and quasi-inverse
have utility beyond just defining the spectrum in not necessarily unital al gebras. No result similar to the following holds for the group of invertible
2.1.3
Definition o f the Spectrum
195
elements in a unital algebra or for inverses, since a homomorphism between algebras does not necessarily respect inverses or invertibility. Only unital homomorphisms do so. However, perfectly arbitrary algebra homomor phisms respect quasi-inverses and quasi-invertibility. 2.1.3 P r o p o s itio n Let A and B be algebras and let tp\ A —*■ B be a homomorphism. Then they satisfy ]
'fi(Aqa) Q BqG
and
V a € A^a-
Thus if B is a subalgebra of A, Bqa is included in A qa . ff A , B and are unital, they also satisfy
and
V o G Ag -
If B is a unital subalgebra of the unital algebra A, B q is included in A g Proof An easy calculation gives the first statement which immediately implies the second. The unital case is also straightforward. □ It is clear that quasi-multiplication is at least separately continuous in a semi-topological algebra. In a semi-normed algebra, the quasi-group is a topological group by the following result o f Kaplansky [1947a]. 2.1.4 P r o p o s itio n In a semi-normed algebra A , quasi-multiplication is jointly continuous and the map which sends an element to its quasi-inverse is a homeomorphism. Proof The first statement is obvious. In order to establish the second, let a and b be two quasi-invertible elements and denote their difference by h. Then we get b o a 9 = a + h + aq — (a + h)aq = h( l — aq) which implies (I - b9) h( l - a9)
=
h(l - a9) - b“h(l - a9)
=
b9 o h(l - a9) - b9 = bq o b o a9 - b9 = a9 - bq.
Denote the difference aq — b9 by k, so we have k = k ( —h(l — a9) ) + (1 + aq) h( l — aq). Hencc if a is an algebra semi-norm and a(h) is sufficiently small, we get a { k ) < [1 + <7(a9) ] 2 [ l - c r { h ) ( 1 + <7(a, ) ) ] ~ 1
This proves that the quasi-inverse is a continuous and hence homeomorphic map on the quasi-group. □
196
2: The Spectrum
2.1.5
Definition and Basic Properties of the Spectrum W ith these preliminaries out of the way, we are in a position to define the spectrum in a not necessarily unital algebra in terms of the quasi-inverse without reference to the unitization. Proposition 2.1.2(c) applied to the embedding of an algebra into its unitization shows how to do this. The present definition agrees completely with the informal definition offered at the very beginning o f this section. 2.1.5 D e fin itio n Let a be an element in an algebra A. Consider the set o f complex numbers {A € C : A ^ 0 and A_1a ^ Ago} The spectrum Sp(a) of a equals this set if A is unital and a is invertible. Otherwise Sp(a) is the union o f this set and {0 }. The spectral mdius p(a) is defined by p(a) = sup{|A| : A 6 Sp(a)}. (7) If the algebra A intended might not be clear, we write SpA (a) and p a ( ° ) The complement in C of the spectrum of a is called the resolvent set of a. In perfectly general algebras the spectrum of an element may be empty or unbounded, as we will now see by example. In these two cases the spectral radius is —oo or oo, respectively. However, in the types o f algebras which are of most interest in this work the spectrum will be nonempty, closed and bounded and hence compact. 2.1.6 E x a m p le Let C (z ) be the function field in one variable, consisting of all rational functions in z. Then any non-constant function has empty spectrum. Theorem 2.2.2 below shows that no element in a normed algebra can have empty spectrum. (Hence no submultiplicative norm can be defined on C (z ).) Many normed (even commutative) algebras contain elements with un bounded spectrum. For instance, consider the unital, commutative, normed algebra C[z] o f all polynomials with the norm ||p|| = sup{|p(A)| : A € C,|A| < 1}. In this algebra, every element has closed spectrum since „ , , f C 5p(p) = \ { A }
if p is not constant if p = A € C.
There also exist unital, normed (even commutative) algebras in which some elements have unbounded and nonclosed spectrum. Consider 71 = {rational functions with poles outside the open unit disc, (A 6 C : |A| < 1 }}. We may give this the norm ||/|| = sup{|/(A)| : |A| < 5 }. The spectrum o f each rational function / G 71 is just {/ (A ) : |A| < 1}.
Definition of the Spectrum
I. ‘ .7
197
Proposition 2.1.11 imparts some order to the situation illustrated by these examples. If every element in an algebra has nonempty, bounded spectrum, then the spectral radius is a non-negative real-valued function on the algebra. W e will show later that in a commutative spectral algebra the spectral radius is also subadditive and submultiplicative and hence is an algebra norm. Commutativity is not enough by itself to give this result, as the next example shows. Theorem 2.4.11 will show that if the spectral radius is subadditive or submultiplicative, then it enjoys both properties and is th-.a an algebra semi-norm. 7 E x a m p le There are unital, commutative algebras in which the spectrum of every element is nonempty and bounded (hence compact), so tha the spectral radius is a non-negative real-valued function, but on which the spectral radius is neither subadditive nor submultiplicative. In C © C (« ) tIk : spectrum of an arbitrary element is given by: rr
«
f {A }
if / not constant if
So ' ve have p{0, l ) = l > 0 + 0 = p{ 0, z) + p( 0,1 - z) p (0 ,1) = 1 > 0 • 0 = p{0, z)p( 0 , 2-1 ). This algebra even has a nontrivial algebra semi-norm:
(a) Any algebra A satisfies
Sp(ab) U { 0 } = Sp(ba) U {0 }
V a, 6 G A.
(8 )
b) Any algebras A and B and any homomorphism
P| SpA (a + 6) U {0 } 6Gker(^>)
V a G A.
(9)
If J ., B and tp are all unital or if A is not unital, this can be simplified to SpB(ip{a)) C SpA (a)
V a G A.
(10)
;c) If B is a subalgebra of A , then SpA (b) C SpB(b) U {0 }
V 6 G B.
( 11)
f *
2: The Spectrum
2.1.9
If B is a united subalgebra of A or if B is not a unital algebra, then we have SpA (b) C 5pg(6)
V b 6 B.
( 12)
Proof (a): If c is a quasi-inverse for A~ 1ab, then A~*6(c — l)a is a quasi inverse for A - 1ba. (b): This is an immediate consequence of Proposition 2.1.3. □ (c): Apply (b) to the embedding map of B into A2.1.9 E x a m p le In the above proposition it is really essential that 0 be treated differently from non-zero numbers in the spectrum. To see this for statement (a), consider the shift S and backward shift 5* on the space X — {/ : N —i• C } of all bounded complex sequences. These operators are defined by f ( n — 1)
if
n > 1
if
n= 1
Thus S 'S is the identity operator, so its spectrum is {1 }, but 5 5 ’ annihi lates any sequence which has its only non-zero value in the first position and thus has spectrum {0 ,1 }. This example is intrinsically infinite-dimensional and it is easy to see that in a finite-dimensional algebra the symbols “u { 0 } ” , may be removed from the statement o f (a ) in the above proposition. To see the special role of 0 in parts (b) and (c) of the proposition, consider C embedded as the upper left entry in the 2 x 2 matrix ring M%. Then 1 G C has spectrum { 1 } in C but spectrum {0 ,1 } in M 2. Elements with only one-sided inverses have special properties: Bernard H. Aupetit and L. Terrell Gardner [1981], James W . Rowell [1984], Scott H. Hochwald and Bernard Morrel [1987]. The Spectral Mapping Theorem The next important result is called the spectral mapping theorem. This theorem includes the following kinds of results: Sp(a2) — Sp(a)2; Sp(a~l ) = {A ~ x : A € Sp(a)} for invertible a; and Sp(aq) — {A/(A — 1 ) : A e Sp(a) } for quasi-invertible a € A. A rational function is, of course, simply a ratio of polynomials (which may be chosen relatively prime). Let p be the polynomial p (A) = ak^k- (W e denote the coordinate function on C by r: r(A) = A VAeC, so this polynomial may be written as p = 5Ht=o Q* z*-) Then for an element a in any algebra A we write p(a) — 5Z*=0<*jfcafc. This agrees with the notation of elementary algebra. (Here a0 denotes the identity in A 1, so that in a nonunital algebra, if the polynomial does not vanish at 0, is in A 1
.'. 1 0
Definition of the Spectrum
199
rather than A ) Clearly, the function p >—>p(a) is an algebra homomorphism of .the algebra o f polynomials into A 1. I f q is another polynomial, we also h&ve ( q o p)(a) = q(p(a)). This action of polynomials on elements o f the algebra is called the polynomial functional calculus. If / is a rational function with no poles in Sp(a), then we may write f = p/q where p and q are relatively prime polynomials and q has no zeroes in.5p(a). As we show in more detail in the proof, q(a) is then invertible, so w$ may define f ( a ) to be p(a) q(a) ~l . This interpretation is so basic that we give no formal definition. Since this rational functional calculus is well defined in any algebra, the following theorem appears to be the natural form o f the universal spectral mapping theorem. Surprisingly, the theorem is Usually restricted to the polynomial functional calculus. 2.1.10 T h e o r e m If a is any element in any algebra A and f is any rational function with no poles in Sp(a), then f ( a ) is defined in A 1. The function f i—> f ( a ) is an algebra homomorphism. Unless Sp(a) is empty and f is a constant, f ( a ) satisfies: Sp{ f{ a) ) = f ( S p ( a) ) where the last symbol is defined by {/ (A ) : A € Sp(a)}. If g is a rational function and either side of the equation (j° / )W
= ff(/ W )
is defined by the above remarks, then both sides are defined and they are equal.
Furthermore, for each a £ A the map f t-> ip ( f ) = f ( a ) is the only algebra homomorphism ip from the algebra of rational functions with no poies in Sp(a) into A 1 which satisfies ip(l) = 1 and tp(z) = a. Proof If Sp(a) is nonempty and / is a constant Ao, then they satisfy / ( « ) = A0 and S p ( f ( a ) ) = {A 0} = f ( Sp( a) ) . Henceforth we assume that / is not a constant and work in A 1. Let p and q be relatively prime polynomials satisfying / = p/q. In order to determine the invertibility of A0 — / (a ) in A 1, we factor q and Aoq — p: q( A) = a(A , - A )(A 2 - A) ■• • (An - A) Ao?(A) - p(A) = 0(ni - A)(/i2 - A) • • ■( nm - A) where a and fi are not 0 (the latter since / = p/q is not a constant). Since q has no zeroes in Sp(a), q(a) = a (A i - a )(A 2 - a) • • • (An - a)
200
2: The Spectrum
2.1.11
is invertible, so f ( a ) = p( a) q(a) ~1 is defined. No Hj appears among the A *’8 since p and q are relatively prime. So ~ f(a) =
~ a)(M2 - a ) • • • (/im - o )[q (A ! - o )(A 2 - a) • • • (An - a )]-1
is invertible unless some f i j is in the spectrum of a; but the f i j ' s are just the roots of Aoq(n) — p{n) = 0. Another way to say the same thing is that Ao is in the spectrum of f ( a ) if and only if an element of the spectrum of a is a zero o f Aoq{fi) — p (| i). But this is what we wished to show. The other remarks are now obvious. □ We will see other functional calculi with corresponding versions of the spectral mapping theorem later in this work. There Eire a number o f inter esting situations where a class of functions / acts on a class of elements a to give elements f( a). If / *—►f ( a ) is an algebra homomorphism, composi tion of functions acts appropriately and a version of the spectral mapping theorem ( S p ( f ( a )) = {/ (A ) : A G S 'p (a )}) holds, then the action is called a functional calculus. There is an intimate association between the spec tral mapping theorem and the action o f composite functions in all cases. In the next chapter we will define the most important functional calculus for arbitrary elements in arbitrary Banach algebras: the holomorphic func tional calculus Theorem 3.3.7. (In that case we must consider an algebra of germs o f functions rather than functions themselves in order to preserve the desired algebraic properties.) A ll our functional calculi are consistent with each other and unique under appropriate conditions. The present spectral mapping theorem produces some order among the examples of poorly behaved spectra given in Example 2.1.6. 2.1.11 P r o p o s itio n If every element in an algebra has bounded spectrum, then every element has closed (hence, compact) spectrum. Proof Suppose the algebra A satisfies this hypothesis but a G A has non closed spectrum. Choose A in the closure of Sp(a) but not in Sp(a). Then Sp{(A - a )-1 ) = {(A — /it)-1 : /x G Sp( a) } is unbounded. □ The Exponential Function and Exponential Spectrum We now introduce a variant of the spectrum which was first considered by Robin Harte [1976]. We begin by deriving the basic properties of the exponential function in a unital Banach algebra. W e define this function in terms o f its power series. (Th e next chapter will contain a systematic exposition of a much more general notion of functions of elements in a Banach algebra.) In a topological group, the connected component which contains the identity is called the principal component. It is always a closed normal subgroup (c/. Hewitt and Ross [1963], 7.1). We will show that the set of
.12
Definition of the Spectrum
201
fmUe products of exponentials is an open and closed normal subgroup of / ' c which equals its connected component. Recall that a group is called ior ion-free if it has no elements of finite order. In the following theorem the second statement under (d) is a result of M i+:o Nagumo [1936], and the last statement under ( f ) is due to Edgar R. L c .jh [1943]. In a recent article, Alessandro Di Bucchianico [1991], like Mi-^umo, emphasizes that the principal component is arc connected and tht. , the easiest way to show that an element is in this component is to find an y c connecting it to the identity. Theorem 2.1.12 below, follows Lorch in f lowing that the kernel of the exponential map (considered as a group homomorphism from a commutative Banach algebra) is just the additive g ro ip generated by {2irie : e is an idempotent in *4}. In contrast to (f), Ver i 1. Paulsen [1982] shows that every finitely generated group arises as •Ag / A gc for some (possibly noncommutative) Banach algebra A. 2.1.12 T h e o r e m Let A be a unital Banach algebra. Define the exponential function exp: A —►A by exp(a) = e°, where °°
e° = £
„n
^
V
a G A.
(13)
n=0
a) Any commuting elements a, b G A satisfy:
(b) For any a G A, ea is invertible and satisfies (* “ ) - * = e - “ . fc) For any a € A , the map t eta is a continuous group homomor phism of the additive group of real numbers into A c id) The set exp(*4) = {e ° : a G *4} includes the open unit ball around the identity element 1. Furthermore, c € A g belongs to exp(.4) if and only if c belongs to some connected abelian subgroup of A g (e) The principal component of A g is the subgroup Ace = { e 0le°3 • • • e°" : n 6 N; ai, a2, ... an 6 A } of j _g generated by exp(*4). It is open in A. Thus A g e ** an open and cios 'A connected normal subgroup of A g ■ (i) Let A be commutative. Then exp(_4) is the principal component of A g , and the quotient group A G / e x p ( A ) is torsion-free. Hence A g has infj itely many components unless it is connected. Provf (a): Since the expansion (13) converges unconditionally, the series for ,xp(a) and exp(6) can be rearranged (or equivalently, multiplied by
2: The Spectrum
202
2.1.12
ordinary Cauchy multiplication) when a and b commute:
(b): This follows from (a). (c): Statement (a) shows that t •—►eia is a group homomorphism. For any s, t G R , the estimate — eto|| <
<
||e*°|| ||1 -
II
exp||*a|| | l-e x p (| f-a | ||a||)
proves the continuity.
(d): I f c satisfies ||c — 1|| < 1, then the series
converges unconditionally in norm. Thus the double series
can be rearranged. Since the series for a is just the Taylor series expansion o f the logarithm, rearrangement leads to the equation e° = c. Result (c) shows that any exponential belongs to a connected abelian subgroup. We defer proof of the converse until we have established the first results o f (f). (e): Let £ be the displayed set, which is obviously the subgroup gener ated by exp(.4), and let Aa? be the principal component of A g - Then £ is connected since t •—* et0let0j • • •eta" for t G [0,1] is an arc connecting an arbitrary element eaie°J • • •e°" to 1. Hence £ is a subset of A a eNext we show that £ is open in A and hence closed in Aa If 6 G £ and a G A satisfy ||6 - a|| < ||6-1 ||_1 , then they also satisfy ||1 - 6-1 a|| < ||6_1 || ||6 - a|| < 1. Hence (d) shows = exp(c) for some c € AThus a - b exp(r) belongs to £, so £ is open in A and hence in A g However an open subgroup is always closed since its complement is a union of (necessarily open) cosets.
2.1.13
Definition of the Spectrum
203
By connectedness this implies A a e = £• It only remains to show that E is a normal subgroup. For any a € A and b G Aa, we have OO
OO
beab~1 = E ( n ! ) -1 6a"6_1 = E ( n ! ) _1(6a6_1) n = exp(6a6_1). n=0
n=0
Hence exp(*4) is invariant under all inner automorphisms. Thus Ace. is also invariant so it is a normal subgroup. (f): When A is commutative, (a) and (b ) show that exp(*4) is a group. Hence (e) implies that exp(*4) is the principal component of A g W e now complete the proof of (d). Suppose c e A belongs to a connected abelian subgroup H of A g - Let B be the (necessarily commutative) closed subalgebra o f A generated by H. Then c is in the principal component of B q and hence is an exponential by what we just proved. Next we show that A g / e x p (^ ) is torsion-free. Suppose a 6 A g and n G N satisfy a " e exp(.A). If b satisfies an = exp(b), then c = a exp(—n~1b) satisfies c" - 1. Hence the spectral mapping theorem (Theorem 2.1.10) shows that Sp(c) is included in the set o f nth roots o f 1. The set {A G C : 1 — A + Ac 6 A g } is connected since its complement is finite. However this set contains 0 (where 1 — A + Ac is 1) and 1 (where 1 — A + Ac is c) so that c belongs to the principal component o f A g - Thus c belongs to exp(*4) by what we have just shown. Hence o = exp(n_1b)c belongs to exp(*4). Thus A g / exp(_4) is torsion free. Since exp(.4) is open, A g has infinitely many components if it is not connected. □ The following concept introduced by Haute [1976] is chiefly useful be cause of the improvement of inclusion (9) of Proposition 2.1.8 which we give in Theorem 2.1.14. We forego giving a definition in the nonunital case and note that the name is only appropriate in the case of Banach algebras. See Murphy and West [1992] for further results. 2.1.13 D e fin itio n Let A be a unital algebra. The exponential spectrum f’Spj, (a) — eSp(a) of an element a G A is the set of complex numbers eSp(a) = {A
G
C : A - a $ Agc} -
The next theorem gives the fundamental properties of the exponential spectrum in a Banach algebra before we have derived the corresponding results for the spectrum itself. For the proof o f non-emptiness, we depend on Theorem 2.2.2 (or Corollary 2.2.8) in the next section. Recall that we use d to denote the boundary of a set in a topological space. The polynomiaily convex hull o f a compact subset K of C is just the complement of the unbounded component o f the complement o f K . Thus it is obtained by “filling in all the holes” of K. We denote it by pc( K ) .
204
2.1.14
2: The Spectrum
2.1.14 T h e o r e m (a ) Let A be a unital Banach algebra. For each a £ A, eSp(a) is a nonempty compact subset of C satisfying deSp(a) C Sp(a) C eSp(a) C p c(5 p (a )).
(14)
(b) Let B be a closed unital subalgebra of a unital Banach algebra A. Then OeSpB(b) C SpA (b) ;
eSpeib) C pc(SpA (b))
V b € B.
(c) Let A and B be unital Banach algebras and let p: A —* B be a continuous unital surjective homomorphism. Then they satisfy p( AGe) = Bae\
eSpB{p(a)) =
f|
e5p(a +
Va
6A
6€ker(<^>)
Proof (a): Prom the definition it is obvious that eSp(a) includes Sp(a), and we will show the latter is nonempty in Corollary 2.2.8. Since A a e is open, eSp(a) is closed. It is bounded since |A| > ||a|| implies that A — a — A(1 - A_1a) belongs to exp(.4) by Theorem 2.1.12(d). We have just noted the middle inclusion of (14). The last inclusion will follow from the first, since it ia obvious that if the boundary o f a compact set K in C is in L, then K is in pc( L) . Let A belong to the boundary of eSp(a). Then we can find a sequence {A n}n€N ° f elements in the complement of eSp(a) converging to A. We conclude |An — A| ||(An — a )-1 || > 1, since otherwise (A - a )(A n - a ) " 1 = 1 - (A „ - A )(A n - a ) - 1 would be in Arje, implying A — a G Ac'e, contrary to fact. This gives IKA - a )(A „ - a ) " 11| < 1 + |An - A| ||(An - a ) " 11| < 2|An - A| ||(An - a ) " 11|. I f A — a had a bounded inverse, this would imply the contradiction 1 < 2|An - Aj ||(A — a ) -1 1|. Hence A is in Sp(a). (b): In the last paragraph, if A - a had a bounded inverse in any larger algebra, the argument would not change. Apply this to 6 G B. (c): Continuity implies p( ea) = e'fii-a> and hence p (A c'e) Q Bae- Con versely, Euiy b G B oe satisfies b = ec>eCa • • •ec" = p( eaieai ■■■e ° ") where ip{a3) = c3 for j = 1 ,2 ,..., n, giving the opposite inclusion. In the second displayed equation in (c), it is obvious that the left-hand side is included in the right-hand side. Conversely if A ^ eSpB(
.15
Definition of the Spectrum
205
.15 C o r o lla r y Let A and B be united Banach algebras. Iftp:A -+ B is a cc tinuous united homomorphism with closed range, the spectrum satisfies: SpB(
P ) SpA (a + b) C pc(5pe (^ (a )))
V a e A.
(15)
6gker(¥>)
Proof The first inclusion was already obtained in Proposition 2.1.8(b). Let C be the closed, hence complete, range o f ip. The theorem shows °) &£'.er(<£)
SpA (a + b ) C
P|
eSpA (a + 6) = eSpc (
fc€ker(y?)
ch concludes the proof.
□
.lansford [1984a] derives most of spectral theory from very simple axioirs without any multiplicative structure. He starts with a normed linear sp : A, a fixed non-zero element 1 6 A and an open subset Q in A conte: :ng 1, excluding 0 and closed under multiplication by non-zero complex mr :bers. I f A is a unital spectral normed algebra, we may take Q to be ei„ -x A q or A a e- Hansford’s “spectrum” is defined to be Spn(a) = {A 6 € i a — A ^ f i } . Other geometrical conditions can be imposed on H. See the reference above, Catalin Badea [1991] and Ignacio Zalduendo [1989]. Relatively Regular Elements and Algebras We conclude this section with a brief introduction to some generalized inverses. John von Neumann [1936b] first introduced regular algebras in connection with his studies of orthomodular lattices and continuous geomet r sj. Kaplansky [1948d] studied weakly regular algebras. We have inserted “relatively” into both names in order to avoid confusion with other uses of the ubiquitous term “regular” . In particular, we use “relatively regular element” in place of “regular element” , since the latter term is used as a synonym for “invertible” . We use weakly regular for another concept below. 2.1.16 D e fin itio n An element a in an algebra A is said to be ( relatively regular / weakly relatively regular ) if there is a non-zero element b € A sat' fying ( aba — a / bab = b ) . In this case, b is called a { relative inverse / wedz relative inverse ) for a. An algebra or ideal is called ( relatively regi tar / weakly relatively regular) if all its non-zero elements are { relatively regi '.ar / weakly relatively regular). Relative inverses are also called generalized inverses (particularly in op erator theory) and pseudo-inverses. The next proposition sums up their rsics elementary properties. 2 .1 1 7 P r o p o s itio n Let a and b be elements in an algebra A. Define p and q by p = ab and q = ba.
206
2: The Spectrum
2.1.18
(a ) I f b is a ( relative inverse / weak relative inverse ) for a, then p and q are idempotents satisfying ( pa = a = aq / pa = paq = aq) . (b ) Any relatively regular element is weakly relatively regular. (c ) A ny relatively regular element a with relative inverse b satisfies aA = p A and Aa = Aq. (d) If the algebra A satisfies ( A l a — 0 / A r a = 0 ), then any a & A is relatively regular if and only if ( aA / Aa ) is generated as a principal ( right / left ) ideal by some idempotent. (e ) Any one-sided ideal which contains some weakly relatively regular elem ent contain s a non-zero idempotent. ( f ) An algebra is weakly relatively regular if and only if every non-zero { right / left ) ideal contains a non-zero idempotent. Proof (a): Multiplying ( aba = a / bab — b ) first on one-side and then on the other by ( b / a } gives these results. (b): Consider (bab)a(bab) = bab. (c): Immediate from (a). (d): Consider the case in which a A has the form p A for some idempotent p. Since p = p2 is in pA, we can write p = ab for some b G A. For any c G A , ac G aA can be written as ac = jkic = abac, so a — aba — 0 under our hypothesis. The other case is similar. (e): Suppose a belongs to a ( right ideal TL j left ideal C ) and satisfies bab = b. Then ( p = ab / q = ba ) belongs to {71 / C ). (f): Result (e) shows that any non-zero one-sided ideal contains a non zero idempotent. Suppose the { right ideal aA / left ideal Aa ) contains the non-zero idempotent { p = ab / q — ba). Then a is weakly relatively regular by the equation given in the proof of (b) above. □ Note that the conditions on the algebra A in (d ) are satisfied if A is unital, semiprime (Definition 4.4.1) or approximately unital (Definition 5.1.1). Von Neumann’s interest in (d) stemmed from the elementary fact that a one-sided ideal I in a unital algebra A is complemented (in the sense that there exists another one-sided ideal J of the same type satisfying 1V\J ~ {0 } and 1 + J = A ) if and only if I is the principal one-sided ideal generated by an idempotent. (In particular if I = pA, then J = (1 — p)A, and the idempotent p is uniquely determined by I and J .) We conclude this brief discussion by quoting Kaplansky’s [1948d] main result without proof. For a proof and an accessible survey of related results see Seiwyn R. Caradus [1978]. See also Harro G. Heuser [1982]. 2.1.18 T h e o re m A relatively regular Banach algebra is finite-dimensional. Other notions of generalized inverses are included in S. T. M. Ackermans and A. M. H. Gerards [1982], Harte [1987], Ackermans [1982], Pietro Aiena [1983], Vladimir Rakocevic [1988] and Danrun Huang [1992].
2.1.19
Definition of the Spectrum
207
Weak Invertibility and the Strong Spectrum The following ideas have recently been introduced by Michael J. Meyer [1992d]. The definition is motivated by the observation that an element a in a unital algebra A is invertible exactly when A is both the smallest left ideal and the smallest right ideal containing a. 2.1.19 D e fin itio n Let A be a unital algebra. An element a £ A is called weakly invertible if A is the smallest ideal including a. We denote the set o f weakly invertible elements in A by A wg - The strong spectrum o f a £ A is the subset SpA (a) = Sps (a) o f C consisting o f all A such that A — a is not weakly invertible. Th e strong spectral radius is defined by Pa ( ° ) = PS( a) = sup(|A| : A £ S f ^( a) } . If A is not necessarily unital, for a € ^ l w e define SpA (a) and pA (a) to be the strong spectrum SpA,( a) and strong spectral radius p^ (a) of a in the unitization. Clearly a £ A is weakly invertible if and only if there exist finite subsets { 61, 62, . . . , 6n} and {c i,C 2, ... , c „ } o f A satisfying 1 = Y *j=i bjOCj. A weak analogue of quasi-invertibility will be introduced in Section 4.5. 2.1.20 P r o p o s itio n Let A be a unital algebra and let be the set of all maximal proper ideals of A. (a) The set A u,g of weakly invertible elements includes all one- or twosided invertible elements so all a £ A satisfy Sps (a) C Sp(a). It is the complement of [JM e =A M . Hence each element a £ A satisfies Sps (a) = {A €
bj cj is an inverse.
208
2: The Spectrum
2.2.1
(c): The equations 1 = bj(ab)cj = 2 " = i bja(bcj) 1 = Y lj=i (bja)bCj prove the first statement. Considering the first case in the second sentence with c as a left inverse for a, we get 1 = £ j = i bjbcj = £ " = i ( 6j c) ( a6) cr (d): Any a
G
A wq satisfies 1 =
bjacj and hence p(a) satisfies
1 = E " = l V >(bj)
(e): Given a, f and A G Sps (a), (a) guarantees a maximal proper ideal M containing A — a. Let p: A —* A / M be the natural map. Then /(A ) = } ( p ( a ) ) = p ( f ( a ) ) is clear and establishes the positive result. Meyer [1992d] contains a simple counterexample to equality. □ If pm '- -A —i>A / M is the natural map, the equation in (a) can be written Sps (a) — [ p M { a) G C : M G S ^ }. In Section 3.3 we will establish a meaning for f ( a ) for larger classes o f functions / when a belongs to certain special types of algebras. The proof of (e) above remains valid in this wider setting. If { 0 } is the only proper ideal o f A, then A wg is just A\ {0 }. Similarly, if A (like B ( H ) ) has a unique largest ideal J then A wq is A \ 1 . In both cases many elements will have empty strong spectrum. On the other extreme, if A is commutative, (b) shows A wg = A g - For the algebra A — C(fl, B) of continuous functions from a compact set fl into an algebra B, A wq is the collection of functions / satisfying f(oj) G Bwq for all u> G Q. For additional results see 2.2.17 and 4.5.7.
2.2
Spectral Semi-norms
In this section we introduce the most important type of algebra semi norms, namely spectral semi-norms. They are characterized by a large number of interesting equivalent properties which show their intimate con nection with the algebraic structure. In the following section we will show that in certain well-behaved algebras all examples o f spectral semi-norms are related. Properties of the Spectral Radius We need the following results on the spectral radius repeatedly. 2.2.1 P r o p o s itio n Let A be an algebra. (a) Any a 6 A and any A € C satisfy p(Xa) = |A|p{a)\
p(an) = p(a)n;
p{ 1) = 1
(1)
where we are assuming that A is unital in the third case. (b) Any a,b G A satisfy: p(ab) = p(ba)
(2)
Spectral Semi-norms
209
unless one product has spectrum { 0 } and the other has empty spectrum. (c) Let A and B be algebras and let tp:A —+ B be a homomorphism. The.. any element a e A with nonempty spectrum satisfies PB(
+ b) : b e ker(v?)} < p^t(a).
(3)
(■i) If B is a subalgebra of A, then any b e B with nonempty spectrum in B satisfies PA(b) < PB(b)• (i.) If A is not unital, then any A + a G *4* satisfies p(A + a) < |A| + p(a) < 3p(A + a). Proof (a): These results all follow from the spectral mapping theorem or even more elementary considerations. (b): Proposition 2.1.8(a) shows this. (c) and (d): The corresponding results in Proposition 2.1.8 give these. (e): Prom the spectral mapping theorem (or the definition), we see Sp(A + a) = A + Sp(a) in A 1. (Hence if either spectrum is unbounded, they both are and the inequality holds in a trivial way.) Since A is not units., 0 belongs to Sp(a). (Hence Sp(a) is not empty, and Sp(A + a) is not empty either since the unitization of a nonunital algebra cannot be a division algebra.) Thus we see |A| < p(A + a) < |A| + p(a). For any e > 0, we ce i find a /x € Sp(a) with absolute value larger than p(a) — e. Hence we get p(a) —e < |/z| < |A + /*| + |A| < p(A + a ) + |A|. The result follows by comb:ning the inequalities and letting e approach 0. Q The reader might compare this result with Theorem 1.1.10. In spec tral algebras, introduced below, the spectral radius has all the properties described in that theorem. It appears reasonable that any two elements in an arbitrary algebra satisfy equation (2), but we have neither been able to rule out the possible restriction mentioned in the statement, nor have we been able to construct an algebra in which this actually occurs. Hence we do not know whether this equation is true in complete generality. A ll elements in algebras with a nontrivial semi-norm satisfy equation (2), as the Gelfand-Mazur theorem (Corollary 2.2.3) will show. Let us restate inequality (3) above in a slightly different form involving an ideal I of A: P A / i ( a + £ ) < *n f{p ^ (a + b) : b e 1 }
V a G A.
(4)
In this form one may study the class of Banach algebras for which this iauqu; lity becomes an equality for all closed ideals J and all elements a e A . This l as been examined by M. R. F. Smyth and Trevor T. West [1975],
2: The Spectrum
210
2.2.2
Gert K. Pedersen [1976] and Gerald J. Murphy and West [1979]. Algebras with this property are called SR-algebras. The last paper shows that C*algebras ( cf. Volume II), commutative algebras A for which the algebra A of Gelfand transforms is dense in C0(T_4 ) (cf. Section 3.1) and Riesz algebras ( cf Chapter 7) are SR-algebras, but that the disc algebra (§1.5.2) is not. It seems likely that this property is fairly common. The Spectral Radius and Algebra Norms The following extremely important result was proved for the algebra of absolutely convergent Fourier series by A. Beurling [1938] and for normed algebras by Gelfand [1941a]. The elementary proof which we give is due to Rickart [1958], Theorem 2.2.5 below shows a case with equality rather than the inequality obtained here. Recall that an algebra semi-norm a on aji algebra A is nontrivial unless A is unital and cr is identically 0. Recall also that if a is an algebra semi-norm, then
2.2.2 T h e o r e m Let a be a nontrivial algebra semi-norm on an algebra A. Then any element a in A has nonempty spectrum and the spectral radius satisfies
< a00(a )a °°(a ~ 1)
so
2.2.3
Spectral Semi-norms
211
Thus (A a)n is quasi-invertible. Define bj 6 A by bj = —[A_,a + (A_,a)2 + • ■■+ (Aj-a)"-1 ], so that for all j it satisfies (A a)n = (Ajd) o bj. By noting Y ^ j-i 0 = 0 and rearranging the double sum we also note £>i + 62 + • • • + bn = 0. Since (A a )" and \j(i are quasi-invertible, we can write f ( Xj ) = bj o ((A a)n) q. Thus equation (4) before Proposition 2.1.2 gives
« _ 1E / ( ^ ) j =1
= =
n - ((f : ^ )o ((A a )T + (n - l)((A a )T ) ' j =1
'
((A a )T .
By uniform continuity, for any positive e we can find a number t close to 1 in the open interval ] 0 , 1 [ for which cr(f(tj) — / (C j)) < £ holds for all n € N and all j = 1 ,2 ,.. . , n . Substituting t and 1 for A in the last displayed equation gives
2: The Spectrum
212
2.2.4
algebra has a ( left / right ) inverse.) The isomorphism is implemented by a •—> A where A is the unique complex number in Sp(a). Proof Suppose every non-zero element of a unital algebra A has a left inverse. Let b be the left inverse of a. By assumption, b has a left inverse which must be a. Thus b and a are invertible and A is a division algebra. The theorem shows that any element a in an algebra with a nontrivial algebra semi-norm has nonempty spectrum. I f A € Sp(a), then AI — a is 0 since it is not invertible. Thus the map a — AI t—> A is a well-defined isomorphism. □ Spectral semi-norms We now come to one o f the most basic concepts of this work: spectral semi-norms. The next theorem will show numerous conditions which are equivalent for an algebra semi-norm. Logically it does not matter which of these is chosen as the definition of a spectral semi-norm. We will select one which is easy to remember and to check. We postpone historical remarks until after the theorem. 2.2.4 D e fin itio n A spectral semi-norm on an algebra A is an algebra semi-norm
V o € A.
If a spectral semi-norm is actually a norm, then we call it a spectral norm. An algebra together with a particular spectral semi-norm or spectral norm is called a spectral semi-normed algebra or a spectral normed algebra. 2.2.5 T h e o r e m The following are equivalent for an algebra semi-norm a on an algebra A. (a) a is a spectral semi-norm. (b) For a £ A, c r ( a ) < 1 implies that a is quasi-invertible in A. (c) The set of quasi-invertible elements of A has nonempty interior with respect to a. (d) The set of quasi-invertible elements of A is open with respect to cr. (e) There is a finite constant C satisfying p(a) < Ccr(a)
V a € A.
( f ) a satisfies p(a) = lim cr(an) l/n fl—*00
V a g A.
I f A is unital, the quasi-group may be replaced by the group in conditions (c) and (d).
2.2.6
Spectral Semi-norms
213
Prcoj (a) => (b): Immediate since p(a) < 1 implies 1 ^ Sp(a). (b) => (c): Zero is in the interior o f AqG(c) => (d): Let c 6 A qG be arbitrary, and let d 6 AqG be an interior point o f AqG- It is enough to show that c is an interior point. Define a map L: A —> A by L( a) = d o cq o a. Clearly L is continuous and maps both c onto d and A gG onto AqG- Hence AqG — L ~ 1( Aq c ) is a neighborhood o f c. (d) => (e): Since 0 is quasi-invertible, there is some e > 0 such that cr(a) < e implies a 6 AqG- Hence cr(a) < e implies A-1 a 6 AqG for all A 6 C with |A| > 1, which in turn implies p(a) < 1. Hence we may choose C tofbe e-1 . (e) => (f): Proposition 2.2.1(a) established p(an) = p(a)n for all a 6 A and n 6 N. Thus (e) implies p(a) = p(an) ltn < C'1/ V ( a n) 1/n
V a 6 A.
Conit ining this with Theorem 2.2.2 gives the desired result. (f ) => (a): Immediate since limcr(an) 1/,n < cr(a). Ta e final remark is obvious.
Q
S; ace 1 is never quasi-invertible, every spectral semi-norm is a nontriv ial algebra semi-norm. Algebra norms with property (d) were first studied by Kaplansky [1947a], who called algebras provided with such a norm, Q-algebras. (He actually considered rings which were not necessarily alge bras.) Ernest A. Michael [1952] and Bertram Yood [1958] first noted the equivalence o f several o f these properties. G iher characterizations o f spectral semi-norms (or norms) will be given below in Theorems 2.2.14 and 2.5.7, Propositions 2.2.7 and 2.5.15 and Coro’lary 2.4.'8. Many examples o f spectral semi-norms are known and wiii t e discussed in this work. Most were listed in the introduction to this chapter. In a spectral semi-normed algebra (A , cr), an element a has spectrum { 0 } if and only it satisfies lim cr(an) 1/,n = 0. We have already called such n—*oo elements topologically nilpotent. We now introduce the symbol AtN for the class of such elements in any spectral algebra A. 2.2.C A N o n -s p e c tra l A lg e b r a N o r m Let A be the disc algebra: the algebra under pointwise operations of functions continuous on the closed unit iliac in C and holornorphic on its interior D. Let S be any subset of D with a limit point in D. Then ||• ||s defined by ||/||g = sup{|/(A)| : A 6 5 } is a norm on A. It is not spectral unless the closure of S includes the bound ary o f the disc. Note that in this example the algebra is a commutative, semisimple (Definition 2.3.2) Banach algebra and thus non-spectral norms can ^ :cur on quite well-behaved algebras. Nevertheless, Proposition 2.5.16 and rxamples (b ), (c), (d), (e) and (h) in the introduction show a number of ca.,es in which every norm on an algebra is spectral.
214
2: The Spectrum
2.2.7
Geometric Series and Spectral Semi-norms When dealing with algebra norms rather than semi-norms, Fuster and Marquina [1984] noted the following equivalence, which can be stated infor mally as: An algebra norm is spectral if and only if all the geometric series which should converge have limits. We remark that this result shows a connection between spectral semi-normed algebras and the more restrictive notion of functional algebras which we will introduce in the next chapter after the holomorphic functional calculus is discussed. 2.2.T P r o p o s itio n The following conditions are equivalent for an algebra norm a on an algebra A. (&) a is a spectral norm. (b ) If the sequence
a( a9 + £
N
a” )
=
v ( ( l ~ aq) ( l - a)(aq + £ a n))
<
(1 + o( aq) ) o ( a 9 — aaq + a — aw+1)
=
(1 + o( aq) ) o ( a N+1).
n —1
n —1
(b ) => (c) => (d): Immediate. (d ) =* (a): If the series converges, then OO
OO
oo
oo
a° ( - E a") = ( - £ a") ° a = a - £ a” + £ «n+1 = °> n —1
n=l
n=l
n=l
so a is quasi-invertible, verifying (b ) o f the previous theorem.
D
Note that the proof o f the implication (a) => (b) remains valid when
^ ( « ° ( - £ a n) ) = °n=l
Since the set of elements on which any spectral semi-norm vanishes is included in the Jacobson radical (Corollary 2.3.4), this implies that a o
2.2.8
Spectral Semi-norms
215
(° n) *s quasi-invertible when a is a spectral semi-norm. Hence the second condition of Proposition 2.2.7 could be included in Theorem 2.2.5 by restating it as: For a 6 A,
2.2.8 C o ro lla ry Any complete algebra norm is a spectral norm. Thus the norm of a Banach algebra is a spectral norm. In particular, the spectrum of any element in a Banach algebra is nonempty closed and bounded, hence compact and the spectral radius satisfies Theorem 2.2.5(f). Any norm on a finite-dimensional algebra is complete and hence is spec tral if it is an algebra norm. For a strong converse see Peter D. Johnson, Jr. [1978] and Robert Grone and Johnson [1982]. Proposition 2.2.7 shows that geometric series play a special role with respect to spectral semi-norms. For future reference, it is useful to list the various geometric series which are commonly used. W e do this in the next proposition.
2.2.9 P r o p o s itio n Let (A ,
or
a°°(aq o b ) < l
implies b 6 A*,g and OO
&« =
a« - ]jn (a « o & )n ( l - a«). n=l
Also, a(aq o b) < 1 implies a(bq ~ a9) < [1 — a(aq o 6)]_1 [1 + £r(a?)]2tr(& — a). (b ) I f A is unital, for any a 6 A g and b £ A , either
or
a ° ° (l - a~lb) < 1
implies b € A g and OO
£>_1 = 5 ^ (1 — a~1b)na~1. n—0
Also, cr( 1 — a~1b) < 1 implies
216
2: The Spectrum
2.2.9
(c) For any a e A , n 6 C \ {Sp(a) U { 0 } ) and A 6
( A - ‘ a )« = M A - M )-1 £ ( A_1( A - /')0*~1® ) T
and
n= 1
( A - a ) -1
5 Z ( / i - A )n( / i - a ) ~ (n+1),
and
T l— 0
a ((A - a ) ~ l - ( n - a ) ~ l) <
—
— A|
(e) For any a 6 >1 and any A 6 C, |A| > ff°°(a ) implies A ^ Sp(a) and OO
( A - 1a )« = - ^ ( A - 1a )nn=l
Also, |A| > a(a) implies
( A - a ) " 1 = ] T A - (n+1>an. n=Q
Also, |A| > a(a) implies a ( ( \ - a ) " 1) < lA l- 'd - |A|-1cr(a))_1.
Proof The idea of the proof is already given in the proof of (a) => (b) in Proposition 2.2.7. Note that (c) and (d) are special cases o f (a) and (b), respectively. For (a), prove the simple identities (1 — a)aq = —a, (1 — a)(aq o b) = b — a and aq o b = (1 — aq)(b — a). The identity (fi~1a)q o A-1a = (A — ii)\~l ( n ~ 1a)9 is useful in deriving (c) from (a). □ Analyticity of the Resolvent Function The last proposition shows that even though a spectral normed algebra need not be complete, it does have a rich supply o f convergent geometric series. We wish to use this to prove one of the basic properties of the
‘
'
Spectral Semi-norms
217
resolvent function A ( A - a ) -1 6 A 1 defined on the resolvent setC\Sp(a) for any element a in a spectral normed algebra. 2.2.10 D e fin itio n Let ( X, || • ||) be a normed linear space and let U be ar open (not necessarily connected) subset o f C. A function f : U X is said to be { weakly analytic / analytic ) on 17 if { for each u> € X * the funcfon uiof: U —> C is analytic on U / for each \ q E U there is a sequence { i „ } . , 6No C X and a positive e satisfying OO
/(A ) = £ ( A - A0) na:n
V A such that |A — Ao| < £ )•
T l= 0
If the complement of U is bounded, a function f : U —* X is said to be ana lytic at infinity if there is a positive constant R and a sequence { x n}neN C X satisfying OO
/(A ) = E
X~nxn
V A such that |A| > R.
n=1
We insert the obvious result called for by this definition, even though it is no: needed for Corollary 2.2.12 which depends on Proposition 2.2.9. 2.2. ? 1 T h e o re m Let X be a Banach space and let U be an open (not necetsarily connected) subset of C. A function f : U —> X is analytic on U if nr.' only if it is weakly analytic on U. Proo. Analyticity clearly implies weak analyticity, so let us assume that / is we -Jdy analytic on U and Ao 6 U. Choose e > 0 so that the closed disc of r:„ iius e around Ao is included in U . Let T be the positively oriented boundary of this disc and let D be the open disc of radius e/2 centered at A0. Consider the formula 1
( / (A l) — /(Aq) _ /(A 2) - / ( A q ) \
Ai — A2 \
Aj — Ao
A2 — Ao
= JL f 271i
/
f W dX
)
Jr (A - A0)(A - A ,)(A - A2 h
If we knew that / was continuous so that the right side existed as a Riemann integral, then this equation would follow from weak analyticity and the Hahr. Banach theorem for any Aj, A2 6 D. Since the norm o f the right side is bounded above by esup{||/(A)|| : A G r}/ (2 / e )3, the left side is bounded pr’jv' lg the existence of
in the Banach space X .
218
2: The Spectrum
2.2.12
In order to avoid assuming the continuity of / or the existence and boundedness of the right side o f (5), we replace / by w o / for an arbitrary ui 6 X * throughout (5). Both sides exist and are equal, and the right side is bounded by the same expression as before with sup{||/(A)|| : A 6 T } replaced by 8up{||wo/(A)|| : A 6 T }. By the uniform boundedness principle, the original right side is bounded in absolute value by some constant. Hence / is not only continuous but actually differentiable.
The definition of weak analyticity makes it clear that /' is also weakly analytic. Hence the argument can be repeated. For higher derivatives, the usual Cauchy formula and the Hahn-Banach theorem imply
i s
/ <
The obvious estimate then shows that / (A ) = converges for |A — Ao| < r.
~ Ao)n/ ^ (A o )/ n ! □
2.2.12 C orollary Let A be a spectral normed algebra. For any a € A, the resolvent function (A - a ) ' 1€ A 1
V A e C \ S p (a )
is analytic on the resolvent set C \ Sp(a) and at infinity. Proof Results (d ) and ( f ) in Proposition 2.2.9 give this.
□
This result will be exploited in the next chapter, where we will show how to use contour integrals in the resolvent set o f elements in a Banach algebra to define a functional calculus involving analytic functions. For more details on vector valued analytic functions, see Hille-Phillips [1957] pp. 92-116. We do not need anything beyond Corollary 2.2.12 for the following result which Bonsall and Duncan [1973b] call the abstract Runge theorem. In the next chapter we will use it to prove the classical approximation theorem of
Runge. 2.2.13 Theorem Let A b e a unital Banach algebra. Let a be an element of A , and let B be a closed unital subalgebra of A which contains a. Then the set Spg(a) \ SpA (a) is a (possibly empty) union of bounded components o f the A-resolvent set of a. I f S contains at least one point from each bounded component of the A-resolvent set o f a and B is the closed unital subalgebra generated by a and {(A —a )-1 : A 6 S }, then the resolvent sets of A with respect to A and B are equal. Proof Let S be the resolvent set of a relative to B, which can also be described as the set {A € C \ SpA (a) : (A - a) 1 € B). Proposition 2.2.9(d)
2.2.14
Spectral Semi-norms
219
shows that S is open. Let u> be an arbitrary element of A " which vanishes on {(A - a )-1 : A 6 S }. I f A 6 C \ 5 p ^ (a ) can be approximated by elements o f 5, the expansion o f Proposition 2.2.9(d) shows that w must also vanish on (A — a )-1 . Since was arbitrary, the Hahn-Banach theorem shows that S is also a relatively closed subset o f the ^4-resolvent set C \ Sp^(a) and hence must be a union o f components. Proposition 2.2.9(f) shows that S includes every complex number with absolute value greater than the spectral radius o f a, so it contains the unbounded component o f the resolvent set o f a in A. Hence the two resolvent sets can only differ by some collection o f bounded components o f C \ Sp^(a). The second statement is now immediate. □ The above result shows that when the spectrum o f an element is changed by calculating it in smaller closed unital subalgebras, it increases only by adding a whole bounded component o f the resolvent set all at once. In the other direction, if the spectrum o f an element is changed by calculating it in a larger algebra in which the original algebra is embedded unitally and isometrically, the theorem shows that the boundary of the spectrum can only decrease by losing the whole boundary o f a bounded component all at once. Thus for 1, a 6 B C A we have dSpe(a) C dSpA(a) C 5 p ^ (a ) C S pg(a). We will extend this observation in Corollary 2.5.8, which is based on an entirely different argument. Stability of Spectral Semi-norms Next we mention the stability properties of spectral semi-norms. We have collected these results together for convenience o f reference even though they involve two definitions which will not be given formally until later in this chapter. Spectral subalgebras and the Jacobson radical are defined in the introduction and more formally in Definitions 2.5.1 and 2.3.3, respec tively. The introduction and Proposition 2.5.3 give long lists o f spectral subalgebras. Properties (d) and (e) are due to Kaplansky [1948b] and (h) is due to Richard G. Swan [1977]. As we have noted, there are many ways to extend an algebra norm from A to A 1 if A is not unital, but all the reasonable ones satisfy condition (g) in the following theorem. The situation is similar for matrix algebras: the condition in (h) is satisfied by all reasonable extensions o f an algebra norm. 2.2.14 T h e o re m The following are equivalent for an algebra semi-norm cr on an algebra A (a ) cr is spectral. (b) The restriction of a to each spectral subalgebra is spectral.
220
2: The Spectrum
2.2.14
(c) The restriction of a to each maximal commutative subalgebra is spec tral. (d ) The quotient semi-norm induced by a on A /X is spectral for each ideal J of A. (e) There is some ideal J for which both a and its quotient semi-norm are spectral on 1 and A/ X, respectively. (When this condition holds for some ideal it holds for all ideals, of course.) ( f ) The quotient norm induced by a on A / A j is spectral, where A j is the Jacobson radical. (g ) Any extension cr1 of a to be an algebra semi-norm on A 1 in which A is closed in .41 is a spectral semi-norm. (h) Any extension of the semi-norm to be an algebra semi-norm on the n x n-matrix algebra over A is spectral if the topology of this extension is that defined by the linear space semi-norm m ax{cr(ajj) : 1 < i, j < n }. (i) cr is equivalent to a spectral semi-norm. Proof (a ) => (b): Since the non-zero spectrum of an element in a spectral subalgebra is the same whether calculated in the subalgebra or in the full algebra, this is immediate from the definition. (b) => (c): Maximal commutative subalgebras are spectral subalgebras by Proposition 2.5.3(b). (c) =>• (a): Each element in the algebra belongs to some maximal com mutative subalgebra, so this also follows from the definition o f a spectral semi-norm. (a ) => (d): The inequality PA/ i i a + X) < PA( a) for all a 6 A from Proposition 2.2.1(c) and '.he definition give this. (d ) =>• ( f ) => (e): Immediate since A j is always spectral. (e) => (a): It is enough to show a(a) < 1 / 3 implies that a is quasiinvertible in A. This inequality implies
Spectral Semi-norms
2.2.1
221
a = (c.ij) be a matrix in li. Then, in particular, by Theorem 2.2.5(b) all its elements are quasi-invertible and the elements satisfy <7 ( 0*1 (1 —a n ) _1a ij) < 2~n. Multiply the matrix I — a on the right by all the elementary matrices which differ from the identity matrix by the insertion o f (1 — a l i ) ~ 1a ij in the 1j position for j > 1. Multiply on the left by all the elementary matrices which differ from the identity matrix by the insertion of <m (l a a ) -1 in the t l position for * > 1. The result will be that there are only zeroes in the first row and column except in the 11 position and that the remai. ling non-zero elements are perturbed by addition o f elements o f the form -,u (l ~ a n ) ~ la ij and thus satisfy cr(6tJ) < 2~n+1. Hence the process can b i continued inductively until J — a has been shown to be similar to a mat :ix o f the form I — b where b is diagonal with each diagonal entry quasi-:nvertible. (h ' => (a): W e can extend a to the matrix algebra as the HilbertSchmidt semi-norm
^(flit
Qij&jj
where k always has the other value from i in {0 ,1 } and 6 is positive if i — j 'nd negative otherwise. A ll the inverses written exist if the matrix is close enough to the identity. Thus M 2{ A ) also has an open set o f invertible elements and inversion is continuous. The method works for arbitrary size matrices, but we stop after giving the result for 3 x 3 matrices. Use the same lotation except this time we assume that {», m, n } is the set {1 ,2 ,3 } in sone order and that m differs from both t and j if they are distinct. Thai?
222
2: The Spectrum btj
— ^[o»j "t- (&tn&nn^nm
&im)(^mm
2.2.15 ^mn^nn^nm)
_ a *j)(®nn — ^nm^mm^ron)
drill
^ni]
( a im d m m Umj — a , j ) ( d n n ~ a nm n m m a m n )
Continuity o f the Spectrum The next result, which is due to John D. Newburgh [1951], shows that for a spectral semi-normed algebra A , Sp(a) C C is an upper semi-continuous function of a e A when we use the Hausdorff metric on the subsets of C. Example 2.2.16 shows that there exists a sequence of nilpotent (hence topologically nilpotent) elements in a Banach algebra which converges to an element with spectrum larger than {0}. However, Corollary 3.1.6 shows that Sp(a) C C is a continuous function of a 6 A when A is also commuta tive. For further results see Newburgh [1951], Aupetit [1975] and [1979a], Gerard J. Murphy [1981] and Laura Burlando [1986], [1990] and [1991]. The last paper gives a criterion for the spectrum to be continuous at an element a in a Banach algebra. The condition is much weaker than Newburgh’s sufficient condition that Sp(a) be totally disconnected. (The notation eCi used in the theorem denotes the closed disk o f radius e in accordance with our convention on closed balls in normed linear spaces.)
2.2.15 Theorem Let (A , cr) be a spectral semi-normed algebra. For any a e A and any e > 0, there is a 6 > 0 so that fo r all b 6 A , cr(a — b) < 6 implies Sp(b) C Sp(a) + eCi and p(b) < p(a) + e. Hence the spectrum and spectral radius are upper semi-continuous func tions. Proof Suppose the inclusion fails for a given a e A and e > 0. We can find a sequence {a„}neN Q A converging to a and a sequence {An} n6N Q C satisfying A„ e Sp(on) \ (Sp(a) + eC x). Since {a „ : n 6 N} is bounded, {A „ : rt 6 N] is bounded. Hence (by taking a subsequence if necessary) we may assume {An} neN converges to some A ^ Sp(a). If A is 0, then A is unital and a is invertible. Since An — a„ converges to —a, An — an is eventually invertible, contradicting the choice o f A„. If A is not 0, then A-1 a is quasi-invertible. Eventually A„ is non-zero and since A "1^ converges to A-1 a, A^'a* is eventually quasi-invertible, contradicting the choice of A„ again. Hence the inclusion is proved by contradiction. The inequality
follows immediately. 2.2.16 Example
□
This example is due to Kaplansky but was first pub lished by Rickart [I960]. It shows that the results of Newburgh’s theorem are restricted to semicontinuity and cannot be extended to give continuity.
2.2.17
Spectral Semi-norms
223
We produce a sequence {T „ }neN of nilpotent operators which converges to an operator having spectrum different from {0}. Let H be £2(N) with {z n}neN denoting the natural orthonormal basis. The operators which we consider are all weighted shift operators defined by S{zm)
SmZm+1 )7l £ N
for some bounded sequence {s m}meN o f weights. The norm of such an operator is easily seen to be ||S|| =sup{|sm|: m 6 N}. For each m, n € N, define tm and tmn to be e~k and (1 —6nk ) e ~ k, respec tively, where k is defined by m = 2fc(2j + 1). Define T and Tn by n) = ^m^m+1 and 'Tn{zm) — t m n Zm + 1
171 G N.
Since Tn - T satisfies (Tn - T ){zm) — 6nke~ kzm+i for all n, m e N and for k as defined previously, we see that the sequence {Tn} neN does converge to T. It is easy to see that each Tn satisfies T^"+1 = 0, since 2n+1 shifts of the index will send any basis vector zm past an index of the form by 2n(2j + 1) where tmn is 0. Hence each Tn is nilpotent and satisfies Sp(Tn) = {0 } and p (T n )
= 0.
Finally, we show that T satisfies p(T) > e 1 and hence has non-zero spectrum. The norm o f T ” is given by ||T"|| = s u p { fc+n _ 1 <m : f c e N } . m =k
For n = 2s — 1, the definition of the weights gives n tm = 3n e x p (-k 2 j - k~1)
m =l
fc=l
and hence ||r” ||1/B > e x p ( - £ - A T). fc=i
Since the right-hand side converges to e~ 1 with increasing j (or n), Corol lary 2.2.8 gives the desired result. 2.2.17 W eak ly Spectral Norms The strong spectrum 5ps (a) and strong spectral radius ps (a) of an element a in an algebra A were introduced in Definition 2.1.19. Meyer [1992d] defined an algebra norm ||•||on A to be weakly spectral if the set Au,g ° f weakly invertible elements is open relative to it. We note some of the parallels with spectral norms.
224
2: The Spectrum
2.3.1
Proposition
Let ||•|| be an algebra norm on a unital algebra A. (a) //H II is spectral, it is weakly spectral. The follomng conditions are equivalent: (b) The norm ||•|| is weakly spectral. (c) All a 6 A satisfy ps(a) < INI so Sps (a) is compact. (d) The identity element 1 is in the interior o f A wg (e) Every maximal ideal of A is closed. (f) For any a e A , ||1 —a|| < 1 implies a £ A wg -
We omit the proofs which are similar to the spectral case. In a nonunital algebra A , the definition and proposition can be modified using the set AwqG o f weakly quasi-invertible elements in place of A wg - (This set is defined in Section 4.5.) In the nonunital case, (e) refers to all maximal modular ideals and 0 replaces 1 in (d) and (f).
2.3
The Jacobson Radical and the Fundamental Theorem
We begin by showing that spectral semi-norms are not unique even on the best behaved algebras. Our next major goal is to prove that if a sequence converges to two possibly different limits in two different spec tral semi-norms, these limits differ by an element of the Jacobson radical. Hence we introduce a preliminary treatment of the Jacobson radical. A few remarks about representations are needed in order to define this concept. Representations and the Jacobson radical will be discussed much more fully in Chapter 4, where all of the current arguments will be elaborated and the reader will find historical remarks.
2.3.1 Spectral Semi-norms are not Unique This is true even in uni tal, semisimple commutative Banach algebras. Consider £l (Z) with convo lution multiplication. Both p and ||•||, are spectral norms. Another easy noncommutative example is the finite-rank operators on a Banach space. In Section 1.7 we described several norms on this set, in addition to the op erator norm. Theorem 2.2.14(b) shows that all of these norms are spectral since the finite-rank operators form an ideal in the various completions. Despite these examples, Theorem 2.3.6 below shows that if a sequence converges in two different spectral norms, then the two limits differ by an element of the Jacobson radical (cf. Definition 2.3.2 and Theorem 2.3.3). Hence in semisimple algebras such limits are unique. Introduction to the Jacobson Radical
2.3.2 Definition A representation of an algebra A on a linear space X is an algebra homomorphism T from A into the algebra C( X) of all linear
2.3.3
The Jacobson Radical and the Fundamental Theorem
225
map.-; (also called operators) of X into itself. For o e A , we write Ta for the operator associated with a. A subspace of X is said to be invariant (or 'I'-ir, variant in cases where the representation might not be clear) if every rep- senting operator Ta maps y into itself. A representation is called irreducible if it has some non-zero representing operators, and X is its only non-zero invariant subspace. The Jacobson radical A j of an algebra A is the atersection of the kernels of all the irreducible representations of A. An cJgebra is called semisimple if A j is {0 } and Jacobson-radical if A j is the •/hole algebra (i.e., if there are no irreducible representations at all). I■’>. order to prove the next theorem we need to know that if T is an irreducible representation of A on X and z is a non-zero vector in X , then any vector x in X can be written in the form Taz for a suitable a e A. To see *his, note that T^z = {Taz : a e A } is clearly an invariant subspace. If it were the zero subspace, then the set of all w for which T^v> equals 0 wou’ d be a non-zero invariant subspace and hence equal to X . Since this woi’ H contradict the existence of a non-zero representing operator, T^z must be all of X for each non-zero z, which gives the desired result.
2.3.3 Theorem In any algebra A the following sets are equal. (a) The Jacobson radical A j o f A . (b) {b e A : p(ab) = 0 for all a e A }. (c) The largest ideal J satisfying any (hence all) o f the following equiv alent properties: SpA/x(a +
£ SPA(a) Q SpA/x(a + 2 ) U {0 }
V a e A",
AqG = {a e A '. a + 1 e (A/X)qG}\ Sp(a + b) = Sp(a) Sp(b) = {0 } p(a + b) = p(a) p(b) = 0
V a e A\ b 6 Z; V b 6 Z; V a 6 A] b e l ; V b e 1.
Proof Let us call the conditions in (c), (cj), (C 2 ), (C 3 ), (C 4 ), (C 5 ) and (ce), respectively. Note (c4)«=>(ce). All ideals satisfy the first inclusion in (ci). The definition of the non-zero elements in the spectrum shows (C])«=>(c2). The following implications are also easy: (c3)=>(c4), (c3)=>(cs), (c4)=>(ce) and (c5)=>(ce). Furthermore (cx) almost implies (C 3 ). Since we may choose an arbitrary representative of the coset a + J in (ci), it is obvious that the spectra of a and a + b differ by at most zero in ( C 3 ) . We will temporarily denote the set in (b) by B. Suppose J is an ideal (not necessarily the largest) with the property described in ( c j and let Z be the largest ideal satisfying (c6). Then, in
226
2: The Spectrum
2.3.4
particular, each element of b G J satisfies SpA {b) = {0 }.
Hence J is
included in J which is included in B (since J is an ideal). We will show that any element b in B is in the kernel of every irreducible representation, which implies J C 1 C B C A j . Suppose b 6 B is not in the kernel of some irreducible representation T -.A -* C(X). Then we can choose 2 € A' so that T^z / 0. By irreducibility we may choose a 6 A satisfying TaTtz = z. Since p(ab) is 0, we may find a quasi-inverse c for ab. But this gives the contradiction: z — z + Tcz —Tcz = Tabz + Tcz —TcTa\,z = Tqz = 0.
We will complete the proof by showing that the Jacobson radical has the property o f J described in (c2) and satisfies (C3) even at zero. In order to do the former it is enough to show that any element which is quasi-invertible modulo the Jacobson radical is quasi-invertible. We will begin by showing that such an element e has a left quasi-inverse. Suppose not. Then e does not belong to the left ideal {ae — a : a £ A ) denoted by A (e - 1). Hence we may use Zorn’s lemma to find a left ideal M which is maximal (under the inclusion order) and includes A (e - 1) but does not contain e. Consider the natural action of A on A / M defined by Ta(b + M ) = a b + M
Vb€M .
The maximality of M shows that this is an irreducible representation. (Any left ideal properly including M would have to contain {e } and thus be the whole algebra A ) Hence the Jacobson radical is included in the kernel of T. Furthermore any element b of this kernel belongs to M since T b(e+ M ) = M implies b = be —(be —b) 6 M . Since we assumed that e was quasi-invertible modulo the Jacobson radical, there is an element c e A with e + c —ce in the Jacobson radical and hence in M . But this implies that e is in M contrary to the construction of M . We have now shown that each element e & A which becomes quasiinvertible in A / A j has a left quasi-inverse c in A . Then c + A j must be the quasi-inverse of e + A j . Thus c + A j is quasi-invertible so c has a left quasi-inverse which must be e. Thus e is quasi-invertible, as we wished to show. Finally, we show that A j satisfies (C 3 ) even for 0 in the spectra. If a is invertible and b belongs to the Jacobson radical, then - a ~ lb has a quasi inverse c so a-1 — ca_1 is an inverse for a + b. Since (ci) implies (C 3 ) for non-zero values, the Jacobson radical satisfies (C 3 ). □ 2.3.4 C o ro lla ry The set on which any spectral semi-norm vanishes is included m the Jacobson radical. Hence, any spectral semi-norm on a semi simple algebra is a norm, and the Jacobson radical is closed in any spectral semi-norm. Any algebra norm on a Jacobson-radical algebra is spectral.
2.3.5
The Jacobson Radical and the Fundamental Theorem
227
Proof If a spectral semi-norm a vanishes on b 6 A , then it vanishes on ab for all a e A , so p(ab) is 0. Hence the set on which a vanishes is included in the Jacobson radical by condition (b) of the theorem. The second statement is an immediate consequence, and the third follows by considering the quotient semi-norm on A / A j, which is spectral by Theorem 2.2.14(d). Since the quotient semi-norm is a norm, A j is closed. The last sentence follows from the definition of spectral semi-norms, since the spectral radius is identically 0 on the Jacobson radical. □ We will eventually see several other proofs that the Jacobson radical is closed in spectral semi-norms. The Fundamental Theorem o f Spectral Semi-norms Theorem 2.3.6 below is derived from Proposition 1.3 of Angel RodriguezPalacio [1985]. It was first stated explicitly in the present form as Proposi tion 3.6 in Palmer [1992] (submitted in 1985). The present proof (Palmer [1989]) is a simple transformation o f Ransford’s beautiful proof [1989] (based on Aupetit [1982a]) of Johnson’s uniqueness of norm theorem. The next lemma, which we call Ransford’s three circles theorem, is a slight extension of his proof [1989] based on Jacques Hadamard. A spectral algebra (Def inition 2.4.1) is simply an algebra on which some spectral semi-norm can be defined. We prefer to give this lemma here where it is needed, with its natural hypothesis, rather than waiting until after the formal definition. 2.3.5 L em m a Let p be a polynomial o f a complex variable with coefficients from a spectral algebra A . Then any R > 0 satisfies: s u p {p (p (C ))2 : ICI = 1 } < s u p { p ( p ( A ) ) : |A| = f l } s u p { p ( p ( A ) ) : |A| = f l - 1 }.
Proof Choose a spectral semi-norm a on A , and let A „ be the ideal on which a vanishes. Then the formula ||a + Aa\\ =
Va £ A
Let n be an arbitrary positive integer and £ be an arbitrary complex number of modulus 1. Choose a norm-one continuous linear functional u on B satisfying w (p (O n + A t ) = ||p(On +Ar|| =
i>y ?(A) = Y lk to 9*-^ where d is the degree of p and each qk is a complex number. Then the Cauchy-Schwarz-Bunyakowski inequality gives:
228
2: The Spectrum
2.3.6
^wcn2 < (H^*0 - (nd + ^QCi*7*!2) /
1
f 2 *
1
q (R e")\ 2dt -
f 2 *
1/2
(n d + l)[-J o
<
(nd + 1) sup{cr(p(A)n) : |A| = R } sup{£r(p(A)n) : |A| = f l - 1 }.
|
Jo
\
<
[q iR -^ d tj
We get the desired result by taking nth roots and letting n approach infinity through powers of two. This makes the convergence monotone decreasing, so that a slight variant of Dini’s theorem (e. g., Stromberg [1981]) shows that the supremum and limit may be interchanged. □ 2.3.6 Fundam ental T h e o re m Let ay and o i be two spectral semi-norms on an algebra A . If a sequence has limits with respect to both semi-norms, then these limits differ by an element o f the Jacobson radical. Proof Let { a „ } be such a sequence. Without loss o f generality we may assume the <7i-limit is 0, and we will denote the other by a. Let 6 in A be arbitrary. Denote the supremum of {<72(6an) } by B. We shall apply Ransford’s three circles theorem with £ = 1 to the first degree polynomials: Pn(A) = A6a„ + ba - ban. This gives p(ba)2
<
sup{p(pn(A)) : |A| = fl}su p {p (p „(A )) : |A| = f l- 1 }
<
s u p { o - i ( p „ ( A ) ) : |A| =
<
[(fl + l)
f l } s u p { < 7 2 ( P n ( A ) ) : |A| =
f l ' 1}
Letting n approach infinity, we get the upper bound <j\(ba)R~'B. Now let R approach infinity, in order to conclude p(ba) = 0. Thus by Theorem 2.3.3, a is in the Jacobson radical, as we wished to show. □ In order to express the next corollary elegantly, we introduce a concept due to Bohdan J. Tomiuk and Bertram Yood [1989] and Michael J. Meyer [1992a], 2.3.7 D efin ition Two (not necessarily complete) norms ||•|| and ||| •||| on a normed linear space A are said to be consistent if the identity map between them has a closed graph. More explicitly, this means that if a sequence { a „ } n€N C A converges to a in j| - 1| and to b in |||•|||, then a = b. Note that if two norms dominate a common norm, or if the closed unit ball in one is closed in the other, then they are consistent. In particular if one norm dominates another, they are consistent. For two complete norms,
? 3.8
The Jacobson Radical and the Fundamental Theorem
229
ths closed graph theorem shows that consistency is the same as equivalence. Corollary 2.3.4 shows that the next obvious corollary captures the full force of the fundamental theorem on semisimple algebras. 2.3.8 C orolla ry co isistent.
Any two spectral norma on a semisimple algebra are
2.3.9 C orolla ry Every surjective homomorphism from a Banach algebra onto a semisimple Banach algebra is continuous. In fact, every surjec tive homomorphism from a spectral semi-normed algebra onto a semisimple spectral normed algebra is closed. Proof We explain the second result first. If
230
2: The Spectrum
2.3.11
[1979a] and gave a simplified account in [1991]. In many respects the culmi nation o f this technique was his influential proof [1982a] [1983] of Johnson’s uniqtieness o f norm theorem (Corollary 2.3.8). A previous version of the present chapter used subhannonic functions extensively, deriving the fun damental theorem by Aupetit’s proof. Ransford’s ingenious proof [1989] has allowed us to return to more standard parts of complex variable theory. We wish to briefly summarize this powerful theory, but the reader should consult Aupetit’s books and papers for more detail. 2.3.11 D efin ition Let U be an open subset of C. We call a function / : U —►R U { —oo} subharmonic if it is upper semi-continuous and for any closed disc D (Ao, r) included in U it satisfies
( 1) The appendix to Aupetit [1991] is a good short list of basic results on subharmonic functions; Hayman and Kennedy [1976] contains full proofs. Obviously if both / and - / are subharmonic, then / is harmonic (Note , / is continuous and equality holds in (1)). It is classical that if / : U —» C is analytic, then A ►-* log|/(A)| is subharmonic. In order to extend this result to Banach algebra valued functions, one uses a technical property of subharmonic functions to show that A >—> log ||/(A)|| is subharmonic for any vector valued analytic function / (Definition 2.2.10). Then Gelfand’s spec tral radius formula (Corollary 2.2.8) and obvious properties of subharmonic functions give Vesentini’s result: 2.3.12 T h e o r e m Let A b e a Banach algebra and letU C C be open. If a function f :U —* A is analytic, then A ►-* p ( f ( A)) and A *-* lo g p (/(A )) are both subharmonic. The following generalizations of the maximum modulus principle and Liouville’s Theorem are almost immediate consequences of basic properties o f subhannonic functions. Recall that pc(S) is the polynomiaily convex hull o f S. 2.3.13 T h e o re m Let A be a Banach algebra, let U C C be open and connected and let f .U -* A be analytic. If A0 € U satisfies Sp(f(X)) C S p ( f ( Ao)) for all A e U, then d Sp (f ( Ao)) C d Sp ( f (A)) and p cS p (/(A )) = pcS p(/(A o)) hold for all A € I/. If, in addition, Sp(f(Xo)) has empty inte rior or C \ S p ( f ( A)) has no bounded component, then Sp(f(X)) is constant for X e U. 2.3.14 T h e o re m If A is a Banach algebra, f : C —> A is analytic and (JAeC S p (/(A )) us bounded, then pcSp(f(X)) is constant.
2.3.15
The Jacobson Radical and the Fundamental Theorem
231
In [1991], Aupetit uses these and similar results to derive numerous results describing or characterizing the center, the Jacobson radical, comxutativity, elements with finite spectrum, etc. See also Aupetit [1982b], Vesentini [1984a] [1984b], Gong Ning Chen [1987] and O. H. Cheikh and Mohamed Oudadess [1988]. Capacity in Banach Algebras Paul R. Halmos [1971] introduced this idea which has close links to subharmonic functions. He was motivated by the desire to find a class of elements related to algebraic elements (i.e., those satisfying a one-variable polynomial) the way topologically nilpotent elements are related to nilpotent elements. This motivates the definition. 2.3.15 D efin ition Let A be a Banach algebra. For any element a € A , define cap„(a) = inf{||p(a)|| : p is a monic polynomial of degree < n } and cap(a) = lim „_ 00capn(a)1/" . It turns out that the capacity of an element depends only on the spec trum of the element and can be calculated by the following attractive (but not easy) alternative formula: cap(a) = limn_ QO6„(Sp(a)) where 6n is the nth spectral diameter defined on a compact set K by
Elements with capacity zero do behave like generalizations o f algebraic elements. It is obvious that elements with finite spectrum have capacity zero, and it is true (but not obvious) that elements with countable spectrum also have capacity zero. See also Aupetit and Wermer [1978]. For algebraic elements of Banach algebras see Nathan Jacobson [1945d], Zem&nek [1978] Bernard R. Gelbaum [1980], Soltysiak [1978], [1982], [1984], Young [1980], Cerach [1982], Aupetit and Zem&nek [1983b], Raul Rodriguez Macias [1983] and Barnes [1985]. .The Approximate Bft-condition There is a very simple concept which forces semisimplicity and has other consequences for spectral norms. The approximate B^-condition was intro duced by Smiley [1955] generalizing a definition o f Frank F. Bonsall [1954a]. It captures a portion of the power of the C*-condition which was introduced in §1.7.17. 2.3.16 D efin ition An algebra norm ||•||on an algebra A is said to satisfy the approximate B# - condition if, for each a € A and each e > 0, there is a
232
2: The Spectrum
2.3.17
(non unique) non-zero element a# € A satisfying ||a#a||“ >||a#|| ||a||(l-e). This metric condition is often satisfied by algebras of operators. Some examples are covered by the following proposition. 2.3.17 Proposition Let X be any normed linear space and let A be a non-zero ideal o f B (X ) or any subalgebra o f B (X ) including Bp( X) . The operator norm on A satisfies the approximate Bft-condition. If H is a Hilbert space, the operator norm on any subalgebra of B(H) which is closed under the Hilbert space adjoint satisfies the approximate B* -condition. Proof Proposition 1.1.18 shows that any non-zero ideal includes the finiterank operators. If T 6 B and e > 0 are given, we can find a non-zero x € X satisfying ||Tx|| > ||T|| ||x||(l-e) and a non-zero w € X ' satisfying u>(Tx) — |M| ||ra:||. Define T * to be x 0 u > so that we have \\T*\\ = |)x|| ||w||; and T * T ( x ) = ui(Tx)x = |M| ||7x||x. The last equation shows that ||w|| ||Tx|| is a characteristic value of T # T so that we have HT^TH00 = pb (X) { T*T) > Hull ||Tx|| > M l lim ||x||(l - e) = ||r#H ||T||(1 - e). The closure of a subalgebra of B(H) satisfying this condition is a C*algebra. D The next proposition, which is essentially due to Bonsall [1954a], shows the power of this apparently weak condition. 2.3.18 Proposition A normed algebra (A, ||•||) which satisfies the ap proximate B*-condition is semisimple. Furthermore, if a is a spectral norm on A which satisfies:
Spectral Algebras
2. .1
2.4
233
Spectral Algebras
In this section we introduce a type of abstract algebra (i.e., one without any definite norm or topology) which provides a natural setting for many results originally proved for Banach algebras. These algebras, which we ca spectral algebras, will be used extensively in the rest of this work. The section also contains the proof of a remarkable theorem which asserts thc t the spectral radius is subadditive if and only if it is submultiplicative, and that both conditions hold exactly when the algebra is a spectral algebra which is commutative modulo its Jacobson radical. In the next chapter, this res alt will be further strengthened by the addition of Gelfand theory which applies to precisely the class o f spectral algebras which are commutative mcdulo their Jacobson radicals. After introducing modular ideals, the section concludes with a few in teresting results using this new machinery. Spectral Algebras The concept introduced here is a key one in this work. It is a purely al gebraic concept which nevertheless implies many basic results in the theory of ianach algebras. 2.4 1 D efin ition An algebra is called spectral if some spectral semi-norm can be defined on it. We wish to emphasize that a spectral algebra is not an algebra together wit i some particular spectral semi-norm, but is rather an abstract algebra on vhich some spectral semi-norm can be defined. Most spectral algebras havs a vast diversity o f spectral semi-norms, and no one spectral semi-norm is r eferred over another. Obviously, Banach algebras are spectral algebras (since the complete norm is a spectral norm), but we will see many other examples of spectral algebras. However, the spectrum in spectral algebras behaves the way it does in Banach algebras. In fact, the main difference between the theory of Banach algebras and that o f spectral algebras is simply the absence o f the holomorphic functional calculus in the latter case. (This functional calculus for Banach algebras will be described in Chapter 3.) Before exploring these ideas, we express the definition in a more geometric way. 2.4,2 P r o p o s itio n An algebra is spectral if and only if there is some balanced, convex, absorbing semigroup included in the set o f quasi-invertible eler. ents. Pro :/ The open unit ball of a spectral semi-norm would be such a set, and r;sely given such a set, its Minkowski functional would be a spectral senv-norm. □
234
2: The Spectrum
2.4.3
Theorem 1.1.10 above shows that for any algebra semi-norm a lim crfa")1/ " has many desirable properties. Theorem 2.2.5(f) shows that the spectral radius in any spectral algebra has all these properties. (This is a nontrivial observation since we have already shown by example that subadditivity and submultiplicativity are not properties of the spectral rar dius (even when it is finite-valued) in arbitrary commutative algebras.) We record this and another similar observation about spectral algebras in the next theorem. 2.4.3 T h e o re m Let A be a spectral algebra. (a) The spectrum of every element of A is closed and bounded, hence compact. (b) The spectral radius in A satisfies all the properties listed in Theorem
1.1.10.
Proof Both results follow from Theorem 2.2.5 applied to any spectral semi norm on the algebra. □ We will need some simple technical results. 2.4.4 P r o p o s itio n An algebra is spectral if and only if its ( unitization / quotient modulo its Jacobson radical ) is spectral. The quotient o f any spectral algebra by any ideal is a spectral algebra. Proof If A is nonunital and o is a spectral semi-norm on A , the semi-norm A^ —* R+ defined by a 1(A + a) = |A| + o(a) on A 1 is also spectral since p(A + a) < |A| + p(a) is obvious. If o is a spectral semi-norm on A 1, then its restriction to A is spectral since the non-zero spectrum of an element in A is the same whether computed in A or A 1 (Proposition 2.1.8). The other statements follow directly from Theorem 2.2.13. □ Some Questions Theorem 2.2.13(c) asserts that an algebra semi-norm is spectral if its restriction to each maximal commutative subalgebra is spectral. We do not know whether an algebra is spectral if each of its maximal commutative subalgebras is spectral. There is some evidence (and several plausible but fallacious “proofs” based on the isomorphism between {1\ + and 2 ^ /(Ii DX?) and Theorem 2.2.13(d)) that in any algebra, or at least in any commutative algebra, the sum of two ideals is a spectral algebra if each of the ideals is a spectral algebra in its own right. A variant of this possible result relates to a semi-normed algebra (.4, cr) and asks whether o is spectral on the sum of two ideals when it is spectral on each ideal separately. This last result would imply that each semi-normed algebra has a largest ideal on which the algebra semi-norm is spectral. The result for commutative algebras implies the existence of a largest spectral ideal in any commutative
2.4.5
Spectral Algebras
235
algebra. One may also inquire whether an algebra is spectral if there is an ideal X in it which is spectral and for which A/X is also spectral. We do not know whether any of these results is true. Modular Ideals Modular ideals play an important role in the project of extending re sults from unital to nonunital algebras. Maximal modular left ideals are also closely related to the Jacobson radical and irreducible representations. Modular ideals were first used (with the name “regular ideal” ) by Irv ing E. Segal [1947a]. The term “modular ideal” was introduced by Jar cobson [1956], Even in nonunital algebras we use the common notations .4(1 —e) and (1 —e)A to denote, respectively, the sets {a —ae : a € A } and {a - ea : a 6 .4}.
2.4.5 Definition Let A be an algebra and let C, H, and J be, respectively, a left ideal, a right ideal and an ideal of A . An element e € A is called a ( right / left / two-sided ) relative identity for ( C. / H / X ) if it satisfies ( A { 1 - e) C C / (1 — e)A C K / .4(1 - e) U (1 - e )A C X ). A oneor two-sided ideal is called modular if it has a relative identity of the type described above. A one- or two-sided ideal of A is called proper if it is a proper subset of A and is called maximal if it is maximal under inclusion among all proper ideals of its type. Obviously if A is unital, every one- or two-sided ideal is modular, with the identity element as a relative identity. As a partial converse, a proper (two-sided) ideal X is modular if and only if A/X is unital. If Z is an ideal of A and e and / are a right and a left relative identity for X, then A/X has both a right and a left identity so that these must agree and be an identity. Hence I is a modular ideal with both e and / as two-sided relative identities. It is also easy to see directly that e o / is a two-sided relative identity for X. For any algebra, the whole algebra is always a modular ideal o f itself, but we will be interested in proper modular ideals. It is not obvious a priori that a maximal modular ideal is a maximal ideal, but the easy proof is included with the following results due to Segal [1947a].
2.4.3 T h eorem Let A be an algebra. (a) A one- or two-sided ideal which includes a modular ideal of the same type is also modular. (b) No proper one- or two-sided modular ideal contains its relative iden tity. If e is the { right / l eft) relative identity for a modular ( left / right) ideal ( C / % ) , then {«.- - b : b G ( £ / 71 ) } is disjoint from AqG(c) A maximal modular one- or two-sided ideal w a maximal ideal of the same type. In fact, if a e A dots not belong to a maximal modular ( left / right )ideal ( C / TZ ), then ( Aa + C / aA + TZ ) equals A .
236
2: The Spectrum
2.4.7
(d) Any proper modular one- or two-sided ideal is included in a maximal modular ideal o f the same type. (e) An element e € A is a { right / left ) relative identity for some maximal modular ( left / right) ideal if and only if e has no ( left / right ) quasi-inverse. Proof We consider only left ideals, since the other cases are similar. (a): The relative identity for the included ideal obviously serves the same role for the larger ideal. (b): If the right relative identity e of the modular left ideal £ belongs to £, then any a € A satisfies a — ae — (ae —a) € £. For 6 € £ , if e —b had a quasi-inverse c, then the following calculation would show e € £ , contrary to what we have just proved: e = c(e —6) — c + 6 = (ce — c) —cb + b € C. (c): The first statement follows immediately from (a). In order to prove the second, let e be a right relative identity for the maximal modular left ideal £. Consider the set {6 € A : Ab C £ }. It is easy to see that this is a left ideal which includes £ but does not contain e. Hence by the maximality of £ , it equals £ . Thus Aa + £ properly contains £ . By maximality again it equals A. (d): The following standard application of Zorn’s lemma proves this. Consider the family of all left ideals which include a given modular left ideal £ but do not contain its right relative identity e. When ordered by inclusion, the union of any linearly ordered subfamily is an upper bound for the subfamily. Hence the partially ordered set has a maximal element, which is a maximal modular left ideal including £. (e): First we show the necessity of the condition. If e is a right relative identity for £ and satisfies hoe = 0, then e. = be —b G £ holds, contradicting (b) above. Conversely if e has no left quasi-inverse, then e does not belong to the modular left ideal >4(1 - e). Thus by (d), e is the right relative identity for a maximal modular left ideal including A ( l — e). □ 2.4.7 T h e o re m Let a be a spectral semi-norm on an algebra A . The closure with respect to a o f every proper modular one- or two-sided ideal of A is proper. Hence, every maximal modular one-or two-sided ideal of A is closed with respect to a. Proof Since ail cases are similar, we consider only left ideals. Let e be a right relative identity for the proper modular left ideal £. We will show that the a-distance from e to any element 6 o f £ is at least 1. If not, a(e — b) < 1 implies that e — b has a quasi-inverse c. But this contradicts Theorem 2.4.6(b). Hence e is not in the closure of C with respect to cr, and thus this closure is proper. The seccnd sentence is an immediate consequence of the first. □
2 4.8
Spectral Algebras
237
2.4.8 C orolla ry An algebra semi-norm is spectral if and only if all max imal modular one-sided ideals are closed with respect to it. Proof If a is a spectral semi-norm on A , the theorem shows that all max imal modular one-sided ideals are
be the usual LP-norm. Define a complete metric OO
p = l
on Lu in the usual way so that convergence in this metric is simultaneous convergence in all IP norms 1 < p < oo. Hence L " has a complete invariant :;;e ric (i.e., d(f , g) = d(0, f - g) for all f , g ) and is thus a Frechet space. Ka ilansky [1948c] showed that Lu has no closed maximal ideals. This corsirasts with the situation in unital semisimple spectral algebras, in which eve-y maximal ideal is closed and there are enough such ideals to have intrr.soction {0}. In [1964] Yood proved the stronger statement that con;ains no closed ideal which is maximal among closed ideals. tii the terminology introduced in Chapter 7, this is a topological annihilacor algebra which is not a modular annihilator algebra and which has scd s zero. This shows that topological annihilator algebras need not re semble normed annihilator algebras, which are automatically spectral. See also Mohamed Oudadess [1985].
238
2: The Spectrum
2.4.10
Commutativity and the Spectral Radius as an Algebra Semi-norm The next theorem shows that if the spectral radius is either subaddi tive or submultiplicative, then it is both and thus is a spectral semi-norm. This happens if and only if the algebra is commutative modulo the Jacob son radical. In this generality, the result is due to Palmer [1992] based on arguments of Jaroslav Zemanek [1977a], [1977b] and Vlastimil Ptak and Zemanek [1977aj [1977b]. For the proof we need the following interesting result of Claude Le Page [1967]. The elegant proof is derived from appar ently independent proofs of similar results due to Isadore M. Singer and John Wermer [1955], Ivan Vidav [1955], F. V. Sirokov [1956] and Marvin Rosenblum [19(58]. See also Victor A. Belfi, and Robert S. Doran [1980], Paula V rbovi [1981], Oudadess [1983b], [1987], Gerd Niestegge [1984], M. I. Karakhanyan, M. I. [1984], Yiannis Taertos [1986], Zalduendo [1989], Bertram Yood [1990] and Badea [1991].
2.4.10 Lemma A unital normed algebra A is commutative if and only if there u a constant C satisfying
||a6|| < C ||6 a ||
V a, 6 € A
Proof If A is commutative, we may choose C = 1. To prove the converse, first note that we may replace A by its completion and thus assume that A is complete. Define eXa by eXa = ^ ^ _g(A a)n/n !, so that by the usual elementary calculation eAo has inverse e a. (See the end of Section 2.1 for a discussion of the exponential function.) Let a, b € A be arbitrary but fixed for the rest of this argument. For any fixed (where A * is the Banach space dual of A ), define / : C —►C by /(A ) = u>(eXabe~Xa). Clearly, for a suitable choice of Cn € A , we can write /(A ) = as an entire function satisfying l/(A)|
^(c^JA"
< |M | |jeAo(6e~Ao)|| < C |M | ||6 e -A“eA“ || = C |M | ||6||.
Hence, by Liouville’ s theorem, / is a constant function. Thus the derivative / ' of / is identically 0. However /'(A ) = w(oeXabe~Xa + eXab ( - a ) e ~ Xa) implies /'( 0 ) = ui(ab-ba). Since u> € A " was arbitrary, we conclude ab = ba by the Hahn-Banach theorem. □
2.4.11 Theorem The following are equivalent for any algebra A in which the spectral radius is finite-valued:
2.4.11
Spectral Algebras
239
(a) There is a constant C satisfying p(a + b) < C(p(a) + p(b))
V a, b €
A.
(b) There is a constant C satisfying p(ab) < Cp(a)p(b)
V a, b G A.
(c) A is a spectral algebra and is commutative modulo its Jacobson rad ical. (d) The spectral radius is an algebra semi-norm and hence the unique smallest spectral semi-norm. Proof Note that the constants C necessarily satisfy C > 1 unless A is a radical algebra. Since that case needs no further consideration, we assume this inequality. (a) => (b): For a, b € A , let Ao € C satisfy |Ao| > 9C2p(a)p(b). Then we may write A0 = with |/j| > 3Cp(a) and \u\ > 3Cp(b). Hence we get p((t* -
=
sup{|A /(/i-A )| : A e 5 p ( a ) }
<
su p{| A /(/z - A)| : |A| < p ( a ) } < (2C ) _1
and similarly p((u — b)~lb) < (2C)~1. Thus Ao — ab = (/z - a )(l + (p. — a )_1a + (v — 6)_16)(«/ — b) is invertible by the spectral mapping theorem, and the following inequality p ((m - <*)-'<* + (" - b)~lb) < C[p((n - a)~la) + p((V - 6)_16)] < 1 implies that the middle factor is invertible. We conclude p(ab) < 9C2p(a)p(b). (b) =► (a): For a, b € A, let Ao € C satisfy |A0| > p(a) + Cp(b). We must consider the unital and nonunital cases separately. If A is unital, Ao - (a + b) = (Ao - a )(l - (Ao - a )-1 6) is invertible, since p((Ao - a ) " 1*) < Cy>((Ao - a)~1)p(b) < C(Cp(b) )~lp(b) = 1 shows 1 £ Sp((Ao —a )-1 6). If A is nonunital, (Ao —a )-1 has the decompo sition A^-1 + c with c £ A 1. Hence Ao — (a + 6)
=
(A0 - a ) ( l - ( A 0 - a ) - 16)
=
(Ao - a ) ( l - Aq lb - cb)
=
(A0 — a )(l — d>(l — Aq 16)_1)(1 — Aq *6)
240
2: The Spectrum
2.4.11
is invertible in A 1, since p(cb( 1 - V & r 1) < Cp(c)p(b( 1 - X o ' b ) - 1) < 1 /C < 1 follows from P(c)
=
p((Xo - a ) " 1 - Ao1) = ^(aAo^Ao - a )- 1 )
<
IV 'Ip W /O A oI - P(a)) < |A0- l \p(a){Cp(b))-x
and p ( b ( l - X ^ lb ) - 1)
=
|Ao|p(6(Ao - 6 ) - 1)
<
|A(71|p(fc)/(|Ao| - p(b)) < |A0|p(6)p(a)“ 1.
We conclude: p(a + b) < p(a) + Cp(b) < C(p(a) + p(6)). (b) => (c): By the last argument we may assume p(a + b) < p(a)+Cp(b) for all a, b € A. Define p(a) = C 2 sup{p(a + b) - p ( b ) : b € >1}
V a € A-
Then p satisfies =
C 2 sup{(p(a + c + 6) - p(c + b)) + (p(c + b) - p(b))
<
p(a) + p(c)
C 2p(a)
<
p(a) < C 3p(a)
p(ab)
<
C 3p(ab) < C 4p(a)p(b) < p(a)p(b)
p(a + c)
V a, c £ A\ V a € A; V a, b € A
and thus p is a spectral semi-norm which also satisfies p(a)2 < C6p(a)2 = C*p(a2) < C 4p(a2)
V a € A.
Hence the ideal on which p vanishes is the largest ideal on which the spectral radius vanishes, so that by Corollary 2.3.4 it is the Jacobson radical. Thus p induces a norm ||•||on A / A j defined by ||a + Aj\\ = p(a) for all a € A. However, this implies
ll(a+ A j ) ( b + -4j)|| =
\\ab + A j \ \ = p ( a b ) < C 3p(ab)
=
C 3p(ba) < Cp(ba) = C||6a + Aj\\
=
C||(fc + A ,)(a + A,)||,
so that A / A j is commutative by the last lemma. (c) => (d): Theorem 2.2.5(f) and Theorem 1.1.10(g) applied to the spec tral norm induced on A j A j show that p i s an algebra semi-norm there.
2,4.12
Spectral Algebras
241
However, Theorem 2.3.3 shows that the spectral radius of a coset a + A j in A / A j equals that of a in A , so that the spectral radius is also an al gebra semi-norm there. Hence, it is clearly the unique smallest spectral semi-norm. (We may, therefore, take C as 1, so the definition of p shows that it is just p.) (d) => (a) and (b): Since p is an algebra semi-norm, we may choose C ~ 1. □ In the next chapter, Theorem 3.1.5 will add additional information to this theorem. It will show that the conditions of Theorem 2.4.5 are also equivalent to the existence of a complete Gelfand theory. In Chapter 4, T h orem 4.2.18 will show that these conditions are also equivalent to uni form continuity of the spectral radius. Yu. V. Turovskii [1985] shows that certain closed subalgebras are com mutative modulo their Jacobson radical. In any associative algebra A we may introduce the Lie product [a, 6] = ab — ba for all a,b € A. A linear subspace closed under this product is called a Lie subalgebra and a Lie subalgebra B is said to be Lie nilpotent if there is a positive integer n such that any ist bo , bi , .. . , b n in B satisfies [6„, •••, [6i, bo] •••]] = 0. The cited paper shows that the norm closure of the associative subalgebra generated by any Lie nilpotent Lie subalgebra is commutative modulo its Jacobson radical. Numerous properties and subsets of Banach algebras have been char acterized in terms of spectral radius criteria. We list the following papers whic i deal with commutativity, the center, the Jacobson radical, finite di mensionality, etc. Since the spectral radius is unchanged by dividing by the Jacobson radical, these properties and subsets are actually determined mod ulo the Jacobson radical. Aupetit [1979a], [1991]; Aupetit and Zem&nek [1978], [1981], [1983a], [1983b] and ZemAnek [1977a], [1977b], [1980] and [1982], Homomorphisms onto the Complex Field The following result, which describes the kernels of homomorphisms onto the complex numbers, plays a crucial role in the next chapter which establishes the Gelfand theory for commutative spectral algebras. 2.4.12 T h e o re m The following are equivalent for a modular ideal 1 in a spectral algebra A. (; ,) A / I is isomorphic to the complex field. (b) I ia maximal as a left ideal. (c) I is maximal as a right ideal. Hence a modular ideal in a commutative algebra is the kernel of a homo morphism into the complex field if and only if it is maximal. Proof Clearly (a) implies the other two properties since in this case I
2: The Spectrum
242
2.4.13
has codimension 1. Suppose (b) holds. Choose a spectral semi-norm a on A ■ Theorem 2.4.7 shows that I is closed, so A/T is a (spectral) normed algebra. We wish to show that every non-zero element of A/T has a left inverse so that A jT is isomorphic to thu complex field by the GelfandMazur theorem (Corollary 2.2.3). Let e be a relative identity for I , so that e + Z is an identity element for AjT. Let a be an arbitrary element not in I , so that a + 1 is an arbitrary non-zero element of A/T. The left ideal A a + T contains a = ea — (ea — a) which does not belong to J, so A a + 1 = AThus a + I has a left inverse, as we wished to show. The case of right ideals is similar. O This last proposition identifies all the kernels of homomorphisms of a spectral algebra onto the complex number field. The next result character izes the homomorphisms themselves. For commutative Banach algebras this theorem was proved by Andrew M. Gleason [1967] and by Jean-Pierre Kahane and Zelazko [1968], independently. Zelazko [1968] extended the orig inal result to noncommutative algebras, and Moshe Roitman and Yitzak Sternfield [1981] gave essentially the present statement and proof. Rele vant counterexamples are contained in the last cited article and in Zelazko [1973a], [1973b]. For other variations see Alberto Arosio and Artur V. Ferreira [1978a] [1978b], Aupetit [1979b], Sergiusz Kowalski and Zbigniew Siodkowski [1980], Chang Pao Chen [1983], S. H. Kulkarni [1984], Regina Wille [1985], Arens [1987], Badea [1991] and Krzysztof Jarosz [1991]. A linear map satisfying condition (b) is called a Jordan homomorphism, and such m aps are studied systematically in Section 6.3 below. 2.4.13 T h eorem Let A be a unital algebra and let ip:A —* C be linear and united (i.e.,
2.4,13
Spectral Algebras
243
Hence we see (p(ab + ba) = 0. By (b) we conclude
=
— 2tp((bab)a + a(bab)) = 4
V a, 6 € A
(d) =$■ (a): Trivial. (d) =>• (e): If a is invertible and ip(a) is 0, we get the contradiction 0 = tp(a~la) — tp( 1) = 1. (e) =>• (f): This follows from
n ”=1(A -
Xj) = A " - n
- ^
( a 2) A " - 2 + •••
and comparing coefficients, we get: n V ' Xj = mp(a) = 0 and £ j= i i
= ^ ^ " ^ (a2)-
However,
0= ( £
j= 1
H
+ 2H
j
J= 1
gives n(n - l ) M a 2)| = 2] ^
j
= I£
j= l
Ail ^ £
lA?l ^ nP(a)2-
j= l
Since n € N was arbitrary and p(a) is finite in a spectral algebra, this gives ‘f ( a 2) = 0.
□
244
2: The Spectrum
2.4.14
In the above theorem, weaken condition (f) to
2.4.14 Lemma Let A be a commutative normed algebra and let A be its completion. If a £ A is quasi-invertible modulo every maximal modular closed ideal o f A, then it is quasi-invertible in A. Proof Theorem 2.4.6(e) shows that if a is not quasi-invertible in A , then a will be the relative identity for a maximal modular ideal M o f A. Theorem 2.4.7 shows that M is closed so that M n A is a closed maximal modular ideal of A. Clearly a is not quasi-invertible modulo this ideal. □
2.4.15
Theorem Let {A, ||•||) be a commutative Banach algebra isometrically isomorphic to C o(ll) for some locally compact space fl. (It is equivalent to describe (A, j| ■||) as any commutative Banach algebra satis fying Theorem 3.2.12.) Then any (not necessarily complete) algebra norm HI •HI on A satisfies IMI < IMII V a £ A. Proof For the proof, we may assume that A actually is Co(fl). Let f l c be the set of points w £ fl such that the ideal Z ( u ) = { / £ A : f{u>) = 0} is closed in the norm |||•|||. We will show that f l c is dense in fl and this will prove the theorem since |||/||| > sup{|/(w)| : w £ f lc } - (Otherwise, we get limn—o o ( / - / M ) £ f c = 0( / / / ( w) ) " = ~ f for so™6 / &
In order to give a proof by contradiction, suppose that f lc is not dense. Let U be a nonempty open subset of fl disjoint from flc - The discussion in §1.5.1 shows that there is a function / : fl —►[0,1] in A which assumes
2.5.1
Spectral Subalgebras and Topological Divisors of Zero
245
the value 1 on some nonempty open subset V of U and vanishes off U. By the last lemma, / is quasi-invertible in the completion of A under ||| •|||. How ver, this is impossible since there is a non-zero function g:fl —♦ [0,1] in A with g(u>o) — 1 for some u>o € V and vanishing outside V satisfying g o f = f and hence g = g o ( / o / « ) = ( j o / ) o / « = / o / * = 0. □
2.5
Spectral Subalgebras and Topological Divisors of Zero
In the introduction we listed a number of sufficient conditions for a subalgebra to be a spectral subalgebra. The easy conditions will be collected in Proposition 2.5.3. The more difficult ones all depend on the theory of topo.ogical quasi-divisors of zero, introduced by Kaplansky [1949a] follow ing Silov [1940b], Topological quasi-divisors of zero cannot be defined in a spectral algebra, since the algebra has no particular spectral semi-norm. However the most useful consequences of this concept are independent of the particular choice of norm, so that they are available in the theory of spectral algebras. We begin with a formal definition of spectral subalgebras, which were informally defined in the introduction to this chapter. Fundamental Properties of Spectral Subalgebras 2.5.1 Definition A subalgebra B o f an algebra A is said to be a spectral subalgebra if it satisfies BqG = B n A qG■
(1)
In the next proposition we characterize spectral subalgebras in terms of conditions on the spectrum o f elements in the subalgebra. Zero plays an exceptional role here as it does whenever nonunital subalgebras are under consideration. Thus we include two additional conditions on the spectra of elements of the subalgebra, each of which is more stringent than the conditions shown to be equivalent to the subalgebra being spectral.
2.5.2 Proposition Let B be a subalgebra of an algebra A . (?) Each of the following two displayed conditions is equivalent to B being a spectral subalgebra o f A : SpB(b) U {0 } = SpA (b) U {0 } SpB(b) C S p A (b) U {0 }
V b 6 B; V ft e B .
(2) (3)
Thus any element b in a spectral subalgebra B of A satisfies PB(b) = pA {b)
(4)
(unless b has spectrum {0 } in one algebra and empty spectrum in the other).
246
2: The Spectrum
2.5.2
In particular, a spectral subalgebra of a spectral algebra is a spectral algebra. (b) The condition Sps(b) = SpA (b)
Vbe B
(5)
holds if and only if B is a spectral subalgebra of A which also satisfies one of the following two mutually exclusive conditions: (b j) B u a unital subalgebm of A , or (b?) B is a nonunital algebra and none of its elements is invertible in A . (c) If B is a spectral subalgebra of A , then the inclusion S p B ( b ) f= S p A ( b )
V fte fl
(0)
follows from any of the following circumstances: (co) Every element of B invertible in A is also invertible in B. ( c !) A is not unital. (c2) B is a unital algebra. (c3) A and B satisfy condition (a), (b), (c), (d), (e) or (f) of Propo sition 2.5.3. Proof (a): The equivalence of equations (1) and (2) follows from the defini tion of the spectrum. Conditions (2) and (3) are equivalent by Proposition 2.1.8(c). The equality for the spectral radii is an immediate consequence of (2), and it implies the last sentence. (We do not know any example in which an element b € B has spectrum {0 } in B and empty spectrum in A although the other possibility is easy to achieve, e. g., in the algebra of Example 2.1.7.) (b): Assume equation (5). This implies equation (2) and thus that B is a spectral subalgebra. Furthermore, if fi is a unital algebra with identity element e, then e must be the identity of A (rather than a nontrivial idempotent) since otherwise it would have spectrum {1 } in B and {1 ,0 } in A . Therefore either B is a unital subalgebra or it is a nonunital algebra. In the latter case, none of its elements could be invertible in A , since they all have 0 in their spectra relative to B and we are assuming (5). Conversely if B is a spectral subalgebra satisfying either (b i) or (ba), then (2) implies (5). (c): Condition (co) is just a restatement of the condition necessary to derive inclusion (6) from inclusion (3) and thus is both necessary and sufficient. Condition (ci) implies that 0 is in the spectrum in A of any element and thus that (3) implies (6). Suppose next that B is a unital algebra (although not necessarily a unital subalgebra) with identity element e. Then b € B is invertible in B if and only if e — b is quasi-invertible in B and thus if and only if it is
2.5.3
Spectral Subalgebras and Topological Divisors of Zero
247
quasi-invertible in A. However if b has an inverse ft-1 in A , then e - eb~1e is a quasi-inverse for e — b. Thus inclusion (6) holds. In the proof of Proposition 2.5.3 we will actually establish (6) rather than just (1), (2) or (3) and thereby establish ( C 3 ) . □ The following example, suggested by Larry B. Schweitzer [1993], shows that not every spectral subalgebra satisfies conditions (4) and (6) above. Let T be the algebra of formal power series defined in §2.9.7 below. Every / € ? with non-zero constant term fo has an inverse in T with constant term l //o - Let A be the algebra of all formal power series which start at some (possibly negative) index. Thus every non-zero element / € A can be written uniquely as / = fnZn = with m e Z and flo = 1 (so a = /?„ = fn+m/fm)- Hence A is a field and can be identified with the field of quotients of T ■ Let B be the subalgebra of A consisting of all power series beginning at a positive index. Note that B1 can be naturally identified with !F C A. It is easy to see that every non-zero element g € B satisfies SpB(g) = {0 } and SpA (g) — 0. Hence g satisfies neither (4) nor (6). 2.5.3 P r o p o s itio n Let A be an algebra. The subsets indicated in (a), (b), (c), (d) and (g) below are spectral subalgebras of A satisfying inclusion (6) above. The sets indicated in (e) and (f) are spectral subalgebras satisfying equation ( 5 ) (which implies inclusion (6) ) above relative to the indicated larger algebras. (a) Any one- or two-sided ideal o f A , any hereditary subalgebra or indeed any subalgebra B for which there is some n € N satisfying BnA B n C B; (b) The commutant of any subset; hence also the center of an algebra and any maximal commutative subalgebra; (c) For any idempotent e € A the “com er” subalgebra eAe; (d) Any finite-dimensional subalgebra or any subalgebra in which each element satisfies a polynomial identity; (3) The range of the left or right regular or extended regular represen tation in jC(A); (f) The image k ( A ) of A in A mm with either Arens product; (g) Any intersection o f spectral subalgebras or any spectral subalgebra of a spectral subalgebra of A . Proof (a), (c): We prove the last portion of (a) first since that establishes all of (a) and (c). If b € B has a quasi-inverse a € A , then they satisfy bnabn~l = bn(ab — b)bn~1 € B. At the nth step in a reduction process we get bua = bn(ab — b) € B. But we may continue bn~la = bn~ l (ba — b) € B until we conclude a = (ba — b) e B. If b € B is invertible in A , then bnb~2nbn = 1 shows that B is a unital subalgebra so (6) holds. (b): The inverse or quasi-inverse of any element in the commutant of a set is obviously in the commutant also.
2: The Spectrum
248
2.5.3
(d): Suppose B is finite-dimensional. Then each of its elements satisfies a polynomial equation of degree less than or equal to its dimension. Suppose b 6 B satisfies some polynomial equation. Since A or B or both may bo nonunital, we will choose p(A) = £)!t=i a^A* to be the monic polynomial, having constant term 0 and lowest degree, which satisfies p(b) = 0. If 6 has a quasi-inverse c 6 A , then the equation (1 - c ) b m = - ( c + &fc) f°r all m G N allows us to rewrite the equation (1 — c)p(6) = 0 as otk)c = - q( b) , where q is a monic polynomial having constant term 0 and degree at most m — 1. Hence both sides of the equation are non-zero, so c is in B. Thus B is a spectral subalgebra. If 6 6 B has an inverse 6_1 € A , then multiplication of p(b) by a suitable non-negative power of b~1 allows us to write 1 € A as a linear combination o f positive powers of fe, no 1 is in B. Hence B is a unital subalgebra and its elements must satisfy (6). (e): We will consider the case of the left regular representation, since the right regular representation is similar and the extended regular rep resentations are easier. Suppose that / - T 6 C(A) is a quasi-inverse for La so that T is an inverse for I — La- Then b = T ( —a) satisfies, b - ab = (I - La)b = ( I - La) T ( - a ) = - a . Hence Lb is a right quasiinverse for La, but in fact it is a quasi-inverse, since La is quasi-invertible. Thus L a is a spectral subalgebra of C(A). Suppose next that for some a £ A the operator La has an inverse S in C(A). Note that La must be injective since it has an inverse. Now define e = S(a) and c = S(e). Any b e A satisfies La(eb - b) = LaS(a)b - La(b) = ab - ab = 0, heitct* eb ~ b in 0 and finally La(Lc(b) - S(b)) = LaS(e)b - LaS(b) = eb - b = 0. We conclude I = Le and S = Lc. Thus equation (5) (with C{A) and LA as the algebra and subalgebra) holds in this case, since L_a is a unital subalgebra if it is a unital algebra. (f): This proof is similar to the last one. Suppose n(a) has / G A " as a quasi-inverse. (This does not depend on the choice of an Arens prod uct.) The equation / — n(a) f = —«(a ) shows that any g € .4** satisfies (I - Lay*( g - f g) = g. Thus (I - La)** maps A ** onto itself, and hence ( / — La ) must map A onto itself. We conclude that there is some 6 G A satisfying b — ab = —a. But this just shows that b is a left quasi-inverse for a. Hence n(b) equals / since / was a two-sided quasi-inverse for «(a). Therefore a is quasi-invertible in A. Finally, we deal with the case in which «(a ) is invertible in A**. Since multiplication on the left by n («) in either Arens product is just (L0)**, this implies that La is a bijection on A. Let e be the element which La
2.5.4
Spectral Subalgebras and Topological Divisors of Zero
249
n aps onto a. Then we have La(eb — b) = ab — ab = 0 for all b € A. Hence Le = I so (Le)** = I and «(e) equals the identity in A * . Thus, if any e.ement of n(A) is invertible, then this algebra is a unital subalgebra of which implies equation (5), as we wished to show. (g): This is obvious. □ We caution the reader that in (e) above we have shown that an element in the range of the left regular representation has precisely the same spec trum whether calculated in this range or in the bigger algebra C(Ai). We do not claim, and it is not true, that an element a € A has the same spectrum in A as La has in C(A). The problem is that the left regular representation has a nontrivial kernel unless the left annihilator A la of A is {0 }. The two-dimensional algebra with basis a, b satisfying a2 = ab = 0, ba = a and b2 = b shows that the two spectra just mentioned may differ without some hypothesis on the algebra A. We have established the following facts. Any a gebra A satisfies Sp(a) = Sp(Lla) V a e A (7) n jd Sp(La) C Sp(a) C Sp(La) U {0 }
VaeA.
(8)
II the left annihilator A la of A is {0 }, then we also have Sp(La) = Sp(a)
V a € A.
(9)
2.5.4 A N on -sp ectra l S u balgebra Proposition 2.5.3 shows that many kinds of subalgebras are spectral subalgebras. The introduction to this chapter listed additional examples. (See also Barnes [1983].) We now give on example of a nmi-npeetral, cloned, unital subalgebra which is a generic ir some sense. Let A be a unital algebra and let B be a subalgebra containing the identity element 1 of A ■ If B is not spectral, there is some element b € B with an inverse b~1 in A but not in B. Thus, the subalgebra generated by b and 1 is also a non-spectral subalgebra o f A and this shows that every unital non-spectral subalgebra includes one of this type. If A is normed and B is closed, then an element b of the type just described can be characterized by the fact that 6_1 exists in A but is not the limit of any sequence of p jlynomials in b (i.e., expressions of the form Ylk=i a kbk). Here is an explicit example. Let H be a separable Hilbert space and choose an orthonormal basis {ip „ : n € Z } parameterized by the set of all it tegers. Consider the operator U € B(T-t) defined by
2.50
2: The Spectrum
2.5.5
which is called the bilateral shift. Clearly U has an inverse defined by
We claim that U l cannot be approximated by polynomials in U. Perhaps the easiest way to see this is to consider the subalgebra B of B(7i) defined by B = { T e B( H) : T H n C Tin n £ Z } where Tin — closed span
: j > n }.
Obviously, B is a closed subalgebra which contains U but not U 1. Topological Divisors o f Zero The next concepts come in versions related to both invertibility and quasi-invertibility. We will give the two definitions in parallel. As usual, we will find the second version more useful, but the first version may be easier to grasp on first reading. We will routinely give proofs for the second version only, leaving it to the reader to reformulate them for the first version.
2.5.5 Definition An element a in any algebra A with an algebra seminorm a is called a left topological { divisor / quasi-divisor ) of zero if there exists a sequence {6n} ngN Q A satisfying both: a(bn) = 1
n £ N and ( a(abn) —>•0 / a(bn - abn) —>0 ).
A right topological ( divisor / quasi-divisor )o f zero are defined similarly. An element which is either a right or a left topological ( divisor / quasidivisor ) of zero is called a topological ( divisor / quasi-divisor )o f zero and an element which belongs to both sets is called a joint topological ( divisor / quasi-divisor ) o f zero if the same sequence { 6 „ } n€N can be used on both sides. These sets are denoted by ( A lz / A^qZ ), ( A hz / Anqz )> ( A z / A^z ) and ( A j z / A Jqz ), respectively. It may help the reader to know that an element a in a semi-normed algebra (A, a) is called a left quasi-dimsor o f zero if there is some 6 G A satisfying cr(b) = 1 and b — ab = (1 — a)b = 0. Hence if the sequence in the above definition of a left topological quasi-divisor of zero had a limit a, then a would be a left quasi-divisor of zero. The utility of these concepts arises from two facts. First, even a (one-sided) topological quasi-divisor of zero cannot have a quasi-inverse in any larger semi-normed algebra, so that it is what has been called a permanently quasi-singular element. Second, in spectral semi-normed algebras the boundary of the set of quasi-invertible
2.5.5
Spectral Subalgebras and Topological Divisors of Zero
251
elements consists of joint topological quasi-divisors of zero, so that such elements are fairly common. These two results will be recorded in the first portion of the next proposition and the following theorem, respectively. The rest of the proposition contains other easy results needed later. Arens [1958], [1960] shows (among other results) that, for any element which is not a topological divisor of zero in a commutative Banach algebra, there is a larger algebra in which the element is invertible. Bonsall find Duncan [1973b] give an example due to Peter G. Dixon showing that an element may be both a right and a left divisor of zero without being a joint divisor of zero. See also Example 3.4.17 below and Zofia Balsam [1980],
2.5.3 Proposition Let (A, a) be a semi-normed algebra. (a) If a € A is a topological { divisor / quasi-divisor )o f zero, then it is not { invertible / quasi-invertible )in any larger algebra to which the algebra semi-norm can be extended. (b) The following set inclusions hold: A o A lqZ Q •A-LqZ
AjiqZ ° -A C AflqZ-
If a, c G A satisfy a o c € (A^qZ / -A r,z ) , then either a o r e belongs tO { -A-LqZ / ARqZ )■ (c )
Proof (a): If the sequence {6n} „ gN C A satisfies the definition for a G A l qz and a has a quasi-inverse in B, then we get the contradiction: 1 = a(b„) = a(aq o a o bn) < a(bn — abn) + a(aq)a(bn —abn) —* 0. The other cases are similar or easier. (b): We consider only the first inclusion. Again let {b„}neN Q -A satisfy the definition for a G A l qz and let c belong to A. Then we get ct(( 1 - (c o a)b„) = a (( l - c )(l - a))b„) < (1 + a(c))a((l - a)bn) -* 0. (c): If the sequence {6 „ } n6N Q A satisfies the definition for aoc 6 A t qz> then we get ff((l - ( ao c))bn) = a(( 1 - a )(l - c)bn) -* 0. If cr((l — c)bn) —* 0, then c is a left topological quasi-divisor of zero. If not, there is some subsequence of {£>„} for which this expression is bounded away from zero. We will denote the subsequence by {6n} again. Define a new sequence {6'n} by Ki = cr((l — c)6 „)_1(l —c)bn. This new sequence shows that a is a left topological quasi-divisor of zero, as we wished. □
252
2: The Spectrum
2.5.7
Richard Arens [1958] showed that in any unital commutative Banach algebra an element was permanently singular if and only if it was a topo logical divisor of zero. In fact for any finite set of nontopological divisors of zero he constructed a unital super-algebra in which they all become invert ible. Recently Antonio Fernandez L6pez, Miguel Florencio, Pedro J. Paul and Vladimir Muller [1990] extended this result to commutative topologi cal algebras. (In this book, multiplication in a topological algebra is only required to be singly continuous. The result is false for the class of topo logical algebras with jointly continuous multiplication, since it is harder to produce super-algebras in this situation.) See also Wieslaw Zelazko [1984], [1986], [1987]. The next theorem has many important corollaries. In it the (less useful) sufficiency of the condition wan firwt noted by Vanin D. Mascioni in [1987], but the necessity of the condition really goes back to Silov [1950]. 2.5.7 T h e o re m Let A be an algebra and let a be an algebra semi-norm on A . Then a is a spectral semi-norm if and only if every element in the boundary o f the quasi-group A qc w a joint topological quasi-divisor o f zero. Similarly, in a unital algebra an algebra semi-norm is spectral if and only if every element in the boundary o f the group A a w a joint topological divisor o f zero. Proof To establish the necessity, suppose it is a spectral semi-norm and {fln}n6N Q AqG converges to a G A . Then o’ o a = (1 —a®)(a —a „) implies a(a% o a) < (1 + a(aq n ))a{a - an). Thus if {a£}n€N has a bounded subsequence, aj* o a will belong to A^c for some n, since 0 is in the a-interior of Aqc- In this case a has a left quasi-inverse. Similarly it has a right quasi- inverse and thus belongs to AqG■ Therefore if a belongs to the boundary of the open set A qc and {on}neN Q AqG converges to a, then ff(a^) approaches infinity. Thus b„ = a(a%) 1a j satisfies the requirements of Definition 2.5.5 on both sides, since, for instance, we get ff(
=
a ( a « ) _ 1a ( ( o - an) a« + a „ a « - a « )
^
Thus if a is in the boundary of AqG, then a is a joint topological quasi divisor of zero. If a is in the boundary of the set of invertible elements, then A is unital and 1 — a is in the boundary of the set of quasi-invertible elements. Hence by what we have already proved, 1 — a is a joint topological quasi-divisor of zero. Therefore, a is a joint topological divisor of zero. For the sufficiency, simply note that if a is not a spectral semi-norm, then A qc >s not tx-open, so some element a of the boundary of AqG belongs
2.5.C
Spectral Subalgebras and Topological Divisors of Zero
253
to A c,g ■ Since we are assuming that a is a joint topological divisor of zero, this ( mtradicts Proposition 2.5.6 without even extending the algebra. □ The Boundary o f the Spectrum, and the Permanent Spectrum We remark that the proof of Theorem 2.1.14(a) above shows that if A is on the boundary of the exponential spectrum of an element a in a unital Banach algebra A, then A — a is a joint topological divisor of zero. Let us specialize to a unital Banach algebra A and call the permanent spectrum of an element a € A the set perSpA (a) = P ){5 p g (a) ■B is a Banach algebra including A }
(10)
where inclusion means isometric inclusion, i.e., the norm of A is just the restriction of the norm of B. Thus we have just shown deSp(a) C perSp(a) and dSp(a) C perSp(a) V a € A (11) The question of when inclusion (11) is proper has been explored by Silov [1947j ], Murphy and West [1980a] and others. The latter paper simplifies an example from the former which shows that proper inclusion can occur. However, it seems to be much more common for the two sets to agree. Results and references are contained in the latter paper. In the next result, note that we do not need to require that the seminorm be spectral on the larger algebra nor that the subalgebra be a spectral subal ?ebra. Any complete subalgebra of a normed algebra and in particular any closed subalgebra of a Banach algebra (cf. Corollary 2.5.10) satisfies the corollary. (Recall that pcS is the polynomially convex hull of S.) 2.5.8 C orolla ry Let a be an algebra semi-norm on an algebra A . If B is a subalgebra on which a is a spectral semi-norm, then it satisfies: dSpB(b) C SpA(b) C SpB(b) U {0 } dSpB(b) C dSpA (b)
Vbe B
V be B
and hence SpA (b) C SpB(b) C pcSpA (b)
V b e B.
Prooj Proposition 2.1.8(c) asserts the second inclusion in the first line for aiy subalgebra. The first inclusion in the same line follows from the theor ;m and Proposition 2.5.6(a). We are using the fact that if ( 0 is / A is a r,on-zero element ) in the boundary of the spectrum of b in B, then ( 6 / ,, ~xb ) is in the boundary of the set of ( invertible / quasi-invertible ) eieme its. For the inclusion in the second line, note that if A is any number in tht boundary of the spectrum of SpB(a), it is the limit of a sequence { A „ n o t in SpB(a) and hence surely not in SpA (a). Hence the first
•254
2: The Spectrum
2.5.9
inclusion shows A is in the boundary of 5p^(a). The final display is an immediate consequence. □ The corollary above can be used to show that certain types of subal gebras are always spectral subalgebras. A number of similar results could be formulated, and we offer this one as an example. Modular annihilator algebras are discussed in Chapter 7. 2.5.9 C orolla ry Any subalgebra of a normed algebra which is a modular annihilator algebra is a spectral subalgebra. Proof The restriction of the norm to the subalgebra is spectral by Theorem 7.4.12. Furthermore, by Theorem 7.5.5 the spectrum of each element of the subalgebra is nowhere dense and hence equal to its own boundary. The result now follows from the previous corollary. □ The next result is particularly important for its application to Banach algebras. Note that while the subalgebra B is a spectral algebra, it need not be a spectral subalgebra of A . We call it a feebly spectral subalgebra. 2.5.10 C o ro lla ry Let A be a spectral algebra. Then the following are equivalent for any subalgebra B. (a) PA(b) = PB(b) V b e B. (b) B is a spectral algebra satisfying dSps(b) C dSPA(b) C SPA(b) C SpB{b) U {0 }
V b e B.
(c) The restriction to B o f some spectral semi-norm on A is a spectral semi-norm on B. (d) The restriction to B o f every spectral semi-norm on A is a spectral semi-norm on B. A closed subalgebra o f a Banach algebra satisfies these conditions. Proof (a) => (d): Immediate from the definition of a spectral semi-norm. (d) => (c): Obvious, since ,4 is a spectral algebra. (c) => (b): This follows from Corollary 2.5.8 and the fact that the spectrum is closed in a spectral algebra. (b) => (a): Obvious. The final remark is obvious by considering conditions (a) or (c). □ Recall the discussion following Theorem 2.2.12. It dealt only with the case of closed unital subalgebras of Banach algebras. However, by applying Proposition 2.5.15 below to the completion of A / A j in some spectral norm and to the closure of the image of B in this completion we can extend the results given there to the setting of Corollary 2.5.10. Hence, the only dif ference between the spectrum of an element b e B when calculated relative to A and B is that certain whole bounded components of the resolvent set
2.5.11
Spectral Subalgebras and Topological Divisors of Zero
255
in the former case may belong to the spectrum in the latter case, or zero may appear if 6 is invertible in A but B does not contain the inverse of b. Adding a bounded component of the resolvent set to the spectrum will remove its boundary from the boundary of the spectrum. The next result is similar to Corollary 2.5.9. 2.5.11 C orolla ry A closed subalgebra o f a Banach algebra is a spectral subalgebra if every element in the subalgebra has nowhere dense spectrum in the subalgebra. Proof Apply the final remark of the last corollary. If Sps(b) (which is closed) is nowhere dense, then it is equal to its own boundary; consequently condition (b) of tho last corollary shows that B is a spectral subalgebra when every element of B has this property. □ Algebras without Divisors of Zero The following two results show two circumstances under which an al gebra without topological divisors or quasi-divisors of zero must be the complex field. In this respect they are related to the Gelfand-Mazur the orem (Corollary 2.2.3). The first is due to Zelazko [1967] improving on Kaplansky [1949a]. 2.5.12 P r o p o s itio n A non-zero normed algebra A is isomorphic to C if and only if it contains no non-zero joint topological divisors o f zero. Proof Clearly an algebra isomorphic to C contains no non-zero topological divisors of zero. Suppose every commuting subalgebra of A is isomorphic to C. Then there is certainly no commuting pair a, b G A \ {0 } satisfying ab = 0. However, if these elements do not commute, then they satisfy 6a / 0 and (6a)2 = 0, which is also impossible. Any two idempotents e, / € A satisfy e ( e f —f e ) e = f ( e f —f e ) f = 0, so they commute. By hypothesis any max imal commutative subalgebra C of A has the form Ce for some idempotent e. Similarly, any element a of A has the form X f for some idempotent / , but since e and / commute, a belongs to C and hence actually has the form Ae. Hence e is an identity element for A — Ce which is isomorphic to C. Therefore, in the rest of this proof we may assume that A is commutative, so that any topological divisor of zero is a joint topological divisor of zero. Case (1): Suppose that A is unital without non-zero topological divisors of zero. If a G A is a topological divisor of zero in the completion B of A, then it is easy to see (by approximating the sequence {6n} „ eN C B by a sequence in A) that a is a topological divisor of zero in A. Theorem 2.2.2 shows that the boundary of the spectrum in B of each element a € A contains some A G C. Theorem 2.5.7 shows that A — a is a topological
256
2: The Spectrum
2.5.13
divisor of zero. Hence by assumption A —a is 0. Since a G ,4 was arbitrary, we have A = C. Case (2): Next suppose A is nonunital. Give A 1 the norm ||A + a|| = |Aj + ||a|| for all A + a e A 1. Then A 1 is not isomorphic to C since A is not zero. Thus A 1 has a topological divisor of zero A + a. Let {An + bn}neN satisfy ||An + fen|| = 1 for a ll n € N and (A + a)(An + fen) —►0. Since {An} nei\ is a bounded sequence we may choose a convergent subsequence and hence assume An —* A0. If both A and A(, are 0, then A has a topological divisor of zero and there is nothing more to prove. Case (2a): Next suppose A = 0 but Ao ^ 0. If some subsequence of {bn }ne* converges to some b € A, then b satisfies a(Ao + b) = 0. Since —AQlb is not an identity element in A, there is some c & A satisfying (Ao + b)c ^ 0. Hence a(Ao + b)c = 0 shows that a is a divisor of zero and a fortiori a topological divisor of zero. If no subsequence of { & n } n g N converges, we can (by selecting a subsequence) assume that ||i>n+i —bn|| is bounded away from zero. Hence the sequence {||bn+i —.M l - 1 (&n+i—&n)}neN shows that a is topological divisor of zero. Case (2b): Finally suppose A / 0 but Ao = 0. As before, the fact that —A-1 a is not an identity element in A gives some c 6 A satisfying c(A + a) ^ 0. Hence c(A + a)bn —> 0 shows that c is a topological divisor of zero. n We now restate the result above for topological quasi-divisors of zero. 2.5.13 P r o p o s itio n A spectral semi-normed algebra A which contains no topological quasi-divisors o f zero (except possibly 1) is either a Jacobsonradical algebra or isomorphic to the complex field, C. The latter case occurs if and only if some element o f A has non-zero spectrum or if and only if A is unital. Proof Suppose some element a G A has spectrum not equal to {0 }. Then dSp(a) must contain a non-zero complex number A, since the spectrum is bounded and nonempty (by Theorem 2.4.3). Theorem 2.5.7 shows that A-1 a is a topological quasi-divisor of zero and hence equals 1. Now let b G A be arbitrary and choose t > tr(b), so that Sp(t — b) contains a non zero element p in its boundary. The argument just given for A- I a implies H~l (t - b) = 1. Hence the map b = (t - fi) 1 >-* t - fi is an isomorphism of A onto C. If every element of A has spectrum {0 }, then A is a Jacobson-radical algebra by Theorem 2.3.3 (b) and thus has no identity element. □ The following result is due to Rickart [1950]. Unlike most of the other results in this chapter, there does not seem to be an interesting formulation in terms of .spectral algebras.
2.5.14
Spectral Subalgebras and Topological Divisors of Zero
257
2.5.14 Theorem Let A be a Banach algebra. Let a be an element o f A having zero in the boundary o f its spectrum. Then either there is a non-zero proper idempotent e (i.e., such that a is invertible in eA e or a is a joint topological divisor of zero. Proof As an element of A 1, a belongs to the boundary of (.41) g - Hence by Theorem 2.5.7, a is a joint topological divisor of zero in A 1 (with the norm ||A + 6|| = |A| + ||6|| if A 1 7^ .4). Thus if A = A 1, the proof is complete. (We have only used the fact that the norm is a spectral norm.) Now suppose A is nonunital and a is not a joint topological divisor of zero. By symmetry we may assume that a is not a left topological divisor of zero. We will prove that there is a left identity e for A such that a is invertible in eAe. 3 y what we have just proved, there is a sequence {An + cn} n6N C A 1 wifi. |An|+ ||cn|| = 1 such that a(An + Cn) converges to 0. Since {An} neN is a bounded sequence, it has a convergent subsequence. By a change of notation we assume {An} n6N converges. It does not converge to 0, since in that case bn = (1 —|An|)_ 1cn (omitting any n with |An|= 1) would put a in A l i - Thus An converges to some A / 0. Then a(—A_ 1c„) converges to a. The sequence { —A_ Icn} „ 6N must be a Cauchy sequence, since otherwise b„ -- ||A~] (cPrx — ) ||—1A- 1 (cp„ — c ,n) for suitable subsequences {cp „} and { c , J puts a into A i z - Thus —A- 1cn converges to a non-zero element e € A which satisfies ae = a. Then any b € A satisfies 0 = a(eb — b). If thers were any b with eb — b non-zero, a would be a left divisor of zero. H en* e is a left identity for A and a fortiori an idempotent. In particular ea = a = ae implies that a belongs to eA e. In this spectral subalgebra (Proposition 2.5.3(c)), 0 is still in the boundary of the spectrum unless a is inve tible in eA e. Since eA e is unital, the first step of this proof shows that a would be a joint topological divisor of zero in eA e (and hence a fortiori in A , contrary to assumption) if it were not invertible in eAe. Since a is non-zero, e is a proper idempotent. □ The Spectral Radius in Subalgebras The following criterion is sometimes useful and in particular it will be usee in the next two results. 2.5.15 Proposition Let || •|| be an algebra norm on an algebra A and let ,1 be the completion o f A with respect to ||•||. Then the following are equivalent: (a) ||•|| is a spectral norm on A. (o) p j( a) = pA {a) a € A. _ (c) A is a spectral subalgebra o f A. Prcx f (a) => (c): Suppose a 6 A has a quasi-inverse k € J Choose a seti. ance {&n}neN in A converging to b. This sequence satisfies a o b„ —►
258
2: The Spectrum
2.5.16
a o 6 = 0 and bn o a —> b o a = 0. Hence a o bn and bn o a are eventually quasi-invertible in A. This implies that a is quasi-invertible in A(c) => (b): => (a): Immediate. n In the list of spectral seini-norms given in the introduction, (b), (c), (d), (e) and (h) .issert that all algebra norms on certain algebras are spectral. Yood [1958] called algebras with this property permanent Q-algebras and noted some of the following equivalent conditions. We call a spectral algebra permanently spectral if all algebra norms on it are spectral. See Michael J. Meyer [1991], [1992b] and Theorem 6.1.5(g) below.
2.5.16 Proposition The following are equivalent fo r an algebra A : ( a ) A w permanently spectral. (b) If B is a Banach algebra and ip: A —> B is an injective homomor phism, then Pb W ( o) ) = pA{a) V a e A. (c) I f B is a Banach algebra and ip: A —* B is an injective homomorphism with dense range, then ip(A) is a spectral subalgebra o f B. Proof (a) => (c): Proposition 2.5.15 applies since we can think of B as the completion of
□
The following result, due to Bertram Yood [1958], provides the simplest examples o f infinite-dimensional algebras satisfying the above proposition.
2.5.17 Theorem Let X be a Banach space and let A be a subalgebra of B( X) which includes Bf ( X) . If A is either closed or a spectral subalgebra of B(X), then any algebra norm on A is a spectral norm. Proof The last result in §1.7.15 shows that any algebra norm on A domi nates the operator norm. Either hypothesis implies that the spectral radius on A is less than or equal to the operator norm. Hence Theorem 2.2.5(e) shows that any algebra norm is a spectral norm on A. □ We have noted that the spectrum in a spectral algebra enjoys the prop erties of the spectrum in a Banach algebra. The next theorem and example show that an algebra is spectral if and only if its semisimple quotient can be embedded as a spectral subalgebra of a Banach algebra, but that the spectral algebra itself may not have an embedding into a Banach algebra.
2.5.18
Spectral Subalgebras and Topological Divisors of Zero
259
2.5.18 Theorem The following are equivalent for an algebra A : (•i) A is spectral. (b) There is some Banach algebra B and some homomorphism with the Jacobson radical as kernel satisfying pB(
A —> B
V a € A.
(c) A modulo its Jacobson radical can be embedded as a dense spectral subalgebra o f a semisimple Banach algebra. Proof (a) => (c): Let o' be an arbitrary spectral semi-norm on A. Corollary 2.3.4 and Theorem 2.2.13(d) show that a' induces a spectral norm |||•||| on A / Aj . Let (.4, HI •HI) l)e the completion of A / A j in this norm. Note that A/A.j is a spectral subalgebra of A by Proposition 2.5.15. The Jacobson radical A j of A has intersection {0 } with A / A j C A, since the intersection is an ideal of the semisimple algebra A/A j and in it each element has spectral radius zero. Note that the quotient spectral norm ||•|| on A / A j when restricted to the isomorphic image of A / A j induces a semi-norm a on A / A j which is spectral, since we have: PA/Aj(a
+ Aj)
+ A j) + Aj)
=
Pj(a + A j ) =
<
||(fl + A j ) + Aj)|| = a(a + A j )
pA/Aj((a
where we have used Theorem 2.3.3 and Proposition 2.5.15. Therefore ~AfAj can be identified with the completion of ( A/Aj , ||•||) and A / A j is a dense subalgebra. Hence Proposition 2.5.15 shows that A / A j is a spectral sub algebra of A /Aj . ’ (c) => (b): Immediate. (b) => (a): Corollary 2.5.10 shows that A / A j is spectral and then Proposition 2.4.4 shows that A itself is spectral. □
2.5.19 Not Every Spectral Algebra is a Subalgebra o f a Banach Algebra Since any Jacobson-radical algebra is spectral by Corollary 2.3.4, the example of H. Garth Dales [1981b] provides the desired example. See also J. Jacobus Grobler and Heinrich Raubenheimer [1991] and Lenore Groenewald and Raubenheimer [1988].
2.5.20
A Simple Unital Banach Algebra with a Non-spectral
N orm We give a Banach algebra A (due to Schweitzer [1993]) which is simple (i.e., it has no non-zero proper ideals), unital and a dense non spectral subalgebra of a C*algebra B called the irrational rotation algebra. Thus the C*-norm on A is a non-spectral norm. For any function F : Z x T —> C and any n G Z, define Fn : T —►C by i*n(C; = F(n, ( ) for all ( 6 T. Consider the Banach space A of all functions
260
2: The Spectrum
2.6.0
F : Z x T —> C such that Fn 6 C (T ) for each n € Z and ||F|| = ^ e |n|||Fn||00< o o
V F e A
n€ Z
This is a Banach algebra under the following twisted convolution product where a is an irrational multiple of 7r F * G(n, C) = Y , F (m - C)G(n - m, e,Qrr*C)
V F, G € A , n € Z; C e T.
mg Z
A Banach algebra defined in this way is called the crossed product of Z with C (T ) induced by the automorphism 0 ( f ) ( Q — f ( e ' aQ of C (T) . Then E ( n <0 — ^n0 (Kronecker delta) is an identity element for A. Consider G(T) as the subalgebra of all functions F € A with Fn — 0 for n / 0. Furthermore, A is a Banach ‘ -algebra under the involution F*(n,C) = F ( - n , e ,Q"C)*
F e A\ n € Z; C € T.
Theorem 10.2.8 shows that any Banach *~algebra has a largest C*-seminorm. It is a norm in this case. Let B be the completion of A in this norm. Let P : A —* C (T ) C A be the contractive linear projection defined by P ( F ) = F0. If 1 were a proper ideal in A, its intersection with C'(T) would have to be {0 }. Schweitzer shows that this implies P(T) = {0 } and then J = {0 }, so A is simple. Consider the following homomorphism T of A into B(C (T)): T f ( 9 )(C) = Y , F (n
V F e A , g € G(T); C € T.
n€Z
Theorem 4.2.10 below and the theory of ‘ -representations of crossed prod uct C*-algebras show that A is not a spectra) subalgebra of B since this irreducible representation is not the restriction of a ‘ -representation of B.
2.6
Numerical Range in Banach Algebras
In this section we will call a normed algebra norm-unital if it contains an identity element which has norm one. We will deal almost exclusively with norm-unital Banach algebras although some of the results also hold for incomplete norm-unital normed algebras. The numerical range of an element in such an algebra is a nonempty, compact, convex subset of the complex plane which contains the spectrum of the element. The numerical range is chiefly useful because it is better behaved than the spectrum under addition and limits. The elements of the algebra which have numerical range included in the real line also have important special properties.
2.?.1
Numerical Range in Banach Algebras
261
The numerical range of an element depends on the precise value of the norm on the two-dimensional space spanned by the element and the identity ele; lent. Therefore, it is not preserved under homeomorphic isomorphism of the algebra, but only under isometric isomorphism. On the other hand, ths numerical range is preserved under unital isometric linear maps which are not homomorphisms. The geometric character of the numerical range has advantages and disadvantages. The properties of Banach algebras in whi :h we are usually most interested are either purely algebraic (preserved under isomorphism) or topologically algebraic (preserved under homeomor phic isomorphism). The numerical range cannot reflect these two types of properties precisely. On the other hand, one might sometimes be able to choose a norm from among several equivalent ones, in order to apply nu merical range considerations to best effect. The excellent expository treatment by Frank F. Bonsall and John Dun can [1971] and [1973a] allows us to limit this soction to those topics actu ally used in subsequent chapters. Thus we omit entirely a discussion of the numerical range of operators on normed linear spaces and of elements in normed algebras which are not norm-unital, despite the historical impor tance of these topics. See also Vasile I. Istra^escu [1981] and its review. A numerical range was first defined by Otto Toeplitz in [1918] for op erators on Hilbert space. We define this concept and derive some useful results about it in Section 9.1 of Volume II. The definition we treat in this section was first given explicitly in full generality by Bonsall [1969a]. Howsver, many of the results and methods trace back to a paper by Gunter Lu.irer [1961], which was in turn based on work by Ivan Vidav [1956] and H. Free eric Bohnenblust and Samuel Karlin [1955]. Other early developments related to numerical range were given by Friedrich L. Bauer [1962] and Palner [1965], [1968a], [1968b] and [1970]. (The letter W in our notation is from the German name Wertbereich for the numerical range.) Defnitions and Basic Properties 2.9.1 D efin ition Let A be a norm-unital normed algebra. The set {w £ A* : w (l) = [Ml = 1} is denoted by A^y. For any element a € A the numerical range W(a) of a is the following set: ^
) = M a ):« 6 4 ].
(1)
The number N w = sup{|A| : A € W( a) }
(2)
is ca’led the numerical radius of a. The Hahn-Banach theorem shows that A ly is never empty. Clearly it is convex and compact in the ,4-topology (i.e., the weak* topology) on A ' ■ If A is the completion of A , then Ay/ and A'w can be naturally identified.
262
2: The Spectrum
2.6.2
Hence for any a 6 A the numerical range is the same whether calculated in .4 or in A Thus Banach algebra techniques can be used to obtain results on the numerical range of elements in a norm-umtal normed algebra. In the next proposition we formally record some more observations which are basic to all that follows. The last result is essentially due to James P. Williams (1967). 2.6.2 P ro p o s itio n Let A be a norm-unital normed algebra. (a) The numerical range of any element of A is a nonempty, compact, convex subset of C. (b) \V(a + b) C W(a) + W(b) V a, b e A. (c) H’ (1) = {1 }. (d) W'(Aa) = AW'(a) V A G C; a G A. (e) If { a „ ) n€N C A converges to a € A , then H '(o) C {lim A„ : A„ 6 W (an)} C n “ , U£LnW (an). (f) If A is complete (i.e., a Banach algebra), then Sp(a) C W(a) C ||a||C,. Proof The map ui ►-* o>(a) is continuous and affine from the nonempty, compact, convex set A'lv with the A topology. Since W (a) is the range of this map, W(a) is a nonempty, compact, convex subset of C. Conditions (b), (c), (d), and (e) are obvious from the definition of the numerical range. In condition (e) we can even take convergence to mean weak convergence. (f) By Corollary 2.2.8, the norm of A is spectral. Suppose A belongs to Sp(a). Then A — a lacks either a left or a right inverse, so that either ,4(A — a) or (A — a )A (or both) is a proper one-sided ideal of A. Let S denote one o f these which is a proper one-sided ideal. Each element of S is singular. Hence Proposition 2.2.9(b) (with a = 1) shows Jjl —6J| > 1 for all b G S. Thus the Hahn-Banach theorem shows that we may choose an w 6 A ' such that w (l) = 1 = ||uj|| (i.e., uj £ -4 ^ ) but u>(b) = 0 for all b € S. Therefore w(X - a) = A — uj(a) = 0 implies A belongs to W^a). The last inclusion follows from |w(a)| < ||w|| ||a|| = ||a|| for all ui 6 A Jy. O The next lemma is essentially due to Bohnenblust and Karlin [1955]. Note that the first thm- expressions in equation (3) are equal even if A is just a normed algebra. This follows from our previous remark on the numerical range in the completion of A. The last two expressions in equa tion (3) are not defined in general unless .4 is complete. The exponential function in a Banach algebra was introduced in Theorem 2.1.12 2.6.3 L em m a Any element a o f a norm-unital Banach algebra A satisfies the following equations.
2.S.3
Numerical Range in Banach Algebras sup{Im(A) : A e W '(a)}
=
in f{i~ 1( II1 — *<“ 11 — 1) : £ > 0}
=
lim t- 1 (||l - ita\\ - 1) t—*0+
= =
263
(3)
lim tt~l log log lie lle 'fa|l
( — 0+
s u p f r 1 log||e_,(a|| : t > 0};
IMIjy = SUP{IA|—1 log ||eAa|| : A 6 C, A / 0}.
(4)
Proof If a is 0 there is nothing to prove so we assume a is not 0. Let The last proposition shows that this is finite and satisfies |s| < ||a||. For any t > 0 and any u> € we have -'iw(a) = t- 1 (w (l — ita) — 1). Hence we see Im(w(a)) = R e(—iu;(a)) < 1(||1 - i£a|| — 1) so s be sup{Im(A) : A € W^(a)}.
s < i n f 1(||1 — ita|| — 1) : t > 0}.
(5)
For any t € R with 0 < t < ||a||_1, the Hahn-Banach theorem gives an W c ^ ' satisfying w (l —ita) = ||u;|| ||1—*ta|| = 1. The map b >-* u;(b(l —ita)) from A into C clearly belongs to A\y. Using this map we get 1 -M 2||a||2
>
||1 + t2a21| = ||(1 -I- ita )(l - ita)||
>
||1 — ita||Re(a>((l + ita )(l — ita)))
>
||1 -*ta||(l - t s ) > (1 -«||a||)2 > 0.
Rearranging the inequality 1 -f t2\\a\\2 > ||1 — ita\\(l — ts) we get ||1 — ita|| - 1 < s + t||a||2 t ~ I — ts for a ll 0 < t < ||a||-1 . Hence, using (5), we conclude s < in f{t- I (||l — ita|| — 1) : t > 0} < lim sup t_1(||l — ita|| — 1) < s. t—o+ Thie confirms the first two equalities of (3). It is easy to see that there is a constant B satisfying |||e- *
for a ll |t| < ||a||-‘ .
Since the second limit is already known to exist, this gives lim «- 1 (||e_i*0||- 1) = lim t-o+ <—o+
1 - tta|| - 1).
Similarly, we can easily find a constant C satisfying |||e~'fa ||— 1| < Ct for £.11 |t| < ||a||_1. Expanding the logarithm function around 1 gives
264
2: The Spectrum
2.6.4
These two equations confirm the third equality of (3). To obtain the last equality of (3), we note that the function f ( t ) = log ||e~'ta|| is subadditive for t > 0. Hence for t > 0 and n G N , we have f ( t ) < nf(t/n). This implies < (t /n )_1/( t /n ) . Thus, given any value of t ~l f ( t ) , there are arbitrarily small values of s > 0 such that s~lf(s) > Since the limit is known to be finite and not greater than the supremum, we conclude lim(__o+ *-1 /(* ) = SUP{*- l / ( 0 : t > 0}Equation (4) can be derived as follows: IWIw
=
sup{|A| : A G W (a )} = sup{Im(A) : A G W (£a);C
=
suP{ r 1log||e-,<‘ “ | | :t > 0 ,C € T } .
G
T}
□
This lemma can be used to justify the remark, in the introduction to this section, that W(a) depends only on the norm on the two-dimensional subspace generated by 1 and a. For any A € C and ( 6 T , Proposition 2.6.2 shows W (A + (a) = A + C,W(a). Thus, we can freely translate and rotate W(a) in the complex plane. Therefore, the second or third expression for sup{Im(A) : A € W (a)} gives a complete description of the support lines of IV(a), find thus a complete description of the closed convex set H^(a). The next theorem is also essentially due to Bohnenblust and Karlin [1955]. 2.6.4 T h e o re m Let A - a norm-unital Banach algebra. The numerical radius is a (linear space) norm on A which is equivalent to the original norm o f A and satisfies P{a) <
lla liw
^
N l
<
e | | a | lw
(6 )
Proof Proposition 2.6.2(f) shows that the numerical radius is a semi-norm satisfying p(a) < HaH^ < ||a||. La order to obtain the final inequality, we let be a primitive nth root of unity (e.g., C,n = exp(27ri/n)). Then 5Z*=i CnJ = 0 f°r any integer j which is not divisible by n. Let r be a positive number. Then we have i
"
(*> 7=0
J
‘
However, by choosing n sufficiently large we can make the right side as close as we please to n -1 ]T]fc=i(Cn/r ) - l e ^'’ " /^ a ( e- 9-> ma^e the difference less than e > 0, choose n so that X ^ l n+1( j!)~ 1(llall/r )n < e /r ). Using equations (7) and (4), we conclude ||o|| < rer for all r > 0. Hence l|a||vv- - 0 implies ||a|| = 0. If ||a||w is non-zero, we choose r = IMI^ to obtain the desired inequality. O We axe now in a position to give an easy proof to the final statement of Proposition 1.1.13 which we left unproved before. The reader is referred
2.6.5
Numerical Range in Banach Algebras
265
to the original statement for notation. We use the numerical range and numerical radius for (A 1, ||•||/<). Since 0 is in the closure of the numerical rang:; of any a € A, we get the elementary estimate ||A+ a||/j > ||A+ a||iv > max |A|, ||a||w/2} which gives
This completes the proof of Proposition 1.1.13. Sym: letric Elements V'e turn now to the theory of those elements in a norm-unital Banach algebra which have real numerical range. Here the terminology is not quite settkd. Vidav [1956] called these elements self adjoint. Lumcr [1961] and Earl Berkson [1963] called them hermitian. Palmer called them symmetric (and with another definition, self-conjugate) [1965; 1968b]. The term “her mitian” was adopted in the exposition by Bonsall and Duncan [1971], and therefore this term should probably be considered standard. However, in this work we use the word “hermitian” , in a different sense. Therefore, we have retained the term “symmetric” in the next definition. 2.6.5 D efin ition An element h in a norm-unital Banach algebra is called symmetric if its numerical range is included in the real axis. The set of symmetric elements in a norm-unital Banach algebra A is denoted by A sN )te that ,4s is a real linear space which is closed in the weak topology and hence in the norm topology o f A. In order to keep the proof of Theorem 2.6.7 elementary we include a lemm i with an elementary proof. The result could also be derived from a (no:i-elementary) application of the functional calculus (Theorem 3.3.7). This lemma is based on one first given by Bonsall and Michael J. Crabb [1 9 7 0 ’ .
In the following lemma we need to know that the power series for arcsir(a) converges unconditionally when a belongs to a norm-unital Bar nach algebra A and sup{||a2n+1 [| : n € N} is finite. The easiest way to show this here is by comparison to the binomial series for (1 + 1 ) 1/ 2, which we show ,o converge absolutely for Irrl < 1 in the discussion preceding Theorem 3.4.5. We have
Denoting the nth coefficient by rn, we see r,
=
§ S + f l ( n + 2l ) | -
T h U S
t h e
power series rnl2n+1 converges absolutely and uniformly for |x| < 1 so that its sum is continuous, which implies = arcsin(l) =
266
2: The Spectrum
2.6.6
Hence if a € A satisfies ||a2n + ,|| < B for all n € N, then the series satisfies
£ h
||rna2"*»|| < £
( 1/2 ) B < 2 5 f ; | f 1/2)
ll' n L „ 2 « + iW
ii
= 25,
"
where the last equality depends on equation (4) before Theorem 3.4.5. We use an elementary portion of Gelfand theory in the following proot, even though that theory is not formally introduced until the next chapter. 2.6.6 le m m a Let A be a complex unital Banach algebra. Let h be an element of A unth spectrum contained in the ojhu internal ] - ^ . * [ o f the. mil axis. Then we get h = arcsin(sin(/i)), where the arcsine and sine functions are defined by their (convergent) power series expansions. Proof We may assume that A is commutative by replacing it with a max imal commutative subaigebra if necessary (cf. Proposition 2.5.3(a)). Let 1' be the Gelfand spate of A. Lot n and t bt; real numbers with |a|p(/i) < |. Thus we have p(sin((s + it)h))
=
sup{|7 (sin((s + it)/i))|} -rer
=
sup{|siii((s-H«)7(/»))|} Trer
— sup{| sin(s7 (/i)) cosh(t7 (/»)) ■>er + icos(s7 (/i)) sinh(<7 (ft))|} <
|sin(sp(/i))| cosh(tp(h)) + |sinh(tp(/»))|.
Therefore there is an open set U containing [-1 ,1 ) consisting of A = s + it in the complex plane for which p(sin(A/i)) < 1. As we have seen, the power series r«^ 2" +1 ^or arcs'n(,l0 converges unconditionally for any h e A with {||/» +1|| : n € N} bounded, and hence for any h € A with p(h) < 1. Thus /(A ) = arcsin(sin(A/i)) is defined by a convergent double series for A 6 [/, so /(A j is the composition of two analytic functions and hence is analytic for A € U. However, if |A| is sufficiently small, XmLo t)r >s less than 1 so the double series for /(A ) converges unconditionally and can be rearranged to give /(A ) = Xh. Thus by the uniqueness of power series expansions, /(A ) = Xh for all A € U ■ Since 1 € (/, the lemma is proved. □ The equivalence of (a), (b) and (c) in the next theorem is essentially due to Bohnenblust and Karlin [1955] but was first explicitly given recognition by Lumer [1961]. Results (f), (g) and (h) were discovered by Vidav [1956]. Condition (e) has a more complicated history. The first equality was proved by Vidav [1956]. At the same time he derived the full equation in a very special case. Allan M. Sinclair [1971] and Andrew Browder [1971]
2.6.7
Numerical Range in Banach Algebras
267
each obtained the full result independently. The present proof is based on a proof by Bonsall and Crabb [1970]. Condition (d) is merely a restatement of some of the other results, 2.6.7 T h eorem Let A be a norm-unital Banach algebra. The following th~?s conditions are equivalent for an element h € A : (a) h e A s . (b) ||e'"* ||= 1 V t G R. (c ) Jim ^ - ‘ (lll - it/»|| - 1) = 0.
Ptirthermore A s and any h, k € A s satisfy: (cl) co(Sp(h)) =■ W(h) C K. (‘0 P(h) = IHIw -- INI( f ) A s n i As = {0 } and A s + *«4s is closed. (K) i(hk - kh) e As(h) ||/»-*Jfc|| < ‘2\\h f iJfc||. Proof The equivalence of (a), (b) and (c) is an immediate consequence of Lemma 2.6.3. For the rest of this proof let h and k be elements of .4s. Theorem 2.6.4 shows that (e) follows from the inequality ||/i|| < p(h) which we will now establish. Let t € R+ satisfy p(i 1/i) < |, so Proposition 2.6.2(f) implies Sp(t_1/i) C ( —7r/2, 7t/2). Since <-1 /i is symmetric, we also get ||sin(t~'/i)|| = 2- I ||e*f 1/1 - e~ft lfc|| < 1. Lemma 2.6.6 then shows r^l/tH = ||arcsin(sin(t_ I /i))|| < r«Hsin(<_I/l)ll ^ f> where ( r" } 58 the sequence of coefficients which was defined before Lemma 2.6.6. We have sho’.vn that p(h) < implies ||/i|| < t%. Hence we conclude ||/i|| < p(h), so that (e) is proved. (a): Since W( h) is a nonempty compact convex subset of R, it has the "or:n [s, £] for suitable s, t € R. Thus (e) shows p(h — s) = ||/» — sH^, = t — s = ||ft — <11^, = p(h — t). This shows that both s and t belong to the spectrum. Since Sp(h) C W( h) was already proved in Proposition 2.6.2, we conclude co(Sp(h)) = W(h) C R. (f): The first result follows from (d) and (e). We must show that A s + iAs is closed. Suppose { h n+ i k n} nen converges to a e A with hn, kn e AsFor any n, m 6 N, Theorem 2.6.4 and (e) show jl^n
^m|l
—
li^n ~ ^ m llw = ®up{|u>(/ln
<
sup{|w((/i„ + ikn) - ( hm + *fcm))| : w € A
^ m )| • ^ e A ^ y }
~
II(hn -f- ikn)
<
\\{hn + ikn) - {hm + ikm)\\.
(hm + ^fcm)l|w
Hence {/» „ } „ e> is a Cauchy sequence in the real Banach space A s- Let its limit be h e As- Similarly {fc„}neN has a limit k 6 As- Thus a = h + ik belongs to .4s + i A s , which is therefore closed.
268
2: The Spectrum
2.6.8
(g): Condition (b) implies Ue,'* e“ />e~ 1*ke~ ,shH = 1 for all s 6 R. By expanding the exponentials, we see that there is a constant B satisfying HI 1 + s2(hk - fcfc)|| - 1| < Bs3
V s; 0 < s < 1.
Letting t be s2 we see lim <~1( ||1 - it(i(hk - kh))\\ — 1) = 0. ( - . 0+
Lemma 2.6.3 shows that no number in the numerical range of i(hk — kh) has positive imaginary part. By interchanging h and k we also exclude numbers with negative imaginary part. Hence i(hk — kh) belongs to ,4s. (h): We have
||A|| = ||A||W
=
s u p { K A )|: We ^ }
<
sup{(w(ft)2 + u)(k)2) 1/2: w € -4 ^ }
=
sup{|u>(/i + ik)\ : u> G A\y}
=
P
+
*fc | ln / <
ll^ +
*fc||-
Similarly we get ||&|| < ||A + t&||, which implies ||A — ifc|| < ||/«|| + ||fc|| < 2\\h + ik\\. a Symmetric Elements in Algebras The following two propositions will be used in Theorem 9.5.8 of Volume II to characterize C*-algebras (cf. §1.7.17). The equivalence of (a) and (b) is proved by an argument in Palmer [1968a], but was first explicitly stated by Ellen Torrance [1970]. The involution in part (c) was first studied by Vidav [1956]. When A v is an algebra, it is a C*-algebra under the involution *: A v ~+ Av2.6.8 T h e o re m Let A be a norm-unital Banach algebra. Denote the set A s + iA s by A v - Then the following are equivalent: (a) A v is an algebra. (b) h 6 -4,s implies h2 G A v (c ) A v w a closed subalgebra tn whtch each element has a unique expres sion o f the form h + ik for h, k G -4.s • Furthermore the map *: A v —* A v defined by (h + ik)m= h - ik for all h, k G A s is a real-linear homeomor phism with norm at most 2 (and an isometry with respect to the numerical radius) which satisfies: (a + b)' — a* + 6*, (Aa)* = A*a", (a6)* = b*ar and (a*)* = a for all a, b G A v and A G C. Proof (c) => (a) => (b): Obvious. (b) => (c): Theorem 2.6.7(f) shows that A v is a closed linear subspace. Let h2 equal k + ij where h, k and j belong to A s- The identity h(k + ij) =
2. >.9
Numerical Range in Banach Algebras
269
(k + ij)h implies hk —kh = i(jh —hj). Theorem 2.6.7(g) shows that hk —kh be’ ongs to both A s and tAs- Theorem 2.6.7(f) shows hk —kh = 0. Thus h ctr imutes with k and hence with j. Proposition 3.1.6 and Theorems 2.1.10 ar( 2.6.7(d) give iSp(j) C Sp(h3) — Sp(k) C Sp(h ) 2 — Sp(k) C R and Sp'j) Q R. so j is 0 by Theorem 2.6.7(f). Therefore h2 belongs to As for a'i ,i 6 ,4s. Let h and k be arbitrary elements of A s; then h + k and therefore hk + kh = (h + k)2 — h2 — k2 belong to A s- This, in addition to Theorem 2.6.7(g), gives 2hk = hk + kh + i[i(kh — A&)] € As + »A s = A v ■ Since Av is the linear span of As, it is an algebra. Theorem 2.6.4 shows As H tA s = {0 } so that the expression h + ik with h, k € As is unique. Thus, as a real-linear space A v has a direct sum decomposition Av = A s® »A s- Hence the map *: Av —* A y is well-defined. It ii obviously real-linear and it is a homeomorphism (with norm at most 2) >y Theorem 2.6.7(h) and an isometry in the numerical radius by the definition. Clearly it satisfies the first, second, and third equations asserted in t iis theorem. To see that it satisfies the fourth equation, it is enough to coii rider h, k 6 A s because of the conjugate linearity of *. However, the resiits we have already obtained show 2(hk)* = hk + kh —i[i(hk —fc/i)]* = hi; t kh + i[i(hk — kh)} = 2kh = 2k*h* so this fourth equation is also satisfied. □ The next proposition is an adaptation to the present setting of argume; ts used by Kaplansky (cf. Joseph A. Schatz [1953]). We will draw the natural conclusion from this proposition in Theorem 9.5.8 of Volume II, after introducing several additional concepts. 2.6.9 P r o p o s itio n Let A be a noTm-unital Banach algebra satisfying the equ valent conditions o f Theorem 2.6.8. Define A v and *: A v —* A v as in that theorem and let A q denote {h € A : W( h) C R+ }. ■a) For each h € A s there are unique elements p, q € A q n {ft}" satis fying h = p — q and pq = qp = 0. Hence we have A s = A q —A q
(b) For each a £ A v ,
and
A q fl (—A q ) = {0 }.
belongs to A q and satisfies
Fro>f (a): It is enough to consider h G A s satisfying \\h\\ < 1. Let h be sue! an element and let C denote {/»}". Then SpA (h2) = Spc(h2) C [0,1] and Theorem 2.6.7(e) show W ( h 2) C [0,1] and ||/i2 — 1|| < 1, since h2 is symmetric. Hence k — 1 + C!,2) (h2 ~ ! ) n belongs to As DC and sat isfies W(k) C [0, 1] and k2 = h2, since the series converges unconditionally (cf. the discussion before Theorem 3.4.5). Define p and q by p = 2~l (k + h) and q = 2~1( k - h ) . (Again we use a simple argument based on the Gelfand
2: The Spectrum
270
2.6.9
theory introduced in Section 3.1.) Each 7 £ IV satisfies iip )
=
2 - i [ f ; ( 1f ) ( 7 ( / * 2 - i n + 7 ( / * ) ] n=0 ' '
=
2 - '( ( 7 {h2) ) l/2 4- 7 W ) = 2 - 1(|7 (/l)| + 7 (fc)).
Theorems 3.1.5 and 2.6.7(d) imply W(p) = co(5p^(p)) - co(Spc(p)) Q R+. Similarly W (q) C R + implies that p and q belong to A q . We conclude 4pq = (k + /i)(fc - h) = k2 - h2 = 0 = (/c - /i)(fc + h) = 4qp. Theorem 2.6.7(d) shows the uniqueness of p and q. (b): The identities W(It) = co(Sp(h)) and ||/i|| = H/iH^ = p(h) for a symmetric element h in A from Theorem 2.6.7, which are used repeatedly in the rest of this proof, will not be cited again explicitly. Suppose 6 £ A v satisfies b = h + ik with h, k £ arid -66* 6 A q . Theorem 2.1.8(a) shows W ( - b ' b ) C co(Sp(~bb') U {0 }) = co(Sp(-b*b) U {0 }) C R+ . The identity 6*6 + 66* = 2(/i2 4- k2), Theorem 2.1.10 and Proposition 2.6.2 show W'(6*6)
C
2 W( h2) + 2 W( k 2) 4- W ( - b b ' )
=
2co(Sp{h))2 + 2co(Sp(k))2 + W ( - b b ' ) C R+ .
Hence W (6 *6 ) C ( - R + )i iR + , so 6*6 is 0. Similarly 66* and, in turn, h? + k 2 are 0. For any w £ A w , w(/i2) £ co(Sp(h))2 and ui{k2) £ co(Sp(k))2 imply 0 < ui(h2) < ui(h2 4- k2) = 0. Therefore p(h) = p(>i2) 1/ 2 = 0 implies that h is 0. Similarly k, and therefore 6, are 0. For any 6 £ A v , conclusion (a) shows 6*6 = p — q with p, q £ A q and pq = qp = 0. Hence -(b q)'b q = —q(p — q)q = q3 £ A q . We have just shown that this implies bq — 0. Hence we get q2 = —b*bq = 0. As before, this implies q = 0 . Thus 6*6 = p belongs to A q . For h £ n A v , t £ R and u> £ A*w , the result just proved implies t2 4- 2tui(h) 4- ui{h2) = ui((t 4- h)2) > 0. Thus the discriminant of this quadratic is nonpositive which implies u>(h)2 < u>(h2). We will use this result with an arbitrary a £ A v which can be written as h 4- ik for h, k £ A s H A v ■ Thus we find ||a*a||
=
p(a*a) = (p(a*a) 4 -p(aa*))/2 = (||a*a|| 4 -||aa*||)/2
>
||a*a 4- aa* ||/2 = ||/i2 4- k2\\ > sup{w(/i2 4- k2) : ui £ A Jy}
=
sup{w(/i)2 4- ui(k)2 : ui £ A l y } = ||a||vv-
^
See also James P. Williams [1967], Crabb and Duncan [1979], Husain and Anitha Srinivasan [1979], Subhash J. Bhatt [1983], Bonsall and Duncan
2.6.10
Numerical Range in Banach Algebras
271
[1980], Roger R. Smith [1979], [1981], J. Martfnez-Moreno, A. Mojtar-Kaidi and Angel Rodri'guez-Palacios [1981], Martfnez-Moreno and Rodriguez?f.’i dos [1985], Thanassis Chryssakis [1986], Crabb and McGregor [1988] and A. K. Gaur and Taqdir Husain [1989], Puuier Bounded Elements We wish to use Crabb’s Lemma 2.6.6 to give some results on power bounded elements. Theorem 2.6.12(a) is contained in Gelfand [1941c] but the very simple proof using Crabb’s leinma is due to Allan and Ransford [1989]. Part (b) was first proved by Katznelson and Tzafriri [1986], but its proof is based on the above paper and Allan [1989]. For extensions of these, results and other proofs, see the above papers Allan, O ’Farrell and Ransford [1987], Esterle, Strouse and Zouakia [1990] and Vu Quoc Phong [1992], 2 .5 .10 D efin ition Let A be a normed algebra. An element ( a € A / a 6 A g ) is said to be ( power bounded / doubly power bounded ) if there is some finite constant B satisfying ||a||" < S for all ( n € N / n 6 Z ). Gelfand’s spectral radius formula shows that the spectrum of a ( power bounded / doubly power bounded ) element a in a Banach algebra is in cluded in ( D / T ). In either case, Proposition 1.1.9 shows that there is an equivalent norm with B = 1. 2.8.11 L em m a Let A be a unital commutative Banach algebra and let a € A be power bounded and satisfy p(a) = 1. Then there is a Banach algebra (B, |||■|||) and a unital algebra homomorphism i p : A —>B satisfying: (a) IIM10III = lim s u p „_ 00 ||a"6|| for all b £ A. (b ) ip(a) is invertible in B and satisfies |||v5(a)||| = |||
272
2: The Spectrum
2.6.12
2.6.12 T h e o re m Let A be a unital Banach algebra. (a) If a € A s » doubly power bounded, then Sp(a) = {1 } is equivalent to a = 1. (b) If a € A is power bounded, then Sp(a) n T C {1 } is equivalent to limn_ 00a” (l - a) = 0. Proof Proposition 1.1.9 allows us to assume ||an|| < 1 for ( n € Z / n € N° ) in case { (a) / ( b ) ). We may also assume that A is commutative by replacing the original algebra by one of its maxima] commutative (necessarily unital) subalgebras including a. (a): Obviously, a = 1 implies Sp(a) = {1 }. Conversely, Sp(a) = {1 } implies Sp( 1 - a) = {0} and 1 - a € A j . Define h to be i $^£1,(1 - a)k/k 6 A j so a — e 'h. For any integer m,sin(mh) 6 A j implies Sp(sin(mh)) = {0}. Lemma 2.6.6 shows h = arcsin(sin(/i)) and the bound on ||an|| gives
Before Lemma 2.6.6 we noted that the coefficients r/t in the Taylor expan sion of the arcsine satisfy J2T= i lrfcl = 7r/^- Hence we get OO
||m/i|| = ||arcsin(sin(m/i))|| < ^
|rfc|||(sin(m/i))t || <
Since m € N is arbitrary, this implies h = 0 and a — 1. (b): For any A 6 Sp(a) n T , An(l —A) € Sp(an( 1 —a)) implies |1 —A| < ||an(1 —a)||. Hence Iim „_00 an( l —a) = 0 certainly implies 5 p (a )n T C {1}. Suppose, conversely, that this inclusion holds. If the intersection is empty, limn-.ct an — 0 so we are reduced to considering the case Sp(a) n T = { 1 } . Apply the last lemma. Then ip(a) satisfies part (a), so it equals 1 6 B. Exjuivalently, ip( 1 — a) = 0 which implies lim „_ 00an(l — a) = 0, as we wished to show. □
2.7
The Spectrum in Finite-Dimensional Algebras
In this section we will study the spectral theory of finite-dimensional algebras. Actually the whole section will be devoted to the spectral theory of linear operators on finte-dimensional linear spaces. This accomplishes the goal stated in the first sentence since any finite-dimensional algebra A may be isomorphically embedded as a subalgebra of an algebra C(A) of all linear operators on a fin’te-dimensional linear space V in such a way that the spectrum of an element a € A is the same as the spectrum of its image in C{A).
2 . 7.1
The Spectrum in Finite-Dimensional Algebras
273
Let A be an arbitrary algebra. Then the extended left regular repre sent;., ion Ll is always an isomorphism and the left regular representation L is ill isomorphism if and only if the annihilator ideal A la is {0 }. After Proposition 2.5.3, we noted that any element a in any algebra A satisfies SpA (a) = Spc(Ai){Li) and
(1)
A la is {0 }, then it also satisfies S p A ( a)
= S p C( A ) ( L a ).
(2)
Thus all spectral questions about an algebra A may be translated into similar questions about the algebra £ (V ) where V is a vector space. When A is finite-dimensional, V may be chosen finite-dimensional, since it can be chosen as either A or A 1. We will give an extended discussion of spectral questions about £(V ) where V is a finite-dimensional linear space. In particular we will show that ail operators in £ (V ) are spectral in the sense of Nelson Dunford [1954] and that the proof of this fact is the major step in the construction of the Jordan canonical form. 2.7.1 C on ven tion For the rest of this section V will be a finite-dimensional linear space with dimension n. The word “operator” will always denote an element of £(V ). In Section 1.7 we reviewed the well-known isomorphism between £ (V ) and the algebra Mn of all n x n matrices, and we will refer to this isomor phism when convenient. 2.7.2 T h e C h aracteristic P oly n om ia l In Section 1.7 we noted that the de .erminant defines a multiplicative map Det: £ (V ) — C which takes the value 0 at an operator T 6 £ (V ) if and only if T is not invertible. Hence we conclude Sp(T) = {A € C : Det(A - T ) = 0}. Any o the usual procedures for calculating determinants in terms of a matrix representation shows that for a fixed operator T, Det(A — T) is a monic polynomial of degree n. We define the characteristic polynomial of T, denoted by chx, by chT(X) = Det(A - T).
(3)
Since the complex field is algebraically closed, the characteristic polynomial a1ways has at least one and at most n roots. Hence Sp(T) is a nonempty set of v or fewer complex numbers.
274
2: The Spectrum
2.7.3 R an k and N ullity
2.7.3
Any operator T 6 £ (V ) satisfies
ranlc(71) + null(T’) = n
(4)
where rank(T) is the dimension of the subspace T(V ) and null(T) is the dimension of ker(T). (This is easily proved by extending a basis for ker(7’) to a basis for V and then considering the image of this basis under T .) In particular an operator is non-invertible if and only if it has a nontrivial kernel. Thus a complex number A belongs to the spectrum of T if and only if there is at least one non-zero vector x satisfying (A - T )x = 0. This motivates the following terminology. 2.7.4 C h aracteristic V alues and C h aracteristic V e cto rs A char acteristic vector for a linear operator T € £ (V ) is a non-zero vector x € V for which there is a complex number A satisfying T x = Ax. A complex number A satisfying this relationship with a characteristic vec tor x of T is called a characteristic value of T, and A and x are said to rorn-hpond to each other. I'he terms “eigenvector” , ami “proper vector” are synonyms for “char acteristic vector” and the terms “eigenvalue” , ami “proper value” are syn onyms for “characteristic value” . In terras of this definition the spectrum of any operator T can be de scribed as the set of characteristic values of T and these are the roots of the characteristic polynomial of T. Since the complex field is algebraically closed, the characteristic poly nomial factors into linear factors m (5 )
where Ai,A2, . . . , A m is a list of distinct characteristic values of T. The exponent nk corresponding to the characteristic value A* is called the alge braic multiplicity of A*. The geometric multiplicity of a characteristic value Aj is defined to be the dimension of ker(Aj — T). Later in our discussion it will be obvious that the geometric multiplicity is less than or equal to the algebraic multiplicity, and they are equal only for the diagonalizable operators we are about to discuss. 2.7.5 D iagonalizable O p era tors The simplest class of matrices to deal with (outside of complex multiples of the identity) is the class of diagonal
n
/a.
The Spectrum in Finite-Dimensional Algebras
275
matrices (i.e., those matrices which have all their non-zero entries on the main diagonal). The product of two diagonal matrices is their pointwise >-3c‘ v and hence any two diagonal matrices commute. An operator S on £ "nite-dimensional space V is called diagonalizable if it is represented by a diagonal matrix relative to some ordered basis. Note than an operator S has a diagonal matrix relative to an ordered basis if and only if each basis vector is a characteristic vector of 5. Hence S is diagonalizable if and only if V has a basis of characteristic vectors for S. Y^hen S is represented by a diagonal matrix, the characteristic values of S, repeated according to their algebraic multiplicity, are just the diagonal entries of the matrix. ” S is diagonalizable, the choice of an ordered basis of characteristic vcc;o.”s is highly nonunique. The order of the basis vectors is clearly in determinate. Of course, any basis vector can be replaced by any of its non-zero multiples. If any characteristic value A has algebraic multiplicity greater than 1, then ker(A — S) may be more than 1-dimensional, and the choice of basis vectors within it is completely arbitrary. Most of this inde terminacy is removed by substituting a list of suitable canonical projection operators onto the linear subspaces of characteristic vectors of S for the choice of an ordered basis of characteristic vectors. In order to study this appro \ch, we need to introduce the concept of the minimal polynomial of a linear operator T on a finite-dimensional linear space V. 2.7.6 M inim al P olyn om ials We could begin this discussion by proving the Cayley-Hainilton theorem which asserts chr (T) = 0
(6)
for all T € £(V ). (What is wrong with the simple argument : “c/it(A) = Det(A T) so chx(T) — Det(t — t) = 0” ?) However it is logically simpler to note that £ (V ) has dimension n2 (since it is isomorphic to the algebra Mn). Hence the powers I, T, T 2, ... , T n of T are linearly dependent so that there is some monic polynomial p satisfying p(T) = 0. The well-ordering of the integers allows us to choose a monic polynomial m ir of smallest degree satisfying • mt7’ (T) = 0. (7) This is called the minimal polynomial of T. It is denoted simply by mi wh?:- T is understood. If p is any polynomial satisfying p(T) = 0, then mix civid2f p. (To see this, note that polynomial division gives polynomials q a-id ~ satisfying p = (mi)q 4- r, with the degree of r strictly less than the degrei of mi. Hence by the minimality of the degree of mi we have r = 0.) This shows that miy is well-defined. 2.7.7 P rop osition / / {Ai, A2, . . . , Am} is the set o f distinct characteristic values of an operator T, then the minimal polynomial o f T has the
276
2: The Spectrum
factorization
2.7.8
m
(8>
>=i where each a} is positive.
Proof In order to prove this, we will assume the truth of (8) with each aj positive and then prove that {Ai, A2, . . . , Am} is the set of characteristic values. First we have 0 = t t i i t { T ) = ( T — Aj ) q j ( T ) for each j and a suitable polynomial qy Since q} ( T ) is not 0 (by the minimality of the degree of mix), we see that there is a non-zero vector x = Qj ( T) y satisfying (Aj —T ) x — 0. Hence Aj is a characteristic value of T. Conversely, if Aq is a characteristic vhJik- <>f 7 with corresponding characteristic vector x, awl if p is any polynomial, then p (T )z = p(Ao)x. Since this equals 0 if and only if (A - Ao) divides p(A), we see that A —Ao divides mix, for each characteristic value Ao of T. □
2.7.8 Ascent o f an Operator In (8) the exponent aj corresponding to the characteristic value Aj is called the ascent of Aj. (The term comes from the fact that the sequence of subspaces ker(Aj — T ) C ker((Aj — T )2) C C ker((Aj — T ) k) is strictly ascending until k = a3 and is stationary thereafter. This can be proved by arguments similar to those just given.) If T is diagonalizable, it is obvious that each characteristic value has ascent one. The converse is easily established after some remarks on projection operators.
2.7.9 Projection Operators Recall that an operator E € £ (V ) is called a projection operator if it is idempotent (i.e., if it satisfies E 2 = E). In this case I —E is also a projection operator and E is said to project V onto the linear subspace EV = {x € V : E x = x } = ker(7 - E) along the linear subspace ker(£) = {x e V : E x = 0} = ( / - E)V. Clearly these linear subspaces satisfy EV fl ker(£) = {0 };
EV + ker(E) = V
so that E determines, and is determined by, the direct sum decomposition V = E V © ker(E). Two projection operators E and F are said to be orthoqonal if they satisfy E F — F E = 0. This is equivalent to the condition EV n F V = {0 }.
2.7.1r
The Spectrum in Finite-Dimensional Algebras
277
2.7.10 Invariant and D e co m p o sin g S u bspaces If E is idempotent, then the subspace EV is invariant under an operator S if and only if E satisf ■; E S E = SE. Hence the direct sum decomposition E V ® ker(.E') decomposes S {i.e., both EV and ker(ij) = (I — E)V are invariant under S) if and only if E and S commute. This is shown by the calculation: ES = E S E + E S( I - E) = S E + E ( I - E) S( I - E) = SE. We are now ready to determine the detailed structure of the diagonalizable operators. This is done here to prepare the reader for the more complicated structure theorem we will later present for arbitrary operators. Any of the implications in the next theorem is easy to establish directly, but we will use a fairly efficient circle of implications. 2.7.11 T h e o re m The following conditions for S € £ (V ) are equivalent. (a) S is diagonalizable. (b) V has a basis consisting of characteristic vectors of S. (c) The minimal polynomial of S has no multiple roots. (d) There exist distinct complex numbers Ai, A2, . . . , Am and mutually orthogcnal non-zero projection operators E i , E 2,. .. ,E m satisfying m
I = ^ Ej i =i
and
SEj = AjEj.
(9)
(e) Condition (d) holds and Ai, A2, . . . , Am are the distinct characteristic values of S. The minimal polynomial o f S is given by: m
miS(X) = Y [ ( X - X j )
(10)
3=1
and the projections in S
E2, ■■., Em art given by the following polynomials m
Ej = Q j ( S ) ,
and satisfy
where
qj(X) = JJ( (A - Afc)/(Aj - Afc)) k=l
(11)
m
S =
Xi Ei j= i
in addition to (9).
and
Ei V = ker(A; "
(12> m
278
2: The Spectrum
2 .7.12
Proof ( a ) o (b ) : Each vector in the basis relative to which the operator has a diagonal matrix must be a characteristic vector, and conversely the matrix relative to a basis of characteristic vectors would be diagonal. (b)=>(c): As usual, we denote the distinct characteristic values of S by Aj, A2, ••■, Am. For each characteristic value A; of S, Aj —S will annihilate each basis vector which is a characteristic vector with Aj as its characteristic value. The other characteristic vectors will simply be multiplied by some non-zero complex number. Hence, applying the polynomial — Aj) to S , gives 0. Since the minimal polynomial must therefore divide this one (it is easy to see that they are actually equal but we do not need this here), the minimal polynomial has no repeated roots. (c)=>(e): Let the minimal polynomial be given by (10) which determines the set Ai, A2, . . . , Am. For each j consider the polynomial q: arid the o(>erator Ej in equation (11). Since the degree of qj is rn — 1, Ej is not 0. However by factoring the minimal polynomial we see that (Aj — S ) E j is 0. Hence each Aj is a characteristic value of S. If A is a characteristic value of S, with characteristic vector x, then mis (S)x = mis(X)x, so A is one of the roots of the minimal polynomial and {Ai, Ai, ••, Am} coincides with the spectrum of S. Similarly, the minimal polynomial shows that E j E k is i) if j and k are distinct. The polynomial 1 — 's 0 since it has degree m - 1 but has the m roots Ai, A j , . . . , Am. Hence we conclude I — anc^ $Ej = X} Ej. If j is a characteristic vector of S. with characteristic value A*, then q3(S)x is 0 if it ^ j and is s if k = j. Hence E, = q} (S) is precisely the projection onto the subspace of V consisting of characteristic vectors with characteristic value Aj. Thus all the statements of (e) are established. (ej=>(d): Obvious. A basis for V ronsistN of the union of the bases for the sub spacer EjV. U The unique representation S = condition (e) called the n]M.trul n:xolution of S. In order to partially extend resentation to non-diagonalizable operators, we need one more commuting diagonalizable operators. This result is true for an family of commuting operators, but we prove it only for a pair.
above is this rep result on arbitrary
2.7.12 Proposition If S and S' are commuting diagonalizable operators in £(V), then there is an ordered basis for V relative to which both S and S' are represented by diagonal matrices. + Proof A proof by induction on the dimension of V starting from the definition is not hard, but we prefer to use the spectral resolution. Let S = AjE j and S' = £ Jj=1 be l 'le spectral resolutions. Since each Ej is a polynomial in S, it commutes with S' and hence with each E'k. Therefore E} E'k = E'kE} is a projection operator for each j and k.
2.7.13
The Spectrum in Finite-Dimensional Algebras
279
Choose a basis for each linear space E jE 'kV. (If E jE 'k is 0, then the basis for 3 j E ' k V = {0 } is the empty set.) It is easy to see that the union of all thes? bases is a basis for V with the desired property. □ ve now generalize, as far as possible, the spectral decomposition to nbn-c iagonalizable operators. The following elementary result shows that any linear operator on a finite-dimensional space is a spectral operator in terminology of Dunford [1954], which we will introduce later. 2 .’*’.'--3 T h e o re m Let V be a finite-dimensional space and let T be a iinear operator on V. Then there exist positive integers m, 01, 02, . . . ,a m, distinct complex numbers Aj , A2, . . . , Am, operators S and N and non-zero operators E \, £ 2, •. •, Em satisfying: (a) T = S + 7V. (b) SN = NS. (c) S is diagonalizable. (d) N is nilpotent. (e) E i, E2, . . . , Em are projection operators. (0 / = E 7= , Bj. ( &) S = Z™= l \j Ej . (h ) EjE/c = 0 whenever j ^ k. (;) The minimal polynomial for T is r i j = i ( A — > Ai,A2, . . . , A m is the list of distinct characteristic values o f T and a i, <12, . . . , arn is the list of corresponding ascents. (j) For each j = 1,2, . . . , m the minimal polynomial for the restriction aj f to E jV is (A — Aj ) a’ . (k) Each j = 1 , 2 ,m satisfies E j V = ker((Aj —T ) a>) and the restric tion o f (Aj —T ) to ( I — E j ) V = (Aj —T ) a’ V is invertible. (!) S, N, Ei, E-2, . . . , Em are all polynomials in T. b\rthermore, S and N are uniquely determined by conditions (a), (b), Sc), end (d). I f S and N are defined by (g) and (a), then (b), (d), (f), and (;i) determine. A1, A2, •••, A„, (provided the numbers in this list are re quired to be distinct) and Ey, E i , . . . , E,n uniquely up to a permutation of {1,2 . . . , m } . If condition (j) is also assumed in the last sentence, then O], a2, . ■., am are also determined up to the permutation of indices. Proof Let v
m m tT - (A ) =
J J (A
— A j ) 0 -1
j= i
be the minimal polynomial for T, so that A j, A2, . . . , Am is the list of dis tinct characteristic values with associated ascents a i , a j , . . . , am according
2: The Spectrum
280
2.7.13
to Proposition 2.7.7. Then the polynomials m
P3W
= I 1 ( A - Afc) 0‘
J = 1 ,2,..., m
have no common divisor, so the Euclidean algorithm guarantees the exis tence of polynomials hj satisfying m
1
= E V ;-
(13)
j= i
Define q} by qj = h} p: and define E j € £ (V ) by E} = Q ] ( T )
j ~ 1 , 2 , . . . , m.
Then (13) shows m
£ > = / .
(14)
J=1 Since the minimal polynomial m ix divides qjqk whenever j / k, we have E jE k = 0
when
j / k.
Hence we get m
E f = E j ^ E k = E} I = E3 k= I
for all j , so that each E : is a projection operator. Thus we have established (e), (f), (h), (i), and part of (1). Define S and N by (g) and (a); then (a), (g), (c), the rest of ( 1) and therefore (b ) are also satisfied. For each j the definition E j = q j(T ) makes it clear that if qo is a polynomial, then q o (T )E j will be 0 if and only if (A — A j)a* divides go. Hence, in particular, (Aj - T ) pE j = 0 is equivalent to p > aj, so that (j) is satisfied. Since (A —Aj ) a> divides each qk for k / j , we conclude ker((A_, - T ) ° J) is included in ker(p/t(T)) for each k / j. Hence (14) shows the first equality of (k). This in turn shows that Aj — T is injective whjn restricted to (/ — E j)V . Since E j is a polynomial in T we have (Aj - T ) ( I - E , ) V = ( I - E j ) ( Aj - T ) V C (/ - E } )V .
Hence (A j — T ) restricted to (/ - E } )V is invertible, since it is an injective linear map of a finite-dimensional space into, and therefore onto, itself.
2.7 13
The Spectrum in Finite-Dimensional Algebras
281
Again using that (A - AJ) “-’ divides each qk for k / j , we get m
{X, - T ) a>V D
E k) V = ( I - Ej ) V.
The opposite inclusion follows from
(/ - E j ) { Aj - T ) ° JV = (Aj - T ) a>( I - E j ) V = (Aj - T )°* V. Thip completes the proof of (k). Equation (g) shows that we may write m
m
jv = T - S = T -
£
A j£ j
=
£
( T
-
A j)E j,
Hence for any positive integer p, we can establish the equation m
by induction, using (h) and the commutativity of T and E j (which follows from (a) and (b) or from (1)). If p satisfies p > m ax{aj : j = 1 ,2 ,..., m }, then N p equals 0 since N p is expressed as a polynomial in T which is divisible by the minimal polynomial o f T. This proves (d) and completes the proof of the twelve properties (a), (b), ... ,( 1). Suppose S ' and N ' are also operators satisfying (a), (b), (c), and (d). Then (a) implies S — S ' = N ' — N. Properties (a) and (b) for S ' and N ' imply T S ' = S 'T and T N 1 = N 'T . Since S and N are polynomials in T , we conclude S 'S = S S ' and N 'N = N N ’. Hence if p equals the sum o f the indices of nilpotence for N and N 1, we get
k=0
Thus S — S' = N ' — N is nilpotent. On the other hand, by choosing an ordered basis relative to which both S and S' are represented by diagonal matrices, we see that S — S' can be represented by a diagonal matrix. Clearly no power of a diagonal operator is 0 unless each of the diagonal entries (and hence the operator) is 0. Hence S — S' = N ' —N = 0, so that S and N are uniquely determined by (a), (b), (c), and (d). S-jppose another family of distinct A,, A j , . . . , A' € C and non-zero op erators EJ, E'2, .. ■, E'r also satisfy (b), (d), (f), and (h). Define S and N by the analogues of (g) and (a) relative to this data. We have already given
282
2.7.14
2: The Spectrum
the argument which shows that ( f ) and (h) imply (e). Hence S satisfies (c). Thus the argument o f the last paragraph shows that S and N have their original meaning. Hence we have Aj E j = S = A'kE'k. By Theorem 2.7.11 we see that Ai, A2, . . . , Am and A',, A2, . . . , AJ. are both the list of distinct characteristic values o f S, and that E\, E 2, . . . , E m and E j, £ $ , ..., E't are both the list o f corresponding projections onto the sub spaces of characteristic vectors. Hence Ai, A2, •. •, Am and E i, E 2, . . . , E m unique up to simultaneous permutations of both lists. Clearly (j) de termines a i,a 2 , . . . , a m. □
are
We remark on one elementary but interesting consequence o f this the orem using terminology introduced in Section 2.3 above. Suppose A is a commutative subalgebra of £ (V ). Then the decomposition T = S 4- N is easily seen to define a direct sum decomposition of A . The second direct summand is an ideal consisting of nilpotent operators and hence, by The orem 2.3.3, is included in the Jacobson radical. The first direct summand is a subalgebra which contains no non-zero nilpotent operator (since every operator in it is a linear combination o f projections) and is thus semisimple. Hence this direct sum decomposition is just the direct sum decomposition o f A into a semisimple subalgebra isomorphic to A / A j and the ideal A j guaranteed by the Feldman decomposition theorem(Theorem 7.1.3 below). In order to improve on Theorem 2.7.13 we need to ascertain the structure of the nilpotent operator N . The next proposition deals with this question. Actually this proposition will not be applied directly to N , but rather to the restriction o f N to each o f the invariant subspaces E kV in the terminology of Theorem 2.7.13. 2.7.14
P r o p o s itio n
Let V be a finite-dimensional linear space with
dimension n. Let N € C (V ) be nilpotent. Lit r and d be the index of nilpotence of N and the nullity of N , respectively. Then there are positive integers rii, 7t2, ■. ■, n ■■■ > rid > 1 and = n’ and vectors x l t x 2 , ■■,x d satisfying N nkXk = 0 for k = 1 , 2 , ... , d , and such that V
=
{ x j , N x j , . . . , A ^"1 ~ *xi, .C2 , N . . . , N n> - 'x 2,
z 2,
N x d, . . . , A T - - 'j r d}
is a basis for V. Proof The proof is by induction on n. If n = 1 there is nothing to prove. Suppose n > 1 and that the result holds for all n' < n. Since jVr ' 1 is non-zero we can find I , € V satisfying N u ' ~ 1X i / 0, where we have set ni equal to r. The space W spanned by W — { x i , N x i , . . . , N n' ~ lx i } is clearly invariant and W is a basis for it. (Linear independence follows from the fact that Ar is the minimal polynomial for N . ) The restriction N w of
I . ” .'A
The Spectrum in Finite-Dimensional Algebras
283
N to W and the operator N w induced by N on V /W ( N w is defined by N w ( ^ + W ) - N x + W ) both satisfy (N \ v )r = 0 and ( N w ) r = 0. Let f and c — ' 33 the index of nilpotence and nullity o f N w . Clearly we have r < r. By our induction hypothesis we may find positive integers n,2 ,n 3 , ... ,rij and vectors x 2, ■■■, x j satisfying r > f =
> «3 > • • • > n j,
n* =
n — r, N Ukik + W = W for k = 2 , . . . , d and such that V
=
{x 2 + W , N
x2 +
W , ... , N n2~ 1x 2 + W ,x 3 + W , N x 3 + W ,
. . . , N n3~ 1x3 + W , . . . , X j + W , N x j + W , . . . , N ni - lx j + W } is a basis for V/W . For any fixed k satisfying 2 < k < d, let A0, A i , . . . , A „ ,_ i satisfy N nkB^ =
* AjJV-'xi. Then N niik —
AJArT*1_T‘k+Jx i vanishes.
Hence N n'x \ — 0 and the linear independence of X\, N x i , ... , N 7ii~ 1X\ :r.r.p'.y Aj = 0 for j < nk- If we define x* by x* = x* X jN i ~ni,x 1, then it therefore satisfies N nkz*. = 0 and N*Xk 4- W = N *&\t 4 - W for , rifc — 1 and for k = 2 ,3 ,..., d. By the linear independence of V
j = 0,1,
' e.nd of >7, there is no nontrivial relation Y lt= i
A; /cArjx )c = 0 o f linear
dependence. Thus d is d and the set V is a basis for V as claimed.
□
Use the notation of the last proposition. For each k = 1,2, . . . , d the restriction of N to the subspace spanned by x k, N x k, ■■■, N nk~ lx k has matrix
(° 1
n (J
o 1
\
0 1
0 •-
•
V
o 1 0
/
relative to this ordered basis. Clearly the vectors x i , ... ,x<j of Proposition 2.7. i4 are not uniquely defined in any sense at all. Nevertheless, it is easy to see £hi.t the form o f the matrix o f N relative to any ordered basis satisfying the conditions on V is completely unique. 7/e now adopt the notation o f Theorem 2.7.13. Let Aj be a characteristic va'^-T of T and let Ej be the corresponding projection. Apply Theorem 2.7.13 to the restriction of N (in the notation of Theorem 2.7.13) to EjV. Then the matrix of the restriction o f N to E j V is the direct sum of blocks
284
2: The Spectrum
2.7.'15
of the form
1
\ Xj
0
Aj
1
1
0
\
Aj 1
X j
Such a matrix is called a Jordan block. A matrix is said to bp the direct sum of Jordan blocks when it has the form
\
/
0
0 V
/
where each square along the main diagonal represents a Jordan block. Combining Theorem 2.7.13 and Proposition 2.7.14 gives the following Jordan canonical form for an operator. Note that the choice of ordered banin it) generally far from unique, but the form of the matrix is completely determined by the description given. 2.7.15 T h e o r e m L e t T be a linear operator on a finite-dimensional linear space V with dimension n. Let Ai, A j , . . . , Am be a list of the distinct char acteristic values of T. Then we can choose an ordered basis for T such that the matrix Jt of T relative to this basis is the direct sum of Jordan blocks. Furthermore we may assume that for 1 < k < j < m the Jordan blocks with diagonal entries A* all occur before any of the blocks with diagonal entries X j , and that the blocks with diagonal entries A* occur in order of decreas ing size. These conditions completely determine J t once the ordering of
A i. A2 , • • •, Am is chosen. Note that we can find all of the spectral properties of T from inspection of its Jordan matrix J = J t - The set of characteristic values of T is the set of diagonal entries of J , and the algebraic multiplicity of any charac teristic value Xj of T is the number of times X3 occurs on the diagonal of J. The geometric multiplicity of Aj is just the number of Jordan blocks
2.7.I f
The Spectrum in Finite-Dimensional Algebras
285
with diagonal entries Aj (since the last basis vector corresponding to any such block is a characteristic vector o f T corresponding to A j). The as cent o f any characteristic value Aj is the length of the longest Jordan block with diagonal entries Aj. Finally, for any characteristic value Aj o f T and any positive integer p, the dimension of ker((Aj —T ) p) is fcoP + X)fc=k0+ i nfc’ where " ii, wa,. . . , is the list o f block lengths in descending order o f blocks with d'agonal entries equal to Aj, and n/t0 is the last block length which is not les.i than p. We have reviewed the details o f spectral theory for linear operators on finite-cimensional spaces as a guide to what follows. However, the the ory can be applied directly to certain situations. Let T be an operator on an infinite-dimensional Banach space X. Suppose that Aj, A2 , . . . , Am is a lisi of distinct characteristic values of T which are isolated points in Sp(T). Suppose that each Xj has finite ascent (i.e., there exists an inte ger p satisfying ker((A; - T ) p) = ker((Aj - T ) p+1), so that the ascent aj of Aj cm be defined as the smallest such integer) and that the dimension of ker((Aj — T ) k) is finite for each j and k. These conditions are satis fied if A i,A 2 , . . . , A m is a list of distinct non-zero characteristic values of a compact operator T . Let T be a contour in the resolvent set of T sur rounding {A i, A2 , • • •, Am} and no other points in Sp(T). For each j let Tj be a contour in the resolvent set surrounding Aj and no other point of Sp(T). Then (anticipating the detailed discussion in the next chapter) for j = l , 2,...,m
E’ =h U X~T)-ldX
“
d
E=2ViJr{X~TrldX
(1 5 )
are commuting projection operators onto finite-dimensional subspaces sat isfying m
E EjX
-
^
=
i= i ker((Aj — T ) a*).
Ej\
EjEk = 0
for
jj= k
Furthermore the restriction of A - T to ( I - E ) X = ( I l j l i (Aj - T ) a’ ) X is invertible for all A € C which are surrounded by T. A ll these remarks are easily verified when integrals such as those occurring in (15) are discussed in the iext chapter. Now Theorems 2.7.13 and 2.7.15 can be applied to the restriction of T to E X and give a complete description of this operator, hence also of the spectral behavior of T on the portion o f the complex plane inside I . It is, o f course, not true that the spectral theory o f finite-dimensional operators is a reliable guide to typical spectral problems for operators on
2: The Spectrum
286
2.8.1
infinite-dimensional Banach or Hilbert spaces. As a matter of fact, it is reasonable to say that the class of compact operators on a Banach space is the most general class of operators which have spectral behavior which is an infinite-dimensional analogue of the spectral behavior of matrices. Even among compact operators there occur operators T (called transcen dental topologically nilpotent operators) which are not nilpotent but satisfy S p (T ) = {0 }.
2.8
Spectral Theory of Operators.
In this section we will introduce some generalities on the spectral theory o f bounded linear operators and then discuss the case o f compact operators in more detail. Let X be a Banach space. The elements of B ( X ) (Definition 1.1.16) will simply be called operators in this example. There are many sources o f additional information, e.g., Dunford and Schwartz [1958], [1963], Henry R. Dowson [1978] and Robin Harte [1988] We begin by giving the classical terminology for the classification of the points o f S p (T ) for T e B (X ). 2.8.1 D e fin itio n
The ( point spectrum / continuous spectrum / residual
spectrum / approximate point spectrum ) of T € B ( X ) is the set:
( p S p (T )
=
{ A 6 C : A ~ T i s not injective} /
c S p (T )
=
{ A s C : A — T i s injective and (A — T ) X
r S p (T )
=
apS p(T )
=
is dense in X but does not equal X } / {A € C : A — T is injective but (A — T )X is not dense in X } / (A 6 C : there is a sequence { x
n }n e N
S X
satisfying ||xn|| = 1 for all n and lim(A - T ) ( x n) = 0} ) In tho terminology introduced in the last section, the point .spectrum is just the set of characteristic values. Thus the spectrum of an operator on a finite-dimensional space equals its point spectrum. Obviously each o f the sets described in this definition is a subset of the spectrum and the first three are disjoint. A point A is in the approximate point spectrum of T if and only if A - T is a left topological divisor of zero. (For sufficiency, consider the one-dimensional operators x n ® x * which are of norm 1 if x ’ 6 X ' is. For necessity, choose a bounded sequence{j/„}n6N such that each x „ = Tn(y n) has norm 1 .) In addition to these subsets, we also wish to consider the boundary of the spectrum, denoted by 0 S p {).
2,8.2
Spectral Theory of Operators.
2.8.2 P r o p o s itio n
287
The spectrum of any operator T € B ( X ) satisfies:
S 'p (T ) = p S p (T ) U c S p (T ) U r S p (T )
(disjoint union).
(1 )
The approximate point spectrum is a closed subset satisfying: p S p (T ) U c S p (T ) U D S p (T ) C apSp(T).
(2)
- :.'cc ’ The open mapping theorem (or its corollary the inverse boundedness thBirsm) shows that a point not in p S p (T ) U c S p (T ) U r S p (T ) belongs to the resolvent set of T , proving (1). T, j point spectrum is included in the approximate point spectrum be cause ~.re may choose x„ = x for all n where T x = 0 and ||i|| = 1 . Since the rp-proximate point spectrum comes from topological divisors of zero, Theorem 2.5.7 shows that it includes the boundary of the spectrum. We may 3esily prove: (a) the approximate point spectrum is the set: {A € C : there is no M € R satisfying ||i|| < M||(A - T)z|| for all x € X } . (b) (A — T ) X is closed if and only if there is an M e R such that for each ; £ (A — T ) ( X ) there is some x e X satisfying y = (A — T )x and
IN < m im . iifeiice the approximate point spectrum includes the continuous spec trum. A standard diagonalization shows that apS p(T ) is closed for any operator T. □ Note that conditions (a) and (b) stated in the proof have independent interest. Defining distance by dist(A, S p (T )) = in f{ |A - /z| : /z e S p (T )}, Proposition 2,2.9 shows ||(A - T ) ~’ || > dist(A, S p (T ))~ l
V A £ S p (T ).
Dual maps were introduced following Definition 1.1.15. The dual T * € B (X m) of an operator T 6 B ( X ) is defined by
rr ( s ' ) ( x )
x*(Tx)
V x ' fc, .V ; x c X.
(3 )
It is easy to see that 7'* does belong to B ( X ' ) and satisfies: ||T*|| = ||T||
V T € B (X )
(4)
so
the map T i—> T* is an isometric anti-isomorphism of B ( X ) into B { X ') . The natural isometric injection k of X into X ** given by
k(® )(® *) = x * (x ) V x£ X , x' e x * I wa.sc also introdum i following Definition 1.1.15. Usually we will regard X as a subset of X " under this embedding. When this is done, an operator
2: The Spectrum
288
2.8.3
T may be regarded as a subset of (the graph o f) T " . viewpoint as we will now show. ,
2.8.3 P roposition
This is a valuable
The dual of any operator satisfies
S p (T ) = S p (r * )
T e B (X ).
(5)
Proof For any A e C and 7 • B {X ), if (A - T ) ~ l exists in B ( X ) then it is trivial to check that ((A - T ) ” 1)* is the inverse of (A - T )* = A — T m. Suppose, conversely, that (A - T *)~ * exists in B ( X *), then by what we have just shown, (A — T * * ) - 1 exists in B ( X '* ) . Hence A — T is injective since it is a restriction o f A - T " . Also A - T is a linear homeomorphism of X onto (A — T ) X which is therefore a closed subspace. If (A — T ) X were not all of X , then the Hahn Banach theorem would provide a non-zero x* € X ' satisfying 0 = x*(A - T ) X = (A — T * ) ( x ') ( X ) . This contradicts the injectivity of (A — T ) ‘ and thus proves that A — T is invertible. □
For any subsets y of X and Z o f X * , we denote the annihilator sets of y in X ' and of Z in X by: y x = {x * 6 X * : x * (x ) = 0
V xey}
and 1Z = {x e X : x * (x ) = 0
V i ' 6 2 }.
These are clearly closed linear subspaces with the former even closed in the X - or weak*-topology. For any T € B { X ) it is easy to show: ker(7“ )
=
( T X )^ ;
T X ~ = x (k e r (T *))
ker(T)
=
-l (T * A ” ) :
T *X *
(6)
C ( k e r (r ))x .
These results imply: p S p (T )
C
p S p (T )* U r S p (T ')
r S p (T )
C
p S p (T *) C p S p {T ) U r S p (T ).
(7)
It is a little harder to show that if T X is closed then T * X * is closed and satisfies T 'X * = k e r (T )1, and it is still harder to show that if T 'X * is closed then T X is closed and satisfies T X — L k er(T *). ( Cf. Dunford and Schwartz [1958], pp. 487, 8 .) See also Seymour Goldberg [1966], II.3.14 and II.4.11 where more detailed results are obtained for not necessarily continuous linear operators defined on dense linear subspaces of a normed linear space X with va'iiex in a normed linear space y and the duals of these operators defined on a linear subspace of y * . Any T e B ( X ) and T! € N obviously satisfy ker(Tn) C ker(Tn+1).
2.8.4
Spectral Theory o f Operators.
289
Moreover, ker(Tn° ) = ker(Tn°+1) implies ker(Tn) = ker(T™°) for all n > n0. " his suggests defining the ascent ct(T ) of T € <S(A’ ) to be the smallest n € M satisfying k e rfT ") = ker ( T n+1), if such an n exists, and oo otherwise. Similarly it is obvious that we get T nX D T n + lX and that T n° X = T n° +1X implies T ^ X = T n° X for all n > n0. This in turn suggests defining the descent 6 (T ) of T to be the smallest '■) satisfying T nX = T n +lX , if such an n exists, and oo otherwise.
11 €
A no itrivial standard result, which we will prove below in Lemma 7.5.2, show:; a (T ) = <5(7’) whenever both ot(T ) and 6 (T ) are finite. In this case, moreover, there is a direct sum decomposition X = T " X ® k e T (T n)
n = a (T ) = 6 (T ) < oo,
(8)
and T restricted to T nX is a bijection onto T nX . P roposition 3.4.1 in the next chapter shows how to construct spectral idempotents with the following properties. Suppose T € B ( X ) has a dis connected spectrum with a C S p (T ) both open and closed in S p (T ). Then there is a projection operator (i.e., an idempotent in B ( X ) ) E = E (a ) satisfying: (a) E € { T } " . (b) S p (T \ E X ) C a and S p (T \ (I - E ) X ) C S p (T ) \ a. This may be a good place to remark that any projection E € B ( X ) satisfies: ( 1 ) E ( X ) = { 1 6 ^ : 1 = E x } = ker(/ — E)\ (2) / — E is a projection, so ( I — E ) ( X ) = { x g X : x = (/ — E ) x } = ker( h); (2 ) X = E ( X ) © (/ — E ) X , where E X and ( I — E ) X are closed. Con versely if a Banach space X is the direct sum of two closed subspaces y and Z , then there is a projection E g B ( X ) satisfying E X = ^ and (I - J )X = Z. The Spectral Theory of Compact Operators.
Compact operators were defined and a few of their basic properties were proved in §1.7.7. We start our discussion by quoting a basic result due to Juliusz Schauder [1930]. 2 .8 . 4 P r o p o s itio n Let X be a Banach space. Then T € B ( X ) is compact if am only if its Banach space dual T ’ g B ( X ’ ) is compact.
For a proof, see Dunford and Schwartz [1958], V I.5.2. If A" is a Hilbert space H , the usual conjugate linear identification o f H * with H shows that this is equivalent to saying that B x ( H ) is invariant under the Hilbert space adjoint. This result is also an immediate consequence of the iden tity £ / f(H ) =
. Still another proof comes from noting that T €
290
2: The Spectrum
2.8.5
B (7 i) is compact if and only if it transforms sequences converging weakly to zero into sequences converging to zero in norm. Hence if K is com pact and { i „ } „ eiv converges weakly to zero, then we have \\K'xn\\2 = ( K K 'x „ , x „ ) < ||A'(K'*a:n)|| ||x„|| -► 0 since {.K ’ XrJneN also converges weakly and {||a;n||}n€N is bounded by the uniform boundedness principle. Thus K mis compact.
2.8.5
P r o p o s itio n
Let K be a compact operator on a Banach space X .
Let A be a non-zero complex number and let n be a positive integer. Then
ker((A — A ' ) " ) is finite-dimensional and (A - K ) nX is closed. Proof Note that (A — K ) n may be written as A " — K P , where P is a polynomial in K . Hence K P is compact and it suffices to prove both statements for the case n = 1 . In order to show that ker(A — K ) is finite-dimensional, it is enough to show that every bounded sequence { i n}neN in ker(A — K ) contains a convergent subsequence. Since we may write xn = A i ( K + (A — K ))x „ = K (X ~ lxn), this follows immediately from the compactness of K . Suppose (A — K ) X is not closed. Then we may find a sequence C X such that { ( X - K ) x n} n€y converges to y $ (A —K )X . Replac ing { x n}n€N by a subsequence, we may assume that no x n lies in ker(A—K ). Define dn by dn = d is t(in,ker(A —K ) ) and choose yn e ker(A —K ) to satisfy
llVn ~ *nll < 2d,»Suppose ||y„ — i„|| is a bounded sequence. Then we may replace it by a subsequence so that { K ( y n - a:n)}n€N converges. In this case
yn - X n
=
A - 1[ ( A - ^ ) + A ' ] ( y n - X n )
=
- X - 1( X - K ) x n + X - l K (y n - x n)
converges to some z which satisfies y — lim (A — K ) ( y n — xn) = (X — K )z . Since this contradicts our choice of y, we may assume that ||yn — ^n|| is an unbounded sequence. Replacing it by a subsequence, we may assume llZ/n - x n\\ oo. Define zu by zn =
II 3/n
^n||
( V n ~~ x n )
so that we have || || = 1 , (A - K )z n = ||y„ - x n| r '(A - K ) ( x n) -> Oy = 0 and = A ' ‘ ( ( A - K ) z n + K z n).
Since K is compact, these facts show that we may replace by a subsequence which has a limit z. Clearly (A — K )z = lim(A - K )z n — 0 implies that ; belongs to ker(A - K ). Define wn by wn = yn -f ||yn —
Spectral Theory of Operators.
2.8.6
291
x n||z e ker(A — K ). Thus we find ||xn — iun|| > dist(x„,ker(A — K ) ) — dn. However, x n - w „ = x n - y n - ||j/n - xn \\z = ||i/„ - xn \\(zn - z ) shows that IJx., — iD„|j < 2d„\\zn - z ||. However this contradicts z = lim z„. Thus we conclude that (A — K ) X is closed. □ 2.8.C P r o p o s itio n Let K be a compact operator on the Banach space X . //{An} „ e N is a sequence of distinct characteristic values of K , then the sequence { A „ } n6|Mconverges to zero. ProcJ Assume the sequence does not converge to zero. Then we may replace it by a subsequence which satisfies |An| > £ for all n where e is some fixed positive number. Tor each n, choose x„ e X satisfying ||xn|| = 1 and K x n = X„x„. W e claim that { x n : n € N } is a linearly independent set. If not, let n be the smallest integer for which x i ,x 2, ... ,x n is not linearly independent so that we can write x n = a xxi + a 2x2 + ---- f- a n- i x n- i
for cij e C. Applying K — An gives 0 = a i(A i — An)x i +
(A 2 — A „ ) i 2 + • • • + a „ _ i ( A „ _ i — An) i n_x.
The linear independence of x i , x 2, . . . , x n_ i and the non-zero character of Aj — An for j = 1 ,2 ,..., n — 1 imply 0 = a i = <*2 = • • • = a n- i which is inconsistent with ||xn|| = 1. Hence { x n : n € N } is linearly independent. Define y n to be the closed subspace span { 2 1 , 2:2 , ■■■, £ „ } and apply the Riesz Lemma (see §1.7.7) to ^ „ _ i % y n to obtain y „ € y „ satisfying ||j/„|| = 1 and dist(j/„,3;n - i ) > 1/2. For n > m we have ^ (A - ^ J
-
J ^ A - V )!!
=
\\Vn - (A - 1 (A „ - K )y n + A - ‘ K y m)|| > 1 / 2
since (A n —K )y n and K y m both belong to y n — 1. However this contradicts the fact that {/v'(An 1yn) ) iiCn must have a convergent subsequence since K is compact and { A ” l j/n} ngN is a bounded sequence. Thus our assumption that An} neN did not converge to zero is untenable. □ 2.8.7 T h e o re m Let K be a compact operator on a Banach space X which is not finite-dimensional. Then S p (K ) contains zero and S p (K ) \ {0 } is either finite or a countable set which converges to zero whenever ordered as a sequence. Any Ao 6 S p {K ) \ { 0 } satisfies:
(a) Ao is an isolated point in S p (K ). (b) There is a projection E\0 6 { K } " such that the subspace E\nX is finite-dimensional, S p(K \E \0X ) = {A o } and A0 ^ S p (K \ (I - E\0)X ). (c) Ao is a characteristic value of both K and K * G B ( X ') satisfying:
1
292
2: T h e Spectrum
2.8.7
0 < dim(ker(Ao - A ') " ) = dim(ker(Ao - A '* )n) < oo
V n 6 N;
0 < a (A 0 - A ') = a (A 0 - A '*) = 6 ( A0 - K ) = 6 ( A0 = A ')*) < oo; E Xo( X ) = ker((A 0 - f f ) ° ) ; ( I - E Xo) X = u Aene
a
=
q (A o -
(Ao -
Em Xu( X ' ) = ker((A 0 - * - ) “ );
A T *;
K ) is the ascent of
( I - E\0) X - = (A o -
(Ao
- AT)**,
K ).
Proof Suppose S p (K ) has Ao ^ 0 as a limit point. Choose a sequence {A n}n€N C S p (K ) \ {0 } of distinct numbers converging to Ao- The last proposition shows that An - K is injective except for a finite set of indices. If (An K ) X = X , the inverse boundedness theorem shows that any injective An - K is invertible, contradicting A„ 6 S p (K ). Since (A „ —K ) X is always closed, this shows that ( An — K ) X cannot be dense. However if (A „ — K ) X is not dense, then the Hahn-Banach theorem gives an x* 6 X * satisfying
■c’ UA,, —K ) X ) = (\ n - K * ) ( x ' ) X = 0 and hence belonging to ker(A„ —A '*). However, since K ‘ is compact by Schauder’s theorem, the last proposition shows that A„ - K ' is injective for all but finitely many indices. This contradiction shows that S p (K ) has no limit points except possibly zero. Hence S p (K ) \ { 0 } contains only isolated points and is either finite or a sequence converging to zero. Let Ao be an element of S p (K ) \ { 0 } and let F be a circle centered at Ao and of such small radius that the intersection of S p (K ) and the closed disc bounded by F is just Ao- Consider F as a positively oriented contour (in the usual sense of complex analysis) and define an operator E\a by
r
as described in Definition 3.3.6. The theorem just mentioned shows that E \o is a projection operator (i.e., an idempotent) satisfying E\0 € { K } " and E\0 = K T \ a where T\0 is defined by
= ^
/ *-■ < * r
Hence Ex0 is a compact operator so that E \a( X ) (on which E\0 acts as the identity element) is finite-dimensional. Furthermore, the spectrum of the restriction A'a0 = a !£a.0a' Of K to E\0X is exactly Ao- The spectral theory o f Theorem 2.7.13 may now be applied to K \ 0. It shows E\0X = ker((Ao - A \ (,)a) where a is the index of nilpotence of Ao — K \ 0 which is the ascent of Ao - A'. Note that Theorem 3.3.7 shows that the restriction of
?.8.7
Spectral Theory of Operators.
293
A( - A' to ( I - E\0) X is invertible which implies ( I - E\a) X C (X 0 - K ) aX . T lis inclusion is an equation since any x 6 X satisfies E \0(Aq — K ) ax = (- 0 — K ) aE\ax = 0. If n is strictly less than a, then (Ao — K ) nX has nontrivial intersection with E\0X so that a is also the descent o f Ao — K . Since the adjoint operation * : B ( X ) —* B ( X ') is a linear isometry, we fir.d
r
S iice K m is also a compact operator, the results already established show that the common value a of ascent (Ao —A"*) and descent ( Ao —K ' ) satisfies E l = kcr((Ao - A * )* );
( I - E'Xa) X ' = (A 0 - K ' ) aX *.
If x * belongs (E \ 0X )* and satisfies :r*(Ao — K )E \ aX = {0 }, then x * o Ex„ belongs to ker(Ao — A"*). I f d\0 and n\0 represent the dimension of Ex0X and of ker(Ao — A \ 0), respectively, then Theorem 2.7.13 shows that
(A-j - K \0)E \ aX has dimension n\0 — d\0 so that the linear space o f x* e (E \ 0X )* vanishing on (Ao — K \a)E \ aX has a basis Clearly B = {x * o E\u, x*2 o E\0 ,x*d o E\0} is a linearly independent set in ker(A 0 - K * ). However we claim that this set spans ker(Ao — K * ) so that we have dim ker(Ao — K ) = d\0 = dim ker(Ao — A '*). To see this, suppose x* e ker(Ao —K * ). Since Ao —K is invertible on I —E \ 0, we have x* = x* o E\0 + x* o I - Ex0 = x* o E\0 + x* o (Aq — K ) o T\0 = x ' o E \a for a suitable T\0. Hence x* can be written a s i ' o E\a for some x * e (E \ 0X ) ' which must satisfy x*(Ao — K )E \ 0 = 0. This shows that B spans ker(Ao ~ A ). The equation (Ao — K ) n = Xq — K again shows that dim ker(Ao — K ) n) = dim ker((Ao — K * ) n) follows for all n £ N from the case n = 1 which we have just proved. This of course shows dim(£'AoA’ ) = dim ((A 0 - K * ) a) = dim(EJ X ) and a (X 0 - K ) = a{Xo - K * ). ° □ Hyperinvariant Subspaces
The spectral results proved above are more than enough to settle a basic question about compact operators. When considering bounded linear op erators on Banach spaces, one naturally wonders whether such an operator pc :essarily has a nontrivial closed invariant subspace (i.e., for T e B (X ), d;_ is there exist a closed subspace y o f the Banach space X satisfying y - {0 }, y / X and T y C ^ ? ). O f course one assumes that X is not onedinensional. Results from the last section give an easy affirmative answer in the finite-dimensional case. If T is an operator on a finite-dimensional
294
2: The Spectrum
2.8.8
space and T is not a constant multiple of the identity (in which case every subspace is invariant), then Theorem 2.7.13 shows that we need only con sider the case in which T has a single eigenvalue and a non-zero nilpotent summand N . Then ker N is a nontrivial subspace (closed with respect to any norm) invariant under not only T but any operator which commutes with T. Infinite-dimensional Banach spaces have been constructed with non-zero operators on them which have no closed invariant subspaces: Per Enlio [1976], [1987], Charles J. Read [1984], [1985], Bernard Beauzamy [1985]. Before these difficult constructions were discovered, Nachman Aronszajn and Kennan T. Smith [1954] showed that any compact operator on a Ba nach space, not of dimension one, also has a nontrivial closed invariant subspace. The Aronszajn Smith result was extended somewhat but its proof remained very difficult for twenty years until a Russian mathemati cian, Victor I. Lomonosov [1974], introduced a new and radically simpler method o f proof. It was quickly realized that the new method o f proof would give much stronger results. The following easy, elementary result is from some unpublished notes of Carl M. Pearcy and Alan L. Shields which incorporate an idea of Hugh M. Hilden. 2.8.8 D e fin itio n An operator K on a Banach space X is said to have a hyperinvariant subspace if there is a closed subspace y of K satisfying y ^ {0 },;y # X , and T y C y for all T e { K } '. 2.8.9 T h e o r e m Any non-zero compact operator on an infinite-dimensional Banach space has a hyperinvariant subspace. Proof Let the compact operator K act on an infinite-dimensional Banach spate X . If K has a non-zero complex number A in its spectrum, then A is a characteristic value and the finite-dimensional subspace ker(A — K ) is hyperinvariant. Hence we may assume that K is topologically nilpotent. We may also assume ||/f|| = 1. Choose x 0 6 X satisfying HK’aroll > 1Define S C X and T C X by S = {x e X : ||x — xoll < 1} = Xo + <*i and T = K S ~ . Neither S nor T contain zero and T is compact. The proof is by contradiction. Suppose that K has no hyperinvariant subspace. Then for each non-zero x € X , the non-zero {AT}'-invariant closed subspace { K } 'x ~ must equal X . Hence { K } 'x is dense for each non-zero x € X . For each T 6 { A } ' , define an open set A/’t by M r = {x £ X :\\Tx - x0\\ < \ } .
Since { K } 'x is dense for each non-zero x, we have
U T€{A)'
M- = *\{0}.
2.8.10
Spectral Theory of Operators.
295
Therefore there is a finite subset T C { K } ' such that U
Mr
T er ' ~:z udes the compact set T . W e will now construct a sequence { T n} n€N inductively where each Tn is chosen from T . Since K x u belongs to T , we car? find Tx e T such that T iK x q belongs to S. Hence K T iK x q belongs ;c T so that we can find Tq € T such that TzK T yK xo belongs to S. Continuing in this way, for any n we find
= TnK T n- i K ■■■T ^K T iK xo in S. Since each Tj belongs to F C { K } ' , we can rewrite this as
= T „ T „ _ i •■•T2TiA'nx0. Define d and M by d = dist(<S,0) and M = max{||T|| : T € F ) . Then we have d < |KI| < ( £
11^11)11^11 ||z0|| < M "||^"|| l|*o||.
i =1 Dividing by ||xo|| and taking nth roots, we get
This is a contradiction since K is topologically nilpotent so that the right side approaches 0. This contradiction shows that K has a hyperinvariant subspace. □ ? . ' 1 Q Essential spectrum W e wish to mention this concept briefly artl give some references. Related results can be found in §1.7.16 and Chf.pter 8 . Let X be a Banach space. The essential spectrum o f an operator T e B ( X ) is the spectrum of its image T + B k ( X ) in the generalized Calkin algebra Q ( X ) = B ( X ) / B k ( X ) . It is closely related to the sometimes slightly larger Weyl spectrum which is the intersection o f S p (T + K ) for all K e B k (X ). These smaller spectra overlook spectral points where the singularity is not “too bad” . A Riesz operator in B ( X ) is one with essential spectrum {0 }. Thus they are the operators with topologically nilpotent iria,<e in the Calkin algebra. Trevor T. West has shown that on Hilbert ppace each Riesz operator can actually be written as the sum of a compact and a topologically nilpotent operator. These terms are sometimes used a more generalized setting with another ideal playing the role of the compact operators or even another algebra playing the role o f B (X ). See
2: The Spectrum
296
2.8.11
the references in §1.7.16, Murphy and West [1980b] and Jaroslav Zemanek [1983], Local Spectral Theory and Decomposable Operators Local spectral theory associates subsets of the spectrum of an operator T e B ( X ) on a Banach space X with subspaces of X .
In [1954] Nelson Dunford introduced the theory of spectral operators in order to extend the theory of normal operators to somewhat more general operators on Hilbert space and to Banach spaces. W e have already shown in Theorem 2.7.13 that every finite dimensional operator is spectral, and it was originally hoped that many operators of analysis (both bounded and unbounded), which were known to have rich spectral theories, would turn out to be spectral operators. Despite the substantial success recorded in Dunford and Schwartz [1971], this important pioneering theory is too restrictive to encompass the most interesting examples. However, Dunford’s concept of the single valued extension property has been basic to all more recent theories. The books by Ion Colojoara and Ciprian Foia§ [1968] and Florian Horia Vasilescu [1982] record other successful attempts at generalization. In [1963], Foia§ defined decomposable operators. To date, this concept has achieved the best balance between strong enough hypotheses to give a rich theory and weak enough hypotheses to contain the most important and interesting examples. We treat oniy the case o f bounded operators. The following definition is a simplification, due to Ernst Albrecht [1979], of the original definition of decomposable operators. Property (/?) and an early version o f (5) are due to Errett Bishop [1959]. 2.8.11 D e fin itio n Let X be a Banach space and let T be an operator in B (X ). I f U and V are open subsets o f C with union C, then two closed T-invariant subspaces y and Z of X are said to decompose T relative to U and V if they satisfy S p (T \ y )C U
S p (T \ z )C V
and
y + Z = X.
The operator T is said to be decomposable if it has decomposition relative to any open cover {U , V } o f C. For any open subset U of C, let 0 ( U , X ) be the Unear space of all analytic functions f : U —> X (cf. Definition 2.2.10). Give this linear space the topology of pointwise convergence which is uniform on all compact subsets o f U , which makes it a FVechet space. Each T € B ( X ) defines a continuous linear map Tu : 0 (U , X ) —* 0 (U , X ). If T v is injective and has cloned range for each open set IJ, then T is said to have property ((5). For each closed set F in C, let Xp be the set of vectors x £ X that are in the range o f Tq \f (in the sense that the constant function x on C \ F is in this range). The operator T is said to have the decomposition property (6 ) provided that Xjy + X y equals X for each open cover {U , V } of C.
2.8.12
Spectral Theory of Operators.
297
T ie class of decomposable operators contains not only normal operators on a Hilbert space but also (on any Banach space) compact operators (in deed, all operators with totally disconnected spectra), spectral operators, generalized scalar operators and surjective isometries. Many interesting partial differential operators are unbounded decomposable operators 2.S. 12
T h e o re m
Let X be a Banach space and let T 6 B ( X ) be an
operator. (a ) T has property ( (0 ) / (b ) ) if and only i f T * e B ( X *) has property
( (S) /(/?)>• (b) T has property ( ((3) / ( 6 ) ) if and only i f T i s similar to the ( re striction / quotient) of a decomposable operator T ( to / b y ) aT-invariant subspace.
(c) T is decomposable if and only if it has both properties (0 ) and (5). F< r proofs and further details see Albrecht and Jorge Eschmeier [1993], Albrecht, Eschmeier and Michael M. Neumann [1986], Kjeld B. Laursen and Neumann [1992], Neumann [1992] and Vivien G. Miller [1993].
2.9
Topological Algebras
This work discusses abstract algebras (i.e., those without topology), normed algebras and Banach algebras in some detail. It has been a deliber ate decision to exclude any general discussion of the many classes o f topo logies ' algebras which have been studied, since otherwise the book would have p’Own beyond any bound. Many results in the book extend routinely from normed algebras to some wider class or classes of topological algebras, but v « will seldom point this out. However, some important applications involve non-normed topological algebras, so we need to give an outline of certain parts of their theory. We shall concentrate on those portions which are cl ssest to Banach algebra theory. The discussion here will be brief and withe it proofs. It was postponed to this chapter because at several points it inv< Ives the hypothesis that the group o f quasi-invertible elements should be op ’ n. 2.9.1 D e fin itio n A Frechet algebra, sometimes abbreviated F-algebra, is a topological algebra in which the topology arises from a complete metric. AI .hough we only require multiplication to be continuous in each vari able separately in a topological algebra, Richard Arens [1947] proved that multiplication is actually jointly continuous in a Frechet algebra. Further more Stefan Banach [1948], with improvements by Wieslaw Zelazko [1960] and by Andrew M. Gleason in his Mathematical Reviews review of that paper, showed that the quasi-inversion map (a ►-* aq for all a € A^ g ) is
298
2: The Spectrum
2.9.2
continuous if and only if the set of quasi-invertible elements is a G g- set. We state these results as a formal theorem. 2.9.2 T h e o re m Let A be a Frechet algebra. Then multiplication is jointly continuous and the topology may be defined by a norm-like function
|| ■||: A —> R+ satisfying: ||a|| = l|a + 6 || < ||Aa||
IIA„a|j
0 if and only if a = 0 ||a|| + ||6 ||
=
j AJ ||a|| if X £ C and |A| — 1
-
0
||Aa„|| - *
V ae A V a ,b e A V a £ A
t/ {A n|„f N C C and|A„| - 0
0 if { a „ } „ eN C A and ||an|| - » 0
A is complete in the metric: 8(a, b) = {ja — 6 ||
V a g i
(1 ) (2) (3 )
(4 )
V A€ C
(5 )
V a, b £ A .
(6 )
The quasi-inversion map (and therefore the inversion map in the unital case) is continuous if and only if the set AqG of quasi-invertible elements is a G { set. Finally, a metric algebra may be completed to be a Frechet algebra if and only if multiplication is jointly continuous.
Following the important and still useful early exposition by Ernest A. Michael [1952J, many authors restrict the term Frechet algebra to the case where the complete topology arises from a countable family of algebra semi norms. We will introduce this class briefly below, but first we discuss another property. A good source for some of this material is Zelazko [1965]. 2.9.3
D e fin itio n In a topological linear space, a set B is said to be bounded if for any neighborhood U of 0 there is some non-zero scalar A such that XB C U. A topological linear space is said to be locally bounded if there exists some neighborhood of 0 which is bounded. ln a normed or inetrizable topological linear space, balls around the origin are obviously bounded neighborhoods, so such spaces are locally bounded. In the complete case the converse is true, and more surprisingly, the metric can always be chosen to be subadditive and to have a strong substitute for homogeneity. If X is a locally bounded topological linear space and U is a bounded neighborhood of 0 , then {-U n
: n e N}
(7)
is a countable basis of neighborhoods of 0 so the linear space is metrizable. In fact, Stefan Rolewicz [1957] shows that the topology can be given by a p-homogeneous norm, which is a norm-like function ||-||: X —►R + satisfying:
Topological Algebras
2S.X
||a|| = ||a + b|| < ||Aa || =
0 if and only i f a = 0 ||a|| + ||6|| |A|p||a||
299 V a€ X
V a ,6 e * V A e C,a e X
for some particular 0 < p < 1. Obviously a p-homogeneous norm is a norm (in the usual sense) if and only if p is 1. Locally bounded linear spaces resemble normed linear spaces rather closely, and for complete ones the riiseiublancc to Banach spaces is even closer. We will now characterize the complete locally bounded algebras, beginning with a brief consideration of general Frechet algebras. 2.8.4 T h e o re m The following are equivalent for a Frechet algebra A : (a) There is a metric 6 defining the topology and satisfying 6(ab,0) < 6 (a , 0)6(6,0)
V a , 6 e ^ t;
( 8)
( b M is locally bounded; (e; For some 0 < p < 1 there is a submultiplicative p-homogeneous norm defining the topology.
Algebras satisfying these hypotheses share most o f the properties of
Banach algebras except that, since they are not necessarily locally convex, the space o f linear functionals on the algebra may be too small to be o f use. The topology of a topological linear space which is both locally bounded and .locally convex arises from a norm. We now explore the locally convex topological algebras. Locally convex topological linear spaces were introduced briefly prior to Definition 1.1.8. Topological algebras in which the underlying topological linear space is locally convex do not have exactly the properties which we desire. Thus, we specialize the concept a bit further. 2.9.5 D e fin itio n A locally multiplicatively convex algebra is a topological algebra in which the topology is given by some collection of (ordinary) algebra semi-norms. If P is a set o f algebra semi-norms on an algebra A , then a net { a ^ i n A converges to a € A with respect to this set if a(as — a) —> 0
V a e P.
(9)
Loyally multiplicatively convex algebras are often called Imc-algeObviously, multiplication is jointly continuous and quasi-inversion is cov.inuous in these algebras. There are many cases in which the colof algebra semi-norms can be taken to be countable. As noted prev/ov^ly for linear spaces, if {
2: The Spectrum
300
2.9.6
The following important result was obtained by Michael in his seminal memoir [1952], where it was used to show that this class of algebras has some of the behavior usually associated with algebras having an open set o f quasi-invertible elements. 2.9.6
T h eorem Every locally multiplicatively convex Frechet algebra is the projective limit of a projective system of Banach algebras. An element in the algebra is quasi-invertible if and only if its representative from each of these Banach algebras is quasi-invertible.
2.9.7 T h e A lg e b ra T — C[[.zj] o f Form al Pow er Series An element of T is a formal power series a nz " without any question o f convergence, so ( a „ } neNo C C is an arbitrary sequence. The algebraic operations are the usual ones for power series: oo
oo
n —0
n —0
oc
n
n=0 k = 0
We will give T the topology of coordinatewise convergence. The following family of semi-norms shows that T is locally multiplicatively convex: co
I
m
I n=0
v
m e N°.
n—0
Theorem
For each m 6 N°, || • ||m is an algebra semi-norm on T . With the topology induced by this family of semi-norms, T is a locally multiplica tively convex Frechet algebra. Furthermore, this is the only Frechet algebra topology on T .
The last sentence is due to Graham R. Allan [1972], O f course this implies that T is not a Banach algebra under any norm. Nevertheless its group o f invertible elements T g — { S ^ L o Q« z " ■ Q° ^ 0 }- *s obviously open. The algebra T also has a decreasing sequence o f closed ideals T m with intersection { 0 }: no
a nzn : a n = 0 for all n < m }
V m e N°.
n=0
Note that the whole algebra is the disjoint union of T g and _F0. In the terminology of Section 6.4 below, it is known that all derivations of T into itself are continuous.
3
Commutative Algebras and Functional Calculus Introduction
In this work we are relatively little concerned with commutative al gebras. The involution in the *-algebras, which are the main subject of stvdy here, is a sort of weak substitute for commutativity. However, some commutative theory is essential in dealing with the more complicated nonco: amutative theory. In particular, commutativity is intimately associated wi h the various functional calculi which we will introduce in this chapter at later. Section 3.1 presents the remarkably simple theory, created by Israel Moise; /ic Gelfand [1941a], which replaces any commutative Banach algebra by a 1 omomorphic image (modulo its Jacobson radical) which is an algebra of complex-valued continuous functions vanishing at infinity on a locally compact topological space. This canonically constructed space is called the Ce'fand space. (In many applications the homomorphism is an isomor phism.) We show that all the, now classical, results in this theory hold for an algebra if and only if it is an almost commutative spectral algebra where ai.”; ost commutative means commutative modulo its Jacobson radical. A number of other equivalent conditions are given in Theorem 3.1.5. They car be briefly summarized by saying that if the spectrum or spectral radius of an algebra has any similarity to the spectrum or spectral radius of an algebra of functions, then the algebra has a full Gelfand theory. Theorem 3.1 5 and Proposition 3.1.7 give some results which imply that a Banach algebra is either commutative or almost commutative. We postpone others to Chapter 4, where we can use Sinclair’s density theorem to give an elegant n r o f (Theorem 4.2.18) due to Jaroslav Zemanek [1982]. Theorem 3.1.11, duo to Georgi EvgenyeviC Silov [1947a], shows that any homomorphism fro a a Banach algebra into a semisimple commutative Banach algebra is au> >matically continuous. The Gelfand theory for double centralizer alge bras, double duals with Arens multiplication and singly generated unital Ba -ach algebras are all explored. Section 3.2 introduces the Silov boundary, hulls, kernels and the hullkei lel topology on the Gelfand space. Completely regular commutative Bp >ach algebras are shown to be those commutative algebras in which 11 ■ hull kernel topology agrees with the original Gelfand topology of the 301
302
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3.0
Gelfand space. Theorem 3.2.12 gives the Gelfand-Naimark characterization of algebras o f bounded continuous functions on a topological space. This is used to construct the Stone-Cech compactification o f a topological space and the Bohr compactification of a locally compact group. It is shown that the latter construction is injective if and only if the group has enough finite-dimensional representations to separate its points. Much of the remainder of the chapter is devoted to functional calculi. These will be discussed at greater length when introduced. We treat two cases in depth in Sections 3.3 and 3.5, respectively. Theorem 3.3.7 details the theory (due to Gelfand [1941a]) in which a meaning is assigned to f (a ) when a is any element in any Banach algebra and / is a function which is analytic on some open set including the spectrum o f a. The uniqueness o f this construction is explored, and it is extended to semisimple spectral algebras in which the algebra of functions which act may be smaller. A spectral normed algebra with a full functional calculus is called a functional algebra. Section 3.4 contains examples and various applications o f the theory de veloped in the previous section to the construction of idempotents, square roots, exponentials and logarithms. Actually, we show how more elemen tary methods can be applied in many o f these situations. Section 3.5 treats the case in which ( a i , a2, . . . , a „) is an ordered n-tuple of commuting elements in a unital Banach algebra A and / is a function of n variables which is analytic in a neighborhood o f the “joint spectrum” of ( a i , a2 , . . . , a „ ) defined in some suitable sense. The proper definition in the mo6t general case is not clear. In the easiest case, A is commutative and the proper meaning o f joint spectrum (Definition 3.5.4) is obvious. Then / (a i,t» 2 , . ■. ,On) also has a standard definition and very useful properties. These results are essentially due to Richard Arens and Alberto P. Calderon [1955] and Lucien Waelbroeck [1954] based on Silov’s proof [1953] for finitely generated commutative algebras. The construction of f (a i , a2, . . . , an) is still technically complicated enough that we do not prove the more recent results and present our proof in such a way that the result of an exterior algebra construction can be taken as an axiom, and its proof omitted by the reader who wishes to do so. (W e do, however, give a complete proof which should be accessible without any previous knowledge of differential geometry.) Theorem 3.5.9 contains the main results. An implicit function theorem (Theorem 3.5.12, essentially due to Arens and Calderon [1955]) and the Silov idempotent theorem (Corollary 3.5.13 and 3.5.15) are among a number o f applications. Theorem 3.5.17 is an important result of Charles E. Rickart [1953] which shows that any (possibly incomplete) algebra norm on a semisimple completely regular Banach algebra is spectral. The ArensRoyden theorem (Theorem 3.5.19, proved by Arens [1963] and Halsey L. Royden [1963], independently) completes our investigation o f the exponen
Gelfand Theory
303
tial 'function begun in Theorem 2.1.12. The section ends with a brief in troduction to Joseph L. Taylor’s construction [1970b], [1970c], [1972a] and [1972b] for commuting n-tuples o f operators. Section 3.6 gives an overview of the duality theory for locally compact abelian groups.
3.1
Gelfand Theory
’ 7e now turn to Gelfand theory. In Section 1.5 we briefly introduced algebras o f complex-valued function under pointwise operations. This is the easiest sort o f commutative algebra to understand. Gelfand theory allows us to study any commutative spectral algebra A in terms o f such an algebra which is a homomorphic image o f the original algebra with kernel equal to the Jacobson radical A j of A . Let A be a linear space o f complex-valued functions on a set fl. Assume that A is closed under pointwise multiplication. In Section 1.5 we agreed to consider a space o f functions such as A to be an algebra under pointwise operations, unless we specifically defined some other multiplication. We also noted there that, if each function in A is bounded, then ll/lloc = su p { | / M I : w e 0 }
is an algebra norm on A . Convergence relative to this norm is uniform convergence, so it is called the uniform norm. Note that the range / (fi) of / is included in S p ^ i f ) (since A — / assumes the value zero at a) € 11 if A = /(w )). Thus P a (J ) > ll/lloo holds for all / € A so that no spectral ?o'T’ :-norm on A can be smaller than ||• H^. Thus, if || • ||oo is a spectral norm, then it equals the spectral radius. Tw o specific Banach algebras o f bounded continuous functions with the !.i iirorm norm as their complete norm were introduced in Section 1.5 and will be used again here. I f O is a locally compact space, ( C(f2) / C 0(f l ) ) is .th e algebra o f all continuous complex-valued functions on fl which ( are bounded / vanish at infinity ). Obviously C 0 (Si) is always an ideal o f C (f l ), and the two are equal if and only if fl is compact. S .i.i
D e fin itio n
Let A be an algebra. Define
to be the set:
r A = {a ll non-zero algebra homomorphisms o f A into C }.
If
is nonempty, for each a e A , define a: a(7) = 7(a)
—►C by evaluation:
V 7 e r*.
The map (‘ ) is called the Gelfand homomorphism and, for any a € A , a is cef^d the Gelfand transform of a. The Gelfand topology is the weakest
304
3: Commutative Algebras and Functional Calculus
3.1.1
topology on in which the maps a are continuous for all a € A . If it is nonempty, the set T^ together with this topology is called the Gelfand space of A . The subset A r of A on which the Gelfand map vanishes (or equivalently on which all 7 G vanish) is called the Gelfand radical of A . The Gelfand radical of an algebra A for which F^ is empty is A itself. We call A ( Gelfand-radical / Gelfand-semisimple ) if A r equals ( A / { 0 } ). Finally we call A almost commutative if it is commutative modulo its Ja cobson radical (i.e., if A / A j is commutative). Note that an algebra homomorphism of an algebra A into C is either identically zero or surjective. The term Gelfand homomorphism is justified in Theorem 3.1.2. In the Gelfand topology on F^, a neighborhood base at an element 70 is defined by {Ujr : T is a finite subset o f A } where Ujr = { 7 6
: |7 (a) - 7 o(a)| < 1
for all a 6 J7}.
(There is no need to consider small positive e’s since the elements in T can be chosen o f arbitrarily large norm.) A net { 7 , , C F^ will converge to 7 € in the Gc-lfand topology if and only if 7 a(a ) = a ( 7 a) converges to 6 ( 7 ) = 7 (a ) for all a € A If (.4, || • ||) is a spectral normed algebra, we will see below that is a subset of the unit ball of the dual space A * of (A , (J • ||) and the Gelfand topology is just the relativized weak* topology The Gelfand space o f A is called the structure space, the spectrum, and the character space by various authors. When A is a spectral algebra which is almost commutative, Proposition 3.1.3 below shows that the map which assigns to each homomorphism in its kernel is a bijection onto the set of all maximal modular ideals of A . Thus, the structure space is also called the maximal ideal space. Many authors refer to the homomorphisms o f A into C as multiplicative linear functionals on A . Section 3.6 shows that when A is the L x group algebra of a locally compact abelian group, then the Gelfand homomorphism corresponds to the Fourier transform. Hence, the Gelfand homomorphism is frequently called the Gelfand transform. We restrict this terminology to calling a the Gelfand transform of a. The Gelfand homomorphism is also often called the Gelfand representation since it represents the algebra (modulo its Gelfand radical) as a more concrete algebra of continuous functions. We note in Theorem 3.1.5(b) that in an almost commutative spectral algebra the Gelfand radical equals the Jacobson radical. In Chapter 4, where we introduce primitive ideals, Theorem 4.1.9 will show that in an almost commutative algebra primitive ideals coincide with maximal mod ular ideals. In Section 4.5, where the Brown McCoy radical is introduced, Theorem 4.5.9 will show that this radical also coincides with the Gelfand radical in almost commutative spectral algebras. Hence, from now on we will speak o f an almost commutative spectral algebra as semisimple if it is
Gelfand Theory
3.1.2
305
Geifand-semisimple. It is interesting to note that Nathan Jacobson [1945a] cites the Gelfand radical as one of the (several) examples which show that his new concept (the Jacobson radical) is useful. In order to prove the following result, and for later use, we need a sym bol fo: the collection of all (not necessarily non-zero) homomorphisms of an algebra A into C. We will denote this set by
Note that
is never
eiupU because it always contains the trivial, identically zero, homomorphis'i! which we will denote by 0. In fact, this augmented Gelfand space i hvav satisfies:
= r * u {o}.
(i)
When it is convenient, we extend a Gelfand transform a to a0 defined on the avgmented Gelfand space by:
The augmented Gelfand space has a Gelfand topology defined by requiring each c ° to be continuous. It is easy to see that the Gelfand topology on r u is just the relativized Gelfand topology o f We have little need for these concepts in the unital case where 1 shows that 0 is an isolated point in the augmented Gelfand space. 3.1.2 T h e o re m Let A be an algebra in which the set is not empty. (a) The Gelfand space is Hausdorff. I f A is nonunital, restriction of each 7 6 ? A i to A
defines a homeomorphism of r A i onto
The Gelfand
homomorphism is an algebra homomorphism of A into the algebra of all continuous complex-valued functions on
I f A is a unital algebra, then
the Gdfand homomorphism is a unital homomorphism. (b) The range of the Gelfand transform a of any element a e A is mcludfd in the spectrum of
a:
a(rA) C
S p (a )
V aeA.
(3)
(c) The Gelfand radical includes the Jacobson radical and contains any commutator: A j C Ar Hence 1 ^,
and
ab — ba e A r
V a,b £ A -
andTA/ Ao are naturally homeomorphic where A
smallest ideal containing all commutators. (d) If cr is a spectral semi-norm on A , then any 7 e [ 7 ( a ) |< p (a ) < er(a)
V a 6 A-
d
is the
satisfies:
(4)
Furth. rmore, the Gelfand radical is closed with respect to a and includes the ...
Aa on which a vanishes.
306
3: Commutative Algebras and Functional Calculus
3.1.2
(e) If A is a spectral algebra, then F^ is compact and is locally compact in the Gelfand topology. The Gelfand transform a of any element a € A is a bounded continuous function vanishing at infinity: O : A ^ C 0( r „ ) . If A or A ) A j is also unital, th. n r A is compact in the Gelfand topology so
C o (r ^ ) equals C (T a ). Proof (aj: For any a,b € A , any A € C, and any 7 6 F.4 , wo have:
(Au) — A7 (a)
( M ) '( l )
-
(a + b )A( j )
=
7(0
(aft)A( 7 )
-
7
7
Aa( 7 );
-f 6 ) = 7 (a) + y (b ) = (a + b )(y );
(aft) = 7(a)7(ft) = (aft)(7).
Hence the Gelfand homomorphism is indeed a homomorphism. The topol ogy of the Gelfand space was chosen to make all the Gelfand transforms a continuous functions. If 71 and 72 are distinct elements in then there is some a £ A with 0 ( 7 1 ) / ^ ( 7 2 )- I f we define e to be the positive distance between these two numbers, then ( 7 6 ^ : |a(7 ) - 6( 71 )| < e/2 } and {*) 6 : |a(7 ) - 0 ( 7 2 )! < e/ 2 } are disjoint neighborhoods of 71 and 7 2 , respectively. Hence F^ is Hausdorff. Suppose A is nonunital. Then 70 : A 1 —<►C defined by 7 o(A + a) = A belongs to F^i and its restriction to A is the zero homomorphism in F^. If the restriction to A of any 7 € F^i is not zero, then the restriction belongs to TA This restriction map is a homeomorphism since it is obviously a continuous bijection of the compact Hausdorff space F^i onto the compact Hausdorff space r ^ . This proves the second sentence. (Note that the inverse homeomorphism of onto T a ' is given by 7 1— ► 7 defined by =y(A + a) = A + 7 (a ) since each 7 e is unital.) If A is unital, then i is the function constantly equal to 1 because the idempotent number 7 ( 1 ) = 7 ( l )2 must be 1 rather than 0 since 7 is not identically zero. Hence the Gelfand homomorphism is unital. (b): If ft is an inverse for 7 (a ) - a in .4*, then 1 = 7 ( 1 ) = 7(ft(7 (a) - a)) =
7( 6)( 7(a )
~
7(a))
= °>
where 7 has been extended (if necessary) to A 1 by 7 (A + a) = A + 7 (a)(c): If an element belongs to the Jacobson radical then its spectrum is {0 }, so by (b) the element is in the Gelfand radical. Any commutator ab — ba for a, b € A and any element o f 7 € r A satisfy 7 (ab -
ba) = ~t(a)y(b) - -y (b )j(a ) = 0 .
If p is the natural map of A onto ( A / A j / A j A n ), then 7 *-* 7 o tp is a natural homeomorphism of onto ( ^ a / A j / ^A/Ai, )■
9 1
2
Gelfand Theory
307
(d): The inequality (4) is an immediate consequence of (b). It shows
that the kernel of any 7 e is closed with respect to a. The Gelfand rad ical is just the intersection of these sets for all 7 e The last statement follows from the inequality. (e): We begin by showing that is compact. Inequality (4) o f (d) shows that for each a 6 A , p (a ) is a finite bound for both |a.(-y)| and ja °( 7 )|. Consider ( ) as a subset of the product space K = fla e ^ P (a)C iThe Gelfand topology on ( / F^ ) is precisely the relativization of the product topology on K . ln this topology, K is Hausdorff and compact by the Tychonoff theorem. Thus if ( F^ / F^ ) is closed in K , it is compact. However, each of the equations which defines whether 7 : A —> C is an algebra homomorphism involves at most throe elements of A . Therefore, is closed since membership in it is determined by these equations. (The set of elements satisfying any equation involving only a finite number of elements of A is closed by definition of the product topology. In the case of three elements, an (e/3)-argument will show this.) Furthermore, each a belongs to C 'o(r^ ), since each d° vanishes at 0 e F^. It follows that r°, is just the one point compactification of so the latter is locally compact. In the unital case the additional restriction 7 ( 1 ) = 1 (where 1 is ar. identity element at least modulo the Jacobson radical) is enough to determine membership in 1'^ , and thus the latter set is closed and therefore compact by our previous remark. □ Note that if (A , || • ||) is a spectral normed algebra, result (d) shows that r .4 is a subset o f the unit ball o f A ' . In this case, the Gelfand topol ogy is just the relativized weak* topology. This situation will be the most common in the rest o f this work. In particular this shows that any homo morphism 7 : A —* C on a Banach algebra A is continuous. Whether this extends to arbitrary Frechet algebras is a question of some interest often celled Michael’s problem from his Memoir [1952] (cf. Jean Esterle [1984], Mohamed Oudadess [1983a], Taqdir Husain [1983], Angel Larotonda and Ignacio M. Zalduendo [1983]). So far we have provided no conditions to force the Gelfand space to bs r.o :i2inpty. Since the Gelfand transform of any commutator is zero, we cannot expect the Gelfand space to be large enough to be useful unless A is “alrrost” commutative. In this case, that turns out to mean that A is commutative modulo its Jacobson radical, and we have already introduced almost, commutative as a name for this condition. Also, T ^ would be empty if any element of A had empty spectrum. In fact, we only get a useful theory whsn there are enough elements in the Gelfand space so that inclusion (3) of °-'!"osition 3.1.2(b) can be reversed. (As usual, zero plays a special r j b ; 1 „he spectrum here, so that zero must be omitted from the reversed inclusions.) If A is almost commutative, the reverse inclusion follows from the assumption that A is a spectral algebra, as we will now show. In
308
3: Commutative Algebras and Functional Calculus
3.1.3
fact, these two conditions are necessary and sufficient ones for a number of interesting results. Before proving this, we establish a simple but powerful consequence of results on maximal modular ideals from the last chapter, particularly Theorem 2.4.12. 3.1.3
P r o p o s itio n
subset M
Let A be an almost commutative algebra.
Then a
of A is the kernel of a non-zero homomorphism onto C if and
only if it is a maximal modular ideal. Proof Clearly the kernel of a homomorphism onto C is a maximal mod ular ideal since it has codimension one and a unital image. Conversely, suppose A I is a maximal modular ideal. First we will show that it in cludes the Jacobson radical. (This will follow from basic results in Chap ter 4, but we need a proof now.) Theorem 2.4.6 shows that M is in cluded in a maximal modular left ideal C. This means that M is included in the kernel o f the irreducible representation of A on A / C defined by Ta(b + C ) = ab + C. for all a, b e A . Hence by maximality, M is this ker nel. Since it is the kernel o f an irreducible representation, it includes the Jacobson radical by Definition 2.3.2. Now the image M / A j o f M in the commutative algebra A / A j obviously satisfies Theorem 2.4.12 and thus is the kernel of a homomorphism of A / A j onto C. Hence M is the kernel of a homomorphism of A onto C. □
3.1.4
Lem m a
Let A be an almost commutative spectral algebra. Let a
be an element in A .
Then a non-zero complex number A belongs to the
spectrum of a if and only if there is a non-zero homomorphism 7 € satisfying: 7 (a )
= A.
Furthermore, 0 belongs to the spectrum of a if there is a non-zero homo morphism 7 6 T satisfying 7 (a ) — 0 , and this condition is also necessary if the algebra A is unital. Proof Suppose 7 : A —* C is a homomorphism onto C satisfying 7 (a ) = A. Theorem 3.1.2(b) shows A € Sp(a). Suppose, conversely, that a non-zero A € C belongs to S p(a). Theorem 2.3.3(c) shows that A belongs to Sp^/Aji0- + Aj ) . Since this quotient algebra is commutative, we may assume A is commutative for the rest o f the proof. Then A ~ la is not quasi-invertible. Theorem 2.4.6 and the commutativity of A show that A 1a is the relative identity for a maximal modular ideal M . Thus Proposition 3.1.3 asserts that M is the kernel of a homomorphism 7 of A onto C. Let b € A be any element satisfying 7 (b ) = 1. Then 0 = -y(b\~l a —b) = 7 (A~*a) —1 holds and implies 7 (a ) = A. If .4 is unital and 0 belongs to S p ^ (a ), then A a is included in a maximal modular ideal I with relative identity 1. Thus, by the same argument, there is a homomorphism 7 o f A onto C with 7 (a ) = 0 . □
Gelfand Theory
3.1.5
309
r lie following theorem combines results from Theorem 2.4.11 and The orem 3.1.2 to give our definitive statement on Gelfand theory. It gives nec essary and sufficient conditions for this theory to apply and shows that an algebra which has even the slightest resemblance to an algebra of functions (modulo its Jacobson radical) in fact enjoys a complete Gelfand theory. 3 .1 . 0
T h e o re m
The following are equivalent for an algebra A .
(a) A is a spectral algebra which is almost commutative. ( b) A is a spectral algebra in which the Jacobson radical equals the Geivind radical. (o) The spectral radius on A is an algebra semi-norm.
(d) The spectral radius is finite-valued and there is a constant C satis fying:
V a,b €
p(a + b) < C (p (a ) + p (b ))
A.
(e ) The spectral radius is finite-valued and there is a constant C satis
fying: p(ab) < C p (a )p (b )
V a, b e
A.
(_") Every element of A has nonempty bounded spectrum and any two elements satisfy: Sp(a + b) C S p (a ) + Sp(b).
(g) Every element of A has nonempty bounded spectrum and any two elements satisfy: Sp(ab ) C S p (a )S p (b).
(h) Either the Gelfand space is empty (which occurs if and only if A is a Jacobson-radical algebra), or the Gelfand space F A is a nonempty locally compact Hausdorff space and the algebra A
=
{a : a € A } of Gelfand
tramforms is a point-separating, spectral subalgebra of C o (T ^ ) satisfying
a (r ^ ) C SpA (a ) C a ( r ^ ) U 0
V a € A
Monover, for each a € A , the supremum Hajlo^ = p (a ) of { |a(7 )| : 7 €
(5) }
is at. lined at some 7 e
/ any of conditions (a), (b), (c), (d ), (e), (f), (g) or (h) hold and A is also unital, then
is compact and each element a e A also satisfies:
a(r A) =
S p (a ).
(6)
Proof (a)=>(h): If A is Jacobson-radical, is empty since each of its elements would define an irreducible representation of A on C. If A is not Jacobson-radical, it contains an element with a non-zero complex number in its spectrum. Hence by Lemma 3.1.4, the Gelfand space of A is not eir.pt *. The rest of the statements follow from Theorem 3.1.2(b) and (e)
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3: Commutative Algebras and Functional Calculus
3.1.5
and Lemma 3.1.4 except the claims that ,4 is a spectral subalgebra and that the spectral radius is a maximum value. Fbr the latter we show that the supremum p (a ) = sup{ |a(7 )| : 7 € } is actually attained. For each n 6 N, choose 7 n € A satisfying |<£,| > p (a ) — 1/n. There is a subnet converging in the compact space but it cannot converge to 0 , so its limit 70 is in T A and satisfies |a(7 0)| = p(a). It only remains to show the image A is a spectral subalgebra o f C o (r A ). If a <= A is quasi-invertible in C o (r ^ ), then 1 does not helong to Sp(7„ (r A)(« )Thus 1 does not belong to a (Y A ). Hence, the second inclusion in (5) implies 1 £ SpA (a ). Thus, a has a quasi-inverse b in A , so b is a quasi-inverse in ,4 for a. (h )= > (f) and (g): This is immediate except that it appears at first that we need to add a “U { 0 } ” to the right side of these inclusions. If A is unital, we do not need to add “u {0 }” since we can use the final equation of (h), and if A is not unital. then zero belongs to the spectrum of every element. The implications (f)= > (d ) and (g)=>(e) are obvious. Theorem 2.4.11 shows the equivalence of conditions (a), (c), (d) and (e). (h)=>(b): The inclusion A j <■ A\- holds for all algebras by Theorem 3.1.2(c), hence it is enough to establish the other inclusion. If A is Jacobsonradical, there is nothing to prove. Otherwise, the inclusions o f (h) show that the Gelfand radical is an ideal in which every element has spectrum {0 }. Thus the Gelfand radical is included in the Jacobson radical by Theorem 2.3.3. (b)=>(a): Theorem 3.1.2(c) shows this. □ Some Consequences of the Gelfand Theory
This is a very powerful theorem with many obvious consequences. For instance, note that condition (b ) shows that an almost commutative spec tral algebra ,4 is semisimple (and therefore commutative) if and only if it satisfies one (and hence all) of the following equivalent conditions: ( 1 ) the Gelfand homomorphism is injective; (2) the spectral radius is a norm; or, (3) the elements of separate the points of A - The inclusions in condi tions ( f ) and (g ) (which will be extended below to commuting elements in an arbitrary spectral algebra) are powerful adds in computation. The only related results for noncommuting elements involve the numerical radius in troduced in Section 2.6 rather than the generally more useful spectrum. Corollary 3.5.14 will show that if ,4 is a semisimple commutative Banach algebra and V A is compact, then A is unital, thus providing a partial con verse for the last statement in (h). We now extend several of our results from commutative algebras to com muting elements in possibly non-commutative algebras. Theorem 2.2.14 showed that the spectrum and the spectral radius were upper semicontinuous functions. The following corollary shows that the spectrum and spectral
311
Gelfand Theory radiu& are uniformly continuous when restricted to commuting sets.
S .I.?
C o ro lla ry Let A be a spectral algebra. Then any commuting ele ments a, b € A satisfy the following results: Sp(a + b)
C
S p (a ) + Sp(b)\
Sp(ab)
c
Sp(a)Sp(b)\
p(a + b)
<
P ( a ) + p( b) ;
p(ab)
<
p{a)p{b);
Sp(a)
c
Sp(b) + p(a — 6 )C i
|o(a) - p(b)\
<
p(a — b) <
C
Sp(b) + a (a — 6 )C i;
a (a — b)
whefe a is some spectral semi-norm in the last two results. Thus the spec trum (in the Hausdorff metric for subsets of C ) and spectral radius are uniformly continuous on commuting elements with respect to any spectral .mni-norm. Proof Propositions 2.5.2 and 2.5.3(c) show that we may replace A by any maximal commutative subalgebra that contains a and b. Theorem 3.1.5 establishes the first four results. The other results follow easily. □ Characterizations of Commutativity
Theorem 3.1.5 gives a number of characterizations of almost commuta tive algebras among spectral algebras. It also shows, as we have just em phasized, that an almost commutative spectral normed algebra (-4, ||• ||) satisf.es: |p(a) - p(b)\ < ||a - b||
V a ,6 € A
This certainly implies that the spectral radius is a uniformly continuous function on such an algebra. Theorem 4.2.15 in the next chapter will give a, .converse. r. We now give additional conditions which imply that a normed algebra is commutative. Note that condition (c) below (which was already obtained in ^ermia 2.4.10) is necessary as well as sufficient. The other two conditions are considered again in Proposition 3.1.8. These results are due to Claude Le Page [1967], Rudolf A. Hirschfeld and Wieslaw Zelazko [1968] and John W . B&’cer and John S. Pym [1971], See also Gh. Mocanu [1967], [1971], C. Cheikh and M. Oudadess [1988], We give related results in Theorem 4.2.15. 3 .1 ." Proposition A normed algebra (.4, |[ ■||) is commutative if there is a constant C satisfying any of the following three conditions.
(a; (b)
||a||2 < C||a2|| ||a|| < C\\a\\°°
V a € A V a € A
312
3: Commutative Algebras and Functional Calculus
3.1.8
(c) ||af>|| < C’ ||6a|| V a, b € A 1 where the norm on A 1 is defined by ||A -I- a|| = |Aj -)- ||a|| for all A £ C and a £ A. Proof Inequality (a) implies
l!a||
<
C 2“ ||a2 | r ‘ < C 2“ C 2"||a 23 | r 2 < •••
<
C 2 ‘ c 2" - . . c 2 '"||a2"||2" "
for any n. Talking the limit as n increases we get (b). Suppose (b ) holds. Lot A be the completion o f A ■ For any a £ A , let { “ n}neN I* 2 a sequence in .4 converging to a. Theorem 1.1.10(h) implies IMI = lim||a„|| < C lim su p ||a„||00 < CHaH00. Then for any A + a £ «4*, Proposition 2 .2 . 1 (e) implies IIA -I- a||
=
|A| -I- ||a|| < jAj -I- C||a||
<
C(|A| + p (a )) < Z C p {\ + a) = 3C||A + a||°°.
Hence for any a, b £ A 1, Theorem 1.1.10(d) implies ||ab|| < 3C||a6||* = 30116011°° < 3C||6a||. Thus we have shown (a) => (b ) => (c ), (with a different choice o f constant in (c)). Now Lemma 2.4.10 completes the proof. □ 3.1.8 P r o p o s itio n The following conditions are equivalent for a normed algebra (A , || • ||). (a ) The norm is complete and satisfies ||a|| < C||a2|| for all a £ A (b ) The norm is complete and satisfies ||a|| < CjlaJI00 for all a £ A . (c ) The spectral radius is a complete norm on A . (d) The Gelfand homomorphism is injective and A is closed in C o (r ^ ).
(e) The algebra A is isomorphic to a closed subalgebra of C o(i'l) for some locally compact space ii. Proof (a)=>(b): The proof o f the last proposition establishes this.
(b )= »(c ): The last proposition shows that A is a commutative Banach algebra, so the spectral radius is an algebra norm. Since the norm in a Banach algebra is spectral, it dominates the spectral radius. Hence, (b) shows that they are equivalent norms. (c )= »(d ): Theorem 3.1.5 shows that if the spectral radius is a norm, then the Gelfand homomorphism is injective. However, the Gelfand homo morphism is an isometry with respect to the supremum norm in C n(T^). Hence, the image A of this map is closed.
Gelfand Theory
3.1.9
313
( d )^ (e ): Immediate. (e)=->(c): If the isomorphism is (a ) = pc 0( f i ) M a )) = llv’ Mlloo holds f r all a € A . (c );> (a ): The identity map from .4 with its own norm to .4 with the •spectral radius norm is a contraction. Hence by the inverse boundedness theorem, the two complete norms are equivalent. Thus, for a suitable con stant C we get ||a||2 < C p (a )2 = C p (a 2) < C||a2||. □ Algebras satisfying the equivalent conditions of the above proposition are usually called uniform algebras. For an introduction to the theory of uniform algebras, including many examples, see the excellent book of Edgar Lee Stout [1971]. Theorem 4.2.26 gives a surprising characterization of uniform algebras. Irving Glicksberg [1964] and Sergei Vitalyevic Kislyakov [1989] show that most uniform algebras are uncomplemented subspaces of the Banach space o f all continuous functions. Quotients of uniform algebras have been characterized by A. M. Davie [1973c]. They are sometimes called Q-algebras, but that term is also used for topological or normed algebras with ar open set o f invertible elements. For a lattice theoretic approach .see Egcn Scheffold [1988] and previous papers. Dual k dps of Homomorphisms
Let A and B be algebras and let ip be a homomorphism o f A into B. Then t e dual map : B* —* A^ maps into since tp* ( 7 ) = 7 o ip is a homomorphism into C whenever 7 is a homomorphism into C. Clearly, ipt need not necessarily take a non-zero 7 € Tb into a non-zero homomorphism in r. 4 , so we must work with the augmented Gelfand spaces in this instance. We will also use
is continuous relative to the
Proof This is immediate.
□
In Section 3.3 we will use the completeness o f a Banach algebra to define a functional calculus. It will then be important to know the relationship between the Gelfand theory of a commutative spectral normed algebra and that of .ts completion. The next proposition gives a very satisfactory an swer: (bey can be naturally identified. 3.1.10 P r o p o s itio n Let (A , ||• ||) be a spectral normed algebra with com pletion A. If ip is the inclusion map of A into A , then tp^-.T-^ —* homeomorphism of T -j onto r. 4 . If A
is {
commutative /
almost commutative )
( comrn.itative / almost commutative ).
then A
is a is also
314
3: Commutative Algebras and Functional Calculus
3.1.11
Proof Proposition 2.1.5 shows that A (or, equivalently, ip (A )) is a dense spectral subalgebra of A and Proposition 3.1.9 shows that
The next result, which is due to Silov [1947a], is related to Corollary 2.3.9. Both results show that certain homomorphisms into Banach algebras are automatically continuous (i.e. they are continuous without requiring continuity a prion). The former result requires surjectivity but does not restrict the target algebra, while the present one makes 110 restriction on the “size” of the range o f the homomorphism, except that it should be in a semisimple commutative algebra. Chapter 6 will contain a number of related results. The last statement in the theorem was obtained earlier by Gelfand [1941a] and is a special case of the Johnson-Rickart uniqueness of norm theorem (Corollary 2.3.10 and Theorem 6.1.1). 3.1.11 T h e o re m Let ( A , a ) be a spectral semi-normed algebra and let (B , r ) be a commutative, semisimple spectral semi-normed algebra. Then r is a norm and ( the graph o f) any homomorphism < p :A —* B is closed. If A and
B are Banach algebrus ( with the latter still commutative
and semistmple), then
the same result since it implies [[61)00 < p b < t for all 6 G B and ||-||oo is a norm on a semisimple commutative algebra.) Let {anJngN C A converge to a € A , and let { v j ( a „ ) } n6N converge to 6 e B. For any 7 € Tg, inequality (4) in Theorem 3.1.2(d) applied to 7 o ip implies that 7 o 1p (a ) = lim 7 o <^(an) = 7 lim
r . ’ .ir- •
Gelfand Theory
315
Y/e can also give a simple proof which does not use the Gelfand theory. Theorem 2.4.3(b) shows that the spectral radius in B is an algebra norm. 's J n c N and b have the same meaning as in the first proof and let c e B be arbitrary. Then these elements satisfy P(cb)
<
p(c)p(tp(an) ) + p(c)p(
<
P(c)[||a„||^ -I- | M a „ ) - 6||b] -»
0
where we used Proposition 2 .2 . 1 (c) to replace p(
Next we turn to showing the connection between Gelfand theory and both double centralizer algebras and Arens multiplication on the double dual of an algebra. Some of the following results were obtained by Ju Kwei Wang [1961] and Laszlo Mate [1965]. Pietro Aiena [1990] contains related spedavized results. 2.1. IS* T h e o r e m Let A be an algebra. . (p.) For any 7 e r A , there is an element 7 € r v {A ) satisfying 7 (L a, R a) = 7
(° )
V a e A
Fence/ ^ v (A ) ** nonempty if is. I f TA is nonempty, restriction of the Gelfand transforms of elements of V ( A ) to {7 : 7 € defines a homcr.'iorphism ip of T>(A) into an algebra of continuous functions on with "z-zmel { ( L , R ) e D ( A ) : L ( A ) -I- R ( A ) C .4r}(b) If A is a commutative, semisimple spectral algebra, then T>(A) is corrrvAative and semisimple. I f ||■|| is a spectral norm on 'D (A ), then each map in the following commutative diagram is contractive and injective. v ( A ) ----------------- - --------------- ► C ( r ^ )
a
------------------- ► Co(r^)
proo/ (a): Choose (L , R ) € V ( A ) and 7 € The double centralizer equation gives:
and a, 6 , € .4 with 7 (a ) ^ 0.
7 ( a ) 7 ( L ( 6 ) ) = 7 (a L (b )) = y (R (a )b ) = j ( R ( a ) ) j ( b ) .
316
3: Commutative Algebras and Functional Calculus
3.1.13
By setting 6 equal to a, we conclude 7 (L (a )) = 7 (R (a )). If 7 ( 6) ^ 0, then
l( L (a))/l(a) = l(R (a))/l
(a) = 7(L(6))/7(6) = 7(#(6))/7(6).
Therefore, for each 7 £ r .4 and ( L , R ) £ Z?(.4) we can define 7
( ( £ , * ) ) = 7 (i(«))/ 7 (a )
for some a € .4 with 7 (a ) ^ 0. It is easy to check 7 £ T d ^ ). Since the same a may be used in the definition o f 7 for all 7 in some neighborhood of each 7 0 , the restriction o f ( L , R ) A will be continuous on { 7 : 7 e r.4}. The kernel o f ip is obvious. (b): The vertical maps in the diagram are just the regular homomor phisms. Since the hypotheses imply A r = {0 }, ip is injective, as are all the other maps. Hence T>(A) is commutative and semisimple. It is also clear that all the maps are contractive. □ When A is a commutative Banach algebra, it would be interesting to know the size o f the subalgebra A o f Gelfand transforms o f elements in ,4 in the algebra C o (r ^ ). An interesting answer is given in Graham R. Allan [1968]: a function / on belongs to ,4 if and only if it is the restriction of a locally bounded analytic function on a weak* neighborhood of r .4 in ,4*. The word “analytic” in this definition has a variety o f natural meanings. Sin-ei Takahasi and Osamu Hatori [1990] have recently found answers to the related question of the size of the subalgebra of the algebra C {T A ) induced by the double centralizer algebra T>( A ). In a number of interesting cases, but not in every case, this is exactly the algebra o f functions which we describe in the next definition. The condition in the definition was first used by Salomon Bochner [1934], Isaac J. Schoenberg [1934] and W illiam F. Eberlein [1955] to characterize those continuous functions on the dual group of a locally compact group induced by measures on the original group. This explains the name. We will show that this algebra is precisely the algebra o f all continuous functions which arise by restricting elements of the double dual -4** to r * c A '. The Image of the Double Dual Algebra
3.1.13 D e fin itio n Let ^ be a Banach algebra with nonempty Gelfand space. A function / £ C (T .4 ) is said to satisfy the Bochner-Schoenberg Eberlein condition if there is a finite constant C so that for any finite list Q i, Q2 , .. •, a n of complex numbers and corresponding list 7 1 , 7 2 , • ■■, 7 n of elements of / satisfies: n
I
n
a) f M J=1
I ^ c \I
"jTj'IU* 1
( 7)
Gelfand Theory
3.1.14
317
For such a function /, the infimum of all possible constants C is called the Bochner-Schoenberg-Eberlein norm o f / and is denoted by ||/||b se The normed linear space of all Bochner-Schoenberg-Eberlein functions is deno,ed by C b s e It is straightforward to show that C b se(T .4 ) >s a subalgebra o f C ( r A ) including A and that || • ||bse is a complete algebra norm on C b s e ^ ^ ) whic; satisfies Halloo < ||a||BSE for all a e .4 . Some of these results are checked in the cited reference. The following result stems from Paul Civin and Lertram Yood [ 1961 ] 3.6, 3.7 and 3.16 as well as Takahasi and Hatori [1990:, Theorem 4. See Section 1.4 for background on Arens multiplication. 3.1.14
T h eo rem
Let A be a Banach. algebra and let K o : A —> -4** and
’. A * —* *4*'* be the canonical isometric linear injections. ( t ) When A ** w provided with either Arens product, maps C A* into C A ***. ( t ) If is nonempty, the set ( r ^ ) 1 is an ideal in *4** with either Aren:■ product and the two resulting quotient algebras coincide and are commuta'ive. Moreover, there is a contractive algebra homomorphism ip which
ki
makes the following sequence exact:
{ 0 } — ►(rA ) ± J £ + A * * - ^ B ( r A ) where
(c) If A is almost commutative, then iP (A ” ) n c ( r A ) = c BSE (r A ) where ip is the homomorphism of (b ) above.
(c )
If A
is commutative, then any two elements f , g €
f9 — 9 ’ f ■ Hence, Arens product.
k0
A ** satisfy
maps A into the center of A ** with respect to either
Proof (a): For any 7 e T A , a e A and / € «4**, the three step definition of *h j Arens products gives a7 = 7 ( 0 ) 7 = 7 a, / 7 = / ( 7 ) 7 = 7 /, and } 9 i l ) = f ( l ) 9 h ) = / ■9 (l ), proving this result. (b): This follows easily from the last result in the proof of (a). (c): It is easy to see that the left side is a subset of the right side. Suppose / is a function in CBSE(r^) with norm ||/||BSE = F . Let B be the collection of all finite subsets of T .4 ordered by inclusion so that it is a directed set. By Helly’s theorem (Yosida [1965], Theorem 6.5, p.109), for each /? S B we may find an element ap satisfying / ( 7 ) = 7 (a p ) for all 7 € /3 and ||a/3|| < F 4 - 1/n where n is the cardinal of /?. Thus {«(a ^ )}/ 36B is a bounded net in A " . By Alaoglu’s theorem, some subnet converges in the
318
3: Commutative Algebras and Functional Calculus
3.1.15
weak* topology to an element g e «4 ** which satisfies g ( 7) = /(7) for all 7 e and H0II = ||/||BSE. (d): These immediate consequences of the definition of the Arens prod ucts are noted here for completeness. □ A unital normed algebra A is said to be generated by one of its elements a if it is the smallest closed unital subalgebra of itself containing a. Not,*'
that this is equivalent to saying that polynomials in a are dense in A In this case, A is said to be singly generated. A compact subset K of C is said to be polynomially convex if it is equal to its polynomially convex hull pcK. This is equivalent to saying that there are 110 bounded components of C \ K . The next theorem shows the relationship of these concepts. It will be generalized to finitely generated algebras in Section 3.5. 3.1.15 T h e o re m
Let (A , ||• ||) be a united spectral normed algebra which
is generated by one of its elements a. Then a is a homeomorphism of Y a onto S p (a ) and the latter set is polynomially convex.
Furthermore, for
any compact polynomially convex subset K of C there is a singly generated algebra A with generator a satisfying S p (a ) ~ K . Proof If two elements 71 and 72 of YA satisfy 71(a) = 72(a)) then they agree on every polynomial in a and hence on A . Thus, a is a one-to-one continuous map from the compact space into the Hausdorff space C. Any such map is a homeomorphism. If there were a bounded component U of C \ S p (a ), we could choose an element Ao from it. For any polynomial p, the maximum modulus principle implies
)p{Ao)|
<
max{[p{A)| : A € d U } < max{|p(A)| : A 6 S p (a )}
<
max{|A| : A e S p (p (a ))} = p (p (a ))
<
||p(a)||.
Given any element b in the smallest (not necessarily closed) unital subal gebra Ao of .4 containing a, we can find a polynomial p satisfying p(a) = b. The above inequality shows that if two polynomials both satisfy this con dition then they agree at Ao- Hence, we can define a homomorphism
3.2.1
Silov Boundary, Hulls and Kernels
3.2
319
Silov Boundary, Hulls and Kernels
Z:. this section we will generally assume that the algebras considered are commutative and spectral. It is of some interest that the definitions of the last section were made in such a way that this assumption is not really ::ecessary. Similarly, we sometimes assume that the Gelfand spaces of the algebras involved are nonempty. This is only to avoid triviality, and if carefully construed, the statements remain true without this restriction. T h e Silov Boundary
We introduce the notion of boundaries and prove the existence of a millin'uni closed boundary for any commutative spectral algebra. The ter minology is suggested by algebras such as the disc algebra /1(D) introduced in 1.5.2 and further explained in §3.2.13. For this algebra, the Gelfand transform z of the coordinate function z on C (i.e., the function satisfy ing 2 (A) = A for all A € C ) sets up a homeomorphism between and the closed* unit disc C j. Under this homeomorphism, the Silov boundary, which we are about to define, is mapped onto the boundary T of the disc. We will use the maximum modulus principle below to show that the inverse image of T under z satisfies the defining properties of the Silov boundary. As the name suggests, the concept is due to Silov [1940a], 3.2.1 D e fin itio n Let ^ be a commutative spectral algebra with nonempty. A boundary for .4 is a subset f l of such that for each element a S A ‘there is a point 7 e f l satisfying l«(7 )l =
N lo o
= ? («)•
The irtsrsection of all closed boundaries for A is called the Silov boundary for J. a id is denoted by dA . Note that a closed subset f l o f is a boundary if and only if it satisfies = ||a||oo- The argument is the same as that given in the proof of Theorem 3.1.5 (a)=>(h) which showed that |a| attains its supremum on F_4 . It is not obvious that the intersection dA of all closed boundaries is non-empty and certainly not obvious that it is a boundary. However, the next theorem, which is due to Silov [1940a], proves these results and thus justifies the terminology “Silov boundary” . The proof we give is due to Lars Hc-rr'.t'.er [1966] when A is a unital Banach algebra. The first inclusion in, (_) gives another interpretation of the term boundary. In relation to the second inclusion, it is of some interest that S p (a ) includes any component of the complement of a(dA ) U { 0 } with which it has nonempty intersection. Otherwise the component would include points in the boundary of Sp(a). 3.2.2
T h e o re m Let A be a commutative spectral algebra. The Silov boundary dA for A is a nonempty boundary for A and hence is the smallest
320
3: Commutative Algebras and Functional Calculus
3.2.2
closed boundary for A . Each a e A satisfies d S p(a ) C a(dA ) U {0 } C S p (a ) U { 0 } where “ u { 0 } ” can be omitted if A is unital.
(1)
If A is nonunital, then th%
map which restricts each element of r .41 to A maps dA \ bijectively onto either dA or dA U {0 }. Proof First we assume that A is unital. We begin by obtaining a par ticularly convenient banc for the topology of T A . Since the topology is the weakest one making each d in A continuous, a base is given by the sets o f the form { 7 e r A : ( 7 ) - bj(ip)\ <
€
: |Oj ( 7)1 < 1
V j = 1 , 2 , . . . , n }.
Call the latter set VJ,1>aa.... . A base for the topology of is given by sets of this form. We will call them special basic open sets. Next we show that if V = VJ,li0i)... a„ is one of these special basic sets, then either B \ V is a boundary for each closed boundary B , or V has a non-empty intersection with each boundary o f A . Suppose B is a closed boundary but B \ V is not a boundary. Then, there is an element b e A such that 1161(3x^)11^ < 1 while integer m, we can achieve l(*m« j ) ' ( 7 )l = W 7 ) mM 7 )l < 1
= 1. For a large enough positive
V j = 1 , 2 , . . . , « ; 7 e B \ V.
Since |(6ma j ) A( 7 )| < 1 also holds on V , compactness of B gives ll(*ma , ) A|B |L < 1 Since f l is a boundary, this implies ||(&maJ ) A ||00 < 1. Thus, 1^(7 ) j must achieve its maximum 1 only when each | ( 7 ) | is strictly less than one. That is, |6 |achieves its maximum only in V . Hence, V must have nontrivial intersection with each boundary of A . Now we prove that dA is a boundary. If not, there is an element a d A satisfying lldla^ll^ < 1 and Hall^ = 1 . Let C be the set { 7 € r .4 : |a(7 )| = 1 }. Then, any 7 6 C does not belong to dA — (~){B : B is a closed boundary for A } . Hence there is a special basic open neighborhood V-, of 7 which has empty intersection with some closed boundary. By the result proved in the last paragraph, B \ V ^ is a boundary for each dosed boundary B . However, {V"-, : 7 € (7 } is an open cover for the compact set C. Let {V i, V2 , . ■•, V-,,} be a finite subcover. Since is a closed boundary, F A \ Vi is a closed
3.2."
Silov Boundary, Hulls and Kernels
321
boundary by the choice o f V\ as one of the V^s. Repeating this argument n — .■ more times, we conclude that \ U"=1Vj C r _4 \ C is a closed bount'ary. However, the element a contradicts this assertion. Hence dA is a boundary. Clearly, it is the smallest closed boundary. Now suppose that there is an a e A and a Ao £ d S p (a ) which does not belong to a(dA ). Since S p (a ) is closed, Lemma 3.1.4 shows Ao = 7 0 (a) for some 70 £ Since dA is compact, the distance from Ao to a(dA ) is positive. Call it e. Choose A 6 C \ S p (a ) satisfying |A — Ao| < e/2. Then the distance from A to a(dA ) is at least e/2. Since A —a is invertible, ((A - « ) “ 1 ) A = (A - a ) ” 1 belongs to A . Since dA is a boundary, we must have !j(A — 1 lloo = ll(^ ~ “ ) _ 1 |a^lloo - 2/e- However, the inequality |(A —7 o (a ))-1 |= |A — Ao|—1 > 2/e contradicts the previous inequality, thus providing the desired inclusion. Now we turn to the case in which A is not unital. We will establish that the Silov boundary is a boundary and the last statement of the theorem first. Suppose B is a closed boundary for A . For each 7 e r A , let 7 * 6 ^ 1 be denned by 7 * (A + a) = A + 7 (a ) so that 7 * is the unique element of he restriction of which to A equals 7 . Define B 1 C by B 1 = { 7 1 : 7 e B } U { 7 0 } where 70 is defined by 7 o(A + a) = A. W e will prove by contradiction that B 1 includes dA i. Suppose not. Then, there is an element b e A 1 and a number t satisfying TA i,
l l ^ l l o o < *
includes the set of restrictions to ,4 o f elements of dA i. Hence dA is a boundary for ,4 since we have already shown that dA i is a boundary for «4 '. Since the set of non-zero restrictions to ,4 of elements in dA i is a closed boundary for ,4, this set includes dA . This completes the proof o f the last stateiient in the theorem. Inclusion (1) for nonunital algebras is now an immef iate consequence. □ Tlnorems 3.2.10 and 3.2.12 will show two important classes of commu tative 3anach algebras in which the Silov boundary is the whole Gelfand spa:* We now give the simplest criterion for this to happen.
322
3: Commutative Algebras and Functional Calculus
3.2.3
3.2.3 P r o p o s itio n Let A be a commutative spectral algebra. If the algebra A of Gelfand transforms is dense in C o (r A ), then the Silov boundary 8a the whole Gelfand space In particular, this happens if for each a € A there is some b e A satisfying 6 ( 7 ) = 0 ( 7 )* for all 7 € Proof Suppose 7 G were not in Da - Then we could find / e C o^^t) satisfying f ( - j ) = 1 and / \qa = 0 . Any element a € A satisfying |(d—/(|oo < 1 / 2 would contradict the fact that dA is a boundary.
If A is closed under complex conjugation, it satisfies all the hypotheses of the Stone-Weierstrass theorem stated in 1.5.1. □ Silov [1940b] introduced his boundary in order to determine which ho momorphisms in the Gelfand space of a closed subalgebra could be extended to be elements of the Gelfand space of a larger commutative Banach alge bra. Recall the results on dual maps in Proposition 3.1.9. In the following theorem, note that if A is any closed subalgebra of a commutative Banach algebra B, then the inclusion map satisfies all the given hypotheses. See also Meyer [1991]. 3.2.4 T h e o re m Let A and B be commutative spectral algebras and let
V a € A.
Then ipHdg) includes dA . Furthermon, the Silov boundary dA is the largest subset of r a with this property for all choices of B and tp. If A is also semisimple, 8a is exactly the set of homomorphisms in which can be extended to any commutative spectral algebra B which includes A as a subalgebra satisfying pB(a) — P a {0 ) f or all a € A . Proof Note that 8b U {0 } is a compact subset o f Fg. Hence its image in
is closed. Thus tp^(8g) is closed in Tb. It is also a boundary since any a € A satisfies
II®I*>'(0b)IIoo = I I M a ) ) A|aBlloo = pe(«p(a)) =
pA
^ )-
Thus, this set includes the Silov boundary o f A . Consider the Gelfand homomorphism of A into B = CoCr^t). This satisfies the hypotheses o f the theorem, and Fg — dg can obviously be identified with under the dual o f the Gelfand homomorphism. Hence, this shows that 8a is the largest subset o f covered by tpHdg) for all choices of B and tp. For the final result, consider tp to be just the inclusion map. Hence, in the first paragraph o f the proof we have shown that each homomorphism 7 in the Silov boundary of .4 can be extended to be an element of the Gelfand space (indeed the Silov boundary) of B. The second paragraph shows that
3.2.5
Silov Boundary, Hulls and Kernels
323
c.ie 1-elfand homomorphism (which may now be regarded as an embedding map) is one in which no element of outside of the Silov boundary can □ be extended. Th%,j£ull-Kernel Topology
'We introduce hulls, kernels and the hull-kernel topology. This last con cept is due to Gelfand and Silov [1941] in the case o f commutative Banach algebras. Marshall Harvey Stone [1937] used a closely related concept for Boolean rings earlier. In Chapter 7 we will study essentially the same con cept in the case o f noncommutative Banach algebras. We note here that the construction we are about to give depends only on the fact that the kernel of ar: element o f is a prime ideal. (This term is defined in Chapter 4.) The hull kernel topology on the set o f prime ideals is called the Zariski topology in Grothendieck’s theory of schemes in algebraic geometry. The hv».” -kernel topology for primitive ideals in a general ring (also defined in Chapter 4) is usually called the Jacobson topology ( cf. Jacobson [1945b]). 3.2.5
D e fin itio n
Let <4 be a commutative spectral algebra. The hull
h (S ) -fi a subset <S o f ,4 is the set: h (S ) = { 7 e r A : 7 (a) = 0
The kernel k (B ) of a subset B of ; A subset B of
k (B )
=
{a
V a e <S}.
is the set:
e .4 : 7 (a ) = 0
V 7 e B} .
which has the form h (S ) for some S c A is called a hull.
^j, i§ sometimes useful to think o f h (S ) and k (B ) described in the fol lowing fashion:
(2) (3) These descriptions make clear the first assertion of the next proposition, wh;c!? records some elementary consequences of the definition. 3.2.3 P r o p o s itio n Let A be a commutative spectral algebra with 'zonsr.zpty. Consider Si C S2 C A and B i C B 2 Q Then h (S i) is a closed set in T A and k (B \ ) is a closed ideal of A when A is furnished with any spectral semi-norm. Moreover h (S 2)
c
h (S i)
Si
C
h k (S i)
h k h (S i)
=
h (S i)
k (B 2) C k ( B l )
(4)
B i C h k (B i)
(5)
k h k (B i) = k (B i ).
(6 )
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3: Commutative Algebras and Functional Calculus
3.2.7
Hence, a subset B o fT A is a hull if and only if it satisfies B — h k (B ). If Sz is the ideal generated by Si, then h (S 2 ) = h (S i). Similarly, if B 2 is the closure of B t in the Gelfand topology, then k (B 2) = k (B \). Proof The first assertions follow immediately from the intersection results which precede the proposition. The four inclusions are direct consequences of the definitions. Together they give the two displayed equations. The first of these equations implies the statement about hulls. The last two assertions are immediate consequences of the original definition. □
3.2.7 T h e o re m
Let A , B andC be commutative spectral algebras. Let 0
— *
A
B
C
— > 0
be a short exact sequence. Let
be the associated dual maps. Then maps Tc homeomorphically onto h (ifi{A )) and ip1 maps Tb \ h i ^ A ) ) homeomorphically onto Proof For 7 e Tc, ^ ( 7 ) = 7 0 ip vanishes on
maps h (ip (A )) onto {0} and S into Suppose that <^(71) = ¥^(72) holds for some 7 1,7 2 ,6 Tg. If 7) does not equal 72, then there is some b e B satisfying 71(6) ^ 72 (b). If 71(6) or 72(6) is zero, we may assume without loss of generality that it is 71(6). Then, 71(d) = 0 and 72(d) = 1 hold when d is defined to be (72(6))~’ 6. If neither 71(6) nor 72(6) is zero, then by choosing d to be A6 + fib2 we can achieve 71(d) = 0 and 72(d) = 1 for a suitable choice of A, fi e C. Choose d as indicated. Then for any a e A , we have 0 = 71 (dip(a)) ~ y2(d
Silov Boundary, Hulls and Kernels
3.2.8
325
For all a e A , we get 7 (a) = 7( 0 , we can find Qo € A so that a > a 0 implies I7 a (ip(e)b) — j(ifi(e)b)\ < e /2 (l + p (b )), and \la{:P{e) ) ~ '7,(v ?( e ) ) l < £! 2- Then for a > a 0, we have \la[b) - 7 W I
=
l7 a (6 )7 (^ (c)) - 7(fc)7(^(c))|
<
l7« (&)l l 7 M c ) ) ~ 7 a M c))l + l 7 o ( M c)) “ 7 (M « 0 )I
<
£.
This shows that
□
The next theorem is frequently proved by an appeal to Kuratowski’s closrre axioms, but we will give a direct proof. Recall that a hull is any subsit H of r A which can be written as h (T ) for some subset I of A . Consequently, a hull H satisfies H — h k {H ). 3.2.C T h e o re m
Let A be an algebra with. r A nonempty. The family
{ r .4 \ H : H is a topology for
is a hull in
This topology is included in the Gelfand topology for
re proof Let T be the family of sets which is asserted to be a topology. Cleanly h (A ) = 0 and /i({0 }) = hold, so that and 0 belong to T . The definition shows that any family { l a : a e A } of subsets of A satisfies
f | h (I a) = { 7 € r A : 7 (a ) = 0 a€A
V a g ( J l a) = h{ ( J I a). 06^4 a€v4
This mplies that T is closed under arbitrary unions. Finally, we must show that T is closed under the intersection of two of its elements. Let Hi and FI2 h >two hulls. Then H
xu
H 2 C h k {H i U H 2) = h (k (H i )n k (H 2) )
is ob ious. Suppose 7 does not belong to H i U H 2. Then for some aj in ,, we have 7 ( aj ) / 0 for j = 1,2. Since the product ai
326
3: Commutative Algebras and Functional Calculus
3.2.9
The topology described in Theorem 3.2.8 is called the hull-kernel topol ogy. We will now show some conditions which are equivalent to the equality
of the hull-kernel topology and the Gelfand topology. The definition is due to Silov [1947a]. who used the term regular. Charles E. Rickart [1960] introduced the term completely regular which is now standard. 3.2.9 D e fin itio n Let fi be a topological space. A subalgebra A o f C (f i) is said to be completely regular on if, for any closed subset K of O and any u^o € fl \ A', there is a function / e A satisfying f(uJo) = 1
and
f\K = 0 .
A commutative spectral algebra A is said to be completely regular if A (= C ( r A ) is completely regular on F^. Note that a Hausdorff topological space will be completely regular fin the topological sense) if and only if C (f i) is completely regular on fi. An easy compactness argument shows that any compact Hausdorff topo logical space is normal, and Urysohn’s lemina shows that a normal space is completely regular. It is clear from the definition that any subspace of a completely regular space is completely regular. Hence, we conclude that any subspace o f a compact Hausdorff space is completely regular. (In par ticular, locally compact Hausdorff spaces are completely regular.) §3.2.15 will show that, conversely, any completely regular Hausdorff space is a sub space of a compact Hausdorff space. The first portion of the following result, is due to Silov [1947a] for com mutative unital Banach algebras. Condition (c) will be used again in a more general setting in Chapter 7. Corollary 3.5.14 will show that a hull in the Gelfand space o f a commutative Banach algebra is compact in the Gelfand (and hence in the hull-kernel) topology if and only if its kernel is modular. This makes condition (c) a bit clearer. 3.2.10
T h eo rem
The following two conditions are equivalent for any
commutative spectral algebra A :
(a) A is completely regular. (b) The Gelfand topology and the hull-kernel topology coincide. The following condition implies (a ) and (b ) and w equivalent to them if A is either unital or a Banach algebra.
(c) The hull-kernel topology is Hausdorff and each point in
has a
hull-kernel neighborhood with a modular kernel. Any of these conditions imply that the Silov boundary of A equals FA . Proof (a) =» (b): Since each hull is closed in the Gelfand topology, we need only show that each subset B which is closed in the Gelfand topology is a hull. Let 7 € FA not belong to B. Then, complete regularity implies that there in an a C A satisfying a("r) = 1 and a\y = 0. Hence, 7 does not
3.2..’ 0
Silov Boundary, Hulls and Kernels
327
belong to h k (B ). Since 7 £ B was arbitrary, this implies h k (B ) C B . The opposite inclusion always holds as shown in Proposition 3.1.4. Hence each C?‘ i.'c.::d closed set B is a hull, so the two topologies agree. ' b ) => (a): Suppose B is a Gelfand closed set, and hence a hull, in £*~;1 ' e r.4 does not belong to B . Since B equals h k (B ), there is some element b € k (B ) satisfying £>(7) ^ 0 . Then a = 6(7)"*6 satisfies a (j) = 1 and d (B ) C { 0 }. (o) (c) (Under extra hypothesis.): Theorem 3 . 1.2 (a) asserts that the Gelfaad topology is always Hausdorff, so (b) implies that the hull-kemel topology is Hausdorff. L' A is unital, any ideal of .4 is modular so that the second condition of (c) is vacuous. Again, if ,4 is unital and (c) holds, then the identity map from with its Gelfand topology onto with its hull kernel topology will be a homeomorphism since it is a continuous bijection of a compact space onto a Hausdorff space. This completes the proof of the equivalence of (b) and (c) when A is unital. Now suppose (c) holds. (We do not need to assume completeness.) It is enough to show that each closed set B C is a hull, and for this it is enough to show that any 70 € which does not belong to B does not belong to h k (B ). Let V be a hull kernel neighborhood of 70 with k (V ) modular. Let
\ v) n P| k (U j) — k (r A \ v u B 0) c k ( B ) . 1"
.
J-i
Kence 70 does not belong to h k (B ). This concludes the proof of (c) => (b) Now suppose A is a Banach algebra and (b) holds. We will construct a hull-kemel neighborhood V for an arbitrary 7 e such that k (V ) is modular. First choose an element b € A such that 70(6) is non-zero, and denote |7o(6)|/3 by e. Then we claim that k (V ) is modular where V is the neighborhood of 70, V = {7 : (7(t>)| > 2e}. Considering V as a subset of ,3(« )C i, we see easily that V is compact. Hence, Theorem 3.2.7 shows that r././*(V) is compact. (Thus, we could refer to Corollary 3.5.14 below vv.;"'. Implies that the semisimple algebra A / k ( V ) is unital so that k (V )
328
3: Commutative Algebras and Functional Calculus
3.2.11
is modular. Instead of using this deep result, we will give an elementary proof, due to Rickart [1960], that k (V ) is modular.) Denote A / k ( V ) by A ' and b 4- k (V ) 6 A ' by a. We take over the notation from the proof of Theorem 2 .2.2 replacing A by A ! and A by e -1 . Then the long displayed equation in the proof becomes
■ ’ =1
where n € N is arbitrary, { ( i = 1,^2.• *s a full set o f nth-roots of unity, and / is an *4'-vali>ed function which is continuous on the circle of radius 1. The right hand side of this expression is a Riemann sum for the Riemann integral 1
/ 0 which exists since / is continuous. Hence, there is an element e' € A ' satisfying e' = lim ((£-- 1a ) " ) IJ. f l —♦ OO
Theorem 3.2.7 shows that any element of r_4 - corresponds to an element of h (k (V )) = V . Hence, for any 7 e we get
7
( e ') = ’
lim ((e - S C a ))" )'’ =
n—»oo
’
Hm >
^ v = 1-
n-^00 ( £ - * 7 ( 0 ) ) " - 1
Therefore k( V ) is modular, since e' is an identity element for the semisimple algebra A ' = A / k (V ). Hence (b ) and (c) are equivalent if .4 is a Banach algebra. Finally, we prove the last assertion in the theorem by contradiction. If the closed set is not all of , complete regularity guarantees an element a £ A with nonvanishing Gelfand transform satisfying a (d X ) — {0 }. This contradicts the fact that dA is a boundary. □ The next result is due to Ernst Albrecht [1982a] and this simple proof was discovered by Jyunji Inoue and Sin-ei Takahasi [1992]. 3.2.11
C o ro lla ry
Every commutative unital Banach algebra A has a
largest closed completely regular subalgebra A reg ■ Proof Let A lrg (with Gelfand space T ) be the closed subalgebra of A generated by all clowed completely regular subalgebras. Let b belong to one of these subalgebras B, which we may assume to be unital. It is clear that the map which restricts 7 e T to B is continuous in the hull-kemel topology from I to Fg- The theorem shows that b is hull-kemel continuous
3.2/2
Silov Boundary, Hulls and Kernels
329
on Ta, so it is also hull-kernel continuous when considered on I\ Hence any element of A reg has a hull-kernel continuous Gelfand transform since ft. is he norm limit of finite algebraic expressions of elements like ft. Thus A reg verifies condition (b) of the theorem. □ N -xt we turn to a condition which is much stronger than complete reg ularity. Algebras satisfying the conditions of the next theorem are called commutative C*-algebras. They will be dealt with at greater length in Vol ume II. The next result was a lemma in the fundamental paper o f Gelfand and Mark A. Naimark [1943], but is often referred to as the commutative Gelfand-Naimark theorem. The complete regularity o f the locally compact Hausdorff space Tc implies immediately that an algebra satisfying the con ditions of this theorem is completely regular. The following result has many consequences some of which will not be considered until Volume II. 3.2.12 T h e o re m
Let (.4 , ||• ||) be a commutative Banach algebra. Then, the Gdfand homomorphism is an isometric map of (A , ||• ||) onto (C o (T A ),
!! ■ [j, ) if and only if a map *: A —» A (called an involution) can be defined satisfying
II
* * <<
+ ft*
1+ 6)* (Aa)* (aft)*
=
a*ft*
(a*)*
=
a
(9 ) (10)
l| a*a| l
=
IM!2
(ID
for all a, b 6 A and A e C. satisfies
=
a*
(7) (8)
When such a map *:.4 —» A is defined, it
a* = (a )*
V a g A.
( 12)
Proof If the Gelfand homomorphism ( ') is an isometric isomorphism of
(-4, || • ||) onto (CoCF^), || • H^), then we can define *: A —» A by a *( 7 ) = ( 0 ( 7 ))*
V7 g
a g A.
It is easy to check that ( * ) has the stated properties. 3r, ipose, conversely, that a map * : A —* A exists satisfying (7) through (11). ;Tor any h e A such that h* = h,\\h\\ =
= ||/»23||2
= ••• =
||/t2 ||2 can be derived by repeated application of (11). Thus Corollary 2 .2.8 nplies IN I - l l f t l f = P(ft)
V h = h- € A .
(13)
Ne :t we will use this equality to show that any h e A satisfiying h = h* also satisfies S p (h ) C R. Suppose not. Then for some s, t in R, k = ah^+th
330
3: Commutative Algebras and Functional Calculus
3.2.13
satisfies * € S p (k ) and k = k“ . W e will derive a contradiction from this. Let r € R satisfy 1 -f 2r > p (k )3. For any positive integer n, define cn € A by c„ = (k + r i )nk. Theorem 2.1.10 shows (1 + r ) n j n+1 e 5 p (c„). Thus, Theorem 3.1.5(c) shows (1 + r ) 2n < p(cn )2 < ||c„||2 = |Kc„|| = p(c’ c „ ) = p ((k 2 + r 2) nk2) < E "= o ( ; y ( n- M k ) 3}+3 = (P (* 2) + r 2) np(k2). Taking
the nth root and then the limit as n increases, we obtain p (k )2 + r 2 < 1 + 2 r + r 2 < p (k )2 + r 2. This contradiction proves that h = h* implies S p (h ) C R for any h 6 A . Thus by Theorem 3.1.5, h = h* implies that h is re*l-valued. Now we show that (12) holds. For a € A , define h, k € A by h = 2-1 (a -+ a *) and k = (2 t)-1 (o — a*). These elements satisfy h = h*,k = k ",a = h + ik and a* = h — ik. Thus for any 7 e we get a * ( 7 ) = h ( 7 ) - tfc(7 ) = (^ ( 7 ) + *fc(7 ) ) ’ = 6 ( 7 )*, establishing ( 1 2 ).
We can now show easily that the Gelfand transform maps A isometrically and isomorphically onto C tl( r A ). For any a 6 A , we have ||a||
=
Ha’ all172 = p (a 'a ) 1/2 = sup{|ar a (7 )|1/2 : 7 e
=
s u p { ( a ' ( y ) a ( y ) ) I/2 : 7 •'
= su p {jn (7 )|
7 e r A ).
Thus (") is isometric, and this together with Theorem 3.1.5 shows that it is an isomorphism into 6’0(1' A ). Since (*) is an isometry, A is complete and hence is closed in C’u (r^ ). However, A is dense by Proposition 3.2.3 and hence actually equals C o (T ^ ), so that the Gelfand transform maps A onto C o (r^ ). □ We now turn to a few examples to illustrate the theory introduced so far in this chapter. 3.2.13 T h e D isc A lg e b r a This algebra j4(D ) was introduced in §1.5.2, which the reader should consult. It is the algebra o f all functions /: D —» C which are continuous on the closed unit disc D = {A € C : |A| < 1} and analytic on its interior D° = {A € C : |A| < 1}. The algebraic operations are pointwise and the norm is simply the supremum norm, which is complete since the uniform limit of analytic functions is analytic. Hence .4(D) is a unital commutative Banach algebra. The first step in studying any commutative Banach algebra is the deter mination o f its Gelfand space. It is often easy to guess the homomorphisms in the Gelfand space but harder to prove that one has guessed all o f them. The present example follows this pattern. For each A s D the map 7 * (/ )
= /(A )
V / G .4(D)
belongs to r*(D )- There are several ways to see that the set { 7 * : A 6 0 } is actually the whole Gelfand space. One instructive way is to note two isomorphic constructions for 4 (D ).
3.S."?
Silov Boundary, Hulls and Kernels
331
L e i P ( 0 ) be the algebra under pointwise operations o f all functions /: D —* C which are uniform limits of polynomials. The norm on P (D ) is, o f ccurse, again the supremum norm which gives the topology o f uniform convergence. Let P ( T ) be the algebra under pointwise operations of all continuous functions /: T —» C which have vanishing negative Fourier co efficients (cf. §1.8.14). The norm is just the supremum norm again. Here we regard T = {A g C : |A| = 1} as the boundary o f 0 and we recall (from §1.8.14) that the nth Fourier coefficient o f / g C (T ) is given by n
J
/ (») = ^
f { e il) e - intdt.
—IT We claim that /1(B) and P (D ) coincide and that restriction to T = c© maps them isometrically isomorphically onto P (T ). Fejer’s Theorem (§1.8.15) shows that the sequence {<xn/}neN of Cesaro means of the partial sums of the Fourier series of a function / 6 C (T ) converges uniformly (i.e., in the norm of C (T )) to /. Hence if / belongs to P (T ), then / is the uniform limit on T o f the sequence {cr„f }n€N of polynomial functions of A = elt 6 T. Now the maximum modulus principle shows that the sequence {<x„/}neN o f polynomials converges uniformly on all bf’ il). (Explicitly: |crn/(A ) - <7m/(A)| < ||anf - amfWoo -*■ 0 holds for all A £ D, where the supremum norm is defined on T J This shows that each / € P ( T ) is the restriction to T o f some function / in P (D ) satisfying l l / l l o o = ll/lloo- In fact, the construction o f the Fejer kernel given in §1.8.15 can be modified to give / explicitly: 7T f ireie) = h
/ (e“ ) Pr ( » - 0 *
j
—IT where P r is the classical Poisson kernel
P { s ) = _____ i z rW
1
i l _____
- 2 r cos(s) + r 2 ’
T ’ o completeness of /1(D) shows immediately that P (D ) is included in /1(D). Hence, in order to show that /1(D ) equals P (O ) and that restriction t c T m ips /1(D ) isometrically isomorphically onto P (T ), it only remains to show :,i>.at restriction maps /1(D ) into P (T ). This is obvious since we have K
f ( ~ n)
=
rlim_ ^
J
f (r e u )e 'nidt
—IT =
lim r - l-
f 2 l t l J |A |=r
f (X )X n~ ldX = 0
332
3: Commutative Algebras and Functional Calculus
3.2.13
for each n g N and / € j4(D). The last equality holds because the integrand is an analytic function. (Note that for any / 6 .4(D) the Taylor series expansion for / around zero converges pointwise to / on D°. However, it need not converge uni formly on D° nor even pointwise at each point of D. This is why the proof of the inclusion A (D ) C P (D ) depended on Cesaro summability.) However, the coordinate function z is a single generator for P (D ), so Theorem 3.1.15 shows that { 7 * : A 6 0 } is the Gelfand space of P (B ) and hence o f A (D ). The same result shows that A <—* 7 ^ and 7 >—> 7 (z ) are inverse homeomorphisms o f D onto T^(d) and of P A(Dj onto D, respectively. The Silov boundary o f the disc algebra is also easy to identify. The maximum modulus principle shows that each / € .4(0 ) must assume its maximum absolute value somewhere on T = cJD. Conversely for each A 6 T, the function 1 -f A *2 assumes its maximum absolute value at A and only at A. Hence, under the identification of with D, the Silov boundary is identified with the topological boundary T. This is, of course, a main reason for the name “boundary” . The last remark and the final result in Theorem 3.2.10 both show that the disc algebra is not completely regular. Hence, the hull-kemel topology on r x(D) is different from the Gelfand topology. In fact, it is clear that no proper subset of D can correspond to a hull if it contains a limit point in D° since no non-zero function in A (D ) can vanish on such a subset. Yood [1963b] noted the following examples of non-spectral algebra norms on .4(D). For any / 6 A (& ), define HI/HI =sup{|/(A)| : A 6 K ) where K is some closed subset of D which has nonempty interior and does not include T. This is a norm since an analytic function which vanishes on K must be zero throughout D. If A is in T \ K , then the function g = ( l + A*c)/2 satisfies the inequality |||g||| < 1 = p (g ) which shows that HI • HI is not a spectral norm. Furthermore, the maximal modular ideal M = ker( 7 >) = {/ € .4(0) : /(A ) = 0 } is not closed and is even dense. To see this, note that for any / 6 y4(D), f — gnf belongs to M but in the norm ||| • |||, lim,,^.*, / - gnf = / holds since |||3 n||| -♦ 0 . A larger algebra related to the disc algebra is H x = {/ : E° —1>C : / is holomorphic and bounded}.
Multiplication is pointwise and || • lloc is the norm, so ^4(0) is a closed subalgebra of H x . It is not hard to see that for each ( 6 T the radial limit } ( ( , ) = limr ^ i- f ( r O exists. Then / belongs to L ° ° (T ) and satisfies ii/ ll* = ll/ IU for all / e ‘ by the maximum modulus principle. The map / •—» / identifies H 00 with the subalgebra of L ° ° (T ) (with convolution
Silov Boundary, Hulls and Kernels
333
multiplication) consisting o f those functions / satisfying
2 tt
[
f ( e it)e ntdt = 0
Vn£N.
s that these are exactly the functions in L ° ° (T ) for which all the negative Fo-'.rier coefficients vanish. Any / G H ° ° can be reconstructed from the co-responding / by the Poisson formula given above. Since C (T ) is closed in 1°°(T), the polynomials are not dense in H °°. In fact, H ° ° is not even separable. Thus the Gelfand space o f H ° ° is much larger than 0 °. The portion of the Gelfand space corresponding to each point on the boundary o f '..he disc is enormous. 3.2.14 E x a m p le Many variations o f the last example have been studied. For instance, one could take any compact set fl in the complex plane, other thf n the closed unit disc D, and consider the algebras A (f l ) and P ( f l ) def.ned in a way analogous to A (D ) and P (D ). (N ote that the proof given above shows that the algebra P ( T ), introduced above, agrees with this notation.) Similarly, one may consider the commutative Banach algebra R (f l ) (under pointwise operations and the supremum norm) o f all uniform limits on fl o f rational functions with their poles off fi. Any compact fi C C, for which the interior o f every component is connected (in order to assure the middle inclusion), clearly satisfies P ( fl) C R (Q ) C A ( f l ) C C (f l ). We ha’ 3 just shown P (D ) = /1(D ) = .4(D) / C (T ) and the case fl — T satisfies P (T ) R ( T ) = j4(T) = C ( T ) where the equality o f R (T ) and C ( T ) follows agi,;.n from Fejer’s theorem. (N . B. The algebra i4(T) equals C (T ) for the tr:- ial reason that T has no interior. Th e notation here conflicts with the inefjiing assigned to /1(T) in Section 1.8 where it represents the functions wit l absolutely convergent Fourier series. Each notation is standard in its own specialized area.) Since i l ( f i ) is generated as a Banach algebra by the set o f functions { 2 } J { ( z —A ) - 1 : A $ Q }, each 7 in the Gelfand space o f i?(Q ) is determined by ohe number A7 = 7 ( 2 ). Since z — A7 is never invertible in R (Q ), each A7 belongs to fl. Hence, as in the argument given before for P (D ), we see that 7 h-t A7 is a homeomorphism o f T o n t o ft. A similar argument shows that the Silov boundary of R (f l ) is always mapped onto the topological boundary of fl under this homeomorphism. For any compact subset f l o f C, let p c (fl) represent the polynomially convex hull o f fl defined before Theorem 2.1.14. Thus p c (fl) is the union o f il with all the bounded components o f the complement of fl. (M ore graphically, p c (fl) is just fl with all its holes filled in.) The maximum modulus principle shows that restriction to fl maps P (p c (f l )) isometrica!! ■ isomorphically onto P ( fl). Runge’s approximation theorem (Theo rem 3.3.3) shows that (z — A) - 1 for A ^ p c (fl) belongs to P (p c(f2 )), so that
334
3: Commutative Algebras and Functional Calculus
3.2.15
P (p c (H )) equals A(pc(ft)). Mergelyan’s theorem (mentioned after Theo rem 3.3.3) shows that P (p c (fi)) actually equals A (p c (fi)). Hence, we have P ( f i ) ~ P (p c (fi)) = fl(p c (fi)) = i4 (p c (fl)). Note that we have proved the special case where fl and pcfi are T and D, respectively. Note also that Runge’s theorem shows that any function in C '(fi) which is the restriction of a function analytic in some neighborhood o f fi actually belongs to R (f l). We have seen examples (i.e., T and D, respectively) in which the in clusions P ( f i ) C R (f l ) and ,4 (fi) C C ( f i) are proper. Certain choices of fl called Swiss cheeses show that R (f l ) C >1(0) may be proper (cf. Theodore W. Gamelin [1969], p.24). Since A ( f i ) satisfies R ( fi) C .A(fi) C C (f l ) where the inclusions are isometries, and since the Gelfand spaces of both /2(S7) and C (f l ) can be identified with fl under the map 7 >—> = 7 ( 2 ), it is not surprising tliat this map is also a homeomorphism o f the Gelfand space o f A (Q ) onto fi for all compact fi in C. Furthermore, the Silov boundary of /I(SI) is always identified by this map with the topological boundary o f S! (t /. Gamelin [1969], p. 31). Before leaving these examples we also mention that if fi is a compact subset of C with connected complement. ;ind connected interior, then re striction to the boundary dfl of il maps P (f l ) isometrically isomorphically onto a maximal closed subalgebra of C (fi). (Errett Bishop [I960]). The classes o f examples discussed above can be extended in various ways to algebras of analytic functions o f several complex variables. The theory, of course, becomes far more complicated in this setting. For further results, we refer the reader to the extensive contemporary literature, starting with the following books: Andrew Browder [1969], Gamelin [1969], Stout [1971], and John Wermer [1976]. An interesting but unrelated example was given by Eva Kallin [1963]. It is a unital commutative Banach algebra A generated by four elements. There is a continuous function / on such that for each point 7 in there is a neighborhood U of 7 and an a e A such that / equals d on {/, but f & A - This contradicted a published theorem of Silov (cf. Arens [1961] for background). The algebra also contains elements a and b such that b vanishes in the neighborhood of each zero o f d, but b does not belong to the ideal generated by a. 3.2.15 Th e S ton e-C ech Com pactification o f a Topological Space Our construction is based on the abstract characterization of the algebras C 0 (f i) given in Theorem 3.2.12, which shows that a commutative Banach algebra is isometrically isomorphic to C 0 (fi) for some locally compact Haus dorff space fi if and only if it is a commutative C*-algebra. Note that fi is compact if and only if C o (fl) is a unital algebra, in which case C o (fi) equals the algebra C (f i) of all bounded continuous functions on fi. Hence, Theorem 3.2.12 also shows that a united commutative Banach algebra A is isometrically isomorphic to C ( f i) for some compact Hausdorff space fi if
3.2.15
Silov Boundary, Hulls and Kernels
335
only if it is a imital commutative C*-algebra. jet f l be any topological space and let C (ft ) be the unital commuta tive Banach algebra of all bounded continuous functions /: ft —►C under ~c':tw ise operations and the supremum norm. It is obvious that =
V / G c (ii);
u e n
defines an involution which makes C (ft ) into a unital commutative C*algebra. Hence, the Gelfand homomorphism ( “) is an isometric isomorphism o f C ( Q ) onto C ( r C (Q )). We will denote r ^ n ) by /?ft. For each uj € ft the map ju,: C (f l) —> C defined by 7«(/
) = /M
VfeC(Q)
clearly belongs to /3ft. Furthermore the map 7 (i.e., the map uj >—> 7 ^) is clearly a continuous map of ft into /3ft. In fact, the range jn of this map is dense in /3ft. To see this, note that a function / € which vanishes on 7 jj is zero since it has the form g for some g € C (ft) which must be identically zero. Assume next that ft is a completely regular Hausdorff topological space (i.e ., a Tychonoff space). Then we claim that the map 7 is a homeomorphism of ft onto its image 7n C /3ft in its relative topology. Since the functions / € C (ft) clearly separate the points of a completely regular Hausdorff space, 7 is injective. The definition of complete regularity shows that for each ljq € ft, a neighborhood base at ljq is given by { { ^ G f t : I / M l > 0} : / € C (ft); / M
= 1).
Since 7 transforms this into a neighborhood base for 7 Wo in 7 n, 7 is a homeomorphism as claimed. Hence, we have shown that any completely regular Hausdorff space ft can be homeomorphically embedded as a dense subset of a compact Hausdorff space /?ft so that each bounded continuous function on ft extends to a continuous function on /9ft with the same bound. In the next paragraph we will show that the embedding into /3ft satisfying tkefce conditions is unique up to an obvious notion of equivalence which we will define there. The essentially unique space /3ft (or more precisely the embedding of ft into /?ft) is called the Stone Cech compactification of ft. 1 Cl is any topological space, then a compactification of ft is a pair (ip, \P) consisting of a compact topological space 'I' and a continuous map ip of fi onto a dense set in 9 . Note that we are not requiring that ip be injective or a homeomorphism onto its image with the relativized topology o f ^ as some authors do. Tw o compactifications (ip, \t) and (ip1, \t') are called equivalent if there is a homeomorphism 6: $ —►\P' satisfying 6 o ip = ip'. To see that the Stone -Cech compactification is unique up to this notion of equivalence, suppose ip: ft —►$ satisfies the conditions for the Stone^ech compactification. For each / e C (ft), let / € C ('I') be its extension
336
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3: Commutative Algebras and Functional Calculus
(That is, / satisfies / o ip = /.)
3.2.16
Then 9: 4* —» Tc(n ), defined by
0( T) ( f ) = f ( r ) f° r all t € 4', defines an equivalence between
and
T .fi —> rC(U) = /?fl. These results, together with the remarks preceding Theorem 3.2.10, show that a topological space fl has a compactification (V, ♦ ) in which ip is a homeomorphism if and only if Q is completely regular. We remind the reader of the algebraic analogue o f the Stone-tech com pactification given in §1.5.1. There we showed that, for any locally compact Hausdorff space, there was a natural identification o f C (f2) with the double centralizer algebra D (C 0( f i ) ) o f C'o(fl). If (ip, ty) is a compactification of fl, then A = {/ o ip : f g C ( ^ ) } is a closed, self-adjoint (i.e., A is closed under complex conjugation of values), unital subalgebra of C (f l ). Conversely, any such subalgebra A C C (f l ) is a unital commutative C*-algebra under complex conjugation as involution, so that the Gelfand transform is an isometric isomorphism o f A onto C ( r A). Clearly, the map -y. fl —* T A defined by t A f ) = /(<*>)
v / e
.4;
uj e f l
is a continuous map. The same argument as before shows that j q is dense in Hence, we have established a bijection between the set of equiv alence classes of compactification of fl and the closed self-adjoint unital subalgebras A o f C (Q ). Note that the algebra A corresponding to (ip ,4*) is the set of functions / £ C (f l ) which can be extended to 'I' in the sense that there is a function t 6 C ('I') satisfying / = g oip. Thus, C (f l ) itself is the algebra associated with the Stoiif Cech compactification. Similarly, if fl tt> locally compact and (ip, 4*) is the one-point compactification of fl, then the associated algebra is C o (fi ) 1 C C ( f 2). We will give another even more interesting example of a compactification in the next example. 3.2.16 T h e B o h r C o m p a c tific a tio n This is a compactification o f a locally compact group which takes into account both the topological and the group structure. Let G be a locally compact group. Recall from Section 1.10 that if /: G —►C is any function, then its left and right translates are defined by uf ( v ) = f ( u ~ lv )
and
f u(v ) = f (v u ~ l )
V v 6 G,
respectively. A function / € C ( G ) is said to be ( left / right / two-sided ) almost periodic if the set
( { „ / : u € G } / {/ „ : u 6 G } / {„ / „ : u, w 6 G } ) has compact closure in C (G ). I f / and g are left almost periodic, then (u (/ + s ) : u € G }~ is included in the (continuous) image under summation
3.2.16
Silov Boundary, Hulls and Kernels
337
of the compact set { u/ : u e G }~ x { „ / : u E G } _ and hence is compact. This shows that the set of left almost periodic functions on G is closed under addition. Similar arguments show that it is a subalgebra of C ( G ) which is even closed under complex conjugation. It is also easy to see that •f {/.t}n€N is a sequence o f left almost periodic functions which converge to / in C (G ), then { uf : u e G } is totally bounded so that / is almost perk die. Thus, the set of left almost periodic functions on G is a closed self-< djoint unital subalgebra of C (G ). It is obvious that similar arguments coulc be applied to the sets of right or two-sided almost periodic functions. However, it turns out that there is an elementary proof showing that left (or right) almost periodic functions are necessarily two-sided almost periodic. (For a proof see Lynn H. Loomis [1953], p. 167, where the present approach to th j Bohr compactification was first given.) Hence, we may speak o f the unita'. commutative C*-algebra of all almost periodic functions and denote this get by A P (G ). C ar previous remarks show that A P ( G ) C C ( G ) determines a compactificat on 7 :G —►bG o f G. Thus 7 is a continuous map of G into bG. Even thouf h we are assuming that G is locally compact, and therefore completely regular, 7 need not be injective nor a homeomorphism. It is not clear a priori that bG has any more structure than that o f a compact Hausdorff space. In fact, it turns out that the preimage under 7 : G —» bG o f the point 7 (e) is a closed normal subgroup o f G which we naturally denote by ker(7 ) and the preimage of each point in the dense set 7 cv is a coset o f ker(7 ). Hence, 7 a carries the structure o f a group. The group operations on 7 a are uniformly continuous in the natural uniformity of bG and hence can be extended to all o f bG. This makes bG into a compr ct topological group. (P roof of these facts may be found in the same reference.) Hence, for any locally compact group G, we have constructed a homomorphism 7: G —►bG onto a dense subgroup of a compact group bG such that each almost periodic function f on G can be extended to be a continuous function / on bG (i.e., / satisfies / 0 7 = /). The extension proc* : is an isometric isomorphism o f A P ( G ) onto C (b G ). The same argurr :nt which showed the uniqueness o f the Stone-Cech compactification up to equivalence shows that 7 : G —* bG is determined up to equivalence by this e; tension property. The present compactification is called the Bohrcornpa :tification since Harald Bohr [1932] first studied it in the special case in which the group G is the group R o f real numbers. T h -3 Haar measure on the compact group bG defines a linear functional on C (v G ) which can be pulled back to A P ( G ) through the isometric isortiorph c extension map / 1—> / of A P ( G ) onto C (b G ). This gives a linear functional M : A P ( G ) —» C satisfying M ( 1)
=
1
\M( f ) \
<
ll/ IL
V / € A P (G )
338
3: Commutative Algebras and Functional Calculus M (n
M {h ) M ( uf v ) )
=
M (/ r
v / e A P {G )
> 0 =
3.2.17
V h € A P {G )\ h > 0 V / € A P ( G ) ; u ,t; e G
M (f )
which is called the t>on Neumann mean. John von Neumann [1934] proved the existence of this mean by an elementary argument showing that if the set of left translates o f a function /: G —►C (not necessarily continuous) has compact closure in the uniform norm, then its closed convex hull includes exactly one constant function, namely M ( f ) . For Bohr’s classical case G = R, the von Neumann mean is given by T
V f € A P (R ).
In this case, there are also two important characterizations of j4 P (R ) in addition to the definition which we have given (which is due to Bochner [1927]). A function / 6 C (R ) is almost periodic if and only if for each e > 0 there is a T > 0 such that in each interval of length T there is an s satisfying ||/ - fsW^ < £■ (This explains the name “almost periodic function” .) Bohr showed that the functions satisfying the above criterion were precisely the uniform limits of functions of the form t >—> - i for c>j 6 C, Sj 6 R. Since these functions obviously separate the points of R, the Bohr compactification "): R - » />R is an injection. 3.2.17 S om e Classes o f G ro u p s The continuous group homomorphism -):G —►bG of the Bohr compactification of an arbitrary locally compact group G is not necessarily an injection. Our description of its construction shows that it is an injection if and only if the functions in A P ( G ) separate the points of G. A locally compact group for which this holds is said to be maximally almost periodic and the collection of all such groups is denoted by [M AP]. Let U . G —►6 (H ) be a continuous group homomorphism of G onto a bounded subgroup of the group of invertible elements in B (H ) where H is a finite-dimensional normed linear space. Such a homomorphism is called a finite-dimensional, continuous, bounded representation o f G . (A n impor tant special case arises when H is an inner product space and each represent ing operator U u for u g G is unitary. Then U is called a finite-dimensional, continuous, unitary representation.) By choosing a basis for 'H, we can write each representing operator Uu(u g G ) as a matrix (U u, i j ) nxn where n is the dimension of H . The multiplicativity of U shows that the trans lates o f each matrix entry function u U u,ij are linear combinations o f the other such functions with a universal bound for the coefficients. Hence, the set o f all such translates is a boundtxl ulmet of the finite-dimensional space
3.2.17
Silov Boundary, Hulls and Kernels
339
spar.ned by {u >—* Uu
(c) There is some injective continuous group homomorphism of G into a compact group.
(d ) There are enough continuous finite-dimensional bounded represen tations to separate the points of G. ( 3) There are enough continuous finite-dimensional unitary representa tions to separate the points of G. P r o o f -Vom previous remarks, the implications (e)=>(d) ^ (a J ^ U O ^ M c) are obvious. If K is any compact group, then the set U of continuous finite-dimensional unitary representations o f K separates the points of K as we will show later by several different arguments. Hence if (c) holds and 7 -+ K is a continuous homomorphism, then the set { U ° y : U £ U ] shows that (e) holds. □
In the above theorem, the implication (d)=>(c) is also obvious directly. (If (d ) holds and U is a point separating set o f continuous finite-dimensional bounded representations o f G, then K — f ] fj eu U g is a compact group, and the map which sends an element of G to its representing operator in each representation of U is an injective continuous group homomorphism of G into K . ) Also, the implication (d)=>(e) follows directly from the simple argument outlined in the next paragraph. Let G be a subgroup of the multiplicative group of C (7 i), where 7i is some finite-dimensional linear space. Choose a basis v l f v 2 , ... ,v „ for H and define an inner product on H by n n n ( ^ 2 XjVj , ^ 2 njVj)o = j=1 j-i j= 1 Assume that G is bounded with respect to the corresponding norm. Since any two norms on H are equivalent, and hence give rise to equivalent norms on C (7 i) = the particular choice of the basis is unimportant. Then G is 9, compact topological group in the relative topology o f B (H ). For any two vectors x, y £ H , define a second inner product by (x , y )=
f
( U x , U y )0dU ,
JU €G
where / ■• •dU represents the Haar integral on G. Then, it is clear that each ~J 6 G is a unitary operator with respect to this new inner product.
340
3: Com m utative A lgebras and Functional Calculus
3.2.18
Thus for any bounded group G in C (H ) (where H is finite-dimensional and G is bounded with respect to any norm on 'H ), there is an inner product on Ti relative to which G is a unitary group. Note that the last portion of this argument shows that any compact group in B (H ) where (H , (•, -)o) is an infinite-dimensional Hilbert space is a unitary group relative to an equivalent inner product. If G was originally the image of another locally compact group under a homomorphism continuous with respect to any norm on 7i, then the homomorphism is still continuous with respect to the new inner product. 3.2.18 G elfa n d T h e o r y and S ilov B ou n d aries fo r P r o je c t iv e T e n sor P ro d u c ts o f C o m m u ta tiv e Banach A lg e b ra s The following re sults are essentially due to Jun Tomiyama [i960]. See also Bernard R. Gelbaum [1962] and Arnold Lebow [1968]. More general results are ob tained by considerably nore complicated arguments in Gelbaum [1959], [1961], [1970] and Kjeld B. Laursen [1970]. Some o f these will be given in Example 7.1.19. T h e o re m .
Let A and B be commutative Banach algebras. Then A&)B is a commutative Banach algebra and its Gelfand space is naturally homeo morphic to r A x r B.
Proof Proposition 1.10.12 shows that AchB is a commutative Banach alge bra. For 7 e and 6 € I'b, the continuous bilinear functional
(a, b) *-* 7 (a )6(b ) obviously induces a well defined function 7 ®<5: A ® B —> C, which is given by 7 & < H E r = iak ® M = Y . k L i 'r M H b k ) when , ||a*|| ||bfc|| < oo holds. This provides a continuous map o f x T s into r ^ H. (This map is just the tensor product of the functions 7 and 6 composed with the multiplication
map A/: C ® C —» C.) Next we wish to define a map from F into x Tb which will be the inverse o f the above map. For each 6 S r ^ B, choose an element c ® d with 0(c ® d) / 0. Replacing this element by a suitable multiple allows us to assume 6 (c ® d) = 1 . Now define ( 70 : .4 —» C / 6g :B —►C ) by ( 70 (a) = (ac ® d) / 6g(b) = (c ® bd) ) for any { a € A / b € B ) . These maps satisfy 70
6g(a
b) = B(ac 0 d )8(c < 8>bd) = 9(ac2 ® bd2) = 9(a g> b),
~ie(aa') = 0(aa'c 0 d )6 (c 0 d) = 0(ac to d )6 (u c ® d) = •yg(a)/yg(a'),
and similarly bg(bb') = 6 g (b )6 g (b ') for all a, a' € A and b, b' g B . Both maps are non-zero since they satisfy 1 -- 0(c
3.3.0
Functional Calculus
341
and u Fb- Hence, we have inverse bijections of x Fs onto In fact, it is obvious that 0 >-> ( 7 g ,6g) is also continuous so these bijections are horaeomorphisms. □ P r o p o s itio n
Let A and B be commutative Banach algebras. Then A<8>B is semisimple if and only if:
(a) Both A and B are semisimple, and (b ) The natural map 6 .A & B —» A ® B is injective.
(This mill hold if
either A or B has the approximation property.) Proof I f A&)B is semisimple, then its spectral radius is a norm. By the last proposition, this means that t t-> sup{|7
V t S A®B
is a norm on A ® B . This clearly implies that p ^ (a ) = sup{|7 (a ) | :7 € T ^ }, = sup{|w <8 w'(<)|: w € A\'.u)' € BJ} are all norms. Hence, A and B are semisimple and 6 is injective. Suppose, conversely, that conditions (a) and (b) hold. Then any t = 1 c„ ® bit which satisfies lla*ll IIM l < 00 and is in the Gelfand radical o f (A ® B ) also satisfies 7 ® S (t) = l(^ k )S (b k ) = 0. Since B is semisimple, this implies that l ( ak)bk is zero for each 7 g Hence for eac.1 7 6 T / and ui' 6 B * , 7($3*^=i w '( 6fc)a*) = 7 ® u>'(t) is zero. Since A is semisimple, this implies that a/ ® u i'(t) = <^C^2kLiu> (bk)ak) is zero. Condition (b ) now shows that t is zero as we wished. □
pB(b ) = sup{j^(fc)|:<5 € r B} and
Not a that this last proposition contradicts Corollary 1 o f Lebow [1968]. G elbai'u [1962], p. 533 notes that the above function 6 >-> ( 7g, 6$) sends the Silov boundary o f A ® B onto the products o f the Silov boundaries of A and B. Tomiyama [1960], Theorem 5 and Gelbaum [1962], p. 530 show that A and B are completely regular if and only if the same is true of A ® B . Several other similar results are proved in one or the other of these papers (e.g., concerning anti-symmetry, analyticity, Wiener’s property.) Finally, in Tomiyema’s paper it is noted that if A and B are semisimple commutative Banach algebras, there is always a cross norm on A ® B such that the completion of A ® B relative to this cross norm is semisimple. (Just pull the norm o f A ® B / ( A ® B ) j (where ( A ® B ) j is the Gelfand radical of A ® B ) back to A ® B, where our previous results show that it is a norm.)
3.3
Functional Calculus
Introdv tion to Various Functional Calculi
A functional calculus is a way o f assigning a meaning to “function f (a ) of an a ement a” for some algebra of functions /: C —» C and some set of
342
3: Com m utative A lg ebras and Functional C alculus
3.3.0
elements a in some class o f algebras. We will use several functional calculi in this work. As the set of functions increases in size, the classes o f algebras and the sets of elements within these algebras to which the functions can be applied will naturally decrease in size. The assignment / i-» / (a ) will always be an algebra homomorphism. The constant function 1 1 (A) = 1
V A€ C
will be mapped onto the identity element in A 1, and the identity function z { A) = A
VAe C
will be mapped onto a. (W e will use z with this meaning throughout the remainder of this section.) These two requirements will determine the ho momorphism, perhaps under mild additional continuity assumptions. Fur thermore, f ( g ( a ) ) will be defined exactly when / o g (a ) is defined (where f o g is the composition o f / and g ) and we will have f { g (a )) = f o g (a ). For each given functional calculus we consider, the question o f whether f ( a ) is defined for a given element a (o f the proper type) will depend only on the spectrum of a. Furthermore, some form of the spectral mapping theorem which asserts S p (/ (a )) = {/ (A ): A G S p (a )} will hold. Finally, the various functional calculi will agree when more than one is defined for a given / and u. We will also consider two functional calculi in which functions are ap plied to ordered n-tuples of elements rather than to single elements. These will be taken up separately, since they are technically much more difficult. The two most elementary functional calculi, i.e., those involving polyno mials and rational functions, were already considered in conjunction with the spectral mapping theorem (Theorem 2.1.10). We briefly review the facts established there. They are defined for any element in any algebra. The functions involved in the first are polynomials. Let a be an element in the algebra A - Let p be a polynomial with complex coefficients, say p = Y l % o n} z}
P(^) =
^or ^
^ e l^')' Then define p(u) by
p(a ) = £ ; =o « jflJ G A 1 where a0 = 1. It is clear that the map p >-> p(a ) is
a homomorphism which also respects composition. Obviously it is the only algebra homomorphism o f the algebra o f polynomials into A 1 which takes 1 onto 1 and 2 onto a. We can easily extend the polynomial functional calculus to a functional calculus in which any rational function can be applied to an element a G A if it has no poles in Sp(a). (O f course, this is not a proper extension if S p (a ) is the whole plane. No element in a spectral algebra has this problem.) Let a S A be given and let p and q be polynomials such that, q has no roots in Sp(a). The spectral mapping theorem shows that zero does not belong to
Functional Calculus
343
the spectrum o f q (a ) in A 1. Hence, q (a ) is invertible in A 1 and its inverse ccmrr” tes with p (a ). Thus, for any a € A and any rational function r = p/q whei”j " and q are polynomials and q has no roots in S p (a ), we can define r (a ) by r (o ) = p (a )q (a ) ~ 1 6 A 1. It is immediately clear that the map r h-» r(a ) does not depend on the particular choice o f p and q (satisfying r = p /q and q has no roots in S p(a ) ) and is a homomorphism. The spectral mapping theorem S p (r (a )) = r (S p (a ))
was proved as Theorem 2.1.10. If a is another rational function, s (r (a )) is defined if and only if the rational function s o r has its poles off Sp(a). It is easy to check that s o r (a ) = s (r (a )) by factoring a and checking the equation for each of the factors A — z and (A' - z ) - 1 which occur. It is also sasy to check that r *-» r (o ) is the only homomorphism o f this set of rational functions into A 1 which takes 1 onto 1 and z onto a. Thus we have another well-behaved functional calculus. Let A be a Banach algebra, and let a be an element in A . Let / — J2 ^ o a nZn be a power .series with complex coefficients having radius of convergence strictly greater than p(a). Then, a " a" converges in norm unconditionally for any extension o f the norm of A to A 1. The set of functions with power series with radius o f convergence strictly greater than p { a) is an algebra and the map / t-> f ( a ) is a homomorphism, since the series converges unconditionally and thus can be multiplied. It is also obvious that this functional calculus extends the polynomial calculus. (We used a special case o f this idea to define the exponential function in The orem 2.1.12.) Rather than discuss uniqueness, composition o f functions or the spectral mapping theorem in the present setting, we go on to a more inclusive functional calculus for analytic functions. 1 'hr 'Analytic Functioned Calculus
We turn to the construction of the most important functional calcu lus for general Banach algebras: that based on analytic functions. This functional calculus was first used by Frigyes Riesz in [1911] for spectral projections of eigenvalues of compact operators but was developed more fully by Angus E. Taylor [1938b], Gelfand [1941a], Edgar R. Lorch [1942] and especially Nelson Dunford [1943], who gave the general spectral map ping theorem. Norbert Wiener’s [1923] use of Banach space valued analytic functions and Marshall Harvey Stone’s book [1932] provided a stimulus to some of this work. Similar ideas were also used in a more concrete setting by Arne Beurling [1938]. The particular construction we will use derives from the first edition (1971) of Wermer [1976] but the idea is probably due
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3.3.1
to Nicolaas Govert de Bruijn [1958]. It was further refined by Bonsall and Duncan [1973b], Walter Rudin [1973] and Robert B. Burckel [1994]. This functional calculus applies to any element a in any Banach algebra A , and the functions for which f ( a ) is defined are those which are analytic in some neighborhood of the spectrum of a. Actually, the set of all functions analytic in some (small, variable) neighborhood of S p (a ) is not an algebra, so that we must turn to the algebra of germs o f functions analytic in some neighborhood o f Sp(a). (The problem arises because we must define the domain o f a sum to be the intersection of the domains of the individual functions, so that (/ + g ) — g will not usually be the same function as /. For the power series with positive radius of convergence considered above, it is not necessary to intersect the domain o f a sum since we may always choose the largest open disc on which the series converges.) We postpone consideration of this complication until our basic construction is complete. The construction depends on certain contour integrals of functions with values in a Banach algebra. We introduce our terminology and notation before explaining the details of the construction. 3.3.1
Definition
Let A be a Banach algebra and let a be an element
o f A ■ Let T a be the collection of all functions / each of which is defined
(uid analytic on some (possibly disconnected) open set Df which includes S p (a ). We will say a contour T in the complex plane is suitable for the pair
(a, /) if T consists o f a finite set {Ti, r 2, . . . , Tp} of disjoint, rectifiable, piecewise smooth, simple closed curves in D f \ S p (a ) such that the winding number of T satisfies 1 f 1 .> _ / 1 2ni Jr (A - Ao) \ 0
if A0 e S p(a) if Aq e C \ D } .
If T is suitable for (a, /), we define an element /£(/) of A 1 by ^ (/ ) —
J f (X )(X — a ) " 1 dX.
( 1)
First we note a way to construct a suitable contour 1’ for a pair /). Let e > 0 be dist( 6 'p (a ),C \ D f ). Choose R > p(a). Cover the compact set R C i \ D f by a finite number of open discs D \, D 2, . . . , D q of radius e/2 centered at points in the set. If the boundaries of some of these discs are tangent, their centers may be moved slightly to remove this problem. Let V be the neighborhood of S p (a ) given by (a ,
i
V = ftCi \ U D k. fc= i Let T be the boundary o f V oriented so that any arc of d R C i in T is counterclockwise and any arc o f dDk is clockwise for k = 1 ,2 ,..., q. It is easy to check that this contour is suitable.
3.3,2
Functional Calculus
The contour integral /„ (/ ) an ordinary Rieinann Stieltjes the norm topology. Explicitly, curves the union of which Stieltjes sums o f the form
345
= ( 2n i) ~ 1 Jr / (A)(A - a ) - 1 dX is defined as integral in the unitization A 1 converging in for each o f the finitely many Jordan closed is T, the integral is the limit of Riemann-
2 ^ E / ( A f c ) ( A fc- a) - 1 ( 7 f c - 7 * - i ) fc=i where the Ajt’s and 7 ^’s belong to T*, 70 = 7 n, and one positive circuit around T ( starting at 70 visits the Ajt’s and 7 k’s in the order 70 i A i, 7 1 , A2 , 7 2 , • • •, 7 n - l, An, 7n*
The Knit is taken as max{e(( 7 fc-i, 7 *): k — 1,2, . . . , n } approaches zero, where i ( 7 * 1 , 7 *) is the diHtaiice along in the positive direction from 7 * _ i to 7 fc. Because the integrand is continuous, the integral exists by exactly the sane proof given in any careful discussion of the Riemann integral for numerical-valued functions. Moreover, the definition shows that any continuous linear functional u) € (-41)* can be moved through the integral sig n to
give:
w(/“(/)) = h
I /(AM(A" a)_1)
{2)
Using the analyticity o f the resolvent function A 1—►(A —a ) - 1 (Corollary 2.2.12) on the resolvent set C \ S p (a ) o f a, we could easily use classical complex analysis, equation (2) and the Hahn-Banach theorem to show that / „(/ ) is actually independent o f the choice of suitable contour T, but we proceed differently. 3.3.2 Lem m a Let a be an element of a Banach algebra A , f a function in T a and T a suitable contour for (a, f ). (a ) If f is identically equal to 1, then (1) equals 1 S A 1. (1>) If f = r is a rational function with its poles off S p (a ), then l l ( r ) = r (a ).
(c)
If { fn }n€N 13 a sequence of analytic functions on D j which converges
pointwise to f and uniformly on compact subsets of D f , then
/ar ( / ) =
lim /ar (/ „).
n —*00
(3)
(ci) If 7 € then i ( I a ( f ) ) — / ( 7 (a ))- Hence, i f T A 1 is nonempty, the Gelfand transform satisfies l a (f ) = f ° a .
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Proof (a): Choose R strictly larger than max{|A| : A 6 T } and let T 0 be the circle of radius R centered at 0 and oriented counterclockwise. Let f be the contour consisting of To together with T with its orientation reversed. Corollary 2.2.12 shows that the resolvent ( A - a ) - 1 is analytic at every point where the winding number o f f is non-zero. Hence, for any u € (-41)* equation (2) shows that w(/£(/ )) = (27ri) 1 JVo f(X)uj((X — a) 1)dX. Proposi
tion 2.2.9(f) shows that the resolvent can be expanded in a uniformly and absolutely convergent Laurent series on To, so we have
I n=0 X > ( a ")A' n~lrfA = 2h rj
w(/* (1)) = i
1o
-
“ (v-
1o
Since u> € M 1)* was arbitrary, (a) is established by the Hahn Banach theorem. (b): Let p and q be polynomials satisfying r = p /q and with q having no roots in Sp(a). By expanding the left side, we see that there are polynomials 9o ,9 i , - - . , g n satisfying n
P(A)fl(C) - P(C)ff(A) = (A - 0 £
9kWCk-
Hence, we may write n
(r(A ) — r (a ))(A — a) 1 = ^
hk(X )a k
k=0
where hk = gk/q and ak = ak/q (a ). Since each complex-valued function hk is analytic inside I\ its integral is zero. Hence, conclusion (a) gives
(c): The point set of the contour T is a compart subset of Df. Hence * { / « } neiv is any sequence of functions satisfying the hypotheses, then it
converges uniformly to / on T. The same proof that works for numerical valued functions shows that the limit of a uniformly convergent sequence of functions with values in A 1 can be interchanged with a Rieinann integral. This gives the desired result. (d): We have already noted that a continuous linear functional like 7 can be moved through the integral sign, so we see
l U a ( f ) ) = 2 ~ i l r } { X )(X ~
The remark about Gelfand transforms is an immediate consequence.
□
3 .3.3
Functional Calculus
347
Runge Approximation Theorem
Lemma 3.3.2 contains enough information about the map /£: —* A 1 so that we may use it to prove the classical Runge approximation theorem [1885]. That theorem will be central to the next step o f our construction, whic’’ will make the rest o f the properties o f our functional calculus, in cluding its uniqueness, quite easy. 3.3.3 H u n g e’ s A p p r o x im a tio n T h e o r e m Let K be a compact subset of C. Then any function analytic in a neighborhood of K can be uniformly approximated on K by rational functions with poles off K . In fact, if A is any subset of the complement of K which includes at least one point from each bounded component of that complement, then only rational functions with poles in A are needed. Hence, if the complement of K is connected, then any function analytic in tie neighborhood of K can be uniformly approximated on K by polyno mials. Prccf Let R ( K ) be the closed subalgebra of C ( K ) generated by the rational functions with poles in A. The abstract Runge theorem (Theorem 2.2.13) asserts that every element of R ( K ) has the same spectrum in R ( K ) as it does in C ( K ) . The identity function z belongs to R ( K ) and its spectrum is exactly K since (A - z )~ * belongs to C ( K ) (and hence to R ( K ) ) exactly when A £ K . If / is analytic in a neighborhood of K , then / belong to T z. CVtoose a contour F suitable for (a, /). Then, l \ ( f ) belongs to R ( K )
and satisfies 1\'(/ ) = / o z by (d). However, the Gelfand homomorphism is injective in this case (since for any A S K we can define a homomorphism 7a e F R ( K ) by 7 a(/ ) = /(A ) for all / 6 R ( K ) ) . Hence / = I $ ( f ) belongs to R (K ). This proves the first statement of the theorem. If the complement of K is connected, it has no bounded components so that. A may be chosen empty. This proves the final statement. □ The final statement o f Runge’s theorem has two extremely important extensions. One is K. Oka’s theorem [1936] which we state as Theorem 3.5.3 below. The other is S. N. Mergelyan’s theorem [1951] which we will not prove. Mergelyan’s theorem asserts that a function / is the uniform limit of polynomials on a compact subset K of C which has connected complement if / is continuous on K and analytic on the interior o f K . Many proofs of this'theorem have been given: cf. Rudin [1974], Stout, [1971]. ’ Range’s theorem greatly increases the importance of equation (3) in Lemma 3.3.2(c). 3.3.-;'.
C o ro lla ry
Let a be an element of a Banach algebra A and let f
he a. function in T a-
Then, it is possible to find a suitable contour T for
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3.3.4
(a, f ) and a sequence { r n} n€N of rational functions satisfying
ll(J) -
l»n r n (a).
n —*oo
(4 )
Moreover, any sequence of rational functions, defined and converging to f uniformly on each compact subset of some neighborhood of S p (a ), satisfies this equation. Proof Equation (4) follows from result (a) and equation (3) o f (c) in the lemma if the sequence satisfies the last sentence in the corollary. We use Runge’s theorem to establish the existence of such approximating se quences. First choose A with exactly one point in each bounded component of C \ Sp(a). Then choose an open neighborhood V of S p (a ) with compact closure V included in D ; \ A. Apply Runge’s theorem to / and V . □
We will use equation (4) to define the functional calculus. The equal ity of the two expressions shows that each is independent of the various choices involved. First, we introduce the algebra of functions (or rather, of equivalence classes of functions) on which the functional calculus is based. The Topological Algebra of Germs of Analytic Functions
Let A be a Banach algebra and let a be an element of A which will remain fixed throughout this discussion. We will define the algebra T a of germs o f functions analytic in a neighborhood of Sp(a). Let M a be the family o f all open subsets o f C which include Sp(a). Note that the elements of M a need not be connected. For each U 6 M a, let T u be the algebra (under pointwise operations) o f all analytic functions f : U —►C. We give T v the topology in which a net { f a }a€A converges to / if f a( A ) converges to / ( A ) for each A S U and the convergence is uniform on each compact subset of U ■ We have already used T a to represent the union of all the algebras T u for U € Afa and D j to represent the domain of / € T a. Our functional calculus is defined for each function in T a but, since T a is not an algebra, we wish to replace it with an algebra. In order to do this, we introduce an equivalence relation: / ~ g
f\u = g\u for some U € M a.
(5)
We will denote the set of equivalence classes by T a. For each U 6 A/a, define p u: T u —►T a as the natural map. For f , g & T a, f g and f + g are the usual pointwise product and sum defined on D f C) D g. It is easy to see that these operations are compatible with the equivalence relation. Hence, they give T a the structure of an algebra. We will not distinguish in notation between a function / € T a and its equivalence class in T aWe wish to give T a the strongest locally convex linear topology such that each map pir.Tii —* T a will be continuous. To do this we define a neigh borhood system at the origin consisting of all convex, balanced, absorbing
3.3.J
Functional Calculus
349
suostts N of J-a such that p'j} 1 ( N ) is open in T u for each U € M a- (In other words, we have defined T a to be the inductive limit (Definition 1.3.4) of { ' } y : U g M a ) under the maps p u v '- T u ? V defined by restriction to V for each pair U, V g N a with V C {/.) 3.3.5 Definition The topological algebra T a which we have just con structed is called the algebra of germs of functions analytic in a neighbor hood of Sp(a). The Homomorphism Ia
For each U € Afa, we will now define a continuous homomorphism It will be easy to see that if /, <7 g T a are equivalent,
If f ’• F j —* A 1 .
then ‘a ’ (/ ) = I a ’ (g )- Hence, the family { Iff : U e M a) defines a homo morphism I a from T a into A 1. Moreover, it is easy to see that l a:T a —» A 1 is continuous. Let U g M a be given. Then, a contour T which is suitable for (a, /) for one / S T u is suitable for all o f them, so we define If f : T\j —►A 1 to be la - Corollary 3.3.4 shows that this is independent of the choice o f T among suitable contours. Clearly, iff is linear and Lemma 3.3.2(c) shows that it is continuous. Equation (4) of Corollary 3.3.4 shows that it is multiplicative, since ihe functional calculus for rational functions is multiplicative. We have already noted that iff ( f ) is invariant under the equivalence relation used to construct _Fa, so for / some 'epresentative /.
g
T a we may define /<,(/) to be I a 1 i f ) for
3.3.6 Definition Let a be an element in a Banach algebra A and let / be a function in T a. In the analytic functional calculus, f ( a ) has the meaning I a( f ) 6 A 1 as defined above. W ) are now ready to state a comprehensive theorem. The statement of the theorem is slightly complicated by the fact that we do not require an identity element. In fact, the reader may wish to note how both the statement and the proof are simplified if A is unital. 3.3.7 Theorem Let A be a Banach algebra. For each a & A , the map / i—> f (a ) constructed above is the unique continuous homomorphism I a- a —* A 1 satisfying: ( a ) / „ ( l ) = l I a{z ) = a. Furthermore, for any a £ A and f g T a, these maps also satisfy: (b) S p (f (a )) = f (S p (a )). (c ) For any g 6 ^7 ( 0)1 9 0 f belongs to T a and satisfies g ° f ( a ) = g (f (a )).
(d ' If either 0 is not in S p (a ) or f ( 0) is zero, then f (a ) belongs to the smallest closed spectral subalgebra of A containing a.
Otherwise f (a )
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3.3.7
belongs to the smallest closed spectral subalgebra of A 1 containing a and 1 . I f a representative of f can be chosen in T a so that D j includes a compact neighborhood of S p (a ) with connected complement, then the word “spectral” can be removed from each of the last two statements.
(e) For any Banach algebra B and continuous homomorphism ip -.A -^ B , both T a C and <£(/(“ ) ) = hold when either (1 ) A , B and p are all unital; or (2 ) S p (a ) does not include zero; or (3 ) f ( 0) is zero.
( f ) I f C is a maximal commutative subalgebra of A and the element a belongs to C, then /(a) =
/ o a
(6)
holds, where ( ') denotes the Gelfand homomorphism on C unless A is not unital and also / ( 0 ) is not zero in which case ( ') denotes the Gelfand ho momorphism on C 1 (embedded in .41).
(g) I f f 18 a rational function, f ( a ) has the same meaning in the analytic functional calculus as it does in the rationed functional calculus. (h) If f has a power series expansion f =
with radius of
convergence strictly greater than p (a ), then f belongs to T a and satisfies OO
f ( a) = Y 1 a " a" n —0
(i) For each f
G
T a, there is a neighborhood V of a such that f belongs
to !Fb for each b G T> and f (b ) is a continuous function of b
G
P.
Proof Throughout this proof, a € A will be arbitrary but fixed. We have already noted that /„ is a well defined continuous homomorphism which agrees with the rational functional calculus if the function is rational. Hence, in particular, conditions (a) and (g) hold. Let l . T a —» A 1 be any other continuous homomorphism satisfying ( 1 ) and (2). Theorem 2.1.10 states that I must agree with the rational func tional calculus for rational functions. Corollary 3.3.4 now shows I — establishing uniqueness. (b): Consider / j G C \ f (S p (a )). Then, g — (fj. — f ) ~ l is analytic on the open neighborhood D f \ { \ : /(A ) = n }. Hence, g belongs to T a and g (a ) is an inverse for n - f (a ), so S p (f (a )) C f (S p (a )). Conversely, consider /z 6 Sp(a). For any A, there is an analytic function g on D j satisfying (n - A)^(A) = f(fj.) - /(A). Therefore (fi — a )g (a ) = f (l * ) — / (“ ) holds. However, this shows that if f (f i ) — f ( a ) had aii inverse then n — a would have one, contrary to assumption. This establishes the reverse inclusion.
Functional Calculus
3.3.'T
351
(c): By (b), we see that g o f belongs to T a if g belongs to F / ( a)- The function defined by 1 (g ) = g o f (a ) is a continuous homomor phism satisfying / (l) = 1 and I ( z ) = } ( a ) . Hence by the uniqueness o f this functional calculus (applied to / (a ) instead of a), we see f o g (a ) = } ( g ( a ) ) . (d ) and (e): Either side of equation (4) o f Corollary 3.3.4 shows that f (a ) belongs to the smallest closed spectral subalgebra of A 1 which contains a and i. The last sentence follows from the last sentence o f Runge’s theorem. We combine the proof of the first sentence with the similar case for (e). Let 3 be another Banach algebra and let ip:A —►B be a continuous hcnyjr’jorphism. Assume first that A , B and
/(a) = d b /
r
f w x~ldx - 2h rj
w
The first integral is zero by the Cauchy integral formula. The second be longs tc any closed spectral subalgebra containing a since all of its Riemann sums do. This proves the first sentence of (d). Furthermore, if
=
- ^ ^ / ( A ) A - 1 ( A - V ( a ) ) 9dA
=
2 ” ^ / ( A ) ( A - y > ( a ) ) - I dA = /^(o)(/ )
where we have used the analogue of (7) stated for
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3: Com m utative A lg ebras and Functional Calculus
3.3.7
we may work exclusively in C. Now (e) with hypothesis (1) applied to the Gelfand homomorphism of C give the desired conclusion. Next suppose A is not unital, so that zero belongs to S p(a). If / (0) = 0 holds, condition (d ) implies /(a) G C. Hence, we can again apply (e) (with either hypothesis depending on whether or not C is unital) to the Gelfand homomorphism o f C to obtain the desired conclusion. Finally if A is not unital and / ( 0 ) is not zero, then we can apply (e) with hypothesis ( 1 ) to the Gelfand homomorphism of C l to prove the last portion of (f). (h): (Recall that (g ) was established earlier.) The partial sums o f the series Qnz" converge to / = a n2" *n under the assumptions o f (h). Hence, the conclusion of (h) follows from applying (g) and the continuity of I a to the sequence of partial sums. (i): Choose a contour T which is suitable for (a, /). Theorem 2.2.13 shows that there is a neighborhood V of a such that each 6 € V has its spectrum in the open set { A € C: A £ T and the winding number of T around A is + 1 }. Hence, each b G V satisfies / G Use the chosen contour T to define h ( f ) = 2V, Jr f W ( X - b )~ 1d\ for each b G V . Proposition 2.1.4 shows that (A — 6 )~* is a jointly uniformly continuous function o f A G T and b € V . Hence, the map b ►-» h ( f ) is continuous by the usual proof from calculus applied to the integral above. □ We mention explicitly three consequences of condition (d) of Theorem 3.3.7. If a G A belongs to some proper closed ideal J o f A (so 0 belongs to S p (a )) and if / (0) = 0 holds, then f ( a ) belongs to 1. Also, f(a )
e {a }"
V a 6 A / G fa
(9)
holds (where { a } " is the double coinniutant of a G A ). Finally, if A is a closed subalgebra o f B ( X ) for some Banach space X and V is a closed subspace of X satisfying a X C X for some a € A , then f ( a ) X C X holds for all / 6 T a for which a representative in T a can be chosen so that D f includes a compact neighborhood of S p (a ) with connected complement. For a long time it had been hoped that for each a € A the homomor phism /„ was unique even without the assumption o f continuity. This was disproved by an interesting construction of Graham R. Allan [1972]. See 3.4.8 through 3.4.16 below for a more detailed discussion of these results. See also Florian Horia Vasilescu [1982], The paper [1989] by John B. Conway, Domingo A. Herrero and Bernard B. Morrel discusses some inter esting extensions. Mohammed Akkar [1978] defines a functional calculus in certain locally convex algebras. We have examined this functional calculus in considerable detail. This is motivated in part by the applications which we wish to make of it, but it is also motivated by our desire to compare this functional calculus with two others which we shall introduce in succeeding chapters. These are the
3.3.8
Functional Calculus
353
functional calculi of continuous functions of normal elements in C*-algebras and Borel functions of normal elements in W*-algebras. Functional Calculus for Spectral Algebras
We would like to extend the functional calculus to spectral algebras but we are only able to do so for Jacobson-semisimple spectral algebras. There are two related problems. Spectral algebras have no topology and we have just noted that there do exist discontinuous functional calculi in some ianach algebras which differ from the standard functional calculus. Thus we need to know that the functional calculus obtained by completing in different spectral semi-norms is always the same. The next proposition removes this difficulty in the semisimple case. In a non-semisimple spectral algebra, we would also face the problem that only spectral semi-norms might be available, so that the completion would actually be a quotient algebra and thus finding preimages in the algebra itself o f the function o f the image of one of its element would cause difficulty. 3.3.8 P r o p o s itio n Let A be a Jacobson-semisimple spectral algebra. Let a be an element of A and f a function in T a. Let o x and <72 be any two spectral norms on A . Let b and c be the element f (a ) defined by the functional calculus in the completions of A with respect to <j\ and 02, respectively. If both b and c are in A , then they are equal. P m of This is a slight variation on the uniqueness portion o f the proof of Theorem 3.3.7. Recall from Proposition 2.5.15 that the spectrum of a does not change in any completion with respect to a spectral semi-norm. Choose a contour T suitable for defining f ( a ) and let K be the compact set T U {/' G C : r has winding number +1 around A }. By Runge’s Theorem (Theorem 3.5.3), we can find a sequence { / „ } of rational functions with poles off K 3 S p(a ) which approximates / uniformly on K . Each f n(a ) is unambiguously defined in A and the sequence {/ n( a ) } converges to b in o\ and t< c in <72) so the Fundamental theorem (Theorem 2.3.6) shows 6 = c since „4 is semi-simple. □
This proposition shows that the following definition is unambiguous and does r.ot depend on the choice of spectral norm. 3.3.9 D e fin itio n Let A be a Jacobson-semisimple spectral algebra. Choose a spectral norm on A and let 3? be the completion. For any a in A and any / G T a if f (a ) € A /(a ) i.; defined in A .
actually belongs to A 1, then we say that
As already noted in the proof of Proposition 3.3.8, the spectrum of a does not change in the completion with respect to any spectral norm.
Hence, if f ( a ) is defined relative to such a completion, and that element
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3.3.10
is in A , then it agrees with f ( a ) defined relative to the completion with respect to any other spectral norm. Thus, the restricted functional cal culus in a spectral algebra A is a homomorphic image of whatever class of functions is involved and satisfies the rules on composition of functions o f any normal functional calculus. It also satisfies the spectral mapping theorem. O f course, we can use the above procedure to define a partial functional calculus in any spectral normed algebra without any restriction o f semisimplicity. 3.3.10 E x a m p le The proposition does not say anything about the exis tence o f / (a ). I f A is the disc algebra and B is the subalgebra o f rational functions with poles off the closed disc, it is clear that B is a spectral sub algebra of A and thus a spectral algebra. For the identity function z in B C A , the functional calculus image f ( z ) belongs to B if and only if / is rational with poles off the spectrum of z which is just the closed disc. Thus, the algebra o f functions involved in the functional calculus is just the algebra o f these rational functions and is therefore as small as it possibly could be. It would be interesting to know what other subalgebras o f T a can arise in different spectral algebras. We will now define and study the class of spectral normed algebras in which the functional calculus is defined for every / 6 T a. Functional Algebras and Subalgebras
3.3.11
D e fin itio n
A functional algebra is a spectral normed algebra
A such that for each a € A and each function / G T a the element f (a ) defined by the functional calculus relative to the completion A o f A is
already in .41. A subalgebra o f a Banach algebra or functional algebra is called a functional subalgebm if it contains / (a ) for each o f its elements a and each / € T a- A normed inductive limit of Banach algebras is called a pseudo-Banach algebra.
The first class of algebras introduced above may in time prove to be of fundamental importance, but we are not sure that the current definition is exactly correct. The class of local algebras, which seems to have been first formally developed by Bruce Black*ttlur in [1986], is our model. Similar ideas have been used chiefly in the K-theory of C*-algebras. The term “local Banach algebra” was intended to suggest that each element individually has a property (viz. a functional calculus) associated with completeness even though the algebra as a whole need not be complete. Since “local algebra” has an entirely different meaning stemming from algebraic geometry, we will not use it. Note that a functional subalgebra is necessarily a spectral subalgebra and that a functional subalgebra of a functional algebra is again a functional algebra. The term “pseudo-Banach algebra” was introduced by Allan, Dales and
Functional Calculus
355
J. Peter McClure [1971]. (The cited paper actually considered only the commutative case.) Their formal definition was in terms of bounded sets, but they showed that their pseudo-Banach algebras are exactly those al gebras which occur as normed inductive limits of Banach algebras. The normed inductive limit construction was described in Definition 1.3.4 and Exar.’ >le 1.3.5. For unital commutative Banach algebras, the reference al ready cited characterizes these algebras among several well known classes of topological algebras (e.g., locally multiplicatively convex, Frechet algebras, etc.) and shows that their Gelfand theory is well behaved. We now note the connection between these two classes below. This is a fundamental result from Blackadar [1986]. 3.3.12
T h eo rem
A normed, inductive limit of functional algebras is a
functional algebra. Hence, in particular, a pseudo-Banach algebra is a func tional algebra. Prccy We will use the notation from Definition 1.3.4. First we note that i'-is r :rm on A is spectral. If a G A satisfies ||a|| < 1, then there is some g and aa € Aa with a = (pa(aa) and ||aQ|| < 1. (Much more is true, of course, but this is all we need.) So aa is quasi-invertible since a Banach f.!geb", is spectral by Corollary 2.2.8. Since a is a homomorphic image, it is also quasi-invertible so the norm is spectral by the same corollary. Let a G and / € F a be given. Suppose a is such that a — <pa(t>) for some b £ .'£ = A a - Let D f be the domain o f /. Then we have S p ^ (a ) C S p s(b), so C = S p s(b ) \ D f may be empty but is certainly compact. If it is not empty, then for each A G C \ { 0 } we can find a f3 > a and a quasi-inverse for / ~ 1ip0a (b ) (since A-1 a has a quasi-inverse in A ). In fact, there is a neighborhood of A in which such quasi-inverses exist. If zero is in C , then A is ii.iital and there is a neighborhood o f zero and a f] so that
We wish to examine the inductive limit of matrix algebras of increasing size over a functional algebra. It is apparently unknown whether a matrix algebra over a functional algebra is again a functional algebra, but the fo!'ov/ing recent theorem of Larry B. Schweitzer [1992] covers a case which ir.icluc.’^s many applications. The applications are of the following type: Let A be the L l-algebra or C*-algebra of a Lie group G. It may be easier t.'i calculate the K-theory o f a dense functional subalgebra consisting o f f’ inc';:ons which are, for instance, C°° on G. The results in this example shov/ that the machinery all works smoothly.
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3.3.13
T h eo rem
3.3.13
Let A be a spectml normed algebra which is also a
Frechet algebra in a topology such that the embedding of A with its Frechet algebra topology into A with its norm is continuous. Let A be the completion of A in its norm. Then A is a functional subalgebra of A and hence a functional algebra. Furthermore, any matrix algebra M n( A ) over A is also a functional subalgebra of M „ ( A ) (when this matrix algebra is given any algebra norm which defines the topology pointwise in terms of its entries) and hence a functional algebra. Proof In Theorem 2.9.2 we have noted that the inverse is continuous in a Frechet algebra. Thus, the integrand / (A)(A — a )~ l in the integral defining f (a ) is continuous. Hence, its Riemann sums converge in the Frechet topol ogy o f A just as in the discussion of the functional calculus in a Banach algebra. Because A (with its Frechet topology) is continuously embedded in A (with its complete norm), the image of f (a ) in A is just f (a ) in A Therefore, A is a functional subalgebra of A . Theorem 2.2.14(h) shows that the extension o f the norm o f A to M n( A ) is still spectral. Hence the image of M n( A ) in M n( A ) is a spectral subalge bra by Proposition 2.5.15. Also, this image is still a continuously embedded Frechet algebra. Thus, it is a functional subalgebra by the same argument as before. □
Blackadar’s [1986] Definition 3.1.1 actually requires that M n( A ) be func tional (in our terminology) for all n G N before he calls it a local algebra. This was necessary for the applications to K-theory, but the theorem above is sufficient for most applications. In terms of Blackadar’s definition, Lothar M. Schmitt [1991] shows that A / T in local whenever A is local and J is a closed ideal. We do not know whether matrix algebras and quotients of functional algebras are always functional without any further restrictions, but this whole theory is in its infancy. See Schweitzer [1991], [1992] and [1993] for examples and discussion.
3.4
Examples and Applications of Functional Calculus
Our careful study of the various elementary functional calculi and the analytic functional calculus is justified partly by their important applica tions and partly by our desire to compare them with two additional func tional calculi which will be defined for certain elements in certain *-algebras. We introduce one of these at the end of this section. We will also explore the situations under which the analytic functional calculus is, or is not, unique when the continuity assumption is removed. However, most o f this section will be devoted to some of the important applications of the an alytic functional calculus. In the interest of giving elementary proofs, we
3.11
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357
will sometimes avoid using this functional calculus in situations to which it c'early applies. We will do this whenever elementary methods yield the sair e results with comparatively little extra effort. In addition to polynomi als and rational functions, the functions which occur most often in practice are the exponential, the logarithm, and the square root. Also, for elements with disconnected spectrum the characteristic function of a portion of the spectrum is frequently useful. Except for this last case, these applications can usually be treated an easily by more elementary methods. R em a rk . We often state theorems about the functional calculus just for Banach algebras, even though they hold for functional algebras also. The reader can easily check whether some small change in wording is needed. Existence of Idempotents
In Section 3.5 we will prove the Silov idempotent theorem (Corollary 3.5.15) which shows the existence o f idempotents in a commutative Banach algebra if its Gelfand space is disconnected. The proof o f that result de pends on the powerful theory developed in Section 3.5. However, we can now prove an easier analogue of this result which is a trivial consequence o f the analytic functional calculus we have already constructed. The next result is due to Gelfand [1941a]. 3.4.1
P r o p o s itio n Let A be a unital, Banach algebra, and let a be an elen-.ent of A . If S p (a ) is the union of a family { K j - . j = 1 ,2 ,... ,n } (n > 1) of disjoint, nonempty, closed sets, then there is a set {e j : j = 1 , 2 , . . . , n }
of o~thogonal non-zero idempotents satisfying e7 = 1 and S p (a e j) = K j U {0 } for j = 1, 2 ,. .., n. Moreover, each of these idempotents is con tain :d in the smallest unital closed spectral subalgebra B of A which contains a, s ; ej € { a } " .
Considering Gelfand transforms relative to the commu-
tathe Banach algebra B, we see that ej is the characteristic function of
Proof For each j = 1 ,2 ,..., n, let f j be a function in T a which is equal to 1 in a neighborhood o f K j and is equal to 0 in a neighborhood o f S p (a ) \ Kj. Define ej by e3 = f j (a ) for j = 1 ,2 ,..., n. Theorem 3.3.7 gives all o f the stated results. □
'.’he above proof is an example of using a disconnected neighborhood of the spectrum and a function which is analytic in the chosen neighborhood but is not globally analytic. It is interesting to write down the idempotents produced in the last proposition explicitly. Let T be a suitable contour in C \ S p (a ) which has winding number +1 around each point of K j and wincing number 0 around each other point in Sp(a). Then we have:
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3.4.2
The Kernel of the Exponential Function
Theorem 2 .1 .1 2 gave the basic results on the exponential function exp: A —* A in a unital Banach algebra A where
V a € A.
If A is commutative, that theorem shows that exp is a group homomorphism from the additive group A to the multiplicative group A g - Following Edgar R. Lorch [1942) and [1943), we now find the kernel of this homomorphism. 3.4.2
T h eo rem
Let A be a commutative unital Banach algebra.
The
kernel of the exponential map exp: A —* A g is the additive group generated by the set {2m e: e is an idempotent in A } . Proof Suppose e is an idempotent in A . Then
OO e x p ( 27ric) = 1 + ^ ( n ! )
1( 27r i)ne = 1 + (e x p ( 27ri) — l)e = 1.
Hence, the kernel of the group homomorphism exp: .4 —* A g contains the additive group generated by { 2jrie : e is an idempotent in A } - Con versely suppose a e A satisfies exp(a) = 1. Then Theorem 3.3.7(b) shows that Sp(a) is a finite subset of the set {27rj'n:n G Z }. Proposition 3.3.1 gives a finite set { e j . j = 1 , 2 , . . . , m } of orthogonal idempotents satisfy ing ej = 1 and Sp(ae} ) = { 2 mn j } for some integer n3. Define b by b= JL i 27rinj e3. Then a —bis zero so there is an element r in the Gelfand radical o f A satisfying a + r = b. Furthermore, the absolute convergence o f the series involved allow us to write 0 = exp( 6 ) —exp(a) = ]C^Lo(n-)~ 1( ( a + r ) " — a " ) = r ^ ^ l ^ n !)- 1 ]C k = i(fc)r * _Iarl~*- The Gelfand transform o f the infinite series is just ]C ^ L i(n-)_ ln “ " 1 = exp(a). Hence, the infinite series is invertible so r is zero. Thus a — b has the desired form. □ Logarithms
We have just determined the kernel of the exponential function on a commutative Banach algebra A - Theorem 2.1.12 shows that, in the com mutative case, the range of the exponential function is the connected com ponent of the identity in A c . However, it is harder to determine whether a given element is an exponential in the noncomutative case. We will consider not necessarily commutative Banach algebras. Clearly any element in the range of the exponential function is invertible. We are about to show that if 0 is in the unbounded component of C \ Sp{a) then a is the exponential of an element log(a). However, the next example shows that this sufficient
2.4.3
Exam ples and A pplications o f Functional Calculus
359
condition is not necessary even in the commutative case. A more subtle condition will be given in the next section. 3.4.3 E x a m p le Consider the commutative unital Banach algebra C '(T) of continuous functions on the 1-torus T. Let g: T —> C be defined by
Clearly, g is continuous so it belongs to our algebra. Just as clearly, exp(p) has the whole of T as its spectrum. £.<'.4
T h e o re m Let A be a unital Banach algebra and let a e A be an element which has 0 in the unbounded component of C \ S p (a ). Then in the smallest unital closed spectral subalgebra of A containing a, there is an element log(a) which satisfies
exp(log(a)) = a. Henpe for any n
G
N, the element a 1/" = exp(^ log(a )) satisfies (a 1/” ) ” = a.
Any element a G A which satisfies cor.c'i '.ion and we can write
||1
— a|| <
1
satisfies the previous
Proof Since the unbounded component which contains 0 is axe connected, ” ’3 "ar. construct a continuous curve connecting 0 to some point on the negative real axis beyond the spectral radius of a. Using this curve, we can
chooso a branch of the logarithm function which is analytic in a neighbor hood of S p (a ) and use it to define log(a). Theorem 3.3.7 then gives the results in the first paragraph. The alternative hypothesis ensures that the series converges, so Theorem 3.3.7(h) gives the final result. □ Note that if S p (a ) is disconnected, then log(a) and a 1/” depend on the choice of the contour in the proof of the theorem. Obviously, any invertible m atrx A G M n for any n € N satisfies the theorem and thus has a logarithm and a-bitrary roots. Square Roots
Let a be an element in a Banach algebra and let b be a square root for a in the sense that b2 — a holds. Then a belongs to any closed commutative subalgebra C containing b, and b is a square root for a in C. The implicit
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360
3.4.4
function theorem proved in the next section (Theorem 3.5.12) will give a strong converse. If a € A is invertible in a closed commutative subalgebra C which is a unital algebra, and if a. has a square root h in C (F c ), then a has a square root in C and hence in A . Furthermore, this square root is unique in C subject to the condition that c — h. This result is, of course, obtained by considering the function F (w , z ) = w2 — z. The main disadvantage of this result is that it only deals with elements which are invertible, at least in some closed commutative subalgebra. Assume now that A is a unital Banach algebra and a is an invertible element in A such that the whole negative real line lies in C \ S p (a ). Then, the usual principal branch o f the square root function belongs to T a so that the analytic functional calculus gives a square root a 1/2 in { a } " C A with spectrum in the open right half plane. We will now show that this is the only square root with spectrum in the open right half plane. If b is any other square root, a 1/<2 and b belong to a common closed commutative subalgebra C. If b has spectrum in the right half plane, then a 1/2 and b must have the same Gelfand transform in C since they satisfy (a 1/2)2 = ( 6 )2. Hence, there is some r in the Gelfand radical o f C satisfying b = a 1/2 + r. Then, 2a 1/2 + r is invertible in C since it has a nonvanishing Gelfand transform. Thus, 0 = b2 - (a I/2)2 = (2a 1/2 + r ) r implies r = 0 . The uniqueness of square roots with spectrum in the right half plane was first obtained explicitly by Einar Hille [1958]. It was rediscovered and given an attractive proof by L. Terrell Gardner [1966]. However, an immediate corollary o f this result, which plays a decisive role in the theory of Banach *-algebras was not known to most workers in that field until discovered by James W . M. Ford in his dissertation [1966], [1967]. We now derive an elementary version of the above result which does ap ply to certain noninvertible elements. Actually, we will deal with elements b satisfying b o b — a. Such an element is called a quasi-square root of a. This is more convenient, particularly in nonunital algebras. Obviously such elements satisfy (1 — b)2 = 1 — a in A 1. We will give a simple derivation of the binomial series expansion for (1 + A )1/2. First, assume the existence of a power series expansion with undetermined coefficients: OO
(1 4 - A ) ’ ^2 = y(A ) = ^ an\ n. n=0
Clearly,
oq
(1)
must be 1. The first equality implies 2(1 + A )y'(A ) = y (X ).
(2)
Substituting the series expansion gives oo
y~^[2 (n + l ) a n+ i + n=0
oo
2 n a n]Af> = ^ ^ a n An . n=0
3.4.5
Exam ples and Applications o f Functional C alculus
361
By oquating the coefficients, we get the recurrence relation an+i (1 — 2 n )( 2 n + 2 ) _ 1an, from which
=
Vn > 0 follows. The ratio test shows that the power series of (2) converges for |A| < 1. Thus, for |A| < 1 we conclude
(3) The ;dentity
is eas ily proved by induction. It shows that
(5) conw rges so that the power series o f (3) converges absolutely and uniformly for jAj < 1. Since it represents a continuous function there, the power series satisfies (3) for all |A| < 1, and in particular (5) converges to 1. 3.4.?
T h e o re m
Let A be a Banach or a functional algebra. Let a € A
satisfy either p (a ) < 1 or ||a||n < 1 for all sufficiently large n. series
Then the
(6) converges unconditionally to an element b € a A fl A a fl { a } " satisfying bob= a
and
p (b ) < 1 .
(7)
i f a satisfies p (a ) < 1 , then b is the only element in A satisfying (7) and, in fact, satisfies 1 — p ( 6 ) < \ /l — p (a ) < 1 . P to<\ If a satisfies p (a ) < 1 , Theorem 2.2.5(f) shows ||an|| < 1 for all sufficiently large n. Thus, under either hypothesis on a G A , the remarks preceding this theorem show that ||( l^2 )a n~ 1 1| converges so that ( 6 ) converges unconditionally to an element b € aAC\ A a n { a } " which satisfies bo b = a. Corollary 3.1.6 shows that each partial sum satisfies
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3.4.6
The last part o f Corollary 3.1.6 then shows p(b) < 1 — y /l — p (a ) since each partial sum commutes with b. The inequality p (a ) < 1 then implies p (b) < 1. When this holds, suppose c G A satisfies c o c = a and p(c) < 1 . Then ca — 2c2 — c3 = ac holds so c commutes with a, and hence with b also. Thus p ( ( b + c)/2) < ( l + p (b ))/ 2 < 1 implies that (6+ c)/ 2 has a quasi-inverse which we denote by d. Hence, we get b—c = 0 o ( 6 - c ) — d o (( 6+ c )/ 2 )o ( 6 - c ) = d o [(b + c + 2b—2 c - ( 62 - c 2 ))/ 2 ] = d o [(6 + c + b o b - c o c)/ 2 ] = d o ((6 + c)/ 2 ) = 0 . □ Continuity of the Spectrum
The following two results are due to John D. Newburgh [1951], Lemma 3.4.6 will be used both in Proposition 3.4.7 and in Proposition 6.1.10. Proposition 3.4.7 gives the same result as the last one in Corollary 3.1.6 with completely different hypotheses. It is also related to Newburgh’s theorem (Theorem 2.2.15). 3.4.6
Lem m a
Let a be an element in a Banach algebra A . Let V be a
bounded open subset of C satisfying V D S p (a ) = V D S p (a ) ^ 0. Then there is a 6 > 0 such that ||6 — a|| < 6 implies Sp(b) fl V ^ 0. Proof Let / be the characteristic function of V . By assumption, / is analytic in a neighborhood o f S p (a ). Theorem 3.3.7 shows that f (a ) is a non-zero idempotent. Hence, Theorem 3.3.7(i) shows that there is a neighborhood a, which we may take to be an open disc o f radius 6 , such that f (b ) is defined for each b in the neighborhood and satisfies ||/(6) - /(o)|| < ||/(a)||. This implies f (b ) ^ 0 and thus, by Theorem 3.3.7(b), Sp(b)C\V / 0 for each b satisfying ||6 — a|| < 6 . □
3.4.7
Proposition
Let A be a Banach algebra. Let a € A have totally
disconnected spectrum. For any e > 0 there is a S > 0 such that j|fe—a|| < 6 implies S p (a ) C S p (b) + e C ),
S p (b ) C S p (a ) + eC i
and
\p(a) - p(b )| < e.
Proof Since S p(a ) Ls compact and totally disconnected, there are a finite number of disjoint closed discs £>_, j — 1 , 2 , . . . , n each containing some point of S p{a), of radius less than e/2 and such that S p (a ) is included in the interior of D = (J"=i ■ Theorem 2.2.15 shows that we can find a 6o > 0 so that ||6 - a|| < So implies Sp{b) C D °. For each j = 1 ,2 ,..., n the lemma gives a 6} > 0 so that Split) n £>_, ^ 0. Define 6 to be the minimum of { 6 , : j = 0 ,1 ,... ,n }. Then any b satisfying ||6 — a|| < 6 has all the desired properties since each point of S p (a ) is in some D ° of diameter less than e, which also contains a point of Sp(b). □
Exam ples and Applications o f Functional Calculus
363
Discontinuous Functional Calculi
VVe will now explore the existence o f discontinuous functional calculi. Theorem 3.3.7 states that for any element a in a Banach algebra A there is a unique continuous homomorphism I a: T a —* A 1 satisfying /„(1) = 1
Ja(z ) = a.
( 8)
:7s :. 'sh to investigate conditions on the Banach algebra A which will ensure that for a e A , any (not necessarily continuous) homomorphism Ja: s i —* A 1 satisfying ( 8 ) equals /„. When this is true for a particular a € A , we say that a has a unique functional calculus and when it is true for all a e A , we say that A has a unique functional calculus. Since /„ is unique among continuous homomorphisms satisfying ( 8 ), a will have a unique functional calculus if and only if every homomorphism Ja: T a —> A 1 satisfying ( 8 ) is continuous. (Thus, this is a type of automatic continuity result similar to those discussed in Chapter 6 .) The main results here are due to Allan [1972], Dales [1973] and Thomas [1978]. First, we will show that not all algebras have a unique functional cal culus. Let T = C[[z]] be the algebra o f formal power series in z introduced in Example 2.9.7. Let A be a Banach algebra and let a be an element in A satisfying S p (a ) = {0 }. Then the algebra T a o f germs o f functions analytic in the neighborhood of S p (a ) — { 0 } can be identified with the subalgebra T {0) o f T consisting o f those series with a non-zero radius o f convergence. Hence if we can embed T into A 1 by two distinct unital homomorphisms ip and ip which agree at z, but disagree at some element o f J-'(o), then we can be sure that the functional calculus is not unique. Thus the search for non-unique functional calculi can begin by studying unital homomorphisms e; T i:ito a unital Banach algebra A . We quote, without proofs, the main theorems from Allan [1972]. Recall that the Jacobson radical and Gelfand radical coincide for commutative Banach algebras by Theorem 3.1.5. £’.4.3
T h e o re m
Let A be a unital Banach algebra and let ip:T —> A be
an injective unital homorphism. Then
3.4.8
T h e o re m Let A be a unital, commutative Banach algebra. Let there be an element b in the Gelfand radical of A such that the descend ing sequence {(.46n)~ } „ 6n of closed ideals eventually becomes constant at non-zero ideal denoted by I . Then I = 0 ^ 1 -4 6 " is dense in T. Fur thermore for any r £ J \ {0 } and any f € T which is transcendental over the polynomials, there are injective unital homorphisms p \ T —* A
and
ip: T - * A satisfying
Hence, any algebra (such as those given in Examples 3.4.15 and 3.4.16)
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3: Com m utative A lgebras and Functional C alculus
3.4.10
satisfying the conditions of Theorem 3.4.9 will fail to have a unique func tional calculus and therefore will have a discontinuous functional calculus in addition to its unique continuous functional calculus. We now give the results due to Marc P. Thomas [1978]. They require some preparation. See also Bade, Curtis and Laureen [1980]. 3.4.10 D e fin itio n Let A be a Banach algebra and let a be an element in A . A subspace D 1 is said to be a-diwsible if it satisfies (A - a ) V = V
V U C .
If X is a subspace of A , denote the largest a-divisible subspace of X by V (a , X ).
The definition means that P (a , X ) is an a-divisible subspace of X which includes all other such subspaces. Hence it is unique if it exists. Since { 0 } is a-divisible and the sum o f any family of a-divisible subspaces is a-divisible, it is obvious that T>(a, X ) always exists. We incorporate this observation in the next proposition. We are primarily interested in the case in which A is commutative and a X C X already holds. 3.4.11 P r o p o s itio n
Let A be a Banach algebra. Let a be an element in
A and let X be a subspace of A . Then T>(a, X ) exists and is unique. It is the largest subspace V of X satisfying (A — a)T>' — V for all A 6 S p (a ). I f A is commutative, then V (a , X ) is included in the Jacobson radical of A . Also for any a, T >(a ,X ) is an ideal of A if X is. Proof We have already shown existence and uniqueness. Suppose V 1 is the l&igest subspace of X satisfying (A - a ) V = V for all A G S p (a ). This implies a V C V since the spectrum is nonempty. It is enough to show (/i — a ) V = V for all /x G C \ Sp(a). Any A G S p (a ) and / iG C \ S p (a ) satisfy (A — a)(/i - a )^ 1! ^ = (/x — a )-1 (A - a ) V = (n — a )~ lV . Hence the maximum nature of V gives (n — a )~ lT>' C V 1. However, this implies t y = (/i - a )(n - a )~ 'D / C (/i — a ) V C D 7 as we wished to show. If A is commutative, let 7 G be arbitrary. For any d G T>(a, X ), there is an e G T>(a, X ) satisfying d = ( 7 (a ) - a)e. Applying 7 shows 7 (d ) ~ 0. Since 7 G was arbitrary, Theorem 3 . 1.5 shows T > (a ,X ) C A j . I f X is an ideal o f A and V C. X satisfies (A — a ) V ~ V for all A G C, then for any b G A , bD1 satisfies tho same conditions. Hence T >(a ,X ) is an ideal. □
This proposition shows that the ideal J of Theorem 3.4.9 is just T>(b, A ). 3.4.12 P r o p o s itio n
Let A be a Banach algebra. F ora G A let I a: T a —*
A ] be the usual functional calculus homomorphism and let Ja: J-a —* A 1 be any other homomorphmm also sutisjymg ( 8).
Define bj\ Jra —> A by
6j ( f ) = Ja( f ) - I a( f ) for all f G JFa. Then 6j is linear and satisfies
3.4. :,3 (u )
E xam ples and A pplications o f Functional C alculus
6Afg)
=
(!-) 6j ( p ) = 0
I a ( f ) 6 j ( g ) + 6 j ( f ) I a( g) + 6 j ( f ) 6 j ( g ) V polynomials p.
365
Vf,g€F.
(■ ) 8j { ? a) C V ( a , A ) . ( ( 1) Sj ( e ) = 0 whenever e is idempotent. ( ( ) I f A is commutative, 6j ( f a) C V ( a , A j ) C A j . Proc ' Properties (a) and (b ) follow from easy computations.
(r ) and (e): For / G T a and A € S p (a ), we can write / = /(A ) + (A —z)g for k me g e T a. By applying 6j , we get 6j ( f ) = (A - a) 6j ( f ) . This show, (A - a) 6j ( J ra) = 6j ( f a) for all A G Sp(a). The last proposition now gives Q T>(a,A). If A is commutative, the final result of the last proposition gives (e). (c ): The elements /0 (e) and Ja(e ) are idempotents. Since /a(e) belongs to the double commutant o f { a } and aJa(e ) = Ja{z e ) = Ja(e)a holds, there is a maximal commutative subalgebra C which includes both Ja{J-a) and «(/■„). This implies 6j ( e ) = 6j ( e )3 = ••• = 6j ( e ) 3" for all n € N. Gclfiiiid’s spectral radius formula now gives ||<5j(e)|| = p (6j ( e ) ) . We may repla :e A by C and then apply (e) to conclude <5j(e) = 0 . . □ Note that e € T a is idempotent if and only if c|sp(o) >s the characteristic function of a clopen subset 5 of S p (a ). For S clopen in S p (a ) we write e£ for such an element of T a. 3.4.13 T h e o r e m Let A be a Banach algebra. Any a a A for which D ( a , .. 4.) is {0 } has a unique functional calcxdus. This holds for all a € A if A is commutative and its Jacobson radical is finite-dimensional. If A is commutative and S p (a ) has at most countably many connected components, then the following are equivalent:
(a)
a has a unique functional calculus.
(h )
V ( a , A ) = { 0} .
(c)
Each connected component S of S p (a ) satisfies
f)
V( a, t
ea T A ) = {0 }.
clopen SCT
Partial Proof We have already established the first paragraph except the last sentence. Suppose A is commutative and A j is finite-dimensional. Then there is a non-zero polynomial p such that the restriction of the left regular representation L a to A j satisfies L p(0) = p ( L a) = 0. Factoring p( a) = a (A i - a )( A2 - a) • • • (A „ - a) shows {0 } = p(a)T>(a, A ) = V( a, A ) . For a proof of the second paragraph o f the theorem see the original paper Thomas [1978] or Dales [1973] and [1994], It is naturally hard to prove the existence o f a discontinuous functional when some component of Sp(a j fails to satisfy (c). The unusual (and not very restrictive) hypothesis
366
3: C om m utative A lg ebras and Functional C alculus
3.4.14
on the topology of the spectrum is used to show that one component S of S p (a ) is clopen. This is crucial in the proof that (c) implies (b). □ We also quote Dales’ main result [1973]. The reader is directed to that paper for further details and for reference to previous work. 3.4.14
T h e o re m
Let A and B be unital commutative Banach algebras
with B semisimple. Then, A has a unique functional calculus if it has an ideal T containing its Gelfand radical and satisfying:
(a ) There is a continuous, unital homomorphism 6 of B into B ( l ) . ( b ) Each a, c € I and b € B satisfy: 6 (b )(a c ) = a O (b)(c). (c ) There is a finite set { f j , c j , . . . , c „ } C T such that each c € T has a
unique expression n
c = ^ 0 (6 j)(c j)
with bj e B.
>=i
If we take B to be the complex field in the above theorem, we see again that a unital commutative Banach algebra with a finite-dimensional radical has a unique functional calculus. (This covers Feldman’s example discussed in Example 7.1.4.) Arens-Hoffman extensions of semisimple commutative Banach algebras, which are discussed in Example 3.4.17, are also covered by the theorem. A simple argument (given on p. 642 o f Dales [1973]) shows that any algebra satisfying the hypotheses of the theorem must have a nilpotent radical. This property o f Arens-Hoffman extensions of semisimple commutative Banach algebras was first obtained by John A. Lindberg, Jr. [1964]. Unfortunately it seems difficult to derive any criterion for general Banach algebras to have a unique functional calculus from this criterion for commutative Banach algebras. 3.4.15 E x a m p le We will exhibit an algebra satisfying Theorem 3.4.9 which also has other interesting properties. Let A be the Banach space L l ([0,1]) with Lebesgue measure on [0,1], This can be made into a Banach algebra by defining convolution multiplication: t
/ * p(0 =
J
f(s)g(t -
v /, g € A\ t e [o, ij.
o An obvious change of variables shows that A is commutative. It is easy to see that the convolution powers o f the constant function 1 are 1 " = xn~ l/ ( n — 1 )! and have norm (n!)'~l . Hence, the constant function 1 is topologically nilpotent and generates the dense subalgebra V of polynomials
15
Exam ples and A pplications o f Functional C alculus
367
ir> j.. (The Stone-Weierstrass theorem (cf. 1.5.1) shows that the algebra V a" polynomials is dense in C ( { 0 , 1 ]) in its norm. A standard argument, basfd on the regularity of Lebesgue measure, then shows that V is dense in A .\ Therefore, the continuity of the spectral radius in commutative spectral normed algebras (Corollary 3.1.6) shows that A is equal to its own Gelfand radical. Another way to see that A is Gelfand-radical is to note that any function which vanishes almost everywhere in some neighborhood o f zero is nilpotent, and that the ideal of such functions is dense in A . Consider the sequence of functions { e n} neM where en( s) equals n for s < 1/n and equals zero for s > 1/n. These functions all have norm one. Any / G C ([0 ,1]) and t > 1/n satisfies
o 1/n
o 1/n
0
<
max{|/(t) — /(r)| : t - ( 1 /n) < r < t}.
Hence, the uniform continuity of functions in C ([ 0 , 1 ]) shows that / = lim e„ * /
(9)
holes pointwise on ]0 , 1 ]. The Lebesgue dominated convergence theorem shows that equation (9) holds in the norm o f A for any / € C ([ 0 , 1 ]). Since (as remarked above) C ([ 0 , 1 ]) is dense in A , (9) holds for all / € A . In the terminology introduced in Section 5.1, { e n} n6N is an approximate identity of riorm one in A . Let {/ n}neN be a countable dense sequence in A (e.g., an ordering of the polynomials with complex rationed coefficients). Corollary 5.2.4 (the Cohen factorization theorem) shows that there is some b e A and some sequence { < ? „ } „ e N C A satisfying ||b|| < 1 and f n = gn *b. Hence .46“ equals A . Therefore, (A b n)~ equals A for any n € N. Hence the unital algebra A 1 and the element b satisfy Theorem 3.4.9, so A fails to have a unique functional calculus. Note that A b = A would imply the existence of an element e to satisfying eb = b and hence ea = ecb = ceb = cb = a for any a = cb in A . Since we havJ shown that A is Gelfand radical, it cannot be unital, so A b ^ A . Ks~.ce, the ideal J = is a proper (dense) ideal. Before continuing the discussion of uniqueness of the functional cal culus, we will mention some other interesting properties of the algebra
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3.4.16
A — Z,‘ ([ 0 , 1 ]). Since the constant function 1 generates A as a Banach algebra, the closed ideals of A are just the subspaces invariant under theVolterra operator t
J
(Tf)(t) =
/(*)
V / e A; t e [0 , 1 ].
( 10 )
o William F. Donoghue, .Jr. [1957] showed that these are exactly the subspaces Zj = { /
G
A: f ( s ) = 0 for almost all s
G
[0, £]}
for t 6 [0,1], Obviously, / 6 J( and g G Jr imply f *g G T t + r ■ A convolution theorem of Titrhrnarsh [ 1920] shows that, conversely, / € J< and / *
OO
ll/ll = ^ l«nl when f i z) = X ] a ” Z"n=0 n—0 Let K be the Banach space C ([0 ,1]) and let T be the Volterra operator defined by (10). Since Sp( T) is {0 }, we may define f ( T ) for any / G U by OC
OC
f ( T ) = V a nT n for f ( z ) = Y a nz " f U f l'0
n
0
3.4.’ ^
Exam ples and Applications o f Functional C alculus
369
and v e will just have the usual functional calculus. Now let A be the Bana h space U ® T l with the norm ||/©a:|| = ||/|| + ||x|| and multiplication i f ® x ) ( g ® y ) = ( Jg , f ( T ) y + g { T ) x )
V /, g e U ; x , y £ T Z .
Clear y, A is a commutative Banach algebra with Gelfand radical {0 } tb 7Z. Furthermore, the isometric isomorphism 6 : U —> A defined by 0 ( f ) --- / © 0 shows that A has a strong Wedderburn decomposition (cf. Example 7.1.4). Nevertheless a discontinuous functional calculus for A is constructed in the reference already cited. The construction is closely re lated -o a construction given in Theorem 6.4.23, which leads to a Banach algebra with two inequivalent Banach algebra norms. These give rise to two (1 fferent. functional calculi —each continuous with respect to one o f the two >;;>nns. Even the exponential function has a different effect in these two functional calculi. Miscellaneous Examples
3.4.1? E x ten sion s and A re n s —H o ffm a n E xten sion s In this example A is a subalgebra of B. Unfortunately there is considerable variation in terminology. Some authors insist that the embedding map be continuous, homeomorphic or even isometric, while otherf make no continuity restrictions at all. Obviously the theories derived depend strongly on these assumptions. If i e A is sufficiently well behaved, there may be a unital algebra B including A in which there is a joint solution to the equations cue = 1 and x a = 1. Similar ideas have been investigated in great detail usually when ‘ihe algebras are assumed to be commutative: Arens [1958], [1960], Ian G. Craw [1978], Vladimir Muller [1979], [1982], [1983], [1988], Gerard J. Murphy and Trevor T. West [1980a], Craw and Susan Ross [1983], Souren Arsha covic Grigoryan [1984], Zame [1984], and Michael J. Meyer [1991]. More generally, let A be a unital commutative algebra and let A\x) be thi algebra of all polynomials in an indeterminant x with coefficients in A . Let p e A[ x] be a monic polynomial of degree at least 2. Let B = A[ x] / pA[ x\ be the quotient algebra of A modulo the principal ideal p>t[ar], and let 6 : A{x\ —* B be the natural map. Clearly, B is a unital commutative algebra and the map a i-* 0( a) embeds A as a subalgebra o f B. Furthermore, 6 ( x ) € B is a solution of the equation p ( x ) = 0. When A is a Binach algebra, Arens and Kenneth M. Hoffman [1956] show how to provide B with a complete algebra norm making B into a Banach algebra. The key idea is the following simple result. B is s lid to be an extension of A if
P ro p : sition.
Let ao, a i , . . . , a „ _ i be elements in a commutative unital
B a n a c h algebra. Then there is a positive number s satisfying
IM
+ ||«i ||s + ' • ■+ ||a„-i ||sn 1 < sn .
370
3: Com m utative A lg ebras and Functional Calculus
3.4.18
For any such number s, the expression n- 1 11Oo + « ! * + ••• + u n _ i z " - 1 + p. 4 [.r]||g = ^ | | a _ ,| | s J
1
defines a complete algebra norm on B.
David T. Brown [1966] and Lindberg [1971] show that if A is semisimple and s satisfies the condition above, then any complete algebra norm on B is equivalent to || • ||,. The norms || • ^ defined above for different values of s (which satisfy the condition) are all equivalent. It has also been shown that all derivations of an Arens-HofFman extension of .4 are continuous if A has the same property (Lindberg [1971], Richard J. Loy [1970a], Julian M. Cusack [1976]). In particular, this holds if A is semisimple. 3.4.18 Ban ach A lg e b ra s o f P o w e r S eries The earliest references to the subject matter discussed here appear to be Gelfand, Raikov and Silov [1946], Lorch [1944] and Silov [1947a], [1947b], Important more recent discussions are Loy [1974b], Sandy Grabiner [1974] and Hade, Dales aud Laursen [1984]. (See also Dales [1994] and Bade [1983].) The terminology in this field is unfortunately not standardized. W e use T = C[[z]] to represent the algebra of all formal power series with complex coefficients (cf. 2.9.7). If Qn ^" happens to converge for some / = a „ z n 6 T and some A e C, we denote it by /(A ). D e fin itio n . A Banach algebra of power series is a Banach space A which is a subalgebra of T containing z such that the coefficient functional OO
C n ( Y a mz m) = a n m =1
OO
V
2
^ ~ otmz m e A m =l
is continuous for each n 6 N°. If A is a Banach algebra o f power series, the closed graph theorem shows that multiplication is separately continuous for each o € A so Proposition 1.1.9 asserts that there is an equivalent norm on A in which it is a Banach algebra. Henceforth, we assume that A is a Banach algebra whenever it is convenient to do so, although the norm given on certain examples may not be submultiplicative until an equivalent renorming is performed. We always consider A ' to be embedded in T in the obvious way. Some authors require that polynomials be dense in a Banach algebra of power series and call algebras t hat lack this property generalized Banach algebras of power series. Banach algebras of power wri
3.4.18
Exam ples and A pplications o f Functional C alculus
371
is dosed under pointwise multiplication and is complete in some submultiplicative norm. Tw o obvious examples are the disc algebra ./1(D) and the algebra //°°(D). (These two algebras are fundamentally different in this theory, because the polynomials are dense in the first case and not in the .second,) Since the functions in A are completely determined by their power series expansion at zero, any such algebra is a Banach algebra o f power se ries. ^valuation at any point of U defines an element of the Gelfand space r * , and since such evaluations obviously separate the points of A these algebras are semisimple. This example is not typical of the type of Banach algebras of power series usually studied. Now suppose the region U is star shaped with respect to zero. Again let A be a linear space o f functions analytic on U, but this time define multiplication by convolution Z
f *a(z) =J f(w)s(z- w)dw 0
and assume that .4 is a Banach algebra relative to this multiplication. Obviously, this does not become a Banach algebra of power series when the Taylor series at zero is associated with each function. Instead, associate with each function OO
f(*) = ] C a"*" the formal series
" ° oo
oo
/ (* ) = £ / (b _ i)( 0 ) * b = $ > „ _ ! ( « - 1 )!* " . n=l
n —I
It is net hard to show
(f*gT=fg f,g£A, so that ,4 is a Banach algebra o f power series provided it contains the con stants (since \( z) = z) and meets the topological conditions. Grabiner [197''], Section 13, shows that many well-known spaces o f analytic func tions (e.g., H p spaces) do become radical Banach algebras o f power series. T i e v'?al structure and other properties of these spaces are determined explicitly. See also Example 4.8.3. As n third example of a naturally defined class of Banach algebras of newer scries, we consider a topologically nilpotent, but not nilpotent op erate' on a Banach space X . Let A be the algebra of all power series such that the series JZ^Li a nTn converges in the weak oper ator topology of B ( X ) , and provide A with the operator norm of B ( X ) . Under suitable conditions on T, A is a Banach algebra o f power series. See Grabiner [1974], Scction 14, for further details and references.
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3.4.18
Our fourth natural example of Banach algebras of power series arises from considering shift operators. Let X be a Banach space. A biorthogonal system in X is a pair o f sequences { z n}neN Q X and {u/n} neN e X* satis fying (jJj(zk) = 6j k. The biorthogonal system ( { z „ } „ ei\i, {u > „}„ei\i) is called a generalized basis if the sequence of functionals separates the points o f X and is called a Markmhevich basis if, in addition, the sequence o f vectors has a dense range. Relative to such a basis, an operator T g B ( X ) is called a weighted shift determined by the sequence o f non-zero complex numbers if each x G X satisfies u n+1 ( T x ) = (An+,/A „)*>„(x) uj0( T x )
=
n> 0
0.
The power series representation determined by T is the map which asso ciates with each x € X the formal power series Y l< ^ =o‘jJn (x )X ~ lzn. The space A 1 of all such power series is a Banach space under the inherited norm. Under this representation of X, T corresponds to the operator of multiplication by z, Anzn corresponds to T n, and A " lusn corresponds to the nth coefficient functional C n on .41. The collection A of power series in .41 without constant terms is frequently a Banach algebra o f power series. We wish to examine our fifth class of examples, the weighted semi group algebras on the semigroup N° of non-negative integers, more closely. These algebras were introduced in §1.9.15. Recall that a weight func tion on N ° is a positive valued sequence {u>„}„ei\i° Satisfying u) q = 1 and for all 772, n e N°. We will denote the semigroup al gebra ( l (N °, u j ) by A . Thus A is the algebra of sequences / :N ° —* C with finite norm ||/||u, = l/ (n)lwn under convolution multiplication / * 9 ( n ) = E l = o / ( m M n - m )- Clearly .4 is a singly generated unital commutative Banach algebra which is a subalgebra of T and in which se quences with finite support (i.e., polynomials in <5j) are dense. The next simple result, published in Gelfand, Raikov and Silov [1946], is fundamental and follows from Theorem 3.1.15. Theorem Let p = ^ (^ (N 0,^;)) be i n f ^ y " } = lim u))Jn . (a) If p is positive, then A is a unital semisimple commutative Banach algebra and A •-» 7 >, is a homeomorphism of { A 6 C : |A| < p} onto where ' y\( f ) = /(A). (b) I f p is zero, then A is isomorphic to the unitization of its radical Aj = {feA :f(0 )=0 }.
This obviously suggests calling a weight function ui radical if p is zero. Recall from Example 2.9.7 that T has a decreasing sequence of closed ideals T m = { / S T : / (n ) = 0 for all n < m }. When we consider A as embedded in T , this gives a sequence of closed ideals called standard ideals. We will call A or its weight sequence lj unicelullar if all the closed ideals o f A
3.^.’ '
E xam ples and A pplications of Functional C alculus
373
are standard. During the 1940s, Silov asked whether every radical weight function is unicellular. Thomas [1983], [1984a], [1984b] finally proved that nonun cellular ones exist. This raises questions about the structure o f A j T when I is not standard. Vv'‘ ■introduce several technical properties o f weight functions and briefly record their logical relations. It is convenient to use t]n for - log so that p = 0 if and only if lim Tjn / n = oo. A weight sequence {u>n }neN° ’s called { convex / star-shaped ) if the sequence ( rjn+i ~ rln / {*?n/«}neN° ) >s increasing. It is called basic if sup{u;m+n+fc/ii;m+*;u>„+*; : m, n S N0} is finite for each k S N°, and ordinary if, for each a & A , there is some k € N° satisfying zk 6 aA. T h e o em
There are radical weight functions on N ° which are not unicel
lular. All the arrows in the following diagram indicate known implications between properties of radical weight functions and none are logical equiva lences
star shaped — convex C
unicellular b a sic--------------► ordinary •
For proofs and related results, see Dales [1994]. The counterexamples needec to show that none of the implications can be reversed are from Bade, Dales and Laursen [1984] and and Thomas [1984b], See also Grabiner [1975], [1974] Section 14, [1984c]; Grabiner and Thomas [1985], Yngve Domar [1983], [1987] Fereidoun Ghahramani and McClure [1990], [1992], Ghahramani, McClure and Grabiner [1990]. 3.4.19 E x a m p le Our next example is a unital, completely regular, semi simple, commutative Banach algebra with a non-maxima) proper closed prime ideal. It is a refinement due to Cusack [1976] of an example of Lorch [1944]. See also Dales and Alexander M. Davie [1973]. Le., K — {K n }n eN be a sequence of strictly positive real numbers satis fying .‘.'o = 1 and K n > ( j ^ K kK n- k
(11)
for rt e N° and all k satisfying 0 < k < n. Let A ( K ) be the Banach algebr * of all / e C ° ° ([ 0 , 1 ]) (the infinitely differentiable functions under pointv, ise multiplication) which are finite with respect to the norm
\\f\\K =
' E n =0
K
1\\f{ n ) \\
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3.4.19
Loy [1974a] shows that these algebras are semisimple if lim(n!/A'n) 1/" is positive. Dales and Davie [1973] show that, if A' satisfies oo
n -1
D
E
n—2
j= l
/ v
( 12 ) '
then (0 , 1 ] is naturally identified with the Gelfand space of A ( K ) . For all r, s satisfying 0 < r < s < 1 define f TS G C °°[0 ,1] by t < r and t > s r < t< s
0
ex p ((r — t ) _1 + (t — s )_1)
Then for any closed w t C r [0,1] and any t C, we can find an /,„ which vanishes on C but is non-zero at t. Hence, if we can choose K satisfying (12) so that A ( K ) contains each function frsy then A ( K ) will be completely regular. An elementary but tedious estimate shows that kn = SUpdl/r?' 11^:0 < r < s < 1} is finite for each n. Hence, it remains to show that for any sequence {A;n} ngN o f positive numbers, we can find a sequence K = { A „ } „ e N satisfying both (12) and also < oo. We may assume kn > 2" for all n 6 N. Define K inductively by K q = K\ = 1 and for n > 2 * " = 2" ( £ Clearly, we have K „ > ( ™ ) K j K „ - j oo
n —1 /
( " ) * > * » - > ) + *»• 0 < j < n,
v
oo
] C Y 1 ( nj K i Kn - > * » ! - E 2 1 < ° ° ’ n= 2 ; - l XJ/ n~2 &ll(i
oo
oo
oo
Y , K - l K < J 2 K 2kn < $ > - " < n=2
n = 2
oo.
n = 2
W ith this choice of the sequence K , A ( K ) is a unital, completely regular, semisimple commutative Banach algebra. For any choice of the sequence K satisfying (11), r = { / & A ( K ) : f ^ ( 0) = 0
n e N 0}
is a non-maximal closed prime ideal. It is non-rnaximal since it has infinite codimension. It is prime since it is the kernel of the homomorphism OO 5 > ! ) - 7 (B)(0)2"
feA (K )
3.4.20
E xam ples and Applications o f Functional C alculus
375
o f A ( K ) into the integral domain T .
3 . 4 .20
P o in t D e r iv a t io n s
Let A be a unital algebra and let 7 be a point
in its Gelfand space which is arbitrary but will remain fixed throughout this discussion. D e f in i t io n
A point derivation o f A at 7 is a linear functional
6 : .4
—♦ C
satisfying
6(a b )
=
6 (a )-y (b )
+ -y (a ) 6 (b )
V a,b&A.
If, ./I is an algebra o f differentiable functions on R and if 7 is evaluation at a point x € R , then evaluation o f the derivative o f a function at x would l>e a point derivation at 7. Thin is obviously the origin o f tho terminology. Lei; M
be the codimension one ideal which is the kernel o f 7 and let
M 2 denote the linear span o f all products o f elements in M , as usual. It is easy to see that a linear functional on A is a point derivation at 7 if and oaly
i f it vanishes on M 2 + C l . Consequently, the linear space o f point
derivations at 7 is in natural bijective correspondence with the algebraic dual o f the quotient space M / M 2. Hence, there are no point derivations at 7 i: M = M 2. If A is a normed algebra and the closure o f M are no continuous point derivations at 7. the cas3 in which M / M
2 is
2 equals
M , then there
Perhaps o f greater interest is
infinite-dimensional in a normed algebra. For
then there are surely discontinuous point derivations at 7.
This has the
following consequence, which is the easiest way to construct discontinuous algebra homomorphisms. P r o p o s it io n Let A be a unital normed algebra with a discontinuous point derivation 6 at 7 e F ^ . Let B be a unital normed algebra (such as the matrix algebra M 2) with some non-zero nilpotent element,. Then there is a discontinuous unital homomorphism o f A into B. P ro o f If B has a non-zero nilpotent element, it has a non-zero element b satisfying b2 =
0.
Define the homomorphism
V ae.
4.
O
It is obvious that for all elements o f the G elfand space o f the algebra
C( Q) Tor a compact H ausdorff space fl, M 2 — M in the notation introduced above, so there are no non-zero point derivations. W e will mention the dif ficult construction of discontinuous homomorphisms on C ( f l ) in Section
6 . 1.
Ir. Volume II we will show that there are no non-zero point derivations
on noncommutative C *-a lge bras either. If A is the disc algebra (§ 3 .2 . 13), there are no non-zero point derivations at points o f the G elfan d space cor responding to the boundary points o f the disc. T h e only point derivations
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3: Com m utative A lgebras and Functional Calculus
3.4.21
at points of corresponding to interior points o f the disc are those arising from differentiation and they are continuous point derivations. Hence, all the point derivations are continuous even though we will show that there are discontinuous homomorphisms from the disc algebra. In Volume II we will show that there are no non-zero point derivations on L l ( G ) for an amenable group or on other amenable Banach algebras. 3.4.21 A Functioned C alcu lu s fo r C on tin u ou s Fu n ction s We will use Theorem 3.2.12 to construct a functional calculus in which continu ous functions act on normal elements in a C*-algebra. This topic will be pursued in much greater depth in Volume II, but the comparison with the functional calculus of this section is instructive. C*-algebras were defined and introduced in §1.7.17. An element { h / c ) in a C*-algebra is said to be ( hermitian / normal ) if it satisfies (/»* = /»/ c*c = cc* ). The set of ( hermitian / normal ) elements in a C*-algebra A is denoted by { A
h / A n )■ Volume II will include many examples of C*-algebras and a much more complete discussion of their properties. The commutative examples are characterized in Theoreom 3.2.12. We need two easy results now. First, note that the inequality ||a*a|| > ||a||2 for all a S .4 is equivalent to the C*condition and implies that the involution is an isometry. To see this, note
that ||a||2 < ||a*a|| < ||a“ || ||a|| implies ||a|| < ||a*||, ||a*|| < ||(a*)*|| = ||a|| and ||a*a|| < ||a*|| ||a|| = ||a||2 < ||a*a||. Second, we note that if A is a C*-algebra, then the unitization .4* is also a C*-algebra when we define (A -f a)* = A* + a* and ||A + o|| = sup{||(A + a ) 6 ||: It e A i }. The C*-inequality follows from ||(A + a ) m(\ + a)H
=
sup{||(A + a )*(A + a ) 6 ||: b e A i }
>
sup{||6 *(A + a )'(A + o ) 6||:6 € -4 i}
=
sup{||(A -f o ) 6 ||2: b € ^4i} = ||A + a||2.
The functional calculus is established in the next theorem. T h e o re m .
Let A b e a C*-algebra. For each c € A n , let «4(c) be the small est norm-closed unital subalgebra of A 1 which includes c and c*. Each ele
ment d of .4(c) is normal and satisfies SpA t (d ) = SpA ^ ( d ) . Furthermore, there us an isometric isomorphism, denoted by f t-* / (c), o f C ( S p A ( c)) onto ^ ic ) which satisfies: (a) z(c ) = c (b) 1 (c) = 1 6 >1 *
(r) / (c) = f o e M) S p A , ( f i r ) ) - f ( S p A (r))
<*■) f ’ ( c) = f ( c ) '
3.4.2 t
(f)
Exam ples and Applications o f Functional Calculus
377
a(f(c)) = a ° f ( c )
for all f e C { S p ( c ) ) and g e C ( S p ( f ( c ) ) ) . Proof For simplicity, we will write T, Sp and C ( c ) for T ^ (c) , 5p^(C) and C (S p 4 (C)(c )), respectively. Since .4(c) clearly satisfies the hypotheses of Theorem 3.2.12, the Gelfand homomorphism is an isometric isomorphism of A { ,) onto C (r). Theorem 3.1.5 shows that c: T —* S p ( c) is a continuous surje. .ion. Next, we will show that c is an injection. Suppose £(71 )j= £(7 2 ) holds for 7 1 ,7 2 S I\ The last remark of Theorem 3.2.12 asserts c * ^ ) = which implies p(c, c’ )~(7 1 ) = p (c ,c * )'( 7 2 ) for all polynomials p in two v-driables. Since elements o f the form p(c, c ') are clearly dense in .4(c), and ( “) is continuous, this implies 7 1 (a ) = 6 ( 7 1 ) = <2( 7 2 ) = 7 2 ( 0 ) for &U 0 € y (c). Hence 71 equals 72 . Thus c:T —♦ S p ( c) is a continuous bijection and hence a homeomorphism. Therefore, we may identify T with Sp( c) , and interpret ( “) as an isometric isomorphism of .4(c) onto C (S p ( c)). The map t-» /(c) is just the inverse map. A ll the stated results will now follow from Theorem 3.2.12 when we establish Sp^i (d ) = S p (d ) for all d 6 «4(c). Proposition 2.1.8(c) shows S p ^ i(d ) C S p ^ c)(d ) = S p (d ) for any d €
.4(c). Suppose A e Sp(d). Then A — d vanishes at some 70 6 I\ Given any e > 0 , there is a neighborhood N o f 70 satisfying |(A — (f)( 7 ) | < e for all 7 6 N . Since (") is surjective, Urysohn’s lemma gives some a 6 A(c) C A 1 satisfying ||a|| = Hall^ = 1 and supp(a) C N . We conclude ||(A — d)a|| =
||(A < e. Since e > 0 was arbitrary, A — d is not invertible in A 1. We have established Sp^i (d ) = S p (d ) for all d G .4(c). □ Wo indicate a few applications of this powerful functional calculus. Oth ers will be given in Section 5.1. Any normal element c in a C*-algebra s a tis fy ||c|| = ||c||oo = p(c) (where (*) is defined relative to -4(c)). Any liermilian element h in a C*-algebra haa its spectrum included in R. Conversel/, if h e A n has real spectrum it is hermitian. If > 6 A n has spectrum included in R + then the square root function opera .es on p so there is a normal element q in p satisfying p = q2. The spectuim of q is also non-negative and hence q is hermitian. For h € A h define h+ and /»_ by h+ — f ( h ) and /i_ = g( h) , where / and g are defined on K >y
Then h+ and h - are hermitian elements with spectrum in R + satisfying: h = /i+ — /»_
h+ h - = h - h + = 0 .
37S
3: Com m utative A lgebras and Functional Calculus
3.4.22
For an example of a Banach algebra with a remarkably large functional calculus see Gilles Pisier [1979]. Algebra Norms on C*-algebras
We now use Theorem 3.2.12 again, this time to get a genuinely noncomrnutative result. The last argument in the following proof is due to Bonsall [1954a]. Related results will be discussed in Theorem 6.1.5. 3.4.22 T h e o r e m Let (A , \\ \\) be a C*-algebra. Any (possibly incomplete) algebra norm ]|| • ||| on A satisfies ||«li2 < ltl«*a||| < |||a||| Hla'IH and ts spectral.
V a e
.4
(13)
The C *-norm || • || is minimal in the sense that if ]|| • |||
satisfies |||a[[| < ||a|| for all a € A then the two norms are actually equal. Proof For any a e A let C be the closed subalgebra generated by a* a. Since C satisfies Theorem 3.2.12, Theorem 2.4.15 gives the inequality. Let B be the completion of (A , ||| • |||) and recall that Proposition 3.2.2 implies PB{ a) < P A ( a) for all a G .4. Replace a by o " in inequality (13), take the nth root, and then let n approach oo. Since a >—> a* is a real algebra anti-isomorphism which sends each complex numer to its conjugate, this gives pAi a )2 < PB (a)Pfi(a*) < pB( a) pA ( a’ ) = pB( a) pA( a) - We conclude Pa(<*) = p s {a ) < j|ja|11 for all a G .4 so that ||| • ||| is a spectral norm. Now suppose that ]|] • |J] satisfies |||a||| < |jo|| for all a S A If there were any a € A satisfying |||a||| < ||a|| then we would get the contradiction I H I 2 < IIH IIII| a *lll< l| a | | l| o *| | =
3.5
N | 2.
□
Multivariable Functional Calculus
Introduction
This section describes functional calculi for finite ordered sets o f com muting elements in unital Banach algebras. Let a = (at, a2, ■■., an) be an ordered n-tuple of commuting elements in a unital Banach algebra A . We wish to define a notion of “joint spectrum” Sp( a) of a which will be a subset of C ". If / is a function of n complex variables which is analytic on a neighborhood of Sp( a) , we hope to define an element f ( a ) of A with the usual properties of a functional calculus. Thus we want f ( a ) to satisfy: (1) polynomials act appropriately, (2) for each a € A, f t—» f ( a ) is a unital ho momorphism from the algebra of functions analytic in a fixed neighborhood U of Sp{a) to A , (3) a spectral mapping theorem holds, (4) composition of functions satisfies / o g( a) = f { g ( a ) ) when either side is defined, (5) / t-i> /(a ) is continuous and unique among continuous functions satisfying (1) and (2). If Sp( a) is chosen quite large, the definition of f ( a ) becomes
3.5.0
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easy, but the spectral mapping theorem and uniqueness will probably fail, while the class o f functions which act will be too restricted. We do not know a completely satisfactory definition in the most general case, so we will give only a partial treatment. See also Wieslaw Zelazko [1979], where the class of all joint spectra satisfying the spectral mapping theorem at least for polynomials is studied, Robin Harte [1972], [1973], [1977], [1981], [1 8 8 8 ], Arne Kokk [1989] and Yu. V. Turovskii [1984a]. Historically the first case to be studied, and the easiest case, is that of a comTnutative unital Banach algebra A . Here the results are essentially due l,o Richard Arens and Alberto P. Calderon [1955] and Lucien Waelbroeck [1954], based on Silov’s proof [1953] for finitely generated commutative algebras. Only Waelbroek obtains a homomorphism. The most natural statements about the uniqueness o f the resulting functional calculus were proved much later by W illiam R. Zame [1979] and Stephen H. Schanuel and [1982]. Bonsall and Duncan [1973b], Chapter 20 gives a beautifully simple treatment o f a functional calculus which is not shown to be a homo morphism or unique. We will construct a system o f homomorphisms with the r/3aker uniqueness property similar to Bourbaki [1967], Theodore W. Gameiin [1969] and Edgar Lee Stout [1971]. We separate the most techni cal portion o f the construction and give an elementary exposition suitable to readers unacquainted with the exterior differential calculus. See also Graham R. Allan [1968], [1970], [1971] and Waelbroeck [1982a], [1982b], [1983a] and [1983b], I f the Banach algebra A is noncommutative, the situation is much more ccrrr-rcated. The difficulty arises first because the proper definition o f the joint spectrum is no longer obvious. In [1970a] Joseph L. Taylor gave a complicated but very natural definition o f the joint spectrum for the special case in which A is the noncommutative Banach algebra B ( X ) o f all bounded operators on a Banach space X. This is now called the Taylor spectrum. In [1970b], [1970c], [1972a] and [1972b] he constructed the functional calculus associated with his notion o f spectrum. Both the definition and construc tion depend on Koszul complexes and are too technical to describe here in detail. Furthermore, basic properties of functional calculi remain unclear and difficult in this case. Alexandr Yakovlevic Helemskii [1981] and [1989b], C 'r.o 'er V I give a somewhat simplified version based on Taylor’s [1972b]. See aiso Vasilescu [1982], Mihai Putinar [1980], [1982], [1983a], [1983b], [1984a], [1984b], [1984c] has apparently solved these problems and given a coherent theory but at the expense of the further complication o f extending the definition from functions analytic in a neighborhood of the spectrum to Stein algebras and less trivial applications of sheaf theory. (See also Jorg Eschmeier and Putinar [1984].) The most detailed exposition o f this theory is still available only in the Rumanian language, but we understand ("e~ 2cnal communication) that Putinar is currently preparing a detailed
380
3: Com m utative A lgebras and Functional Calculus
3.5.0
book-length exposition in English. Given these two special cases (ordered n-tuples in commutative alge bras and commuting ordered n-tuples in B ( X ) ) , there are some obvious ways to try to define the joint spectrum and functional calculus for com muting ordered n-tuples in arbitrary unital Banach algebras A. If C is any commutative unital subalgebra of A including {a 1 , 02 , . . . , an}, then the commutative theory can be applied to o = ( a i , a2, ■■■, an) considered rela tive to C. Obviously the theory depends on the choice of C. In order to have as many functions as possible in the functional calculus, one must choose C as large as possible. The only natural choice is the double commutant {o i, 02, . . . , 0 , , } " o f {a i , a 2, . . . , a „ }. This procedure is satisfactory in the very special case in which a 1 , 02 , ■■. ,a „ belong to the center A z o f A and A z is chosen as C. Unfortunately, in essentially all other cases examples are known in which no choice will give a satisfactory spectrum or one as small as Taylor’s spectrum (see Taylor [1970a], Zbigniew Slodkowski and Zelazko [1979]). This procedure is also incompatible with a satisfactory spectral mapping theorem. Another approach would be to apply Taylor’s theory to the left regular representing operators { L a i, L a j L an }. This also presents many problems in most cases, although not for C*-algebras (Raul E. Curto [1982]). Before introducing our notation and proceeding with the detailed expo sition, we wish to mention, but not pursue, the idea of a functional calculus for several noncommuting elements. One could begin such a discussion by considering some kind of generalization of polynomials in several noncom muting indeterminants. This approach has been pursued by Taylor [1972a], [1972b] and [1973b], Harte [1972] and [1973] and Alan James Pryde [1988]. Another approach is to abandon the homomorphism property o f a func tional calculus and use ordinary analytic functions o f several variables, sym metry considerations and a suitable idea o f joint spectrum. Albrecht has developed that approach. His valuable survey [1982b] cites the relevant literature. For some related ideas see N. C. Luong and Arpad Szaz [1979] and Denis Luminet [1986] for a special case. We now return to the main subject o f this chapter: a functional calculus for several elements in a commutative algebra. We need a little notation. An element of C " will be denoted by a single lower case Greek letter such as A and its components will then be denoted by subscripts, e.g.,
(1) For each j = 1,2,... , n the function zy. C " —> C will be defined by 2 j ( A ) = 2 j ( A 1, A 2 , . . . , A n ) =
AJ
V A G C n.
(2 )
Then z = ( z \, 22,. • ■, zn) is the identity function or the coordinate function on C ": r(A) = A
V A e C n.
(3)
M ultivariable Functional Calculus
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Polynomials and Polynom ial Convexity
Lt t P ( C " ) be the algebra, under pointwise operations, o f polynomial functions with domain C " and range in C. O f course, V ( C n) is generated as a unital algebra by its subset {z i , z2, ■. ■, z „ }. Let p S V { C n) be expressed by Qj"*J'a....■*" ^
P =
z" " '
0<jl+j3 + - -+jn
If a = (o j, 02 , . . . , an) is any ordered n-tuple o f commuting elements in any algebra A , we define p( a ) by P (a) =
X !
....>» ai ’ a23 • •' <
■
(4)
0<ji +jaH---
For ai y n € N, this clearly defines a homomorphism o f V ( C n) into A . Fur thers ore if q = (? i, 9 2 , • • •, 9 m ) is an ordered m-tuple o f polynomials from 7>(Cn ' and p belongs to P(
Hoc
V p 6 P (€ " ).
A set K C C ” is said to be polynomially convex if it satisfies K = pc ( K) . A compact polynomial polyhedron in C " is a set of the form {A e C B:|Aj| < M, \pk( A)| < M for j = l , 2 , . . . , n and k = 1 , 2 , . . . , m } where M is a constant and each Pk belongs to V ( C n). Fcr any compact K C Cn, we can write Pc ( K ) =
f|
{A e
cn:|p(A)| <
II p\K Hoc}-
p € P (C ")
Hence, pc ( K ) is again compact. It is also immediate that a compact poly nomial polyhedron is polynomially convex. We now show that the converse is air- ost true in a certain sense.
382
3.5.3
3: Com m utative A lg ebras and Functional Calculus
Lem m a
3.5.2
If K is a compact polynomiaily convex set in C " and
U C C n is a neighborhood o f K , then there is so m e com pact polynom ial polyhedron included in V and including K
in its interior.
P r o o f Choose M sufficiently large mj that the compact set K is included in the interior of the closed unit ball B of radius M centered at the origin. For each (, 6 B \ U , the polynomial convexity o f K C U allows us to choose a polynomial p w h ic h will satisfy |p“’ (C)l > M > Up*-||oo- For each £ G B \ U , let be the open set { A G C ": |p^(A)j > A/}. Then { U^: £ 6 B \ U } is an open cover for the compact set B \ U. Let U ‘-1, C/4*2, . . . , [/‘•m be a finite subcover. Then {A G C n:|Aj| < A/; |p‘**(A)| < A/;j = 1 , 2 ,...,n; k — 1 , 2 , . . . , m } satisfies the conclusion of this lemma. □
In addition to polynomial polyhcdra, certain other types of compact subsets of C " are known to be polynomiaily convex (Eva Kallin [1965]). Any compact convex set is polynomiaily convex. Hence, the polynomiaily convex hull of a compact set is included in the convex hull. Surprisingly, the union of any two disjoint compact convex sets is polynomiaily convex, but the union of three disjoint closed polydiscs (i.e., sets of the form { A G C ": |Aj - Cj| < r j , j - 1 ,2 ,... ,n } where C € C " and r = ( r i , r 2, ... , r „ ) is an ordered n-tuple of positive numbers called the polyradius o f the polydisc) is not always polynomiaily convex. The union of up to three disjoint balls {A € C " | A , - < i l a ) 1 / 2 < r } is polynomiaily convex but the situation for four balls is unknown. For n > 1, polynomial convexity is not very well understood. However, for n = 1 it is not hard to show that a compact set K C C is polynomiaily convex if and only if C \ K is connected. A t this point we will state an important approximation theorem due to Oka [1936]. 3.5.3 T h e o r e m
Let K C C " be a compact polynomiaily convex set. Any
function analytic in a neighborhood of K can be uniformly approximated on K by polynomials.
The proof of this theorem will be deferred. It will be proved in the course of establishing the existence of the multivariable functional calculus just as Runge’s approximation theorem, Theorem 3.3.3, was proved in the course o f establishing the one-variable functional calculus. The special case of Theorem 3.5.3 for n = 1 is just the last sentence of Theorem 3.3.3. The Joint Spectrum
We now introduce a notation for the linear space o f ordered n-tuples of elements of an algebra and define the joint spectrum. 3.5.4 D e fin itio n For any algebra A , let be the set of ordered n-tuples o f elements in A . We will consider A {rl} as a vector space un der pointwise operations, but never as an algebra (unless n = 1 in which case we identify and A ). For any unital algebra A and for any a =
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(o i, <J2 , . . . , a „) € -4
Lem m a
element a of =
Let A be a unital, commutative, spectral algebra.
Any
satisfies: (A t C
.
{Aj
, A 2 ~v ti-2j • • * > An ~~ On }
is included in some proper ideal 1 of .4} n
• =
{A G C ":
no element b in
satisfies 1 = ^ ( A j — a,j)bj} i=1
=
{ 7 (a ) : 7 G T ^ }.
Proof The first two sets obviously equal Sp( a) . I f A belongs to the second set, then Theorem 2.4.6 gives a maximal modular ideal A which includes {A i — 0 1 , A2 — a2, ■.., A„ — a „ }. Proposition 3.1.3 and Lemma 3.1.4 then show that there is a homomorphism 7 € satisfying 7 (a) = A. Thus, the s'jcoiil set is included in Sp( a) . The opposite inclusion is established by choosing 1 = ker(7 ) where 7 G r * satisfies A — 7 (a). □
The first statement in the following result is a generalization of a special case of Corollary 2.5.8. The rest o f this result is a partial generalization of Theorem 3.1.15 on singly generated unital spectral normed algebras. Let ( A, || • ||) be a spectral normed algebra. Let a t , a3, . . . , an belong to J.. ~f A is the smallest unital closed subalgebra o f itself which includes {ay, a.2, . . . , a „ }, then this set is said to generate A . If such a finite set of generators can be found, then A is said to be finitely generated. 3.5.6 P r o p o s itio n Let (A , ||• \\) be a unital commutative spectral normed algebra. Let a = ( 0 1 , a2, ■■■, an) belong to and let C be the unital closed subalgebra of A generated by { 0 1 , 02 , . . . , a „ } . Then a satisfies SP a ( « ) ^ Spc( a) C p c(S p ^ (o )).
(5)
7 A itself is generated by { 0 1 , a2, .. ■, a „ }, then the map d :T ^ —♦ C n defined by a( 7 ) = ( 7 ( 0 ! ) , 7 ( 02 ) , . . . , 7 (o « )) is a homeomorphism onto its range SpA ( a) which is polynomiaily convex.
384
3:
3 .5.6
Com m utative A lgebras and Functional Calculus
P roof T h e following trivial observation proves the first inclusion S p A (a )
=
{7 (o) :7 e
C
{ 7 (a ): 7 f I ' c ) = Spc ( a) .
= {(7 lc)(o) :7 e
T h e second inclusion is established by the following computation where A G S p c ( a ) and p e V ( C n ) are arbitrary: |p(A)|
<
m a x {| p (7 (o ))| :7 G Tc } = m a x {| 7 (p (a ))| :7 G Tc } = p c ( p ( « ) )
=
f ' A ( p ( “ ) ) = m ax{|7 (p (a))|: 7 G F .* } = m ax{|p (7 (o))|: 7 G
=
llPU’p^fa) Hoc-
Suppose
.4
is generated by { a 1 ,0 2 ,... , a n }.
A n y element 7 G I.4 is
determined by its values on these elements so that a is one-to-one. Thus, a is a homeomorphism since it is a continuous m ap from a compact space to a H ausdorff space. A ny polynom ial p G V ( C N ) and A0 G p c S p (a ) satisfy |p(Ao)| < m ax{|p(A)| : A G
5 p (a )}
= m ax{|p(7)| : 7 G F ^ } < ||p(a)||.
Hence, just as in the proof o f Theorem
3 . 1. 15,
we can define a hom om or
phism
□
The Algebra P ( K ) T h e algebra P ( K ) o i all uniform limits of polynom ials on a compact sub set K of C " , with pointwise algebraic operations and the supremum norm, is an exam ple o f a finitely generated unital com m utative Banach algebra. This algebra is generated as a unital Banach algebra by the components o f the coordinate function z = (21, 22, . . . , zn). Thus, the last proposition shows that the m ap 2 is a homeomorphism o f r p (h ') onto the polynom i aily convex set S p( z ) . S p ( z ) includes p c ( K ).
Each point A G K clearly belongs to S p ( z ) .
Hence
However, any A not contained in p c( K ) satisfies
|p(A)| > ||p|jf||oo for some polynom ial p. T h e right side o f this inequality is simply the norm o f p considered as an element o f P ( K ) . Sp ( z ) , this contradicts Theorem js homeomorphic to
3 . 1.2 (d ).
If A belongs to
Thus, S p ( z ) equals p c( K ) which
under the m ap A 1 —►7* where 7 a ( / ) = / (A ).
N ow suppose K is already polynom iaily convex. For any polynom ial p, considered as an element of P ( K ), we have P = p( z) = p o z
?.5.7
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since ( ') is a homomorphism. Since ||/|| = ||/"||" holds for any / € P ( K ) and r.ny n 6 N, the Gelfand map ( ') is an isometry and hence injective. Thus, the density of the polynomials in P ( K ) proves / = /
o
z for all / S P ( K ) .
We summarize these results and a few additional comments in a proposition. 3.5.7
P r o p o s itio n
Let K be a compact subset of C " and let z be the
coordinate function on C n considered as an element in P ( K ) ^ nK Then z is a } omeomorphism of T P ( Kj onto p c ( K ) . If K is polynomially convex, z satisfies
f = fo z
V feP(K ).
f = fo (z)-
Furthermore, a unital Banach algebra A
is isometrically isomorphic to
P ( K ) for some polynomially convex set K if and only if
A
is finitely gen
erated and satisfies ||a||2 = ||a2||for all a £ A . Proof We have just proved the first two statements o f this proposition. Clearly any Banach algebra which is isometrically isomorphic to P ( K ) sat isfies ihe stated conditions. Suppose A is a finitely generated unital Banach algebra and satisfies ||o||2 = ||a2|| for all a e A Then A is commutative by Proposition 3.1.8 and satisfies ||a)Joo = p (o ) — Jla ll° ° = llal! for aJJ a 6 A . Hence, the Gelfand map is an isometric isomorphism of A onto A Let A be generated by {o i, a2, . . . , a „ } and let a S be the element a = ; 0 ] , 02 , . . . , a n). Then Proposition 3.5.6 shows that a is a homeoinorp iism of onto the polynomially convex set Sp( a) . Hence, the map
C c(r^).
c —* c o (a*- )
c e .4
is an isometric isomorphism of A onto P ( S p ( a ) ) .
□
The Arens-Calderon Theorem
We need a little more technical notation. For any unital commutative Banach algebra .4, let A ^ ' 1 be the disjoint union of for n £ N. We regard each .4
ana lot
A ^
^"m,n (**)
—* A^m~n^ be defined by
^"m,n (®1 ’ **2 i • • • , Am)
(**n+ 1 j Qn+2 * ■• ■* Om )•
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3: Commutative Algebras and Functional Calculus
3.5.8
We regard C " as embedded in *4 (n> so that for any A € C m, we have 1*
<^2> • - • t *^m) = ' ^2i • * * » ^ m )
*^2i • • • j An)»
and
(-^n + 1) ^ n + 2 i • • - t ^ m )*
The following proposition is due to Arens and Calderon [1955]. It will play a key role in our proof that the multivariable functional calculus has the desired properties. 3.5.8 A re n s—C alderon Theorem Let A be a unital commutative Banach algebra. Any m, n € N (satisfying m > n) , a G . 4^") and b G satisfy n,nt„Sp( a, b) = Sp( a) ,
and
Sp( a) C 7rmi„ p c ( 5 p (o , 6)) C pc(S p(a)).
also
(6 ) (7)
Furthermore if n G N ,a G -4*n) and a neighborhood U of Sp( a) in C " are given, it is always possible to find m G N and b G _4<m n) satisfying 7rm,np c(S p (a , 6)) C U.
(8)
Proof The first claim follows from the observation Sp{a)
= { 7 ( ° ) ; 7 € r ^ } = wmi„ {7 ( a , 6 ) : 7 G T ^ } = 7rmi„S p (a , 6)
where n, m, a and b satisfy the hypotheses. From this result we can conclude ^"m,n( Sp( a, b) ) C ■Km npv( Sp( a, b)). Now for any compact set K C Cm, we have Sp( a) —
^m.np c(A -)
=
7rm,n{ A G C m:|p(A)|<||p|K | | o o fo r a llp G P (C m) }
C
7rm,n{ A G C m:|p(A)| < I H a -IU
for all p = q o 7rmT1 for q G ^ ( C " ) } =
p c( 7rm n(A ')).
Consideration of the case K — Sp(a, b) gives TTm.nPC( Sp( a, b) ) C pc( nmrlSp( a, b) ) = pc(S p(a)).
We now prove the last assertion of the proposition. Let n G N, a 6 .4*"^ and U € M a be arbitrary. By (7) we need only choose b to exclude each C G pc ( Sp( a) ) \ U from 7rm „ p c ( Sp( a, b) ) . Suppose such a C is given. Since £ does not belong to U, it does not belong to Sp( a) . Hence, Lemma 3.5.5 shows that there is an element 6 G -4*n^ satisfying n
£ (0 - a ^ J= 1
= l.
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387
Kence, ( does not belong to the spectrum of a in the unital subalgebra C geneiated by { a i , a2, ■■., a„, b\, b%,. . . , 6£ }. This, together with the first inc;r£:cn of (5) and ( 6 ), implies ( $ * 2n,nSpc(a, &*■)• However, the second paragraph of Proposition 3.5.6 shows Spc(a, b( ) = pc (Sp^(o, t f ) ) so ( is not contained in n2n,nV>c(Spjl(a, &<•)). This set is closed since 7T2n,n is continuous and pc;Sp^(a, tfi)) is compact. Thus the complement £/*• of this set is a neighborhood o f ( . The family { U ( :£ € p c(S p (a )) \ U } is an open cover for the compact set pc.(Sp(a)) \ U. Choose a finite subcover { U ^ : j = 1,2,./., p}. We will now show that m = n (p + l) and b = ( b*•', b^2, . . . , S j X m -’-.j satisfy ( 8 ). Any £ e pc ( Sp( a) ) \ U belongs to some U ik. Thus, C does not be long to 7r2„,„p c ( Sp( a,b^k) ) . Let C* be the unital closed subalgebra o f A generated by { « i , a2, ■■■, , bj*, . . . , } and let C be the unital closed •subalgebra of .4 generated by {a i, a2, . . . , an, blt b2, . . . , &m_ „ } . Then, £ does not belong to Tr2n,nSpck(
We generalize the notation introduced in Section 3.3. For each a G let T a be the set {/ : / is analytic on its domain D f which belongs to JVa}. Note that if a 6 *4^°°* belongs to A ^ , then S p(a ) is a compact subset of C ". Define a relation ~ on T a by: / ~ g
there exists a U e M a satisfying f\u — g\u-
This is an equivalence relation and the set of equivalence classes is denoted by T a- As before, we do not distinguish in notation between / S T a and a representative o f / in T a. It is again easy to check that T a is a j algebra under the operations inherited from the following operations in T a\ f+9
fg
=
f\D,nD„ + 9\Df nDg
=
(/iD/noJlfflo/nD,)
where the right hand side has the customary (pointwise) meaning. Findly, we make T a into a topological algebra by defining f a —►/ to ’near1 that there is a neighborhood U of Sp( a) such that for some ao and szr' , '-opresentatives / and f a in T a ' t
U
c
D,
n
f l
D fa
a>ao
'an '. 'd f a converges to / pointwise on U and uniformly on any compact
388
3: Com m utative A lg ebras and Functional Calculus
3.5.9
The following theorem is essentially due to Waelbroeck [1954]. A theory with dome o f tho same featured wan developed by Silov [1953] and Arens and Calderon [1955]. However, their construction was not shown to define a homomorphism except in the case of semisimple Banach algebras. The proof that we give is based upon Arens [1961], Nicolas Bourbaki [1967], and Gamelin [1969].
Note that we have not generalized the first two properties o f (d) in The orem 3.3.7. Moreover, the uniqueness proved here is definitely a property of the family { I a:a £ >1°°}, whereas in Theorem 3.3.7 each separate map I a was proved unique. Zame [1979] has shown that the individual m are unique in this multivariable functional calculus, but the proof is not included here since it requires substantial preparation. If we restrict atten tion to the functional calculus in a semisimple commutative Banach algebra -4, any homomorphism from T a into A composed with an element of is necessarily continuous, so the closed graph theorem shows that the orig inal homomorphism was continuous. Zame [1979] introduces a transfinite construction to extend this uniqueness result somewhat. Angel Larotonda and Ignacio Zalduendo [1984] give an easier construction. See also Sc’nanuel and Zame [1982] for a functorial interpretation. 3.5.9 T h e o r e m
For any unital commutative Banach algebra A , there ia a unique family { I a:a € «4*oc*} of continuous linear maps
(a) I a( p) = p( a )
v n e N; a £ A
Furthermore, for each n £ N, a £ satisfies:
and f
£ T a, this family also
(c ) I a is a continuous homomorphism. (d) I f C is another unital commutative Banach algebra, then any con tinuous unital homomorphism tp: A —* C satisfies * (/ «(/ )) = ('-•) /<,(/) = / oo. (f) Sp ( I a( f ) ) = f ( Sp( a) ) . (g) For all h £ (cf. Definition 3.5.4,) and g e to T u and natinfien Ia( g o ft) = //„(/,)(
9 ° h belongs
(h ) If a representative of f can be chosen in T a so that D j includes a polynomially convex compact neighborhood K of Sp( a) , then I a( f ) belongs to the unital closed subalgebra generated by { a i , 02 , ... ,a „ }- If, in addition,
3.5A’
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389
I< ca .tains the origin 0 in C " and f ( 0 ) = 0 , then I a( f ) belongs to the unital close: subalgebra generated by { o i , 0 2 . , a„} . 0 , Each m G N , b G and g G J~b satisfy
V , 6)(h) = Ia {f)h (g ) where h ( \ i , X2, . ■., An^.m J — f ( A i , A2, . . . , Xn) g ( \ n+ 1, An+ 2t . . . , An+ m).
(j) Each b G A and g G
satisfy
h(g) = 9(b)P ro of For this proof we fix n G N, a Banach algebra A satisfying the hypotheses, and a G _4'n'. For any neighborhood U o f Sp( a) , we make the proof depend on the existence o f a continuous (actually C ° ° ) .4-valued function v with compact support in U , which is independent of U and of the (many) choices made in its construction in the following weak sense: I f U is another neighborhood of Sp( a) with U C U and if v is constructed from U by the same procedure (which includes various choices) by which v was constructed from U, then
j
J
fvd V =
C"
fvd V
(9)
C"
holds m t all functions / analytic on U where the product f v is the pointwise scalar product and d V represents 2n-dimensional Lebesgue measure. We then cefine I a: T a —♦ A by / .(/ ) = / f v d V
( 10 )
C"
where we have chosen a representative / G T a of / G T a and constructed v from D f . This is well-defined (as a function o f / G T a) by (9). It obviously makes /0 a continuous linear map. The construction o f v shows that /„ satisfies (e), (i), and ( j ). The construction o f v and the verification o f (e ), ( i ) , and (j) depend on Stokes’ theorem for exterior differential forms on Cn and lie entirely outside the realm of Banach algebra theory.We postpone these steps. Readers who wish to avoid them may simply assume the existence of a function v with the above properties. We shall now prove Theorem 3.5.9 startirg from the following axiom to be verified later. A x io m . For each a 6 there is at least one continuous linear map la-fo. —* A which satisfies properties (e), (i) and (j). N , e first that, by definition of the joint spectrum, (f) is an immediate consequence o f (e). Furthermore, (j) implies /Qj( l ) = 1 for j = 1 ,2 ,... ,n.
390
3: Commutative Algebras and Functional Calculus
3.5.9
This result together with repeated application of (i) implies /0 (1) = 1. From this, we can conclude (b ) as the special case of (i) when the function g is 1. Condition (j) also implies I aj { z) = aj for j = 1,2,... ,n where z is the coordinate function on C. This result, our previous results, and (i) imply Ia(zj ) = aj for j = 1,2,..., n where z = ( zi, z2, .. . , z„) is the coordinate function on Cn Now, this result together with (i) implies Ia( p ) = p ( a )
(1 1 )
where p is any monomial. The linearity of I a now gives (11) for any poly nomial p € V ( C " ). We have proved (a). We have now proved or assumed that {/ Q:a € A
onto K.
Thus, property (e) shows I z( f ) = f ° z. Proposition 3.5.7 now
shows that / = I z( f ) o ( z " ) belongs to P { K ) . This completes the proof of Theorem 3.5.3. U Proof of Theorem 3.5.9, continued. We now prove the uniqueness asser tion. Suppose { I a.a € A <00>} is smother family of continuous linear maps I a.Ta A , satisfying (a) and (b). Let / € T a be arbitrary but fixed.
Choose a representative of / in T a. By Theorem 3.5.8, we can choose satisfying
b € A
{'m~n'1
Sp( a) C 7rm,np c( 5 p (a , 6)) C D f .
By Lemma 3.5.2 we can find a polynomial convex set K in C m satisfying pc(Sp(a, b)) C K ° and nm,n( K ) C Df . By Theorem 3.5.3, we can find a sequence { Pk}ket « ° f polynomials converging to J o n,njl uniformly on K . Continuity and (b) imply i ai f )
h{f °
=
/ * m.„ (a ,6 )(/ ) =
=
/ .(/ o ir m,„) = /„(/).
TTm.n) =
Hm
/a(Pfc)
=
lim
I a( pk)
r .-'.f
M ultivariable Functional C alculus
391
Hence, {/ 0: a 6 -4^°°^} is unique, subject to the restriction that each /„: T a —► j- :r s continuous linear map satisfying (a ) and (b).
The family of maps {tp o I a:a € -4^°°^} satisfies the properties of {/^(oj: a 6 ,A(oo)} except that each tpo Ia:^Fa —» A
is defined on the algebra which is possibly smaller than the algebra on which /^(o) is defined. Since T a includes the polynomials, the above uniqueness argument proves (d). (c): O f course, only the multiplicative property of each I a remains to be pro^eci. Let f , g e T a be given. Choose representatives o f / and g in .F0. By Theorem 3.5.8, choose b 6 satisfying nm^ p c (S p (a , b )) C D f n D g. Use Lemma 3.5.2 to choose a polynomially convex set K in C m satisfying pc{5 p (a,6 )) C K ° and nm,n( K ) C D f n D g. Now, Theorem 3.5.3 shows that v/e can choose sequences {p/t}/teN and {
= ~
,
=
s ) — J(a,b)(f8 0 I(a,b)({f 0 ^m,n){9 0
v
lim
'
k —*oo
=
lim
fc—»OO
I(a,b)(PkQk)
/(a,6)(Pfc) Jim l ( a, b) ( Qk) = l ( a , b ) { f ° k —*oo
(fif 0 ^ m . n )
This proves (c). (h): The first sentence is sin immediate consequence of Oka’s approxi mation theorem and the continuity of /0. The second sentence follows since its hypotheses guarantee that the approximating polynomials can be chosen with constant terms equal to zero. i (g): Let h 6 and g e F i a(h) be given. For the present argument we will use 7r and 7r' to denote the maps 7rn+m,n and 7r^+m „. Let k: C n+m —►C be defined by ‘ i * , ,’ k ^ g o n o n —g o n , i. e ., /v{Xi, A 2 , . . . 1 An+ m ) = g ( h ( A i , A 2 , . * ■, An )) — 5(An+i> An +2, • *. >An + m ) .
Then k belongs to f ( a,ia(h)) and satisfies (12)
k o ( z , h o n ) = g o h o w —g o h o n = 0
where z is the coordinate function on C ". k : C n+m —> C be defined by .'xj
—
k O (zi, Z2 , . . . ,
—
g o h o i r — g o ( zn+i, zn+ 2> • ■• 1 zn+jt
For j
=
0,1,2, . .. ,m , let
hj + l O 7T, /lj-f-2 O 7T, . . . , hm O 7r) 0
^ j + 2 ° tt, . . . , hm o 7r).
392
3: Com m utative A lgebras and Functional Calculus
3.5.9
Note ko = k o (z, h o n) — 0 and km = k. Thus, we can write
k=
- V O
= £ ( z" + j - hi 07r) (\ z ~ — > n+j-h jo n J
where each fraction is actually analytic in a neighborhood of Sp(a, I a( h) ) since it can be written as (Zn+j — /lj O 7r) °
^j + l ° 7T, ^ j + 2 ° 7T, . • • , hm O 7r)
[—5 ° ( 2'n4 - l i 2'n+ 2 )--->
( ^ n + 1 » ^ n + 2 »
■ * • »
— 1 j
°
? r, ^ j +
1
°
7T, • • • ,
°
^ ) ]
in which the two terms in the brackets differ only in the n + j position with Zn+j in one term and hj o n in the other. We already know that /(„,/„(/>))
is a homomorphism and that it satisfies A “ ./«(h))(Zn+> —
° w)
=
I(aj a(h))(zn+j) — I(a,I„(h)){hj ° 7r)
=
I a( hj ) - U h j ) = o
where we have used (a) on the first term and (b ) on the second. Hence, we conclude 0 = / («,/ „((.»(*) = I ( aj a( h) ) ( g ° h o tt) - /(o,/„(/,)) (s 0 * ' ) = Ia(9 ° h) — I i j h ) { g ) , where on the first term we have used (b) and on the second term an analogue of (b ), which can be proved from (i) just as (b ) was. This concludes the proof of (g), and hence of the whole theorem, except for the construction o f the function v satisfying ( 10 ) and the verification o f the properties (e), (i), and (j). □ Construction of v: Exterior Differential Calculus We turn now to the construction of the function v associated with a neighborhood U o f S p (a ) in C " which will allow us to establish the Axiom on which the proof of Theorem 3.5.9 has depended up to now. The next 13 pages are essentially an elementary appendix and can be skipped with little loss o f continuity. We begin with a brief review of the differential and integral calculus o f C °° exterior differential forms on an open subset V of Cn As a source for further information on this subject at about the level' needed here, we mention the elementary book by Nickerson, Spencer, and Steenrod [1959]. Let C ° ° ( V ) be the set of all complex-valued functions on the open subset V o f C " which have continuous partial derivatives of all orders with respect to the 2n real coordinates in V . The identity function on V will be denoted by z = ( z i , z 2, ... ,zn), where z( A) = A for all A = ( A j, A2, . . . , A „) 6 V C C". (Thus for j = 1 ,2 ,... ,n, Zj-.Cn —* C satisfies z^(A) = Xj for all A 6 V . ) The partial derivatives with respect to the real and imaginary parts of z} will be denoted by g— and . respectively.
3.5.9
M ultivariable Functional Calculus
393
A derivation of C °°(K ) is a linear map D : C ° ° ( V ) —►C ° ° ( V ) which satisfies D(fg) = D(f)g + fD(g) V /, g e C ° ° { V ). The set o f all derivations on C ° ° ( V ) will be denoted by A (K ). Pointwise oper?' ;ons on A ( K ) make it into a linear space. For / G C ° ° ( V ) and D e £ ( V ) , we define f D by (fD)(g) = f(D(g))
VgeC°°(V).
This makes A ( K ) into a C,00(K)-module. It can be shown that the set - j — 1,2, ...,r a } is a free module basis for A ( V ) over C ° ° (V ). However, we will usually use another module basis. For j = l , 2 , . . . , n , define elements gf- and of A ( V ) by gf- = 5 ( g f- + 7 gpj) and ^ = 5 ( 3“ - 7 gpj). I f / is analytic on V , an easy calculation shows is the ordinary complex derivative with respect to the jth-coordinate. Furthermcre, the equations = 0 for j = 1 , 2 , . . . , n are equivalent to the
Cauchy-Riemann equations. Hence, these equations hold if and only if / is analytic on V by Hartog’s theorem (which asserts that a function on C " is analytic if and only if it is analytic in each variable separately, cf. Hormander, [1966]). From previous remarks
= !> 2» • - - » « } is
clearly a free module basis for A ( K ) over C ° ° { V ). Thr- space T>l ( V ) of differential I-form s on V is by definition the module dual o f A (K ). That is, an element a o f 2? 1 (V ) is a linear function a: A ( V ) —» C ° ° ( V ) satisfying, a(fD) = fa(D)
V / 6 C ° ° (V ); D e A (V ).
For any / 6 C ° ° { V ) , the map D ^
D(f)
V D e A (V )
obviously belongs to 2?1 (K ). It is denoted by df. Hence the set { dzj , dzj - . j = 1 .2 .... ,n } is a free module basis for T>l ( V ) which is dual to the module basis { jjfr'.j = 1 ,2 ,... , n } o f A ( V ) (i.e., these bases satisfy d z j ( g=-) = ) = = * ^ > (^ ) = for all j, fee {1 ,2 ,... ,n }). Some times i; will also be convenient to use the module basis {d x3, dy3 \j = 1 .2 ...., n } for T>l ( V ) . These basis vectors are related by the usual equa tions: dzj = d i j + idyj,dzj — dxj — idyj. Hence, this second module basis is dual to the basis { g f r , g ^-:.7 = 1,2, . . . , n } for A ( V ) . In terms of ihes ’ bases the differential df of any / 6 C ° ° ( V ) can be written as
The next step in our review is quite abstract. For q > 1, an elementary • • • A <7 , where / is a C ° °
q-form is an expression o f the form f<J\ A a? A
394
3: Com m utative A lgebras and Functional C alculus
3.5.9
function on V and each a3 is a 1-form on V . For q > 1, a q-form on V is a finite sum o f elementary <j-forms. Tw o <j-forms are identical if one can be transferred into the other by a finite sequence of identities of the following two types: So 1 A <72 A • ■• A o} - \ A [ gaj + /icr') A <jj+ 1 A ■• • A aq
= S9a i A <72 A • • ■A Oj -1 A Oj A 0j +i A • • • A aq + f ha\ A <72 A ■• • A
/<7i A <72 A • • • A a} A <7J+i A • • ■A <7, = —fo\ A <72 A • • • A <7^+1 A <7j A • • • A <7, where /, g and h belong to C ° ° ( V ) and <7 ' and each <7*. belong to 2 ?'(K ). Thus, we require o\ A <72 A • ■• A aq to be linear over C ° ° (V ) in each factor separately and to be anti-commutative in each pair o f adjacent positions. In particular, the 9 -form <7] A <72 A • • • A <7, is zero if equals <7*. for any distinct integers j and k. Let Z>‘, (V') be the C ° ° ( V ) module of (/-forms on V . Since X>'(K) had a free module basis with 2n elements, and since an elementary q-form <j\ A <72 A • • ■ A <7, is zero unless {
V <7 € V q( V) ' , p € D P( V )
also follows. Clearly, the notation A for the exterior product is compatible with our previous notation for 9 -forms. It is convenient to denote the regular C °°(V )-m odu le C ° ° ( V ) by T )° (V ) and to call the functions which belong to it 0-forms. The exterior derivative d is a collection of linear maps d- . Vt{ V ) - * V , + l ( V )
q = 0 ,1 ,2 ,...
defined as follows. For each / 6 V ^ ( V ) = C ° ° ( V ) t df has the meaning already assigned as an element of D '(V '). For the elementary q-form <7 = fd a t A ■• • A daq,
*
M ultivariable Functional Calculus
395
da has the meaning
da = df A dai A • • • A daq.
The definition for other g-forms is obtained by the postulated linearity It is sasy to check that for any q-form a and any p-form p we get d(a A p ) = (d a ) A p + ( —l ) qa A dp.
Another important property of exterior differentiation is expressed by
These are all the results we need on the exterior differential calculus of forms. Note that all the results are o f a computational nature. The integral calculus of forms gives a little more substance to the theory. The main result is Stokes’ theorem which asserts that any “ ((/+ l)-dimensional” , compact subset K o f C " with suitably smooth “(/-dimensional” boundary d K and any (/-form a satisfy (13)
To make sense of this formula, we must define what a “(/-dimensional” , compact subset o f C " is and define the integral of a (/-form over such a set. In general, it is not necessary to require that a “(/-dimensional” , compact subset actually be a ^-dimensional submanifold o f C ", but merely that it be an image under a (possibly highly singular) C°°-mapping o f some piecewise smoothly bounded compact subset o f R 9 which is the closure o f its interior. This can be systematically expressed in terms o f C°°-singular simplices and chains. (This leads to deRham’s theorem, which identifies the cohomology groups based on the exterior derivative as the duals (under integration) o f t’/:3 l ingular homology groups.) However, we will only need to apply StoVoV theorem to (2 n - l)-forms on the boundary o f a 2n-ball B centered u i Lbs c -igin, so we cam use simpler definitions. An elementary 2n-form on V ctvi >e written as a = fd x i A dyi A dx2 A dyi A ■• • A dxn A dyn for a suitable C°°-function /. (Note that the order is important in order to get tbo eigr. correct.) Then f B a is defined to be the ordinary 2n-fold integral Jg f dxidyidxidy2 • • ■dxndyn. An easy calculation shows dzi A :’zi A dzi A dz2 A • • • A dzn A dzn = (2 i)ndxi A dyx A dx 2 A dj/2 A • • ■A dxn A dyn.
A {2n — l)-form can be written as
f j Oj where each a} is o f the form
dx 1 a c p i A dx 2 A dj/2 A • • • A dx„ A dy„ with the jth factor removed. In order
396
3: Commutative Algebras and Functional Calculus
3.5.9
to define the integral of this form over the boundary dB of B, we need a suitable C °° parameterization of aB, say F: R —> o B where R is a compact rectangular parallelepiped in R 2" -1 . Then we define f t
f
r
.
OB
Jt
^ (* 1 .1 / 1 . •••.*n .l/ n )
/ f } ai = / f } o F ~x77— i
j*
7---- r - dt i dt 2- - dt 2n - i ,
R
where in the numerator o f the Jacobian the jth component is omitted. The orientation of the parameterization is important in general, and we could describe this by giving an explicit parameterization (in terms of trigono metric functions). However, since all the boundary integrals we need will be zero, we omit this. We have described a standard form of the calculus of forms. However, in fact, the functions and forms we use will mostly have values in A or A ^ rather than in C. The partial derivatives of an A - or _4(n)-valued function are defined in the usual way. We use C x‘(C n, A ) and C ° ° (C n, .4-n)) to denote the C’ 'x'(C ",C )-inodu le o f C °° functions on C " with values in A and in A ln\ respectively. There is no difficulty in extending the rest o f the theory. The reader who wishes to check some particular point will find that evaluation of elements o f A * or ( A " ) ) * on both sides of an equation usually reduces the problem to the classical complex-valued setting. Construction of v in the One-dimensional Case Before constructing the function v in the n-dimensional case, we will show how the one-variable functional calculus could have been defined in terms o f differential forms. This result will be used to check property (j) in the theorem. Recall that for any a 6 A and / € T a we defined / „(/ ) by
/o (/)= 2
h ri
where F was sin admissible contour. We would now drop the dummy vari able A and write this as W
) = h
j [ z ~ a )~ lfd z r
where z is the coordinate function on C. Let U be the open neighborhood of Sp(a) defined by U = (A e C :T has winding number +1 with respect to A }, and let K be the compact set K = U = U U T. Choose a function w in C °°(C ) satisfying w = 1 in a neighborhood of Sp(a) and having compact support in V . Define a function u 6 6’00(C, A ) by _ W( A) ) ( A _ a ) ” ‘ U(A) = { 0
x $ s P (a) A 6 5p(a).
3.5. V)
Multivariable Functional Calculus
397
Stokes’ theorem and the other properties of forms give
=
b
j
jK d W
* dz = b , j l ^ w
- d i r ' dz
j 1{\%)dv
= K
C
(Here we have used the fact that the integral f r ■dz corresponds to the boundary integral in Stokes’ theorem.) This shows that we could have used the lest expression to define /<»(/)■ An Avvihary Function u YV( now wish to generalize this construction to elements a € and functions of n -variables which are analytic in a neighborhood of Sp(a). To do this, we note that we could have started from the penultimate line of the last equation to get
hu) = h ! j }dufKdz=h ! fduAdz K
B
where B is a closed ball centered at the origin and satisfying K C B. To handle the general situation, we start out with a technical lemma about the construction o f an auxiliary function u € C co(C n, A ^ ) and a form based upon this function. 3 .5 ,i( L e m m a Let n 6 N, a 6 and U 6 M a be given. Then there are: (a) a C °° -function w: C " —►R with compact support in U such that w — 1 holds in a neighborhood of Sp(a), and (b ) a C ° ° -function u: C " —*
satisfying
n Y l ^ Z j-a j) = l-w.
(14)
.7 = 1
When w and u satisfy these conditions, the 2n-form fi = du\ A dz\ A du2 A dz2 A • • A du„ A dzn has compact support in U . Moreover, if U € M a with U C U is given and w, u, p. are any elements satisfying the same conditions relative to each other and to U as w, u and n satisfy relative to each other and to U , then there is an (n — 1)-form r supported on U such that H — ji = d r A dzi A dz 2 A • • • A dzn .
398
3: Commutative Algebras and Functional Calculus
3.5.10
(For the cognescenti we remark that we are claiming that p. is uniquely determined as an element in the Dolbeault cohomology group H ~'n o f (the germ of neighborhoods o f) Sp(a) with coefficients in >4^"*.) Proof The existence of a function with the properties of w is well-known. Choose such a function. We turn to the construction of the functions Uj for j = 1,2 ,... , n. For each £ € C " which does not belong to Sp(a), choose 6“* 6 A ^ satisfying n
Y . b% 1
- « , ) = !•
Choose an open neighborhood (/*• of £ on which the continuous affine func tion n -a ,) j=1 has invertible values. For each k = 1 ,2 ,..., n let by n
be the function defined
ui = bU Y l b^ z} ~ a^ " l j - 1
on U "v Then each is analytic on [/**. The family € C " \ S p (a )} is an open cover for the paracompact space C ” \ Sp(a). Therefore, we can choose a locally finite C °° partition of unity subordinate to this cover. Tliat is, we choose a family { h„: a 6 A ) of functions so that each function ha belongs to C oa(C n \S p(a )), has range in [0, l] and has support in U^a for some ( n e C " \ Sp(a). Furthermore, near any point A 6 C " \ Sp(a) there are only finitely many non-zero members of the family {h a:a € >1} and these satisfy YlaeA /‘l« ( A ) = 1. Now choose uk for k = 1 , 2 , . . . , n to be the functions defined by ( «* (• * ) =
£ (1 - w ) ( - M M ^ ) « ! H * )
<
aeA
[
0
if A 6 C ” \ Sp(a) if A € Sp(a)
The sum is a finite sum for any particular A 6 C " \ Sp{a). Clearly, each uk belongs to C °°(C ",> 4). Hence u = (u i, u2, . . . , u „) satisfies the condition specified for u € C ° ° (C n,^4^n)). We check that the form p. behaves as desired. Since J^"= 1 uj ( z: —a>) = * holds on the complement V of supp(tu), exterior differentiation gives n
n -
j= i
a^duj + Y l uJdzj = 0 >=i
(15)
2.5.10
Multivariable Functional Calculus
399
on V. Taking the exterior product with dui A du2 A •■■A duk-\ A duk+i A • • • A du„ A dz\ A dz2 A • • ■A dzn, we get (zk - ak)dui A du2 A • • • A
A dzi A dz2 A • • • A dzn = 0
on V . Multiplying by ufc, summing and applying (14), we find that /z is zero on V . Hence, fi has compact support in U. Now let U C U be smother neighborhood of Sp(a)._ Let w , u and p satisfy the same conditions relative to each other and to U that w, u and fi satisfy relative to each other and to U . In particular, we have n - a>) = 1 i
We want to show that p — n has a special form. To simplify our discus sion,'we will write a for the n-form dzx A dz2 A • ■• A dzn. Then we want to find ar (n — l)-form r supported on U such that p — fi = dr A cr. (The'point is that Stokes’ theorem will then show that the integrals with lespeet to y. and p of an analytic function over a ball which includes U will have tlie same values.) For each k = 1 , 2 , . . . , n, we have n:-.-uk
n n ( ] T ^ i ( z » - aj ) + w ) u h - ( Y uj j z j - a, ) + w ) uk
=
j * n
J ‘ ~ aj ) ( u]V k - U j U k) + w u k - w u k .
= i=i
Defining pj
~ ai )P i * +9fc
* = l,2 ...,n .
(17)
i=' Sim;ia.’ Iy we get n w -w
=
-
- afc)(txfc - ufc) fc=i n
=
- X ](z fc -
fc=i
n
n
ak) ( Y i ( zi - ai)Pj.k + ?fc) = - Y , ( Zk ~ a^ qk j =i
fc=i
400
3: Commutative Algebras and Functional Calculus
3.5.10
since phk — -pk.j for all j, k = 1 , 2 , . . . , n. Now define u 6 C ° ° (C n, .4*")) by n
Wfc = Wfc + ^ ( 2 j - a j p j . f c j= i This function satisfies
k = 1 ,2 ,... ,n.
n
] r ( u j - flj) = i - w j= i since p} k = —Pk,j for all k, j = 1 ,2 ,..., n. Let ft be the 27i-form ft = diii A dz\ A dfi2' A dz-j A • • ■A dun A dz,,.
Obviously, it is enough to show that both ft - ft and ft — n can be written in the form dr A. it for suitable choices o f an (n — l)-form r supported on U . We shall write each of the Ti-forms ft - ft and ft —/.< a.s a sum of n-forms which we will show explicitly can be written as dr A a with r supported in U.
For each k = 0 ,1 ,2 ,..., n, let uk € C ° ° (C ", uk J
f + Qj \ u
be the function
‘f 3 < * if j > k
so that u° = ti and un = u hold where in the last case we have used (17). For each k = 0 ,1 ,2 ,..., n, define p.k by /x* = du* A dz\ A du* A dz -2 A ■• ■A du* A dz,, so that fx° = ft and fxn = ft both hold. Then we get /x* —
= du\ A d z j A
du 2 A d^ 2 A ' ' ' ^ dzk - 1 Adq/t A d z* A - • • ditf\dzn = ( d r ) h a where r is ±<7*d u i A dua A- ■ A d u * _ i A d u * +1 A- • A du„. Thus, /i —/i =
~ M * 1) has the
desired form since qk = urn* — urn* has support in s u p p (tu )U s u p p ((tu )) C t/.
For distinct integers i and A: from {1 ,2 , ..., ti}, let u’ ’* 6 C ° ° {C n, ^ " ^ ) be defined by
{
Uj uk + (z, - a,)p,'k
if if
j^ ij^ k j = k
u, + ( zk - ak) p k,i t
if
j = »•
Then p,tk = ~Pk,■ implies n ^ « j ,lc( 2j ~ a j) = 1 - w. j-l
(18)
3.5.11
Multivariable Functional Calculus
401
Define a 2n-form p,l'k to be p,l,k = du\k A dz\ A d u j * A dz-i A • • •A du'^k A dzn and deine a (2n — 2)-form p to be the form du\ A du2 A • • • A dun with and d's . omitted. We get dukk
=
du* + Pi,*dzi + ( Zi ~ ai )dPi,k
du\’k
=
du{ - Piikdzk - (zk - ak)dpitk
which Tiplies diL^k ' du\'k A dzk A dzi — duk A dux A dzk A dzi -■ ( Zi - ai)dpitk A dui A dzk A dzi - ( zk - ak)d u k A dpi}k A dzk A dz*.
For a s litable choicc of sign, this in turn implies p l'k - p. -
Nolr
± ( ( 2j - at)dpitk A dui - ( zk - ak)d u k A dpi}k) A p A o .
n
n
- d w = ^ ( z , - a j ) d Uj + ^ U j d z j
i =l
j=i
which implies —dp%,k A dw A p A a — dpi
- ((z 4 - a jd p it A dui - (zfc - ak)duic A dpiik) h p h a . Hence, we can write p.l,k - p, as ± d r A
M hsis the form vdV where
v is sorne C°°-function (a Jacobian determinant), and where dV is 2ndimensianal Lebesgue measure. This is the function v mentioned at the beginning of the proof o f Theorem 3.5.9. We assume vdV — (2” ])n M from now on. 3.5.11 L em m a Let U ,w ,u ,p and v be as defined in Lemma 3.5.10 and above. For each f 6 T a, define I a( f ) to be
W >=(2 ^
J
! d “ = J ! '‘dV B
C"
(19)
402
3: Commutative Algebras and Functional Calculus
3.5.11
where f is some representative of f inj~a,fi and v are constructed relative to U = D f, and B is a closed ball centered at the origin which includes D f. Then T a —» A is a well-defined continuous linear map satisfying (e), (i), and (j) o/Theorem 3.5.9. R em a rk . Despite the fact that / may not be defined on all of B or C ", the integrals are unambiguous since /z and v are zero off the domain o f /. Proof We check first that / „(/ ) is well-defined (i.e., it depends only on a € A and / £ !Fa). Suppose / and / are two representatives in f a of / € T a. Without loss of generality, we may assume D j C D j. Let /j and fi be any two 2n-forms constructed as in Lemma 3.5.10 from D j and D j, respectively. Then Lemma 3.5.10 asserts that there is an (n — l)-form r with support in D f such that fj. = p + drAa where a — dz\ Adz? A ■• ■Adzn. For an analytic function /, d (f r A a ) — df A t Aa + fd r A a — fd r A a holds. Hence Stokes’ theorem shows
J B
fdfi —
J B
fdp, =
J
fd r A a =
B
J B
J
d (f r A a) =
fr Aa = 0
OB
since r = 0 in a neighborhood of dB. Thus T a —> A is well-defined by (19). Clearly Ia is a continuous linear map. The remarks given before Lemma 3.5.10 show that I a satisfies (j) (when n is 1 ). (i): Let n, m 6 N, a 6 A^n\ b 6 A^mK f £ T a and g € Tb be given, and let h be the function specified in condition (i). To simplify notation, let ■ k and 7r' represent the maps 7rn+m n and 7rJ,+m n. Then the function h of condition (i) is simply (/ o 7r ) ( g o 7r'). Let w e C ° ° {C n),u £ C ° ° (C n,>4*n^) and /j be constructed as in Lemma 3.5.10 so that they are suitable for defining I a( f ) . Let w 6 C ° ° (C m),u 6 C °°(C m, A ^ ) and p. be constructed as in Lemma 3.5.10 so that they are suitable for defining Ib(g). To simplify notation, from now on we will write /, w, u, g, w and u to represent the functions f o n ,w o n ,u o n ,g o n \ w o n' and u o n ', respectively. Similarly, we use n to represent the 2n-form on C '“+m, which has the same formal expression as ^ on C ", and we use /^to represent the 2m-form p. = dui A dzn+1 A du2 A dzn+2 A ■• ■A dum A
just
(20)
? .5 ."
Multivariable Functional Calculus
403
Also the equations n
m
$ > ,- (* ;- a ,- )l- i«,
and
^
U j ( z n+i - b j ) =
j= i
1 -
w
i=i
imply m
m - aj ) + ^ 2 w ^ j ( zi+ n - b j ) = 1 - w + w ( 1 -
w ) = 1 - ww.
j- 1
i=\
Hence, we can use wu, e C ° ° (C n+m) and (u,wu) 6 C°°(Cn+m, .A<"+m)) to define /(„,(,)(/*) = /(„,6) (/
/(o,i)(/i) =
J f w * d ( w u i ) A d z n + i A d ( w u 2) A d z n + 2 A A d(w^iin) A
However, for each
= 1 ,2 ,..., m, n
d{%DJ.j)
=
Uj dw
+
ivdHj — U j ( — £
n
> fc~
o.k)duk —
k- 1
Ukdzk)
+
wdiij
fc=i
implies /z A d ( w H j ) = w(fj. A d iij).
Thus, we conclude
f .. m
(n + m )!
. -
Stokes’ theorem and (20) will give the desired equality if we can find an (n ->■il)-form r supported on ( D j , D g) and satisfying [ ( n + m)\wm — n\m\\fi = dr A a
(21)
where a is the form a = dz\ A dz2 A ■■• A dzn . We prove (21) by writing the left side as a sum of terms, each o f which is sho’.vn to have the correct form. For j = 1, 2 ,. .., n, let Hj be the (2n —1)fcrm which is equal to /j except that the factor duj is removed. For k = 0 , 1 , 2 , . . . , n, let Tk be the (n — l)-form satisfying n rjt A a = w k ^ T u j H j -
->=1
404
3: Commutative Algebras and Functional Calculus
3.5.11
Since d u } A Hj = ft for j = 1, 2 , . . . , n, we see
n drk A a = kwk~ 1
u} dw A fij + nu)fcfi. >=i
Using the now familiar expansion for —dw and noting again that dujAfij = /j, we get rt dr* A cr
=
~kwk~l
Mj(zj — a3)n + nii)fc/i j= i
=
- u>) + nwfc]/z = [(n + & )«/ - fciyfc_1 ]/j.
Thus, defining r by
E
(n + k - l)\m\
it!
fc= 1 we get
dr A a
=
^
Tfc’
(n + A: - l)!m ! /
fc
------ jtr-------- v " + v ^ / ( n + A)!m! §
1
*!
*
)
fc A
-
(n + f c - l ) ! m ! ------- ( t ^ T j !
fc_ A “
T
[(n + m)\wrn - n!m!]/z. This concludes the proof o f (i). It only remains to show that our definition of /„ satisfies condition (e). This amounts to showing 7 (/ « ( / ) ) = / ( 7 ( « ) )
for each 7 € and / 6 T a. Let w, u and /z be defined as in Lemma 3.5.10 so that they are suitable for defining /0(/)- Let ti 6 C 00(C n, C^n^) be defined by u} = 7 0 Uj, and let /} be defined from u as /x was defined from u. Since 7 is a continuous homomorphism, we get
= (2H i r ! ld i‘ This reduces the problem to a calculation relative to the algebra C. We can apply the available results on the functional calculus to the case A = C. Let '/(a ) be the element (7 (0 !), 7 ( 02) , . . . , 7 ( a „ )) in C ,n*. Then / belongs to T >(a) and w,u,p. are suitable for defining / - , ( / ) ■ By our uniqueness result, we could also obtain
(/ ) from functions w £ Croo(C n),ii 6:
3.5.12
Multivariable Functional Calculus
405
6 ,a-(('n, C (n)) and an associated form fl supported in a closed polydisc cen tered at ~f(a) on which / has a uniformly convergent power series expansion. Sine; ,ve know that satisfies (j) and (i), the argument given previously establishes that *r
, ( W ) ) " (2 S r f , d > ~ T & r / m
=m a))'
This concludes the proof of this lemma, and hence, the complete proof o f Theorem 3.5.9. O Having proved this theorem, we are justified in simplifying our notation by replacing / „(/ ) by V n 6 N; a 6 A (n>; / 6 ? a.
/ (a ) = / „(/ )
We leave it to the reader to translate the various equations in the statement of Theorem 3.5.9 into this notation. Implicit Function Theorem The following implicit function theorem is essentially due to Arens and Calderon [1955], However, they obtained the full result only for semisimple comrratative Banach algebras or for functions / with well-behaved power series expansions. The particular version given here is found in Gamelin [1969]. Its proof is based on the results o f Theorem 3.5.9. This theorem can in turn be developed into a functional calculus of multiple valued functions (i.e., functions defined on a Riemann surface with more than one sheet). For this viewpoint, which we will not develop here, see Michel Bonnard [1969], Allan [1970], Edgar Lee Stout [1971], and V. Ja. Lin [1973]. 3.5.1^ T h e o r e m Let A be a commutative unital Banach algebra. Let n 6 K and a 6 A ^ be given. Let F be a complex-valued function which is analy ic on an open subset U of Cn +1 satisfying Sp(a) £ < + i , 1U. Suppose that jC — never vanishes on U. If there is an h € C (T ^ ) satisfying Z\ F (h ,a ) = Q (i.e . F (h (y ),a 1 (y ),d 2 (y ), ■■ ■,an(y )) = 0 for all 7 e T ^ ), then there is a unique element c 6 A satisfying
c= h
and
F (c , a) = 0.
Proof Let z and w be the coordinate functions on C " and on C, respectively. The ordinary implicit function theorem for analytic functions (cf, e.g.,
406
3: Commutative Algebras and Functional Calculus
3.5.12
Hormander [1966)) shows that for any ip € Ta there will be constants > 0 and 6^ > 0 so that the equation F(A ,/j) = 0 has only one solution with A in the disc D * = {A: |A - h(xp)\ < e ^ } for each p in the polydisc P v = { n G C ": \nj - a3(x}>)\ < 6* for j = 1, 2, . . . , n} . Let V' 1’ C VA be the open neighborhood o f t}> defined by V',‘ = {7 6 Va If‘ h ) K ' P )I < £*/2 and |aj(7) — ^(VOI < t>'t’/2 for j = 1, 2, . . . , n } . Choose a subcover { K ^ 1, V . . . , V * * }. Define e > 0 by e = mi nf e^ : j = 1, 2, . . . , p} , and define V C x by V = {(71,72) G I'a x r^:|/i(7i) - /t(72)| < e/2}. Suppose (71,72) G V satisfies 6(71) = a( 72). For some j , 71 belongs to VAV-’ so that h ( j i ) find both belong to D ^ J and 0(71) = a( 72) belongs to P * > . Also, F(h(~lk) ) d(~lk) ) is zero for k = 1, 2. Thus, /i(7i) equals /i(72) by the uniqueness of solutions of F( \ , f_t) — 0 with A G for H G P * J. Therefore, V is an open neighborhood of the diagonal in x such that (71,72) G V and 6(71) = 6(72) imply h( 71) = h( 72). For any (')'i,72) G x \ V, choose 6 G A satisfying 6(71) ^ 6(72) and then choo6e a neighborhood V b of (71,72) in F.a x T ^ such that (^11 ^2) G V 6 implies b(ip 1) ^ 6(^2)- The family o f all such V 6 forms an open cover for the compact net Ya x F^ \ V. Let { V b>, V bj , , V h,,‘ " } be a finite snbcover and define b G by b = (61,62,..., bm- n). Then 7(0,6) = ip(a,b) implies /i(7) = h(ip) for all 7, ip G Hence, we obtain a well-defined function H on Sp( a, b) from the definition f f ( 7 (a ,6) ) = /»(7 )
V7 G ^ .
(22)
We wish to show that this function H is continuous. Suppose 7*. (a, 6) converges to 70(0,6) for some sequence {~tk}k€N C It is enough to show that /i(7/t) converges to /i(7o)- If this is false, we can choose a neighborhood U o f /i(7o) and a subsequence o f {/i(7fc)}fceN which remains completely outside U. Since is compact, we may choose a subnet o f { 7 * } fcen which converges to some 7^ G 1^. Then we have ~tb(a,b) = 70(0,6) so that h { l 0) = h( l o) - This contradiction shows that {h(-yk)}keN converges to h(~i0) as we wished. Now we claim that H can be extended to a function which is analytic in a neighborhood of Sp(a, 6). To simplify notation, let F be F 0 7Tm+i,n+i defined on (U ,C m~n). Since H satisfies F (H ,Z i,z 2 , . . . , z m) = 0
(23)
on Sp(a,b), and the partial derivative of F with respect to its first variable never vanishes on (U ,C m~n), the implicit function theorem for analytic functions shows that for each A G Sp(a, 6) there is sin open polydisc P x C C m centered at A and an open disc D x C C centered at H { A), such that for each /z G P x there is a unique solution G D x of the equation F (w ,n ) = 0. Furthermore, the function H x: P x —» D x defined in this way is analytic and is completely determined on any connected open subset of
Multivariable Functional Calculus
407
P x by its value at a single point and the requirement that it be a continuous solution of F ( H X, zt,z 2, . . . , z„) = 0. Since H is continuous, we may assume that P x is chosen small enough so that H maps P x into D x. Assume that P x, D x, and H x satisfying these conditions have been chosen for each A G Sp(a,b). If A and fi belong to Sp{a, b), the uniqueness o f H x and H** show that either A 6 or ft 6 P x implies H x — on P x fl P^ ■ For any A € Sp(a,b), let be the polydisc centered at A with polyradius equal to half the polyradius of P x. If V x n V> is nonempty for any A, / 1 6 Sp(a,b), then A or ft belongs to P x D P^ so that H x and agree on V x D V^. Hence, we can extend i f to be an analytic function on the neighborhood V = UAeSp(0ii))K >l o f Sp(a,b). Let c be the element H (a ,b ) 6 A defined by the multivariable func tional calculus. Theorem 3.5.9(e) and equation (22) give c = H (a, b j = h. Theorem 3.5.9(b) and (g) and equation (23) give F (c, o) = F o 7rm + 1,n+1 (c, a, b) = F ( H (a, b), a, b) = 0. This proves the existence of the element c. Suppose d were another element satisfying the same conditons as c. Then d = h = c would imply the existence of an element r in the Gelfand radical of A satisfying d = c + r. We wish to show that r is zero. Let G be the function defined on C n+2 by
dF
1
G(u,w,z) - - j [ F { w + u , z ) - F ( w , z ) - u ^ ( w , * ) ] where z is the coordinate function on C ". This function is analytic on a neighborhood o f Sp(r, c, a). Also by assumption we have F ( c + r,a) = F( c, a) = 0. Hence, when u\uG(u, w, z) + ^~(w , z)] = F ( w + u, z) —F( w, z) is applied to (r, c, a) 6 -4
and
/ ( c ) = 0.
408
3: Commutative Algebras and Functional Calculus
3.5.13
To prove this, we merely apply Theorem 3.5.12 to the case n — l,a = 0, U = V x C and F = / o It is this case n = 0 which we will use in our first application. The next result and Corollary 3.5.15 are both known as the Silov idempotent theorem. They were first proved by Silov [1953). We will use the implicit function theorem to give an extremely simple proof. Usually they are derived from the multivariable functioned calculus directly. No proofs are known which avoid this heavy machinery, although Silov’s original proof depended only on his imperfect (although still highly nontrivial) multivari able functioned calculus. 3.5.13 C o r o lla r y 'Let A be a commutative Banach algebra. If K is a compact open subset of F t h e n there is a unique idempotent e 6 A such that e is the characteristic function of K . Proof If A is unital, apply the theorem to F (w ) = w2 — w = 0 and its solution h, the characteristic function o f K in C ( r ^ ) . If A is nonunital, replace A by A 1. Theorem 3.1.2(a) shows that restriction to A defines a homeomorphism o f onto F^, which is the one point compactification of TA . Hence, the element e cam be defined as before. It belongs to A since e clearly vanishes at infinity. □ We give two important corollaries o f the Silov idempotent theorem here. (W e will also use the idempotent theorem in the proofs of Lemma 3.5.16 and Theorem 3.5.19.) The first gives a partial converse o f the easy assertion that r A is compact if A is a commutative unital Banach algebra. This shows that a semisimple commutative Banach algebra A is united if and only if VA is compact. Easy examples show that the hypothesis of semisimplicity cannot be dropped. This result is due to Silov [I953J. 3.5.14 C o ro lla ry Let A be a commutative Banach algebra. (a) The Gelfand spaa; of A is nonempty and compact if and only if A modulo its Gelfand radical ui unital. In this case A is algebra direct sum of of the closed ideals eA and A j . (b) A hull H in F > is compact if and only if its kernel k (H ) is a modular ideal. Proof (a): Denote the Gelfand radicad of A by A j . Theorem 3.1.2(e) asserts that is compact if A / A j is united. In this case A j is modular, so Theo rem 1.2.24, Proposition 3.1.3 and Lemma 3.1.4 show that is nonempty. Hence, is nonempty and compact. Conversely, if is nonempty and compact, Corollary 3.5.13 shows the existence of an idempotent e such that e is the identity function on Hence, e + A j is an identity element in A / A j and eA is an ideal satisfying e A C iA j = { 0 }. (b): Theorem 3.2.7 now gives this. □
Multivariable Functional Calculus
409
I he next result is also due to Silov [1953]. 3.5.: 5 C o r o lla r y
The following are equivalent for a unital commutative
Banach algebra A. («.) r A is disconnected. (b) There is an idempotent in A other than 0 and 1 . (-:) A is the direct sum of two closed non-zero ideals. (<1) A is the direct sum of two non-zero ideals. Proof (a) => (b): I f is disconnected, let it be the disjoint union o f two nonempty open and closed sets K\ and K 2. These sets are compact so Corollary 3.5.13 gives an idempotent e which is neither 0 nor 1 since its Gelfand transform is the characteristic function of K \ . (b) => (c): Let e be an idempotent not equal to zero or one. Then eA -■ {a 6 A: ea = a } and ( 1 — e )A = {a 6 A: ea = 0 } are non-zero closed idea's satisfying A = eA © (1 — e)A . { ■■) => (d): Obvious. (d) =t- (b): If Zi and I 2 are non-zero ideals satisfying A = I i © I 2. let 1 = e\ © ej be the corresponding direct sum decomposition o f 1. The identity e\ © e 2 = 1 = l 2 = e\ ®e\ implies that ei and e2 are idempotents. They are both non-zero since T\ and I 2 ^ non-zero. (b) => (a): I f e is a nontrivial idempotent, e 6 C ( r A ) is a nontrivial cont nuous characteristic function, so is disconnected. □. I s we have seen, the Silov idempotent theorem lies very deep (at least in our present state of knowledge). However, in certain circumstances the ex istence o f nontrivial idempotents can be proved in a much more elementary way. See Proposition 3.4.1. We now wish to prove an important theorem o f Charles E. Rickart [1953], It depends on a lemma which uses the Silov idempotent theorem. It also depends on the concepts o f hulls and kernels and completely regular algel>ras introduced in Section 3.2. Irving Kaplansky [1949a] proved a result similar to Theorem 3.5.17 in which the algebra A was assumed to have the forir C (f2) for some locally compact Hausdorff space fi. 5. .6 L em m a Let A be a commutative Banach algebra. Let I be an idea of A and let H be a compact hull in which is disjoint from h (J ). Th?:there is an element 6 6 I which satisfies b (H ) = { 1 }. prc:/ Corollary 3.5.14 shows that k (H ) is a modular ideal. Hence, L + k (H ) is ais-o a modular ideal. It satisfies h (I + k (H )) = h (T )n h k (H ) = h(l)C\H = i) a n i hence is not included in any maximal modular ideal. Theorem 2.4.6 ibus shows J + k (H ) = A. Corollary 3.5.14 then shows that there is some eieivsnt a 6 A satisfying 7 (a ) — 1 for all 7 6 i f . Let a = 6 + c be a decomposition (guaranteed by A = l + k ( H ) ) satisfying b 6 I and c 6 k (H ).
410
3: Commutative Algebras and Functional Calculus
3.5.17
Since 7 (c) = 0 holds for all 7 6 H , the equation 7 ( 6) = 7(6 + 0) = 7 (a ) = 1 shows that 6 has the desired property. □ 3.5.17 T h e o re m Let A be a completely regular, semisimple commutative Banach algebra and let B be a commutative spectral algebra. Let ip :A —>B be an injective homomorphism. (a) The associated dual map tp^: Tp —+ maps Tg onto (b) Every element a 6 A satisfies S p ^ (a )u {0 } = S p s (y ? (a ))u {0 }. This result also holds if B is a normed algebra instead of a spectral algebra. (c) Any (not necessarily complete) algebra norm ||| • ||| on A is spectral. Proof (a): Let B denote the image of Fg in under tpK Suppose B is not all of and choose 70 € \ B. Then for each 7 € Tg, there is an element a-, € A satisfying 0^ (70) f ( ^ ( 7 )). For each 7 € Tg, define e7 and t/7 by e7 = 2 - 1 |“ 7 (7 o )- a - r(y>t (7 ))| and Uy = {ip 6 |d7 ( 7 o ) - o 7 (v>, (V’))l < £7 } so that U1 is an open set in containing 7 . Choose a finite subcover U u , U-n >• • • >^ 7n ° f the compact set T g and denote each alj and £lj by a3 and tj, respectively. Let £ be m in{£1 ,£2, ... , £ „ } and let V be the open neighborhood V = {7 € r i:| a j(7 o )- O j(7 )| < e
V j = 1 ,2 ,... ,n }
of 70 in which is disjoint from B. Choose an open neighborhood W of 70 with compact closure in V. Lemma 3.5.16 shows that there are elements 6, c 6 A satisfying 6(70) = 1, 6(7 ) = 0 for all 7 ^ W, c (y ) = 1 for all 7 € W and c(7 ) = 0 for all 7 ^ V. Therefore, 6c equals 6, and since the Gelfand radical is zero this implies be = b. Since ip(c) vanishes on Tg, p (c ) has a quasi-inverse d in B. However, this gives
:> k - g
Multivariable Functional Calculus
411
PrvoJ I f a is spectral, Theorem 3.1.5(b) shows A j — ,4r and Corollary 2.?./ shows that this set is cr-closed. Conversely, suppose that A j is aclosed. Let a be the norm induced on A / A j . Since A / A j satisfies the hypotheses of Theorem 3.5.17, <7 must be spectral. Thus, any a £ A satisfies p(a) = p(a + A j ) < a(a + A j ) < <x(a). Hence a is spectral. □ The Exponential Function and the Group A g / exp(A)
T i e exponential function was introduced in Theorem 2.1.12 where it was shown that in a commutative unital Banach algebra A, exp(«4) is just the connected component o f the identity in the group A g o f invertible elements. Theorem 3.4.2 gave the kernel o f the homomorphism exp: A —> A g and Example 3.4.3 showed that Theorem 3.4.4 could not completely describe the range. Now, with the help o f the multivariable functional calculus, we can answer the resulting question and show relations to a number of interesting topological concepts. Consider first the case A = C ( X ) where X is some compact Hausdorff space. W e recall the definition of the first cohomotopy group of a topological space X . (For further information on this subject see Sze-Tsen Hu [1959].) The set G of continuous functions from X to T is obviously a group under pointwise multiplication. Tw o functions in G are said to be homotopic in G if one can be continuously deformed in G into the other. More precisely, we say -/a and f\ in G are homotopic if there is a continuous map F : X x [0,1] —> T (called a homotopy) satisfying F (x , 0) = fo (x ) and F (x , 1) = /i(x ) for all x 6 X . Obviously, this is an equivalence relation on G. It is also obvious that it respects multiplication in G. Hence, the set o f homotopy classes o f G is again a group. It is denoted by 7T1 ( X ) and is called the first cohomotopy grov? of X. 7ot any / € C (X )g , the function //|/| is defined and belongs to G. (In fact, f (x , t ) = f ( x ) / ( l — t + <|/(x)|) is a homotopy of / into //|/| in the set C (X )a - ) Let 0 be the group homomorphism o f C ( X ) g onto 7r * (X ) defined by setting 0 (f ) equal to the homotopy class o f //|/| in G. W e claim that exp( C { X ) ) is the kernel o f 0 so that ( C ( X ) )g / e x p (C (X )) is isomorphic to n \ X ). •"or any g 6 C (X ), exp(g)/| exp(g)| is just exp(tlm (g)). Thus, the ho motopy G (x , t) = e x p (i(l — t)Im (p ))
V x € l ; (€ [ 0 , 1 ]
between exp(g)/| exp(g)| and the constant function 1 shows that 0 (exp (g)) is the identity element in irl (X ). Hence, e x p (C (X )) C ker(0) holds. conversely, 0 (f ) is 1, then there is a homotopy F: X x [0,1] —►T sii'slyin g F (x ,0 ) = /(x)/|/(x)| and F (x , 1) = 1. Since t >-+ F (-, t ) is a continuous map of [0 ,1 ] into C (X )a , f/\f\ belongs to the principal compoof C (X )a - Similarly, the homotopy between / and //|/| noted above
412
3: Commutative Algebras and Functional Calculus
3.5.19
shows that / belongs to the principal component o f C ( X ) a • Hence, The orem 2 .1 . 1 2 shows that / belongs to e x p (C (X )). This establishes that d induces a group isomorphism C W c / e x p ^ X ))- ^ * ).
(24)
We now consider A g / exp (.4) for an arbitrary, unital, commutative Ba nach algebra. The Gelfand space is an invariant o f a commutative spectral algebra. If the algebra is determined up to isomorphism, then the Gelfand space is determined up to homeomorphism. However, many nonisomorphic commutative Banach algebras have the same Gelfand space up to homeo morphism (e.g., the disc algebra (§3.2.13) and the algebra of all continu ous functions on the disc). Nevertheless, certain algebraic properties of a commutative Banach algebra depend only on its Gelfand space. Corollary 3.5.14 provides an example. It shows that a commutative Banach algebra is unital modulo its radical if and only if its Gelfand space is nonempty and compact. We will now show two more cases of this phenomenon. In fact, the next theorem shows that the zero and first order Cech cohomology groups o f the Gelfand space with integer coefficients arise as natural alge braic invariants of the algebra. We will return to this interpretation after stating and proving the theorem. The first half o f the next theorem is a reformulation o f the Silov idempotent theorem. The second half is known as the Arens Royden theorem. It was discovered by Arens [1963] and, partly independently, by Halsey L. Rpyden [1963]. See also Rickart [1969]. The proof we give is due to Gamelin [1969]. It is based on Oka’s [1936] solution for open polynomial polyhedra of a problem solved by P. Cousin [1895] for polydiscs. We will now state this Cousin problem. For a proof, see Gunning and Rossi [1965]. Let {V a : a G A ) be an open cover for an open polynomial polyhedron V C C n. Cousin data for { V0: a € A } is defined to be a family o f functions { 9a,p- (<*, P ) G A x A } such that each ga,p is analytic on Va D Vp and the family satisfies 9a,0 + 9g~, + 9-ya - 0
on
Va D V J D
for all q , /?, 7 G A. Oka’s theorem asserts the existence of a family of functions {
9a,0
for all q, /? € A. 3.5.19 T h e o r e m Let A be a unital commutative Banach algebra. Let H ° (A ) be the additive subgroup of A generated by the idempotents in A
3.5.19
Multivariable Functional Calculus
413
and let H l (A ) be the multiplicative group A ?/ ex p («4 ). The Gelfand homorn jrphism induces an isomorphism of H ° ( A ) onto H ° (C (T a )) and an isomorphism of H 1 (A ) onto H 1 ( C ( r ^ ) ) . Proo.' The idempotent elements in C (T X ) are exactly the characteristic functions o f open and closed subsets. Hence the Silov idempotent theo rem ^Corollary 3.5.13) shows that the Gelfand homomorphism restricted to is an isomorphism onto H ° (C (T X ))It is obvious that the Gelfand homomorphism maps A g into C ( T X ) g and cxp(«4) into exp(«4). Hence, ( “) does induce a homomorphism of H l (A ) into .¥ * (C (r ^ )). The implicit function theorem (Theorem 3.5.12) applied to F (w ,z ) — ew — z, shows that this homomorphism is injective. It only remains to show that it is surjective. In other words, we must show that for any y € C (T X ) g , there is a c e A g such that //c belongs to e x p (C (r ^ )). Let / € C { T a ) g be given. By the Stone-Weierstrass theorem (1.5.1), the algebra {]C ^ = i bjb^+y. n G N, bj € >1} is dense in C ( r ^ ) .
Hence, we
can find an element b = (i>i, &2 . •. • ,&2n) G A 2"-* satisfying
n i - r l E M ; + ji u < i . 3 -1
Theorem 3.4.4 shows that there is a function g G C (T A ) satisfying n
r
lY
bi bn + } = e X P(9)-
J- 1
Thus, it is enough to find elements c G A g and h G C '(r ^ ) satisfying n c -^ ^ C ^ e x p W ,
(25)
since we will then have //c = exp(h — g). Choose a neighborhood U of Sp(b) C C 2n such that n
3= 1 does not vanish on U where z — (z x, Z2 , ■■■, Z2n) is the coordinate function on C 2". Proposition 3.5.8 asserts the existence o f an element d € ,4(m- 2n) such that 7rm,2nPc(Sp(fc, d )) is included in U and includes Sp(b). Lemma 3.5.2 shows that we can find an open polynomial polyhedron V C C m satisfying pc(Sp(b,d)) C V C Xm,2n (U ) and hence also satisfying Sp(b) C 7rm,2ri( V ) Q U. Then, p defined by p = p o nm 2n does not vanish on the
414
3: Commutative Algebras and Functional Calculus
3.5.19
open polynomial polyhedron V . Let j Vu: a 6 A } be an open cover of V such that, on each Va , p has a continuous logarithm pa which varies in absolute value by less than it. For each ordered pair (a , /3), define pajj by Pa,0 = Pa - Pa- Since exp(pai/3) = exp(pa )e x p (- p ^ ) = 1 holds, and |pa — pg| varies by less than 2 n, each pa ^ is a constant function which is an integer multiple o f 2ni. Furthermore, we have Pa,Q + P/3,7
Pi,a = 0
for each triple Hence, the {p Q ^ :a , 0 £ A ) are Cousin data for the open covering {V a : a € A } of the polynomiaily convex set V . Thus, by the theorem of Oka stated above, there are analytic functions {g a : a e A } such that each qa is defined on Va and satisfies qa — qp = p„,p on K. H Vp for each pair (a , 0 ). Hence, there is a well-defined analytic function q satisfying q\Va = exp (qa) for a e A . Note that q is never zero on V since it is locally an exponential function. Furthermore, since pa — qa = pp — qp holds on Va n Vg for any pair (a , /?), there is a continuous function h on V satisfying fc|Va = Pa —qa for all a. Therefore, q~lp = exp(h) holds on V since it holds on each Va. Define c € A to be q{b,d). Then c is invertible (since q was) and satisfies c~l bjb^+ j = c~lp(b) = (q (b ,d )~ 1p (b ,d )) = exp(h (b,d )). This is what we wished to show. □ We now interpret H ° ( A ) and H l (A ) as cohomology groups. For details on sheaf theory, see Roger Godement [1958]. For any commutative unital Banach algebra A, we have an exact sequence
0
—-
H °(A )
^
A
— ► H l (A )
Ag
— * 0.
For the special case A — C ( X ) , this sequence has a sheaf theoretic inter pretation. (In fact a similar interpretation is valid at least whenever A is semisimple and completely regular.) Let Z be the constant sheaf on X o f integers under addition. Let C and Ca be the sheaves on X of germs of continuous functions under addition and of germs o f nonvanishing contin uous functions under multiplication. Then multiplication by 2 ni and the exponential map induce a short exact sequence of sheaves 0
— >Z
^
C
^
CG
— > 0.
This gives rise to an exact sequence of cohomology groups 0
— *
-1 *
H ° (X , Z )
H l (X , Z )
^
H ° (X ,C )
—
H l (X , C )
^
H ° (X , C a )
Multivariable Functional Calculus
415
Hiivvt ver, the zero order cohomology group with coefficients in a sheaf is just the group o f global sections o f the sheaf and H l(X , C ) — 0 holds, since C is a fine sheaf. Clearly, the groups of global sections o f Z, C, and C q are, respectively, H ° ( C ( X ) ) , C ( X ) and C ( X ) g ■ Therefore, the cohomology exact sequence becomes 0
H ° (X ,Z )
™
H \ X ,Z )
C {X )
— *
^
C (X )G
0
where we could substitute H ° ( C ( X ) ) and H l ( C ( X ) ) for H ° ( X , Z ) and //: (X , Z ) if we wished. The Arens-Royden theorem gives the identification of H ‘}(A ) with H ° ( C ( X ) ) = H ° (X , Z ) and of H l (A ) with H l ( C ( X ) ) = H - (X , Z ). Since we have already identified the cohomotopy group 7r* ( X ) as C ( X ) g / e x p (C (X )), we have established a natural isomorphism o f H 1 (X , Z ) onto 7T1 ( X ) for any compact space X . Recall also that Theorem 2.1.12 shows that C (X )G / e x p (C (X )) is torsion free. Edward K. Blum [1953] gives two interesting arguments, which are in dependent o f the Silov idempotent theorem, showing that H ° (A ) is the fundamental group tti(G ) of the principal component G of A g Taylor ([1971], [1972a], [1975] and [1976]) defines natural variants of H °(C < (X )) and H 1 ( C ( X ) ) for a pair Y C X of compact spaces and shows that the resulting structure satisfies the axioms of cohomology theory. From the uniqueness o f structures satisfying these axioms, he infers that these groups are identical to the Cech cohomology groups with integer coeffi cients. The axioms allow him to identify H ° ( M ( G ) ) (the result is essen tially Paul J. Cohen’s idempotent theorem [1960a], [1959b]) and / fI (M (G )) where M (G ) is the measure algebra on a locally compact abelian group (cf. :hc next section). (This general result gave concrete new information on which invertible measures on R are exponentials.) Taylor also obtains re■suits in the more general case of convolution measure algebras. Along simi'nes, he defined and studied the K-theory o f a commutative Banach algebra [1973a] using the results o f Arens [1966], For some related results the reader may wish to consult Rickart [1969], Iain Raeburn and Taylor [1977], and Kallin [1963]. The last named article contains an example related to the first article o f Arens. The literature of commutative Banach algebras is enormous. For further general information see the following books: Browder [1969], Gamelin [1969], Stout [1971], and Wermer [1976]. Taylor Spectrum and Functional Calculus in B (X ) We give only the briefest introduction to this subject and refer the reader to HelemskiT [1981] and [1989b] Chapter V I, the original articles
416
3: Commutative Algebras and Functional Calculus
3.5.20
Taylor [1970a], [1970b], [1970c], [1972a] and [1972b], Vasilescu [1982] and the forthcoming book by Putinar for further details. If A" is a Banach space, S = (Sy, S2, ■• •, Sn) and A = (A i, A2, . . . , An) are ordered n-tuples o f commuting operators in B (X ) and of complex num bers, respectively, then A belongs to the Taylor spectrum if the Koszul complex K (X , S — A) is not exact. For lack o f space, we shall not define the Koszul complex (see Helemskii [1989b], V.1.1). Its exactness implies a < * < „ k e r ( S * ) = {0 } and £ £ = 1 S k (X ) = X. The other conditions in volved in proving exactness are o f a similar nature. (For instance if n = 2, the only other condition is that S2Xj = S1 X2 should imply the existence o f a z e X satisfying Xi = S\z and x 2 = S2z.) Thus, although the Taylor spectrum is hard to define, it is not hard to compute. In spite of the fact that we have not even explained the definition of Taylor’s joint spectrum adequately, we believe it will be worthwhile to state the resulting theorem which relates it to the functional calculus. Th e proof may be put together from Helemskii [1989b] and Taylor [1970b], We begin with a simple definition of a functional calculus. 3.5.20 D e fin itio n Let a = (a i, a2, . . . , an) be an ordered n-tuple of commuting elements in a unital Banach algebra A. Let a" denote the double commutant of the set {a j, a2, . . . , an} in .4. Let U be an open subset o f C n. We say that a has a functional calculus on U if there is a continuous unital homomorphism I„ : T\j —> a " C A satisfying /a/( ^ ) = afc
V fc = l,2 ,...,n .
3.5.21 T h e o r e m Let ?i be a Banach space and let S = (S i, S2, • .., Sn) be an ordered n-tuple of commuting operators in B (X ). (a) The Taylor spectrum SpT(S ) of S is a nonempty compact subset of Cn satisfying |Afc| < p(S fc) for all \ = (A i, A2, . . . , A „) € SpT (S ) and k = 1 , 2 , . . . , n. (b ) If A is any closed unital subalgebra o f B (X ) including {S i, S2, . . . , S „ } in its center, then SpT(S ) is included in S p ^ (S ). There do exist examples where this inclusion w proper for all choices of A. (c) The ordered n-tuple S has a functional calculus on every open subset U of C n which includes SpT(S ). (d) If U is a domain of holomorphy, then S has a functional calculus on U if and only if SpT (S ) C U. In this case the functional calculus for S on V is unique. (e) Let f = (/ 1 ,/2, •. • ,/m) be an ordered m-tuple of functions analytic on an open set U which includes SpT(S ). Then SpT( f ( S ) ) — f(S p T(S )) where we have written f ( S ) for 1$ (/ ). Hence i f U and V 3 SpT ( f ( S ) ) are domains of holomorphy and g is analytic on V, then go f ( S ) is defined and equals g (f (S )).
3.6.0
Commutative Group Algebras
3.6
417
Commutative Group Algebras
Section 1.8 described the group algebras associated with the compact abelian group T in some detail. This involved some results about group algebras of the discrete abelian group Z. Section 1.9 gave basic informa tion about the Banach algebras L 1 (G ) and M (G ) for an arbitrary locally compact group G. In particular, 1.9.13 identified M ( G ) with the double centralizer algebra of L l (G ). For locally compact abelian groups, 1.9.9 showed that both algebras are commutative. The reader may wish to con sult those sections for background information and notation. A ll notation is summarized in Table 1, p. 144. In the present section we will show that the Gelland space of L l (G ) can be identified with the locally compact group G of a 1 continuous homomorphisms o f G into T, with pointwise product and the compact open topology. The group G is called the character group of G. W e then prove that the natural map of G into G (the character group o f G ) is a homeomorphic group isomorphism. The proof, which depends on showing that L 1 (G ) is completely regular, is essentially due to David Abramovic Ralkov [1945] (written in 1940), Henri Cartan and Roger Godeinent [1947]. We also give a brief survey o f some results o f Joseph L. Taylor [1965], [1973a] on the Gelfand space o f M (G ). The section concludes by mentioning some other group algebras. We begin with a simple special case o f the result we will prove in com plete generality below. Let G be a discrete abelian group and define G to be the set of group homomorphisms from G to T. W e will regard G as a group under pointwise multiplication: hk(u) = h (u )k (u )
Vh,k € G; u € G
and as a topological space under the topology o f pointwise convergence. (If we identify each homomorphism h € G with its graph in IlueG ^ = T 0 * this becomes the product topology.) If {h a} a^A is a net in G converg ing In this topology to an element h 6 T G, then h satisfies h (uv) — lim ha(u v) = lim ha(u)ha(v ) = lim /*a (u )lim ha(v ) = h (u )h (v) for all i., v € G. Thus G is closed in T 0 and hence is compact. We claim tha G is a topological group. Continuing with the same notation, we find Sim h~l (u ) — lim ha(u )“ = /»(«)* = h~l (u). If {fca } Qgyi is another net in C converging to an element k € G, then it satisfies lim ha(u)ka(u ) = !im iQ(u )lim ka(u ) = h (u)k (u) = hk(u). This verifies our claim, so G is a compact abelian group. T h e o re m
Let G be a discrete abelian group. Let G be the compact abelian
grui-v described above and let IV (c > be the Gelfand space of the commuta tive unital group algebra Then the isometric linear isomorphism
418
3: Commutative Algebras and Functional Calculus
uj'. t 00(G ) -
3.6.0
( \ G ) ' defined by Vft 6 f » ( G ) ; / € e '(G )
« h { f ) = Y . / (« )M w _1) u6 G
sends G C £°°(G ) (with its topology of pointwise convergence) homeomorphically onto r/i(G) C ( l (G )* (with its Gelfand topology). Proof Only the last sentence remains to be proven. Any h € G and any /iff € £l (G ) satisfy h (u~l ) = /i(u)* and hence UhU + 9)
=
£ / u6G
ueGveG
v e G ue G
=
£
£
v6 G
u6G
s (v - 1 « ) M v - I « ) * = Uk(f)u;h{g).
Hence u; sends G into r />(gj - The map u> is well known to be a linear isometry o f t ° ° (G ) onto t l (G )* . Thus if 7 belongs to the Gelfand space there is a unique h 6 t ° ° (G ) 1 satisfying 7 = w/,. This function then sat isfies 7 ( 6U) = /»(u)* for any u € G. For u,v € G we get h (u )h (v ) = 7 (^u)*7 (^u)* = 7 (<5u * <5t))* = h(uv), so A belongs to G. This shows that u> is a bijection o f G onto r h ( g )- Since both spaces are compact and Haus dorff, u> is a homeomorphism if either it or its inverse is continuous. It is not hard to check that ui is continuous, but we take this opportunity to note that the inverse map is given by the obviously continuous map 7 >-►/i7 defined by h~,(u) = 7 (fiu)* for all 7 € ^e'(G) and « € G. □ Now let G be an arbitrary locally compact group. The Banach space dual L l ( G ) ‘ o f L l (G ) can be identified with L ° ° (G ). (See Hewitt and Ross [1963], Section 12.18, for a proof in the generality needed here.) W ith each h € L °°(G ), we will associate the linear functional u>^ € L ‘ (G )* defined by «fc (/ ) = j G f (u )h ( u - l )du = j
fhdX
V / € L ‘ ( G)
(1)
(where h is defined by h(u) = h (u~l ) for all u 6 G ). The twist built into this definition gives the following formulas - * ( / * s ) = « , . * ( / ) = wfc. A /(ff)
V
A
e m
f , g € L l (G )
(2)
where the convolution products are defined and belong to L ° ° (G ) even when G is a not necessarily unimodular group. In the unimodular case (a fortiori, in the abelian case), these reduce to «* (/ • »)= «,.* (/ )= < * * ./ (»)
V / »6 L 00(G ); f , g e L '( G ) .
(3)
Commutative Group Algebras
419
O f course, the proofs depend on Fubini’s theorem. N.>xt suppose that G is a locally compact abelian group and note that any 7 in the Gelfand space T g of L l (G ) belongs to the unit ball of L l (G )* and hsnce can be associated with a function /i7 6 L °°(G )\ satisfying
and
J
f * 9 (ii)h *,(u 'l )dit —
J
f(u )h ^ (u
x)du
J
g(u)h-l(u ~ 1)du
(5)
tor ai’ /, g 6 L 1 (G ). It is easy to see (just use the definition of convolution multiplication and Fubini’s theorem) that the second formula will hold if h-f is -nultiplicative h^(uv) = /j7 (u)/j7 (v )
V u .dgG
and the integrals exist. Thus we are led to consider bounded group homo morphism of G into the multiplicative group of C. In fact, by considering h(un) — h (u )n for u 6 G and n € Z we see that such a bounded group o r:orphism h into the multiplicative group o f C must be a group ho momorphism into T = {A € C: |A| = 1}. W e give a formal definition. 3.8.1 D e fin itio n Let G be a group. A homomorphism of G into T is calied a character of G. If G is a topological poup, then the collection of all continuous characters o f G is denoted by G. This set is endowed with pointwise multiplication and the compact-open topology. It is called the character group or dual group of G. It is obvious that G is a group under pointwise multiplication. cc7."ji ,ct-open topology is defined by the basic open neighborhoods N (h 0, K , £) = {/ »€ G: \h(u) - Ao(u)| < e for all u € K }
The
(6)
where ha € G, K a compact subset o f G and e > 0 are arbitrary. It is easy to see that G is a topological group under this topology. Note that any character on any (not necessarily topological) group satisfies
3o
h(e)
=
1
\h(u)\
=
1
VueG
h (u ) = /»(u_ I )
=
h (u )m= h(u)
Vu6 G
h(uv)
=
h (u )h (v)
V u, v e G.
far we have noted that the map h
(7) (7)
u>h defined by
((88)) (9) (9)
((10) 10)
420
3: Commutative Algebras and Functional Calculus
3.6.2
maps G into Tg- This map is injective since distinct continuous functions give rise to distinct elements of L ° ° (G ) and hence to distinct elements in L 1 (G )" . More is true. 3.6.2 P r o p o s itio n Let G be a locally compact abelian group. Then the map h >—* u>h defined by ( 1 1 ) is a bijection of the set G onto the Gelfand space T c of L l {G ). Proof It only remains to show that the map is surjective. The best way to prove this is to show that for any 7 e T &• the function h1 € L ° ° (G ) satisfying (4) actually belongs to G. For any g € L l (G ), consider the function u >—* 7 ( ug) from G to C. We have already noted in Section 1.9 that u 1—> ug is a continuous inap o f G into L 1 (G ) so that u >-t 7 ( ug) is continuous and is bounded by HffH,. We claim that this function satisfies
7(fl)Mu~*) = 7 ( u 9 )
(12)
as an element of L ° ° (G ) (i.e., almost everywhere). To establish this, it is enough to show that J 7 (g)h y(u ~ l )f(u )d u — J 7 ( Ug )f(u )d u holds for all / € L l(G ). However, Fubini’s theorem and (4) give J ~ t(g )h -,(u ~ l )f(u )d u
= = = =
'r (g h i f ) ~ ~t(f * g)
J /i7 (t;-1 ) J f(u )g (u ~ 1 v)dudv J / ( « ) J h-y(v~l) Ug(v)dvdu J f(^ h (u g )d u
as we wished to show. Choose g 6 L 1 (G ) to satisfy 7 (g ) ^ 0 and change h~, on a set of measure zero so that it satisfies ( 1 1 ) everywhere and hence is continuous on all o f G. This gives 7
=
y(uvg) = 7(u(vff))
=
7 ( « . s ) M u - 1 ) = y(9)h-r( v ~ 1)hy( u - 1)
for all u, v € G, so that h~r is indeed a continuous character on G.
□
We have identified G and T g as sets. However T g is a locally compact topological space under the Gelfand topology and G is a topological abelian group under pointwise multiplication and the compact-open topology. It is not immediately obvious that the group operations on G are continuous in the Gelfand topology transferred from T g , nor is it clear that the compactopen topology on G is locally compact. However, we will prove both o f these
Commutative Group Algebras
421
ai o ice by showing that the bijection h >—►u>h of G onto T g is actually a homomorphism. 3.6.3 T h e o r e m Let G be a locally compact abelian group. Then the map h >--> ujh defined by ( 1 1 ) is a homeomorphism of G with its compact-open topology onto T g with its Gelfand topology. Hence G is a locally compact abelian group and a net {h a } ae/i Q G converges to ho € G if and only if lim j ha(u ~ l )f(u )d u = f ho(u~1 )f(u )d u holds for all f e L l (G ). Further more the map (h, u) ►-» h(u) V h € G; u € G of G x G —> C is jointly continuous. Proof First we show that if {/»Q} o6yi C G converges to ho € G then the last statement o f the theorem holds. For any given / € L l (G ) and e > 0 choose a compact set K C G satisfying f G\K |/|dA < e/4. Then
ha € JV(/i0, K - i ,£ /( 2 ||/||1 + 1)) implies
|J
ha { u ~ l ) f ( u ) d u
-
/
<
(J + J
<
(e/(2||/||1 + l))||/||j + 2(e/4) < e
h o ( u ~ 1) f ( u) du\
k
)\ha(u ~ l ) - h0(u ~ l)\ \f{u)\du
g \k
as desired. Hence h u>h is continuous from G to TG. To complete the proof, we must show that the image in T g o f any compact-open neighborhood N (h o ,K ,e ) C G includes a Gelfand neighbor hood. In a moment we will show that (u>h,u) t—» h(u) is jointly continuous from T g x G to C with the Gelfand topology on r G. Assuming this, we see that for each u € K we can choose a Gelfand neighborhood Uu of w/,0 in F -3 and a neighborhood Wn o f u in G so that u>h € Uu and v € Wu imp'/ |ft(v) — ^o(«)| < £/2. Since K is compact we can find a finite set Ui, U2 , ■. ■,u„ satisfying K C U "= i WUJ. Let U be the Gelfand neighborhooc of uJh0 defined by U = f l " = i UUj. Then any satis y
e U and v e K
\h0(v ) - h (v)| < \h0(v ) - h0{uj)\ + |/»0(u>) - h(v)\ < e for the j satisfying v € UUj. This shows U C N (h 0, K , e) as we wished. It remains to show the joint continuity of the map (u>f,,u) * h(u) for h e G and u 6 G. Let 70 € T g and uo € G be given and choose an / 6 L i (G ) satisfying 70(/ ) / 0 and a Gelfand neighborhood U o f 70 such that. Oh £ U implies U h {f) / 0. Then all u>h € U satisfy i'.(u)
=
h (u )
J
f (v )h (v
l )d v /u h( f ) =
j
f (v )h (u v l )dv/ujh( f )
422
3: Commutative Algebras and Functional Calculus =
J
n v u - 'W u ) d v M
3.6.3
f ) = ujh( f u)/ujh( f ).
Hence it is enough to show that (7 , u) 1—* y (f u) is continuous on U x G. For any e > 0, choose a neighborhood W of uq 6 G satisfying ||/u - /Uolli < e/2 and the Gelfand neighborhood Uo C U o f 70 defined by |7 (/U0) —7 o(/u0)l < e/2 for ail 7 € U0. Then any ( 7 , u) G U0 x W7 satisfies \y(fu) - 7o(/uo)l < |7(/u) - 7(/uo)l + W /uo) - 7o(/uo)l < II/u - /uolli + e/2 < £■ Hence the map is jointly continuous. □ Given any locally compact abelian group G, we have constructed an other locally compact abelian group G. The construction can be repeated to construct the character group G of G. There is a natural map o f G into G defined by 0 (u )(h ) = h(u)
V u e G; h e G .
(13)
The joint continuity o f (h, a) *-* k (u ) shows that 8 (u ) is a continuous char acter on G. Obviously 0 is a homomorphism. W e now wish to prove the Pontryagin duality theorem which asserts that 0 is a homeomorphic isomor phism of G onto G. This theorem establishes a complete duality between G and G in that each is the character group of the other (up to homeomorphic isomorphism). It is, perhaps, the most fundamental result of commutative harmonic analysis. Since the duality map 0:G —+ G (defined by (13)) is obviously a homo morphism, the duality theorem will be proved when the following results have been established: (a) 0 is injective. (b) 0 is a homeomorphism onto its image. (c) 0 (G ) is closed in G. (d) 0 (G ) is dense in G. Establishing (a) is equivalent to showing that G separates the points of G. Since we have identified G with T g , this is obviously related to (and in fact follows easily from) showing that r g separates the points of L l (G ). The latter is precisely the question o f whether L l (G ) is semisirnple. This would be the next natural question to ask in the study o f the commutative Banach algebra L ’ (G ) after finding a representation of its Gelfand space. We will give an easy proof o f the semisimplicity of L l (G ) in Volume II, but here we will prove that G separates the points o f G directly. Result (b) can be proved by showing that when G is considered as a family of maps o f G into C via the map 0, then the topology of G agrees with the compact-open topology. Then result (c) will follow from the elementary fact that any locally compact subgroup of a locally compact group must be closed (Hewitt and Ross [1963], 5.11).
Commutative Group Algebras
423
Result (d) is an immediate consequence of any proof that L l (G ) is a completely regular commutative Banach algebra whenever G is a locally compact abelian group. (See Jean Esterle and Jose E. Gale [1982] for an extension.) When this result is applied to L 1 (G ), it easily shows that the image of 6 (G ) is dense in G identified with the Gelfand space of I/1 (G ). For the.second time we see that a major step in establishing the Pontryagin duality theorem is a natural step in studying the commutative Banach algebra L l (G ) (viz. determining whether it is completely regular). The proofs o f (a), (b), and (d) can all be based on a Fourier inversion theorem. First we introduce the Fourier-Stieltjes transform. Let G be a locally compact abelian group. The Gelfand transform is a contractive homomorphism o f L l (G ) into Co ( r e ) . The identification o f TG with G given by ( 1 1 ) allows us to make the following definition. D e fin itio n Let G be a locally compact abelian group. The Fourier transform is the contractive homomorphism ('): L l(G ) —* C o (G ) defined by f ( h ) = uk( f ) =
J
fhdX =
J
fhdX =
J
f(u )h (u )*d u
V / € L l (G )\ h € G.
(14)
The range of the Fourier transform is denoted by A (G ) and is called the Fourier algebra on G. Similarly, the inverse Fourier transform is the con tractive homomorphism ('): L 1 (G ) —> C o(G ) defined by 3(u) = J g O (u )d X = J g (h )h (u )d h
V g € L ^ G ); u € G
(15)
where dX and dh represent Haar measure on G. Its range is denoted by A.(G) and called the Fourier algebra on G. This definition generalizes the classical concept of (coefficients o f) Fourier series introduced in Section 1.8 where G = T and G ~ Z. It also generalizes the Fourier integral transform where G = R and G 2; M. Th e calculation ( / ) C 1) = / f (u ~ 1 )*h(u)*du = (/ f (v )h (v ~ 1 )d v )’ = /(/»), which holds for a,1 J e L l (G ) and all h € G, shows that the Fourier algebra is closed under complex conjugation. ■ Since L l (G ) is frequently regarded as an ideal in the measure algebra M (G ) under the map / 1—* fdX (Section 1.9) this suggests the following more general definition. 3.9.5 D e fin itio n For a locally compact abelian group G. the contractive homomorphism ( (~ ): M (G ) —> C (G ) / ('): M (G ) —> C (G ) ) defined by ( A(h )
=
J hdy. = J hdp
/ i(«)
=
J
0 (u)dn =
J
V fc € G; /i € M (G ) /
h(u)dn(h)
(16)
V y, 6 M (G ); u e G ) (17)
3: Commutative Algebras and Functional Calculus
424
3.6.6
is called the ( Founer-Stieltjes / inverse Fourier-Stieltjes ) transform and its range { B (G ) / B (G ) ) is the Fourier-Stieltjes algebra on ( G / G ). Note that the inverse Fourier and inverse Fourier-Stieltjes transforms differ from the direct transforms by the omission of a complex conjuga tion. This prevents the perfect symmetry which the duality theorem shows exists between G and G from being extended to the direct and inverse trans forms. This difference is necessary if the inversion theorem is to hold. It is quite easy to show that the inverse Fourier-Stieltjes transform is an injec tion. The calculation given above to show that the Fourier algebra A (G ) is closed under complex conjugation easily extends to show that B (G ), A (G ) and B (G ) are all closed under complex conjugation. Now we may state a standard inversion theorem. 3.6.6 Inversion Theorem Let G be a locally compact abelian group. If a function belongs to B (G ) n L 1 (G ), then its Fourier transform belongs to L l (G ). Moreover the Haar measure on G and G can be chosen to satisfy
v
/ = /
/ e B (G ) n L l (G ).
O u t proof o f this theorem will depend on the theory o f positive definite
functions and on Bochner’s Theorem. 3.6.7 Definition Let G be a group. A function f-.G — * C is said to be posi tive definite if for any finite sets { « , , u2, . . . , C G and { Aj , A2, . . . , An} C € it satisfies n
n
k~l j= 1 3.6.8 P roposition Let G be a group and let f: G —* C be a function. (a ) Then f is positive definite if it it is a character or if G is a locally compact group and f has the form g * § for some g € L l (G ). (b) If f is positive definite then it is bounded and satisfies f — f, I l f Hoc < f ( e ) and \ f ( u ) - f ( v )\2 < 2 f ( e ) R e ( f ( e ) - f ( u v - 1) )
V u ,v e G .
(18)
Proof (a): The two cases have very similar proofs. Suppose a character h, ■■■,Un} Q G and {Ai , A2, . . . , An} C C are given. Then we get n
£
n
£ a * a ; h(uku; ' )
n
=
n
Y , 5 > fcA ;ft(Ufc) M « j ) *
Commutative Group Algebras
3 .3 .C
425
Replace h by g * g in the above calculation to get n
n
n
n
r / n / ( £
\ I n
3 \k-l
J
\j=i
M ’
d v > 0.
(b): For the first two results choose n = 2, Ui = e, u2 = « , Ai = 1 and A2 = A in the definition, to get /(e) + |A|2/ (e ) + A / (« ) + A * / ( O > 0 . The c loice (A = l / A — i ) shows that { f ( u ) + f ( u ~ l ) / i f ( u ) —i f (u ~ l ) ) is real, proving / = /. I f A is chosen to satisfy A/(u) = -|/(u)|> the displayed inequiJity gives |/(u)| < / (e ) for any u e G. For (18), we may assume f (u ) ^ f ( v ) and choose n = 3, Ui = e, u2 = u, «3 = v, Aj = 1 and for any real t, X2 = ~ A 3 = t|/(u) - f (v )\ / (f (u ) - / (« )). This gives 2 R e (/ (e )—f (u v ~ 1 ))t 2 + 2 \f(u ) —f {v )\ t + f (e ) > 0 . This quadratic has no real roots, so its discriminant is negative, giving (18). □ For locally compact abelian groups we can characterize the positive definite functions. Readers will notice that the proof involves ideas from Hilbert space theory. When we look at unitary group representations in Voluire II, we will see a close connection with positive definite functions. This will explain the origin of the following proof and the appearance of Hilbert space ideas in it. 3.6.9 B och n er’s Theorem Let G be a locally compact abelian group. The following assertions about a function f: G —> C are equivalent: (a) / is continuous and positive definite. (b) / belongs to B (G ) and there is a non-negative measure fi e M ( G ) satisfying f = /». Proof (b) => (a): This “easy” direction is proved in the same way as (a) in Proportion 3.6.8:
n
n
I. ( £
/•6 (: \k=i
Xkh(uk)
^ A jh (u } ) j =l
d fi> 0 .
(a) =>• (b): Let p :G —> C be continuous and positive definite with p(e) = 1 . Let / e Coo(G) have compact support K . Inequality (18) shows
426
3: Commutative Algebras and Functional Calculus
3.6.9
that K can be partitioned into a finite number o f Borel sets B i , B 2, . .., B „ so that the function (u, v ) •—* f ( u ) f ( v ) ’ p (u v~l ) varies is little as we please on each Bk x B} . Hence the integral
[ J
f f ( u ) f ( v ) ‘ p (u v~1)dudv
g
J
(19)
g
can be approximated as closely as we please by finite sums of the form
£ £ / ( « * ) / ( « , r p ^ u ; 1 )A (fl* )A (B ,) > 0 fc=i ]= i which are non-negative by positive definiteness. This shows that the inte gral (19) is non-negative. In fact (19) is non-negative for any / in L l (G ), since Coo(G) is dense in L *(G ). We use the positive definite function p to define a linear functional r)p and a (possibly indefinite) inner product (•, )p on L 1 (G ):
i ) p( f )
=
/
J
/ (u )p («)d u
(/ » s )p
=
'? p ( /
•$)
V f , 9 € L l ( G)
(2 0 )
g
Obviously ( f , g )p is linear in the first variable and conjugate linear in the second. The poeitivity o f the inner product follows from that o f the integral (19): (f , 9)P
/
=
f f (u )g (u -'v Y p (v )d u d v = f
JgJg
=
[ J
[ f(u )g (v Y p (u v )d v du
JgJg
(21)
f f ( u)g (v )’ p (v v ~ l )dv du.
g
J
g
This is enough to derive the Cauchy-Schwarz inequality: |(/,s)p|2 < ( f , f ) P(g ,g )P
V f,g e L l (G ).
(22)
Next we consider the case in which g is the characteristic function of some compact symmetric neighborhood V o f e in G multiplied by A( V) ~*. Equa tion ( 2 1 ) gives (/
= =
(fl,fl)p- 1 =
[
J
M V ) 1p (u v " 1 )d u d u - f f(u )p (u )d u
/ (“ ) [ g
J
J
v
/ / (u )A (V X
Jg
A{ V
)~2[
g
1 / ( p ( « ^ _ I ) - p(u))dvdu Jv
f ( p ( u t r ' ) -p (u ))d v d u .
Jv Jv
Commutative Group Algebras
3.6.3
427
The continuity of p together with inequality (18) show that we can make ikesn expressions as small as we please merely by requiring V to be small. Thus (22) yields W / )| 2 < ( / , / ) p = tp( / * / )
V /€L»(G ).
(23)
Let k represent / * / and let kn be the nth convolution power of k. Then iteration o f the last step gives
M / )| 2 <|rp(fc2) r ' < . - - < | r p(fc2")|2‘ ". Recalling ||p| !«, = 1 and applying Gelfand’s spectral radius formula as n increases and our identification o f the Fourier transform with the Gelfand transform, we get
w /)i 2< p m G M = nfciu = iiTT/iu = ii/iiLThus rp is linear functional on the Fourier algebra A (G ) on G with respect to the s'.ipremum norm with bound 1. We can extend rp to a linear functional on Cq(G ) with norm 1 . Hence the Riesz representation theorem gives a measure /x € M (G ) of norm 1 satisfying ,r(u)p(u)du
-
f
Tp( f ) = f f ( h )d f i ( h )- f Jg
j g
=
[
f(u)fL(u)du
f(u )h (u )d u d fi(h )
JgJg
V / e L l (G )
JG
Since Coo(G) C L l (G ) D C 0(G ) is dense in Co(G ), p equals fi at least almost everywhere. Since both p and jj, are continuous, they are equal. It only remains to show that is positive. This follows from l = p ( e ) = j ^ dn (h ) = n{G)<\\n\\ = l.
□
Bochner’s theorem and the Jordan decomposition theorem for measures shov/s ‘ hat any function / in the Fourier-Stieltjes algebra B (G ) is a linear combination / = f\ — fi + i(fz — f t ) o f four positive definite functions. We can now prove the inversion theorem. Proof of inversion theorem First we consider two functions / and g in B(G.) n L l (G ) and a function k € L l (G ). Let n j and fj.g be measures in M (G ) satisfying f = [if and g = jj.g. This notation gives i;* / (e )
=
=
f k ( u ~ 1) f ( u ) d u = f f c ( « - ' ) f h ( u ) d n f ( h ) d u Jg Jg Jg
I
( k(u~l )h{u)dudiif(h) =
JGJG
f k (h )d n f(h ) J
6
428
3: Commutative Algebras and Functional Calculus
3.6.9
from which we derive
f
k (h)g(h)dnf (h )
=
Jg
I k * g(h)dnf (h ) = k * g * / ( e ) = k * f * g(e) / * *fWdMg(h) = f k(h )/(h )dfig(h ). J J&
Jo
=
g
Since B (G ) = {k : k 6 Z / ( G ) } is dense in C o (G ), this proves gdnf = fd »„
V / ,f l e B (G )n L \ G ).
(24)
We will define a linear functional fl on Coo(G) which will turn out to be a multiple o f the Haar functional. In this paragraph and the next let k belong to G o o (G ) and let K be the support o f k. We want a function / 6 B (G ) n L l (G ) such that its Fourier transform does not vanish on K. For any ho 6 K we can find some function f0 6 C o o (G ) such that /o(/»o) is non-zero. Then / 0 * fo (h ) = /o(^)/o(M* is a non-negative-valued function which is positive at h — ho. Since K is compact, we can find f \, f 2, ■■., /n 6 L 1 ( G ) such that the Fourier transform o f / = f x* j x+ f 2 * f 2 H------1 - f n * fn does not vanish on K . Proposition 3.6.9 shows that / € G o o (G ) C L l (G ) is positive definite and hence belongs to B (G ) also. Thus we can define «(* ) =
f
j ~ d „ f (h ). Jg f (h )
(25)
Equation (1) shows that this expression is actually independent of / (so long as the integrand is defined and / 6 B (G ) fl L l (G )). Therefore fi is a linear functional on G o o (G ) since / € B (G ) D L l (G ) can be chosen to be positive on the supports o f any two functions ki and k2 in Coo(G) and used to define f}( a k t + 0k2) for a, 0 6 C. Since / in (2) is positive definite,/ and [i j artf both positive. Hunci: { } iw a positive linear functional on Cuo(G). Next we will show that fi is left invariant so that it is a multiple of Haar measure on G . Let ho € G be arbitrary but fixed. Let k and / be as in the last paragraph except that we require / to be non-zero on both K — supp(fc) and hoK. Let g be the pointwise product g = h^1 f so we get /(/») = g lh ^ 'h ) and dfxj(h) = d^if(h ^h ). This gives
Jg
-
f (h )
= Jg f
g(h0 ft)
'h)
L m ^ w=sm'
so n is left invariant. To see that il is not zero, we fix a non-zero continuous positive definite function / in L l (G ) for a moment, and choose a function
«.t>.10
Commutative Group Algebras
429
k 6 Coo(G) so that f kd/j.f is non-zero. This gives Sl(kf) = [ k f /gdfig = [ kdnf ? 0 . J6 Jd
(26)
Thus we may choose Haar measure dh on G to satisfy fi(fc) = f k(h)dh J
V k 6 Coo(G).
(27)
g
Combining (3) and (4) gives [ J
kdpf = ft (* / ) =
f k (h )f(h )d h J
g
V k g Coo(G); / e B (G ) D L l {G )
g
(28) and ihus the measures satisfy d » f (h ) = /(fc)dfc
V / € f l (G ) n L l {G ).
Since /i/ is a finite measure, / belongs to L l (G). / € 5(G ) D L l (G ) and a € G equation (6) gives / (« ) = (£ / )(«) =
f
Hu)dnf{h) =
f
(29)
Furthermore for any
f ( h ) h ( u ) d u = / (u )
as we wished to show.
□
The power o f Bochner’s theorem is the ability to produce a function in -B(G) merely by choosing a continuous positive definite function. By combining the inversion theorem with Bochner’s theorem, we see that a conti i u o u s positive definite function in L 1 (G ) is the inverse Fourier trans form of its non-negative-valued Fourier transform. W e illustrate the utility of this starting with a lemma. Theorem 3.6.11 is a special case o f the Gelfand-Raikov theorem proved in Volume II. 3.6.10 Lem m a Let G be a locally compact abelian group and suppose U is a r.eighborhood of e. Then there is a continuous positive definite function g in I 1 (G ) fl B (G ) satisfying g (e) = 1 and g(u ) = 0 for all u € G \ U . Proof Choose a symmetric compact neighborhood V of e satisfying V 2 C U. DeiL j g by u »-► \ ( V ) ~ lXv * X v (u ), where x v is the characteristic function o f V . This expression shows that g is positive definite. It is easy to show g (u ) = A ( V ) _ 1 A ( V D u V ) which gives the continuity o f g. □ 3.6.11
Theorem
Let G be a locally compact abelian group.
Then G
separates the points of G, so 0:G —+ G is injective. If G is compact, the span if G is dense in C (G ).
430
3: Commutative Algebras and Functional Calculus
3.6.12
Proof It is enough to show that for any u / e in G there is some h £ G satisfying h(u) / 1 . By the lemma and the Hausdorff property o f G, choose g € B ( G ) n L l (G ) satisfying g (e) — 1 and g(u) = 0. The inversion theorem gives 0 = g(u ) = (g )\ u ) = and
1=
j g(h)h(u)dh
9(e)- (9)te) = J g{h)h(e)dh - j g{h)dh
from which the existence of such an h e O is obvious. If G is compact, then the span sp((7) o f G is a point separating sub algebra o f C (G ) which is closed under complex conjugation. Hence the Stone-Weierstrass theorem (§1.5.1) shows that sp(G ) is dense. □ We state a partial result. 3.6.12
P r o p o s itio n
If G is a locally compact abelian group, then 9 is a
homeomorphism of G onto its range in G. Proof Recall that G has the compact open topology as a set of maps from G into T. Hence it is enough to show that the sets U (K , e) = {u £ G : \h(u) — 1 | < e for all h e K ), with K C G compact and e positive, form a neighborhood base at e. First we show that each of these sets is a neighborhood of e. The joint continuity o f ( h, u) >—> h(u) shows that for each k € G there are neighbor hoods Uk Q G and V* C G o f e and k, respectively, satisfying \h(u) — 1| < e for all h 6 Vk and u € Uk- Choose an open cover , F*a, . . . , Vkn o f K . Then D”= l Uk, is obviously a neighborhood of e included in U {K ,e ). Next we show that if U is any neighborhood o f e in G then there is some K and e satisfying U {K ,e ) C U. Use the lemma to find g 6 B (G ) satisfying g(e) = 1, g > 0 and vanishing off U . The inversion theorem gives 1 =
9(e)= (
Choose a compact set K satisfying
J
\g(h)\dh <
G\K
Then the neighborhood U (K , j ) is included in U since the inversion theorem gives 1 -
3.6.1.3
Commutative Group Algebras
431
3.6.13 Plancherel Theorem Let G be a locally compact abelian group. The map f t -* f is an isometry o f (L 1 (G)r\L2 (G ), || • ||2) into (L 2 (G ), || • ||2). This map extends to an isometry of L 2 (G ) onto L 2 (G ). Proof For any / e L x(G ) Pi L2 (G ), it is easy to check that g = f~ * f is continuous and positive definite. Therefore we may apply the inversion theorem to get II/II2
=
J f(u )f(u )d u = f - * f ( e )
-
(g )'{c ) =
=
J
j
g{h)h(e)dh
gdX = J ( f ~ * fYdX = J \ f \ 2dX = ||/||2.
Ker.ce (~) is an isometry of L l (G ) n I 2 (G ) into L 2 (G ). We must still show that t,he image is dense. It is enough to show that any g € L 2 (G ) satisfying
j
V/
fgdX = 0
G L 1 (G )
n
L 2 (G )
(30)
is zero. Note that / e L 1 (G )H L 2 (G ) and u e G imply /„ G L 1( G ) f l L 2(G ) aiic! f u(h ) — f(h )h (u '~ l ) so that any g £ L 2(G ) satisfying (18) also satisfies (/5 )'(w _1) =
J
/{tygityh iu '^d h , -
j
f u(h)g(h)dh = 0
for al! / G L 1 {G ) and u G G. The injectivity of the inverse Fourier trans form now gives fg = 0 (as a function in L 1 ( G ) ) for all / G L 1(G ) D L 2(G ). Lei f be the characteristic functions o f open set V with compact closure. / belongs to L 1(G )H L 2(G ) and for any h G G, f (h ) is just f y h (u )mdu. L ettin g V vary, we conclude g = 0 in L 2 (G ).
For any / G L 2(G ), we may define / G L 2 (G ) by / - lim /„ where {/ „ }„ g N G L 1 (G ) n L 2 (G ) is any sequence converging to / in the L 2-ncrra. This gives an isometry / i-> / of L 2 (G ) onto L2 (G ). □
T ’ -c map / »-♦ / of L 2 (G ) onto L 2 (G ) just defined is called the FourierPlancherel transform. The polarization identity applied to Plancherel’s theerora now gives Parseval’s identity (f , g ) = (f , g )
v / , 9 g i 2( G)
(3 1 )
432
3: Commutative Algebras and Functional Calculus
3.6.14
where the inner products are those of L 2 (G ) and L 2 (G ), respectively. Also many identities such as
immediately extend from functions in L l (G ) to functions in L 2 (G ). Parseval’s identity also implies fg = f * 9
V / , 5 6 £ 2 (G )
(33)
which implies A (G ) = L 2 (G ) * L \ G ).
(34)
3.6.14 T h e o r e m For any locally compact abelian group G, the commu tative Banach algebra L y(G ) is completely regular. Hence the ( Fourier / Fourier-Stiltjes ) algebra { A (G ) j B (G ) ) is dense in ( C U(G ) / C (G ) ). Proof Let k be a point in G and let U be a neighborhood of k. We must construct a function / € L l (G ) satisfying f (k ) = 1 and f (h ) = 0 for all h £ U. Let V be a compact symmetric neighborhood of 1 6 G satisfying V 2k C U . Then the characteristic functions x v and \Vk belong to L 2 (G ) so that there is a function / G L 1 (G ) satisfying / = H V ) ~ 1X v * Xvk-
As before, we find f (h ) = A( v ) ~ l J x v {h g )x v k (g ~ l)dg = X ( V ) - lX (v h n V k ) which gives the desired result. We have already noted that both A (G ) and B (G ) are closed under complex conjugation, so in both cases complete regularity gives ali the □ hypotheses of the Stone-Weierstrass theorem. As noted previously, the results we have just derived from the inver sion theorem, together with Hewitt and Ross [1963], 5.11, establish the following. 3.6.15 P o n tr y a g in D u a lity T h e o re m For any locally compact abelian group G, the natural map 9:G onto G.
G is a homeomorphic isomorphism
3.6.16
Commutative Group Algebras
433
Proof The only part of the proof which might not be perfectly clear is the density of $ (G ) in G. I f 6 (G ) were not dense, the complete regularity of l } ( G ) would give a non-zero function g € L 1 (G ) satisfying g (6 (u )) = 0
VueG.
By the uniqueness of the inverse Fourier transform, g is non-zero. However this gives the contradiction g(u )
=
J h ( u ~ 1) ' g(h)dh = J 6(u ~ l)(h )* g(h)dh
=
g (0 ( u - l ) ) = 0
VueG.
□
The duality theorem now allows all o f the results proved previously only for G co be reinterpreted as statements about arbitrary locally compact abelian groups. For instance (22) becomes A (G ) = L 2 ( G ) * L 2 (G ).
(35)
We remark that it is easy to show that if G is a compact group then G is discrete, and if G is discrete then G is compact. The Haar integral on a discreia group G is simply summation
j fd\=Y /(«)■
J
u€G
Then the Haar measure A on G satisfying the inversion theorem and/or Planch irel’s theorem is the one normalized to satisfy A (G ) = 1. 3.6.16 Id e a l T h e o r y Throughout this subsection, G will denote a locally compac t abelian group. We are interested in the closed ideal structure of Li1 (G ). We will use the identification of L ° ° (G ) as the dual Banach space of the 70 up algebra L l (G ). We consider the elements o f G as continuous characisrs which are elements of L ° ° (G ) and also as multiplicative linear functionals on L 1 (G ) which are elements of I 'i h g )■ This identification agrees vith our previous notation since when ft G G is thought of as a functiti in L ° ° (G ) the corresponding element o f F£,i(C) is just Uh 6 L l (G )*. We alsj use the complete regularity of L l (G ). Since L l (G ) is Gelfand semisimple, { 0 } is the intersection of maximal modular ideals, a condition which ’ /ill be defined as strong semisimplicity in Section 4.5 below.
434
3: Commutative Algebras and Functional Calculus
3.6. lfi
Let J be a subset of L l (G ) and W a subset of L ° ° (G ). We are interested in their annihilators in L '^ (G ) and L l (G ), respectively: I x
Wi
=
{ h e L ° ° (G ) : u h( f ) = 0 for all f e 1 }
=
{h € L X (G ) : j
=
{ f e L l (G ) : u)h( f ) = 0 for all h e W }
=
{/ e L l(G ) : f f (v )h (v ~ l)dv = 0 for all h e W } . Jg
f ( v ) h ( v 'l)dv = 0 for all f € 1 }
Clearly T L is a weak* closed linear subspace o f Z/*1 (G ) and W i is a (norm) closed linear subspace of L l (G ). If W and I already satisfy these additional conditions, then the Hahn Banach theorem shows ( I ± )± = 1
and
(W X)-L = W.
Now suppose X is a norm closed linear subspace which is invariant under translation by elements o f G. Then I 1 is also translation invariant, but more interestingly it can be expressed as l x
= =
{h € L ° ° (G ) : [ f (u v )h (v ~ l )dv = 0 for all / e l ; u e G } Jg { h e L ° ° (G ): f * h = 0 for all f e l } .
FYom this, it is not hard to see that a closed linear subspace of L l (G ) is an ideal if and only if it is translation invariant. We can apply the theory o f hulls and kernels to L l (G ) and G (considered as the G e lfa n d space of L 1 (G )). The question of spectral synthesis is: When is a closed ideal I o f L l (G ) completely determined by its hull h ( l ) l We say that a (necessarily closed) subset S of G is a set of spectral synthesis if its kernel k (S ) is the only ideal with hull equal to S. To explain this terminology further, let S be a wubset o f G and let VV(.S') be the weak* closed linear spam o f S in L °°(G ). Since spS is invariant under translation by element of G, VV(S) is also invariant. For any weak* closed, translation invariant, linear subspace W of L ° ° (G ), call S(VV) = W f l G the spectrum ofW . Clearly, VV(S( W ) ) is the smallest weakly closed, translation invariant subset W of W with spectrum S. Obviously, S (W ) is always closed and, conversely, it is true that every closed subset of G is S ( W ) for some W satisfying our conditions. This follows from complete regularity and the fact that if W = Z x , then S^W ) = h (l ). Then S is a set of spectral synthesis if and only if W (S ) is the only weakly closed, translation invariant subset of L °°(G ) with spectrum S. In this case, any function in any W with S (W ') — S can be approximated by linear combinations of elements of S (i.e. synthesized from S).
3.6.17
Commutative Group Algebras
435
If G is a compact abelian group so G is discrete, then every subset of G is a set of synthesis. However if G is locally compact abelian but not compact, Paul Malliavin [1959] showed that there are always subsets of G which are not sets of synthesis. Norbert Wiener’s Tauberian theorem { i 932] is essentially the statement that every closed ideal is included in some maximal modular ideal which is equivalent, in the present setting, to the density in L l (G ) of the set of elements / for which their Gelfand transform / has compact support. Suppose S is a closed subset of G. Let J ( S ) be the closure in L l(G ) of the set of elements with compactly supported Gelfand transforms vanishing on some neighborhood of S. It is easy to see that J ( S ) is the smallest closed ideal of L V(G ) with hull S just as k (S ) is the largest such ideal. Hence S is a set of synthesis if and only if these two ideals coincide. Stated another way, S is a set of synthesis if and only if every / e L l (G ) for which / vanishes on S can be approximated in L l (G ) by elements with Gelfand transforms vanishing in a neighborhood of S. See Rudin [1962], Chapter 8 for examples of sets of synthesis and nonsyrthesis. Chapter 7 will extend a number of these questions and results to noncommutative groups and noncommutative Banach algebras and will give proofs in that setting. 3.6.17 T h e M easu re A lg e b ra M (G ) The algebra M (G ) is so large that it is still not well understood even for locally compact abelian groups. Cohen’s idemopotent theorem [1959b], [1960a] identifies the idempotents ir I f.'G ). Let A be Haar measure on a compact abelian subgroup and let Xi> X2 , ■■■, Xn be a finite list of distinct continuous characters on H . Then M = ‘(X i + X2 + ■■■+ Xn)A is an idempotent measure on G and every such measure can be constructed by a finite number o f operations of the form H>~* 1 — H and (fii, H2 ) >—►fi\ * fi2 starting from idempotents of this special form. (Note m + ~ Mi * M2 = 1 ~ (1 — Mi) * (1 — M2)-) another theorem by Cohen [1960b] describes all homomorphisms from L 1 (G ) into M ( H ) where G and H are arbitrary locally compact abelian groups. Every such homomorphism can be extended to a homomorphism o f M ( G ) into M (H ) , but not uniquely. See Jyunji Inoue [1971] for an extension. Taylor [1965], [1973a] gives by far the most detailed information on the structure of M (G ). First he identifies an abstract class o f commutative con volution measure algebras based on the theory o f complex ordered L-spaces. A convolution measure algebra is simply a commutative Banach algebra M. which is an L-space and for which multiplication is an L-homomorphism from /A ® M to M . The dual of such an algebra has the form C (O ) for a suitable (large) compact space ft. Let A be the closed linear span o f the Gelfand space Tm ° f M in M * — C (fi). Then A has the form C (S ) for e compact commutative semigroup S and convolution multiplication in M (S ) agrees with the original multiplication in M . Thus his class of ab stract convolution measure algebras really deserves its name. The Gelfand
436
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3.6.18
space r.M is also a semigroup which has its own natural topology in which it is a topological semigroup. Certain idempotents in this semigroup are identities for locally compact subgroups of F>|. They are called critical points and from them much of the structure of M can be calculated. Tay lor characterizes those convolution measure algebras which are isomorphic to L l (G ) for some locally compact abelian group G as those semisimple algebras which include no proper L-subalgebra. He is able to describe the invertible and exponential elements rather concretely, and these results were new even for the case G = R. See the second reference above for an expository presentation, 3.6.18 T h e B e u rlin g A lg e b r a ^ ( Z , w ) Beurling algebras were defined in §1.9.15 but we wish to give a simple example. D e fin itio n A weight function u; on Z is a positive valued function w : Z - * R “ satisfying wq = 1
and
V m, n 6 Z.
wm+n < u;mu;n
If o' is a weight function, then the Banach space OO
* ‘ (Z,u,) = { / : Z - » C
: H/IL =
l/ (")k .< °o }, n — —o o
provided with convolution multiplication, is called the Beurling algebra on Z defined by
u j.
Denote lim „ _ ±oo^n/n by
The inequality for the weight function
p ±
u j
= p ± (i l (Z,uj)). shows that the limits above
exist and that p± agrees with ± in f{± u > ^ / "} (cf. Theorem 1.1.10). P roposition For any weight function u j , f (Z, u j ) is a unital semisimple commutative Banach algebra. For any A G C satisfying p~ < A < p + , the series f ( n)^ n converges. If we denote its sum by /(A ), then f is a homeomorphism of the annulus {A : p_ < A < p+ } onto I V ( z iUJ). Proof It is easy to show convergence, that i l (Z, u j ) is a unital commutative Banach algebra and that / is continuous into I V (z,^)- Consideration of 7 ( 6± ) for any 7 e ^ 1 ^ ) (which satisfy 7 (<5_ ) 7 (6+ ) = 1 ), shows that the map is onto and hence a homeomorphism. Thus it is obvious that t l (Z, u>) is semisimple. □
4 Ideals, Representations and Radicals
Introduction In any mathematical theory it is important to compare the general ab stract structures which arise with comparatively well understood concrete examples. Representation theory is a systematic attempt to do this. We have already seen one example of a representation theory in the Gelfand thecry for commutative spectral algebras given in Chapter 3. Any com mutative spectral algebra modulo its Gelfand radical is isomorphic to a spec oral subalgebra o f the algebra o f all continuous functions vanishing at infirity on a locally compact Hausdorff space. Since algebras of continuous functions are comparatively concrete and can perhaps be somewhat better understood a priori than general commutative spectral algebras, this is an important (and characteristic) representation theory. Note that the class of all commutative spectral algebras is too large to be successfully represented. Only the semisimple commutative spectral algebras can be faithfully rep resented. Fortunately the pathology o f the non-semisimple algebras can be neat y excised by dividing out the Gelfand radical. I i this work we will study several other representation theories in considerab e detail. This chapter deals with the representation of general algebras and part of the second volume is devoted to the representation theory of *-alg;ebras. In both these cases the phenomena noted above occur. Not all of the objects (algebras or *-algebras) can be faithfully represented, but the pathological part cam be divided out. The first two sections of this chapter deal with the representations them selves, postponing until the later sections a discussion of how the patholog ical portion splits off. Sections 4.3 to 4.7 introduce the theory of radicals whic’i explores the latter question. H setting up a representation theory for general algebras, it is first necessary to choose a class o f algebras which are concrete and comparatively well understood. O f course the algebras chosen must be sufficiently general to represent many of the most interesting examples. One’s first choice might be the algebras of finite-dimensional matrices studied in Sections 1.7 and 2.7. Historically these were used through the first half of this cenirry. However, for the type of algebras in which we are principally interested, this choice is much too restrictive. Typical and well-behaved algelras would have no non-zero representations. Thus it turns out that
437
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4.0
the class of algebras most suitable for our representation theory consists of the algebras of all linear maps of some (not necessarily finite-dimensional) linear space into itself. We will study homomorphisms of general algebras into algebras of linear maps on linear spaces. These homomorphisms will be called representations of t he algebra in question and the linear spaces will be called representation spaces. When the spaces are not finite-dimensional, we will usually need to consider norms on them and confine our attention to bounded linear maps. Representations can be broken down into simpler pieces by finding in variant subspaces of their representation spaces. The atoms under this process o f decomposition are called irreducible representations. They play a fundamental role. Unfortunately it is not usually possible to decompose a general representation into the sum of irreducible representations. Much o f the first two sections will deal with those ideals which are the kernels o f irreducible representations. These are called primitive ideals. They are examples of a class of ideals called prime ideals which will also be studied. We also introduce ultraprime ideals. The first section deals with ideals and with the purely algebraic theory of representations while the second section explores representations more fully and the phenomena related to norms. The case o f spectral algebras and spectral normed algebras will be considered most intensively. It turns out that in order to study irreducible representations it is useful to consider a wider class o f representations, called cyclic representations. Each cyclic representation can be constructed (up to equivalence) from the algebra itself. We will see this same situation again in the second volume when we consider representations of *-algebras. Representations on reflexive spaces play a role intermediate between the general theory o f this volume and the more special theory of the next volume. Some basic results about them are given at the end of Section 4.2. Historical Remarks Theorem 4.2.13 asserts that any irreducible representation of a spectral algebra is strictly dense. For Banach algebras this result was proved by Charles E. Rickart [1950]. The concept o f strict density and the principal techniques and results on this subject are due to Nathan Jacobson [1945c], This paper brought the theory of irreducible representations to its present central position. The theory of representations is quite old. For a discussion of its early history see Thomas Hawkins [1972]. Representations of algebras were con sidered by Charles Saunders Peirce [1881] and a more systematic study was begun by Theodor Molien [1893]. However, the representation the ory of finite groups, which was also considered by Molien [1897] but was mainly developed by G. Frobenius, soon eclipsed the representation theory
Introduction
439
for algebras. Much of this theory is contained in the second edition of W. Burnside’s classic book [1911]. Representations of algebras are mentioned in Leonard Eugene Dickson’s book [1927]. They are treated in a quite mod em way by Emmy Noether [1929] and in Adrian A. A lbert’s book [1939]. Since early discussions of algebras dealt with finite dimensional concrete algebras, the representation theory was not very explicit. However, even before algebras were generally thought of abstractly, their representations were studied implicitly since it was obvious that the given presentation of ■<:. hypercomplex system (an algebra presented as a linear space with a particular basis and a multiplication table for the basis elements) was not necessarily the most advantageous presentation. Since the time when Noether [1929] began to develop the general the ory of modules, the language of module theory has tended to supersede the entirely equivalent language of representation theory. In this work we will primarily retain the older language for two reasons. First, our emphasis will always be on the algebras rather than the representations themselves. Sec ond, the appropriate modules for Banach algebras are rather special, and the added complication of a norm on the algebra makes the language of modr.les less convenient. Nevertheless, the reader should be aware that the theories of modules and o f ( normed / Banach) modules are entirely equiva lent to the theories presented here of representations on linear spaces and of continuous representations of ( normed / Banach ) algebras on ( normed / Banach ) spaces. We adopt language o f modules occasionally (cf. §4.1.17). Radicals In the rest of this chapter we will introduce various concepts o f a radical for an algebra. By far the most important is the Jacobson radical which we will call simply the radical and explore in detail in Section 4.3. (It was briefly introduced already in Section 2.3.) In Sections 4.4 and 4.5 we wiH consider the Baer radical (or prime radical) and the Brown- McCoy or strong radical. Most of the material dealing with these three radicals could be developed for general rings, but we prefer to work in the setting o f algebras. a" Section 4.6 subdirect product representations are considered corre sponding to each o f these radicals. Spectral algebras are also characterized in terms of their natural representation as a subdirect product relative to ths Jacobson radical. We then discuss in Section 4.7 a theory of radicals in an arbitrary semi-abelian category. This theory provides a common ab stract generalization of the three radicals studied in the previous sections and of the reducing ideal and Leptin radical of a ‘ -algebra which will be introduced in Volume II. A t the end of the chapter we will discuss exam ples ;md the history of radicals. We prefer to postpone the history until the technical notions involved have been introduced.
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4.1
4.1.1
Ideals and Representations
As usual we start by defining the basic concepts. Irreducible represen tations were already briefly introduced in Section 2.3. 4.1.1 Definition Let A be an algebra and let X be a linear space. A ( representation / anti-representation } T of A on X is a { homomorphism / anti-homomorphism ) a Ta of A into C (X ). A subspace y of X is said to be T-invariant if Tay belongs to y for all y G ^ and all a € A. The { representation / anti-representation ) T o f A on X is said to be: (a) Faithful if the( homomorphism / anti-homomorphism } is injective. (b) Trivial if Ta — 0 for all a € A. (c) Irreducible if {0 } and X are the only T-invariant subspaces, and T is not trivial. (d) Cyclic if there is a vector z G X (called a cyclic vector) satisfying X = {T az : a e i } . (e) Equivalent to a ( representation / anti-representation ) S of A on a linear space y if there is a linear bijection (called an equivalence) U : X —* y satisfying SaU — UTa V a G A. (1) We make a few comments on the formal side of this definition before beginning to discuss the concepts systematically. We will sometimes use the word “ representation” to cover both representations and anti-representations and will seldom explicitly state results for anti-representations. If trivial representations were not specifically excluded in the definition of irreducible representations, the trivial representations on zero- and one-dimensional spaces would satisfy the definition of an irreducible representation for an essentially trivial reason. However, the trivial representation on a onedimensional space has quite different properties from those of irreducible representations, although some authors (possibly through inadvertence) al low it as an irreducible representation. The trivial representation on a zero-dimensional space is excluded simply for convenience. Its inclusion or exclusion makes little difference to the theory. Note that a (necessarily trivial) representation on a zero-dimensional space is cyclic. Although the definition of equivalence in (e) is stated asymmetrically, note that —* X establishes an equivalence between S and T. Thus it is immediate that equivalence is an equivalence relation. Note also that equivalence of representations clearly preserves all of the other properties defined here. If A is an algebra and S :A —♦ C (X ), T : A —> C(y) are two represen tations of A, we will write Equiv(S, T ) for the set of all equivalences of S and T. Hence Equiv(.S', Y j C C ( X , y ) is nonempty if and only if S and
4.1
Ideals and Representations
441
T are equivalent. Its elements are linear bijections V : X - * y satisfying TaV - V S a for all a € A. 7 henever we speak of a representation T o f an algebra A without spec ifying the linear space on which the maps {T a : a € A } act, we will denote this linear space by X T . Prom now on we will speak of the dimension of X 1 e s the dimension of T. T ie left regular representation (cf. Definition 1.2.1) L o f an algebra on its urderlying linear space is an example of a representation. A subspace £ of an algebra A is L-invariant (i.e., invariant under the left regular rep resentation) if and only if it is a left ideal. Our next results will show that this example is particularly important. Construction of New Representations from Invariant Subspaces If T is a representation of A on X and ^ is a '/’-invariant subspace, then
a^
(2 )
is a representation of A on y. This representation is called the restriction of T to y and is denoted by T y . (In this case T y is also called a subrep resentation of T .) There is another notion of restriction which one must carefully distinguish from this one. I f B is a subalgebra o f A and T is a rep resentation of A on X, then the homomorphism a i—►Ta can be restricted to B. When we need to distinguish these two notions of restriction we will call the first (where each Ta is restricted to y ) the restriction of the representa 'ion T, and we call the second (where the map T itself is restricted to B ) the. restriction of the homomorphism T. G ven a representation T of A on X and a T-invariant subspace ^ there is another natural, and even more important, way to construct a new representation. For each a € A, define T ? , y (x + y ) = Tax + y Since
is T-invariant,
by vx + y e x /y .
(3)
is a well defined element of C( X/ y) . Thus
a i—►T a ^ is a representation of A on X/ y. It is called the reduction of T by y. We will use L A! C to denote the reduction of the left regular representation L by a left ideal £. Internal Models for Cyclic Representations The last construction gives a standard model for all cyclic represents tions is shown in the following proposition. This result seems to have first appe? -ed explicitly in Jacobson [1956]. For unital algebras, the result can be stated without mentioning modular ideals and in that form it was known much earlier.
442
4: Ideals, Representations and Radicals
4.1.2
4.1.2 P r o p o s itio n Let A be an algebra and letT be a representation of A on X . Then T is cyclic if and only if T is equivalent to L A/C for some modular left ideal C of A. If z is any cyclic vector for T, then C can be chosen as C = { a e A : T az = 0}. Proof Suppose 2 is any vector in X . Then C = {a e A : Taz = 0 } is a left ideal. Now assume that z is a cyclic vector. Then there is some e £ A satisfying Tez = z. Clearly e is a right relative identity for £., so that £ is modular. For a £ A, let U (a + C ) = Taz. This is well-defined by the choice of C and hence is a linear injection of A / C into X . It is surjective since z is a cyclic vector. Finally it is easy to check that TaU — U L ^ C holds for all a € A. Thus U is an equivalence. This completes the proof of the necessity of the condition. Now suppose £ is a modular left ideal with right relative identity e. Then L ^ C(e + C ) — a + C holds for all a £ A. Hence e + C is a cyclic vector for L Since equivalence preserves cyclicity, this completes the proof o f the sufficiency of the condition. □ Irreducible Representations Having obtained a model for each cyclic representation we now turn to the problem o f picking out the irreducible representations. They are special cases o f cyclic representations. The next theorem, which is essentially due to Jacobson [1945a], shows that any irreducible representation is equivalent to L A^M for a maximal left ideal M . However, modular ideals had not yet been introduced by Irving E. Segal [1947a] and the satisfactory result just quoted made it difficult to see that theory should be based on only certain distinguished maximal left ideals. ( I f M is a maximal left ideal, then it is easy to see that is either irreducible or trivial. The latter case is equivalent to A 2 C M . Thus if L a/m is irreducible, there is a maximal modular left ideal M ' with L a/ m ' equivalent to L A^M .) Although Segal [1947a] and Einar Hille [1948] use modular ideals cleverly in close proximity to their discussion of the irreducibility of L A /M for M a maximal left ideal, they seem to have overlooked the fact that M may always be chosen modular. Hence again it seems that the next theorem first appeared in full generality in Jacobson’s book [1956]. 4.1.3 T h e o r e m Let A be an algebra and let T be a representation of A on the linear space X . Then the following are equivalent. (a) T is irreducible. (b) Every non-zero vector in X is a cyclic vector, and X is not zero dimensional.
Ideals and Representations
443
(c) There is a maximal modular left ideal C so that T is equivalent to l a
/c
Proof (a) => ( b): The set, {z £ X : TA z = { 0 } } is a T-invariant subspace and hence equals {0 } or X . Since T is nontrivial this subspace is {0 }. For any z £ X, TA z = {T az : a € .A} is a T-invariant subspace and hence equals { 0 } or X . For each non-zero z € X , TAz equals X so z is cyclic. ( 6) =► (c): Proposition 4.1.2 shows that T is equivalent to L A! C for some modular left ideal C. If £ is proper but not maximal choose a proper left ideal K, with K, properly including C. Then K./C is a non-zero and proper L-^/£-invariant subspace. This is impossible since every non-zero element of K /C is a cyclic vector for by the equivalence of h All~ and T. (c) => (a): If C is a maximal left ideal and ^ is an L^^-invariant subspace o f A /C , then {a £ A : a -I- C € 3^} must be a left ideal of A which includes C. Hence y is either { 0 } or A /C . Thus if C is modular, both L A/ C and its equivalent representation T are irreducible. □ Decomposition of Representations Suppose T is a representation of an algebra A on a linear space X which satisfies
x = y ©z
(4)
w h e rry and Z are both T-invariant. When this happens the pair (y , Z ) is said to decompose T. In this case the reduction of T by y is equivalent to the restriction of T to Z . Given a T-invariant subspace y , only in very special cases will there be a T-invariant subspace Z so that ( y , Z ) decomposes T.‘ Obviously irreducible representations are indecomposable. The next ';b.eoTCTC>. will state an interesting and useful intermediate condition. The ir.ecrem is essentially due to Issai Schur [1905] and is one of the oldest results in this whole theory. It was stated in the present form in Albert’s [1939]. 4.1.4 Theorem Let T be an irreducible representation of an algebra A on a linear space X . Then {Ta ) ' = { S e C (X ) : STa = TaS,
a £ .4 }
(5)
is a division algebra. I f (T X )' is a division algebra then T is indecomposable. Suppose S belongs to ( T * ) '. Then S X is a T-invariant subspace. Sines S X is { 0 } if and only if S is zero, S X equals X for all non-zero S. Similarly { x : Sx = 0} is T-invariant and must equal { 0 } unless S is zero. Thus every non-zero S £ (T X )' is invertible in C (X ), and its inverse must beioag to (T ^ )', so (T a ) ' is a division algebra. Suppose the representation is decomposable. Then the projections onto the zwo subspaces both belong to ( T * ) ' since the subspaces are invariant.
4: Ideals, Representations and Radicals
444
4.1.4
However these projection operators are nontrivial idempotents, and thus (TjCf is not a division algebra. □ The usual representation on C " (n > 2) of the algebra of strictly lower triangular n x n matrices is an example o f a representation which has a nontrivial invariant subspace but no nontrivial decomposition. In this case {Ta ) ' is not a division algebra. I f one represents the algebra of all N x N matrices which have only finitely many non-zero entries on the vector space o f all complex sequences, it is not hard to see that the commutant is just C and hence i# a divwion algebra, but the subspace 3^ of all sequences with only finitely many non-zero entries is an invariant subspace (which does not give rise to a decomposition). Thus we have seen that none of the implications in the theorem are logical equivalences, ln Volume II we will see a theory in which the corresponding concepts are all equivalent. I f T is a representation of an algebra A on X , then the decomposition of X by the T-invariant subspaces y and Z corresponds to an internal direct sum decomposition of th. :; corresponding .4-module. W e will briefly indicate the construction of the corresponding external direct sum. Suppose S and T are representations of A on y and Z , respectively. Then we define a representation S 0 T o f ^ o n [ V © 2 as the linear extension to all of ^ © -Z o f the following expression: ( S © T )a(y © z) = Say 0 Taz
Va
G
A\ y
G
[V; z
G
Z.
Clearly this can be extended to define the direct sum o f a finite number of representations. If { T " : a G v4} is an infinite family o f representations of an algebra A on linear spaces { X a : a G A }, then we define an algebraic direct sum a®/4T Q o f {T** : a G A } on the direct sum (cf. Definition 1.4.2) a l U * Q. by (© o€/4T “ ) a(© a€/4x “ ) = (&a€AT?xa where a G A is arbitrary and a >—►xa is any function in which is equal to zero except for a finite set o f a € A. If { T a : a G A } is an infinite family of representations of A on Banach spaces { X a : a € A }, then it is sometimes desirable to define the topolog ical direct sum of the representations on the topological direct sum o f the representation spaces { X a : a G A }. This is not always possible but when we need the construction we will discuss it. Intertwining Operators We will also state a variant of the first portion of the last theorem. Let A be an algebra and let S and T be representations of A on linear spaces X and y , respectively. We wish to consider the set o f intertwining operators o f S and T consisting of all linear maps V: X —» y which satisfy: TaV x = V S ax
x fc X\ a € A.
5
Ideals and Representations
445
T h u is obviously a linear space under pointwise operations. 4.1 5 P r o p o s itio n All non-zero intertwining operators between irreducible representations are equivalences. Proof We use the notation established just before the statement of the proposition. It is enough to show that each non-zero element is a bijection. Hot'ever (just as in the proof of the last theorem) if V is a non-zero inter twining operator, then V X is a non-zero T-invariant subspace o f y which mufit equal jy, and ker(K) is a proper S-invariant subspace of X which must equ.il { 0 }. O Ideals and Irreducible Representations Next we investigate the connection between irreducible representations and ideals. These results were at least essentially known to Jacobson when be wrote [1945a]. 4.1.S T h e o r e m Let T be an irreducible representation of an algebra A on c linear space X and let X be an ideal in A. (a) The restriction of the homomorphism T to X is either trivial or an irreducible representation of X. Furthermore every irreducible representa tion ofX arises in this way. (b) I f X C k er(T ) holds, then Ta+I = Ta
V a G .4
( 6)
defines an irreducible representation T of A /X . Furthermore every irre ducible representation of A /X arises in this way. Proof (a): The set { x G X : T jx = { 0 } } is a [T-invariant subspace and hence must be { 0 } or X . I f it is X , the restriction of the homomorphism T te I is trivial. I f it is {0 }, then for each non-zero x € X the T-invariant subtspace T jx must be X . Hence the restriction o f the homomorphism T to I is an irreducible representation by Theorem 4.1.3(b). How suppose T is an irreducible representation o f X. I f z € X , b, d € X and a G A satisfy l\z = 0, then T/TabZ = T ^ z = T ^T tz — 0 holds. Thus {T a:yz : a G .4 } is included in the T-invariant subspace { x € X : T jx = { 0 } } , which must equal { 0} . Hence we can choose any z G X T \ { 0 } and use T aTbZ = TabZ
V a € A,
bGI
to define T a uniquely on X = T jz. Clearly the restriction of the homojnor jhism T to I is T and T is an irreducible representation of A . This proves that every irreducible representation o f X arises by restricting the hoix omorphism of some irreducible representation o f A-
446
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4.1.7
(b) Clearly T is well-defined and hence is an irreducible representation of A jT when I C k er(r). On the other hand, if T is an irreducible repre sentation of A j T and tp: A —> A jT is the natural map, then T is obtained in the indicated way from the irreducible representation T o ip q{ A. □ Primitive and Prime Ideals We comment on the history of the next definition. Jacobson [1945a] introduced the term “primitive algebra” defining it to be an algebra in which {0 } satisfies condition (a) of our Theorem 4.1.8. In the same place he introduced the concept and notation of the quotient J : A of an ideal. The term “primitive ideal” was introduced by Jacobson also but in [1945b]. Prime ideals were defined by Wolfgang Krull [1928], cf. Neal H. McCoy [1949]. We have remarked earlier on the origin of the word “ideal” from the concept o f an ideal number in an algebraic number field. These were introduced in order to restore unique factorization, and thus the concept of a prime ideal was associated with the very origin of ideal theory. The reader should note that primitive ideals are defined in an asym metrical fashion. Since representations correspond to multiplication on the left, a primitive ideal is not necessarily primitive (although it is an ideal) in the reverse o f the algebra. This is the reason for the asymmetry of the notion o f quotient introduced in the next definition. 4.1.7 Definition An ideal is called primitive if it is the kernel of some irreducible representation. A proper ideal T is called prime if whenever Zi and I 2 are ideals satisfying I i J2 C I , then either I ( C I or 2j> C I . The set o f primitive ideals of sin algebra A will be denoted by I I * and the set o f prime ideals by P a If £ is a left ideal, then the set C :A = {a e A :a A C £ }
(7)
is called the quotient of £. In Chapter 7 we will introduce a topology on the sets I I * and P * , and thereafter they will be considered to be topological spaces. I f I is any ideal of A the symbols hn (T ) and hp ( l ) denote the sets hn (J ) = { V € l l A : l C T } hp ( l ) = { V 6 P a : I Q 'P } which are called the hulls o f J in IT * and in P a , respectively. These hulls, which will be the closed sets in the topology introduced in Chapter 7, will be studied in more depth there. It is easy to see that the quotient £ : A of £ is a (two-sided) ideal. If £ is a modular left ideal with right relative identity e, then any a € C : A,
4.1,8
Ideals and Representations
447
satisfies a = at ~ (ae — a) G £. Hence £ : A C £ holds. We refine this observation in the following theorem, which is essentially due to Jacobson [1245a], although of course he did not mention modular ideals. 4.1,8 T h eorem Let A be an algebra. (a ) An ideal V of A is primitive if and only if it has the form C : A for some maximal modular left ideal £. In this case V is the largest ideal included in £. (b) A primitive ideal is prime. (In fact, if A is the completion of a normed algebra B, then the intersection of a primitive ideal of A with B is a prime ideal of B .) (c) A primitive ideal V satisfies V = f l { £ : C is a maximal modular left ideal satisfying V = £ : A ). '(d) An element a € A is quasi-invertible if and only if it is quasiinvertible modulo each primitive ideal. (e) If I is an ideal of A , then the map V i—> V fl I is a bijection of II 4 \ hn( l ) onto I I j and the map P n P / I is a bijection of hn ( l ) onto /ZProcf (a): Suppose V is the kernel of the (non-trivial) irreducible repre sentation T : A —> C (X ). Choose 2 € X \ { 0 } and define £ by £ = {a € A : Tai ~ 0}. Then £ is a maximal modular left ideal and T is equivalent to IjA/c ^ s(i0wn jn Proposition 4.1.2 and Theorem 4.1.3. Clearly V is the kernel of and this kernel is £ : A. We have noted £ : A C £. Any ideal I C £ clearly satisfies X A C £ and hence is included in £ : A. (c): Since £ : A C £ holds for any modular left ideal £, we know that V is deluded in f l { £ : £ is a modular left ideal with V = £ : .4 }. To prove the opposite inclusion suppose V is the kernel of the irreducible representation T: A —> C (X ) and a belongs to A \ P . Then Taz is non-zero for some z G X. Thvs a does not belong to £ = {a € A : Taz = 0}. However, we have just neied that £ satisfies V = £ : A. (b): We will show that if V is a primitive ideal o f A then VC\B satisfies a condition formally stronger than our definition of a prime ideal o f B. Suppose £ 1 and £ 2 are left ideals of B satisfying C 1 C2 C. V = C : A, where £ is a maximal modular left ideal of A. Note that £ is closed by Theorem 2.4.7 and hence V = C : A is also closed. Since B is dense in A, the closures £ r sad £ 2 of £ 1 and £ 2 are left ideals in A. I f £ 2 is not included in £ : A, then £ 2^4 + £ properly contains the maximal left ideal £. This implies A = C2 A + £. Thus C y A = C ^ C 2A -I- £ f £ C V A + £ = £ implies £ 1 C £ : A = V. (d): If a is quasi-inveritible in A it is certainly quasi-invertible modulo each primitive ideal. In order to establish the converse, we will assume that o G A is quasi-invertible modulo each primitive ideal of A. First we show that a has a left quasi-inverse. Suppose not. Then Theorem 2.4.6 shows
448
4: Ideals, Representations and Radicals
4.1.9
that a is the right relative identity for some maximal modular left ideal, £. By assumption a is quasi-invertible modulo the primitive ideal C : A. Let 6 be a quasi-inverse for a modulo C : A so that b — ba + a = boa belongs to £ : A C £. Since b — ba belongs to £ this implies a € £ which contradicts our choice of £. Therefore a has a left quasi-inverse. If we denote the left quasi-inverse of a by b, then b + V must equal ( a + P ) v for each primitive ideal V . Hence b has a left quasi-inverse modulo ' each primitive ideal. The argument just given shows that b has a left quasi- < inverse. Hence b is quasi-invertible and a is its right quasi-inverse, so a = bq is quasi-invertible. (e): Theorem 4.1.6(a) shows that the rnap V *—» V H I is a surjection of n A \ hn(T ) onto n z . In order to show that it is an injection, suppose V U V 2 e l\A \ h P (l ) C PA \ hp ( l ) satisfy P , f l X = V 2 D l . Then P 21 C Vi H I C V 2 and T <£V2 imply ~P\ C V 2. A similar argument gives V 2 C V\ so that the map is a bijection. Theorem 4.1.6(b) shows that the map V i—>V / 2 is a surjection of h P (l ) onto HA/x, and this map is obviously an injection. □ Primitive and Maximal Modular Ideals The next result was noted by Jacobson [1945b] for unital algebras. 4.1.9 T h eorem A maximal modular ideal is primitive. Hence every proper modular ideal is included in some primitive ideal. In a commutative algebra every primitive ideal is maximal modular. Proof If M is a maximal modular ideal, then by Theorem 2.4.6 there is a maximal modular left ideal £ which includes M . Then M A C M C £ implies M Q C : A . Hence maximality implies M . — £ : A . Thus M . is primitive by Theorem 4.1.8. The second statement now follows by Theorem 2.4.6. If £ Is a maximal modular left ideal in a commutative algebra, then it is a maximal ideal so that £ C £ : A (which is clear for any two-sided ideal £ ) implies £ : A = £. Since any primitive ideal has the form £ : A for some maximal modular left ideal, this proves the last statement. □ Prime Ideals We will have much more to do with primitive ideals than with prime ideals, but the next theorem, which is due to McCoy [1949], will find a use. Note that it shows that the condition established for primitive ideals in the proof of Theorem 4.1.8(b) is not properly stronger than the definition of a prune ideal. 4.1.10 T h eorem The following conditions on a proper two-sided ideal V in an algebra A are equivalent. (a) V is prime. (bj For all a, b G A,aA b C V implies a G V or b €~P
4.1.10
Ideals and Representations
449
(c) For all ( left / right } ideals C\ and C2 of A , £ 1 £ 2 C V implies C i Q V or C 2 C V . Furthermore ifX is an ideal of A , then the map V i-> VC\X is a bijection of PA \ hp (X ) onto Pz and the map V i-> V /X is a bijection of hp ( I ) onto P.A/lProof (a) => (b): The inclusion aAb C V implies A a A A b A C V . Since V is prin:, we conclude that either A a A C V or A b A C P . I f the first inclusion holds, then (^41 a^41) a C A a A C V . However, since V is prime this implies o e t 'a ^ 1 C V . If the second inclusion holds, the analogous argument gives b £ A 1 bA 1 C V. ( b) =» (c): Suppose C \£ 2 C "P holds but C 2 is not included in V . Choose 6 e £ 2 \ V . Then any a € £ 1 satisfies aAb C £ j £ 2 C V . Thus (b) implies a € V since b <£ V . Hence C\ is included in V. (c) =► (a): Obvious. Finally we prove the last sentence. If V is a prime ideal of A which does aot include 1, then V fl X is a proper ideal of I . Suppose X\ and X2 are iceals in I satisfying X\X2 C V n l . Since V is a prime ideal o f A and satisfies (IX i)(X X 2) C V , either H i C V or I I 2 Q ~P must hold by (c). Simil",rly since either T (T \A l ) — [ l l \ ) A l C P o r J (J 2A 1) = (T I 2 ) A l C V holds and I C V does not, either T\ C V n I or X2 C V n I holds. Thus V fl J is a prime ideal of X. In order to prove that P ^ P n l i s a surjection onto the set of prime ideals of X , suppose that V is a prime ideal of X. Let V be the ideal of A defined by V = {a G A : XuX C V } . Since V is prime we conclude V = {a € A : Xa C V }. The inclusion V C V n l is obvious. To obtain the opposite incision, note that any b G V f l l satisfies I ( 6I 1) = (Xb) ! 1 C V X 1 = V Since P is a prime ideal o f X this implies b e bXl C V . In order to see that V is ;, prime ideal of A , let Xi and X2 be ideals o f A satisfying XiX2 C V. Then (XX\)(X12) C XX\X2 C V implies eijther XX\ C /P or X I 2 C V since V is a p ime ideal of X. Hence either I i C V or X2 C V holds, so that V is a prime ideal of A which clearly does not include X. The proof of Theorem 4.1.8(a) shows that V 1-* P n J is injective on PA \hp ( I ) , so it is a bijection. Lei
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4.J.11
nition 1.3.8) which is a branch of model theory sometimes associated with nonstandard analysis. We shall not investigate these roots but will use Mathieu’s characterization an a definition. This sterns from work of Gunter Lumer and Marvin Rosenblum [1959]. 4.1.11 D e fin itio n A normed algebra A is called an ultraprime algebra if there exists a positive constant k satisfying
llifcjy > *IHI Nl
Vfr,ceA
A constant k for which the above condition holds is called an ultraprime constant for A. A closed ideal J o f a normed algebra A is said to be an ultraprime ideal of A if A / I is an ultraprime algebra in its quotient norm. To paraphrase this definition, a normed algebra A is ultraprime with a (positive) ultraprime constant k if for any b, c G A there is some a G A satisfying Htacll > k\\b\\ ||a|| ||c||. To further simplify we often restrict a, or b and c, or all three to have norm 1. Note that the supremum of all ultraprime constants need not be an ultraprime constant. A (necessarily closed) ideal V is an ultraprime ideal o f a normed algebra A if there is some positive constant k such that for any 6, c € A there is some a £ A satisfying ||6oc + V\\ >
+ V\\ ||a + V\\ ||c + V\\.
Obviously an ultraprime ideal is a closed prime ideal. Unfortunately, not all primitive ideals are ultraprime ideals (Pere Ara and Mathieu [1991]). An example is the ideal { 0 } in the algebra Bns i ' H) ° f Hilbert-Schmidt operators. (L et b = c be a projection onto an n-dimensional subspace.) If X is a normed linear space, any (necessarily primitive) subalgebra A of B ( X ) which includes B f ( X ) is an ultraprime algebra and any positive number less than 1 may be used as an ultraprime constant. To see this, let S, T G A o f norm 1 and 1 > e > 0 be arbitrary and choose x G Xy and x * G X ( to satisfy ||S(x)|| > 1 — e and ||T*(x*)|| > 1 — e. Then ||S* 0 ** T|| = | | 5 (* )® T * (x ’ )|| = ||5(x)|| ||T*(®*)|| > (1 - e f . George A. W iliis [1989b] shows that for a discrete group G, (G ) is ultraprime if and only if G is an ICC-group (i.e., a group in which all conjugacy classes except { e } are infinite). As we shall see, any prime com mutative normed algebra (not isomorphic to C ) is an example of a prime but not ultraprime algebra.
•M 12
Ideals and Representations
451
<..2,12 T heorem Let A be a normed algebra. (a) A is an ultraprime algebra i f and only i f one (hence all) o f its dense subnlgebras is an ultraprime algebra. Hence the completion o f an ultraprime normed algebra is ultraprime. (b) I f A is an ultraprime algebra, then all o f its ideals are ultraprime algebras. (c) A is an ultraprime algebra i f and only i f A 1 is an ultraprime algebra. (d) I f A is an ultraprime algebra, then its center is trivial. Proof (a): Let B be a dense subalgebra. If B is an ultraprime algebra with an ultraprime constant k and b, c G A are arbitrary with norm 1, choose b', d and a in B satisfying ||6'|| = 1, He'll = 1, ||&—6'|| < k/3, ||c—c'|| < k/3 and H&'ac'H > fc||a||. Then bac = 6'ac' + (6—b ')a c+ b 'a (c—cf) implies ||6ac|| > ||^||-||(6-V)oc||-||6'a(c-c')|| > k||a||-(k/3)||a||-(k/3)||a|| = (fc/3)||a||. Hence A is ultraprime. Conversely, if A is an ultraprime algebra with an ultraprime constant k and b, c G B are arbitrary, choose a £ A satisfying ||6ac|| > k\\b\\ ||a|| ||c||. 3 y approximating a by an element in B, we see that B is an ultraprime algebra.
(b): Since A is an ideal of itself, it is enough to consider an arbitrary ideal I in an ultraprime algebra with an ultraprime constant k. Let b, c G 1 be arbitrary with norm 1 . Then we can find a £ A o f norm 1 satisfying |[6ac|| > k. Again we can choose d € A o f norm 1 satisfying \\bacdc\\ > fcj|6ac||. Hence acd G J satisfies ||6(acd)c|| > ft||6ac|| > k2. Hence J, an arbitrary ideal, is an ultraprime algebra. 'c): If A 1 is an ultraprime algebra, then A is also by (b). Suppose A is & norsunital ultraprime algebra with ultraprime constant k. W e give A 1 the "ora i ||A + a|| = jA| + ||a|| as usual. Let p, + b and v + c in^t1 be arbitrary and choose a G A to satisfy ||6oc|| > k\\b\\ ||a|| ||c||. Then we also have I I M llcll ^ k\M lla ll llcll so I I H > *||6||||a|| and similarly ||ac|| > fc||a|| ||c||. Hence the inequality II (m + b)a(v + c) || > >
||/iac|| + ||i/6a|| + ||6ac|| fc||/i + 6||||a||||>/ + c||
shows that ^4 * is ultraprime. (d): The center A z is called trivial if it is zero- or one-dimensional. If A z is not trivial, Proposition 2.5.12 shows that it has a topological divisor o f zero. Suppose c G A z and the sequence {& „}„gN C A z satisfy ||c|| = 1, i|6n|j = 1 for all n G N and bnc —* 0. Then any sequence { a n} „eN C A of ■norm 1 elements would satisfy bnanc —►0 which contradicts the ultraprime character of A. □ Mathieu [1989] shows that a normed algebra is ultraprime if and only if 3 j.ii \ hence every, ultrapower is a prime (or, equivalently, an ultraprime)
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4.1.12
algebra. He uses this to prove several o f the above results and also that every matrix algebra M n( A ) over an ultraprime algebra is ultraprime. We now indicate the construction of the normed symmetric algebra of quotients of an ultraprime normed algebra. For details, see Mathieu [1991]. We could carry out the construction as a direct limit, but we will proceed in another way.
Mathieu [1989] constructs a larger one-sided algebra of
quotients, but we judge the symmetric version to be of more general interest. It generalizes the double centralizer algebra. Let A be an ultraprime normed algebra with k as an ultraprime con stant. Consider the set T of triples (J, L, R ) where I is a non-zero ideal and L, R: I —* A are continuous linear maps satisfying the double centralizer equation: aL(b) = R (a )b V a ,b e J . (8) Thus these maps satisfy L(ac) = L (a )c and R(ba) = bR(a) for all a € J and 6,c G A by the argument o f Theorem 1.2.4, since even in a prime algebra the annihilator ideals are zero. (Note that the results o f Proposition 1.2.3 are also available.) We introduce an equivalence relation on T by (J , L , R ) ~ (J ' , L ', R ') <=*• L|znj' = L '\ in r and R\m i' = R'\ira'Because A is a prime algebra, no product or intersection o f non-zero ideals is the zero ideal. Also if there is any non-zero ideal C C I n l ' o n which L agrees with L ' and R with R ', then (J, L, R ) and ( I ' , V , R ') are equivalent. This establishes the transitivity o f the equivalence relation. Denote the equivalence class o f (X, L, R ) by [X, L, R\. As a set, let B be the set o f these equivalence classes o f triples. The algebra operations on B are defined by [ X . M l + C T .Z ', # ]
=
A[X, I , R]
=
[ l n T , L + L ', R + R!) [X, XL, AR]
[1 , L , R )-\ T ,L ',R !]
=
[ I T P i T l ,L L ',R !R }.
For any triple (X, L, R ) in T and b, c G X, we can find a € A satisfying ll&ll I lill Hall IMI > m a c )|| = \\R(b)ac\\ > fc||A(6)|| ||a|| ||c||
which implies ||Z|| > fc||fl||. Similarly we find ||fl|| > fc||L||. Again, for any pair of equivalent triples (X, L, R ) ~ (T ', L ', R !) and any b G X and c G X', we can find an a € A satisfying Ill'll ||6|| ||a|| ||c||
>
||i'(6ac)|| = ||L(6ac)|| = ||L(6)ac||
>
fc||i(6)||||a||M
which implies ||L'|| > fc||L||. Obviously, ||i?'|| > fc||/i||, ||Z|| > k\\L'\\ and ||i?|| > fc||H'|| also hold. Hence if we define a norm on B by ||[ I , L, R] || = inf{max{||L'||, H^H} : ( I , L, R ) ~ (X', V , & ) } ,
4.;..3
Ideals and Representations
453
we co get an algebra norm rather than just a semi-norm. Finally, we can defh 3 a homeomorphic isomorphism of A into B by a i-+ (A , L a, R a). We wil! state the resulting theorem, but leave the rest o f the details to the reader who may consult Mathieu [1991],
4,1.13 Theorem Let
B be the normed symmetric algebra of quotients as
defined above. Then B is a unital ultraprime normed algebra containing a hom eomorphic and isomorphic image of A . It is the largest normed algebra extension of A satisfying: (a) For each b G B there is a non-zero ideal 1 of A satisfying b l C A and lb C A. (b) If b € B and a non-zero ideal 1 of A satisfy b l — { 0 } or l b = { 0 } then 6 = 0 . If A is an algebra, then an operator T: A —» A is an elementary operator if it has the form T = i L aj Rt,. for some n € N and a j , bj € A. The? a is an obvious homomorphism 6 o f A ® A R (where A R is the reverse algebra o f *4) onto the algebra £ l (A ) o f elementary operators on A. Usually this lomomorphism has a nontrivial kernel. Ultraprime normed algebras provide an exception. The elementary proof is given in Mathieu [1989]. (See Erling St0rmer [1980] for related interesting ideas.) 4.1.24 P roposition Let A be an ultraprime normed algebra. With the notation introduced above, 0 is an isomorphism of A ® A R onto 81(A). Ideals and Arens Multiplication The following results o f Paul Civin and Bertram Yood [1961] describe the behavior of ideals in relation to the Arens products on the double dual algebra. Nilpotent ideals are introduced in Definition 4.4.1. 4.1.15 P roposition Let A be a Banach algebra and let 1 be a ( left / right / two-sided ) ideal of A. L et1 be the weak* closure of k (1 ) in A**. (a) I is a ( left / right / two-sided ) ideal of A ** with respect to both Arens products. (b) If 1 is a proper modular ideal, 1 is also a proper modular ideal with respect to both Arens products. (c) If 1 is a primitive ideal, 1 is included in a primitive ideal of A " with respect to both Arens products. (6 ) If 1 is nilpotent, 1 is nilpotent with respect to both Arens products. P r o o f (a): We consider only the case in which J is a left ideal since the other cases are similar. Let ( / = lim K.(aa) / g = lim n(bfj) ) be an arbitrary elerr-fcnt o f ( .4** / I ) with ( aa / bp ) elements of ( A / J ) and the limit ii the weak* topology. We use the continuity properties (14) and (15) from Theorem 1.4.2. Consider the first Arens product and a fixed a. Then
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4.1,16
* ( aa)
Ta (TL(b)X - TL(c)y)
=
TaL(b)X - TaL(c^y = T R^ bx - T fi(a)cj/
=
T R(a)(T bx - Tcy) = 0.
The irreducibility of T implies T i ^ ) x = T :t > (A ) - * C (X ) by T (LJt)x - T L(c)y for any x
Therefore we may define G
X and ( L , R )
G
V (A )
where x — T cy■ (When convenient, we could also write T ^ ^ x = T ^ e)X where x = Tex .) This obviously defines a representation extending T , since it is well defined by the calculation above. It is also clear that any extension o f T would have to agree with T . Finally, T is irreducible since it extends an irreducible representation. D
4.1.17
Ideals and Representations
455
Modules, Normed. Modules and Bimodules We close this section by formally defining modules and describing the :der'^cation between the theory o f representations and the theory o f mod ules mentioned in the introduction. 4.1.17 D e fin itio n Let A be an algebra and let X be a linear space. Then X is called a left A-module if there is a fixed representation T : A —> C (X ) and Tn(x ) is denoted by ax for all a S A and x £ X . H A and X are both normed and the representation T : A —> B (X ) is continuous, then the corresponding module is called a normed left A-module. The norm o f T (as a bounded linear map from A to B ( X ) ) is called the the bound o f the normed module X . In the above situation, if both *4 and X are complete, the left module is called a left Banach module. I f X is a linear space and S: A —►C ( X ) is a fixed anti-representation o f A on X with its action denoted by xa = Sa(x ) for all a G A and x 6 X , then X is called a right A-module. Normed right A-modules and right Banach A-modules are defined analogously to left ones. If X is both a left find a right .4-module, and if the two module actions satisfy a(xb) = (ax)b V a, b G A ; x G X then it is called an A-bimodule. (This means that the related representation anti-representation have commuting actions.) If both module actions are normed module actions, then the bimodule is said to be normed. The larger of the bounds for the left and right module actions is called the bound for ih;3 bimodule action. If, in addition, both A and X are complete, the bimoc ule is called a Banach bimodule. Note that a left ,4 -module is a linear space X with an action of A , denoted by juxtaposition, and satisfying (a + b)x = ax + bx\ (X a )x — X (ax) = a(Xx)\
a (x + y) = ax + ay, and
(ah)x = a(bx)
for all a, b G A , x, y G X and A G C. For right modules,the very last of these conditions must be replaced by x(ab) = (xa)b for all a, b G A and x € X. l f T : A —> C ( X ) is any representation, it defines a left ^4-module struc ture on X . Conversely, if X is a left ^4-module, then the map T : A —» C (X ) defined by Ta(x ) = ax V a e A; x e X is z representation o f A on X . Clearly these are inverse constructions. A ll terminology for representations is extended to modules in the obvious way.
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4: Ideals, Representations and R adicals
4.2.1
A left A-module action o f a normed algebra A on a normed linear space X defines a normed left .A-module if there is some finite constant M satis fying ||ox|| < M\\a\\ ||a:|| V a € A\ x e X . The infimum o f all constants M having the above property is the bound of the normed module X .
4.2
Representations and Norms
Definitions We now introduce norms into these purely algebraic considerations. The reader may wish to compare the following definitions with those given in Definition 4.1.1. Although both irreducible representations and topologi cally irreducible representations have been studied since the beginning o f Banach algebra theory, sometimes no distinction in terminology was made. As we will see in Chapter 9 o f Volume II topological irreducibility playS a key role in the representation theory o f *-algebras while irreducibility is nor mally more important in ether contexts. The terminology we adopt seems to have been first used systematically in Rickart’s important book [I960]. Some authors use “algebraically irreducible” or “strictly irreducible” for what we call simply “irreducible” . Sometimes they then use “irreducible” for what we are calling “topologically irreducible” . The same remarks hold for “cyclic” and “topologically cyclic” . The other terminology of this defi nition also seems to have been standardized by its use in Rickart’s book. 4.2.1 Definition Let T be a representation o f an algebra A on a normed linear space X . Then T is called: (a) Normed if Ta £ B (X ) for each a € A . (b) Topologically cyclic if there is a vector z £ X (called a topologically cyclic vector) such that T A z = { T az : a € A ] is dense in X . (c) Topologically irreducible if {0 } and X are the only closed T-invariant subspaces, and T is not trivial. (d) Topologically equivalent to a representation S o f A on a normed linear space if there is a homeomorphic linear bijection U (called a topo logical equivalence) o f X onto which satisfies SaU = U T a
V a £ A.
(1)
I f ( A , a ) is a semi-normed algebra then a representation T of A is called: (e) Continuous if it is normed and continuous as a map from ( A , a) to B( X) .
( f ) Strongly continuous if a •—►Tax is continuous as a map from (^4, a) to -V for each fixed x £ X ,
4.2,2
Representations and N orm s
457
We will informally record a number o f simple consequences o f this definitk a. Note that it is easier for a representation on a normed linear space to b-3 topologically cyclic or topologically irreducible than to be cyclic or irrecucible, but that it is harder for it to be topologically equivalent to another representation than to be equivalent. 1' T is a normed representation, it is obvious that the closure o f every T-invariant subspace is T-invariant. Any restriction of a normed, continuous or strongly continuous represen tation has the same property. Reduction by a closed T-invariant subspace also preserves each o f these properties. 13t A be a spectral normed algebra and let £ be a modular left ideal of A . 1'hen L ^ c will be irreducible if and only if £ is maximal. However it is also easy to see that L A! C will be topologically irreducible if and only if £ is maximal among closed ideals. Since the two concepts (irreducibility and topological irreducibility for L A/C) agree for any closed left ideal such as £ , then L a ! c is topologically irreducible if and only if it is irreducible. We will find the same result in some other contexts, notably for ^-representations o f C*-algebras, but the two concepts do not usually agree. Strongly Continuous Representations Note that a continuous representation o f a semi-normed algebra on a norsnsd linear space is always normed and strongly continuous. The con verse is not always true. However, the uniform boundedness principle gives a simple result in that direction for Banach algebras. 4 .2 .' P r o p o s itio n A normed strongly continuous representation o f a Bantch algebra on a normed linear space is continuous. P roo f Let T be a strongly continuous normed representation o f a Banach algebra A on a normed linear space X . Apply the uniform boundedness principle to {S * : x € X i } where (as usual) X\ is the unit ball o f X and Sx {a) = Ta(x )
V a e A x e X .
Strong continuity shows that each linear map Sx is continuous from A to X . The fact that T is normed shows: ||5s (fl)|| = ||ra*|| < IIT J 11*11 < IIT J Hence, there is some M satisfying IIS'* II < m which implies
V x e-V j
V a G Ai x € -V.
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4: Ideals, Representations and Radicals
so that T is continuous.
4.2.3
□
We now obtain the analogue of Proposition 4.1.2 which first occurred explicitly in Rickart [I960]. Closely related results were discussed in Segal [1947a] and Hille [1948]. N ote that the next two results deal with cyclic rather than topologically cyclic representations. 4.2.3 T h e o r e m Let T be a strongly continuous, cyclic representation o f a semi-normed algebra (A , cr) on a normed linear space X with cyclic vector z. Then L A^C is equivalent ( under a continuous equivalence) to T where C is the closed modular left ideal C = {a G A : Taz = 0 }. I f A is a Banach algebra and X is a Banach space then L A/C and T are topologically equivalent. P roo f The proof o f Proposition 4.1.2 shows that the map U :A / C —» X defined by U (a + C ) — Taz establishes an equivalence between L A^C and T . If T is strongly continuous then C is certainly closed so the quotient semi-norm o ' is a norm. Let M satisfy |[Ta,z|| < M a (a ) for all a G A . Then ||i/(a + £)|| < M o '(a + C ). Thus U is continuous. Suppose in addition that A is a Banach algebra and X is a Banach space. Then A/C is a Banach space under its quotient norm. Since U : A/C —►X is a continuous bijection it is a homeomorphism and hence a topological equivalence. □ Proposition 4.2.2 recorded that the uniform boundedness principle shows that a normed strongly continuous representation of a Banach algebra is continuous. In the following corollary, which does not seem to have been ex plicitly noted before, we do not require that the representation be normed. 4.2.4 C o r o lla r y Any strongly continuous cyclic representation o f a Ba nach algebra on a Banach space is continuous. Any equivalence between two strongly continuous cyclic representations o f a Banach algebra on Banach spaces is a topological equivalence. P roo f The first statement follows from the last statement o f Theorem 4.2.3 since L A>C is obviously continuous. To prove the second statement, suppose T '.A —* C ( X ) and S : A —> C ( y ) are strongly continuous cyclic representations o f the Banach algebra A on the Banach spaces X , y. Let U: X — ► y be an equivalence. Let z € X be a cyclic vector for T. Then U z is a cyclic vector for S and we may define C by C = {a G A : Taz = 0 } = {a € A : SaU z ~ 0}. Theorem 4.2.3 asserts that the maps a + C i—> Taz and a + C *—* SaU z are topological equivalences. Thus T az >—►SaU z is a topological equivalence. However, this map is just U since SaU z = U T az holds for all a G A . □ In Corollary 4.2.16 results similar to those of the above corollary will be obtained again under different hypotheses using the much deeper Theorem 4.2.15.
Representations and N orm s
459
Topologically Irreducible Representations
The next result contains a partial analogue o f Theorem 4.1.3 and Propo sition 4.1.5. Conclusion (a) is an exceptional result because it draws an algebraic conclusion from a partly topological hypothesis. 4.2.6 P r o p o s itio n Let T be a normed representation o f an algebra A on a,.normed linear space X . The representation T is topologically irreducible i f and only i f every non-zero vector z e X is a topologically cyclic vector. I f these conditions hold, then: (a) The kernel o f T is a prime ideal. (b ) The restriction o f the homomorphism T to an ideal 1 o f A is either trivial or topologically irreducible. (c) I f I is an ideal included in k er(T ), then Ta+ x — Ta
V a6 A
defines a topologically irreducible representation o f A / T . Every topologically irreducible representation o f A / 1 arises in this way. P roo f The equivalence o f the two conditions is proved as in Theorem 4.1.3 the easy fact that the closure o f a T-invariant subspace is T-invariant. (a): To prove this statement, suppose C\ and C2 are left ideals satisfying C 1 C 2 Q ker(T) and C2 9? ker(T). Then = span(Tc2X ) is a nontrivial T-invariant subspace. Hence y is dense in X . Let a belong to C\. Then C ker(T) implies Tay — {0 }, and this implies Ta = 0 since y is dense. Therefore L\ is included in ker(T). This proves that k er(T ) is prime. ! (b): The set { x G X : T jx = { 0 } } is a closed T-invariant subspace and hencs must be {0 } or X. I f it is X, the restriction o f the homomorphism T to I is trivial. I f it is {0 }, then for each non-zero x e X the closed T-invariant subspace (Tjx)~~ must be X . Hence in this case the restriction of tbs homomorphism T to I is topologically irreducible. (e): Clearly the two sets of operators {T a : a G .4 } and {T a+ i : a + 1 G A / T } are equal when T and T satisfy the equation in (c). Hence if either set is topologically irreducible, the other is also. □ Note that the proofs o f (b) and (c) above are essentially the same as the proofs o f (a ) and (b) o f Theorem 4.1.6. However, we cannot extend the sacor d sentence o f Theorem 4.1.6(a) to the present situation. Here is the tremble. i S-ippose T is an ideal o f A and T : I —» C (X ) is a normed topologically irreducible representation o f I on X . Then { x € X : Tb(x) = 0 for b G 1 } is a closed T-invariant subspace which must be {0 }. Suppose 6i , 62, • • •, bn € I and z\,z 2 , . . . , z n G X satisfy ^ 7 = o ~ Then any a G A and d e l
4: Ideals, Representations and Radicals
460
satisfy
n
n
n
Td y * Ta)>i Z-, — y * Tdab, zj = Tda y *
i=o
j =0
4.2.5
Zj = 0.
J=0
Hence 53"_ 0 Tab1Zj belongs to the subspace which we have just shown to be {0 }. Therefore we may define T G £()>)> where n T a (^ 2 ,T b ,Z ]) =
j =0
= span (TjA '), by
n Tab,Zj
V a € A , n G N ; bj £ T ; Zj G X .
j =0
Clearly T is a topologically irreducible representation o f A on y which ex tends the restriction of the representation T to y . In general there seems to be no reason for T a to be a bounded operator on y and hence no way to extend T to be a normed representation of A on X. (If I is an approx imately unital Banach algebra, T is continuous and X is a Banach space then Corollary 5.2.3 gives an easy proof that the extension is possible.) However, we do not actually know of any topologically irreducible normed representation T o f an ideal J in an algebra A which cannot be extended to a normed representation of the whole algebra A . O f course, the most interesting counterexample would have A a Banach algebra, Z closed, the representation space a Banach space and T continuous. Topologically irreducible representations are not well understood and they can behave very badly. For instance let T € B ( X ) be one of the oper ators without invariant subspaces discussed before Definition 2.8.8. Then the identity representation of the commutative unital subalgebra A o f B( X) generated by T is a continuous topologically irreducible representation. The inclusion A C A shows that this representation has a nontrivial commutant which is not a division algebra by the Gelfand-Mazur Theorem (Corollary 2.2.3). Compare Schur’s lemma (Theorem 4.1.4). A normed representation T o f an algebra A on a normed linear space X is said to be topologically completely irreducible if for any S G C( X) , any e > 0 and any elements x l , x 2, ■■. , x n G X there exists an a € A satisfying \\TaXj - Sxj\\ < e
V j = 1 ,2 ,.. .,n .
It is not known whether continuous topologically irreducible representa tions of a Banach algebra on a Banach space are topologically completely irreducible. For some partial results in this direction involving the existence of finite-rank operators, which are due to J. M. G. Fell and J. Dixmier, see Example 4.8.8. See also Wieslaw Zelazko [1991].
Spectral Sem i-norm s and N orm s on Representation Spaces We now invcHtigate wmte of these concepts for spectral semi-nornied algebras and spectral algebras. The following result is implicit in Irving Kaplansky [1947a], Segal [1947a] and Hille [1948],
4,2.
Representations and N orm s
4.2, ' Proposition is closed.
461
A prim itive ideal in a spectral semi-normed algebra
P roo f Suppose V is a primitive ideal in a spectral semi-normed algebra (A ,cr). Theorem 4.1.8(a) shows that there is some maximal modular left ideal C satisfying V = C : A . Theorem 2.4.7 shows that C is closed. The definition o f C : A now shows that V is closed. □ V/e next generalize Theorem 4.2.3 and also restate the conclusion in a way that will be useful. Theorem 4.2.8 will then show that this generalized theorem applies to irreducible representations o f spectral semi-normed al gebras. The following two results are due to Rickart [1950] in the case of Ranc.eh algebras.
Theorem Let T be a cyclic representation o f a semi-normed algebra (A , cr) on a linear space X . I f z is a cyclic vector fo r T and i f {a 6 A : Taz “ 0 } is closed then 4 .2 .7
||x||z = inf{
V x e X
(2)
defines a norm on X such that T is a continuous representation on (X , ||• ||2) satisfying ||Ta||z < cr(a) fo r all a 6 A . I f A is a Banach algebra, then II ||z) is a Banach space. I f in addition X is already a Banach space under some norm || • || and T is strongly continuous with respect to || • ||, then !| • || and ||• ||z are equivalent. Proof Let C = {a G A '■Taz = 0 }. Then ||• ||z is simply the quotient norm o f A/C transferred to X by the equivalence established in Proposition 4.1.2. The inequality ||Ta||z <
462
4: Ideals, Representations and Radicals
4.2.10
The following result o f Larry Schweitzer [1992] will be used later. 4.2.10 T h e o r e m Let A be a spectral normed algebra. Let B be a normed algebra and let
G
) } = C.
(c) For any irreducible representation T o f B there is a representation S o f A and a linear injection V : X T —> X s satisfying: V T bx = Sv W V x
V b £ B ,x € X T.
When these conditions hold: (d) The closure in A of the image under (p of each proper modular ideal of B is a proper modular ideal of A. (e) The representation S can be taken to be irreducible. ( f ) The linear map V is continuous and V ( X T ) is dense in X s in the norms introduced in Theorem 4.2.7 fo r the representation spaces X T and X s.
Before giving the proof o f this theorem we wish to point out explicitly that we will use weaker hypotheses than given in the formal statement of the theorem. For all the results except ( f ) it is enough to assume that A is a topological algebra with an open set o f quasi-invertible elements. For the whole theorem it is enough to assume that 5 is a topological algebra. In (f) the norm on X T can be considered as arising from either the norm giving the topology o f B if there is such a norm, or from the norm induced by ip from A W e preferred to use norms as usual in the formal statement, but we will use these weaker hypotheses to generalize some results in Volume II. In many applications B is actually a dense subalgebra o f A. Proof We will actually prove this result under the above weaker hypotheses which were originally used by Schweitzer: Let A be a topological algebra with AqG open and let B be a topological algebra. We will use jC to denote the closure in A of (£)■ (a ) => ( b): Let e be a right relative identity for C in B. By Theorem 2.4.6(b) we conclude that {e — b : b G £ } is disjoint from BqG- Since A^ g n p (B ) = ^ (B q c ) and AqG is open, AqG is disjoint from {
< .2.11
Representations and Norm s
463
the maximality of £ .) Thus £ is a proper left ideal in B including C which must therefore equal £ by maximality. (o) => (c): (Condition (b) is now interpreted to say that £ equals £.) Let T be sin irreducible representation o f B on X and choose a non-zero 2 e . Define £ = {b € B : Tbz — 0 }. Then £ is a maximal modular left idea! in B and the map b + C i—►Tbz provides an equivalence between L B^C and T by the proof o f Proposition 4.1.2. Now use the notation already established. Note that £ is a proper modular left ideal in A by the density of ip{3) in A . Hence we can take S to be the representation L A^ on A/C and V to be the (well-defined) map TdZ i-+
— S cS p ^ Z ■ Scz — Scz
Scz — 0.
To prove the last statement in the theorem we may use all three condi tions and the construction in the paragraph above. Thus conclusions (d), {'s) wxi (f) follow easily. □ Strict Density o f Irreducible Representations The next theorem, which was proved in the case of Banach algebras by Fickart [1950], is an important consequence of Schur’s lemma (Theorem 4 .1 .4 ) and the Gelfand- Mazur theorem (Corollary 2.2.3). 4.2.1 1
T h e o re m
Let T be an irreducible representation o f a spectral
algebra A on a linear space X . Then
(T a Y = { $ e C {X ) : ST a = TaS a e A } is the set o f complex multiples o f the identity map on X . Proof Choose a spectral semi-norm a on A . Let z € X be non-zero and let jj ■lj2 be as defined in Theorem 4.2.7.
Suppose S belongs to ( T 4 )',
464
4: Ideals, Representations and Radicals
4.2.12
Then for x = Taz we get S x = S T az = TaS z which implies j|Sx||2 < ll^olljll^llj, <
A representation T of an algebra A on a linear space
X is called strictly dense if whenever x x, ar2, .. ■, x n is a finite list o f linearly independent vectors in X and y i , y t , . - - , y n is a list of vectors in X there is an element a £ A with T ax } ~ j/j for j = 1 , 2 , . . . , n. The definition can be rephrased by saying that, for all n € N, TU is nfold transitive on linearly independent sets of vectors in X. The left regular representation o f the quarternions (a real rather than complex algebra) is irreducible but not 2-fold transitive, and hence not strictly dense. However, for (complex) spectral algebras we have the following remarkable theorem. This important result, which was mentioned in the introduction to this section, is due to Rickart [1950], A version for a closed irreducible algebra o f operators on a Banach space was obtained by Yood [1949]. The refinement stated second in the theorem is due to Allan M. Sinclair [1976] for Banach algebras. The result was known earlier for C*-algebras. Recall that a functional algebra is simply a spectral normed algebra in which the analytic functional calculus is prnHorvwd. 4.2.13 T h e o r e m Any irreducible representation o f a spectral algebra is strictly dense. I f T is an irreducible representation o f a functional algebra A and X i, x 2, . . . , x „ and j/i, y2, .. ■, yn are both finite lists o f linearly inde pendent vectors in X , then there is an element a £ A satisfying Te*Xj = yj fo r j = 1 , 2 , . . . , n . P roo f Let T be an irreducible representation o f a spectral algebra A on a linear space X. We use induction on n > 2 in the following statement: For any list x i , x 2, . . . , x n of n linearly independent vectors, we can find an a € A satisfying Tax} = 0 for j = 1 ,2 ,..., n — 1 and Tax„ ^ 0. First we show that the statement is true for n = 2. Suppose not. Then there are linearly independent vectors y, z £ X such that for all a € A , T ay = 0 implies Taz = 0. For any x £ X, choose b (x) £ A so that T ^ y = x and define S: X —» X by S x = T blx)z. Then S is well-defined by the assumption on y and z. Thus S is linear. We now show that S belongs to (T A )'. For any a £ A and x £ X , b(Tax ) = ab(x) follows from TaT ^ x )y = Tax. Hence, ST ax = T ab^ z = TaT ^ x)Z = TaS x shows S £ (T A )'. Theorem 4.2.11 now shows that S must be just multiplication by some scalar, A £ C. Then T^z = S l\ y = X7\(x)y holds for all b £ A - However this implies TA (z — Xy) — { 0 } and hence z ~ Xy. This contradicts the independence of y and z and shows that the
4.2.14
Representations and Norm s
465
statement holds for n — 2 . Now assume the statement for some n > 2 and let x lt x 2, ■■■, x n+i be a list of n + 1 linearly independent vectors. Choose a 6 A satisfying Tax i = Tax 2 — . . . — Tax n- 1 = 0 and Tax n+1 ^ 0. If Tax n is zero, then a has the desired property. If T ax„ and Tax n+i are independent, choose c - A so that T cT ax n = 0 and T cT ax n+1 ^ 0. Then ca has the desired property. Thus we may suppose Tax n+i = XTax n for some A € C. Choose 6 in A satisfying T bi i = T bx 2 = ■■• = T bx „ -\ = 0 and T b(x n+1 - Axn) ^ 0.
I f T:,xn is zero, then b has the desired property (since Tbx n+i ^ 0).
If
TbXn and Tbx n+X are independent, then a c € A can be chosen as before so that cb has the desired property. Thus we may suppose T bx n+y = fiTbx n for some /x G C. Clearly fi does not equal A. Choose d € A satisfying T/£'iXn = Tax n . Then a - db has the desired property since Ta-dbXn+ i = XTa.;n —TdTbx n+l = XTax n —TdTbfix n = (A —fi)T ax n 0. This completes the Induction. How suppose x i ,x 2 , ■■■,xn are linearly independent and yi,ya,...,yn are arbitrary. By what we have just proved we can find a} e A satisfying T 0j x.j ^ 0 and Ta.Xi — 0 when i ^ j for j = 1 , 2 , . . . , n. Use the irreducibil ity c f T to find bj € A satisfying TbT aj Xj = y} for j = 1 ,2 ,..., n. Then a = h
4 .2.14
Prooj By Theorem 4.2.13, we can successively choose ap + i, ap+2, ... such that ,;a„|| < 2 ~n, Ta„X 1 = Ta„X 2 = • • • = TanXn_ i = 0,
466
4: Ideals, Representations and Radicals
4.2.15
and Tanx n
span{T(,nX i, TJ,nx 2, . -., TJ,nx n}
all hold where bn is defined by bn = ap+ 1 + ap+2 + • • • + an_ i. Let a be an. Give X the complete norm described in Theorem 4.2.7 ho that T becomes continuous. Then Tax m = E ^ L p+ i Tanx m — 0 holds for m < p and Tax q = E ^ = p + i T« » x
x g = T° qx
holds for q > p. Also Tax k = £ “ =p+1 Ta„ x k = E™=pVi T*nx k = Thmx k holds for any p < k < q. Hence Tax q = Taqx q + Tbqx q is independent of {T ax k : 1 < k < q — 1} = {T^X k : 1 < k < q — 1} for all q > p. □ Continuity o f Irreducible representations We now derive the most important consequence from this circle o f ideas. The following result is a slight restatement o f Barry E. Johnson’s main re sult in [1967a], This was the final step in his original proof o f the uniqueness of the Banach algebra topology o f a semisimple Banach algebra (Corollary 2.3.10). We will discuss this and other important consequences and exten sions in Section 6.1. Our proof is essentially Johnson’s original one. 4.2.15 T h e o r e m Let A be a Banach algebra. Any irreducible normed representation o f A on a normed linear space is continuous. P roo f Let V be the kernel of the irreducible normed representation T of A on a normed linear space X . Then V is primitive and hence closed by Proposition 4.2.6. Thus we may, and do, replace A by A / V and assume that T is faithful. To avoid double subscripts we then consider A as embedded in B ( X ) as an irreducible algebra of operators with elements A , B , C , etc. However we shall continue to use ||yl|| to represent the norm o f A as an element of A . By Proposition 4.2.2, it is enough to show that T is strongly continuous (i.e., for each x G X , A —> A x is continuous on A ). If X is finite-dimensional, then ,4 C B ( X ) is also finite dimensional. Hence the linear map A —» A x would be continuous on A for each x G X . Thus we may assume that X is infinite dimensional. Furthermore, the commutator A ' o f A in B ( X ) is C l by Theorem 4.2.11. Thus we may choose a sequence { x n} ngN of vectors in X which are linearly independent over A ' and satisfy ||a;n|| = 1 for all n G N. Suppose x G X is an element such that the map A h-» A x is continuous on A . Then for each fixed B e A , the map A >—> A B *-+ A B x would be continuous for all A G A. However, the irreducibility of T shows { B x : B G A } — X for any non-zero x € X . Thus either the map A ^ A x is continuous on A for all x G X or it is discontinuous for all non-zero x G X . By deriving a contradiction from the second possibility we will establish the theorem. Assume A >—» A x is discontinuous for each non-zero x € X . W e first show that for each K > 0, e > 0, and m e N there is an A G A satisfying:
4.2.16
Representations and N orm s
467
(a) ||4| < e; (b) A x i = A x 2 = • ■• = A x m- i = 0 ; (c) ||Arm|| > K . Let K , e and m be fixed for the rest of this paragraph. Let M n be l.bc set {A 6 A : A x n = 0 } . Then each M n is a maximal modular left ideal o f A . Define 72 and £ by 72 = M \ D M 2 D • • • D M m-\ and £ R -f M m- Theorem 4.2.13 shows that there is an element B G A satisfying 3 x i = B x z = • ■• = B x m- 1 = 0 , B x m = x m ^ 0 . This element B belongs to % C £ but not to M m so that the maximality o f M m implies £ = A . Thus addition defines a continuous linear map of 72. © M m onto A . By the open mapping theorem there is a constant 6 > 0 such that for any B 6 A satisfying ||B|| < 8e there will be elements A € 72 and C G M m satisfying B = A + C and ||A|| < e, ||C7|| < e. Since A —> A x m is discontinuous, we can choose an element B G A satisfying ||B|| < 6e and ||J3xm|| > K . Now choose A and C as just indicated. Then A satisfies (a), (b), and (c) (since C :;m = 0 ) as required. . By induction we can choose a sequence {j4 n}neN ° f elements o f A sat isfying: (a’ ) \\An \\ < 2 - » ; (b’) A nx i = A nx 2 =:•■■ = A nx n- i = 0; (c’ j j|A nx n|| > n + ||i4ixn + • • • + i4n_ 1 x„||. Define Bk G A by Bk = J2n>k Since each A n belongs to the closed ideal M k for all n > k, Bk belongs to M k - Thus for each k G N, we get ll-BoXkH
=
WA^k + A 2x k + ■■■+ A kx k + B kx k\\
>
IMfcXfcH - ||i4ix* + - " + A fc_ix*||
>
fc = fc||xfc||.
Scwever, B 0 is a bounded operator since T is a normed representation. This contradiction establishes the theorem. □ 4.S. 16 (a) then X (b) tations
C o r o lla r y Let A be a Banach algebra. I f T '. A —» C (X ) is an irreducible representation on a linear space, has a unique Banach space topology relative to which T is normed. An algebraic equivalence between two normed irreducible represen o f A on Banach spaces is a topological equivalence.
P roo f (a): Theorem 4.2.8 gives such a Banach space norm on X . I f X has any Banach space norm relative to which T is normed, Theorem 4.2.15 shows that T is continuous. Hence Theorem 4.2.8 shows that the given complete norm is equivalent to the one assigned by construction (using any non-zero z G X ). (b): Theorem 4.2.15 shows that the representations are continuous. Hence Corollary 4.2.4 applies. □
468
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4.2.17
Characterizations of Commutative Banach Algebras W e will now show the power o f Sinclair’s refinement in Theorem 4.2.13 by giving Jaroslav Zemanek’s remarkably simple proof [1982] o f spectral criteria for commutativity (or more precisely, almost commutativity) in Banach algebras. A number of results equivalent to those in Theorem 4.2.18 were obtained in Theorems 2.4.11 and 3.1.5 for spectral algebras. Recall that if a functional algebra A is nonunital, the exponential function is defined in «4l . 4.2.17 T h e o r e m
Let A be a functional algebra. F or any a € A , consider £(a) = sup{p(a - e~baeb) : b € .A}.
The following three conditions are equivalent fo r any c € A (a) £(c) is finite. (b ) C ( c ) = 0 . (c) c + A j is in the center o f A / A j. I f in addition to these conditions the spectral radius o f c is zero, then c is in the Jacobson radical o f A . P roo f (a )= »(c ): We may assume £(c) < 1. Let T be an arbitrary irreducible representation of A . If A is nonunital, T can be extended to A 1 by T\+a = X I + Ta for all A + a € -41 . We want to show T c = ATj = XI. If not, there is
some vector x in the representation space with T cx — y linearly independent from i . By Theorem 4.2.13 we can therefore find 6 € A satisfying T ebX — x and Te»j/ = x + y. We conclude T c - e - ' c e * 1 = r e-<’ ( T 'e'>c-cel' ( a:) ) = T e-<‘ ( x ) = x ^ 0.
However, this implies that 1 belongs to the spectrum of c — e~bc e b, con tradicting our assumption. (c)=>(b): If (c) holds, then c — e~hc e b belongs to the Jacobson radical so its spectral radius is zero. (b)=?-(a): Obvious. The final remark follows since the Jacobson radical of the set {a G A : a + A j is in the center of A / A j } is the subset o f elements with spectral radius zero. ,□ 4.2.18 T h e o r e m The following conditions are equivalent fo r a functional algebra A. (a) The spectral radius is uniformly continuous on A . (b) A is almost commutative. P roo f (a)=>(b): Choose an e > 0 so that for any two elements a, c G A , ||a - e|| < e implies |p(a) - p (c) \< 1 . Then any a, b e A satisfy |p(a - e~baeb) - p(a)\ = \p ( a - e baeb) - p (e~ baeb)\ < ||a||e_1
4.';.*9
Representations and N orm s
469
which implies £(a) < p (a) + e _ 1 ||a|| < oo. Hence (b ) holds by the last theo-em. (b )= »(a ): B y Theorem 3.1.5, the spectral radius is an algebra semi-norm which is certainly less than or equal to the norm of the functional algebra .4, s'nce that norm is spectral. Hence we conclude \p(a)—p(b)\ < p ( a - b ) < jja -■ 6[| for all a, 6 G A . D 'em anek’s [1982] paper includes many other simple consequences of Theorem 4.2.17. Let us denote the set o f topologically nilpotent elements in a Banach algebra A by A tN - We know this set includes the Jacobson radical o f A . I f A tN is closed under either sums or products then it is closed under both and thus equals the Jacobson radical which is also closed in the norm. Similarly an element b G A belongs to the Jacobson radical if and only if it satisfies b + A tN £ A tN ■ W e give another important consequence o f Theorem 4.2.11, which can also be obtained from Theorem 4.1.9 and Proposition 2.4.12. 4.2.19 T h e o r e m Every irreducible representation o f a commutative spec tral algebra is one-dimensional. P roo f (First P roo f) Let T be an irreducible representation o f the commu tative spectral algebra C on a linear space X . Then To is commutative and hence satisfies Tc £ T^. Since Theorem 4.2.11 asserts that the lat ter r pace is C l, and since Tc is irreducible, we have Tc = C l and X is one- limensional.
□
P roof (Second Proof) Use the same notation. Theorem 4.1.9 shows that ker(T) is a maximal modular ideal in C. Proposition 2.4.12 shows that C/ k :r(T ) is isomorphic to C. Hence again T is one-dimensional. □ L seems to be unknown when the above result holds for topologically irrecucible representations in place o f algebraically irreducible representa tions. In Volume II we describe a special case in which this is true. Representation on Reflexive Banach Spaces The extended left regular representation shows that any Banach alge bra lias a faithful (even isometric) representation on a Banach space, and Corollary 4.2.9 shows that any primitive Banach algebra has a faithful con tinuous irreducible representation on a Banach space. Representations on reflexive Banach spaces are rarer, but we can elucidate when they occur. The results in this subsection are due to Nicholas J. Young [1976] and Sten Kaijser [1981]. We begin with a construction which will yield all normed irrfcv acible representations o f a Banach algebra on a reflexive Banach space. It generalizes the Gelfand Nalmark-Segal construction which will be stud ied in Section 9.4 of Volume II.
470
4: Ideals, Representations and Radicals
4.2.20
4.2.20 D e fin itio n Let A be a Banach algebra and let u be a linear functional on A . Let ( A ^ / ^ A ) be the ( left / right ) ideal ( Au, = { b e A : u(ab) = 0
a e A } / WA = {6 e A : iu(ba) = 0
ȣ / }).
Define a norm on ( A / A * / A / „ A ) by
=
sup{|w(a&)| : a € A i } /
\\b + uA t
=
sup{|w(6a)| : a G A i } )■
Denote the contractive { representation / anti-representation ) o f A on ( A /A u j A / u A ) ) induced by the ( left / rig h t) representation o f A on A by ( L u / R “ ). Denote the completion o f ( (A / A , , || • ||w) / (A/u,A, || • I D ) by ( X w / y * ) and the extension by continuity of each ( L " / ) to ( X w / y ^ ) by ( }. Finally, in order to obtain a representation from i?“ , we let R“ ‘ : A ^ B ( ( y u’)* ) be defined by R f = (/ ft)*. Call u> ( left / rig h t) autoperiodic if ( X w / 3 ^ } is reflexive. A linear functional which is either left or right autoperiodic is called autoperiodic. Young shows that autoperiodic linear functionals are weakly almost pe riodic (defined before Theorem 1.4.11). He characterizes when u> € A*w a p is { left / right ) autoperiodic in terms of a geometric condition on the weak closure of ( { aw : a £ A i } j {u>a '■ a G -4 i} ) and in terms of the equality of the mixed limits o f certain double sequences He also gives a number o f examples and determines the autoperiodic linear functionals in a few cases. For instance, if A = C(D.) where H is a locally compact Haus dorff space, the left ( = right) autoperiodic functionals on A are those of the form: / ►-» f° r n € N , « j 6 C and Uj e fi. In other words, the set of left autoperiodic functionals in this case is just the linear span of the Gelfand space r * . However, the most important examples of autoperiodic functionals arise as follows. Using the notation of Definition 4.2.20, suppose that A is unital. Let z 6 X w be the element 1+ A ^ . Then z is clearly a topologically cyclic vector for L " . Moreover, a + A ^ •-* w (a) defines a continuous linear functional on A /A u, so there is an element z* e X u* satisfying u>(a) = z * (Z £ (z ))
V
a e A.
(These ideas were already included in the important paper Bonsall and Duncan [1967].) We now show that in the present context there is a much stronger converse. 4.2.21 T h e o re m Let A be a Banach algebra and let T : A —> B ( X ) be an irreducible normed representation o f A on a reflexive Banach space X . Choose non-zero vectors z G X and z’ € X * and define u G A * by w (o) =
4.2.22
Representations and Norms
471
z * (T a( z )) fo r all a 6 A . Then ui is right autoperiodic and satisfies = A/u,A (i.e., no completion is necessary). Furthermore, the map V : A /WA —* ifined by V (a + „ A ) = T Z ( z * ) VaeA is g well-defined homeomorphic linear isomorphism and its adjoint is a topological equivalence between T and R ? * . Hence every irreducible normed representation o f a Banach algebra A on a reflexive Banach space is topologically equivalent to B?* fo r some right autoperiodic functional u> € A * . P roo f Theorem 4.2.15 shows that T is continuous. Let C be the left ideal {6 e A : Tb(z ) = 0 }. In Theorem 4.2.3, we have already noted that A/C with its quotient norm is homeomorphically linearly isomorphic to X . Hence for x * G X * the expression x* >—►sup{|x*(Tt( 2 ))| : b € A \ } is equivalent to the original norm. So !!« J- „ A \ r = sup{|w(a6)| : b e A i } = su p{|V (a + wA )(T b ( 2:))| : b e A \ }
shows that V is a well-defined homeomorphic linear isomophism. Thus its range is norm closed. Therefore, its range (which is clearly a linear subspace) is both weakly closed and weak = weak* dense. This implies that A/wA = is a reflexive Banach space, so ui is right autoperiodic. It only remains to show that V * is an equivalence between T and /?"*. Tfca following calculation, which holds for till a,b,c 6 A , proves this: V '( T a(T b( z ) ) ) ( c + „ A )
=
V (c + u, A )(T ab(z )) = z * (T cab(z ))
=
V (c a -h uA )(T „ (z ))
=
V ( T b(z ))(B Z (c + uA ) )
= J % ' ( V ( T t ( z ) ) ) ( c + „A).
□
Essentially the following result was obtained by Kaijser [1981] as well as Young [1976]. Although their proofs differ, both involve the theorem on weakly compact operators of Davis, Figiel, Johnson and Pelczynski (Diestel ,an4 Uhl [1977], VIII.4.8). Because of the technicalities involved, we give only a sketch of the more difficult direction in Young’s proof. 4.2.22 T h e o r e m A normed algebra A has a ( faithful continuous / iso m etric } representation on some reflexive Banach space i f and only i f the weakly almost periodic functionals o f norm 1 on A ( separate the points o f A / determine the norm o f A ) . P roof Suppose T : A —* B (X ) is a ( faithful continuous / isometric ) representation o f A on a reflexive Banach space X . Choose norm 1 vectors z £ X and z * 6 X *. As before, define u € A * by u>(a) = z * (T a( z )) for
472
4: Ideals, Representations and Radicals
4.2.23
ail a € A . Then u> is weakly almost periodic by direct application of the definition since bounded subsets o f reflexive spaces are weakly compact. As z € and z* £ Af,' vary, this collection of weakly almost periodic function als of ( bounded norm separates the points / norm at most one determines the norm ) of A . (Sketch.) In order to prove the converse, Young shows how weakly almost periodic functionals can be approximated by autoperiodic ones using the construction o f Davis, Figiel, Johnson and Pelczynski already cited. We have shown how to obtain a representation on a reflexive space from an autoperiodic linear functional. The direct sum of all these representations on the ^-direct sum of the reflexive spaces then provides the ( faithful continuous / isometric ) representation needed to complete the proof. □ We now turn to Kaijser’s approach in terms o f factoring a weakly com pact map through a reflexive space. 4.2.23 T h e o re m Let A be a unital Banach algebra and let uj e A " be a weakly almost periodic linear functional. Then there is a representation 7W o f A on a reflexive Banach space Z “ , and topologically cyclic vectors z " € Z * and z“ * £ Z “ * satisfying II*"II 11*" *11 < IMI
and
w (a) = z - ( 7 y z “ )
V a £ A.
(3)
P roo f Theorem 1.4.11 shows that the map P^: A - * A *, defined by Pw(a ) = for a € A , is weakly compact. Hence the last theorem in 1.7.8 shows that there is a reflexive Banach space a map S € B (A , Z u ) with range dense in 2 “ and an injective map Q € B ( Z “ , A * ) satisfying Pu = Q S, ||S|| = ||PJ1 = |M| and IIQH < 1 . If, as usual, L and R represent the left and right regular representations of A on itself, then P u inter twines L and the representation a —> (R a )' of A on A * in the sense that Pu>(La(b )) = (R a )* (P w(b )) holds. Hence by the result already cited, there is a representation T * of A on Z u satisfying S(ab) = S (L a(b )) = T % (S (b )), Q ( T ? ( x ) ) = (K a )* (Q (x )) and hence a6u> = P ^ a b ) = Q ( I ? ( S ( b )) ) for any a,b e A . Thus we have u>(cab) = aiw(c) = Q * (K (c )){ T ? (S (b )))
V a ,b ,c e A
(4)
Taking z“ = .5(1) and zw* = Q * (« ;(l)), gives the desired result. The injective map Q is the dual o f Q ' o k which therefore has dense range in Z u *. This shows that both these vectors are topologically cyclic. □ 4.2.24 C o r o lla r y Let A be an Arens regular unital Banach algebra. Then there u a homeomorphic faithful representation o f A on a reflexive Banach space Z . Moreover the Arens representation o f A ** on Z is also faithful and homeomorphic.
4.2.25
Representations and N orm s
473
Proof For each w £ A * of norm 1 , construct the representation on 2 “ as in the theorem. Consider the £2-direct sum o f these representations on th •; f 2-direct sum Z of the family Z ^ . For each a £ A , there is some u of lorrn 1 satisfying ||a|| = w (a) = (T ^ (z u>) ) < Hit'll < ||Ta||. This shows that the continuous representation T is homeomorphic and faithful. For t e Arens representation T 1 o f Definition 1.4.14 and / € A **, we find UT)1!; > suP { ( T - ) } :u ; e A * ; ||w|| = 1} > s u p {| ^ * (T “ ) } ( ^ ) | : w € .4*; j'w|| = 1} > sup{|/(w)| : w £ A *; |M| = 1 } = ||/|| where z " and z " * are defined as in the theorem relative to each particular w. Thus T 1 is also home amorphic and faithful. □. Injective Banach Algebras This notion was introduced by Grothendieck and has been put in a more modern setting by Kaijser [1976], which is our matin reference. For each w in tho dual Banach space of a Banach algebra A , the map P ^ '-A —* A * defined by P u (a ) = ao> was introduced before Theorem 1.4.11, and uniform algeb as were defined following Proposition 3.1.8. Integral operators were discussed in §1.7.12 and the injective tensor product in Section 1.10. See also Anna Maria Mantero and Andrew Tonge [1979]. 4.2.25 D e fin itio n A Banach algebra A is said to be { injective / geomet rically injective) if it is unital and its multiplication defines a { continuous / contractive ) map from A ® A with its injective tensor norm into A. O ' course, multiplication always defines a continuous map with respect to the projective tensor norm on ,4 ® A . It is easy to see that A is { inje c tiv 1! / geometrically injective ) if and only if for each u £ A * the map Pu is an integral operator and the map w n Pu from A * to B j ( A , A * ) is ( continuous / contractive ). Kaijser gives a new proof o f Grothendieck’s criterion that A is injective if and only if for every Banach algebra B the in jective tensor product A ® B is a Banach algebra under the product induced by the products in A and B. 4.2.3 3 T h e o re m Every injective Banach algebra has a faithful representatior as a norm-closed subalgebra o f B(Ti.) fo r some Hilbert space H and hence is Arens regular. Every geometrically injective Banach algebra is a uniform algebra and hence is commutative and semisimple. Proof (Sketch of the first portion) Injective maps such as P u can be fac tored through a Hilbert space, just as weakly compact ones can be factored through a reflexive space. The analogue of the last theorem in §1.7.8 still holds. Hence the proofs o f Theorem 4.2,23 and Corollary 4.2.24 do not need any essential change. □
474
4: Ideals, Representations and Radicals
4.3
4.3.1
The Jacobson Radical
This section studies the most important of till radicals, the Jacobson radical. In the introduction to this chapter we explained that represen tation theories ordinarily cannot provide information on perfectly general mathematical objects but that the pathological portion can often be neatly split off. The Jacobson radical is the intractable portion of an algebra rel ative to irreducible representations. This radical was briefly introduced in Chapter 2 (Definition 2.3.2) since it plays a fundamental role with respect to spectral theory as well as representation theory. It was also pointed out in Chapter 3 that the Gelfand radical introduced there equals the Jacobson radical, at least for spectral algebras which are commutative modulo either of these radicals. The next definition is due to Jacobson [1945a], although he used prop erty (b) o f our Theorem 4.3.6 as his definition. 4.3.1 D e fin itio n The radical, or Jacobson radical, of an algebra A is the intersection o f the kernels o f the irreducible representations o f A . It is denoted by A j . An algebra is said to be semisimple if its radical is zero and is said to be Jacobson-radical if it equals its Jacobson radical. In this definition we are following the common convention that the in tersection o f an empty family o f subsets of a set is the set itself. Thus an algebra is Jacobson-radical if and only if it has no irreducible represen tations at all. We could rephrase the definition by saying that A j is the intersection of all the primitive ideals of A . O f course the Jacobson radical is an ideal. We shall see that semisimple algebras Eire comparatively well be haved. (Th e fundamental theorem of spectral semi-norms (Theorem 2.3.6) was an early example of this.) The next theorem shows how to construct a (largest) semisimple quotient algebra from any algebra. 4.3.2 T h e o r e m
Any algebra A and any ideal X o f A satisfy:
(a) X j = X n A j . (b ) X C A j implies (A / X )j = A j/ X . (c) A/X semisimple implies A j = X j. Thus in particular A j is radical and A / A j is semisimple. P roo f (a): Theorem 4.1.6(a) asserts that the irreducible representations of an ideal of A are precisely the restrictions to the ideal (o f the homomorphisms) of the irreducible representations of A . This implies X j = X fl A j and hence ( A j ) j = A j . (b): Theorem 4.1.6(b) asserts that if J is an ideal o f A which is included in A j , then the irreducible representations of A are precisely those which can be written as T o tp for some irreducible representation, T , of A/X where
£ C- O
T h e Jacobson R adical
475
simple, Theorem 4.1.8(e) shows X = C\{V : V € hn (X )} which implies i j = i n A j = n { V : V e n A } = A j.
□
n immediate consequence o f this theorem is the following simple but useful result. i-X .S
C o ro lla ry Any algebra A satisfies: A j = ( A 1)/.
P ro o f The theorem gives A j = (-4.1) j fl A . If the algebra is unital there is reihin g to prove, so we assume otherwise. Since A + a >-* A defines an irreducible representation o f A 1 on C with kernel A , we conclude (.A1) j C A which implies A j = (.4 1 ) j fl A = (^4x) j . □ Quasi-regular Ideals The following definition is due to Reinhold Baer [1943]. The idea stems from Sam Perlis [1942], 4.3.4 D e fin itio n A one-sided ideal or ideal is called quasi-regular if it contains a quasi-inverse for each of its elements. Because a one-sided ideal or ideal is a spectral subalgebra, it is quasiregular if each of its elements has a quasi-inverse in any algebra in which it is embedded. Note also that a one-sided ideal or ideal £ of an algebra A is quasi-regular if and only if it satisfies S p A (a) = S p c(a ) = { 0 }
V a e £.
The 3quations obviously imply that £ is quasi-regular. To see the opposite implication note that A ~ ‘ a € £ C AqG
implies SpA (a ) C Spc.(a) C {0 }.
V a e£ ;A eC \ {0 }
If A is not unital these inclusions are
obviously equalities, but if A is unital then 1 does not belong to £ C A qa
so that SpA (a ) equals { 0 } in this case also. The first result in the following lemma is due to Baer [1943]. The second is due to Jacobson [1945a]. 4.3.5 L em m a A left ideal is quasi-regular i f every element is left quasiinvertible. An element which belongs to a quasi-regular left ideal belongs to ’every prim itive ideal. Proof Suppose £ is a left ideal and every element of £ is left quasi-invertible ( i,e,, lias a left quasi-inverse). Let b be the left quasi-inverse for a € A. Then o — ba — a belongs to £. Thus b has a left quasi-inverse c as well as iis right quasi-inverse a. This implies c = c o (b o a ) = (c o b) o a = a. Thus a is quasi-invertible with quasi-inverse b. Hence £ is quasi-regular.
476
4: Ideals, Representations and Radicals
4.3.6
To prove the second statement suppose that an element a does not belong to some primitive ideal. Then there is an irreducible representation T satisfying Ta / 0. Choose z S X T satisfying T az / 0 and then choose b e A satisfying T(,Taz = T fjnz = z. If c is a left quasi-inverse for ba then c o (ba) = 0 holds and so Tbz = T ^ ^ z = T cz + Tfoz — TcTbaZ = z is a contradiction. Thus if an element a does not belong to some primitive ideal, then ba is not quasi-invertible for some b G A and hence a does not belong to any quasi-regular left ideal. □ Diverse Characterizations o f the Jacobson Radical In the next theorem (a) and part of (d) are due to Hille [1948], while (b) and (c) are due to Jacobson [1945a]. In [1947a] Segal defined an algebra to be weakly semisimple if the maximal modular left ideals have intersection {0 }. He was apparently unaware o f Jacobson’s paper [1945a]. Part o f this theorem was already obtained in Theorem 2.3.3. 4.3.6 T h e o r e m Let A be an algebra. (a) The Jacobson radical is the intersection o f all maximal modular left (o r right) ideals o f A . Hence it is the same in the reverse algebra. (b ) The Jacobson radical is a quasi-regular ideal which includes all quasi regular one- or two-sided ideals o f A . (c) The Jacobson radical is the set {a G A '. A 1a C (d) The Jacobson radical is the largest ideal I satisfying any (hence all) o f the following equivalent conditions S p A / i(a + Z ) Q SpA (a ) C SpA/x(a + J ) U { 0 }
Vae^;
■AfG = {a € A : a + T € (A / 2 )qa}\ Sp(a + b) = Sp(a) Sp(b) = { 0 } p(a + b) = p(a) p(b) = 0
V a G A\ b G X; V b e I; V a G .4; b € I . V b GI.
(e) Any spectral subalgebra B o f A satisfies B fl A j C B j. P roo f We first prove (c), (b ) and the first sentence of (a); the latter two only for left ideals. Theorem 4.1.8 shows that the intersection A j o f all the primitive ideals of A equals the intersection of all maximal modular left ideals of A (even if one, and hence both, of these families is empty). Lemma 4.3.5 shows that every quasi-regular left ideal is included in the intersection A j of all the primitive ideals. If e G A is not left quasi-invertible, then
T h e Jacobson Radical
4.3.
477
there is a maximal modular left ideal £ with e ^ £ by Theorem 2.4.6. Thus e does not belong to A j by what we have already shown. Now the first sentence of Lemma 4.3.5 shows that A j is quasi-regular. Clearly any element belongs to a quasi-regular left ideal if and only if it belongs to the sat defined in condition (c). Thus we have established (a), (b) and (c) except that we have not yet shown that A j is the intersection of all maximal modular right ideals and contains till right quasi-regular ideals. However, (A + b)a £ AqG is equivalent to a(A + b) £ AqG by Proposition 2 . 1 .8 . We conclude A j = {a £ A : a A l C A qc } = {a £ A : a belongs to some quasi-regular right ideal}. This completes the proof of (b ) and shows that the radical of the reverse of an algebra is the same as the radical of the £ gebra itself. This fact completes the proof o f (a ) also. (c ): This was proved as Theorem 2.3.3(c). (e): This follows from (b ) since B fl .4 is a quasi-regular ideal o f B. □ Aa remarked after the definition o f quasi-regular ideals, an ideal or one sided ideal is quasi-regular if and only if each element has spectrum equal to {O '. Hence if the set of elements with spectrum { 0 } is an ideal, then that ideal is the Jacobson radical. (Usually, however, there are many elements with spectrum {0 } not contained in the Jacobson radical. The various niloe ,ent elements in a full matrix algebra are examples.) As noted below, commutative spectral algebras always have the property that the set of elements with spectrum {0 } is an ideal. An algebra, A , with this property has the further remarkable property that any subalgebra, B, satisfies Bj C A j (1B and any spectral subalgebra, B, satisfies B j = A j D B. Algebras which have no non-zero elements with spectrum {0 } form an extreme case. Commutative semisimple spectral algebras are an example. For finite dimensional algebras the Wedderburn structure theorems show that any algebra satisfying this condition must be commutative. However, we w ll give a family of infinite-dimensional noncommutative Banach alge bras satisfying this condition in Example 4.8.5. Any subalgebra o f such an algebra is semisimple. 4.3.7 C o r o lla r y tvio-.' .ded identity.
Every one-sided identity in a semisimple algebra is a
Pw o, Suppose A is S34: rT = {ey — y : y a quc si-regular ideal inclined in A j . If A
an algebra and e is a right identity for A . Then the £ A } satisfies A I = {0 } and T A C X so that it is in which each element satisfies bq = —b. Hence X is is semisimple, then X is zero. □
, 3 will now record the beautiful way in which norms, spectral norms and complete norms relate to the Jacobson radical. The next four corollaries are
478
4: Ideals, Representations and Radicals
4.3.8
satisfies a °°(a ) = 0. (This concept is also due to Jacobson [1945a] who called these ideals generalized nil ideals.) In a spectral semi-normed algebra this is equivalent to Sp(a) = { 0 } for all a € 7. Thus in this case an ideal is topologically nil if it is quasi-regular.
4.3.8 Corollary Let {A , a ) be a semi-normed algebra. Then cr°°(a)
— 0 holds fo r each a € A j so that the Jacobson radical is a topologically nil ideal.
P roof This follows immediately from Theorems 4.3.6(b), 4.1.6(b) and 2.2.2.
a
The second statement in the next corollary was obtained independently by Max Zorn and Einar Hille about the same time (cf. Hille [1948], pp. 475, 476) as Jacobson [1945a], 4.3.9 C o r o lla r y The Jacobson radical o f a spectral semi-normed algebra ( A , o ) is a topologically nil ideal which contains every topologically nil one sided ideal or ideal. I t also equals the set {a € A : lim a ((a b )n) = 0 for all b £ A 1}. n •—*oo
Proof The first statement follows immediately from Theorems 4.3.6(b) and 2.2.5 and the remark following Definition 4.3.4. To prove the second statement we note that a e A j and b e A ' imply ab G A j which implies 0 = a ^ (a b ) = limn_ oc,((7 ((a 6) n) ) 1^n. Hence lim ^ tx , a ((a b )n) = 0 holds for any a € A j and b e A 1. Conversely, if <7°°(a ) > 0 holds for a 6 A , then lim „_oo a(ab)n = 0 fails for any b = A e C satisfying |A| > cr00(a )_1. Thus a ^ A j implies that lim ^ o o a(ab)n = 0 fails for some b e A 1. □ The following simple results are o f tremendous importance, so we repeat them despite their earlier appearances in Chapters 2 and 3. 4.3.10 C o r o lla r y The Jacobson radical o f a spectral semi-normed algebra is closed. Hence a spectral semi-norm on a semisimple algebra is a norm. Proof Theorems 4.3.6(a) and 2.4.7 prove the first statement. The second is an immediate consequence of the first. □
4.3.11 Corollary The Jacobson radical o f an almost commutative spectral algebra equals the Gelfand radical. P roo f Recall that an algebra is almost commutative if its quotient modulo its Jacobson radical is commutative. Theorem 3.1.5 shows that the state ment o f this corollary is equivalent to almost commutativity for a spectral algebra. We record this fact again here for completeness. □
4.3.12
T h e Jacobson Radical
479
Jacojson Radical and Idempotent Elements the next proposition we collect some results on the Jacobson radical a:::' ;dempotent elements. Result (c) is due to Kaplansky [1949a], A pe rusal o f the proof o f Theorem 2.5.14 shows that (c) remains true for unital spectral semi-normed algebras. The proof of (d) is a slight modification of an argument due to Joseph H. Maclagan Wedderburn [1907]. He made no topological assumption on A but assumed that A j was a nilpotent ideal. (Nilpotent ideals and nil ideals will be defined below.) Whenever A j is nil, the power series used in the proof terminate, so that no limits are involved in summing them. Since the existence o f these limits is the only issue in the proof, this result holds for functional algebras as well as Banach algebras. Result (e) was obtained by Chester Feldman [1951]. It will be used later (Theorem 7.1.3) to prove one of the main results of his paper. 4.S.L2 P r o p o s itio n Let A be an algebra. >,) A j contains no non-zero idempotents. (b ) Any idempotent e in A satisfies (e A e )j = eAe n A j — e A je . Hence, the Jacobson radical M n( A ) j o f an n x n m atrix algebra over a
unital algebra A is just M n( A j ) . Let A be a Banach algebra. (c ) Every element o f A j is a two-sided topological divisor o f zero. (d) Any idempotent in A / A j can be written as e + A j , where e is an idempotent in A . (e) Any finite or countable set o f orthogonal idempotents in A / A j may be written as {e + A j : e £ £ }, where £ is a set o f orthogonal idempotents in A . P roo f (a): If e is an idempotent element in A j , then it is quasi-invertible and hence e + eeq — e(e + eq) = e(eeq) = eev implies e = 0 . (b): Proposition 2.5.3(e) gives eAe fl A j C (e A e )j. Let a £ A 1 and b ^ (e A e )j be arbitrary. Then ab = aeb holds and eaeb € (e A e )j has a c^asi-inverse c in eAe. We have c o a b — ab + c — cab — aeb + c — ceaeb = ( ; — e)aeb + c o eaeb = (1 — e)aeb. However (1 — e)aeb has square zero so that it is quasi-invertible. Hence ab is left quasi-invertible. Since a £ A 1 was arbitrary we conclude that b belongs to A j by Theorem 4.3.6(c). This concludes the proof of the first equality since b e (e A e )j was arbitrary. The inclusion e A je C eAe fl A j is obvious and any b € eAe fl A j satisfies b = 3be € e A je . The remark about matrix algebras now follows from Theorem 1.6.10. (c): Corollary 4.3.8 and Theorem 2.5.14 show that each element of A j is either a two-sided topological divisor of zero or is invertible in eAe for a
4: Ideals, Representations and Radicals
480
4.3.12
suitable proper idempotent e. However, the latter condition is impossible since it would imply that A j contains the idempotent e. (d): Let a + A j be idempotent. We will actually show that there is a (necessarily unique) idempotent e in the smallest closed spectral subalgebra of A containing a (e.g., it is in the double commutant { a } " of { a } in A ) satisfying e + A j = a + A j . Define j by j = a2 — a € A j . Then —4 j ( l — ( ~ 4 j ) q) = 6 is defined and belongs to A j fl { a }" . Thus b satisfies ||6jj°° = 0 so that the series OO
converges absolutely to an element c which belongs to both A j and any closed spectral subalgebra of A containing a, and which satisfies 4c2+4c = b. Define e by e = a + 2ac - c € a + A j . Then we get e2 - e
=
a2 + 4a2c^ -t- c2 + 4a2c — 2ac - 4ac2 - a - 2ac + c j + 4j c 1 + c2 + 4j c + c = j + jb + 4 xb j - (4j + l)jr'(l - ( - 4 j ) q) = 0 .
(e): Suppose that {a ; + A j : 1 < j < n } is a set of orthogonal idempotents in A / A j. By induction using (d) we may assume that a set {e j : a < j < n — 1 } of orthogonal idempotents have been found satisfying ej + A j = a3 + A j . Define an idempotent e € A by e = Y^=i ■ Then the orthogonality of {a j + A j : 1 < j < n } implies (1 - e )a „ (l — e) + A j — an + A j . Let en be the idempotent constructed as in (d) from (1 — e)an( l - e). The construction gives an element in the closed spectral subalgebra (1 —e).A (l —e) of A . Hence {e ; : 1 < j < n } is a set of orthogonal idempotents satisfying e3 + A j = aj + A j for 1 < j < n. Clearly the argument covers the case o f a countably infinite set of orthog onal idempotents. □ The perturbation set V (S ) o f a set S in an algebra A is the set {a G A : a + S C S }. Clearly this set is always closed under addition. Suppose A is unital and S = A g - Since A g is closed under multipli cation by non-zero numbers, a e V ( A g ), b € A g and A / 0 implies Aa + 6 = A(a + A-16) G A g - Hence V ( A g ) is a linear subspace. I f ev ery element of A can be written as the sum o f invertible elements, then V (A g ) is an ideal. To see this, consider a € V ( A g ), b € A g and c G A with c = Cj where the c} are all invertible. Then ac3 + b = ( a + b c j l )c] is invertible for each j, so acj and hence ac = Y7j = i aci belong to the pertur bation set. Since multiplication from the left works similarly, V ( A g ) is an ideal of A as claimed. We wish to work in not necessarily spectral algebras, so we must consider the perturbation set o/ A q in A : P a ( A q ) = ACXP(A q ). The next result is due to Arnold Lebow and Martin Schecter [1971] for uni tal Banach algebras. See also Jurgen Schulz [1985].
4.3. 3
T h e Jacobson Radical
481
4.2. 3 P r o p o s itio n In any spectral algebra A , the Jacobson radical is the j “.rturbation set in A o f the group A q in the unitization: A j — {fl G A : a -f- A q C A @ }' Proof Denote the perturbation set by V = V ^ (A q ). The third condition under Theorem 4.3.6(d) shows A j C V in any algebra. Since the spectrum is bo-jnded in a spectral algebra, any element is the sum of the two invertible elements A and —(A —a) for any A with absolute value strictly greater than p(a). Hence the perturbation set V is an ideal by the argument given above. In this case, Theorem 4.3.6(d) shows V = A j . □ T. Keith Carne [1981] studies irreducible representations of general al gebra tensor products and gives conditions for their semisimplicity. The nuclearity (Definition 1.10.23) o f the tensor norm is a key factor. This im proves previous results of Bernard R. Gelbaum [1962], Kjeld B. Laursen [1970] and Jun Tomiyama [1972].
4.4
The Baer Radical
W e will now turn to the study of another radical— the Baer radical. It is o f : auch less importance in our theory mainly because it is not necessarily closed in spectral normed algebras. Nevertheless the algebras for which this radical is zero have a number of useful properties. Definitions T i e terms “ nilpotent” and “nil” introduced in the next definition were used by Wedderburn [1907], and Gottfried Kothe [1930], respectively. Kothe provided an example which showed that they did not coincide. The concept of a nil ideal was exploited by Elie Cartan [1898]. The term “semiprime” was introduced by Masayoshi Nagata [1951], but the concept was already considered by Baer [1943]. Ii should be noted that, unlike primitive ideals, all the concepts pre sented here and for the rest o f this section are invariant under reversal of the order of multiplication. Thus, anything proved for left ideals will hold (peri aps with an obvious reflection in the statement) for right ideals. 4.4.1 D e fin itio n Let A be an algebra. Then A is called: (?) nil if each element is nilpotent (i.e., for each a G A there is some n f N satisfying an = 0 ); (b) nilpotent if there is some n € N satisfying A n = { 0 } (i.e., there is soire n 6 N such that for any a i , a2, ■■■, an G A the product aia2 ■■■an is zero j , (<) semiprime if it has no non-zero nilpotent ideal.
482
4: Ideals, Representations and Radicals
4.4.2
A n ideal X of A is called nil or nilpotent if it is nil or nilpotent when it is considered as an algebra in its own right. However, an ideal X of A is called semiprime if A /X is semiprime. The two systems for applying the above terms to algebras and to ideals are slightly confusing, but we are following well established tradition. The terms “prime” and “primitive” are applied to algebras and ideals according to the same system as “semiprime” . One must be careful to note that a semiprime ideal of an algebra is not usually semiprime when considered as an algebra in its own right. N il and Nilpotent Ideals In Theorem 4.4.11 below we will give results of Sandy Grabiner [1969] and Peter G. Dixon [1973b] which assert that a nil Banach algebra is nilpo tent and a semiprime Banach algebra has no non-zero nil one-sided ideals. Most of the following results can be found in Cartan [1898] and Wedderburn [1907], 4.4.2 P r o p o s itio n Let A be an algebra. (a) I f A is nil or nilpotent, then every subalgebra o f A and every homo morphic image o f A has the corresponding property. (b) I f A has an ideal X such that X and A/X an: both nil or both nilpo tent, then A has the corresponding property. (c) The sum M o f all the nil ideals is a nil ideal and A / M has no non-zero nil ideals. (d ) The sum o f a finite set o f nilpotent ideals is nilpotent. The sum of all the nilpotent ideals is a m l ideal which contains every nilpotent one-sided ideal. (e) I f some ideal X o f A has a non-zero nilpotent ideal, then there is a non-zero nilpotent ideal o f A . Hence any ideal o f a semiprime algebra is a semiprime algebra. P roo f Both (a) and (b) are obvious, (c) and (d): First we note that the sum of two nil or nilpotent ideals has the corresponding property. Suppose Xi and I j are both nil or both nilpotent ideals in A . Then ( I ; + J2) /J i is isomorphic to Xa/(Xj fl J 2) which is nil or nilpotent by (a). Thus (b) shows that X\ + X2 is nil or nilpotent since both (J i + X2 ) /X\ and X\ have the corresponding property. By induction both these proportion are preserved under finite sums of ideals. If an element a € A belongs to the sum o f all nil ideals, then a belongs to the sum of finitely many nil ideals and hence is nilpotent by what we have just proved. Hence the sum of all nil ideals is nil. Since the sum of till nilpotent ideals is an ideal included in the sum of all nil ideals it is a nil ideal also. Let M be the sum of till nil ideals. We will show that A / M has no non-zero nil ideals. If J is the preimage in A of a nil ideal of A / M , then
T h e B aer R adical
483
1 / M . and M are both nil so J is nil by (b). Therefore 1 is included in M by ;h-3 definition of M . Thus A j M has no non-zero nil ideals. Finally, we note that the last claim o f (d) follows from the fact that any nilpotent one-sided ideal is included in some nilpotent two-sided ideal. For instance, if £ is a left ideal satisfying Cn — {0 }, then £ is included in the idea! C A 1 which satisfies (£^41)n = C ( A l C )n~ l A 1 C CnA l — {0 }. (e) Let ,V be a non-zero nilpotent ideal of 1. The ideal M = A l M A 1 of A. satisfies M 3 C 1 M 1 C M . Thus M is nilpotent, since Af is nilpotent. The last sentence follows immediately from the first. □ The sum of all the nil ideals of an algebra is called the nil radical. This was the first radical considered for algebras. It was introduced by Cartan [1898]. We denote the nil radical o f an algebra A by _Anu. Proposition 4.4.2(c) shows (■/4/'/4ml)ml ~
&nd
(^4nil)ni! “
Thiwe are the two most basic properties which any radical should possess. For finite-dimensional algebras nil ideals are nilpotent. Wedderburn [1907] introduced and concentrated on the nilpotent ideals. Their theory is un fortunately even harder than the theory o f nil ideals to generalize beyond finite-dimensional algebras and algebras with the descending chain condi tion on left ideals (where again nil ideals are nilpotent). Semiprime Ideals The next results are mainly due to Nagata [1951]. The only nontrivial step is the proof that each semiprime ideal satisfies (e). This theorem is very similar to Theorem 4.1.10. 4.4.3 T h e o r e m Let 1 be an ideal in an algebra A . The following are equivalent: (a) 1 is a semiprime ideal o f A . (b) aAa C 1 implies a e l fo r all a £ A ; '(c ) I f M is an ideal o f A , then M 2 C 1 implies M C I . (d) I f £ is a one-sided ideal o f A satisfying Cn C l fo r some positive integer n, then £ satisfies £ C l . (e) Z is the intersection o f some set o f prime ideals o f A . Proof (a ) => (b): If aAa C 1 holds, then ( A 1a A 1/ I ) 3 equals { Z } in A/1. Thus ( A 1a A 1) / l = { Z } holds in the semiprime algebra A / l. However, this implies a £ 1. => (c): If M 2 C Z and a £ M hold, then aAa C M 2 C 1 implies a £ J. Since a £ M was arbitrary, M is included in Z. (c ) => (d): Let £ be a left ideal satisfying £ n C Z. Then C A 1 is an ideal c£,t:sr/:ag (C A l ) n C CnA l C 1 A 1 = 1. Suppose that m is the smallest
484
4: Ideals, Representations and R adicals
4.4.4
integer satisfying ( £ A l ) m C J. I f m > 1 held, then ( ( £ A 1) m~ 1) 2 C X would hold. Then (c) would imply (£.A 1 )m~ 1 C X. This contradiction shows m = 1 so £ C £ A l C X holds. ( d) => (a): We must show that A/X has no non-zero nilpotent ideals. If (Af/X)n is the zero ideal (i.e., it equals { J } in A / X ), then Af n is included in X. This implies A/’ C I s o that A f /X already equals { X ) in A/X. (e ) => (c): If M 2 C n{Ptl : q G A } holds for any set of prime ideals { V a : q G ^4}, then for each q G A , M 2 C V a implies M C V a by the definition of a prime ideal. Hence M is a subset of C\{Va : a G /I}. (b) => (e): If (e) is false there is an element ai which belongs to the intersection o f all the prime ideals o f A which include X ), but cti does not belong to X. We can inductively choose an G an_i.A an- i \X by (b). Define 5 as the set {a n : n G N }. Consider the family of ideals which include X but do not intersect S. It contains X. When this family is ordered by inclusion, Zorn’s lemma shows that it has a maximal element which we will call V . W e want to show that V is prime. This will prove the implication (b) => (e) by contradiction since the choice o f ai then shows that fl] G 5 belongs to V . Suppose a, b G A do not belong to V . We will show 2 V and thus establish that V is prime by Theorem 4.1.10(b). By maximality of V there are elements a,-, aj G S satisfying Aj G V + A 1aA 1 and aj G V + A 1b A l . By symmetry we may suppose i < j . This implies
4.4.5
T h e B aer R adical
485
Hence if V is a semiprime ideal of X, then V includes V A and A V so V is an ideal o f A . I Tow assume that I is a semiprime ideal of A and let V be a semiprime idea of X. Then we have just shown that V is an ideal of A . We will use Theorem 4.4.3(c) to show that V is a semiprime ideal o f A . Suppose Af is an ideal of A satisfying A f2 C V ■ Then A f2 C P C I implies A f Q X. Hence A f is an ideal of X. Since V is a semiprime ideal of X which includes A P it includes Af. Hence V is a semiprime ideal of A. (b): We will use Theorem 4.4.3(c) and (d). Suppose A f is an ideal of X satisfying A P C V fl X. Then A 1A f is a left ideal of A satisfying ( A l J -f f = A 1A f (A l A f )A 1A f c A 'A f X A l A f = A l A f(X A l )A f c A l AfXAf C A 1Aj '2 C A l (V fl X ) C V . Since V is & semiprime ideal of A it includes A f C A 1Af. Since A f is an ideal of X we conclude A f C V ft I . This proves that V fl X is a semiprime ideal of X. (c): Suppose A f is an ideal of A/X which satisfies A f2 C V/X. Let AA. be its complete preimage in A . Then M 2 is included in V + X = V . Since V is a semiprime ideal o f A we conclude M Q V and hence A f = M / X C V/X. Theorem 4.4.3(c) shows that V / l is semiprime. (d): Let M be the complete preimage in A of V and let A f be an ideal of A satisfying A f2 C AA. Then (A f / X )2 C M / X = V implies Af/X C V since V is a semiprime ideal o f A/X. Hence A f is included in M . Therefore M is semiprime. □ Rsinhold Baer [1943] called any ideal J o f an algebra A a radical ideal if X is a nil ideal and A /X contains no non-zero nilpotent one-sided ideals. He proved that the intersection o f all radical ideals is a radical ideal (now called the Baer lower radical) and the sum of all radical ideals is a radical ideal (now called the Baer upper radical). He made the easy observation that the Baer lower radical is (in our terminology) simply the intersection of the semiprime ideals since this intersection is automatically nil. The Baer Radical The following definition is due to McCoy [1949], Jacob Levitzki [1951] and Nagata [1951] independently showed that the ideal defined here, which McCoy called the prime radical, coincides with the Baer lower radical. We follow Nathan J. Divinsky [1965] in calling this simply the Baer radical. Theorem 4.4.6 below shows that this agrees with Baer’s original definition. 4.4.5 D e fin itio n The Baer radical, or prime radical, of an algebra, A , is the intersection of all the prime ideals of A . It is denoted by A b - The algebra is called Baer-radical if it equals its own Baer radical. Tc agree with systematic usage we should call an algebra Baer-semisimple if its 3aer radical is zero. However, the next result shows that the term semip -ime already covers this situation. (Nagata [1951] invented the term
486
4: Ideals, Representations and Radicals
4.4.6
semiprime to signify that such an algebra could be written as a subdirect product o f prime algebras. He suggested the term semiprimitive to replace Jacobson semisimple. His systematic notation has not been adopted by others.) The (easy) fact that the Baer radical (as defined here) is semiprime was noted by McCoy [1949], The fact that it is the smallest such ideal was proved by Nagata [1951], The following theorem shows that A b is Baer’s original lower radical. 4.4.6 T h e o r e m The Baer radical o f an algebra is a semiprime nil-ideal which is included in every semiprime ideal. Hence the Baer radical is {0 } if and only i f the algebra is semiprime. The Baer radical o f a commutative algebra is the set o f nilpotent elements and hence is the nil radical. Proof The fact that the Baer radical is a semiprime ideal which is included in every semiprime ideal is an immediate consequence of Theorem 4.4.3(e). Since {0 } is a semiprime ideal if and only if the algebra is semiprime, the second sentence follows from this. Proposition 4.4.2(c) shows that the sum, M , of all nil ideals (the nil radical) is a nil, semi-prime ideal. Hence Proposition 4.4.2(a) shows that (he Baer radical is nil since it is included in M If A is commutative the set M of nilpotent elements is clearly an ideal. It includes A b by the first result of this theorem. I f this inclusion were proper, A / A b would include the non-zero nilpotent ideal A l b/AB for some b € M \ A b ■ Since A / A b is semiprime, we conclude that Af equals the Baer radical. □ Having characterized the Baer radical as an intersection of ideals and characterized the algebras with the smallest possible Baer radicals, we turn our attention to the Baer radical itself and characterize it as a sum o f ideals. 4.4.7 T h e o r e m An algebra is Baer radical if and only if every non-zero homomorphic image o f it has a non-zero nilpotent ideal. The Baer radical of any algebra is a Baer radical ideal which includes all Baer radical ideals and all one-sided nilpotent ideals. P roo f Let A be an algebra. By the last theorem A b is the smallest ideal o f A such that A / A b is semiprime. Hence A — A b holds if and only if every non-zero homomorphic image o f A contains a non-zero nilpotent ideal (i.e., is not semiprime). Theorem 4.4.6 shows that A b is a semiprime ideal of A and that ( A b ) b is a semiprime ideal o f A b - Hence Proposition 4.4.4(a) shows that ( A b ) b is a semiprime ideal of A. Theorem 4.4.6 then shows A b Q ( A b ) b - Since the opposite inclusion is trivial, A b is a Baer radical ideal. Now suppose X is any Baer radical ideal of A . If 2 is not included in A b , then 2/(X n A b ) is a non-zero homomorphic image which must
4-J,. 8
T h e B aer R adical
487
contain a non-zero nilpotent ideal by the first statement o f this theorem. However 1/(1C\A b ) is isomorphic to (J + A b )/ A b which is an ideal o f the seiriiprime algebra A / A b and hence contains no non-zero nilpotent ideal by Proposition 4.4.2(e). This contradiction shows that X (which was an arbitrary Baer radical ideal) is included in A b - Finally Proposition 4.4.2(d) shows that A b includes all one-sided nilpotent ideals since it includes all Baer radical ideals and hence all nilpotent ideals. □ Note that the first sentence of the last theorem shows how to character ize Baer radical ideals in terms of nilpotent ideals. One might be tempted to continue the process and define a new concept by the same construc tion starting with Baer radical ideals. However, an application of Theorem 4.4.3(e) shows easily that we simply get the concept Baer radical again by this construction. This remark is due to Adam Sulinski, Robert F. V. Anderson, and Divinsky [1966] whose work suggested the treatment of Theorem 4.4.7 above. Next we prove the analogue o f Theorem 4.3.2. The two assertions in the last sentence have each been proved already (in Theorems 4.4.6 and 4.4.7 respectively) but are included for comparison. Property (a) was established by McCoy [1949]. T h e o re m Any algebra A and any ideal X o f A satisfy: (a) I g = i n A b ■ (b) J C A b implies (A / 1 ) b = A b f l ic,) A/X semiprime implies A b = 1 b Thus in particular A/ A b is semiprime and A b is Baer radical.
4 .4.8
arocf The proof o f Theorem 4.3.2 may be copied, merely substituting “prirr.o” for “primitive” and “the laat sentence of Theorem 4.1.10” for “The orem 4.1.8(e)". W e give alternative proofs for (a) and (b) also. (a): Proposition 4.4.4(b) shows that 1 n A b is a semiprime ideal of 1. Hence Theorem 4.4.6 implies 1b Q 1 fl A b T o prove the opposite inclusion, suppose a is an element o f 1 \ 1 b - Then there is a prime ideal M of I which does not contain a. Define S to be the set 1 \ M which contains a. Consider the family of ideals o f A which do not intersect S. This family is not empty since { 0 } belongs to it. Zorn’s lemma applies to this family when it is ordered by inclusion and gives a maximal element V . We repeat the argument given in the proof of Theorem 4.4.3 (b)=>(e), which shows that V is a prime ideal. Suppose 6i and 62 are any elements of A satisfying b i, 62 ^ V . Then the maximality of V gives elements C\ and C2 in S satisfying c.j £ V + A 1b jA 1 for j = 1 , 2 . This implies c\Ac 2 Q ("P + A 1b iA i )A ('P + C V . However this contradicts the fact that C ilc 2 has nontrivial intersection with 5. Hence V is a prime ideal which does not include a. Since a was an arbitrary element of I \ 1 b and
488
4: Ideals, Representations and Radicals
4.4.9
since I ft A b equals f l { I n P : V is a prime ideal of A } , we conclude that I D A b is included in 1 b - This proves (a). (b): Proposition 4.4.4(c) and (d) prove (A / T ) b
=
n { P : P is a semiprime ideal of A / 1 }
=
n { P / l : P is a semiprime ideal of A }
=
A s / I.
□
The following two conditions which imply that an algebra is semiprime may have some interest. The second condition significantly strengthens Yood [1972], Corollary 2.7. 4.4.9 P r o p o s itio n An algebra is semiprime if either: (a) I t has a separating family o f topologically irreducible normed repre sentations; or (b ) I t is a dense subalgebra o f a semiprime topological algebra. In fact, the intersection o f a closed { prime / semiprime ) ideal o f a topological algebra with a dense subalgebra is a { prime / semiprime ) ideal o f the subalgebra. P roo f (a): This follows from Proposition 4.2.5 and Theorem 4.4.3(a). (b ): This follows from Theorem 4.4.3(b). The last remark follows from Theorem (4.1.10 (b) / 4.4.3(b) ).
□
The intersection of the kernels o f the continuous topologically irreducible representations of an arbitrary Banach algebra A would seem to be an interesting object. It is a closed semiprime ideal, but not much more seems to be known about it. The following result is chiefly o f interest to us through its use in Theorem 4.4.11. It shows that a nil algebra with a uniform bound on the degree of nilpotency o f its elements is nilpotent. It is due to Nagata [1952]. Graham Higman [1956] gave a more general version. The simpler proof which we use here is due to Philip J. Higgins and appears in Jacobson [1964]. 4.4.10 P r o p o s itio n Let A be an algebra. Let n be an integer such that a " = 0 holds fo r all a £ A . Then A is nilpotent and satisfies A N — { 0 } where N = 2" - 1. P roo f The proof is by induction on n. The case n = 1 is trivial. Let a " = 0 hoki for all a € A . For a, b e A and A € C, the polynomial p (A) = (a + A6)n vanishes identically. Hence the coefficient qn(a,b) = ]C>=o a ’ fw™-7'- 1 of A is zero for a, b 6 A . For all a, b, c G A , we see n —1
0
=
n —1 n — 1
5 3 g n(a ,c ^ )6 n- J- 1 = 5 3 ] T V c ^ a n- fc- 16n- J- 1 j —0 k=0 j~0
1
Th e B aer R adical
489
n—1
=
^ a*c^n( 6, a " - * -1 ) = an~ 1c q „ (b ,l) = nan~ l cbn~ l . k= 0
We conclude an - 1 c6n_1 — 0 for all a, b, c € A . Let A f be the ideal which is the linear span of {abn~ 1c : a, c £ A l ,b £ A }- Then we have just shown A fA A f = {0 }. Also each a + A f £ A / M sat isfies (a + Af ) n~ l — Af. By induction we may assume (A /.\f)M = A f where M = 2n_1 — 1. Thus A M C A f holds and this implies A N = A M A A M C A fA A f = {0 }. This proves the proposition. □ N il and Nilpotent Ideals in Banach Algebras In the next theorem (b) is due to Grabiner [1969]. Conclusion (c) and the first sentence of (a) are due to Dixon [1973b] with proofs based on that of the previous reference. The second sentence of (a) is due to Grabiner [1976]. Conclusion (d) is an easy consequence of (b). Prom (c) we conclude that the Baer radical is the smallest interesting radical for Banach algebras. Note that a Banach algebra is Jacobson semisimple if and only if it has no non-sero topologically nil ideals and it is semiprime if and only if it has no non-zero nil (equivalently, nilpotent) ideals. 4.4.11 T h e o r e m Let A b e a Banach algebra. (a) Every nil left ideal o f A is included in a sum o f nilpotent ideals. Every finitely generated nil left ideal is included in a nilpotent ideal. (b) I f A is nil, then it is nilpotent. (c) The Baer radical o f A equals both the sum of all the nilpotent ideals o f A and the sum. o f all the nil left ideals o f A . Hence A b is the nil radical o f A and A / A b has no non-zero nil left ideals. (d) The Baer radical is nilpotent i f and only i f it is closed. Proof (a): W ithout loss o f generality we may assume that A is unital. For each b € .4 let Ab and A b A be, respectively, the principal left ideal and ihe principal ideal generated by b. I f A b is nil we will show that A b A is nilpotent. Hence any nil left ideal £ is included in the sum o f the collection {,46.4 : 6 € £ } of nilpotent ideals. I f £ = A b i + Ab? H------ f- Ab„ is a finitely generated left ideal, then it is included in the nilpotent ideal A b iA + 4 .62^4 H---- + AbnA which is nilpotent by Proposition 4.4.2(d). Assume that Ab is nil. For each n € N define 5 n to be the closed set {a G A : (ab)n = 0 }. Since Ab is nil, A equals U ^ =1 <Sn. The Baire category theorem shows that there is some <S„ with nonempty interior. Thus for some ao £ A and some e > 0 , ((ao + a ) 6) n = 0 holds for all a € Ae- Let a t . li and uj e A * be arbitrary. Since the polynomial w (((ao + Aa) 6) n) is zero for all A € C with |A| < e, it is zero for all A € C. Since w was arbit ary ((ao + Aa)6) n is zero. Hence ao + Aa belongs to <Sn for all a € A i and „ e C . In other words Sn — A and each element ab of the principal
4: Ideals, Representations and Radicals
490
4.5.1
left ideal Ab satisfies (ab)n = 0. Proposition 4.4.10 now shows that Ab is nilpotent. Since ( A b )N = { 0 } implies (A b A )N C (A b )N A = { 0 }, *46.4 is also nilpotent. (b): For each n € N let Sn be the closed set {a € A : an = 0}. Since A is nil, it equals U” = 1 <S„. Hence by the Baire category theorem there is some ao 6 A and e > 0 satisfying (a© + a )n = 0 for all a € A e. Arguing as before, we conclude A = S n. Proposition 4.4.10 shows again that A is nilpotent. (c): Theorem 4.4.7 shows that A b is a nil ideal which includes the sum of all nilpotent ideals. Hence (a) shows that A b equals the sum of all nilpotent ideals which equals the sum of all nil left ideals. Proposition 4.4.2(c) shows that A j A b has no non-zero nil ideals. (d): If A b is closed, Theorem 4.4.6 and (b) above show that it is nilpo tent. If it is nilpotent then there is some N satisfying A b Q {a . aN = 0}. Hence the closure of A b is included in the same set and is thus nilpotent. Hence A b is closed by Theorem 4.4.7. □
4.5
The Brown-M cCoy or Strong Radical
We will now discuss one more radical. This radical was defined by B. Brown and McCoy [1947] and is sometimes called the Brown-McCoy radical. Segal [1941] defined a unital algebra to be semisimple if its maximal ideals had intersection {0 }. Later [1947a], he changed the term to strongly semisimple. Brown and McCoy originally defined this radical in terms of G-regular ideals which we introduce with a different name in Definition 4.5.5. Michael J. Meyer [1992d] establishes some deep analogies with the Jacobson radical. This radical is closed in spectral algebras and is quite well behaved in Banach algebras. However, it is o f less importance in functional analysis than the Jacobson radical. The Jacobson radical o f familial Banach alge bras is usually a pathological portion. This is not true o f the strong radical. For instance, if H is a separable Hilbert space, then the strong radical of B (H ) is the ideal o f compact operators.
4.5.1 D e fin itio n
Let A be an algebra. The intersection o f all maximal modular ideals o f A will be called the Brow n-M cCoy radical, or strong radical, of A and will be denoted by A m - An algebra is called strongly semisimple if A m is zero and is called Brown-McCoy-radical if A m equals A . The set o f maximal modular ideals of an algebra A will be denoted by If I is a subset of A , then /»“ ( ! ) will denote the hull o f I in E A : hs ( I ) = { M 6 Ea : I C M ) . We follow our usual convention on the intersection o f an empty family o f sets, so that A m is A if and only if A has no maximal modular ideals.
4.5.2
T h e B ro w n -M cC o y or Strong R adical
491
Then an algebra A is a Brown-McCoy radical algebra if and only i f it has no maximal modular ideals and hence, by Theorem 2.4.6, if and only if it hes no modular ideals at all. We need a simple result analogous to Theorem 4.1.8(e) and the last sentence of Theorem 4.1.10.
Theorem
Let A be an algebra and let J be an ideal o f A -
The
map M
>-> M f l l
is a bijection o f E A \ /i“ (Z ) onto E j and the map M i—> M / X is a bijection of hB ( X ) onto E a /X P i oof Let M be a maximal modular ideal of A which does not include X. Then X + M equals A by maximality. Thus X / ( M Pi I ) which is isomorphic to {'£ + M ) / M = A / M is a simple unital algebra. Hence M fl I is a maximal modular ideal of X. The last sentence of Theorem 4.1.10 shows that M M Pi I is injective. Now suppose A f is a maximal modular ideal o f I . Let ip: I —* I / M be the natural map and let e € I be a relative identity for M so that ifi{e) is the identity element in I / M . Consider the map ip: A —> X / M defined by ip(a) = ip(eae) for all a £ A - It is easy to check that ip is a homomorphism. For instance for a, b £ A , we have ip(ab) — ip(eabe) =
The following result is the analogue of Theorems 4.3.2 and 4.4.8. It was essentially proved by Brown and McCoy [1947]. 4.5.3 Theorem
Any algebra A and any ideal 1 o f A satisfy:
(f.) Xm = X C \ A m ■
t
( b ) I C A m implies ( A / X )m = A m / 1 (c) A/X strongly semisimple implies A m = Xm - Thus in particular A / A m is strongly semisimple and A m is Brow n-McCoy radical. P r o of The proof of Theorem 4.3.2 may be repeated, merely replacing “prim
itive” by “maximal modular” and “Theorem 4.1.8(e)” by “Theorem 4.5.2” . D
492
4: Ideals, Representations and Radicals
4.5.4
The following simple but important result was noted by Segal [1947a] for complete algebra norms. 4.5.4 T h e o r e m spectral norm.
The strong radical is closed in any spectral or weakly
P ro o f This follows from Theorem 2.4.7 and Proposition 2.2.17.
□
We now give some characterizations o f the strong radical similar to the characterizations o f the Jacobson radical in Theorem 4.3.6 and Proposition 4.3.13. They are due to Brown and McCoy [1947] and Michael J. Meyer [1992d]. Definition 2.1.19 introduced the notion o f weakly invertible el ements, but we deferred defining “weakly quasi-invertible elements” . By the general convention on the use of the prefix “quasi-” , an element a € A should be weakly quasi-invertible if 1 —a is weakly invertible in A 1. Weakly quasi-invertible elements were called G-regular by Brown and McCoy [1947] who used the criterion o f Theorem 4.5.7 as the definition o f their radical. 4.5.5 D e fin itio n An element a in an algebra A is called weakly quasiinvertible if the ideal n %(a)
=
{(6 - fa ) + (c - oc) + y ^ e j - d j a e j ) :
<=i n € N, d, c, di, e; € A
for
i — 1 ,2, . . . , n}
is all o f A . The set o f such elements is denoted by AwqG- An ideal is called weakly quasi-regular if each o f its elements is weakly quasi-invertible. 4.5.6 P r o p o s itio n The following are equivalent for an element a in an arbitrary algebra A . (a) a is weakly quasi-invertible. (b) a belongs to 1 (a ). (c ) 1 — a is weakly invertible in .41. (d ) a is not the relative identity for any maximal modular ideal o f AP roo f Let J be the ideal of A 1 generated by 1 — a so (c) is equivalent to 3 — <41. It is easy to check that J ~ 1 (a ) + C(1 — a). Note also that a is a relative identity for an ideal if and only if the ideal includes 1 (a ). (a) => (b): Obvious. (b ) => (c): Since 1 — a € J , (b ) implies 1 = 1 — a + a € .7(c ) =► (d ): I f a is the relative identity for a maximal modular ideal M o f A , then 1 (a ) C M implies J C M + C(1 — a) ^ A 1. (d) => (a): Theorem 2.4.6(d) shows that a is a relative identity for some maximal modular ideal which includes 1 (a ) whenever 1 (a ) is a proper subset of A . O
4.5.7
T h e B ro w n -M cC o y or Strong R adical
493
4.5.7 Theorem Let A be an algebra. Then the strong radical A m is a weakly quasi-regular ideal o f A which includes all weakly quasi-regular ideals o f A . Hence it is the largest ideal 2 satisfying any (hence all) o f the follom ng equivalent conditions Sps A / I(a + X ) C Sps A (a ) C Sps A /J(a + 1 ) U { 0 } a + Z € ( A / X ) WqG & fl € AwqG
Sps (b + a) = Sps (b) Sps (a ) = { 0 } ps ( b + a ) = p s (b) ps (a ) = 0
V a € A;
V a € A;
V b € A', a € T; V a G I; V b € A ;a € X; V a 6 X.
P roo f I f a is an element o f a weakly quasi-regular ideal I , then every element in A l a A x C X is weakly quasi-regular. Thus an element a € A belongs to some weakly quasi-regular ideal if and only if ^4 * 0^41 is a weakly quasi-regular ideal. Therefore the theorem will be proved if we can show that a € A belongs to A m if and only if A 1aA 1 is a weakly quasi-regular ideal. Suppose A 1aA 1 is not weakly quasi-regular. Then there is some b € A 1a A l which does not belong to 1(6). Thus 1(6) is a proper modular ideal and by applying Zorn’s lemma to the set o f ideals including X(b) but not containing b we see that b is the relative identity of some maximal modular idea! M . Thus b does not belong to M and hence b does not belong to A m However this implies a $ A m since a g A m would imply b 6 A 1aA 1 C A m Conversely, suppose A 1a A 1 is a weakly quasi-regular ideal of A and M is a maximal modular ideal o f A . Then (A / M )(a + M ) { A / M ) is a weakly quasi-regular ideal o f A / M . However since A / M is simple, either a belongs to M or ( A / M ) ( a + M ) ( A j M ) equals A / M . The latter is impossible since 1 £ A / M cannot be weakly quasi-regular. Thus a belongs to M . Since M was an arbitrary maximal modular ideal o f A , a belongs to A m A non-zero number A € € belongs to the strong spectrum if and only if A " 1 a is weakly quasi-regular. I f an element a belongs to a weakly quasiregulu ideal, then Aa is weakly quasi-regular for all A € C, so Sps (a ) = {0 }. This, together with the arguments in the proof of Theorem 4.3.6, establishes the second sentence. □ 4.5.8 Proposition In any spectral algebra A , the strong radical is the perturbation set in A o f the set A\,g o f weakly regular elements in the unitization: A m = {a € A : a + A ^ q £ •''Cg)-
496
4: Ideals, Representations and Radicals
4.6.1
representation. An algebra A is called simple if it has no ideals except { 0 } and A. Obviously an algebra is both unital and simple if and only if { 0 } is a maximal modular ideal. Simple unital algebras are the constituents of the subdirect product decomposition of strongly semisimple algebras. Recall that Corollary 4.3.10 asserts that any spectral semi-norm on a semisimple algebra is a norm. Theorem 4.5.7 shows that the same is true for a strongly semisimple algebra. Hence without loss o f generality we speak o f spectral norms instead o f spectral semi-norms in the next theorem. 4.6.1 Th eorem A n algebra A is semiprime, semisimple, or strongly semisimple i f and only i f it is isomorphic to a subdirect product o f prime algebras, prim itive algebras, o r simple unital algebras, respectively. I f A is a spectral normed algebra which is either semisimple o r strongly semisimple, then the subdirect product representation may be chosen normed with, the isomorphism contractive. P roo f Let { 1 ° : a € .4} be the set o f prime, primitive, or maximal modular ideals o f A depending on whether A is semiprime, semisimple, or strongly semisimple. Then n { I “ : a € A ) is {0 }. So the map a n a where a (a ) = a + l a is an isomorphism o f A onto a subdirect product o f { A / l a : a € .4}. This proves the necessity o f all the purely algebraic conditions. I f A is a spectral normed algebra, then each T a in the decomposition by primitive or maximal modular ideals is closed. Thus o i - t a i s a contractive isomorphism onto the normed subdirect product when each A / l a has its quotient norm. Suppose A is a subdirect product o f { A a : a € A ], where each A a is a prime algebra, a primitive algebra or a unital simple algebra. Then the ideals 1 “ = {a e A : a (a ) = 0 } satisfy A a = A / T a and nQgA^a = {0 }, so that A is semiprime, semisimple, or strongly semisimple respectively. This proves the sufficiency. □ A Subdirect Product Representation fo r Double Centralizer Algebras We remark on an alternative direct product decomposition in terms of simple unital direct factors which is sometimes available. Let A be am algebra and let J be an ideal of A . Theorem 4.5.2 shows that the map M >-* M Pi J is a bijection o f \ h ~ ( l ) onto S j. If M is any ideal in H i let M € \ h ~ { I) be the corresponding ideal satisfying M Pi T — M . Since M is maximal and does not include I we have A / M = ( M + I ) / M ~ T / M Pi I = X / M . Hence the direct factors in Y[~M£-=.A\hs-(T )(A / M ) are isomorphic to the corresponding direct factors in X\m€3 i @ /M ) . Therefore there is a homomorphism of A into Y Im c -z (A / M ) with kernel H m z S iM — This is a subdirect product decomposition of A over a smellier space of maximal modular ideals. representation will be larger.
In general the kernel o f the
Subdirect Products
497
Now replace A and J in the last paragraph by T>(A) and A , respec tively, where A is a strongly semisimple algebra. Theorem 4.5.8 shows HMeEAM = n ^ ieS = { 0 } so that the subdirect product de composition is now an isomorphism. Hence we have established the first portion o f the next result. 4.6.2 C o r o lla r y I f A is strongly semisimple, then there is a natural isomorphism o f V { A ) onto a subdirect product included in (A / M ). This subdirect product is given explicitly by: (L , R ) ^ ( M ^ L (e M ) ) where fo r each M €
V (L , R ) e V (A )-, M e ~ A
eM is a relative identity fo r M .
Proof It only remains to justify the explicit formula. The calculation a ( ( L . R ) — L (e M ))b = a (L (b ) — I/ (e ^ ) 6) = R (a )(b — e ^ ( 6) € M for all a, b ~ A shows that ( L , R ) = L (e ) + ( ( L , R ) — L ( e ^ ) ) is a decomposition of (L , R ) € T>(A) corresponding to T>(A) = A + M . This shows that the subd rect product decomposition is given by the indicated formula. D A Ch aracterization o f Spectral Algebras V/e wish to improve Theorem 4.6.1 to give a characterization o f spectral algebras. For this we need to introduce a definition. 4.6.S D e fin itio n Let A be a subdirect product o f algebras { A a : a 6 X }. Then A is called ample if an element a e A is quasi-invertible whenever a (a ) is quasi-invertible in A a for each a e A. 4.6.4 T h e o re m The following are equivalent fo r an algebra A . (a) A is a spectral algebra. ( t ) A / A j can be embedded as a dense subalgebra o f a semisimple Banach algebra B such that the spectral radius o f A / A j is the restriction o f the spectral radius o f B. (c ) A / A j is isomorphic to an ample, normed, subdirect product o f strict ,y dense spectral normed algebras o f bounded operators on normed lin ear spaces. Proof (a ) => ( b): Let a ' be an arbitrary spectral semi-norm on A . Theorem 2.2.1* (d) and Corollary 4.3.10 show that a ' induces a spectral norm |||-||| on A / A j. het (A , IjHH) be the completion o f (A / A j , ||H||). Let ip: A -> A / A j and yj-.A —* A / A j be the natural maps. Any a € A satisfies pip(A )ifi{a) = lllvH3 ^!!00 = P^(v,( a)) 80 we simply write p(
498
4: Ideals, Representations and Radicals
4.7.0
A second application of Corollary 4.3.10 shows that a induces a norm on A / A j . The completion of A / A j in this norm is clearly A / A j - This is the
semisimple algebra B of (b). By Theorem 4.3.6(d) and previous remarks we get p(
4.7
Categorical Theory of Radicals
We will now investigate some more fundamental common properties of all radicals. A number of general theories o f radicals have been advanced: e.g., Shimshon Amitsur [1952], Vladimir A. Andrunakievic [1961], Divinsky [1965], Gray [1970], Aleksandr Gennadievic Kuros [1953], Adam Sulinski [1958], Barry James Gardner [1989]. The theory which most conveniently covers the three radicals introduced in this section and the radicals which will be introduced in Volume II is that of Mary W . Gray [1970]. In order to give this theory we must introduce a small part of the language o f categories and express some of the results of this section in that language. Many aspects o f the subject will be ignored. However, we will show that several of the categories which we study in these notes are semiabelian. This type of category was introduced by Gray in part to give a setting for a very general theory of radicals [1967], [1970], For further information on the more traditional approach to radicals of algebras and rings, see Divinsky [1965]. We give only a superficial introduction to category theory in func tional analysis here. For more details see Boris Samuilovic Mitjagin and Albert Solomonovic Svarc [1964], Sten Kaijser [1976], A. W . M. Graven [1979], Yurii Vasilyevic Selivanov [1978], Michael Grosser [1979a], Johann Cigler, Viktor Losert and Peter Michor [1979], Michor [1978] and Alexandr Yakovlevic Helemskii [1989b].
4.7.1
Categorical T heory o f Radicals
499
<1.7.- D e fin itio n A category C consists of: (a) A class C of objects. (b) For each ordered pair ( A , B ) o f objects from C a set H o t t i q ( A , B ) . (c; For each ordered triple (A ,B ,C ) o f objects from C a map (called composition) (o): H o m Q (B ,C ) x H o m ,Q (A ,B ) —> H o it iq (A ,C ). These are subject to the following restrictions: (d ) Each ordered quadruple (A ,B ,C ,T >) of objects from C and each cp € H orriQ (A , B ), tp € H om .Q (B, C) and 0 € H orrifj(C , V ) satisfy 6 o (tp o tp) = (0 0Xp)0tp. (e) For each object A in C there is an element I a € H o7t iq ( A , A ) such that each object B in C, each tp € H om .Q (A , B ) and each tp € H orriQ (B , A ) satisfy I a ° iP ~ ip
and
‘P ° I A — 'P-
An object 0 in a category C is called a zero object if for each object A in C, N a m t j(A , 0) and Hom,Q(0, A ) are both singletons. I f ( A , B ) is an ordered pair c f objects in a category C, an element 6 € H o m .Q (A ,B ) is called an istmorphism if there exists an element 6~l G H o m ,Q (B ,A ) satisfying 0~Lo 9 = Ia and 6 o6~l = I B. Objects A and B are said to be isomorphic if H o m c (A ,B ) contains an isomorphism. The elements of H o m ,Q (A ,B ) will be called morphisms. ?rom now on we write H o m (A ,B ) instead o f H ottiq (A , B ) if there is no danger of confusion. It is convenient to insist that the sets H o m (A , B )
in a given category be disjoint. W e make this requirement (although our notation for zero-maps is at variance with it). Thus for any < p e H o m (A , B ) end -p € H o m (C , T>), tp o tp is defined (uniquely) if and only if T> = A. In all the categories which we will consider, the class o f objects will be so large that it is a proper class rather than a set, the objects themselves will be sets (with additional structure) and each set H o m (A , B ) will consist of mpps with domain A and range in B. Composition o f functions will agree with composition as defined in the category. These are the most commonly considered types o f categories. A ll our categories will also contain zero objects. It is easy to see that for each object A in any category there is only one t'.s r'.s :.'.! in H o m (A ,A ) with the properties o f I a - Furthermore, if A and B are objects and 0 € H o m (A ,B ) is an isomorphism, then 0~l is uniquely determined and is also an isomorphism. Similarly any two zero-objects in a category are isomorphic. Thus there will be no confusion if we denote every zero object by 0 and also use the syrrboi 0 to denote the unique element o f H o m (0, A ) and H o m (A , 0) for all J. :st the category. Finally, if A and B are any two objects in a category
500
4: Ideals, Representations and Radicals
4.7.2
and / € H o m (A , B ) can be written a s / = 0 o 0 f o r 0 € H o m (A , 0) and 0 € H om (0, B ), then we denote / by 0 also. Then 0 € H o m (A ,B ) is uniquely determined independent of the choice o f zero objects used in its definition. The notation has the advantage of making / oO = 0 and Oo/ = 0 hold whenever the composition is defined. We will now introduce the examples with which we are mainly concerned in this chapter. In each case we define H o m (A , B ) for an ordered pair (A , B) o f objects, merely by a qualitative description o f the kind o f maps which belong to H o m (A ,B ). I f these maps are identified (as usual) with their graphs in the Cartesian product A x B, then the sets H o m (A ,B ) will not be disjoint. Disjointness may be achieved by an easy indexing scheme but we ignore this technicality. 4.7.2 Definition Sym bol A SA SSA NA BA GNA GBA
The following table describes seven categories.
O bjects
algebras spectral algebras spectra) semi-normed algebras normed algebras Banach algebras normed algebras Banach algebras
M orphism s homomorphisms homomorphisms continuous homomorphisms continuous homomorphisms continuous homomorphisms contractive homomorphisms contractive homomorphisms
These categories are called, respectively, the category o f algebras, the cate gory o f spectral algebras, the category o f spectral semi-normed algebras, the category o f normed algebras, the category o f Banach algebras, the geometric category o f normed algebras, and the geometric category o f Banach algebras. Example 4.8.11 will show a substantial difference between the topolog ical and geometric categories of Banach algebras. We now introduce cate gorically defined kernels, cokernels, images, and coimages. The verifications which are left to the reader are all easy. A monomorphism k in a category is simply a morphism which can be cancelled from the left of an equation (i.e., k o f = k o g => / = g). A subobject in a category is an equivalence class o f monomorphisms, where k\ and jfc2 are equivalent if there exist morphLsrns / and g satisfying ki = k2o f and k 2 — k} o g. In particular a subobject is not an object in the category, and although it corresponds to a monomorphic image of an object it need not itself correspond to any object. For instance, the embedding o f C ([0 ,1]) in Z-1 ([0,1]) defines a subobject o f L ‘ ([0, lj) in each o f the two categories of Banach algebras listed in the last definition. However, since the image of C ([ 0 , 1 ]) is not closed in . ^ ( [ 0 , 1 ]) it does not correspond to any object in these categories. This complicates the application o f this definition to such examples as the category o f Banach algebras.
4.7.2
Categorical T heory o f Radicals
501
Let fci and k2 be monomorphisms representing two subobjects TZi and n 2. Then H 2 is said to include TZi if there is a monomorphism k satisfying k 2 o r = fci. A subobject is said to be a zero subobject if one (and hence ail) of itf representing monomorphisms is a zero morphism. A morphism / is said to have a kernel if there is some monomorphism k satisfying: (a) / o k = 0 ; (b) I f g is any morphism satisfying f ° g = 0, then there is a unique morphism h satisfying k o h = g. The subobject which is the class o f all monomorphisms equivalent to k (w iich is also the class of all monomorphisms satisfying (a) and (b )) is called the kernel o f f . Ip all of the categories listed in Definition 4.7.2, every morphism has a kernel and the kernel of / € H o m (A ,B ) is simply (the subobject rep resented by the injection into A o f) the set theoretic kernel ker(/) of /. Thu? in the first three categories any ideal is a kernel and in the last four categories any closed ideal is a kernel. An epimorphism u in a category is a morphism which can be cancelled from the right o f an equation (i.e., f o u = g o u => f = g ). N ote that in the last five categories listed in Definition 4.7.2 any morphism with dense range is an epimorphism. A quotient object is an equivalence class o f epimorphisms where ui and u2 are equivalent if there exist morphisms / and g satisfying ui — f o u 2 and u2 = g ° « i • A morphism / is said to have a cokemel if there is an epimorphism u satisfying: (a ) u o f = 0 ; (b) I f g is any morphism satisfying g o / = 0, then there is a unique morphism h satisfying h o u = g. The quotient object which is the class o f all epimorphisms equivalent to u (which is also the class of all epimorphisms satisfying (a) and (b )) is called the cokemel o f f . Ir- the categories A and S A , the cokernel o f / € H o m (A ,B ) is repre sente'] by B B/ ideal generated by f ( A ) . In th-j categories S S A , N A , B A , G N A and G B A , the cokernel of / e Harr (A , B ) is represented by B i-» B/ closed ideal generated by f ( A ) . A fnorphism / is said to have an image if there is some monomorphism i satir.'ying: { i ) There is a morphism g satisfying i o g = /; ( ) If j is a monomorphism such that there is a morphism h satisfying j o h = f , then there is a monomorphism k satisfying j o k = i.
502
4: Ideals, Representations and Radicals
4.7.3
The subobject which is the class o f ail monomorphisms equivalent to i (which is also the class of all monomorphisms satisfying (a) and (b )) is called the image o f f . In all o f the categories o f Definition 4.7.2, the image o f / € H o m (A , B) is represented by the map A / k e r (f) i-» B induced by /. A morphism / is said to have a coimage if there is an epimorphism u satisfying: (a) There is a morphism g satisfying g o u = /; (b ) If i! is an epimorphism such that there is a morphism h satisfying h o v = f , then there is an epimorphism w satisfying w o v — u. The quotient object which is the class o f all epimorphisms equivalent to u (which is also the class o f all epimorphisms satisfying (a) and (b )) is called the coimage o f f . The coimage o f / € H o m (A , B ) in all o f the categories o f Definition 4.7.2 is represented by the natural morphism A
_4/ker(/).
4.7.3 Definition
A category with a zero object is called semiabelian if: (a) Every morphism may be factored into a representative o f its coimage followed by a representative o f its image, and (b) Every morphism has a cokernel. A category with a zero object is called cosemiabelian if it satisfies (a) and (c) Every morphism has a kernel.
4.7.4 Proposition A ll o f the categories o f Definition 4.7.2 are both semi abelian and cosemiabelian. P roo f This follows from the remarks above.
□
A category B is called a subcategory of a category C if every object o f B is an object o f C, any ordered pair (.4, B ) of objects o f B satis fies H o m ^ (A , B ) C H o m Q (A ,B ) and H o m ^ (A , A ) contains the identity IA £ H o t u q IA , A ). A subcategory B of C is called fu ll if it satisfies H o m ^ (A ,B ) ~ H o m Q (A ,B ) for all ordered pairs ( A , B ) o f objects of B. For instance, S B A is a subcategory of N , A , S N A , and B A , and it is a full subcategory o f S N A . In the following theorem we use the notation A n to represent a sub object o f an object, A . Recall that by definition A n is not an object but an equivalence class o f monomorphisms. However, in all of the cases under discussion here, A n is easily and naturally associated with a definite object.
Categorical T h eo ry o f Radicals
503
4.7.5 D e fin itio n Let C be a semiabelian category. A radical subcategory of C is a full subcategory R satisfying: (e.) If A is an object in R and i e H om (X , B ) represents the image of / S H o m (A , B ), then I is an object in R ; (b) I f A is an object in R and k e H o m (T l, A ) represents the kernel of / e H o m (A , B ), then 72. is an object in R ; (c) For each A € C , there is unique subobject .A ft o f A which satisfies: (c i) .A ft is a kernel; *(<*) .A ft is represented by a monomorphism with an object o f R as dofciain; 1 (C3 ) A n includes any subobject of A which is a kernel and is also represented by a monomorphism with an object of R as domain; (d) I f u € H o m (A , B ) is a representative o f the cokernel o f a represen tative of .Aft, then the subobject 6 f t is the zero-subobject o f B. W hat we call a radical subcategory here (following Gray [1967]) corre sponds to a hereditary radical in the terminology of some writers. See, for instance, the excellent book [1965] by Divinsky. 4 .7'.’6 ' T h e o re m The fu ll subcategory defined by the condition that all its sbjscis satisfy (A = A j ; A = A m ) a radical subcategory o f each o f ',13 categories A , S A , S S A ,B A , and S B A . The fu ll subcategory defined by the condition A = A b w a radical subcategory o f each o f the categories A,' 3 A , and S S A . Pro?/ To prove (a ) in the definition o f a radical subcategory, note that X css be chosen as A j ker(/) for all o f the categories we are considering. Heace (a) follows from Theorems 4.3.2, 4.4.8, and 4.5.3. Similarly, to prove (b) oae notes that K may be taken as ker(/) which is an ideal o f A . The sac?3 three theorems complete the argument. The subobject .A ft of (c) is represented by A j —* A , A m —> A and A b —1 > A in the three cases under discussion. Thus (ci ) follows from Corollary 4.3.10 and Theorem 4.5.4 and the fact that A j , A m , and A b are always ideals. Finally ( C 2 ) , ( C 3 ) , and (d) follow from Theorems 4.3.2, 4.4.8, and 4.5.3. □ In N A and G N A , A j , A b and A m are not necessarily closed so that A/'A j ,A / A b and A / A m are not normed algebras.
4.8 4 .8 .1
History of Radicals and Examples
H istory o f Radicals
Because o f the central importance o f radicals in the theory o f algebras, discuss their history in detail. In order to simplify the discussion as inucli as possible, we will concentrate on radicals o f algebras, mentioning ~:2
504
4: Ideals, Representations and Radicals
4.8.1
only briefly the important advances which related solely to the extension of the concepts from algebras to more general rings, and we will oinit any discussion o f radicals other than those considered in this work. For infor mation on them, see Divinsky [1965]. We will also transpose notation and terminology to agree with that used in this work whenever it is convenient. In particular, we make operators act on the left and speak o f left modules although many authors have used the opposite convention. Again we emphasize the root idea o f radicals. When trying to establish a structure theory for a wide class o f abstract mathematical objects, such as algebras, it is usually necessary to discard certain pathological cases. It frequently turns out, as it does in the case of algebras, that in each object the pathological part can be excised by judiciously choosing the kernel of a morphism which contains the pathology and studying only the image of that morphism which is healthy. The trick is to identify the pathology so that: (a) the above construction is possible; (b ) a significant theory can be developed for the images; and (c) the excluded kernels are not too large, i.e., they do not contain the really interesting examples. It is a curious historical fact that the strategy just outlined was first applied to a class o f non-associative algebras— the Lie algebras. These algebras arise naturally in the study of differentiable groups initiated by Sophus Lie. They were first seriously studied by Wilhelm Killing [1888] although the term “Lie algebra” was not introduced until 1933 in lectures by Hermann Weyl at the Institute for Advanced Study. Killing’s work, although very powerful, was not rigorous and his proofs were not accepted by most contemporary mathematicians. In his thesis [1898], Elie Cartan gave rigorous proofs o f Killing’s ideas and extended them to a theory ift which the radical o f a Lie algebra plays a central and very explicit role. (For a Lie algebra, the radical is the sum o f all solvable ideals which is itself a solvable ideal. The quotient modulo this ideal has no non-zero solvable ideals.) One o f his most notable contributions was the discovery of what has come to be called “Cartan’s criterion” — that a Lie algebra is seuiisimple if and only if its Killing form is nondegenerate. The thesis gave a complete classification of complex simple Lie algebras (slightly correcting Killing’s results) and showed that any complex Lie algebra in which the radical is zero is the direct sum of (finitely many) simple Lie algebras. The term semisimple ( halbeinfach) had been introduced by Killing to describe a Lie algebra which is the direct sum of simple Lie algebras. In [1898] Cartan applied the theory o f radicals, which had been so successful in his treatment of Lie algebras, to the treatment o f finite di mensional real or complex associative algebras. The distinction between algebras which contained quarternionic components and those that did not was at the forefront of hi , discussion and makes it look somewhat odd to the modem reader. Furthermore, Cartan did not use the recently devel
4.8.1
H istory o f R adicals and Exam ples
505
oped idea due to Theodor Molien [1893] o f an algebra A / 1 defined as the quotient of an algebra A modulo an ideal I . However, except for these points the results will look familiar to any mathematics graduate student. He shows ([1898b], pp. 57-59) that every algebra decomposes into a di rect r am o f a nil ideal and a semisimple subalgebra where again the latter term signifies a direct sum o f finitely many simple unital algebras. Fur thermore, all o f the simple summands are full matrix algebras over R, C or H (tre quarternions). (The last part is not expressed quite so explicitly, and the quarternions cause considerable difficulty.) This is the full set of struc ure theorems usually called the “Wedderburn structure theorems” . Carti n showed the power of the concept o f a radical for associative alge bras-) ;,nd also set the theory o f finite-dimensional associative algebras over the r ;al and complex fields on a very firm footing indeed. Jcseph H. Maclagan Wedderburn’s fundamental paper [1907] general ized he structure theorem just given to finite-dimensional algebras over v b it ary fields. (Wedderburn had just completed a definitive study of the finite fields.) The paper is important for many reasons, one o f which is that It seems to have been much more widely read than Cartan’s paper. It is i very modern paper which can be read without any culture shock by a oresent-day mathematician. The methods are much more general and abstr act than those used previously, and he uses Molien’s [1893] idea of quotient algebras very effectively. He shows quite directly that there is a largest nilpotent ideal (this result is o f course special to finite-dimensional algebras) and that the quotient modulo this ideal has no non-zero nilpotent ideals. He uses these properties to define the term semisimple. Wedderburn does not use the term radical, but it was apparently stan dard by the time o f Leonard Eugene Dickson’s book [1927]. O f course the term arose from the fact that the radical either consists of the set o f roots o f zei 3 (if it is thought o f as a nil ideal) or is the ideal root o f the zero ideal (if it Is thought o f as a nilpotent ideal). These results were so impressive that many mathematicians tried to extend their validity. After Emmy Noether had introduced the idea of chain conditions on ideals o f a ring, Emil Artin [1926] proved that a ring satisfying both ihe ascending and descending chain conditions on left ideals satisfies the Wedderburn structure theorems. Later C. Hopkins [1939] showed that the dascending chain condition implies the ascending chain condition so that, >nly the former is needed. A rtin’s result, with this improvement, is frequently called the Artin-Wedderburn theorem. This success further encoi raged mathematicians to try to extend these results to more general rings. For instance, see Kothe [1930]. It gradually became apparent that the definition of the radical for more general rings was the central problem. For rings satisfying the descending chain condition, the largest nil ideal (called the nil radical) is nilpotent and satisfies various other properties
506
4: Ideals, Representations and Radicals
4.8.1
which make it the obvious choice to play the role of a radical. However, for more general rings, including infinite-dimensional algebras, the various desirable properties o f the nil ideal belong to different ideals, each one of which can be considered as a radical in some situations. Reinhold Baer [1943] proposed to call an ideal J o f a ring A a radical ideal if X were nil and A jX had no non-zero nilpotent left ideals. He showed that any ring contained a largest and a smallest radical ideal. (The latter is the Baer lower radical introduced in Definition 4.4.5.) He showed by example that not every intermediate ideal has to be a radical ideal. He also observed (although he attributes this to the referee) that the largest or smallest radical ideal o f A/X is zero when X is, respectively, the largest or smallest radical ideal of A . From his definition it is obvious that X itself is its own largest or smallest radical ideal. Baer’s largest radical ideal had been considered previously by Kothe [1930a] and Hans Fitting [1935]. Sam Perlis [1942] showed that the nil radical o f a finite-dimensional alge bra is the largest quasi-regular ideal o f the algebra. Baer [1943] introduced the term “quasi-regular ideal” and showed that in any ring the sum o f all quasi-regular ideals is a quasi-regular ideal, and that the quotient modulo this ideal has no non-zero quasi-regular ideals. However, Nathan Jacobson [1945a], in a paper the great importance o f which was immediately recog nized, showed that a significant theory could be based on the use of this ideal as a radical. Indeed, except for the role played by modular ideals, Jacobson’s original paper, together with his paper [1945b], contains the complete theory very much as it is known today. Jacobson noted that his radical agrees with the nil-radical in rings satisfying the descending chain condition on left ideals, and agrees with the Gelfand radical for commuta tive Banach algebras. Several o f the results on the Jacobson radical were obtained indepen dently and simultaneously by Max Zorn and Einar Hille, cf. Hille ([1948], pp. 475-6). Hille had access to preliminary versions o f Irving E. Segal’s paper [1947a] in which modular ideals were first defined (with the name regular ideals). He seems to have been the first mathematician to recog nize the important connections between modular ideals and the theory of the Jacobson radical. Most o f these connections are presented in his book (Hille [1948]). However, an exposition of the theory containing all elements o f our present day theory does not seem to have been given until the book of Jacobson [1956]. We have already mentioned the important references relating Baer’s def inition of his lower radical to the prime radical defined by McCoy [1949], These are Jacob Levitzki [1951] and Masayoshi Nagata [1951]. A transfinite construction o f the Baer radical due to Baer [1943] was considerably simplified and generalized in Sulinsky, Anderson and Divinsky [1966]. In ring theory the Baer radical is associated with a successful structure the
History o f R adicals and Exam ples
507
ory due to Alfred W . Goldie [1960] for rings satisfying the ascending chain condition. The strong radical was defined by B. Brown and Neal H. McCoy [1947] and [1948], while strong semisimplicity was independently introduced by Segal [1941], [1947a]. The discovery of these and many other radicals for rings suggested cre ation of a general theory o f radicals. This was carried out in different ways in e. number o f papers which were mentioned preceding Definition 4.7.1. Ths book by Divinsky [1965] summarizes a number o f these theories. How ever, we have found that the theory due to Mary W . Gray [1970] is most suitable to the application which we wish to make throughout this work. Tbs radicals satisfying her definitions are called hereditary radicals by most au: jprs. Some non-hereditary radicals for rings have been widely consid ered. The distinction relates to property (a) of Theorems 4.3.2, 4.4.8, and 4.5.3 which is not enjoyed by non-hereditary radicals. For further discussion see D vinsky [1965], especially page 125. Sxatriples ? collect several interesting examples which illustrate the theory of the chapter. Our first examples show that the Jacobson, Baer, and strong radicals are all distinct. 4 . 8 . 2 E x a m p le We have already remarked that when X is an infinite dimensional Banach space, the algebra B a ( X ) o f approximable operators is a semisimple (even primitive) Banach algebra which is Brown-McCoyradicel This algebra has many other desirable properties illustrating the general fact that Brown-McCoy-radical algebras, unlike Jacobson-radical algebras, are often well behaved from the analysts’ viewpoint. Similarly 3 [ s' ) is semisimple and primitive, but Ba( X) is included in B{ X) mApart from a few examples, Jacobson-radical algebras received little attention until the 1980s. Even now, little seems to be known about noncoftimutative Jacobson-radical algebras. However, the 1981 conference held in Long Beach, California sparked renewed interest. Examples 3.4.15 and £ .4 ,IS deal with radical Banach algebras, but we will have relatively more to say about them, so we will simply list a number o f additional references: Grebiaer [1971], [1973], [1974], [1981], [1983], Jean Esterle [1981a], [1981b], Grg.rJ.aek [1982a], [1983], Bachelis [1983], Bade [1983], Laursen [1983b], Grabiner and Thomas [1985].
V7
?eobson-radical.
-i.3,r T h e D isc A lg e b r a u nder C o n v o lu tio n This example and some var;*7 iis are mentioned in Georgi EvgenyeviC Silov ([1947a], p. 104) and Hilie ([1948], p. 478). Let the underlying Banach space o f A be the same as
508
4: Ideals, Representations and Radicals
4.8.4
the disc algebra (§3.2.13), but let multiplication be defined by convolution A
/ * aW = f
1
/{*■ -
= J / (A (i - s))g(\s)\ds.
o
u
Then any f € A satisfies
l/2(A)| = | / * / ( A ) | <
j
i ||/||2|A|dS = ||/||2|A|
0 and hence by induction satisfies j ir (A )|
<
J\\f\\ | | / ir1| A r-a( ( n - 2 ) ! ) - 1|A|a"-2d»
0 ii/ ir iA i" - 1 (n - 1 )! This shows that each / € A is topologically nilpotent which implies A — A j . On the other hand the Titchmarsh convolution theorem ([1926], p. 286) shows that A has no non-zero divisors of zero so that it has no non-zero nil (or nilpotent) ideals and hence is semiprime. For further details and some generalizations see 3.4.18 above and Grabiner [1974]. 4.8.4 A n o th e r Jacobson-radical Sem iprim e A lg e b r a This example is due to Dixon [1973b]. It is separable and possesses a bounded approxi mate identity (see Chapter 5 for the definition) so that it satisfies Graham R. Allan’s [1972] criterion for possession o f a discontinuous functional calcu lus (Theorem 3.4.9 above). The algebra A is the set o f (almost everywhere equal equivalence classes o f) measurable functions on [0 , oo[ satisfying OO
11/11 = j
e ~ ^ '\ f{t)\ d t< o o
0 with convolution multiplication
f * 9 (t) —
j
t f ( s ) g ( t ~ s)ds.
o Arguments similar to those given above show that each element is topolog ically nilpotent and that there are no non-zero divisors o f zero, so that A is Jacobson-radical and semiprime. The sequence o f functions {xx(o,n-']}n eN
4.8.
History o f R adicals and Exam ples
509
(where x s is the characteristic function o f some subset S o f the real line) form 'in approximate identity for A . A s noted earlier, sets o f nonsynthesis in commutative algebras give rise to radical quotients: Gregory F. Bachelis [1983]. 4.8.E T h e A lg e b r a o f the Free Sem igroup on T w o G enerators Let X be an arbitrary nonempty set. A word formed from X is a finite sequence w = x i x 2 - - - xp o f elements Xj from X where the Xj are not necessarily distinct. The length o f a word w, denoted by n(w ), is the length o f the sequence o f symbols (i.e., n (x ix 2 • • ■x p) = p). We allow the empty word 0 with length n(0) = 0. The unital free semigroup F S ( X ) generated by X is the collection o f all words from X made into a semigroup by the juxtaposition product: ( x i x 2 ■■-x p)(y iy 2 ■■■yq) - x i x 2 ■■■x pyiy2 ■■ yq. Clearly the empty word is an identity element in this semigroup and the length function is additive n(w z) = n(w ) + n (z )
V w, z € F S ( X ) .
The eet X is called the alphabet for F S ( X ) . It is clear that the cardi nal o ' X determines F S ( X ) up to isomorphism of semigroups. Thus we write F S (n ) for F S ( X ) if X has cardinal n G N. For n > 1, F S (n ) is noncommutative. Clearly F S ( X ) is countable if and only if X is finite or countably infinite. For n > 1, F S (n ) includes a subsemigroup isomorphic to any other countable unital free semigroup. (T o see this, one chooses two distinct elements x ,y from the alphabet o f F S (n ) and then considers the subsemigroup generated by a suitable subset of {x y kx : k € N }.) Hence, in a certain sense, all countable unital free semigroups are equally com plicated. For simplicity and definiteness we will concentrate attention on F S {2 ) although the reader can extend most o f the results to other cases without difficulty. For the rest o f this example (except Proposition 2), we use F S ( 2) to de note ?5({a:, j/}). It is sometimes convenient to have a particular sequential order ng for F S { 2). We define { w p } p 6 n to be the sequence
0 , x, y, x 2, xy, yx, y2, x 3, x 2y, xyx, xy 2, yx2, etc. so th.it znzn- 1 ■• •z2z\ = wp is equivalent to p = l + E 2>_1 + E 2
510
4: Ideals, Representations and Radicals
4.8.5
where n = n(w q) is the largest integer satisfying 2” < q. In particular wpwq comes before wr w, in the sequence if the indices satisfy p < r and q < a (unless p — r and q = s). Now let S be any semigroup. Let *4o(S) be the set o f functions f : S —* C which axe zero except at a finite number o f elements o f S and let -4(5) = t l (S ) be the set o f functions /: S —> C satisfying ll/lli = £ l / ( s ) l < oo. a£S Then .4o(5) and A ( S ) are both algebras under the convolution product / * g{ s) —
f ( u ) a( v) Utl=l u, v € 5
Furthermore (-4 (5 ), || • ||j) is a Banach algebra and ^4q(5) is a dense sub algebra so that -4(.S’) can be identified with the completion of the normed algebra (.4o(S), II • Hi). We call ^4q(5) the algebraic semigroup algebra o f S and .4 (5 ) the t 1-semigroup algebra of S. Note that the convolution product is obtained by identifying a function / with the formal sum f ( s) sWe will represent the elements o f > t(5 ) by such formal sums whenever it is convenient to do so. We denote A o (F S (2 )) and A ( F S ( 2 ) ) by A q and A , respectively, in the rest o f this example (except the last Proposition). The following result gives the most interesting property of the Banach algebra A . The author learned this result from W illiam L. Paschke in 1972. P r o p o s itio n 1 . The Banach algebra A has no non-zero divisors o f zero and no non-zero topologically nilpotent elements. Proo f Suppose / and g are non-zero elements of A and p and q are the smallest indices satisfying f (w p) / 0 and g(w q) / 0, respectively Then / * g(wpwq) = f (w v)g(w q) ^ 0 shows that f * g is not zero. Furthermore for any n € N, n m 1/n > i r « ) i l/n = i / K ) i implies p ( f ) > |/(jop)| > 0. Hence / is not topologically nilpotent.
□
Clearly the center o f A is the set o f multiples o f the identity element. Although A is highly noncommutative it does have a large set of algebra homomorphisms onto C. For any word w € F S (2 ), let nx(w ) and ny(w ) be the number o f occurrences o f x and y, respectively, in w. For any (A ,n ) € D x D \ {(0 ,0 )}, the function j x ^ - A —> C defined by OO
OO
7 A „ . ( $ > n« ;n) = 5 > n=l
n=l
nA "‘ ( " ''V * < '‘'“ >
H istory o f R adicals and Exam ples
511
belongs to and it is easy to see that these are the only elements o f r ^ . •I'ruce A. Barnes and John Duncan [1975] show that every element of A which is not in the center has connected spectrum with interior. A stronger result was proved for elements o f A o by Abdullah H. A l-M oajil [1975]. See also'D.uncan and Williamson [1982]. The first reference cited also contains the following result. P r o p o s itio n 2 . Let A be the i 1-semigroup algebra o f the free group on { x n : n € N }. There is an unbounded linear functional on A which is bounded on each commutative subalgebra. P"coJ Let B be the closed subaJgebra generated by all words o f length at least two. By choosing a Hamel basis we may define a linear functional ui tc satisfy w (xn)
=
n
w (B)
=
{ 0 }.
Giee. ly w is unbounded on A. Let C be a commutative subalgebra o f A and choose an element c = 53 7nXn + d of C with 7 „ e C, d € B and c not equal to d. (I f no such element exists there is nothing to prove.) If a — 22 a nx n + b with a n e C and b 6 B commuting with c, then it is easy toisee that J2 anXn and Y 2 ln x n commute so that a n-ym = 7 na m holds for all n, m € N. Hence there is some j3 € C satisfying ctnx n — /3^7„a;nHence the estimate K «)l
,
=
| w ( ^ Q nXn)| = |/?w(]T7nZn)|
=
ll/?X^7nZ„||(|w(^7nZ„)|/||^7nZnll)
=
(Kc)|/^|7„|)||a||
shows that ui is bounded on C.
□
4.8.6 Noncom m utative Sem iprim e b u t Jacobson-radical Banach .A ite b ra s The following examples which are weighted cross products are iaker.-. from Julian M. Cusack [1976]. They seem to have been suggested by Rudi A. Hirschfeld and Stefan Rolewicz [1969] and by weighted semigroup algebras as described on p. 8 of Bonsall and Duncan [1973b], Lat A be a Banach algebra, let 5 be a semigroup and let VT be a real valued weight function on S satisfying: ' 1 ) W (a ) > 0
s€ S
*■■■ (b ) W{ at) < W ( a ) W{ t )
V a, t € S.
512
4: Ideals, Representations and R adicals
4.8.6
We use f 1 (S, A , W ) to denote the Banach space of all functions from S into A which are finite with respect to the norm: ll/ ll = H
l l / ( ^ ) l i ^ ( s ) < oo.
This is a Banach algebra under the convolution product f* 9 (s )= Y ,f(r)g (t)
V f , y * t l (S,A,W y,a€S.
rt = a
Let V' be a semigroup homomorphism of S into the group of isometric automorphisms of A. Define a twisted convolution product on t x{ S, A, W ) by / * v 9(*, = £
/ (f) Kg(t).
rt~ a
Since each Vr is an isometry, the norm is still submultiplicative for this product, and since it is an automorphism the usual calculation establishes the associative law. We denote £*(5, A, W^) with this new product by el ( S , A , W , V ) . For any t € S and a € A , denote the function s >—> 6sta by k(t, a). These elements satisfy k( t , a) * v k(r,b) = k (tr,a V t (b ))
V t, r € 5; a, b € A
and hence their linear span is a dense subalgebra of i 1( S , A , W, V ). P r o p o s itio n With the notation as given, t 1( S , A , W ) and t l (S, A , W, V ) are Banach algebras. I f W satisfies the two additional conditions: (c) W (s t) = W ( t s ) V s , ( € S; (d) lim lV (s " )1/n = 0 VseS; in addition to (a) and (b), then £' (S, A , W, V ) is Jacobson-radical. P roo f It is enough to show that each k( t,a) , t £ S, a € A, belongs to the Jacobson radical. Hence we will show that for any / £ ^ (S , A, W, V ) , t 6 S , and a £ A, the element g = f * v k{t,a) is topologically nilpotent. Denoting the nth convolution power by gn we find fl(s) = $ 3
VseS
rt~a
(where t occurs only in the summation index) and < / " (* )-
,//(*„) *
V s € S; n > 2.
A
H istory o f R adicals and Exam ples
513
From this we obtain iisnn
= s€S
* «es E<• I i E ft(E h/mi ndw*) j - I n= i j = l r j t —Bj
By o< nditions (b) and (c) on W we have W(.s) < W ( t n) r i j = i W ( rj ) which gives:
||5»||
<
||a|rw(«~) 5 3 »€5
<
£ r i t r a( - r „ < = »
fil/ M M r ,) >=1
i M r w i m r
Hence assumption (d) shows that g is topologically nilpotent so that k(t, a) belongs to the Jacobson radical. □ Clearly this algebra is noncommutative if 5 or A is noncommutative or V is nontrivial. Now let F S ( 2) be the free semigroup on two generators described in the last example. For each word s € F S ( 2), define W (s ) by W (s ) = (n (s )!)- 1 when n (s) is the length of s and define V, to be the identity map on C. Then £ '(F S (2 ),C , W, V ) is a noncommutative radical Banach algebra by the r suits we have just proved. Arguments similar to those given in the iast example show that i 1( F S ( 2 ), C, W, V ) has no non-zero divisors o f zero. Hern; it is prime (and thus a fo rtio ri semiprime). A lother class o f examples starts with a unital, prime commutative Ba i l ac': algebra A and a nontrivial isometric automorphism V o f A (e.g., take 4 to be the disc algebra A ( D ) discussed in §3.2.13 and define V by V ( f ) ' X ) = /(£ A) for some fixed f € T \ { 1 } and all A € D ) . Let S be the semigroup N o f natural numbers and define W and V by W(n) = (n !)- 1 and Vn — V n. Consider the Banach algebra i 1(N, A, W, V ). I f / and g are two non-? 3ro elements in this algebra with m and n, respectively, the smallest integ rs in N at which each is non-zero, then f * y g ( n + m ) = f ( n ) V nf ( m ) is non-vero. Hence i 1(N, A, W, V ) is another noncommutative radical Banach algeb -a with no non-zero divisors of zero. T ie last example constructed by this method has non-zero nilpotent elements although it is still a semiprime, Jacobson-radical Banach algebra. Let 5 be N and let A be the algebra C ( [ —1,1]). Define W and V by W { n ) = ( n ! ) - ! and Vnf ( t ) = f ( ( - l ) nt) for all n € N, / € A, and t € [-1,1 ]. In order to show that the algebra ^‘ (N ,^ , W, V ) is semiprime, we will aho'.v that if / * v 9 +v f = 0 holds for all g then / is zero. Suppose / is iitei-, t o and let n be the least integer at which it is non-zero. Using the functions k(s,a) introduced earlier, we get / k(m , 1 ) / ( 2 n 4- m ) =
514
4: Ideals. Representations and Radicals
4.8.7
/ (n )V "+m/(n) = 0. By choosing m to be 1 or 2, we get the contradiction /(n) = 0 since /(n)2 = 0 holds. Hence the algebra is seiniprime. However, we note that A;(l, /) has convolution square zero if / satisfies /(£)/(—£) = 0 for ail t e [ - 1 , 1 ].
4.8.7 A Jacobson Semisimple Banach A lgebra w ith a Dense N il Subalgebra We use the terminology introduced in Example 4.8.5. Let S be the free unital semigroup with generators {x „ : n € N } and let A = >1(5) be its i l -semigroup algebra. Let I be the closed ideal of A generated by the set of words xSlxja • ••x,n in which xp occurs more than p times, where p is defined by p = m ax{*i, t2, .. ., » „ }. Let B be A / I and let Bo be the dense subalgebra of B which is the image of .Ao = .Ao(S) under the natural
map of A onto B. I f a e Ao equals Y2k=i and p is the largest index of any Xj occur ring in any of the words wk, then aqp+1 belongs to J. Hence Bo is a nil algebra. To show that B is semisimple we show that for each non-zero element b € B there is some product bd which is not topologically nilpotent. Let b = $3 ^ 1 a gwq + ^ be an expansion of 6 as a series of words wq in S and assume ajW i ^ J. For simplicity we call such an expression an infinite linear combination of words. Let p be larger than the index of any x3 occurring in tin- Define d to be 2~kxp+k +%■ The element (bd)n will be an infinite linear combination of words of the form Wj l XP+klWj l XP+k2 ' ' ' wjnxp+k„A word of the above form with each j k equal to one will be called special. We wish to show that there is no cancellation between special words and other words of the above general form, and we consider two cases. First, suppose one of the words \vJx contains an x r with r > p. Then the total number of letters x r with r > p occurring in the general form would be at least n + 1 , and since each special word contains exactly n such letters no cancellation could occur. Second, suppose no in the general form contains an x r with r > p. Then equality forces each wJt to be wi. Since the special words are obviously distinct for different choices of p + k x, ... ,p + kn we see that there is no cancellation. Now consider the special word with the k, chosen by ki = max{<7 : pq divides i} -*> that this special word dor* not belong to X. For n ~ rhV + no with 0 < n, < p, k, equals an s satisfying 0 < s < q exactly n, + n ,+ i(p - 1) + nJ+2(p - l)p + •• • + nq(p - l)p ,_ a _ 1 times. Hence the coefficients of the special word with this choice of fcj is 2 ~r |ai|" with r - (p - 1) + •• • + » i ( p - 1)] < (p ~ l ) - 1 n. From this we
H istory o f R adicals and Exam ples
4.3,E
515
obtain the estimate
||(6d)"||1/n > 2 “ 1/(p- 1) |a,| which, proves that bd is not topologically nilpotent. Hence the perfectly arbitrary non-zero element b does not belong to A j so A is semisimple. . 4':.£.C Topologically N ilpoten t Banach Algebras These were intro duced by J. K. Miziolek, T. Muldner and A. Rek [1972] as a class of topo logical algebras. Recently their study was revived by Peter G. Dixon [1991] and continued by Dixon and Vladimir Muller [1992], Dixon and George A. Willis [1992]. We briefly introduce the subject and refer the reader to the - r i- r al papers for more detail. In particular, we omit most statements about actual rates of decrease contained in the original papers. D efinition
A Banach algebra A is said to be topologically nilpotent if
lim sup{||aia2 • •-anil1/" : aj € A i for j = 1 , 2, . . . , n } = 0. Tl—+00
It is called uniformly topologically nil if lim sup{||a"||1/" : a 6 A }
n —»oo
= 0.
The following implications are obvious topologically nilpotent
=>
uniformly topologically nil
=>
Jacobson radical.
(1 )
The next theorem combines results from Dixon [1991] and Dixon and Muller [1992]. Note that (f) is simply the assertion that A is uniformly topologi cally ail with a particular rate of decrease. Th eorem The following are equivalent for a Banach algebra A. (a) A is topologically nilpotent. (b ) Every sequence {a n}n€N £ A\ satisfies lim ||ai
For every closed ideal X of A , T and A j T are topologically nilpotent. (d) For some closed ideal I of A, T and A jT are topologically nilpotent. (e) For every element a e A, there is some finite constant C = C (a) satisfying ||an||I/,n < Cn~3'2 /" for all n 6 N. j (f) There is some finite constant C satisfying (c )
supdla"!!*/" : a e A } < C r T 32"/"
V n e N.
If A is commutative, these are equivalent to A being uniformly topologically nil, but there, are noncommutative Banach algebras fo r which this fails.
516
4: Ideals, Representations and Radicals
4.8.9
The implication (e) => (f) depends on a Baire category argument, but the rest of the proof is a series of careful estimates. Perhaps the most interesting is the proof that in a commutative algebra uniform topologically nil implies topologically nilpotent. The proof depends on writing ai
4.8.9 Theorem Let A be a topologically nilpotent Banach algebra and let T : A —> B (X ) be any continuous representation of A on a non-zero Banach space X . Then TA (X ) / X . In particular, Ai2 7^ A. Proof Assume to the contrary that TA (X ) — X. Then the map T from the projective tensor product A ® X to X defined by ® xj ) —
4.8. 3
H istory o f R adicals and Exam ples
517
Taj(xj) for dj g A, Xj e X and ||aj|| ||xj|| < oo is a continuous surjective linear map between Banach spaces. The open mapping theorem and ' heorem 1.10.8 give a constant K such that any x G X can be written as x - Yl'jLi Taj(xj) for YlJLi lla j l l llz jll ^ ^ I M I - Iterating this construc tion by applying it to each Ta) (xj) instead of x, and applying induction, allov/d us to write for any n € N oo
x =
/
\
n
y i ( n Ta" . i ) j= i \k=i )
oo
i n
\
where n ( n 11°*../i i ) ii^ ii ^ ^ niix iij= i \k= i )
Denote the nth supremum in the definition of topological nilpotence by S (n ) so that lim S (n ) — 0. In the display above we write as Tai jo 2lj ■•■<>«.> which gives the estimate oo
llx ll
<
5 z l l r lllla l'Ja2^ ' “ an'>llllx-/ll i= i
<
^||T||5(nn|a1,,||||a2,J||..-||a„,J||||xi ||
<
r ||( S ( n ) * T ||* ||.
oo
For any non-zero x, this is a contradiction, since S (n ) —>0.
□
4.8.10 Corollary Let A be a commutative Banach algebra. I f its Jacobson radical is both modular and topologically nilpotent, then A is unital if and only i " A 2 = A. In particular, if A has a topologically nilpotent ideal o f finite codimen sion, hen A is unital if and only if A 2 = AProof Any unital algebra A obviously satisfies A 2 = A. Suppose A j is modular. Proposition 4.3.12(d) shows that an idempotent e may be chosen for its relative identity. Consider the algebra direct sum decomposition: A = A e (B A ( 1 - e). If the second ideal is zero, e is a multiplicative identity for A. Otherwise A ( 1 —e) is included in A j so it is topologically nilpotent. The theorem shows (> l(l — e ))2 ^ ,A(1 — e) so we conclude that A? = (A e )2 © (> l(l — e ))2 is a proper subset of A. If T is any ideal of finite codimension, the Wedderbum theorem (The orem 3.1.1) shows that there is an ideal J including J so that A / J is semisrnple and hence unital. Furthermore, J / I is finite-dimensional and radica. so that it is nilpotent. Hence, it is easy to see that J = A j is topolo ;ically nilpotent. Hence we are in the situation already considered.□
4.8.1:* Differences between the Topological and Geometric Cate gories o f Banach Algebras In the geometric category if
518
4: Ideals, Representations and R adicals
4.8.11
is an exact sequence, tp must be an isoinetry and C must carry the quotient norm. The following result is called the short five lemma. Let
0 —
A
1* 0 —► A"
V>1
B
ya.
c
—► 0
lfls
^2. c~
—►0
be a commutative diagram with exact rows. If 0i and 0$ are isomorphisms, then 02 is also. In the topological category of Banach algebras this was proved infor mally just before Theorem 1.1.16. However, it fails in the geometric cat egory of Banach algebras as shown by the following easy example due to Robert C. Busby [1971a]. In the above diagram let A = = C = C~ = C and B = B~ = C x C. Let the vertical maps be the identity maps and
5 approximate Identities and Factorization
Many Banach algebras which occur naturally are not unital. However, of them do contain a sequence or net of elements which behaves like a ■’ implicative identity in the limit. Perhaps the most elementary example Is algebra Co(R) of continuous functions vanishing at infinity on the real iiv .3 . Choose a sequence of functions {/ „}n e N € C o (R ) each with values in the unit interval [0,1] and with /„ having the value 1 on all of [—n, n]. A moment’s thought shows that any function g € Co(R) satisfies lim f„ g = g
n —*oo
where the limit is in the norm ||•||oo of Co(R ). The Fejer kernel, introduced in §?..8.15, provides another interesting example of the same phenomenon in L 1, C, A C or C 1, as defined there. This chapter treats such approximate identities. The first section pri marily shows the relationship between various possible definitions. The main result may be informally summarized as saying that the weakest defi nition implies the strongest. Section 5.1 also provides additional examples. In a unital algebra every element a has the trivial factorization a = la = al. Paul J. Cohen showed in [1959a] that every element in an algebra with a suitable approximate identity can also be factored. Allan M. Sin clair ([1978], [1979] and [1982]) showed the existence of a highly structured analytic semigroup of divisors. Sections 5.2 find 5.3 give versions of this theory. Section 5.3 also considers factorization results which depend only on separability of the algebra rather than the existence of an approximate identity.
' I
5.1
Approximate Identities and Examples
In ‘ his Section we will discuss various definitions of approximate iden tities. We also introduce the strict topology and show that a nonunital Banach algebra A which has an approximate identity is dense in its double centre iizer algebra in the strict topology. As a consequence, the double qe^traiizer algebra is the completion of A in the strict topology. We begin bXjCjcPning an approximate identity. For a more complete discussion of the material in this section, see Robert S.Doran find Josef Wichmann [1979]. 519
520
5: A pproxim ate Identities and Factorization
5.1.1
In the literature, the terminology for approximate identities is some what variable. We will use the strongest common condition as a definition and then show that various, apparently much weaker, conditions imply the existence of a bounded approximate identity in this strong sense. Recall that the weak or ^'-topology on a normed linear space A is the weakest topology in which each element of A ' is continuous. 5.1.1 D efinition Let A be a normed algebra. A ( left / weak left ) approximate identity for A is a net {e Q} ag/1 C A such that, for each a € A, { « o tt}o € 4 converges in the ( norm / weak topology ) to a. A ( right / weak right } approximate identity is defined similarly and a ( two-sided / weak two-sided) approximate identity is a net which is both ( a left and a right / a weak left and a weak right) approximate identity. An approximate identity {ea}af_A Q A (of one of these types) is said to be bounded by M e R+ if the set {jjed| : a e j4} is bounded by M and is said to be bounded if it is bounded by M for some M € R + . Similarly, an approximate identity is said to be countable or commutative if {ea : a € j4} is countable or commutative, respectively. If the net is a sequence, the approximate identity is said to be sequential. A normed algebra is said to be approximately unital if it has a bounded two-sided approximate identity. It is obvious that an approximate identity of any type is a weak ap proximate identity of the same type. Approximate identities which are not bounded are not particularly useful. See Dixon [1992] for some of the pathology in incomplete normed algebras. If any type of approximate iden tity for a non-zero algebra is bounded by Af, then it is clear that M is greater than or equal to 1. We will use this observation without comment in this section. Approximate identities were apparently first considered explicitly by Irving E. Segal [1947a], who proved that any norm-closed self adjoint subal gebra of the algebra of bounded linear operators on a Hilbert space contains an approximate identity. The group algebra L l (G ) for any locally compact group also contains an approximate identity; special cases of this fact were used informally long before abstract approximate identities were defined. See, for instance, Hermann Weyl and F. Peter [1927], John von Neumann [1934], and Andre Weil [1940]. An approximately unital algebra A obviously shares some of the proper ties of a unital algebra. For instance, the regular homomorphism of A into ’D iA ) and both the left and right regular representations are homeomorphic isomorphisms. If A has an approximate identity bounded by M , we see ^ < | | IJ < IM |
V a e i
and similarly for R^. Hence if M = 1, the three maps just mentioned are
5.1.2
A pproxim ate Identities and Exam ples
521
isometries. Also, any element belongs to the closed principal (right / left / two-sided ) ideal it generates. Similarly, we will show that any maximal left ideal in an approximately unital Banach algebra is closed. This is true in i, unital algebra because the ideal would then be modular. 'n a unital algebra, every element a is trivially a product: a = la = al. Fro n the beginning, approximate identities have been used to extend this useful result which is no longer trivial if the algebra is only approximately unital. We will pursue this in the next two sections sifter showing some other properties of approximate identities. We begin by showing how weaker conditions imply the existence of approximate identities. Equivalent Conditions in the next theorem, part (b) is due to Peter G. Dixon who has made a penetrating study of related results and counterexamples [1973a]. The proof given for (b) shows that a Banach algebra with a bounded left approxi mate identity and a right approximate identity has a two-sided approximate identity. Such an algebra need not have a bounded right or two-sided ap proximate identity, but a neat uniform boundedness argument shows that all sequential right approximate identities must be bounded. There is also a Banach algebra with a left and a right approximate identity (even with other desirable properties) but with no two-sided approximate identity. In the same paper, norms of approximate identities are studied. Part (c) was discovered independently by Mieczyslaw Altman [1972] and Joseph Wichmam [1973], and part (f) is essentially due to Teng Sun Liu, Amoud van Roo’j and Ju Kwei Wang [1973], Lemma 12. Barry E. Johnson [1972a], Proposition 1.6, gave a proof for (e) which was actually strong enough to establish (c). Th eorem Let A be a normed algebra. (>.) There is a ( left / right) approximate identity fo r A if and only i f for ever e > 0 and every finite subset { a i , aa,.. ., a „ } of A there is an element e e A satisfying { \\eaj —a^-1| < e / \\aje - aj\\ < e ) fo r j = 1 ,2 ,... ,n. (:>) I f A has both a left and a right bounded approximate identity with bounds M and N , respectively, then it has a two-sided bounded approximate identity with bound M N + M + N . (•-•) The following condition implies that A has a { left / right ) ap prox mate identity bounded by M : There is a number M € R+ such that, fo r every a € A and every e > 0, there is an e e A satisfying ||e|| < M and { ||c;a —a|| < e / ||ae —a|| < e ). (
522
5: A pproxim ate Identities and Factorization
5.1.2
llej || < M , ||e21| < M , l ^ a - a|| < e, and ||ae2 - a|| < e. (e) I f A has a bounded weak ( left / right / two-sided ) approximate identity, then it has a bounded approximate identity o f the same type. (f ) A closed commutative subalgebra C of A includes a bounded left ap proximate identity fo r A if fo r each a € A there is a bounded net {e Q} aex C C (with both the net and the bound possibly depending on a) such that {e aa}a6A converges to a in the weak topology. Similar statements hold fo r right and two-sided approximate identities fo r A included in C. Proof (a): Suppose the “left” hypothesis holds. Let A be the set of finite subsets of A ordered by inclusion. For each a € A, choose ea € A so that ||eQa — a || < n_1 holds for all a € a where n is the number of elements in a. Now it is clear that {e a } QgA is a left approximate identity for A. The converse is obvious. By symmetry, the same construction is possible if we replace “left” by “right” . (b): Let {eQ} ae/( and {fp}pf_B be, respectively, a left and a right bounded approximate identity for A with bounds M and N . Let A be A x B ordered by setting (a ,0 ) > (a ',0 ') if a > a ' and 0 > 0'. Define a net {gs}seA by setting g(Q,p) equal to /g o ea. An elementary calcula tion shows that {gsjseA is a two-sided bounded approximate identity with bound M N + M + N. (c): We have assumed that for any a £ A and any e > 0 there is an e € .A satisfying ||e|| < M and ||(1—e)a|| < e. Let {a i, a 2, .. ., a „ } be a finite set in A and let e be a positive number. Successively choose ex, e2, .. ., e„ in A satisfying Jje^|J < M and
||(1 - e ,)(l - e ^ , ) •' • (1 - e1)aJ|| < e(2 (l + for j = 1,2,... ,n. Define f £ A b y I - f = ( I - e „ ) ( l - en~ i) •• •(1 - ei). Then for any j between 1 and n, we have ll/flj —fljH
=
| | (1 -/ )a j
=
UK1 - e« ) ( l - en -i) • •• (1 - eJ+i)][(l - ej) • •• (1 - e i ) ] a j
<
(1 +
- e i) ■■■ (1 - ex)aj || < e(2 (l + M ) ) ~ l .
Choose e € A satisfying ||e|| < M
and
||e/ - /|| < e(2 max{||a_, || : j = 1 ,2 ,..., n } + I ) -1.
Then for each j between 1 and rt, these inequalities give IN j ~ M
<
M f “j ~ « j ) l l + l l (e / - f ) a , || + ||/a, - a j M\\fa} - a j ||+ ||e/ - /|| ||a.j|| + ||/a j - a j
<
M e(2 (l + M ) ) - 1 + 2~1e + e(2 (l + M ) )~ l = e.
<
A pproxim ate Identities and Exam ples
523
The proof of (a) now establishes (c) since ea can be chosen with norm beviiided above by M . % ): Results (b) and (c) together imply (d). (e): Let {e Q} Qgx be a bounded weak left approximate identity. Since the norm closure of the convex hull C = co{eaa : a € j4} is closed in the weak topology (Dunford and Schwartz [1958], Theorem V.3.13), it contains a. Since C — Ca where C = co{eQ : a £ A } is bounded by the bound for {e Q} 0gx, (e) follows from (c). The other cases are similar. (f): As in the proof of (e), we may assume that for each a there is a bounded net such that ea a converges in norm to a. For each n € N, let A (n ) be the closed set A (n ) = {a € A : there exists a net in C bounded by n which converges to a in norm}. Since A equals U^Li^4(n) aad each .A(n) is norm closed, the Baire category theorem asserts that some includes an open ball. The set {a j — a 2 : a i,a 2 € A ( N ) } includes a neighborhood of zero and is closed under scalar multiplication, so it must equal A. For any ai — a 2 € A with alt a2 € A (N ), let {eaj}aeA satisfy lirr. jitaja j —fljH = 0 for j = 1,2. Then {e Ql oeQ2} aeA C C is a net bounded by N 2 + 2N which satisfies lim ||(eQi o ea2 )a — a|| = 0. Hence, (c) shows that A has a bounded approximate identity and the proof of (c) shows that it m ay be chosen in C. The proof for right approximate identities is similar. Use (b) for two-sided approximate identities. □ If V is a dense set in an algebra A with a ( left / right / two-sided ) approximate identity or bounded approximate identity, then each element in this approximate identity can be approximated by an element in J> giving an approximate identity in T>. Hence any separable approximately unital normed algebra has a countable bounded two sided approximate identity. This is an important result since Sinclair’s theorem (Theorem 5.3.2) shows that much more is true in this case. Dixon [1987] shows that a left, right or two-sided approximate identity in a Banach algebra can be approximated by another of the same type in which all elements have their spectrum as close as desired to the unit interval [0,1] in C. The following result is due to Hans Reiter [1971].
I be a closed ideal of a normed algebra A. (a) If A has { left / right / two-sided ) bounded approximate identity, then A/1 has an approximate identity of the same kind. (b) I f T has a ( left / right / two-sided ) bounded approximate identity, then A has a ( left / right / two-sided ) bounded approximate identity if and only if A / I has a bounded approximate identity of the same kind.
5.1.8 Proposition Let
Proof (a): If { ea }aeA is any kind of approximate identity in A, it is obvious that {"a + Ija e A is an approximate identity of the same type in A/T with ,l :• otient norm.
524
5: A pproxim ate Identities and Factorization
5.1.4
(b): For the converse we will use (c) or (d) of Theorem 5.1.2. We consider the typical case of left approximate identities. Let M be the bound for the approximate identity in J. Let e > 0 and a G A be arbitrary. Using the approximate identity in A/2 with bound N , we can find an e in A and a 6 in 7 satisfying ||e|[ < N and ||ea — a + 6|| < e/(2M + 2). Using the approximate identity in J, we can find / € A satisfying ||/|| < M and ||/6 - 6|| < e/2. This gives Ho - (/°e)a|| < ||(/- l)(e a —a + 6)|| + e/2 < e. Since \\f o e\\ is bounded by M N + M + N , this verifies the hypotheses of Theorem 5.1.2(c). The case of right approximate identities is similar and then Theorem 5.1.2(d) handles the two-sided case. □ Dixon and George A. Willis [1992] prove an interesting result on approx imately unital extensions which is quite unrelated to the last proposition. See Example 4.8.8 for the definition of topological nilpotence. A topologi cally nilpotent ideal is never approximately unital. P rop osition Let A be a commutative Banach algebra satisfying A 2 = AThen A is approximately unital if and only if it has some topologically nilpotent ideal 2 with A/2 approximately unital. It is easy to see that a direct product of finitely many Banach algebras is approximately unital if and only if each factor algebra is. The same result holds for tensor products. Richard J. Loy first established a similar result in [1970b] and the following proposition is due to James R. Holub [1972]. 5.1.4 Prop osition Let A and B be Banach algebras and let A ® aB be their complete algebra tensor product with respect to a reasonable norm o . Then A ® ff B is approximately unital if and only if A and B are approximately unital. Proof The result actually holds for bounded one-sided approximate iden tities. We will give the proof for left identities. First suppose {e Q} Qe/* and { f p j p e B are bounded left approximate iden tities in A and B with bounds M and N , respectively. Let A be A x B with the usual product order. We will show that {ea ® /a}(a„3)eA *s a bounded left approximate identity for A ® a B. For any t = X !j= i ai ® bj in the algebraic tensor product, we get n
t - { e a ® f 0)t
=
n
5 3 (a j
ea a j )
-
3=1
®
bj +
^
aj
®
( fcj
-
]=1
n
—^ ^(flj ” Caaj ) & (bj —f(ib3)> 3= 1
fpbj)
A pproxim ate Identities and E xam ples
525
and hence n
(ea
®
f(j)t)
<
n
£ V ( ( a j - eQa j) ® b3) + ^ < 7(0, ® (6, - /gftj)) i= i >=i n + ] C a ^ aj _ e“ aj ) ®
_
j=1
<
116,11 + £ > J 1 | 6j — fpbj\\ i- l
j= i
n +
_ e° a^
~
i= i This estimate shows that {ea ® /a}(a,/3)gA is an approximate identity bounded by M N in the algebraic tensor product. However, A <8>a B is just the completion of this algebraic tensor product in the norm a, so that we hf,ve a bounded approximate identity on the complete tensor product. Conversely, suppose {ea}aeA is an approximate identity for A B. Let u> be an arbitrary element of A* and consider the function w ® I of the algebraic tensor product A ® B into B. This linear map is continuous in the injective tensor norm and hence in cr so we extend it to the complete tensor product. For any a,c € A, recall the notation cw(a) = u>(ac) and note ( ( cuj ® I )(a ® b))(d) = (u> ® I ) ( ( a ® 6 )(c® d))
V a,c € A ;b,d € B.
Linesrity and continuity allow us to replace a ® b by any tensor t € A ® a B. Choofe w and c so that u j ( c ) = 1 = ||u;|| ||c|| so that we get (cw ® I ) (e a)(d ) = (u ® I ) ( e a(c ® d)) for ali d e B and ea in the approximate identity. This shows that { ( co; ® c,£A is a bounded approximate identity for B. Obviously, A and rigo„ pproximate identities can be handled similarly. □ The
trict Topology and Double Centralizers
\: will now introduce the strict topology. It was first defined by R. Creighton Buck [1958] for the special case of C0(il) C C (ft), where ft is a locally compact space. The general definition given here is due to Robert C. B'-ijby [1968], who also proved Propositions 5.1.6 and 5.1.7. See also Dona! \ Curtis Taylor [1970]. 5.1 D efinition Let A be a normed algebra and let J be an ideal in A. The s rict topology of A defined by I is the topology defined by the family
526
5: Approximate Identities and Factorization
of semi-norms { A
5.1.6
6 G 1 } defined by At (a)
=
||<xi||.
Pb(a)
=
\\ab\\
Va€ A
Thus, a net {a Q} QeA in A converges to a G ^4 in the strict topology defined by J if and only if lim ||6aa - 6a|| = lim ||aQ6 —ai>|| = 0 holds for all b G J. Clearly, the strict topology is locally convex. It is Hausdorff unless there is a non-zero element a € A such that ab — ba = 0 for all b G J. It is also clear that the multiplication of A is continuous in each variable separately in the strict topology. 5.1.6 Prop osition Let A be a Banach algebra with a two-sided approx imate identity. Then all double centralizers are continuous so T>b (A ) is a Banach algebra which equals V (A ) as an algebra. Furthermore, A is dense m V (A ) in the strict topology of T>(A) defined by A. Proof The first result follows from Theorem 1.2.4. Let {e Q} Q6* be a two-sided approximate identity for A I f d G V (A ) and a G A, then dea belongs to A for each a € A. Also both Aa(d —dea) = ||(a
5.1.8
Approxim ate Identities and Exam ples
527
we have aL{b) — lim aLa(b) = lim R a(a)b = R(a)b. Hence, Theorem 1.2.4 shown that (L, R ) is a double centralizer. Clearly, { ( L a, R a) } aeA converges to (L, R ) in the strict topology. Hence, T>(A) is complete in the strict topol ogy defined by A. The final remark follows from the proof of Proposition 5.1.8. □ Approximate Identities and Arena Multiplication Recall the definition of the two Arens multiplications given in Section 1.4. Part of the following result was first noted by Paul Civin and Bertram Y o o i [1961], Lemma 3.8. Also recall that a mixed identity for A** is a right identity for the first Arens product which is also a left identity for the second Arens product. 5.1.8 Prop osition Let A be a Banach algebra. (a) A has a bounded ( left / right / two-sided } approximate identity if and only if A ** has a { left / right / mixed ) identity fo r the ( second / first / both ) Arens multiplication(s). (b) If e e >1** is an identity fo r the { first / second ) Arens multiplica tion. then e is also a ( left / right) identity fo r the ( second / firs t) Arens multiplication, and the canonical image in A ** of any bounded weak ( left / rig h t} approximate identity fo r A converges to e in the weak topology. Froof Theorem 5.1.2 shows that a normed algebra with a bounded ( left / right / two-sided ) weak approximate identity has a bounded approximate identity of the same kind. Suppose {e Q} Qg * is an arbitrary subnet of a bounded weak left approximate identity for A. Alaoglu’s theorem shows that a subnet {/c(e7) } 7gr of {/c(eQ) } QgA converges to some e € A ** in the a pology where k is the natural map. Any a £ A and £ A ' satisfy w 'c) = lim w(e7a) = lim au(e7) = e(au ) = we(a). Therefore, any / e A** and us € A * satisfy e • /(w) = f(u ie) = f{io). Hence, e is a left identity for the ssscnd Arens multiplication. If A** has an identity for the second Arens mv.lt:plication, e must be this identity. In this case, the canonical image of any bounded weak left approximate identity converges to e in the weak topology, since we have shown that some subnet of an arbitrary subnet converges to e. An entirely analogous argument shows that the canonical image in A ** of a bounded weak right approximate identity for A has a subnet converging to some right identity for the first Arens multiplication. Sirr.iiax'.y, if A ** has an identity e for the first Arens product, the net itself converges to e. Conversely, suppose A** has a left identity element e for either Arens multiplication. Since « (> li) is dense in in the ^4*-topology, there is a bounded net {ea}aeA such that {n (ea) } aeA converges to e. All w € A* andYi C A satisfy lim ui(eaa) = lim « ( e Qa)(w) = lim « (e Q) ( 0w) = e(auj) = "1 .;a;L-1— e«(a)(ui) = e • «(a )(w ) = «(a )(w ) = ui(a). Hence {e Q} Q6A is a
528
5: A pproxim ate Identities and Factorization
5.1.9
bounded weak left approximate identity. Similarly, if e is a right identity element for either Arens multiplication, then it is the limit of the canonical image in A " of some bounded weak right approximate identity for A. □ 5.1.9 A pproxim ate Identities in G roup A lgebras We now dis cuss approximate identities in L l (G ) where G is a locally compact group. Certain concrete approximate identities were considered long before the ab stract definition was introduced. In 1.8.15 we showed that the Fejer kernel, for example, is an approximate identity for L l ( T). We will now construct a simple two-sided approximate identity bounded by one in L 1(G). For terminology, see Section 1.9. Let U be the collection of all compact neighborhoods of the identity in G ordered by reverse inclusion. For each U £ U, let ey be X (U )~ 1 times the characteristic function of V . Then each ey belongs to L l (G ) and satisfies ||efy||x = 1. Hence, {eu}uau in a net in L l (G ) bounded by one. Theorem 20.4 in Hewitt and Ross [1963] asserts that, for any e > 0 and any / € lJ (G ), there is a neighborhood V £ U satisfying ||v/ — /||, < £ for all v 6 V. Hence if U £ U satisfies V C V, then Fubini’s theorem gives ‘ \\ev * f ~ / I I ,
=
<
j
Ij
G
U
1u) dv - f ( u) \du
j
J \ vf ( u ) - f ( u ) \ d v d u
G
=
=
U
A( U ) - 1
A(t/)-1
J
J \ vf ( u ) - f ( u ) \ d u d v
U
G
J
\\vf — f \hdv < e.
u For any e > 0 and any non-zero / € L '(G ), the same theorem guarantees the existence of a V € U satisfying both ||/u — /||, < e/2 and |A(t>)-1 —1| < e/(2||/||j + 1) for all € V. Then for any U £14 satisfying U C F , Fubini’s theorem gives 11/* et/ -
/||,
=
J |J G
<
& ( v ) ~ l f ( u v ~ l )dv - f (u) \du
U
I
A( U y 1 J
G
U
W
|A ( v y ' M u ) - f( u)\dvdu
y 1J J w v y ' M u ) - f ( u ) \ d u d v U
G
5.1. .’
A pproxim ate Identities and E xam ples
<
IA H - V v H - Mu)\du
M U ) - 1J V
+J
529
G
l/ u (w ) -
/ (« )| d w ^ ^
G
=
A( U ) - 1J ( | A ( i , r - 1| IIMIj + ||/„ - fW ^dv < e.
v Hence, {eu}ueu is a two-sided approximate identity bounded by one. We have constructed the simplest approximate identity bounded by one for L l (G ) when G is an arbitrary locally compact group G. However, certa:n special approximate identities may prove useful in various circum stances. First, we mention another one which can be defined for any locally compact group G. Let V be the collection of all compact symmetric (i.e., u 6 V «> u~1 € V ) neighborhoods of the identity in G. For each V € V, let f v ba A -1/2 times the characteristic function of V. Then define ev to be ll/vll. f v so that {e v 'lv e v is a net of elements of norm one in £*(G ). Ar guments entirely similar to those given above show that this net is again a two-b ded approximate identity bounded by one. It has the advantage that its elements are invariant under the natural involution on L l (G ), which will be in roduced in Volume II. Sf me groups G have approximate identities in the center of L l (G ). Thesr are called central approximate identities. D efinition Let G be a locally compact group. A subset of G is said to be invariant if it is invariant under all inner automorphisms. The group G is said to have ( an invariant neighborhood / small invariant neighborhoods ) (denoted by ( G e[IN] / G e[SIN] )) if ( it has / included in every neigh borhood of the identity it has ) a compact neighborhood U of the identity which is invariant. Prop osition Denote the class of locally compact abelian groups by [A], the class of compact groups by [K], the class of discrete groups by [D] and the class of locally compact unimodvlar groups by [Um], Then these classes satisfy [A] U [K] U [D] C [SIN] C [IN] C [Um], The algebra L 1(G) has a central approximate identity (which may be chosen to be >/ norm 1) if and only if G e[SIN]. Proof If G is abelian, every neighborhood of the identity is invariant. If G is compact, the remark after Theorem 4.9 in Hewitt and Ross [1963] shows that G belongs to [SIN]. If G is discrete, {e } is a compact invariant neighborhood of e. Thus the first inclusion holds. The second inclusion is
5: Approxim ate Identities and Factorization
530
5.1.9
obvious. Suppose U is a compact invariant neighborhood of e in G. Then A(U ) is finite, non-zero and satisfies \ (U ) =
J
ufd\ =
J
f ud\ = A {u )\ {U )
VueG
where / is the characteristic function of U. This gives the final inclusion. We shall only establish that L ‘ (G ) has a central approximate identity of norm 1 when G €[SIN]. (The opposite implication is proved in Robert D. Mosak [1971].) Let U be the set of all compact invariant neighborhoods of e ordered by reverse inclusion. For each V €14, let eu be A(U )~ l times the characteristic function of U. Since this choice of 14 is cofinal in our earlier choice, {eu ju eu is an approximate identity. To see that it is in the center, let / € L 1(G ) and u € G be arbitrary and consider eu * f ( v )
=
j e u { u v ) f ( v ~ l )dv = J ue u {v )f{v ~ l )dv
=
j
(e u )«(v )f (v ~ 1)dv =
J
A (v )f (v ~ l )eu(vu)dv = f *e u (u ).
□ Many other relations are known between these and other classes of lo cally compact groups. A locally compact group G is said to belong to [FC]~ if each of its conjugacy classes (Cv = {u -1uu : u e G } for v € G) has compact closure, and to belong to [FIA]- if the closure of the group of inner automorphisms of G is compact in the topological group Aut(G) of all homeomorphic isomorphisms of G. (The natural Braconnier topol ogy of Aut(G) is described in Hewitt and Ross [1963], 26.3. It is a slight variation on the compact open topology.) Since the closure of a conjugacy class is included in the continuous image of the closure of the group of inner automorphisms, the inclusion [FIA]~ C [FC]~ is clear. An easy argument, (which we omit) also shows [FIA]- C [SIN]. The following important the orem was proved by Siegfried Grosser and Martin A. Moskowitz in [1967]. T h eorem A locally compact group belongs to [FIA]- if and only if it belongs to both [FC]_ and [SIN]. The Freudenthal-Weil theorem (c.f. e.g., Grosser and Moskowitz [1967], Theorem 4.3) asserts that a connected group lives in [SIN] if and only if it belongs to the class [MAP] introduced in §3.2.17 above, and that such a group is necessarily homeomorphically isomorphic to the direct product of R " and a compact group for some n. (It is immediately obvious that such a direct product belongs to both [MAP] and [SIN].) We mention also the inclusion [FC]- C [IN] and refer the reader to Grosser and Moskowitz [1971a] Palmer [1978] and Volume II of the present work for detailed refer ences, examples and many related results.
5.]. 10
A pproxim ate Identities and Exam ples
531
5.1.10 T h e sem igroup algebra L 1 (R + ) This algebra plays a special role relative to the theory of Section 5.3, so we will also discuss it briefly. If /. Is a Banach algebra with a countable approximate identity bounded by 1 , then there is contractive homomorphism of L *(R + ) into A and the ocncrcte analytic semigroup of L 1 (R + ) is mapped onto the more abstract semigroup of A constructed in Section 5.3. From another vantage ooini,, L 1(R + ) has the same relationship to the classical Laplace transform ■: '!!) has to the Fourier integral transform and L * ( T ) has to Fourier ssr3p. Ivlore specifically, in each of these three cases the Gelfand transform can be identified with the classical transform mentioned. This example is related to Examples 3.4.10, 4.8.3 and 4.8.4. As a Banach space, /y‘ (R + ) is the usual space of equivalence classes of absolutely integrable functions on R + = [0, oo[ with respect to Lebesgue rriepsire. The norm of / € L l ( R+ ) is given by ||/||j = /0°° \f(t)\dt, and the conpcution product o f f ,g € L l (R + ) by
f
/ * 9 (t) =
J ( t - s)g(s)ds
V t€ R +.
Jo
A change of variables shows that L 1 (R + ) is commutative. . Let H represent the open right half plane {A € C : Re(A) > 0}. By identifying the closed right half plane H with the Gelfand space of under the map A *-+ 7 * defined by 7a ( / ) =
r
f ( t ) e - Xidt
V / € L \ R+ )
Jo
v/e see that L 1(R + ) is semisimple but nonunital. We also see that under this idersti'ication the Gelfand transform / of / € L *(R + ) is just the Laplace transform: £ (/ )(A) = / ( 7 a ) , as already claimed. The Laplace transform £ (/ ) is analytic on H as well as continuous on all of H. The proof given in Example 3.4.10 shows that the sequence of functions {en}r»cN (where en(« ) equals n for 3 < 1 /n find equals zero for s > 1 /n) is a sequential approximate identity of norm 1 for L 1(R + ). The Titchmarsh convolution theorem [1926] shows that L l ( R + ) has no non-zero divisors of zero. Johnson [1964a] extended Wendel’s theorem to identify the double centraUzer algebra of / ,1 (R , ) as the convolution algebra M ( R+). As a Banach space this is the linear space of complex regular Borel measures on R + with the total variation norm. The convolution product is given by I ■
<*> = / / Xe { s + t)dfi(s)dv(t)
V Borel sets E.
' Toe basic analytic semigroup of L 1 (R + ) is closely related to the gamma fuhctnin:
532
5: A pproxim ate Identities and Factorization
5.1.11
(which is most famous for extending the factorial: T(n + 1 ) = n!). For each A G H, define e(A) G L *(R + ) by e (A )(0 =
tA" 1e - ‘/r(A)
V
t €
R+.
Sinclair [1982] proves that this is an analytic function which is also a semi group with interesting growth conditions. If A € H converges to 0 in any reasonable way, then e(A) becomes an approximate identity. For related results see Grabiner [1981], Dales [1983] and Dales and McClure [1987].
5.1.11
C*-algebras are Approxim ately Unital C*-algebras were defined in §1.7.17. Proofs of all the following remarks will be found in Volume II. The functional calculus introduced in Example 3.4.17 is the chief tool in the proofs. In a C*-algebra A, the set A h of hermitian elements (i.e., those elements h € A satisfying h’ = h) plays the role of the set of real numbers in the complex numbers. Each hermitian element has its spectrum in R. (The converse is only true in the commutative case.) The set A + of hermitian (or, equivalently, normal) elements with spectrum in R + then plays the role of non-negative real numbers. These elements are actually called positive elements. In particular, the set of positive elements defines a partial order on A h h < k <=> k - h G A + . In a nonunital C*-algebra A, it turns out that A + O A\ is an approximate identity (obviously bounded by 1) in its own order. This approximate identity consisting of positive elements is attractive because of its natural order. In the case of separable C*-algebras, it is easy to choose a sequential ap proximate identity. Johan F. Aarnes and Richard V. Kadison [1969] proved that any separable C*-algebra has a commutative approximate identity. We will use Sinclair’s analytic semigroup construction in Section 5.3 to show that any separable approximately unital Banach algebra also has a com mutative sequential approximate identity.
5.1.12 Approxim ate Identities in B k (X ) Let A* be a Banach space which is not finite-dimensional. Then the ideals B f (X ), B a (X ) and B k (X ) of finite-rank, approximable and compact operators, respectively, are all nonunital. The following results are from Dixon [1986] and Niels Gronbaek and George A. Willis [1993]. In the latter paper, versions of some of the results are stated for any subalgebra of B ^ (X ) which includes B p (X ). See Definition 1.1.17 and §1.7.7 and compare Doran and Wichmann [1979] , §30. We say that a normed algebra A has a ( left / righ t) approximate unit if for every a € A and every £ > 0 there is tin element « satisfying ( \\ue - e\\ < £ / ||e« — e|| < £ ). Theorem 5.1.2(c) shows that if there is a uniform upper
'2
A pproxim ate Identities and E xam ples
533
bound for the norms of the elements u in a left or right approximate unit, the" the algebra has a bounded approximate identity of the same type, but t h l ails if there is no such upper bound. T h eorem Let X be a Banach space. (a) B f ( X ) always has a left and right approximate unit. ( d) There is an example of a Banach space X fo r which neither BA(X ) nor 3k {X ) has a left approximate unit. (; ) ( B f (X ) and BA(X ) have / Bk ( X ) has ) a left approximate identity if X has the { approximation property / compact approximation property ). (d) ( B f ( X ) and BA(X ) have / Bk {X ) has ) a bounded left approximate identity if and only if X has the ( bounded approximation property / bounded compact approximation property ). (e) ( B p (X ) / BA(X ) / B k ( X ) ) has a bounded right approximate iden tity f and only if it has a bounded two-sided approximate identity and this occurs if and only if the identity operator in B ( X ') can be approximated in the Grothendieck topology by the adjoints of operators in some bounded subs ",t. of { Bf (X ) / BA(X ) / Bk ( X ) ). I f X has a shrinking basis, then the associated sequence {P n} of basis projections in B p (X ) is a bounded two-sided approximate identity fo r each of these algebras. Proof (a): The collection of all finite-dimensional projections can serve as the approximate unit. (b): The example is Szankowski’s subspace of i 1 from [1978]. Details are given in Dixon [1986], §3. (;): We use the criterion of Theorem 5.1.2(a). Let K\, K%, . . . , K n be an arbitrary finite set of operators in ( B f ( X ) / BA( X ) / B k (X ) ). Then T j(X i) is compact so under our hypotheses, for any e > 0, we can find an operator E € ( B f ( X ) / BA( X ) / Bk ( X ) ) satisfying \\EKj - Kj\\ < e as we wished to show. (d): The existence of the bounded approximate identity follows from our last argument and Theorem 5.1.2(b) since the operator E can be chosen of bounded norm. The converse is an easy consequence of the proposition in §1.7.10. The existence of a bounded approximate identity with bound B in one of these algebras immediately implies that for any e > 0 and any finite subset { x i , Z2, .. ., x „ } of X there is an operator T in the algebra, satisfying ||T|| < B t,: d [|Txj — XjH < e for j = 1 ,2 ,..., n. (;■): The Banach space adjoint operation provides an injective isometric anti-aomomorphism of B (X ) into B ( X ') which sends BF( X ) into B p (X * ) anr. J^ {X ) into B k (X ')- Hence each of the algebras in (e) has a bounded rigb- approximate identity if and only if the algebra of all its Banach space x\y its in B k ( X ') has a bounded left approximate identity. The argun.f 3 given in the proof of (d) establish the equivalence of the existence of
534
5: A pproxim ate Identities and Factorization
5.2.1
these bounded left approximate identities and the corresponding approx imation properties on X ' . However, these approximation properties on X ' imply the corresponding approximation properties on X and hence the existence of bounded left approximate identities for the original algebras. Now apply Theorem 5.1.2(b). □
5.2
General Factorization Theorems
We use the next simple proposition in both Theorem 5.2.2 and 5.3.2. It has been stated in a very general way to cover tiny possible application. We denote the closed linear span of a subset S in a normed linear space by Span(S ). 5.2.1 P rop osition Let A be a normed algebra with a bounded { left / right / two-sided ) approximate identity {e a } Qej*. Let .. , , T ^ be continuous ( representations / anti-representations / representations and anti-representations) on normed linear spaces X l , X 2, .. . , X n, respectively. Consider {a i,a 2, . -. ,Op} C A and fo r each j = 1,2,... ,n, arbitrary finite sets {x ji, x.,2 , . . . , xiq ]} C spa^\(TAX y). Then fo r any positive number £, there is an ao € A such that ( Jjeaa_, - aj || < e
Vj = 1,2,... ,p /
ea - a j < £
\\eaaj - a j || < e and ||a_j e„ - a , || < e \\Tli)x}k ~ *>*11 < £
Vj = 1,2,... ,p /
Vj = 1,2,... ,p )
and
V j - l,2 ,...,n ; k = 1 ,2 ,...,^
hold fo r a lia > aoProof Suppose the approximate identity is a left one bounded by M . First, we choose a representation T : A —* B (X ) and x € span(T^A') from among all the given data. Since x is in span(T^A'), we can choose bj £ A and x} € X so that ||x—£ ”1, ^i>jx>ll < £(2 + 2A/||T||)-1 . To simplify notation, denote the difference x - Yl'JLi Tb,Xj by z. Choose ao £ A so that a > a0 implies ||ert6, - 6J < £(2||T|| £ "1 ^ ||xj + l ) -1 for j ^ 1,2, ...,m . Then, for q > q0 we have m
l|rea* - * | |
<
||re,>(z)|| + ^ | | r eor 6j* j - r 6,xj || + ||-z|| j= i n
<
Pr„|| N I +
<
(||r||M+l)£(2 + 2M ||r||)-1
», II I M
m
+ 11*11
m
+ £||T||(2||T||^||xJ||+ l ) - 1^ | | x J||= e. 3=
1
J= 1
General Factorization Theorem s
535
If the approximate identity is right or two-sided, the argument for tin antirspreaentation would be similar. The approximation was achieved simply by choosing a large enough ao to approximate a finite list of elements from A sufficiently well. Obviously, givezi a whole list of representations and vectors, we will still only have to approximate a (larger) finite list of elements from A. Since {ea} aeA is a -let, this can be done. □ a normed algebra A has either a left or a right approximate identity, sach element a 6 A is the limit of a net (actually a sequence) of prod ucts of elements in A . Paul J. Cohen [1959a] proved that every element in a 3anach algebra with a bounded approximate identity is actually the ~rod.uct of two elements (rather than just the limit of products) and that these elements can be chosen to satisfy useful additional conditions. We shall prove a version of this result discovered independently by Edwin He witt [1964] and by Philip C. Curtis, Jr. and Alessandro Fig&rTalamanca [19S6]. This version of the theorem is usually stated for Banach modules. Since we usually use the language of representations rather than the equivaler.i language of modules, we use it here also. A number of simpler proofs have been offered for theorems similar to Theorem 5.2.2. However, none of them establishes a sharp bound for ||a||. (See Miecslaw Altman [1971], Vla3timil Ptak [1974] and Sidney L. Gulick, Teng Sun Liu and Amoud van Rooij [1967].) A simpler proof of the full results has been given by Paul J. Xco3:s [1964] when the approximate identity is two-sided and bounded by 1. See also Section 5.3 below, Doran and Wichmann [1979], Ptak [1979a] and Florian A. Potra and Ptak [1984], Chapter 8. If
5.2.2 Th eorem Let A be a Banach algebra with a { left / right) approxi mate identity bounded by M . Let T be a continuous ( representation / antirepresentation ) of A on a Banach space X . Then fo r each y € span(T^A') arA each e > 0, there are elements a € A and z € X which satisfy: • (a) y = Taz; ( b ) ||a|| < M ; (c )
||y - z|| < e;
(d) z e ( T Ay)~. Hencs, the set {T az : a € A, z € X } is precisely the closed linear subspace spim(7AX ) of X . ' -j ’
:t is enough to prove the first caso, since it gives the second when ,:d to the opposite algebra. vv'j make A 1 into a Banach algebra by defining the norm as follows:
||A+ a|| = |A| + |M|
VA+ae/
We 'define a representation of A 1 on X which extends T, and which we will
536
5: A pproxim ate Identities and Factorization
5.2.2
denote by T again, by +a%
Ax
T aX
V A - f f l £ A} ' , X £ X .
Note that the norm ||T’|| of the bounded linear map T : A l —* X is the maximum of 1 and the norm of the original map T \ A —* X . It is convenient to use P to denote the number 2 A f(l -I- 2M ) ~ l . Note P < 1, P = 2A/(1 - P ) and 2M = P(1 - />)“ > = , P ". The idea of the proof is to construct a convergent sequence of invertible elements 6n = P n + an in A 1- Then we set a = lim bn = lim a„ € A and z = lim T ^ y where d„ — b~l . Thus {Tdny } converges even though {lir ^ H } approaches infinity. Let i en be a left approximate identity for A bounded by M . We use £ to denote the set {ea : a 6 A }. We will always denote elements of £ by “ e” , often with some subscript. For any e € £, Proposition 2.2.9(e) shows that P + (1 — P )e is invertible in A 1. We will denote its inverse by c i.e.
c = ( P + (l- P )e )- 1
Veef.
(1)
Proposition 2.2.9(f) also shows c - P ~ l € A and ||c|| < 2 P " 1
V e e £.
(2)
Hence any e € £ and any a € A satisfy: l|ca - a||
<
||c|| ||a - (P + (1 - P)e)a||
<
2 P _1(1 — P)||a — ea||.
Thus, the net {ca } a€/1 C A 1 satisfies the defining properties of a bounded left approximate identity in A except that the elements ca are in A 1instead of in A ■ It is easy to check that the proof of Proposition 5.2.1 remains valid, so that for any finite sets { a j , a3, . . . , an} C A and {jfi , I / a , . . . , ym} Q span(T^A') and any 6 > 0, there is an e 6 £ such that llc a j- a j < 6 \\TcV] - y} \\ < 6
j = l , 2 , . . . , n and j = 1,2,... ,m
hold simultaneously. Let y e span(T’^ A') and e > 0 be arbitrary. These will remain fixed throughout the rest of this proof. We will inductively choose elements f „ € £ so that bn = P n + (2A /)"1 Y X = i P kek satisfies bn e ( A ') g
(4)
and so that the inverse dn of bn satisfies \\Td n y - T dn_ i y \ \ < 2 - n£
( 5)
5.2.1
General Factorization Theorem s
537
for all n > 1, where do = 1. We choose ex so that \\Tc, y - y \ \ < 2 - l e. Then, bx = P + (2 M )~ l P e x = P + (1 - P )e x is invertible with ci = dx as its inverse. Thus, ex satisfies (4) and (5). Now suppose e i,e 2, . . . , e m have been chosen satisfying (4) and (5). We will use (3) to find em+ x 6 S which satisfies (4) and (5). Let 6 and S' be the positive numbers S = (2 m+1||r||(2P-1||j/||+ ||dm||)+l)-15and<5/ = min{||dm||-1,^(||dm||2+(5)-1} so ||dm||26 '(l — S ')~ l < 6 . The inequalities of (3) show that we can find em+i € £ so that the following inequalities hold: IlCm-f l^fc
efc|| ^
\\Tcm+ly-y\ \
<
^
k — 1 ,2 ,..., 771
6.
w
We show first that 6m+i is invertible. We have Cm+lbm+X- b m
=
Cm+X( P m+1 -f (2 M )_ l P m+1em+i) m
m
+ (2 W )- 1J P * < w 1el - P m - (2 M )~ 1' £ , P kek fc=i fc=i r -C m + liP + (1 - P > m+1) - P m
=
(7)
+ (2 M ) 1Y ^ P k(cm+xek - e k) k=l m
=
(2 M )~ l Y P fc(cm+iefc - et) k=l
which gives m
||cm+i6m+, - bmII < (2 M ) - l Y , pk6' ^ 6' < ll^m ir1k=l The induction hypothesis shows that bm is invertible with inverse dm. Henc3, Proposition 2.2.9(b) shows that Cm+Xbm+X is invertible and its in verse satisfies ||(cm+,6m+, ) - 1 - dm|| < H dJIV (1 - 6 ') - 1 < 6.
(8)
Thus em+i satisfies (4). We now check (5). Using (8), (6) and the definition of 6 .ve get \\Td^+ly - Tdmy\\
=
||T(Cm+1fcm+I) - i Cm+1y - Tdmy\\
<
llr [(cm+l6m+1)- '- d micm+iy|! + WTdniTc^ +iV ~ y)|| m f 1)
1 - d n . i l ||CW.+ 1|| M l
+||r|| Hdn.ll \\TCm+1y-y\\ <
||T||(WP-1||l/|| + ||dm||5) = £2-<"*+1).
538
5: A pproxim ate Identities and Factorization
5.2.3
This completes the proof of (5) and hence the inductive construction of the e„ € £. We now construct the elements a € A and z e span(T^A') which satisfy (a), (b), (c), and (d) of the theorem relative to our choice of y € span(T4A') and £ > 0. For m > n, (5) gives W T ^ y -T ^ y i <
m £
\\Tdky - T dk^y\\<
fc = n + l
m £
2 ~k£ < 2 ~n£.
fc = n + l
Hence, is & Cauchy sequence. Denote its limit by z. Then using the case n = 0 in the above inequality, we get \\y — z\\ < £. Since each € £ has norm satisfying ||en|| < M , the series OO
(2 M )- 1^ P " e n n—1
converges. Denote its sum by a. Then a is also given by a = limn_ 00 6n. Since T is continuous, we see y = lim ^oo T>,nTdny = Taz. Thus, a and z satisfy conditions (a) and (c) of the theorem. We also have OO
||a|| < (2M )->
p n llenll < (2 M )" 1P(1 - P ) ~ l M = M, n~ I
so a satisfies (b). Finally, lim TCay = y shows that y belongs to (T Ay )~ . Since each dn belongs to A l ,T ^ y also belong to (TAy)~. Therefore z = limn_ ocTdny belongs to (T Ay )~ ■ This completes the proof of (d). The last sentence follows immediately since we have shown that any y € span(TUA') has the form Taz for some a 6 A and z € X . □ We have proved Theorem 5.2.2 for factoring single elements. However, in some applications it is necessary to deal with the simultaneous factoriza tion of more than one element. The first extensions of this sort were given independently by Nicholas Th. Varopoulos [1964a] and Johnson [1966], both of whom used their extensions to prove important theorems which we will present later. They proved the “algebra” (rather than the “representation” ) form of Corollary 5.2.3(c) below. The form we give was first established by Marc A. Rieffel [1969a], whose proof we use. The “algebra” form of Corollary 5.2.3(b) was established by Ian G. Craw [1969] for Frechet alge bras. Corollary 5.2.3(a) is a slight extension of his result in the Banach algebra case. Similar results have also been obtained by Heron S. Collins and William H. Summers [1969], 5.2.3 C orollary Let A be a Banach algebra with a ( left / right ) ap proximate identity bounded by M . Let T be a continuous { representation / anti-representation ) of A on a Banach space X .
General Factorization Theorem s
539
(a,. I f S is a countable union o f compact subsets of span(T ^ X ), then ther? is an a € A so that S C TaX . ' I f S is a compact subset of span(T^A’) and e is a positive num ber, Lhen there is an element a 6 A and a continuous function z :S —> X satisfying: r °i) y = T« ziv ) v y^S; (ba) ||a|| < M ; (b3) tlv - *(y)|| < £ V yeS; ;b 4) z(y) € (TAy)~ V y e S. (c) I f {j/n}neN is a sequence in spein(TAX ) converging to zero, and £ is a positive number, then there is an element a € A and a sequence {Zn}ngN converging to zero and satisfying: (c i) yn = Tazn V n € N; (ca) ||a||
v a e A; f e y -,s e S.
It is claar that T is bounded with |jr|| = ||!T||. Wo now apply Theorem 5.2.2 to A, y , and T. We choose y 6 y to be the Identity function y(s) = .1 for all s £ <S. We must show y € span(TAy ). For ary £ > 0, choose a finite cover B (s i, e/2), B (s2, e/2),.. ., B(sk,£/2) gi <5, by e/2-balls and a partition of unity g i, g2 .. ., gx subordinate to this cov3:'. (That is, for k — 1 , 2 ,K , gf. S —* [0,1] is continuous, and satisfies gk(s) ~ 0 for s ^ J3(sfc,e/2) and S jtL i 9k = 1- Cf. Rudin [1974].) Then frorp the approximate identity, choose e*. satisfying ||Ttfcs*. — s*|| < e/2 for k = 1 , 2 Thus, fk(s) = gk(s)sk belongs to y and K
\ \ y - ^ T 'j k W *=1
K
=
s u p {| | s -]T 0 k(s )r etsk||} ,65 *=1
<
s u p {y 'p fc(s)(l|s-«fcll + IIs* - T^SfclQ}
K
*e 5
<
fc=i
sup{ ,e S
9k^ e + ||«— * k | | < e / 2
^ ell»- n ll> e / 2
540
5: A p p ro x mate Identities and Factorization
5.2.4
Hence y € span(T^>’)Now let £ > 0 be arbitrary. Then we find a e A and z 6 3^ so that T az — y, ||a|| < M , \\y - z\\ < e and z e (T Ay )~ . We get conclusion (b) by evaluating y and z at each point s 6 S in the expressions T az = y, ||y—2 || < e and « 6 T A (y). (c): We specialize to the case where the compact set <S is {yn : n € N }u {0 }, where {jfn} „ eN is a sequence converging to y0 = 0. We consider the subspace Z of y consisting of all / € y such that /(0) = 0. Clearly, this is a ckeed T-invariant subspace. Let T be the restriction of the representation T to Z. Apply Theorem 5.2.2 to A, Z and T . Again, we choose y 6 Z to be the identity function y(s) = s for each s 6 <S. Let e > 0 be arbitrary. The theorem guarantees a z € y such that Taz = y, ||a|| < M, ||y — z\\ < e and 2 6 (TA y)~. Set z„ equal to z(yn) for each n. Clearly, zn satisfies the desired properties. This proves (c). (a): Suppose S is the union U^LjSn where each <S„ is compact. Since the norm is a continuous function, B „ = sup{||s|| : s € Sn} is finite for each n. Consider the set <S = U ^ , (njBn) -' 1<Sn u {0 } where (n B „)~ 1S„ is just {(nJ3n) -1s : s € <Sn}. Then <S is compact. (To see this, it is enough to check that each sequence in S has a convergent subsequence. If it contains a subnequence moving through the sequence of <Sn, we are done. If it does not, then there is a subsequence included in some (n B n)~ 1Sn. Since this is a compact set, there is a convergent subsequence.) Apply (b) to S. □
5.2.4 C orollary Let A be a Banach algebra with a { left / right ) approxi mate identity bounded by M . //{an} n€N is a sequence in A which converges to zero and e is a positive number, then there is an element e 6 A and a sequence {ftn}ngN A converging to zero satisfying (a) ( On = e6„ / o„ = b„e ) Vn€ N (b) ||e|| < M (c) (la,, - 6n|| < e Vn€ N (d) < bn e (A o n )- / bn e (On.4)- ) V n e N. Proof Apply Corollary 5.2.3(c) to the left or right regular representation. □ We now state Cohen’s original [1959a] factorization theorem together with its converse, which is due to Altman [1975], Note that not all of (b) is needed to establish (a).
5.2.5 C orollary The following two conditions are equivalent fo r a Banach algebra A(a) A has a { left / righ t) approximate identity bounded by M . (b) There is a constant M such that fo r each a £ A and e > 0 there are elements e € A and ( b € (-Aa)~ / b € (a^4)~ ) satisfying (a = eb / a = be ), ||e|| < M and ||a < £.
5.2.e
General Factorization Theorems
541
In particular, when these conditions hold, each element o f A can be written as a product. Proof (a) => (b) Immediate from Theorem 5.2.2 or Corollary 5.2.4. (! ) => (a) Given a 6 ^ and e > 0, choose e,b 6 A satisfying (a = eb / i — b e ), ||e|| < M and ||b—a|| < e/M. Then ||a — ea|| = ||e£>—ea|| < ||e!| |j >—a|| < e. Hence (a) follows by Theorem 5.1.2(c). □ C inverses of various forms of this theorem have been sought by a number kers. Donald C. Taylor [1968], F. Dennis Sentilles and Taylor [1969] and jiu, van Rooij, and Wang [1973] found partial converses. William L. Pasc ke [1973] showed by example that the converse of the final (weaker) state nent in Corollary 5.2.5 is false. More recently, Dixon [1990] gave a four-dimensional algebra A in which every element may be factored since A is ;he direct sum of two algebras, one with a left identity and the other with a right identity, but which obviously lacks an approximate identity. He also noted the existence of (non-separable) commutative semisimple Ba nach algebras with factorization but without even unbounded approximate identities. However, he showed that every separable commutative Banach aige a with ( left / right) factorization of sequences converging to zero has a (possibly unbounded) approximate identity. By ( left / righ t) factoriza tion of sequences converging to zero, we simply mean that for any sequence {« n } .6N Q A converging to zero there is an element e 6 A and a sequence {&n}»eN Q A converging to zero satisfying ( a„ = ebn / a„ = b„e ). Ad ditional results and examples on converses to the factorization theorem, particularly for commutative algebras, are given in Willis [1992b]. Note that if a normed algebra A has a right approximate identity which is ( 1 ounded by 1 / bounded / arbitrary ), then the left regular represen tation of A is ( isometric / homeomorphic / faithful). The factorization theorem gives new information on double centralizers. Thecrem 1.2.4 shows that, if A is a Banach algebra and A la = A r a = {0 } (e.g., if A is a semiprime Banach algebra), then any pair of maps L : A —* A and I: A —* A satisfying c f ■:
aL(b) = R(a)b
a, b 6 A
will ave both L and R linear and continuous. Of course if A has a right appr >ximate identity, then A l a — {0 }. We now establish an improvement of Theorem 1.2.4 which is due to Johnson [1966]. 5.2.C P rop osition Let A be a Banach algebra with a bounded { left / right) approximate identity. Let ( L / R ) : A —>A be a map which satisfies ( L(ab) = L(a)b / R(ab) = aR(b) ) Ther. (L / R ) is linear and continuous. cea>"ilizer.
V a .k A
Thus it is a ( left / right )
542
5: A pproxim ate Identities and Factorization
5.2.7
Proof We prove only the first case since the second follows by considering the reverse algebra. Let ai, a2 6 A and A, n € C be arbitrary. By Corollary 5.2.4, we can find elements e, 6i, 62 6 A such that a\ = ebx and a2 = eb2. Therefore, we get L ( Aid] + A2fl2)
~
L(e(Ai6] + A2&2)) = L(e)(X\bx 4- A262)
=
XiL{e)b\ -(- X2L(e)b% = X iL(eb i) -I- XiLifib^)
— A ]L (a i) + A2L (a 2). Hence L is linear. Thus it is a left centralizer. We will now prove that L is continuous. Suppose {a „ }n£N is a sequence in A which converges to zero. Corollary 5.2.4 shows that there is a sequence {bn}n€N converging to zero and an element e 6 A such that o„ = ebn for all n. Thus, limn_ 00 L(aT>) = lim „_00 L(ebn) = limn_ 00 L(e)bn = 0. Hence L
is continuous.
□
Rieffel proved a module version in [1969a], but we shall not need his result. Maximal One-sided Ideals in Approximately Unital Banach Algebras In a unital Banach algebra every one- or two-sided ideal is modular. Hence, Theorem 2.4.7 shows that every maximal one- or two-sided ideal is closed. Michael D. Green [1976] has shown that the same conclusion holds true in an approximately unital algebra, at least for one-sided ideals. This is an example of ways in which approximately unital algebras resemble unital algebras. 5.2.7 Th eorem Let A be a Banach algebra with a bounded { right / le ft) approximate identity. Then every maximal ( left / righ t) ideal is closed. Proof Consider the case in which A I is a maximal left ideal. Let { 6„}„e N be a sequence in M converging to b € A. We wish to show b € M . Let bo = 0 and consider the sequence { b — which converges to zero. Corollary 5.2.4 gives a sequence { c „ } neNO converging to zero and an element d satisfying b - b„ = c^d for all n € N°. To simplify notation write c = Co and for positive n c n = c - c n , so b„ = Cnd, Cn —* c and b = cd. Consider the left ideal £ = { a € . 4 : a d € A 1 } . Since it contains each c„, it is enough to show that C is closed. We may assume that £ is not equal to A. The maximality of M shows that Ad + M equals A. Hence, we may write d = ed + m for some e € A and m € M . Then any a € A satisfies (a — ae)d — am € A'f. This shows that £ is a modular left ideal. We will now show that £ is also maximal. If AC is a left ideal which properly
C ountable Factorization Theorems
5.3.1
543
inciades C, then the maximality of M implies A = Kd + M . As before, we may choose e € K. satisfying e = ed + m for some m 6 M . Hence, tiny € A satisfies a = ae + (a —ae) € K + C = K. Since any strictly larger left ideal equals A , we conclude that M is a maximal modular left ideal and thsrc-fore closed by Theorem 2.4.7. □
• •'
5.3
Countable Factorization Theorems
1: this section, we prove an important theorem, due to Sinclair, about 3a:iach algebras with countable approximate identities. This theorem holds in ery separable approximately unital algebra. We also prove some weaker factorization-like theorems which hold in any separable Banach algebra, v/hether or not it has an approximate identity. The main theorem (Theorem 5.3.2) is a simplified version of the main theorem in Sinclair’s book [1982] which was based on his papers [1978] &nc' ]1979]. By omitting a number of deeper properties, we are able to ~ "i i simpler induction argument as noted by Sinclair [1982], Both the proparties we prove and those we omit are useful, but Corollary 5.3.4 below is certainly the most important consequence of this theorem. For interesting applications of analytic semigroups in L l (G ) for a locally compact group see Jose E. Gale [1991]. 5.8.1 D efinition Let A be a Banach algebra. Let H be the open right half plane {A 6 C : Re(A) > 0} and, for each a satisfying 0 < a < 7t/2, let S (a ) be the sector S (a ) = {A : |arg(A)| < a; 0 < |A| < 1} C H. A function e: H —* A is called an analytic semigroup in A if it is an analytic function satisfying e(A + n) = e(X)e(n)
V A, /j € H.
An analytic semigroup is said to be bounded if for each a satisfying 0 < q < vr/2, the set {||e(A)|| : A € S (a )} is bounded. Note that if e: H —» A is an analytic semigroup, then {e (l/ n )}n€N is a sequential commutative approximate identity. If the semigroup is bounded, then this approximate identity is bounded by the bound for any sector 5 (a ). 5.3.2 Th eorem Let A be a Banach algebra with a countable two-sided bounded approximate identity. Let ( T: A —* B (X ) / S :A —+ B (y ) ) be a continuous ( representation / anti-representation ). Then fo r any x € and y € (5 ^ (y ))_, there are: ; bounded analytic semigroup e: H —» A,
544
5: A pproxim ate Identities and Factorization
an entire function x :C —* X ivith x(0) = x, and an entire function y: C —* y nnth y(0) = y satisfying: (a) Te{\)(x(n + A)) = x (n ) V A € H;/i € C StO>)(y(p + A)) = y(n) VAeH^eC. (b) (e (A M )-= M e (A ))-= > t V A 6 H. (c) z(A ) € (T a (x))~ and y(A) € (S A (y )y V A€ (d) For each a satisfying 0 < a < 7t/2 and a £ A,
5.3.3
C.
a = lim e(X)a = lim ae(A) A—0
A—0
when A is restricted to the sector S (a ). (e) // the original approximate identity has bound 1, then all positive real t satisfy J|e(i)|| < 1. Proof We will inductively choose a sequence {/ n}neN from an approximate identity for A with bound M . For each A € H, we will then construct an exponential element e„(A) £ .41 by n e„(A ) = e x p ^ A ^ / j - 1)) (1) j =i and define e(A), x(X) and y (A) by e(A) = lim e„(A ), n-+
oo
x(X) = lim Ten(_ A)(x) and y (A) = lim Ten(_ A)(y). n —»oo
'
n —»oo
v
'
The convergence of these limits depends on the careful choice of /„ so that for each n the “factor” exp(A(/n —1)) makes the elements which it multiplies very much smaller. (When the algebra is commutative, this really is a factor in the expression for en, but in the noncommutative case we must estimate how far it is from being ail actual factor.) The limit of the sequence of exponentials is in A rather than A 1 since each en(A) obviously has the form e„(A ) = e~nX + c„ where c„ is in A. The group properties of the exponential function yield the semigroup properties of e: H —» A. Although the calculations involved are not really difficult, the notation is sufficiently complicated that we will state a purely computational lemma. 5.3.3 Lem m a Let A be a Banach algebra and let T : A —» B ( X ) be a continuous representation or anti-representation. Let n + a £ .A1, M £ R + , / € A m and x € X be arbitrary. To simplify notation, write N = M + Hall + 1 and P = ||(/- l)a|| + ||a(/- 1)||. (a) All k £ N satisfy \\(a + f - l ) k ~ a k - ( f - l ) k\ \ < N kP. (b) All k 6 N satisfy I
I
- r a*(x)|| < N k\\T\\ (ll^llP + \\T,{x) - *11).
5 .? .'
C ountable Factorization Theorems
545
(.:) All A € C satisfy ||eA(o+/-l) _ eXa _ eA(/—1) + JJI < (eN|A| _ y p (<1) For any positive B e R, there is a constant C, depending on ||a||, B and M , such that any A € B Cj satisfies ||eA(/j+a+/-l) __ eA(/i+o)|| < eR*(A(i)^e|A|(M+l) _ J + C P ). 1 ) For any positive B e R, there is a constant D depending on ||a||, \n\, 3 and M such that any A 6 BC\ satisfies J^exp(A(/i+a+/-l)) (* ) - Texp(A(/1+o))(*)|| < D(\\x\\P + \\Tf x - z||).
Proof (a): Expanding (a + (/ - l ) ) fc without using the commutative law gives
<
(tJajl + M + l ) kP = N hP.
( j ) : Since N is greater than both 1 and \\f — 1||, (a) gives ||T(a+/_ 1)k(x) - r ak(x)||
<
||T(/_ 1)k(a;)|| + ||T|| \\x\\NkP
<
liriliV *-1! ! ^ ^ ) - a;|| + Ill’ ll \\x\\NkP
<
N fc||r||(||x||P+||T/(x )-x | | ).
): Expanding the three exponentials and using (a) gives ||eA(a+/-l) _ eAa
_
CA(/-1)+ 1 ||
t i
<
k'
j r ^ N kP = ( e N M - 1 ) P .
(d): By using (c) at the first inequality, we get
jj„A(/i+a+/-l) _ eA(/i+a)||
_
eE*(A/i)||eA(a+/-l) _ eAa||
<
eRe(A/i)^||eA(-^—’ ) - 1|| + (eN|A| - 1 )P ).
546
5: Approximate Identities and Factorization
5.3.3
Hence, we may choose C — eNB — 1. (e): Expand the exponentials in series to get ll^ e * p { A ( / * + a + / — l ) ) ( x )
—
^ e x p ( A ( / i + a ) ) ( :c) l l
^
00 a/* eB|Ml £ n k r - n r ii (ii»iip + i i w
<
D(\\x\\P + \\Tf ( z) -x\\) .
where D = e ^ + I ^ H T ll.
- *n)
□
We now resume the proof of the main theorem. Note that in (c), (d), and (e) of tbe lemma, the right hand side of the inequalities have factors of the form P = ||(/ —l)a|| -I- ||a(/ - 1)|| or ||T/(x) —x|| which can simultaneously be made as small as we please by suitable choice of / from an approximate identity. This is the key to the construction. Let {gn '■ n € N } be a complete list of the elements in a countable bounded approximate identity. (We are not claiming that the approximate identity is sequential, so the order of this list is in no way related to the structure of the approximate identity as a net.) We will now inductively choose the sequence { / „ } neN from this or tinother countable approximate identity bounded by M to satisfy the following five conditions for all A 6 n C i:
- r en(A)(xj||
<
2 n
(3 )
<
2 n
(4 )
Up* e „ - , ( A
e
e -(n-lJite(A) ^JA|(M+ i) _ }
IS <
<
CO 1 A
c
:~n
||en_,(A ) - en(A)||
2-n-"1' 1(2)
<
2“ ”
k< n
(5 )
) - gk e „ ( A ) || <
2~”
V k < n.
(6)
Consider the case n = 1. Define t’o(A) to be 1 € A 1. We use (d) and (e) in the lemma with fj, a, and hence P all equal to zero. Then (d) shows that (2) holds no matter how we choose f x. Lemma 5.3.3(e) (with the indicated choices) and Proposition 5.2.1 applied to the representations T, 5, L and R (where the latter two are the regular representations) show that we can choose f i from the approximate identity to simultaneously achieve (3), (4), (5) and (6). Now suppose we have already chosen / i, /21 • ■•, /n- We will use (d) and (e) of Lemma 5.3.3 again, but this time with fj. = —n and a = f i + h + ■•• + /„. Then, as before, (d) gives (2) and we apply Proposition 5.2.1 to T,
C ountable Factorization Theorem s
547
S, L and R to achieve (3), (4), (5) and (6) simultaneously with (2). This completes the construction of the sequence of exponential functions en. '^equality (2) shows that for any A 6 H e(A) = l+ _ lim j V e f c M - e f c - j f A ) ) km 1 V
(7)
ccrvcrges. For |A| < 1, it also gives the estimate c(A)|| < 1 +
el*|(M+i) _ i g-rrp + I _ I _— e-E*(A)
2-1
'
For A in the sector S (a ), this estimate becomes eW(M+i) _ i He(A)ll -
2 +
j _ e _ C08(a )|A|
has the limit 2 + (M + 1 )/ cos(a) as |A| approaches 0. Hence, e(A) is bounded in S (a ) for each 0 < Q < tt/2. A s noted earlier, e(A) is in A not just A 1 and the semigroup property follows from that of the exponential function. Moreover, our proof shows that convergence is uniform on tiny sector o f the form {A : 0 < |A| < n;arg(A) < a } for any n € N and 0 < q < 7t/2. Since every element of H is included in the interior of a sector of this type, e is analytic on H since it is the limit of the sequence of analytic functions e „ . (a): Equations ( 3) and (4) show that we may define which
x(A) = lim Te (_ A)( i ) n~>O G
v
and
y (A) = lim Seri(_ A)(y)
'
n —» o o
since ihese limits exist. Then property (a) of the theorem follows directly from the group property of the exponentials A en(A). Similarly, inequalities (5 ) and (6) show that for all m € N the limits lim e „ (- A )gm n —» o o
and
lim
g m en ( -
A)
n —» o o
oxis'.. Let us temporarily call them hm( A) and km( A), respectively. These functions satisfy the analogue of condition (a) of the theorem. Hence for each A € H and each m e N, we have grn € e(A)^4 and gm € ^4e(A). Let a € A be arbitrary. In the proper order of the approximate identity we have a = lim gma = lim a gm, establishing (b). (c): Proposition 5.2.1 shows that ( x / y ) belongs to { ( T A ( x ) ) ~ / (Ss.{y))~) since by hypothesis it belongs to { ( T A ( X ) Y / ( S J\ ( y ) Y ) . Hence, Tfn‘; \( x) e ( T a ( x ) ) ~ and SCn(\)(y) € (S a (v )T are both obvious and imply (c). '
548
5: Approxim ate Identities and Factorization
5.3.4
(d): Let a € A, 0 < a < n/2 and e > 0 be arbitrary. Define B to be sup{||e(A)|| : A € S(oc)}. Property (b) allows us to choose an element c in A satisfying ||a — e(l)c|| < e/(2 + 2B ). By the continuity of the function e on H, any A e S(a) sufficiently close to 0 satisfies ||e(A)a —a||
<
||(e(A) - l)(a - e(l)c)|| + ||e(A + l)c - e(l)c||
<
(||e(A)|| + l)e/(2 + 2fl) + e / 2 < e .
Since e > 0 was arbitrary, this proves (d). (e): If the bound M of the approximate identity from which we chose the /„ is 1, then for positive real t we get n
n
IM O II < e - B,l|exp« £ / * ) ! ! < e- " ‘ e x p (^ | | M | ) < e - ntent = 1. *=i *=i
O
The following corollary summarizes the most important consequence of this theorem. See Sinclair [1982] for other consequences and additional properties which the analytic semigroup possesses if it is constructed by a still more careful induction. 5.3.4 C orolla ry Any separable approximately unital Banach algebra or any Banach algebra with a countable bounded two-sided approximate iden tity has a commutative sequential two-sided approximate identity bounded by 1 in an equivalent algebra norm. I f the original approximate identity was bounded by 1, no renorming is necessary. Proof We have already noted that if a normed algebra is separable, then tiny bounded twosided approximate identity can be approximated by elements ill the countable dense set, thus giving a countable approximate identity. Now suppose A satisfies the hypotheses of Theorem 5.3.2. Let e: H —» A be the resulting analytic semigroup, and let S be the restriction of this-, semigroup to the sector S (a ) for some 0 < a < n/2. Apply Proposition 1.1.9 to get an equivalent algebra norm in which S is in the unit ball. Then n€N is a commutative sequential approximate identity bounded by 1. If the original countable approximate identity was bounded by 1, result (e) of the theorem shows that no renorming is needed. □ Dixon [1973a] shows that even in a commutative algebra with a bounded (uncountable) approximate identity of bound M there may be no approxi mate identity of norm less than M in any equivalent norm. He also exhibits tin approximately unital Banach algebra which lacks any commutative ap proximate identity and an incomplete normed algebra with a sequential two-sided approximate identity but no commutative (even one-sided) ap proximate identity.
5.3,5
Countable Factorization Theorems
549
Factorization in General Separable Banach Algebras We now wish to discuss some results of J. P. Reus ChriBtensen [1976] and Richard J. Loy [1976] which give (among other consequences) a weak factorization theorem for representations of separable Banach algebras on separable Banach spaces. The proofs of the next two theorems depend on some simple results from the theory of analytic spaces. A topological space is called an analytic space if it is Hausdorff and is the continuous image of a separable complete metrizable space. The collection of analytic spaces is close i under forming continuous images, closed subsets, countable products and countable unions. Hence, the linear span of an analytic subset Q of a topo Kgical linear space is analytic since it equals U~=1{ E ; = i V j : ^ 6 Q}. We need two fundamental results about analytic spaces. Pettis lemma: If B is an analytic second category subset of a topological iinea- space, then B — B includes a neighborhood of zero. Gpen mapping theorem: A continuous linear surjection of tin analytic space onto a Banach space is an open map. For proofs of these results, see either Christensen [1974], the appendix to Treves [1967] or Dales [1993]. The next result is due to Loy [1976]. We will give some less abstract consequences after we prove the theorem. 5.3.5 Th eorem Let X i, X2, .. . , X n and y be Banach spaces with all but the lo. it separable. Let T: X i x X2 x • • •x Xn —» y be a continuous multilinear map. Let Z be a closed subspace of y which is included in the linear span of the range of T . Then there exists an integer m such that each z 6 Z can be written in the form m
z= ^
T ( x\ j , x 2j , .. ., xnj )
for sv table Xij € Xt. Furthermore, the expression ||z||T defined by | m
m
j
T ( x ij, x 2j , ■■■. x n j ); x i:j € A ; I
in f I ^ | | z i j | | ||i2>H---t|iCnj|| : * =
i= i
(i= i
J
gives i complete norm on Z equivalent to its original norm as a closed subspace of y. Pnxif The continuity of T shows that there is a constant M ' satisfying ||*|| < M'\\z \\t
VzeZ.
5.3.6
5: A pproxim ate Identities and Factorization
550
The product space X\ x Afc x ••• x X„ is analytic, and its injection into the algebraic tensor product is continuous when the latter carries its projective norm
IMI
=
inf \
11*0II lla;2j||---||a:nill :
I*-' x —
m
1
x nj ; m e N ; X j j e Xi > .
i=i
J
Hence, the range { x t ® x2 ® ® x n ■. x, € X i} of this map is tin an alytic set. Therefore, its linear span is analytic. Now T can be uniquely extended to a continuous map T :(3 '^ 1Xl ■—> ^V. Since Z is closed in y . T ~ { Z ) is analytic and hence the open mapping theorem shows the existence of a constant M " satisfying ll2llr <
m"\\z\\
V z 6 z.
Let M = max {A/', 1 + A/"}. Now for each positive integer p, define a set p
By
=
{ z € Z : z = Y T ( x u , x 2}„ . . , x nj); 3=
1
||x2j||... ||x„j|| < M||z||}.
XijZXi-, j= i
P
n
Since Bp is the continuous image of a closed set in r i m
X i), it is analytic.
} = i i=i
However, Z equals U“ ,S p so the Baire category theorem shows that some Bq is second category. Hence, the Pettis lemma shows that Bq —Bq contains a neighborhood of zero in Z . However, B2q includes Bq - Bq and (like Bq - B q) is closed under scalar multiples so we conclude Z — B2q. Hence, we have proved the theorem with m = 2q. Cl Next we interpret this as a factorization theorem. 5.3.6 C orollary Let T be a continuous representation or anti-representation of a separable Banach algebra A on a separable Banach space X , and let Z be a closed subspace of span{Tax : a € A; x € X ) . Then a positive integer m and a real number M exist such that each z € Z can be written in the form m z=
xj ■>=>
Countable Factorization Theorem s
551
fo r aj € A and Xj € X satisfying m
£ | M
\\xj\\ < M\\ z \\.
3= 1 Proof Apply the theorem with X x = A, X2 = X = y , Z = Z and T ( a ,x ) = Tax. □ The following remarks will be useful for our next application. Let A be an algebra. Recall that n
A n = span{
aj :a j € A } j= l
is an ideal in A for each n = 2,3,... and that these ideals satisfy • •• c A n c ■■• c A 3 c A 2 c A. We will be interested in the case in which some .4" has finite codimension in A- In this case, A 2 has finite codimension in A. Assume that A 2 has codimension p in A, and choose ej, e2, ..., ep € A so that each a e A may be written in the form a = Ylk=i aka'k + ]Cv=i A; eji with q 6 N, Aj € C and eik,a'k e A. Expand each a* and each ak in the above expression into a similar expression. By collecting terms, we can write m
a= E k= 1
P
^kb'kb't. + ^ i~ 1
P
^
p
Aijeiej + ^ 2 Aj ej
j —1
> —1
with Aj, Aij € C and 6*, 6^., bk € A- Proceeding in a similar fashion, we see that for any n, A n has codimension at most p + p2 -1------\-pn~ 1. Similarly, if A 2 has at most countable codimension, then A n has at most countable ccciirr.ension for n — 3,4,... also. Hence A n has at most countable codi mension for one integer n > 2 if and only if A n has at most countable codimension for all integers n > 2 . Now we specialize the above considerations to separable Banach alge bras and give a simple but fundamental result which combines work of Christensen [1976] and Loy [1976]. 5.3.7 Th eorem Let A be a separable Banach algebra in which A p has at mosi countable codimension fo r some integer p > 2. Then fo r each integer n > 2, A n is closed and has finite codimension in A. Furthermore, for each integer n > 2, there is a positive integer m = m( n) and a real number M - M ( n ) such that each element of A n can be written in the form m
n
E n t= 1
m
n
Oij with Oij € A satisfying ^ j = 1i= l
llayll < M||a||.
552
5: A pproxim ate Identities and Factorization
5.3.8
Proof By the remarks preceding this theorem, A ” has at most count able codimension for each integer n > 2. The subspace A n in its relative topology is analytic since it is the countable union of the analytic spaces *p = { £ Z = i i i ; = , a0k '■djk £ A }. Let X be an algebraic complementary space for A n in A. Then X with its relative topology is analytic, since it is at most of countable dimension. Hence, the map 6 : A n x X —* A de fined by B(a, x ) = a + x is a continuous linear bijection from the analytic space A n x X with its product topology onto a Banach space. The open mapping theorem shows that 6 is a homeomorphism. Hence, if a sequence {bfcHeN Q. -4n converges to a € A satisfying a = b + x with b 6 A n and x £ X , then x is zero and the sequence converges to b. Hence A n, and similarly X , are closed in A. However, the Baire category theorem shows that a closed subspace of at most countable dimension of a Banach space is finite-dimensional. Hence, each A n has finite codimension and is closed in A. To obtain the remaining results in this theorem, we apply Theorem 5.3.5 with X, = A , y = 2 = A n and T ( a i , a 2 , ■■■,an) = a\U2 ■■•a„. □ In Volume II we will actually need a slight extension of the above theo rem which we will now state. Its proof is entirely similar and will therefore be omitted. 5.3.8 C orolla ry Let A be a separable Banach algebra with center A z I f A z has countable codimension m A , then A z A n is closed and has finite codimension in A fo r all positive integers n.
6 Automatic Continuity
Int
iuction
m is chapter discusses various linear maps between Banach algebras whiclji preserve some aspect of the multiplicative structure. It investigates when' algebraic restrictions make these maps automatically continuous and when they can be described more precisely or explicitly than assumed a priori. The first two sections deal with automatic continuity of homo morphisms. Jordan homomorphisms between associative algebras can be thought of as algebra homomorphisms which simply ignore noncommuta tivity. Explicitly, a Jordan derivation is a linear map <£>:A —> B between algebras which satisfies ip(ab + ba) — ip(a)ip(b) + tp(b)
V a, b e A.
They are studied in Section 6.3. A derivation is a linear map 6 defined on an algebra A which satisfies the derivation equation 6(ab) = 6(a)b + aS(b)
Va,beA.
Sectit n 6.4 is devoted to the study of derivations and shows their several in timate connections with homomorphisms. Finally, Section 6.5 deals briefly with Jordan derivations. Banach algebra theory is a hybrid between algebra and analysis. In the most favorable situations, analytic proofs can be used to establish that certain desirable topological or geometric conditions follow from purely al gebraic hypotheses. A Banach algebra A is said to have a unique Banach algek: a topology if any two complete algebra norms on A are equivalent, so that t he norm topology given by a Banach algebra norm is unique. An even more stringent condition is that A have a unique normed algebra topology in the sense that any two (not necessarily complete) algebra norms must be equivalent. Finite-dimensional Banach algebras have this property trivially. Meier Eidelheit [1940a] showed that for any Banach space X , B ( X ) has a unique Banach algebra topology (c /. §1.7.15). Following Barry E. Johnson [1967c], we will show in Corollary 6.2.8 that B ( X ) has a unique normed algebra topology if A” is linearly homeomorphic to A”® A\ Israel Moiseevic Gelfand [1941a] showed that any commutative semisimple Banach algebra has a unique Banach algebra topology (our Corollary 3.1.11). 553
554
6: A utom atic Continuity
6.0
Charles E. Rickart made a penetrating study of the uniqueness of norm question [1950] and his results were completed by Johnson [1967a]. The resulting Johnson-Rickart theorem (Theorem 6.1.1) asserts that any semi simple Banach algebra has a unique Banach algebra topology. This theorem includes the two earlier results mentioned above as well as many other par tial results that were obtained before 1967. It is a cornerstone of general Banach algebra theory and will be used repeatedly in the sequel. We have already derived this result as Corollary 2.3.10 of the fundamental theorem of spectral norms, but because of its importance, we will also derive it, as Johnson did originally, from his theorem on irreducible representations (Theorem 4.2.15). This uniqueness of norm theorem shows that if a semi simple algebra A has any complete algebra norm, then the Banach algebra topology can be described purely in terms of the algebraic structure of AAn actual description would be most interesting. Not all Banach algebras have a unique Banach algebra topology. A simple example can be obtained by the following rather silly construction. Let ( X, ||■Hj) and (y , ||• ||2) be two Banach spaces which have the same algebraic dimension but are not linearly homeomorphic (e.g., £l and t 2). Use Hamel bases for X and y to define a linear isomorphism of X onto Make X into a Banach algebra by defining all products to be zero. Then ||■||, and ||v?( )||2 are two inequivalent complete algebra norms on X . A much more interesting example, which is due to the combined work of Chester Feldman [1951] and William G. Bade and Phillip C. Curtis, Jr. [1960b] is given in Example 8.1.5. This example is commutative and has a one-dimensional nilpotent Jacobson radical. Theorem 6.4.23 gives other algebras with inequivalent norms due to Richard J. Loy [1974a]. None of these counterexample algebras are semiprime. (O f course the JohnsonRickart theorem shows that they are not semisimple.) Thus a number of authors have asked whether all semiprime Banach algebras have a unique Banach algebra topology. The answer is still unknown, but Julian M. Cu sack [1976], [1977] has shown that the existence of a semiprime Banach algebra which does not have a unique Banach algebra topology implies the existence of other weird mathematical structures. (See Example 6.4.24 for a discussion of these results.) Let A be a Banach algebra under two norms ||• ||j and ||• ||2. If the iden tity map from (A , ||■||j) to (A , ||• ||2) is continuous, then the open mapping theorem shows that the two norms are equivalent. Hence if a Banach alge bra fails to have a unique Banach algebra topology, then it is both the range and the domain of a discontinuous surjective isomorphism between Banach algebras. Conversely, if a Banach algebra is either the range or domain of a discontinuous surjective isomorphism between Banach algebras, then it can obviously be given two non-equivalent complete algebra norms (its own and the one induced by the isomorphism). Hence the question of whether a
6.1.0
A utom atic Continuity o f Homomorphisms into A
555
Banach algebra A has a unique Banach algebra topology can be generalized by poking about the continuity of homomorphisms from A into, or onto, anoLier Banach algebra or from another Banach algebra into, or onto, A. Positive results of this type are said to deal with the automatic continuity of the maps involved. Theorem 6.1.5 collects a large number of similar re lationships between various uniqueness of norm and automatic continuity conditions and shows their logical relationships in a diagram. This shows tb?A , the fundamental Johnson-Rickart result is implied by many conditions which have also been studied and shown to hold under various hypotheses. Following the statement of Theorem 6.1.5, we gather references to some of these results.
6.1
Automatic Continuity of Homomorphisms into
A
The next two sections contain several major results on automatic con tinuity of a homomorphism tp:A —►B between Banach algebras. In the present section we consider some general techniques and the results putting restrictions on the target algebra B. Theorem 6.1.3 states that any surjective homomorphism of a Banach algebra onto a semisimple Banach alge bra is continuous. Corollary 6.1.14, which is due to Rickart [1950], states that any homomorphism of a Banach algebra onto a dense subalgebra of a strongly semisimple Banach algebra is continuous. Corollary 6.1.15, due to Georgi Evgenyevic Silov [1947a], asserts that any homomorphism of a Banach algebra into a commutative semisimple Banach algebra is continu ous. Theorem 6.1.18, taken from Nicholas P. Jewell and Allan M. Sinclair [1973], is too complicated to state here, but shows that all surjective homorrcrphisms of a Banach algebra onto certain radical Banach algebras are contiruous. Example 6.1.19 gives algebras satisfying the hypotheses of this theorem and related results. A simple example shows that at least density of the range is necessary in the first, second, and fourth results mentioned above. Let A be any infinite-dimensional Banach space made into a Banach algebra by defining all products to be zero. Let u be a discontinuous linear functional on A. Then the map if: A —* M 2 defined by
¥>(«)= (o
" (oo))
V a g .4
is a discontinuous homomorphism into the finite-dimensional strongly semis'rnple algebra M 2. Note that this construction depends on nothing except iae.iort that the matrix unit e.\%2 (with a 1 in the first row and second col umn) satisfies (ei,2)2 = 0. Hence there is a discontinuous homomorphism : "t"> ' :y algebra with a non-zero nilpotent element. Recall that §3.4.20 conuiris a discussion of the existence of point derivations. For certain in
556
6: A utom atic Continuity
6.1.0
teresting and nontrivial commutative Banach algebras, discontinuous point derivations are easily constructed. We show there how this gives rise to dis continuous homomorphisms into any unital normed algebra with a non-zero nilpotent element. It is unknown whether the hypothesis in Theorem 6.1,3 that the tar get algebra be semisimple can be relaxed to substitute semiprime for semi simple. In fact we do not even know whether prime could replace semisimple in the hypotheses. Recall that Theorem 4.4.6 shows that a commutative algebra is semiprime if and only if it has no non-zero nilpotent elements. We have already raised the question of whether semiprime Banach alge bras have a unique Banach algebra topology. See Cusack [1977] for further discussion of these questions. That paper shows that if any of the above questions have negative answers there is a topologically simple radical Ba nach algebra. No such Banach algebra is known. There axe a number of known results on uniqueness of norm and on automatic continuity of homomorphisms which we do not present in this section. Some relate to types of algebras which we have not yet studied. (Theorem 8.7.15 gives such a result.) Among results that assert that a ho momorphism A —* B between Banach algebras is continuous or “almost continuous” if A satisfies certain conditions, we will only give a few of the easiest. Also results on *-homomorphisms will be discussed in Volume II. However, many other important results will not be given anywhere in the work. We feel this omission is justified because H. Garth Dales, a preemi nent expert in the subject of automatic continuity, is currently completing a monograph [1994] on the subject. In his seminal paper [1950], Rickart introduced the fundamental notion of the separating ideal which is expounded in this section. He proved that strongly semisimple Banach algebras have a unique Banach algebra topol ogy and conjectured that any semisimple Banach algebra has this property. He showed how to reduce this question to the question of whether all primi tive Banach algebras have a unique norm topology. (It was this special case which Johnson [1967a] settled.) Rickart [1950} also showed that a primitive Banach algebra with minimal ideals has a unique Banach algebra topol ogy. He remarked that the last two results had been noted independently by Irving Kaplansky [unpublished]. Finally Rickart noted the continuityof-homomorphism results which corresponded to his uniqueness of norm results. In the interim between Rickart’s conjecture [1950] and Johnson’s proof [1967a] that every semisimple Banach algebra has a unique Banach algebra topology, a number of partial results were obtained which have now been subsumed under the general theorem. We will not bother to refer to these results. For some related results, see the excellent short book by Sinclair [1976].
6.1.1
A utom atic Continuity of H om om orphism s into A
557
Uniqueness of Norm \7e begin this section with the most important automatic continuity result—Johnson’s uniqueness of norm theorem. The next three results, originally due to the combined work of Rickart [1950] and Johnson [1967a],
worii already stated as Corollaries 2.3.9 ami 2.3.10 where they were derived froin the fundamental theorem of spectral semi-norms. We shall now de rive |;hem, as Johnson did originally, from his result on irreducible normed representations (Theorem 4.2.15).
6.1.,
Theorem A semisimple Banach algebra has a unique Banach al-
gebri. topology. Proof This simply means that any two Banach algebra norms tire equivalent so that the topologies they induce are equal. It is an immediate consequence of the next theorem. □
6.1.2 Theorem Let (^4, ||• ||,) and (.4, ||• ||2) be Banach algebras. Then the quotient norms induced by ||• ||x and ||• ||2 on A / A j are equivalent. Proof Again this follows from the next theorem. First set A equal to (.4, ||• ||j) and B equal to A / A j with the quotient norm defined by ||• ||2. Then reverse the roles. □
6.1.3 Theorem Let A and B be Banach algebras with B semisimple. Any surjective homomorphism t p : A —* B is continuous. Proof Suppose t p : A - + B satisfies the hypotheses but is not continuous. Then it is not closed so that there is a sequence {a n}„gN of elements in A whicl converges to zero but for which the sequence {v?(an)}neN converges to a ron-zero element b € B. Since B is semisimple and b is not zero, there is some irreducible representation T of B satisfying 7), ^ 0. Theorem 4.2.8 shows that X T may be given a norm relative to which T is continuous. Since tp is surjective, T o tp is an irreducible normed representative of A so that it is continuous by Theorem 4.2.15. Thus the continuity of T shows that the sequence {T v,(an) } ngN converges to the non-zero operator Tb, but the continuity of T o tp shows that Tv(an) = (T o
6.1.4
Theorem Let A be a spectral normed algebra, and let B be an algebra. Let tp: A —* B be a surjective homomorphism. Then tp maps the closure of the kernel of tp into the Jacobson radical of B. In particular, if B is semisimple, then the kernel of tp is closed.
558
6: A utom atic Continuity
6.1.5
Proof Let £ be a maximal modular left ideal of B. The preimage of £ is a maximal modular left ideal of A and hence is closed in A by Theorem 2.4.7. Thus, ip maps the closure of the kernel of ip into C. Since C was arbitrary, ip maps the closure of its kernel into the Jacobson radical of B. □ Each of the following conditions has been studied and shown to hold for some significant class of Banach algebras. Recall that a norm ||| - ||| dominates a norm ||• || if there is some finite constant M satisfying ||a|| < Af|||a||| for all a 6 A and that two norms on the same linear space are consistent if the identity map between them has a closed graph. Conditions (d) and (e) are stated in the geometrical category of normed algebras while all the rest are stated in the (more fundamental) topological category. We give most conditions a brief name. 6.1.5 Th eorem The arrows in the diagram below show the implications known to hold among the following conditions on a Banach algebra (A , ||-||). (a) A has a unique algebra norm topology: The || ■||-norm topology is the only algebra norm topology on A (i.e., any (not necessarily complete) algebra norm on A is equivalent to the norm || • || and hence is actually complete). <=> Any injective homomorphism o f A into a normed algebra is a homeomorphism onto its image (which is therefore closed). <=>• Any isomorphism from a normed algebra B onto A is a homeomorphism (so, in fact, B is a Banach algebra). (b ) The norm topology of A is minimum: The ||• ||-norm topology is the smallest algebra norm topology (i.e., any (not necessarily complete) algebra norm ||| • |j| dominates the norm ||• ||j.
6.1.5
A utom atic Continuity o f Homomorphisms into A
559
algebra norm |||■||| which is dominated bp ||•j| is actually equivalent to ||•||/ <=> Any continuous injective homomorphism of A into a normed algebra :,z zci'ially a homeomorphism onto its image (which is therefore closed). (g) A is permanently spectral: Any (not necessarily complete) algebra norm is spectral. •*=> The range of any injective homomorphism into a Banach algebra is a spectral subalgebra. (h ) Any algebra semi-norm on A is continuous. <=>■ Any homomor phism of A into a normed algebra is continuous. (i) Any homomorphism from a Banach algebra into A is continuous. (;) Any homomorphism from a Banach algebra onto a dense subalgebra of J. is continuous. (k ) Any surjective homomorphism from a Banach algebra onto .4 is continuous. (1) Any surjective homomorphism from a Banach algebra onto A has a closed kernel and thus defines an isomorphism from the quotient Banach algebra onto A. (m ) Any algebra norm on A is continuous. 4=s> Any injective homo morphism of A into a normed algebra is continuous. (n ) The norm ||• || dominates every Banach algebra norm on A . <=> Every surjective homomorphism from A onto a Banach algebra is continu ous. Any two algebra norms on A are consistent with each other. (p ) Any algebra norm is consistent with the complete norm | |• ||. If A is also semisimple, then (c ) and (k ) hold, (g ) implies (o ) and hence ( d ) A ( g ) is also equivalent to ( f ) A ( p ) , ( f ) A ( o ) , ( d )A (p ) and (d )A (o ). ((a )A (h )) ^ ( ( b ) A ( h ) ) ^ ( ( f ) A ( h ) )
560
6: Autom atic Continuity
6.1.5
Before giving the easy proof, we will gather references to examples satis fying these conditions. Johnson [1967c] and [1969b] showed that condition (h) holds for any unital C*-algebra without proper closed cofinite ideals, for B ( X ) and for B ( X) /B k ( X ) if the Banach space X has a continued bi section, and for B k ( X ) if in addition B k ( X ) has an approximate identity (c/. Theorems 6.2.3 and 6.2.6 and Corollary 6.2.7). Theorem 3.4.18 (due to Bonsall [1954a]) and §1.7.15 (due to Yood [1958]) show, respectively, that any C*-algebra satisfies (d) and both B ( X ) and B k {X ) (for any Banach space X ) satisfy (b). Hence C*-algebras without proper closed cofinite ide als, B ( X ) (for any Banach space X with a continued bisection) and B k ( X ) (for any Banach space X with a continued bisection and the compact ap proximation property (§5.1.11)) satisfy (a)A(h). Meyer [1992c] shows that the same is true for the generalized Calkin algebra B( X) /B k ( X ) for some classical sequence Banach spaces X. Sandra B. Cleveland [1963] used the main boundedness lemma of Bade and Curtis to show that any C*-aJgebra actually satisfies condition (b). Dales [1989] and Angel Rodriguez-Palacios [1990] give distinct but closely related proofs which use the above rela tively elementary property of C*-algebras and Corollary 6.1.12 to obtain Cleveland’s theorem much more easily. We give a version of this argument in Theorem 6.1.16. Cleveland’s result extends Kaplansky’s [1949a] discov ery that commutative C*-algebras satisfy the stronger condition that any algebra norm actually majorizes rather than just dominates the C*-norm (our Theorem 2.4.15). In fact, Kaplansky’s result is used in proving (d) for arbitrary C*-algebras. Albrecht and Dales [1983] studied condition (n) and showed that (if one assumes the continuum hypothesis), it holds for a wide class of C*-algebras including the commutative ones for which (h) fails. Conditions (o) and (p) and results about them tire due to Bohdan J. Tomiuk and Bertram Yood [1989] and Michael J. Meyer [1992a], All those Banach algebras which we know to satisfy condition (c) also satisfy (k). These include all semisimple Banach algebras (Johnson [1967a] and Theorem 6.1.1 above) and various radical algebras of power series. Condition (e) was proved for approximate B # -norms (which need not be complete or spectral themselves a priori) by Smiley [1955] following work of Bonsall [1954a], See Proposition 2.3.16. Since the complete norm on any C*-algebra and on any subalgebra of B ( X ) which includes B p ( X ) (for any Banach space X ) satisfies the approximate B^-condition and condition (b), it also satisfies (d)A(g). Condition (g) was proved for completely regular semisimple commu tative Banach algebras by Rickart [1953] (our Theorem 3.5.17) and for semisimple modular annihilator normed algebras by Yood [1958]. Theo rem 8.4.12 removes the condition of semisimplicity. Condition (g) holds for radical Banach algebras trivially as noted in Corollary 4.3.9.
6 1.5
A utom atic Continuity o f Homomorphisms into A
561
Corollary 8.1.15 shows that semisimple commutative Banach algebras have property (i), as first noted by Silov [1947a]. Corollary 6.1.14 is Rickart’s result that strongly semisimple Banach algebras satisfy (j). In [1967b] Barnes showed that semisimple modular annihilator Banach algebras also satisfy (j). In Theorem 8.7.15 we obtain the same result for the wider class of senisimple Duncan modular annihilator algebras. Theorem 6.1.4 has just given the elementary proof that semisimple algebras satisfy condition (1), brt this follows from (k) which they also satisfy. Theorem 6.1.18 gives the result of Jewell and Sinclair [1976] that a wide class of (often Jacobsonradical) Banach algebras satisfy (k). Example 6.1.19 gives specific cases. Prooji First we establish the equivalence of the various alternate descriptions of th^’given conditions. (a !: Let
6.1.6
6: A utom atic Continuity
562
ip is » homeamorphism, so its dense range is closed, and thus equals B/Bv . Since |- 1majorizes the norm on B/Bv , it is a spectral norm. (d) => (f): This implication was first noted by Rodriguez-Palacios [1990]. Suppose (A , || ■ ||) satisfies (d) and ||| • ||| is an algebra norm satisfying IIMII < A/]Ja|| for all a 6 A- Then the closed unit ball A\ of A relative to ||| • HI is a semigroup which is bounded relative to || • ||. Proposition 1.1.9 gives an algebra norm a equivalent to || • || in which every element of A\ has norm at most 1. Thus
6.1.6 Proposition I f J is a closed ideal of a Banach algebra A and there is a discontinuous homomorphism from A/T, then there is a discontinuous homomorphism from AProof Since the quotient map 9: A —* A/2 is open, tp-.A/T —* B is contin uous if and only if o 9 is continuous. □ Separating Spaces Having restated, rederived, and investigated the relationship among these most fundamental results, we begin a more systematic study of au tomatic continuity. We introduce a notion (due to Rickart [1950]) which will be our major device in this study. The normed linear spaces of the definition will usually be Banach algebras in our applications. However, we will also use the case in which they axe Banach modules, i.e., Banach spaces together with a fixed continuous representation of a Banach algebra, and perhaps, a commuting continuous anti-representation also.
6.1.7 Definition Let X and y be normed linear spaces and let ip: X —* y be a linear map. Define av : y —* R+ by ° A v ) = H rffN I + Hi/- v W II - x e x }
Vpey.
S.’ .S Tost*
Automatic Continuity of Homomorphisms into A is called the separating function oftp. The sets X*
=
{a; € X : av (tp(x)) = 0}
yv
=
{ i / e ) ' : <M y) = ° }
563 ,
.
and
ejre called the separating spaces of tp in X and y, respectively. IT ;here is no possibility of confusion, we write
—
{v € y '■ there is a sequence {a ;„}neN C X satisfying lim x n — 0 and lim ^ ( i „ ) = y} n —»o o
x*
n —*oo
(1)
=
6.1.8 P rop osition Let X and y be normed linear spaces, and let tp: X —* y be a linear map. (a) Then a is a semi-norm on y satisfying: o{v) < llj/ll
and a(
V x e X; y e y .
Hence X v and y v are closed linear subspaces of X and y, respectively. Also a is a norm if and only if y v = {0 }. (b) I f S € B( X) and T e B( y) are bounded linear maps satisfying *p o S = T o 1p, then a satisfies cr o T( y) < max{||S||, ||T||}
Vy6y
which implies T( yv) C y v and S( XV) C X * . (c ) I f W i s another normed linear space and i p : W —* X is a bounded linear map, then the separating spaces satisfy y v C yv. (d ) I f X = A and y = B are normed algebras and tp is a homomorphism or anti-homomorphism, then they satisfy cr(
Va £ ^ | be B
a ( M « ) ) < *(&) max{llalli M<*)l|}
V a e A\ b e B.
Thus, A 9 is a closed ideal of A and Bv is a closed ideal of ip(A )~. I f ip(A) is dense in B, Bv is a closed ideal in B. Proof (a): The statements about the separating function tire easily verified and imply the statements about the separating spaces. (b): Any y e y satisfies ooT(y)
<
inf{||5i|| + ||y - v?(Sx)|| : x e X}
<
inf{||S|| ||x|| -I- ||T|| ||y — v?(a:)|j : x e X }
<
max{||5||,||T||}cr(y).
6.1.9
6: A utom atic Continuity
564
This immediately implies a o ip o S(x) < max{||S||, ||T||}<x o
lim ipoip(w„) = y. Then lim rji(wn) = 0 shows that y belongs to y *.
n —*00
n — »oo
(d): The two inequalities are special cases of the inequality in (b) when S and T tire chosen from the regular representation operators L a, L ^ a ), Ra and R^{a) ■ The results about the separating spaces follow easily. □ The following properties show the usefulness of the separating spaces. 6.1.9 Prop osition Let X and y be ( normed linear spaces / Banach spaces ), and let ip: X —* y be a linear map. (a) Then,
IIV'OV’II <
M\\ip\\.
Proof (a): Display (1), after the definition, shows that
6.1.10
A utom atic Continuity o f Homomorphisms into A
9 o ip o ip has a closed graph. This in turn implies 9(Z^,ov) = {0 }. conclude Z ^0
565
We □
We now give other technical results which will be used later. The first is due to Barnes [1967b]. 6.1.'.0 P rop osition Let A and B be Banach algebras and l et < p : A- * B be a homomorphism. Every element of B^ has connected spectrum containing zero. In particular B,p contains no non-zero idempotent. P ro tf Suppose there is some a € Bv such that S C Sp(a) is a nonempty oper and closed subset of Sp(a) which does not contain zero. Then we can choose an open set V C C satisfying dist(V, 0) = 6 > 0 and VC\Sp(a) — VO Sp(a) = S. Hence Lemma 3.4.6 asserts that there is a neighborhood V of a such that each 6 6 2? satisfies S < p(b). However, since a belongs to Bv there is a sequence {a n} neN £ A satisfying lim an = 0 and lim
p b W { o)
~b)
VaeA:
b e B^.
Under the hypotheses of the theorem, V ( ^ ) = v ( A ) n B v CBtN.
Proof The corollary follows by taking 6 =
566
6: A utom atic Continuity
6.1.13
\np(a*}+ip(a —o »). If we estimate the spectral radius of p„(A) in the norm of B, we get PBi PnW) < |A| ||v>(a„)|| + ||v>(a - a„)||. Using the norm in A, we get PBiPrxW) < *u(A a„ + (a - On)) < (|A| + 1) ||a„|| + ||a||. Substituting these estimates into Lemma 2.3.5 with arbitrary positive R gives Pb M * ) ) 2 < ( ( * + l)||fln|| + ||o||) ( « - , ||v (a„)|| + ||vp(a-an)||). When we first let n and then R approach oo we conclude Pb M * ) ) 2 < INI ||^(a) - fc|| < ( p a ( 0 was arbitrary, the proof is complete.
□
It is unknown whether Corollary 6.1.12 can be improved to state that C Bts which would imply C B j by Corollary 4.3.9. In fact this improvement is logically equivalent to the conjecture that every homomor phism with dense range of a Banach algebra into a semisimple Banach algebra is continuous. This conjecture combines the good features of The orem 6.1.3 and Corollary 6.1.14. To see the equivalence, first suppose that Bv C BtN always holds. Then Bv would be a topologically nilpotent ideal and hence included in B j = {0 } whenever
6.1.13 Proposition Let A and B be spectral normed algebras and let t p : A —* B be a homomorphism or anti-homomorphism with dense range. (a) p{c) < < t ( c ) fo r all c in the center of B. (b) BV C B M . (c) I f B is strongly semisimple, then
8.1.14
A utom atic Continuity o f Hom om orphism s into A
567
.7J'.i-i is the composition of
6.1.14 Corollary Let A and B be Banach algebras with B strongly semisimple. Then, any homomorphism t p : A - + B with dense range is continu ous. Proof Proposition 6.1.13(c) and the closed graph theorem prove this.
□
6.1.15 Corollary Let A be a Banach algebra and let B be a commuta tive 3emisimple Banach algebra. continuous.
Then any homomorphism
Proof We have already proved this result as Theorem 3.1.11, but we will give a different proof here. Any closed subalgebra of a commutative semi simple Banach algebra is strongly semisimple by Theorem 4.5.8. Thus we apply Corollary 6.1.14 to ip: A —* p (A )~ . d The next result was first established by Cleveland [1963] with a much nors difficult proof. Proofs similar to the present elementary proof were discovered by Dales [1989] and Rodriguez-Palacios [1990]. • ^ I-
3 ' .23 Th eorem The algebra norm topology on a C*-algebra (A , ||• ||) is ":rhirvj.m, that is, for any (possibly incomplete) algebra norm ||| • ||| on A there is a finite constant M satisfying ||a|| < A/|||a|11
VaeA
Proof Let B be the completion of (.4, ||| - HI) and let tp:A —» B be the embedding homomorphism which is injective and maps onto a dense subeigsbra. Note that y?(*4) is a spectral subalgebra of B by Theorem 3.4.18 ,arvd Proposition 2.5.15. Corollary 6.1.12 shows that A f is an ideal of A ir.cludad in A tN ■ Hence A v is included in the Jacobson radical of A which
568
6: A utom atic Continuity
6.1.17
is just {0 } by Proposition 2.3.18. Hence the continuous homomorphism &-.A —* B/Bf of Proposition 6.1.9(b) is actually injective. The algebra norm defined by a >-* |||<^(a)||| satisfies |||<^(a)||| < |||i/>||| ||a|| for all a € A and hence is equivalent to ||• || by Theorem 3.4.18. However, this norm on A is certainly less than or equal to ||| • |||, so the conclusion follows. □ We conclude this section by proving a rather technical criterion, due to Jewell and Sinclair [1976], for a Banach algebra to have a unique norm topology. Example 6.1.19 will show that this result applies to several in teresting Banach algebras including the Banach algebra L 1(0,1) which was shown to have a discontinuous functional calculus in Example 3.4.10. The next lemma is a slight improvement of a lemma of Sinclair [1975]. Actually the original form of this lemma would suffice in the proof of Theorem 6.1.18, but we will use the present form of Lemma 6.1.17 again in Theorem 6.3.17 to prove the continuity of all derivations of an algebra satisfying the hy potheses of Theorem 6.1.18. In the reference already cited, Lemma 6.1.17 is also used to give additional proofs of Theorem 4.2.15 and Corollary 6.4.16. If X and y are normed linear spaces, recall that C { X , y ) and B (X , y ) denote the linear space of all linear maps of X into y and the normed linear space of all continuous linear maps of X into respectively. 6.1.17 Lem m a Let X and y be Banach spaces. Let {S „ }n € N C B ( X ) and { r „ } n€N C B(y) and (p e C( X, y) satisfy Tnip - tpSn £ B(X,y) fo r all n 6 N. Then there is an integer N satisfying (T i T3 ■■■Tny v r
= (T i T2 ■■■T Ny v ) -
Vn>JV.
Proof First, we note that the identity TnTn+i ■■■Tm
-
-
+ 1 " 'Tk-l(Tlc
proves that the operator on the left is continuous for each m. Proposition 6.1.8(b) shows Tm+ i C for all m 6 N. Hence
(T,r2••■Tm+ly j -
C
(T XT3 ■■■Tmy v )~
for all m € N. Suppose this inclusion is strict for infinitely many m. By our first remark, we may group the T j’s and S} ’s into finite products cor responding to the intervals of constancy of (T\T2 ■■■Tmy v )~ and hence assume that the inclusion is strict for all m 6 N. By scaling, we may also assume ||Sm|| < 1 for all m € N. For each m € N, let Q,n. y —►>^/(TiT2 ■■-T my v )~ denote the natural map. Proposition 6.1.9(c) shows that Q mT\T2 ■■-T mip is continuous but QmT\T2 ■■■ is discontinuous for each rn e N.
6.1. "3
Automatic Continuity of Homomorphisms into A
569
Inductively, choose a sequence {z „ }„ e N satisfying ||a;n|| < 2_n and
WQn'7 Ti •■-Tn_il/)In||—\\Qn{TiT2 ■• Tn-i
y. S1 S2 ■• • Sfc-iJfcll *=1
+ (IIQ n (T 1T 2 • • - TnV? - ^ 5 , 5 , • • - 5 n )|| + ||<5„T1T2 • ■■TnvW)
for al" n 6 N. Let z denote S\S% •• ■Sfc-ix*. Since each tail of this series has norm at most 1, each n € N satisfies H ^ l
>
||Qn¥>*ll
>
\\QnvSiS2 • • • Sn- i x n II - IIQ n
n —1
Jfc=l
| j
-
||<3nV?5i52---5„ f \Jb=n+l
5n+i5 „+2••-Sic-iXk J ||> n. /
V
This contradiction proves the lemma.
□
6.1.18 Th eorem Let B be a Banach algebra satisfying: (a) B has no non-zero finite-dimensional nilpotent ideals; (b) For each infinite-dimensional closed ideal 1 of B there is a sequence {bn}neN ^ B such that the sequence {(6162 •••6nZ)~ } ngN of closed right ideals is constantly decreasing. Then any surjective homomorphism of a Banach algebra onto B is con tinuous, so B has a unique norm topology. Proof The second portion of the conclusion follows from the first by The orem 11.5. Let A be a Banach algebra and let tp\A —* B be a surjective homomorphism. Proposition 6.1.8(d) shows that B^ is a closed ideal. It is enough to prove Bv — {0 } by Proposition 6.1.9(a). Su >pose B>p is infinite-dimensional. Choose a corresponding sequence { 6n} „ n as in condition (b). Choose { a „ } ngN C A satisfying tp(an) = b„ for ea:h n € N. We obtain a contradiction by applying Lemma 6.1.17 to X = A, y = B, tp = tp, Tn(b) — bnb for all b 6 B and Sn(a) = ana for all a 6 A Hence, Bv must be finite dimensional. Then the spectrum of eaca of its elements is a finite set. Proposition 6.1.9 then shows that each clement has spectrum equal to {0 }. Since we are assuming that Bv is finite-dimensional, this shows that Bv is nilpotent. Hence, (a) gives Bv = f0} as desired. (Section 1.6 contains the well known arguments that each element in a finite dimensional algebra has a finite spectrum and that such an element is nilpotent if its spectrum is {0 }.) n
6: Automatic Continuity
570
6.1.19
6.1.10 Exam ples Satisfying Theorem 6.1.18 We note that Theorems 6.1.18 and 6.4.20 have the same hypotheses, so that our examples satisfy both. Let B be the Banach algebra i 1([0, lj) introduced in Example 3.4.10. R ecall that multiplication is convolution t
f
*
J }{»)9{t~ »)ds
V/j€6;
t€ [0,
l]
o and that this algebra is commutative and neither semisimple nor semiprime. Indeed its Baer radical B b is dense. Furthermore, Theorem 3.4.9 shows that B does not have a unique functional calculus and does contain a discontinuously embedded irnage of the disc algebra /1(D). Hence B is a very badly behaved Banach algebra. Nevertheless it does satisfy Theorem 6.1.18. Example 3.4.10 shows that the closed ideals in B have the form Z0 = {/ 6 B : f — 0 a.e. on [0, a]}
V a € [0,1].
All these ideals (except Zo = B) are nilpotent, but none (except 1\ — { 0} ) are finite-dimensional, so B has no non-zero finite dimensional nilpo tent ideals. Now let Z0 be an infinite-dimensional closed ideal in B so that a is strictly less than 1. For each n 6 N, let bn be the characteristic function of [1 — 2~"(1 — a), 1]. Then it is easy to see that (&1&2 ' ' ' &nZ»)
=
—2- n(1- u)
is a strictly decreasing sequence of closed ideals. Hence B satisfies hypoth esis (b) of Theorem 6.1.18. We conclude that all homomorphisms of a Banach algebra onto B and all derivations of B (see Theorem 6.4.20) are continuous. Hence Kamowitz and Scheinberg [1969], whose results were originally stated for the continuous case only, give all the derivations of B and all the automorphisms in the connected component. John C. TVipp [1983] provides two attractively stated theorems more special than Theorem 6.1.18, which still cover most known examples. We omit the proofs. 6.1.20 Th eorem Let
{o}-
Automatic Continuity of Homomorphisms from A
571
1 .:.Jorem 6.1.3 can be derived from Theorem 6.1.18. Obviously a semisimple Banach algebra has no non-zero nilpotent ideals— finite-dimensional or oihsrwise. Furthermore, by applying some of the considerations used in the proof of Theorem 4.2.15 to a closed infinite-dimensional ideal of a semisimple Banach algebra A, first under the hypothesis that it is not an"iiila.ad by some infinite-dimensional irreducible representation of A and then, under the hypothesis that it is annihilated by all such representation, A can be shown to satisfy the hypothesis of Theorem 6.1.18. For details see Jewell and Sinclair [1976]. Let A C C[[t]] be a generalized Banach algebra of power series as de scribed in §3.4.18. If J is any non-zero ideal and a is a non-zero element of X, then {atn : n € N } is a linearly independent set in X. Hence A has no non-zero finite-dimensional ideals— nilpotent or not. Suppose X is any closed non-zero ideal. Then {tnl } ntN is a constantly decreasing sequence of closed ideals. Hence any such algebra satisfies the hypotheses of Theo rems 6.1.18 and 6.4.20. The conclusions of these theorems are known to hold on somewhat more general algebras of power series including some which are jVachet algebras rather than Banach algebras (Loy [1969], [1970a]). We remark here that Arens-Hoffman extensions (Example 3.4.13) tire know;':, to have a unique norm topology (David T. Brown [1966] and John A . Lh-dberg, Jr. [1971]). Other related results will be found in various examples, particularly Theorem 6.4.23 and Example 6.4.24 and Loy [1983] (cf. Lawrence Stedman [1983]). Many interesting results on continuity and compactness of module maps are known. We mention Gerhard Racher [1981a], Johnson [1985] and Willis [1289a].
;
Automatic Continuity of Homomorphisms from A
a homomorphism >p:A B, we now investigate conditions on the algebra A which make ip automatically continuous. This theory is considerably more complicated than the corresponding theory (discussed above) of conditions on the final algebra B. Since a comprehensive account of this theory will appear in H. Garth Dales [1994] about the same time .that the present work is published, we will refer the reader to that account, Sinclair [1976] or other sources for a number of the more complicated facts. ,Kers tre will prove some basic results and outline the more recondite por tions cf the theory. The following definition and proposition are essentially due to Johnson [1969b]. 6.2.1 D efinition Let
6: Automatic Continuity
572
6 .2 .2
defined by
( Z4fo>) = {a € A :
= {0 } } / l R(v ) = {a € A :
The two-sided continuity ideal 1(ip) of ip is I
l
= { 0 } } ).
W ) H T r ((p ).
6.2.2 Proposition Let i p : A —* B be a homomorphism between normed algebras. The continuity ideals are ideals of A . algebras, they satisfy
I f A and B are Banach
Z l ( v>)
=
{a € A : b hh>ip(ab) is continuous from A to B };
1r ( v )
—
{<* G A - b h-+ >p(ba) is continuous from A to B };
=
{a € A : b h-* ipiab) and b » - » tp(ba)
J(ip )
and
are both continuous from A to B }.
Proof Each continuity ideal is an ideal since B^ is an ideal of ip{ A) . The alternate descriptions of the continuity ideals follow by applying Proposition 6.1.9(c) to the regular representation maps L v (a), R^^y. B —* B. □
We give two of Johnson’s results ([1969b], [1967c]) using these ideas.
6.2.3 Theorem Let A be a C*-algebra. I f A is unital and has no proper closed cofinite ideals, then any homomorphism ip:A —►B into a normed algebra is continuous. If A is not unital, the same result holds if the double centralizer algebra V ( A ) of A has no proper closed cofinite ideals.
The proof of this depends on the following more technical theorem. It is enough to consider the case of double centralizer algebras since V ( A ) is always unital and it equals A if and only if A is unital. We use a variant of the left continuity ideal, introduced above, for this argument. 6.2.4 T h eorem Let A be a C*-algebra and let
The principal device in the proof of Theorem 6.2.4 is the following lemma. If the target algebra B of the homomorphism ip: A —►B is just a normed algebra, we may replace it with the closure of ip(A) and then complete it. This does not affect any of our results, so we may always assume that B is a Banach algebra in which ip(A) is dense.
6.2.5 Lemma Let ip: A —* B be a homomorphism between Banach algebras and let X'L (ip) be the ideal defined in Theorem 6.2.4. If { f n j n e n> {fln}n€N
Automatic Continuity of Homomorphisms from A
6.2.5
573
are sequences in V ( A ) satisfying 9m9n = 0 then f n
e
Vm ^ n€ N
and
for all but finitely many n
f ngn = /„
e
V n € N,
N.
Prc )f If the result were false, we could choose a subsequence {/ n}neN which was never in 2'L(tp), so we assume this for the original sequence. Define W „ = { i € B : ip (a f„)b = 0 for all a € .4} and denote the quotient map B —►B/W n by rpn. Proposition 6.1.9(c) shows tha. ipn oipis discontinuous for each n. Hence we can find hn <EA satisfying ||An,| < 2_n ||(/„||_1 and \\ipn °
establishing the lemma.
□
Pro >f of Theorem 6.2.4 At the end of §1.7.17, we showed that the double centraiizer algebra of a C*-algebra is again a C*-algebra. In order to sim plify notation we will henceforth denote the ideal 1 L ' (
574
6: Automatic Continuity
6.2.6
continuity points for h. Now we apply the functional calculus for continuous functions to h. For any Ao € C \ F w e can find a continuous function / with /(A) = (A — Ao)-1 in a neighborhood of F. Then g (A) = (A — Ao)/(A) — 1 is zero in a neighborhood of F. Hence g(k ) belongs to I so f (k ) + 1 is an inverse for k — Ao- Thus we have shown Sp(k) C F. The proof of the theorem is now completed by noting that a C*-algebra (such as T>/T) is finite-dimensional if each of its elements has finite spec trum. This result can be found in many books on C*-algebras and a proof is included in the second volume of this work. □ Proof of Theorem 6.2.3 We may assume that B is the closure of tp(A). By hypothesis, the closure of TL(ip) is either { 0} (in which case T>(A) is finite-dimensional so that the theorem is trivially true) or it is all of A. But this implies l 'L(ip) = V (A ), so the identity element 1 of T>(A) belongs to the ideal. But this implies B ^ is {0 } which gives the desired result by Proposition 6.1.9(a). □ The above result applies to B (7 i) for any Hilbert space 7i. It also applies to B k (H ) since we showed in §1.7.14 that T>{Bk(‘H )) = B(7i). Proposition 6.1.6 shows that it also applies to the Calkin algebra B (J i )/ B ^ (H ). We wish to extend these results to Banach spaces more general than Hilbert spaces. Stefan Banach [1932] noted on p. 245 that many classical Banach spaces X are linearly homeomorphically (or even isometrically) isomorphic to X (b X . Whenever this is true, we can iterate the process to write X as the direct sum of 2” subspaces each isomorphic to X and hence to each other. Fix this continued bisection for the rest of the next proof. FVom it we can construct many sequences {£n}n€lN° of projections in B (X ) satisfying: (1) Eo = I and (2) for each n € N, E n = E n- i E „ = i is am equivalent projection to E „ -\ — E n by an element Un € B (X ) which is an isometric map of X onto itself. (We require E n = - E „ )U n.) Any such sequence will be called a complete chain in the following proof.
6.2.6 Theorem Let X be a Banach space satisfying X = X ® X . Then any homomorphism ip: B (X ) —►B of B (X ) into a normed algebra B is continuous. Proof As noted above, we may as well take B to be the closure of tp (B (X )) in the completion of the original B. Lemma 6.2.5 can be modified to show that only finitely many complete chains can miss the left continuity ideal T i ( r ) - We will call these complete chains exceptional Note that when E n belongs to an ideal then the rest of any complete chain containing E „ must belong to the ideal. Conversely, if E n does not belong to the ideal then at least one of E „ + i and E „ - E n+i must not belong to the ideal. Hence any projection from the continued bisection which is not in belongs to an exceptional complete chain.
5.2.7
Automatic Continuity of Homomorphisms from A
575
Since there are only finitely many exceptional complete chains we can chopse p so large that each exceptional chain has a distinct projection as its pLi entry. Hence if Ep comes from any complete chain at least one of E p+1 and Ep —Ep+1 belongs to However, if one does the other does also, so they both do. Now there are only 2P projections which can serve the :o".e _'f Ep+i in any chain, so the sum of all of them is the identity operator I. This implies / € which gives Bv = {0 } and ip continuous. □ 5.2,7 C o rollary Let X be a Banach space satisfying X = X © X . Then any .wmomorphism of the generalized Calkin algebra B (X )/ B k (X ) into a ncnr.ed algebra is continuous. If B k (X ) has a bounded left or right approximaii identity, then any homomorphism of Bk [ X ) into a normed algebra is, continuous.
The first result follows from Proposition 6.1.6. 'The proof of the last theorem can be modified as in the proof of Theorem 3.2.4’ to show ip(B k (X))B< p — {0 }. Now for b € Bv , let { K n} be a sequence in B n (X ) which converges to zero and satisfies lirn
3.2.S C o rollary Let X be a Banach space satisfying X — X ® X . Then any (not necessarily complete) algebra norm on B (X ) is equivalent to the operator norm and hence is actually complete. If B k (X ) has a bounded left c r right approximate identity, then the same result holds for B k (X )Proo f This is just property (a) of Theorem 6.1.5. Thus the corollary follows froi:i the above results and the last theorem of §1.7.15. □
Michael J. Meyer [1992c] extends the above corollary to certain Calkin algebras of classical Banach spaces. In [1989] Charles J. Read produced 'Lrst example of a Banach space X for which a discontinuous module derivation and hence (Proposition 6.4.4) a discontinuous homomorphism from B (X ) into a Banach algebra could be constructed. His space was sim ilar to the James space J , so it is of interest that George A. Willis [1993] 1 has shown that all homomorphisms from J are automatically continuous even though J does not satisfy the hypotheses of Theorem 6.1.18. If the continuum hypothesis holds, Dales, Loy and Willis [1992] produce a more . fundamental example of a Banach space X for which there is a discontinu ous homomorphisms on B (X ) even though all derivations defined on it are continuous. We will discuss this unusual hypothesis below.
576
6: Automatic Continuity
6 .2 .8
Homomorphisms from C (C l)
Not surprisingly, the first attempts to prove the continuity of all homomorphisms from a Banach algebra A into another normed algebra were directed towards commutative algebras and, in particular, towards the al gebra C (Q ) of all continuous complex-valued functions on a compact Haus dorff space fi. (The important paper by Gelfand and Nalmark [1943] has a misstatement in Lemma 2 which, if accepted, would imply the continuity of such homomorphisms.) To avoid triviality, in the following remarks we will always assume that Cl is not a finite set. In [1949a] Kaplansky proved (Theorem 2.4.15 above) that any algebra norm on C (Q ) is necessarily larger than the uniform (or supremum) norm ||•Hoc- This result, together with the open mapping theorem, shows that any complete algebra norm is equiva lent to the uniform norm. Hence C(Cl) has an incomplete algebra norm if and only if it has an algebra norm inequivalent to ||• Hoq. In 1948 Kaplan sky asked whether all (not necessarily complete) algebra norms might be equivalent to ||• H,^. In [1960] William G. Bade and Phillip C. Curtis, Jr. published a fundamental paper directed towards proving this. The basis of their technique was a “main boundedness theorem” , similar to Lemma 6.2.5, which applied in any commutative Banach algebra. It shows inter alia that if tp: A —* B is any homomorphism from a commutative Banach algebra to a normed algebra, then for any sequence {e n} n€N of orthogonal idempotents (i.e., those satisfying enem = 6nmen) there is a constant M satisfying M e n)|| < M\\enf V n € N. For semisimple completely regular commutative Banach algebras A, they showed that the discontinuity of any homomorphism from A was concen trated at a finite set F (called the singularity set of tp) in the Gelfand space of A. In particular, ip is continuous on the ideal of functions vanishing on F. They were able to construct discontinuous homomorphisms (using point derivations as in §3.4.16) on certain semisimple completely regular commutative Banach algebras, but remarked that they knew no discontin uous homomorphisms when A = C'(fi). They showed that all homomorphisms from C (fl) are “almost” continu ous in a rather technical sense which we will state after a bit of preparation. For any uj € Cl we define the largest and smallest ideal corresponding to u j . The traditional notation is
3w
=
{/ e C (f l ) : f(u>) = 0}
=
{/ 6 C {i l ) : f = 0 on some neighborhood of w}.
and
Our results in Example 3.2.15 show that each is a maximal ideal o f C (fl) and that every maximal ideal arises in this way from some u j . Corol lary 7.3.3 shows that X is the smallest ideal which is included in but
Automatic Continuity of Homomorphisms from A
577
in no other maximal ideal. If F is a finite subset of SI, let M (F )
=
{ / e C (S l ): f (w ) = 0 V w € F )
T>(F)
=
{/ e C(Sl) : f is constant in some
and
(variable) neighborhood of each
oj
€ F }.
Then M ( F ) is an ideal and V ( F ) is a dense subalgebra of C(Q). We now state (without proof) the main result of Bade and Curtis for a homomor phism onto a dense subalgebra of a Banach algebra essentially as given in Sinclair [1976], Theorem 10.3. If ip: C (S l) —►B is any homomorphism into a norm 3d algebra B, then B may be taken as the completion of the closure of i^ (C (l)) in B, so this theorem makes no essential restriction on the range of the homomorphism. For hulls see Definition 3.2.5. 6.2.8' Theorem Let SI be a compact Hausdorff space and let C (fl) —►B be a discontinuous homomorphism onto a dense subalgebra of a Banach algeb n B with Jacobson radical B j . Then: (a) The hull h(l(
where 6j (C (S l ))6k(C (S I)) = { 0} for j ? k.
For j = 1 ,2 ,..., n, the direct summands satisfy 6] (C (S l))ip (M ulj) = { 0 }.
I'Vie that each of the non-zero homomorphisms 6jIm wj is zero on the dense subalgebra J u)j and hence is discontinuous. Thus this theorem shows that i; there is a discontinuous homomorphism from C (Sl), then there is a discontinuous homomorphism from some vanishing on and with range dense in a radical Banach algebra. Jchraon [1976] extended the results of this theorem by considering the locally compact space SI' = S1\F and the algebra Cq(SV) which is just the set of restrictions of functions in M f - Each function in Co(n') has a unique ex tension to a continuous function on the Stone-Cech compactification 0(SI') of SI'. Since the discontinuous portion 6 of the above homomorphism is a homomorphism on M f , it can be re-analyzed as a homomorphism from
578
6: Automatic Continuity
C (0 (i l ')). For each point ui‘ in
6.2.9
\ H', let
be the intersection with Cb(fi') of the ideal defined in C (0 (i l ')). He proved that the disconti nuity of the original homomorphism can be confined to a finite set of ideals for J € Discontinuous Homomorphisms and the Continuum Hypothesis
Until the beginning of 1976, it seemed that a slightly more subtle ap proach might prove that all homomorphisms from any C (fi) were contin uous. During 1976 Jean Esterle ([1977], [1978b], [1978c]) and H. Garth Dales ([1979a], [1979b]) (also see their joint announcement [1977]) inde pendently constructed discontinuous homomorphisms from C (fi) for any infinite ft. Both their constructions, which took somewhat different ap proaches, used the continuum hypothesis. This appeared briefly to be an artifact of their methods of proof. However in 1976, while their papers were being completed or awaiting publication, Robert M. Solovay used a construction due to an undergraduate at California Institute of Technology, W. Hugh Woodin, to show that there are models of set theory in which no discontinuous homomorphisms from any C (f l ) exist. In these models, ev ery algebra norm on C (fl) is equivalent to the uniform norm. Woodin soon simplified and amplified the result. He wrote them out for his doctoral thesis at the University of California, Berkeley. Unfortunately, none of the original work was ever published, so the book by Dales and Woodin [1987] is the only accessible source. We refer the reader to that book for details of the information we are about to briefly summarize. First we need to address the seeming contradiction in the results we have just quoted. Godel’s incompleteness theorem shows that no system of axioms strong enough to define the positive integers can prove its own consistency. The standard axioms for set theory are named for Zermelo and Franke] and are denoted by ZFC. Despite Godel’s result, it seems reasonable to hope that no inconsistent results will be derived from these axioms. (Like most mathematicians, I believe that even if such an inconsistency were discovered, logicians would quickly find a new fundamental axiom system.) Because of Godel’s result, the best that can be proved for axiom systems which include ZFC is relative consistency. This simply means that if the new system leads to inconsistent results, then ZFC must already be internally inconsistent. These proofs are accomplished by a technique called “forcing" which was initiated by Paul J. Cohen. It essentially uses ZFC to construct a model that satisfies the axiom system in question. Thus if ZFC is consistent, so is the new system. Here is the example which led us into this discussion of logic. The countably infinite cardinal is denoted by No and it is elementary to show that the cardinal number of the set R of real numbers is 2H° . Thus this cardinal is called “the cardinal of the continuum”. It is easy to show that the cardi-
Jordan Homomorphisms
s .r .”)
579
o f the continuum is strictly larger than No- The continuum hypothesis (denoted by CH) is the statement that the cardinal of the continuum is the ne::t cardinal number after Ko- This next cardinal is denoted by Kj. The symbol -iCH stands for the statement “the continuum hypothesis is false” or equivalently “2H° ^ Ni” . It is known that both of the axiom systems ZFC + CH (the Zermelo-Frankel axioms together with the continuum hy pothesis) and ZFC + -iCH (the Zermelo-Frankel axioms together with the denial of the continuum hypothesis) are relatively consistent. Thus a math ematician who “believes” (or at least accepts) the Zermelo-Frankel axioms is fres to either believe the continuum hypothesis or believe its denial. In the -‘
6.3
Jordan Homomorphisms
Jordan homomorphisms (under another name) were considered by Ger man Ancochea [1942] and [1947]. Other early papers were by Kaplansky [1947b] and Loo-Keng Hua [1949]. In two fundamental papers [1950] and [1952] Jacobson and Rickart introduced the term “Jordan homomorphism” and made the first studies similar in spirit to the approach taken here. The name “Jordan homomorphism” comes from the generalization of quantum mechanical formalism introduced by Pascual Jordan [1932] and later stud ied in a classic paper by Jordan, John von Neumann and Eugene P. Wigner [1934]. (See also Harald Hanche-Olsen and Erling St0rmer [1984].) This generalization is based on the “symmetrized” Jordan product a o b = ab + ba
wher2 a and b are elements of an associative algebra. A Jordan homomorphis^i is simply a linear map which preserves this product. Our most definitive results on Jordan homomorphisms are in Volume ” deal with Jordan ’ -homomorphisms between C*-algebras and other
580
6: Automatic Continuity
6.3.1
special ‘ -algebras. Here we give a fairly detailed analysis of Jordan homo morphisms between algebras and between Banach algebras. We list a few of the main results. If A and B are both commutative algebras, it is obvious that a Jordan homomorphism
V a, b € A.
( 1)
For certain calculations with Jordan homomorphisms, it is convenient to use the Lie bracket notation: [a, i] = ab — ba
V a, b € A.
(2 )
We begin with some trivial algebraic identities which we need later. 6.3.2 Lem m a Let A and B be algebras and let
Jordan Homomorphism*
6.3.3
581
(e) <^([a, b]2) = [<^(a),^(6)]a V a, b 6 A . (f) v>([[a, 6], c]) = [[
implies <.c'ab + ba) = ip((a + ft)2) -
(b': The sufficiency of (b) is now obvious.
If
By inc uction (b) holds. (c): Note 2
(d): This follows from (c) and abc + cba = (a + c)b(a + c) — aba — cbc. (e); This follows from (d) and [[a, 6], c] = abc + dba — ( bac + cab). (f). This follows from (c), the definition of a Jordan homomorphism and the identity [a, fc]2 = a(bab) + ( bab)a — ab2a — ba2b. (g): Using (a), (d), (a) and (c), we get: (
ip(a)ip(b))(ip(ab) -
—
ip(abab) - ip(ab)ip(b)ip(a) - tp(a)tp(b)tp(ab) + ip(a)ip(b)2ip(a)
=
ip(abab) — >p(ab2a + abab) + ip(ab2a) = 0 .
□
6.3.3
Lem m a Let A and B be algebras and let t p :A —> B be a Jordan homoT.iorphism. If e is an idempotent in A and a is an element of A , then tp(e) is an idempotent and (a/ ea = ae=> ip(a)
(a)
Lemma 6.3.2(e) asserts 0 = [[v’(a ),^ (e )],^ (e )] and this implies
ip(a)ip'e) - ip(e)ip(a)ip(e) = ip(e)ip(a)ip(e) - ip(e)tp(a). Multiplying on the left b> ip(e) gives 0 = ip(e)>p(a)ip(e) — ip(e)ip(a). Therefore,
Thus (a) is proved. The rest of the statements in the proposition are now immediate. □ In the next result, (a) and (b i) are due to Nathan Jacobson and Rickart [1950], while (b2) is due to Wieslaw Zelazko [1968].
6: Automatic Continuity
582
6.3.4
6.3.4 Th eorem Let A and B be algebras with B semisimple. Let ip-.A—* B be a Jordan homomorphism. (a) For a, b € A,ab = ba implies ip(a)ip(b) = ip(b)tp(a) if either: (&i)
J = span {a + be : a, c € Bz are nilpotent and b € B }. It is easy to check that this is a nil ideal and hence a quasi-regular ideal. Therefore I = {0 } by Theorem 4.3.6, so there are no nilpotent elements in Bz except zero. (b): If (b i) holds, the conclusion follows immediately from (a) and the definition of a Jordan homomorphism. Suppose (b2) holds. Let 7 be an arbitrary element in Tb- Theorem 2.4.13 shows that 7 o ip is a homomor phism. Since the elements of
2tp(a) + 2ip{av) = ip(aaq + aqa) = 1p(a)
(3)
Hence the proof will be complete if tp(a) and ip{aq) commute. Multiply ing (3) on the left by 1p(a) then multiplying it on the right by tp(a) and subtracting the results gives (2y?(o) - ip{a)2 )
(4)
Note also 2ip(a)3
+
=
i^(2a2 + aaq + aqa — ( a2aq + a ’ a2)) = 0.
(5)
8.2.3
Jordan Homomorphisms
583
Adding the right side of (3) to the left side of (5) gives . 1 'p(a)2
+
( 2
=
2
(6)
Substituting (4) in (6) and dividing by 2 gives
+
(2
=
However this can be rewritten as ip(a) o ifi(a) o ip(a9) =
Hence we get >p(a9) o ip(a) — ip(a9) o ip(a) o ip(a) ° ^>{a9) =
wfe:cb is equivalent to the commutativity we wished to establish.
□
Recall that an algebra A is called prime if {0 } is a prime ideal. From Theorem 4.1.10 it follows that A is prime if and only if it satisfies one (hence all) of the following implications a,Ab = {0 }
=*•
(a = 0 or b — 0)
V a, b € .4;
h:Z2 = {0 }
=>
(Ji = {0 } or J2 = {0 })
C i C 2 = {0 }
=>
(£ i = {0 } or C 2 = {0 }) V left ideals C u C 2 of >1.(9)
V ideals
(7) of A ; (8)
primitive algebra (i . e . , an algebra in which {0 } is a primitive ideal) is prime by Theorem 4.1.8. Condition (8) shows that a simple algebra which is not the one-dimensional zero algebra is prime. We need two other simple properties of prime algebras which we include in a lemma. A
0.3." Lem m a Let A be a prime algebra. (a) If elements u, v € A satisfy ucv + vcu = 0 for all c € A , then either v, — C or v = 0. ■ (o) If A is a topological algebra, every dense subalgebra is prime. Proof Replace c by due. Then uduev + vdueu = 0. However uev = —veu and ndu = —udv so that —udveu — udveu = 0 . Since d and e are arbitrary, uAvAu = {0 }. Thus either u = 0 or uAv = {0 }. In the latter case, either u - 0 o r v = 0 so that (a) is proved. S>>tement (b) follows immediately from Condition (7) above. □
Tkv; next result is a fundamental result of Israel N. Herstein [1956]. We ’ use it later to obtain results on Jordan homomorphisms into more genertl algebras.
6: Automatic Continuity
584
6.3.7
6.3.7 T h eorem Let A and B be algebras mth B prime. Let ip :A —* B be a Jordan homomorphism such that either: (a) ip(A) - B , or (b) B is a topological algebra and ip(A) is dense in B. Then tp is either a homomorphism or an anti-homomorphism. Proof Let u = tp(ab) - < p(a)ip(b) and v — tp(ab) - < p(b)ip(a). We will use various formulas from Lemma 6.3.2 to show that u and v satisfy the criterion of Condition (a) in the last lemma. For any c = ip(d) in ^ (A ) we have ucv
=
ip(ab)ip(d)tp(ab) + ip(a)ip(b)
=
Similarly we have vcu
=
tp(abdab) + tp(badab) —
Therefore we conclude ucv + vcu
=
ifi(2abdab + abdba -(- badab) ~ [ M a)¥>(6) +
=
ifi(2 abdab + abdba + badab) -ip[(ab + ba)d(ab) + ( ab)d(ab + ba)] = 0.
If ip(A) is dense in the topological algebra B, the result above holds for all c € B. Hence by Lemma 6.3.6(a), either v or u is zero. For a £ A , define 7ia = {b € A :
Let A and B be algebras and let tp :A —* B be a Jordan homomorphism. Then ker(^j) is an ideal if either (a) B is semiprime and ^ ( A ) = B, or (b ) B is a spectral normed semisimple algebra and <^(.4) is dense in B. Under either hypothesis, if A is spectral normed and B is semisimple, then
ker(v?) is closed. Proof Let { P a : a € A } be the set of prime ideals of B. For each a € A, let ipa be the natural map ipa: B —» B /V a and let <pa = V\» ° 'P- Then by
'•.3.'
Jordan Homomorphisms
585
the theorem, ker(i^Q) is an ideal for each a. Hence ker(^j) = n aeyi ker(i£a) is an ideal. I" B is a spectral normed semisimple algebra, we make the same ar gument replacing prime ideals by primitive ideals. By Proposition 4.2.6 each primitive ideal is closed. Hence each B/Va is spectral normed and tpa{ s ) is dense. Thus the argument is completed in the same way using the s’.ternative topological hypothesis of Theorem 6.3.7. Ii A is spectral normed and B is semisimple, each tpa (with V a primitive) has ; primitive kernel in A (if it is a homomorphism) or in the opposite algebra of A (if it is an anti-homomorphism). Since the opposite algebra is also t pectral normed, each ker(<^a ) is closed. Hence ker(
Let A and B be algebras and let ip: A —►B be a Jordan home morphism with ip(A) = B. ( t ) If ker(¥?) C A j , then
Proof (a): Let xp:B —> B / B j be the natural map. Then ker('0 o ip) =
6.3.7. However, whenever
586
6: Automatic Continuity
6.3.10
is a homomorphism onto a simple unital algebra } C fl{ker(^>Q) : a € A } = n{ip~l ( M a) : a € A } = <^_ 1 (D{A^a : Q € A } ) = ip ~ 1( B m ). Hence v’ M jw ) Q BmNow let 0:B —+ A / A m b® the Jordan homomorphism induced by ip~x. The same argument shows that B m Q ker(0) = ip( A m )■ a Part (a) of the next theorem is due to Sinclair [1970b]. Part (b) is due to Civin and Yood [1965], 6.3.10 T h eorem
Let A and B be Banach algebras and let p: A —+ B be a Jordan homomorphism. Then tp is continuous if either: (a) B is semisimple and
Proof (a): Let V be a primitive ideal of B and let rp:B —* B / V be the natural map. Theorem 6.3.7 shows that ip o ip is either a homomorphism or an anti-homomorphism. I f it is an anti-homomorphism, replace A by its reverse Banach algebra. Then ip 0
Part (a) of this theorem shows that the set of Jordan automorphisms of a semisimple Banach algebra A is a subgroup of B (A ). Sinclair [1970b] shows that the connected component (in the norm topology) of this group consists of automorphisms. To obtain this, he uses a generalization to Jordan automorphisms of a theorem of Georges Zeller-Meier [1967] which we prove in the next section (Proposition 6.4.21). The generalization shows that each Jordan automorphism which is sufficiently close to the identity is the exponential of a Jordan derivation. Then Proposition 6.4.21 shows it lies on a norm continuous one-parameter group of automorphisms. Hence the subgroup generated by such one-parameter groups is a connected open (and hence closed) subgroup which must be the connected component of the identity. The next theorem is a slight generalization of a theorem of Sinclair
Jordan Homomorphisms
587
[1970a], Note that in this theorem conditions are placed on B in order to s ' io w that a Jordan homomorphism tp'-A —» B is the direct sum of a homomorphism and an anti-homomorphism. In Theorem 6.3.12 we will reach the same conclusion from restrictions on A. Th eorem Let A and B be algebras. Let B be unital and satisfy one of the following hypotheses: (a) B is semiprime and has no non-zero commutative prime algebra as a homomorphic image. (b) B is semisimple and has no irreducible representations with commu tative image. (c) B is strongly semisimple and has no non-zero commutative simple homomorphic image. If ip. A —* B is a surjective Jordan homomorphism, then there is a direct sum decomposition B — l\ ® T i of B and surjective linear maps ipj-.A—* Jj for j = 1 , 2, such that tfii is a homomorphism, <02 is an anti-homomorphism, and ip = tfi + tf2 Proof Let { V a : a € A } be the set of prime, primitive or maximal modular iderl? of B, depending on whether B satisfies (a), (b), or (c), respectively. For each a € A, let ipa be the natural map B —* B /V a and let
, .One can easily prove (as in Sinclair [1970a]) that the decomposition given in the last theorem is unique. If the second half of conditions (a), (b), and (c) are omitted, this uniqueness will obviously fail. However, Sir.cif,ir gives an example to show that even the theorem fails without this rss^i’ lion. We reproduce this interesting example. >a r, C = C ( [ —1,1]) and let Z> = {/ € C : /( 0) = 0 }. Let A be the algebra of 2 x 2 matrices with entries from C on the principal diagonal and entries from V in the other positions. This is obviously an algebra under
6: Automatic Continuity
588
6.3.12
the usual matrix operations. Let ip:A A leave the principal diagonal entries fixed, but replace eexA of the other two entries by a function which agrees with it on [—1,0] but agrees with the opposite entry on [0,1], It is clear that ip is a Jordan isomorphism and a little calculation shows the lack of ideals with the properties of I\ and J2. We now turn to a theorem which derives essentially the same conclusion as Theorem 6.3.11 from very different hypotheses. We begin by recalling some facts about matrix algebras. Given any algebra V , the family o f n x n matrices with elements from V forms an algebra 2?® M „ under the customary matrix algebraic operations. It is natural to identify each d e 2? with the diagonal matrix with d in each position on the diagonal. We will assume from now on that V is unital. Denote the matrix with 1 in the ith row and jth column and zero in all other entries by E '* € V ® Mn. These elements satisfy: =
bjk E *
=
1
=
E %jd
V 1 < i,j,k ,t < n
n
«=i dE,J
( 10) V d e V-, 1 < i , j < n
under the above embedding of T> into V ® M „. In fact it is easy to see that an algebra A is isomorphic to 6 0 M „ for some algebra B if and only if it contains a subalgebra V isomorphic to B and elements E '} for 1 < * ,j < n such that these identities hold. (Theorem 1.1.20 contains an easier characterization.) In this case every element of A can be written uniquely as
01 ) t= l
j = 1
for suitable d,} 6 T>. It is easy to check that such an algebra is isomorphic to the algebraic tensor product V ® M „ as a linear space with multiplication given by (d\ ® b i)(d 2 ® 62) = d\d2 ® b\t>2
V d\, d2 e 2?; 6j, 62 £ M n.
We say that ^ i s a n n x n matrix algebra over T> when it is isomorphic to V ® M „ . In this case, we usually consider V as embedded in A as indicated above. The following important theorem and its proof are due to Jacobson and Rickart [1950], 6.3.12
T h eorem Let A and B be algebras, and let A be a full n x n matrix algebra for n > 1 over a unital algebra "D. Let
S.i...
Jordan Homomorphisms
589
Then here are orthogonal central idempotents gi and g? in B such that g\B and g-j_B are full matrix algebras over y{T>) and the maps
=
v (d E 'i E JJ + E 33E '3 d)
=
=
v i d n v ^ M E ” ) + v { E » M E '’ ))
=
Since -/e have proved this for i = j already, we have
V d € P ; 1 < i , j < n.
Hencti any element of A written as in (11) satisfies n
n
n
n
( 12 ) «=1 j = 1
i= l j = 1
We now introduce two orthogonal sets of matrix units {gij : 1 < i, j < n } and [hi] : 1 < i, j < n } in B. For t ^ j define and
gij =
hi} =
(We define g « and ha a little later.) Note that E 'J = E iiE iiE ii + E 3i E i* E ii implies
=
^ (£ "M ^ ) = ^
hij
=
v ( E » M 2 5 * ) = * > (£ > > (£ « ).
M
E
” )
For j j k, this shows gij9kt = 0. For i ^ j ^ k ^ i, it also shows = '
=
^ M
^ g j k
= ‘t ( E '3 )g]k = v ( E '3 M E 3kM E kk)
E * ) + v ( & k)
=
ip(E'3E jk + E ]kE iJM E kk)
=
590
6: Automatic Continuity
6.3.12
where we used Lemma 6.3.3 and E kkE 'J = E ' 3 E kk = 0 for the fourth equality. Thus, any i,j,k ,£ satisfy 9 ij9 k t =
(14)
6 jk 9 it-
Similarly, any t, j, k, I satisfy (15)
hijhkl = Gjkhil-
Now for each i chooee j ^ i and define 9n — 9 ij9 ji
ha =
h ijh ji.
Then gti is independent of the choice of j ^ i since 9ik9k, = 9,j9jk9ki = 9ij9ji when i ^ j ^ k ^ i. Similarly ha is well defined. It is easy to check that (14) and (15) now hold for all choices of indices. Furthermore for j ^ i,
n
n
n
(16) n
n
n
n
*=i
j— i
*=i
j= i
For o, 6 € V and i ^ j , we have ip(ab)ip(E!it) + tp(ba)
In this section, we have studied homomorphisms preserving the Jordan product a o b = a i + ba for a,b in an associative algebra A. Along with
6.3.13
Jordan Homomorphisms
591
this symmetrized product, it is natural to consider the anti-symmetrized Lie product
[a, i] = ab — ba
V a,b e A
and the Lie homomorphisms which preserve it. However, the most signifi cant r 3sults on this product require some sort of left-right symmetry in the algebra, such as that given by an involution. Since we are postponing the study of algebras with an involution until Volume II, we will postpone the cisc'/ssion of Lie homomorphisms. The reader may wish to consult papers by Wallace S. Martindale, III [1969], [1977] and Richard A. Howland [1969]. Hers is a typical result from the first reference. ®.3.M
Th eorem Let ip :A —* B be a Lie isomorphism where A is unital, prime and has a nontrivial idempotent and B is unital and simple. Then tfi is th e sum
6.4
Derivations
In our discussion of derivations we will use the language of bimodules, normed bimodules and Banach bimodules introduced in Definition 4.1.17. We au’./ays regard any algebra A as a bimodule under the regular mod ule ac ion under which ab has the same meaning whether considered as a product in A , the left module action or the right module action. I f A is a ( nonr.ed algebra / Banach algebra ), this regular bimodule is a ( normed bimodule / Banach bimodule ). 6.4.1 D efinition
Let A be an algebra and let X be an .4-bimodule. A bimodvle derivation of A into X is a linear map 6: A —* X which satisfies 6(ab) = 6(a)b + a6 (b)
a, b e A.
The se'; of derivations of A into X is denoted by A (.4, X ). If A is a normed a - g s b a n d X is a normed .4-bimodule, then the set of bounded derivations, A ( A X ) n B (A , X ), is denoted by A B(^4, A'). When X is just the regular .4-bimodule A itself, we speak of an algebra derivation of A and denote the sets by A (.4) and A b (A ). , Bimodule derivations arise naturally in a number of situations (cf. Ringross%[1972a], Johnson [1972a]). In particular they are at the center of the cohomology theory of Banach algebras given in Volume II. When no con fusion is likely, we will usually shorten “bimodule derivation” and “algebra derivr.t'on” to just “derivation” .
592
6: Automatic Continuity
6.4.2
It is clear that A (A , X ) and A b (A , X ) are linear spaces under pointwise operations with the latter normed and complete under the norm of B (A , X ). It is also trivial to check that both A(_4) and A b ( A ) are closed under the Lie bracket
[61,62] = 6162 - M i
V fc .fc e A .
( 1)
Readers acquainted with Lie algebras will recognize these spaces as (often infinite-dimensional) Lie algebras. The following example sxplains the term “derivation” . Let A be the set, C °°(R ), of infinitely differentiable complex-valued functions on R made into an algebra by pointwise operations. Define 6: A —►A by setting 6f equal to the derivative of / for any / € C °°(R ). It was, of course, in this context that Gottfried Wilhelm Leibniz discovered the formula
6"(afc) = £
( " V - fc(a)6fc(6)
*=o '
(2)
'
for repeated differentiation. Since the usual inductive proof depends only on 6 being a derivation, we will use Leibniz’ formula for derivations without further comment. The formula for the derivative of a polynomial allows derivations to be defined on algebras over any field, if the elements of the algebra can be interpreted as polynomials. A similar remark holds for algebras of formal power series over any field. We have little to say about these purely alge braic types of derivations although they have important applications. Our next example, which is fundamental to our subsequent discussion, can be defined in any algebra. 6.4.2 D efinition Let A be an algebra and let X be an .4-bimodule. For any element x e X , the map 6X: A —►X defined by 6x(a ) = xa - ax
a€A
is called the inner derivation determined by x. A derivation S of A into X is called an inner derivation if it is equal to 6X for some x € X . The set of inner derivations is denoted by A / (A , X ). If X = A , we write A j (A ). It is trivial to check that an inner derivation is a derivation. In fact, the set A i (A ) of inner derivations is a Lie ideal of the Lie algebra A (.4). This means that [6,Sa\ belongs to A ; (.4) for all 6 6 A (.4) and a € A and it follows from the useful formula [6, 6a} = 66a - 6a6 = « 4(a)
a e A, S e * { A ) .
(3)
If A is a normed algebra and X a normed bimodule, then inner derivations are clearly bounded. Hence for a normed algebra A, A/(,4) is a Lie ideal of
Derivations
6.4.?
593
A b U )■ Note also that the map x >—>Sx is a linear map of X into A j(A , X ). The kernel of this map is the center Xz of X where the center of an A bimo 'ule is the set {x € X : ax = xa for all a G A }, so that the center of the regular .4-bimodule is just the usual center of the algebra A. We will show that all derivations on certain algebras are inner. For a coran.utative algebra the only inner derivation is the zero map. This special case will also prove important. We need to consider the separating space of a derivation. Proposition 6.1 .8(d) shows that the separating space B^ of a homomorphism (p : A —* B is a closed ideal of
V b, c € A
gives he desired result.
□
Homomorphisms constructed from Derivations
There are two major connections between derivations and homomor phisms. We discuss the more elementary case first. Let A be an algebra and let X be an ,4-bimodule. Then A x X can be made into an algebra by pointvdse linear operations and the multiplication (a ,x )(b ,y ) = (ab,ay + xb)
V a,b € A; x,y € X .
(4)
We denote this algebra by A ® X . If A is a ( normed algebra / Banach algebra ) and X is a ( normed / Banach ) .A-bimodule, then it is clear that these operations make A 0 X into a ( normed algebra / Banach algebra ) under the algebra norm ||(o, x)|] = ||o|| + |||x|||
V a € A\ x e x
where 111• 111 is the equivalent norm on X defined by i||x||| = sup{||x||, ||ox||, ||x6||, ||ax6[| : a, 6 € A i ]
V x € X.
Note I hat A ® X is just the normed algebra semidirect product of the subalg;bra {(a, 0) : a £ A } which is naturally isometrically isomorphic to A m d the ideal {(0, x) : x € X } which is naturally homeomorphically linear!;r isomorphic to X considered as an algebra with all products zero. f
6.4.4
6: Automatic Continuity
594
6.4.4 Prop osition Let A be an algebra, X an A-bimodule and A (& X the algebra defined above. If 6: A —> X is a derivation, then iptm . A —* A ® X defined by y>«(a) = (a, 6(a )) is an injective homomorphism. If A and X are normed, then the separating spaces satisfy ( A (b X ),P6 — {0 } ® Xt so that tpt, is continuous if and only if 6 is continuous. Furthermore,
||a|U = ||o|| + |||«(a)|||
V ae A
is an algebra norm on A which is equivalent to the original norm (or, equivalently, continuous) if and only if 6 is continuous. Hence if every injective homomorphism of a Banach algebra A into a Banach algebra u continuous, then every derivation of A into a Banach A-bimodule is continuous. Proof Straightforward.
□
The last statement of the above proposition extends several of the re sults in Section 6.2 about automatic continuity of homomorphisms to cover derivations. For instance, Theorem 6.2.6 shows that all homomorphisms from B (X ) are continuous when X is a Banach space linearly homeomorphicaily isomorphic to X x X . Hence, all derivations from B (X ) into Banach B(A”)-bimodule8 are continuous when X has this property. Dales, Loy and Willis [1992] give the first example of a Banach space X for which B (X ) is known to have discontinuous derivations but no discontinuous homomor phisms. (See §1.7.8 above, and compare Loy and Willis [1989], Read [1989] and Willis [1993].) Here is a very simple concrete construction of an inequivalent algebra norm using the process described in the proposition. Let A be the disc algebra of §1.5.2 with its usual supremum norm || • ||d on D. Then the second norm is just ll/IU = II/IId + I/'(1)I
v
/ ga
Th is norm arises by letting C be a Banach .4-bimodule under the action
f \ = A/ = / (1 )A and considering the bimodule derivation 6( f ) = /'(1).
Later we will extend these remarks about inequivalent algebra norms to obtain two inequivalent complete norms on a commutative Banach algebra. One I'aramr.tr.r Croups of Homomorphisms and Derivations
We now turn to the second connection between derivations and homo morphisms. It only makes sense in the case of derivations with range in the original algebra which must be normed. The interesting case is that of a continuous derivation on a Banach algebra. Let A be a Banach algebra and let 6 be a derivation of A. If the series E r = o ( » !) " ^ n(a) converges for all a e A (which is certainly true if 6 is
Derivations
3.4.5
595
bounded), then we will denote the sum by e6(a). For any a, b € A and any A G C, Leibniz’ formula gives
n=0 ~
k=0 ' '
y , y , Xn~k6n~k(a) Xk6k(b) y , Xm6m(a) A n _ jfc! jfci “ 2 -, rn! ^
n = 0 fc=0
=
n=0
m =0
a k6k(b) Jfc!
fc=0
eX6{a )eX6{b).
Thus for each A G C, ext is an automorphism which is continuous if 6 is. We vsill further investigate the case in which 6 is continuous. It is clear that j h* e(< is a continuous group homomorphism from the additive group of E : to the group of automorphisms of A in B (A ). In order to fully descri'3 3 this interesting situation we need more terminology. For related cer.c;;: ,s see Jerome A. Goldstein [1985]. 8.4.S Definition Let 6 be an algebra. A one-parameter group (of invertible elements) in B is a group homomorphism a from the additive group of R into the multiplicative group of B. If B is a topological or normec algebra, the terms continuous one-parameter group and bounded one-parameter group have the obvious meaning. Let a be a one-parameter group in a topological algebra B. When the limit limt_,o(Qt — 1 )/ t exists, it is called the generator of a. Of course, B is unital if there is a one-parameter group in B. We have checked that if £ is a continuous derivation of a Banach algebra .4, then t I—►et0 is a norm continuous one-parameter group of automorphisms of A in S (A ). As arguments from elementary calculus should suggest, the generator of this group is 6. The case in which B is B (X ) with its strong operator topology for some Banach space X is important. For our present purposes, it is enough to consider the case in which B (usually B (A ) ) is a Banach algebra. 6.'4.6 Proposition Let B be a Banach algebra and let a be a norm ccni'.r.v,oiu one-parameter group in B. (a) Then the generator of a exists. (b ) If the generator is denoted by 6 € B, then it satisfies a t = ets for all i f 1 . (c) 4 ny 6 G B satisfies 6 = limt_ 0(e<# ~ 1)AThus 6 *-» (t *-» elt) is a bijection of B onto the set of norm continuous one-parameter groups in B. Proof Since a is norm continuous, there is some £ > 0 such that |f| < £ implies ||1 - a t|| < 1. For any such t the series — —««)" converges to an element that we denote by S(t) = log(l —( l —a t )) = log(at).
596
6: Automatic Continuity
6.4.7
We note that any t G R and n G N satisfying |nt| < e must also satisfy 6(n t ) - log(ant) = log((Q( )n) = n log(a() = n6(t). Hence any m € N and any a satisfying |s| < e must also satisfy m ~1 6(s ) = 6(m - I a). This implies 6(r t) = r6(t ) for any rational number r satisfying 0 < r < 1 and any t satisfying |t| < e. However 6( - ) is a continuous function of t for |t| < e since t >—►Qj is continuous and the series defining 6( - ) converges uniformly. Hence 6( 3 t) = s6(t ) holds for all s € [0,1] and t satisfying |t| < e. Define 6 by 6 = s~l6(s ) for some (arbitrary) ,s p]0,e[. This implies a, =
(6)
for all t e]0,e[. Now fo. any t e R ,t/n belongs to ]0, e[ for some n G N. Hence (6) holds for any t € R. Thus 6 = limt_ o < - I (eW — 1) = lim(_ o ~ 1 ) holds. For any 6 € B, we have ..
et 6- \
I
*
lim — -— = lim I 6 + 1 > t-o t <-o \
r - 26n \
c
---- ;— I = 6. n! /
This, together with (6), proves that <5»—►( t »—►etS) is a bijection of B onto the set of norm continuous one-parameter groups in B. □ We mention a suggestive example for which continuity and convergence to the generator are not in the norm topology. Let A be C 00(R) and let a :R —►£ (C °°(R ))g be the translation map defined by (< *tf)(s) = f (t + s)
V / G C °°(R ); s, t e R.
Then the generator (in a generalized sense) of a is the map 6 € £ (C °°(R )) defined by ( 6f ) ( s ) = lim
= / '(«)
VaeR.
Thus 6 is the ordinary derivative map given as our first example of a deriva tion. This example points up the fact that Proposition 6.4.6 covers a very special case of a more general situation. In fact, the unbounded generators of groups and semigroups of operators continuous in the strong operator topology have received more attention than the norm continuous ones stud ied here. We will not explore the more general situation here. We turn now to the analogue of Proposition 6.4.6 in which we are chiefly interested. 6.4.7 Th eorem Let A be a Banach algebra. If 6 is a bounded derivation of A, then t >—►eu and t *—* e' t6 are norm mntinuous one-parameter groups of automorphisms of A in B (A ). If a is a norm continuous one-parameter
6.'*.p
Derivations
597
group of automorphisms of A in B (A ), then its generator is a bounded derivition of A. The maps S »-► (t *-» et6)
a ■—►(generator of a )
and
(7)
are rr utually inverse. They establish a bijection between A b (A ) and the set of norm continuous one-parameter groups of automorphisms of A. Proof Calculation (5) shows that 1 i-> ets and 1 1—* e 'ts are norm continuous one-parameter groups of automorphisms of A in B(A). Suppose, conversely, that a is a norm continuous one-parameter group of automorphisms of A in B (A ) and that 6 is its generator. Then 6 is a bounded linear map and for any a, b e A, we get
<(rf) v
_
=
Un, a W - g
<->o
t
4-.0
t-*o
t
t—0
t
a6(b) + 6(a)b.
Hencs 6 is a bounded derivation. Proposition 6.4.6 shows that the maps of (7) a."3 mutually inverse. Thus they obviously establish a bijection between ) and the set of norm continuous one-parameter groups of automor phisms of A □ 6.4.8 Prop osition A. Any t
Let A be a Banach algebra. Let a be an element of satisfy
€ R and any b e A
(8)
ets‘ (b) = etabe~ta.
Cone jrsely, if t *—> ut is a norm continuous one-parameter group in A and a e A is its generator, then 6a is the generator of the norm continuous one-r irameter group 1 1 —> (b Furthermore, the inclusions SpB(A)($a)
Q
{A - f l : A, /i € S p (a )}
S P B ( A ) ( a a)
C
{e * - " :
A,
fi
e
and
S p (a )}
hold ■ohere aa € B (A ) is the map defined by a a(b) = eabe~a
V b e A.
Frooj By induction one shows ( 6* ) n(b) = ' p ( - l ) k( j ^ a n- kbak
V b e A.
6: Automatic Continuity
598
6.4.9
A calculation parallel to that given in (5) above gives (8), establishing the first statement. The second statement follows from the first, equation (6) and the uniqueness statement in Theorem 6.4.7. Finally, we turn to calculating the spectra. Note that the second in clusion follows from the first together with the spectral mapping theorem, Theorem 3.3.7(b). Let a e A b e arbitrary but fixed. Note that La and R a commute where L and R are the left and right regular representations of A. Then Corollary 3.1.6 and Propositions 2.5.3 and 2.5.2 give SpB(A )(6a)
=
SPB (A)(La ~ -Ra) C SpB(A )(L a ) ~ SpB(A )(Ra)
=
SpA (a ) - SpA (a)
=
{A - n : A, fj, e 5 p ^ (o )}.
□
Having established the connection between bounded derivations and norm continuous one-parameter groups of automorphisms, we turn now to some simple properties of derivations before showing that any derivation of a semisimple Banach algebra is continuous. 6.4.9 P rop osition Let 6 be a derivation of an algebra A. Then (a) 6(e) = 0 for any central idempotent e. In particular 6( 1 ) — 0 if 1 is an identity element in A. (b) 6 maps the center of A into itself. (c) I f l i s any ideal satisfying X2 = X, then 6 maps T into itself. Proof (a): Any central idempotent e satisfies 6(e ) — 6(e2) = S(e)e + eS(e) = 2e<5(e). Multiplying by e and subtracting gives e6(e ) = 0. Hence 6(e ) = 2e6 (e ) is zero. (b): Any central element z and any element a satisfy 6(z)a = 6(za) — z6(a ) = 6(az) - 6(a )z = a6(z).
(c): Finally if a belongs to I = J 2, then there are elements Oj, bj £ I
n ctjbj. Thus 6(a) = ^ (d(aj)bj + aj6(b j)) belongs to I }=t
satisfying a = again.
D
We mention an interesting connection between derivations and the func tional calculus. Let 6 be a bounded derivation of a unital Banach algebra A. Let a be an element of A commuting with 6 (a) and let / belong to T a, so that /(e) is defined by the functional calculus. Then we get 6( f (a ) ) = f '( a ) 6(a).
To prove this, note first that any A € C \ Sp(a) satisfies
0 = 6(1) = 6((A - a)(A - a)-1 ) =
- 6 ( a ) ( X - a ) - 1 + ( X - a ) 6 ( ( X - a ) - 1)
Derivations
599
This implies <5((A — a) *) = (A - a) 26(a). Hence for any u € .4*, we get
w (6(/(a))) =
/(A)(A ~ a )~2dA)’
where Rg(a) is t,he adjoint of the right regular representation Rs(a) of 6(a). Expanding the integrand in the neighborhood of each A € T and applying the residue theorem and the Hahn-Banach theorem, we obtain the desired result. Let A be an algebra and let 6 be a derivation of A. Then 6 belongs to the algebra £(.4) of linear maps on A . We may consider the inner derivation 6s of £(.4) determined by 6. For any o G A, L a and R a belong to C (A ). The obvious computation shows 6s(L a) = C S(a)
and
6s( R a ) -
Rs{a)
a € A.
(9)
This computation is the main part of the proof of the next simple result which shows that any derivation of a “good” algebra A is “almost” an inner derivation when extended to 2?(«4). For some algebras which have outer derivations (e.g., simple nonunital C*-algebras) the “almost” can be dropped. In this case T>(A) is an algebra with only inner derivations. 6.4.20 Prop osition Let A be an algebra and let 6 be a derivation of A. Let'O: A —*' T>(A) be the regular homomorphism and let 6s be the derivation on 'D 'A ) defined by 6S(L, R ) = ( 6f L, 6sR ) = ( 6L - L 6, 6R - R 6 )
V (L , R ) € V (A ).
Then 6 satisfies (a) 6 o 6 — 6s o 0 . (b) 6 maps the annihilator ideal A a into itself. (c) When Aa vanishes and A is regarded as embedded in V (A ) , then 6s 'is1a' derivation o f V (A ) which extends the derivation 6 of A (d) If A a = { 0} and A 2 = A both hold, then the map 6 >-* 6s is a linear bijection of A ( A ) onto A ( V ( A ) ) . Proof Result (a) is a restatement of (3). Since ker($) = A a , (b) follows from (a). (The calculations 0 = 6(ab) = 6(a)b + a6(b) = a6(b) and 0 = h(b?,) = 6(b)a + b6 (a) = 6(b)a for any a € A and b £ A a prove (b) directly.) For any a, b G A and (L , R ) 6 T>(A), we get 6s (L )(a b )
=
6(L (a b )) - L ( 6(ab))
=
6(L (a )b ) - L ( 6(a)b + a6(b ))
=
6(L (a ))b + L (a )6(b) - L (6(a ))b - L (a )6(b)
=
( 6L - L 6)(a )b = 6s(L )(a )b .
600
6: Automatic Continuity
6.4.11
Hence 6t (L ) is a left centralizer. Similarly Sg{R) is a right centralizer. With the same notation we also find aSt (L )(b )
=
a((SL — L6 )(b )) = a6 (L (b )) — a L(S (b))
=
a6 (L (b )) - R (a )6(b) + 6(R {a )b ) - S(aL(b))
=
6(a )L (b ) - 6(R (a ))b = R (6(a ))b - 6(R (a ))b
=
S6(R )(a )b .
Hence 6g maps V ( A ) into V (A ). It is a derivation since it is the restriction of the inner derivation determined by 6. These facts together with (a) prove (c). Suppose A a = {0 } and A 2 = A both hold. Regard A as embedded in V (A ). Then any derivation of V ( A ) maps A into itself by Proposition 6.4.9(c). Thus restriction to A maps A ( V ( A ) ) into A (.A) and this map is the inverse of 6 ■—►6f. Hence the latter linear map is a bijection of A (.A) onto A (P («4 )). □ The next proposition is essentially due to Ivan Vidav [1955], Feliks Vladimirovic Sirokov [1956] and David C. Kleinecke [1957], independently. 6.4.11 Prop osition Let (A , ||■||) be a normed algebra and 6 a continuous derivation on A. If 6(a) commutes with a for some a € A , then ||6(a)H00 0. Proof For a, b e A suppose 6t,(a) = —Sa(b) commutes with a. Then ( 6a) 2(b) — 0 holds and thus Leibniz’ formula (2 ) shows 62 (b2) = 2( 6a(b ))2. By induction, we will prove 6£(bn) = n!(60(6) ) n for all n € N. Leibniz’
formula and the induction hypothesis give
%(bn)
ll«* (a )ir
=
^(bn~l)b + n S r 1(bn-1)Sa(b)+ 0
=
tfa((n - l)\ (6a(b ))n- l )b + n ((n - l ) ! ) ( 60(6) ) " - ^ 0(6)
=
0 + n!(6a(6))"
=
\\<>a(b)\r = nlim ||M&)n||I/n —»oo
<
and
Um (n !)- 1/,,||60(6)|| = 0.
f l —*00
Thus we have proved the proposition for any inner derivation Si,. However if 6 is an arbitrary derivation of A and S(a) commutes with a for some a 6 A, then (3) shows that St(La) = J^«(o) commutes with L a. Hence H-I^o)!!00 = 0 holds by the first paragraph of this proof. This implies \\S(a)\r = 0, since ||(6(a) ) "+ 1 |l”
< ll(^(a))||’* ‘ ||«5(a)||n"
holds.
□
The next important result is due to Isadora M. Singer and John Wermer [1955]. See also Kaplansky [1958a], [1958b]. As they noted, it implies that
Derivations
601
there is no complete algebra norm on C °°(R ). For an extension of this result see C lang Pao Chen [1978]. Marc P. Thomas [1988] verified their conjecture that ' le word “continuous” could be removed from the hypotheses. We will outlire his quite technical proof below. 6.4. 1 2
C orollary The range of any continuous derivation of a commu tative spectral normed algebra is included in its Jacobson radical.
Proof For a commutative spectral normed algebra, 116(0)11°° = 0 implies that C(a) belongs to A j by Theorem 4.5.8. □
T'.e original proof for Banach algebras given by Singer and Wermer [1955] is so attractive that we will give it here although it is less elementary than ,he proof above. After Proposition 6.4.16, we will indicate still another attractive proof which is due to Sinclair [1969]. Let 6 be a continuous derivation of the commutative Banach algebra A. Then ext is an automorphism for each A € C by the calculation (5). Hence, if 7 is an element of (i.e., any homomorphism of A onto C), then 7 o ext is also an element of Theorem 3.1.5 shows that |7 (eAS(a))| < ||a|| holds for each a e A. Thus A *-» j ( e xs(a )) is a bounded analytic function, so it is a constant. This implies 0 = lim,\-.o A- 1 [7 (eA*(a )) — 7 (a)] = 7 (6(0)). Since 7 € r * was arbitrary, Theorem 4.5.8 implies that a belongs to A j. Therefore 7 maps A into A j. We remark that this proof for commutative Banach algebras can easily be extended to commutative spectral normed algebras. Suppose 6 is a bounded derivation on a commutative spectral normed algebra A. Let ^ be the completion of A and let 6: A - * A be the extension of 6 by continuity. Clear y 6 is a derivation of A so that 6(^t) C A j by the proof just given. Hence 6(A) is included in A D ^ ij. However A D jI j is an ideal in A which consists of topologically nilpotent elements. Hence An^& j C A j holds by Theo: am 4.3.6(b). Thomas [1988] settled the more them thirty year old conjecture of Singer and Y/ermer [1955] that the last corollary remains valid even for not nec essarily continuous derivations. Martin Mathieu and Volker Runde [1992] show that if 6: A —* A is a derivation on an arbitrary Banach algebra sat isfying [a, 6(a)] € A z for all a € A, a centralizing derivation, then its range is in the Jacobson radical. See also Mathieu and Murphy [1992] and Yood [1984a], 6.4.13 Th eorem Let A be a commutative Banach algebra. The range of any derivation 6: A —* A is included in the radical of A. Outlv. -.e of Proof Since a derivation always sends idempotents to 0, we may assume that A is unital. In [1969a] Johnson showed that if this result fails for e. >ommutative unital Banach algebra A, then there is a finite nonempty set of orthogonal idempotents {e i, e2, ■■■ ,en} such that the restriction of 6
602
6: Automatic Continuity
6.4.14
to B = (1 —ei —e2 --- — en) A has range in B j. Moreover 6 sends each ekA into itself and each of these algebras has the form 72* where 72. is a radical algebra. Hence, if the theorem fails, there is a derivation 6:721 —* 721 and an element b € H with 6(b) 0 72. Thus 6 (b) is invertible. Let its inverse be c. Then 6 = c6 is a derivation on 721 with 6(b) = 1 for some radical Banach algebra ft and some 6 € 72. It is enough to show that this cannot happen. In the notation of 3.4.10, let T> — 'D {b ,'R }) be the largest 6-divisible subspace of H 1. The proof shows that b is not nilpotent and that for some to 6 N the closure V of the ideal V coincides with the closure of bm'R} / {0 }. The quotient algebra 'R } / V is isomorphic to the algebra C[[z}} o f formal power series over a certain unital commutative algebra C. The isomorphism sends b to z and sends the derivation 6 to the ordinary derivative of the formal series. Thomas discovers intricate structure in this algebra which he calls recalcitrant systems. They preclude the possibility that 6(b) = 1, thus concluding the proof. See the original paper, Volker Runde [1990] and [1991] or Dales [1994] for full details. D The following important corollary is due to Singer and Wermer [1955] for bounded derivations. We prove the result for arbitrary derivations. It was first proved in this form by Johnson [1969a]. See also Curtis [1961] where it was established for completely regular algebras. 6.4.14 C orollary algebra is zero.
Any derivation of a commutative semisimple Banach
Proof This follows immediately from Theorem 6.4.13 or from Corollary 6.4.12 and Theorem 6.4.18. □
6.4.15 C orolla ry Let A be a Banach algebra with a semisimple center. Then any derivation is zero on the center. Hence any norm continuous one-parameter group of automorphisms leaves each element of the center fixed. Proof Proposition 6.4.9(b) shows that a derivation of A maps the center A z of A into itself. Hence the restriction of the derivation to A z is zero by
Corollary 6.4.14. This proves the first statement. The second then follows from Theorem 6.4.7. □ We turn now to a result of Sinclair [1969). The proof is similar to the proof of Kleinecke [1957] given in the proof of Proposition 6.4.11. We also obtain a third proof of Corollary 6.4.12 from it. 6.4.16 Prop osition Any continuous derivation of a spectral normed al gebra A maps each primitive ideal of A into itself. If A is a semisimple
Banach algebra, this is true of any derivation.
Derivations
603
Proof Let 6: A —* A be a derivation and let P be an ideal. First we show by induction on n that any b G P satisfies 6n(bn) + P = n ](6(b ))n+ P . The case 1 !s trivial. For any 0 < j < k repeated application of Leibniz’ formula shows 6J(6fc) can be expressed as a sum of products (b)&* (b) ■■■6^>(b), where each jk is a non-negative integer. Since some ji must be zero in eec’’. product, & (bk) belongs to P . Now Leibniz’ formula and the induction hypothesis show 6n+1 (bn+1) + P
=
^ n(6n6)) + P = 6( ^ ( ,n) ^ ( 6" ) 6n~J'(6)) + P j =0
=
£
( " ) [ ^ ( b ^ - ^ b ) + 6’ (bn) 6n- i + 1 (b)} + V
j —0
=
n6n(bn) 6(b) + 6” (bn) 6(b) + P
=
(n + 1) 6n(bn) 6 (b) + P = (n + l)!(£ (6))n+1 + V.
If 6 is continuous, Stirling’s formula gives for each b € P
11(6(6) + f > n r “ = [(n !)_1||6"(6") + P ll]"-1 < (e/n)H«|| ||ft||. Hence each element of ( 6 ( P ) + P ) / P is quasi-nilpotent. The definition of & de:: ation shows that ( 6( P ) + P ) / P is an ideal. Hence ( 6( P ) + P ) / P is a quas!-regular ideal and thus is contained in the Jacobson radical of A / P by Theorem 4.3.6. Finally, if P is primitive ( 6( P ) + P ) P = {0 } so 6 maps V into itself. The second statement follows from Corollary 6.4.19. □ Here is the third proof of Corollary 6.4.12 which was promised above. Let 6 be a continuous derivation on a commutative spectral normed algebra. We replace A by ,4*, defining 61 :^41 —» A 1 by <5*(A + a) = 6(a) for ail A + c € A 1- (If A is already unital, this is consistent by Proposition 6AC;£,).) N o w Theorems 4.1.9 and 2.4.12 show that A / P is isomorphic to C for each primitive ideal P . Hence any element a € A can be written as A -(- b with A € C and b £ P . Thus 6 (a) = 6(b) belongs to V by Proposition 6.4.16. Since V was an arbitrary primitive ideal, 6(a) belongs to A j . Since a was arbitrary, 6 maps A into A j . ' We now turn to Johnson and Sinclair’s proof [1968] that every derivation of a semisimple Banach algebra is continuous. They actually prove more. Since we see only a small simplification in proving Corollary 6.4.19 directly of deriving it from Theorem 6.4.18, we follow Johnson and Sinclair’s original procedure. Thus for the next few results we will deal with a map b\ A —> A which satisfies 6(a + b) = 6(a) + 6(b) and 6(ab) — 6(a)b + a6 (b) for all a, b € A, but which is not necessarily linear. We call such a map an additive derivation. O f course, an additive derivation is linear over the ratio’id numbers, and it is therefore bounded if and only if it is continuous. Coniii uity also implies real linearity.
604
6: Automatic Continuity
6.4.17
Theorem 6.4.18 and Corollary 6.4.19 were conjectured by Kaplansky [1958b], He also conjectured Corollary 6.4.19 for the special case of C*algebras [1958a], The latter conjecture was verified by Shoichiro Sakai [I960]. We will prove this special case directly in Volume II. 6.4.17
Lem m a Let A be a Banach algebra and let 6 be an additive derivation on A. Let T be a bounded representation of A on a Banach space X . (a ) I f T i s irreducible and X is infinite-dimensional, then the map A i-> b {A )x
A €A
is continuous on A for each x € X . (b) If T = ®^L1 7m ts the direct topological sum of infinitely many irreducible subrepresentations on finite-dimensional T-invariant subspaces { X n : n G N } and no pair of the T " are equivalent, then the above map is continuous for each x in each X n except for a finite set of n € N. Proof In order to avoid large numbers of double subscripts, we suppress the symbol T and denote elements of A by capital letters as if they were operators on X. (This notation is not logically defensible since T may have a nontrivial kernel I , and it is not clear a priori that T is 6-invariant. (After Corollary 6.4.19 haf been proved, Proposition 6.4.16 will show that J is 6-invariant.) The reader can check that no confusion arises.) The proof is by contradiction.
In case (a), suppose xo € X is a non-zero element such that A I—►6(A)xo
is continuous on A. Let x be any other element of X. Since T is irreducible, there is an element B G A such that B x o = x. Hence A >—> 6(A )x — 6(A )B x o = 6(A B )x o - A 6(B )x o is continuous on A . Thus if j4 i—►6(A )x is discontinuous for any x G X , it is discontinuous for all non-zero x € X. We will derive a contradiction from assuming A i—►6 ( j4 ) x is discontinuous for all non-zero x. This will prove the lemma in case (a). In case (b), the above argument shows that A —» 6(A )x is discontinuous for all non-zero x € X n if it is discontinuous for any xq G X n. For the sake of proof by contradiction, we may therefore assume that A <5(A)x is discontinuous for each non-zero x in each X n. (Any X n for which this does not hold is simply omitted from the list.) Since the X n are all closed and .4-invariant, it follows that A S (A )x is discontinuous for all non-zero x € X. In case (a ) we choose a sequence {x „ }„ e N of linearly independent unit
vectors. Corollary 4.2.14 shows that we can choose a sequence of elements
Derivations
6.4.::'
605
{C n} „ eiN in A satisfying C nC n- 1 • • •C ix n_ i = 0 { C nC n- 1 • • •C\Xj : j > n } is linearly independent
(10)
(The main point of this choice is simply to make C„C n_ i • •■C\xn non-zero for each n € N.) In case (b) we will find {x n} n€N in X and {CnJneN *n satisfying these same conditions. Choose each xn to be a unit vector in X n. For each n € N, consider the image of A under T 1 ® T 2 ® ••• ® T ". This is a subalgebra of B (X l ) ® B (X 2) ® ■• • ® B (X n) which includes each direct summand. Hence by Wcdderburn’s Theorem (Theorem 8.1.1), it is isomorphic to a direct sum o ' the algebras B (X k). Since the T k are all inequivalent, it must be the direct sum of all of them. Hence we may choose an element C n in A such i lat C nXj = 0 for j < n and the restriction of C„ to X n is invertible. Clearly these C „ can be chosen to satisfy ||C„|| < 2“ ". Then they satisfy all the conditions of ( 10 ). By assumption A —►6(A )x is discontinuous for each non-zero x € X , so we can choose elements A\, A 2, ■■■ in A satisfying ||>U < m in{l + |M C j C j ^ • • •C 0 H "1 : j = 1,2,..., n} H lS(An)C nC n. l - C i x n\\ > n + £
(11)
M A j C j C j - ! • • -COII-
j= i The restrictions on the norms of the C „ and A n allow us to define elements arc D n in A by
D
OO
(12) i =i OO
Dn
—
^ n + l
^ 3
A . j C j C j - 1 • ■•C n+ 2.
j= n + 2
These elements satisfy IIA.II \\6(C n+iC n ■■■Ci)|| < 2
(13)
S ince OO
oo
<
( l + ||«(Cn+1 Cn - - . C1 )||)-1 (l+
53 j= n + 2
\\Cj Cj . 1 . - - C n+2\\)
6: Automatic Continuity
606
6.4.18 OO
<
( l + ||6(C n+1 Cn . . . C I )||)-1 ( l +
£
2 ~3)
j=n + 2
holds. For any n > 2 we have D
n—1 ' A j C j C j - i •• •C 2C j ) + AnCnCn-i •• •C 2C\
—
+ D nC n+ iC n • ■■C 2C\
which implies n—1 6(D )x n
=
S (J ^ A i Ci C i . l - - - C 1 )x n + S[An)C nC n. l - - - C 2C ix n
1 + 6{D n )C n+ iC n • • C iC \xn + A „ 6(C nC n~i ■••C 2C i )x n + D n6(C n+1 C n • • • C 2C i )x n.
However, equations (11), (10) and (13) give ||.4„6(Cn •• •C2C i)x n|| < 1, 6( D N )C n+ 1 - C 2C lxn = 0 and \\Dn6 (C n+l ■■■C 2C { )x n\\ < 2 . These in
equalities together with ( 1 1 ) give ||6(Z?)||> ||6(£>)*„||>n-3 This contradiction proves the lemma.
V n > 0. □
In the next theorem it is not assumed that A is unital. The notation (1 — e )A simply denotes the set {a — ea : a £ A ). 6.4.18 Th eorem
Let A be a semisimple Banach algebra, and let 6 be an additive derivation of A. Then A contains a central idempotent e satisfying: (a) eA and (1 —e )A are invariant under 6 . (b) 6 restricted to (1 —e )A is continuous. (c) eA is finite-dimensional.
Proof For each primitive ideal V of A, let ip-p: A —* A / V be the natural map, and choose an irreducible representation T v of A on a linear space X-p with kernel V. Theorem 4.2.8 shows that X-p can be given a Banach space norm so that T v is continuous. Assume for a moment that a *-* Tf(a)x
a€ A
is continuous for each x G X. Then this rational linear map is real linear. Thus (tT ^a) - Tf^taj)x = 0 holds for all t G R, a G A and x G X. This implies tT^a^ — T ^ta^ = 0, so tS(a) —S(ta) belongs to V for each t G R and a e A Hence tp-p o 6 is real linear. We will now prove that it is continuous
6 . 4 *? 9,
Derivations
607
by the closed graph theorem. Suppose {a n} „ eN C A converges to zero and {rp-p(6(an) ) : n € N } converges to b + V in A / V . Then all x € X v satisfy TTx - lim T ^a^ x = 0, so = 0. Thus b + V = V holds. Hence rpp o 6 has a closed graph and is therefore continuous. " ' A / V is infinite-dimensional, then X-p is infinite dimensioned (since >./■' i3 isomorphic to a subalgebra of B (X p )). Hence Lemma 6.4.17(a) s’:c a 3 that our assumption above (on the continuity of the map a »—►T ^ a^x) is correct in this case, so ip-p o 6 is continuous. Next we consider the case where A / V is finite-dimensional. I f our as sumption (on the continuity of the map a >—> T^a^x) were false for infinitely many choices of V , we could construct T = ® as a direct topological sum contradicting Lemma 6.4.17(b). Thus our assumption above is correct except for finitely many such V ■ Let V i , V 2, ■■■,V n be these exceptional coSmte primitive ideals. Let Vo be the intersection of all primitive ideals of A such that ipj> 06 is continuous and hence real linear. Let ipo:A —» A / V o be the natural map. Ther. rpo° 6 is real linear, and a now familiar closed graph argument shows that it is continuous. If Vq is {0 }, then the theorem is proved with e = 0. Hence we assume Vo is not zero. Theorem 4.3.2 and the semisimplicity of A show that the ideal Vo is semisimple. Also {0 } = fl"=0P j = r\"=x(Vo n V j ) holds. Thus the map b >-> (b+VonVi b+v 0nv2, • • •, b+VoCtVn) is an injective homomorphism of V 0 |cto V o /{V 0C\Vi) x V 0/ ( V 0n V 2 ) X • • ■x V o/iVoH Vn ) which is isomorphic to {Vo + V i )/ V i x (V () + V 2 ) / V 2 x • • • x (Vo + V n) / V n which, in turn, is a subalgebra of the finite-dimensional algebra A /V \ x A / V 2 x ••• x A / V n. Hence Vo is a finite dimensional semisimple algebra. Theorem 8.1.1 shows that Vo contains an identity element e. For any a £ A, ea and ae belong to Vo and thus satisfy ea — eae = ae. Thus e is a central idempotent so 6( e) = 0 follows from Proposition 6.4.9 (which obviously does not depend on linearity). Therefore 6(ea) = 6(e)a + e6(a) — e6(a ) holds, so eA = Vo and (1 - e )A are 6-invariant. The natural map tpo'- A —* A /V o induces e continuous isomorphism 0 of (1 — e )A onto A / V o- The open mapping theorem shows that G~x is also continuous. The restriction of 6 to (1 —e )A is continuous since it can be expressed as the composition 9~l o (rpo o 6 ) of the two continuous maps 9~l and rpo 0 6. □
,
'An additional argument proves that 6 is actually complex linear on ( l * - e)A. We refer the reader to the original paper for this (Johnson and Sinclair [1968]). C orollary
A derivation of a semisimple Banach algebra is con
tinuous.
Since e A is finite-dimensional and a derivation is linear, its restriction
608 to e A is continuous.
6: Automatic Continuity
6.4.20 □
Small modifications of the proofs of Lemma 6.4.17 and Theorem 6.4.18 show that these results hold for additive left centralizers on semisimple Banach algebras. In fact such a map L on A is continuous on all of A since L (a ) = L(ea ) = L (e )a for any a € eA. We have already noted that such maps are linear. We will now prove the theorem of Jewell and Sinclair [1976] on the conti nuity of derivations on certain not necessarily semisimple Banach algebras. The hypotheses on the algebra are exactly the same as in Theorem 6.1.18. See Example 6.1.19 for a description of some Banach algebras to which this theorem applies. 6.4.20 Th eorem
Every derivation of a Banach algebra A is continuous if A satisfies: (a) A has no non-zero finite-dimensional nilpotent ideals. (b) For each closed infinite-dimensional ideal I of A , there is a sequence { “ nJneN - A such that the sequence } „ eN of closed right ideals is constantly decreasing.
Proof Let 6 be a derivation of A and let As be the separating space of Definition 6.1.7. Propositions 6.1.8 and 6.1.9 show that As is a closed linear space which equals {0 } if and only if 6 is continuous. If a € As and {onJngN are as in the definition and b € A is arbitrary, then the limits ab = lim^OnJb + On6 (b )) = lim 6(anb) and ba = lim(6(6)an + b6 (an) ) lim^fbon) show that As is an ideal. Suppose As is infinite-dimensional. Choose a sequence satis fying condition (b) in the statement of the theorem. Lemma 6.1.17 applied to X = y = A, ip = 6, Sn(a) = Tn(a) = a„a for all a € A gives a contradiction. Hence As must be finite-dimensional. Therefore, the linear map 6 is continuous when restricted to As- We will show that A j is zero. Let a, b € A t be arbitrary and let {a n} „ eN be a sequence converging to zero such that { 6(On)}neN converges to a. Then we have ab = lim6(an)fc = lim(<$(an£>) —a„6 (b )) = 0. Since the finite dimensioned ideal A t is nilpotent, it is zero. Hence 6 is continuous by Proposition 6.1.9. □
We remark that all derivations are known to be continuous on certain other non-semisimple Banach algebras. Loy [1970a] proves this for certain Banach algebras of formal power series. Loy [1970a], John A. Lindberg, Jr. [1971], and Julian M. Cusack [1976] prove the continuity of every derivation on an Arens-Hoffman extension of a commutative Banach algebra with the same property, and in particular on an Arens-Hoffman extension of a commutative semisimple Banach algebra (cf. Example 3.4.13). Cusack [1976] shows that if there is a discontinuous derivation on a semiprime
8.4.2'
Derivations
609
Banach algebra, then there is a topologically simple radical Banach algebra. It is not known whether such a Banach algebra exists. See Examples 6.4.24,
6.4.25 and 6.4.26. The next result is due to Georges Zeller-Meier [1967]. The proof uses an argument due to L. Terrell Gardner [1965]. Jacques Dixmier [1957], p. 314, gives a similar result, where a is assumed to be an isometry and the constant \/3 replaces 1 . He attributes the proof to Jean-Pierre Serre. A similar result for *-automorphisms of C*-algebras is due to Richard V. Kadison and John R. Ringrose [1967]. Johnson [1975], p. 88 shows by a three-dimensional example that Dixmier’s result [1957], which states that an all omorphism a satisfying Sp(a) C {A € C : arg(A) < 27r/3} is the exponential of a derivation, is the best possible result of its type. 6.4.2"
P rop osition Let a be a continuous automorphism of a BaXiach igebra A. If a has spectrum in the open right half plane or satisfies ! ! « - - '!! < l, then a belongs to a norm continuous one-parameter group of automorphisms.
Proof If ||a - /|| < 1, then the spectrum of a lies in the open right half plane, so we need only consider the latter hypothesis. If we extend a to A 1in the natural way (i.e., a(A + a) = A + a(a )), then the spectrum of a still satisfies the hypothesized restriction. Thus we may assume that A is unital. Le•; 6 = L og(a ), where Log is the principal branch of the logarithm function. We will show that 6 is a derivation so that a = es lies on the norm continuous one-parameter group (t •—» ets). Let 0 :B (A ) —* B (A ) be the automorphism defined by
/?(T) = aTa - 1
V T € B(A).
For any a, b € A, we get 0 (L a)b — a L 0a _ 1 6 = a (a a _ 1 (6)) = a(a)b = L a(a)b. Hence L_\ is a closed invariant subspace of (3. Note that /? is exp(Sf), where &s £ B ( B (A )) is the inner derivation defined by 6 € B(A). The s jectral restriction on a shows Sp(6) C {A € C : |Im(A)| < n/2}. Proposition 6.4.8 shows Sp(6s) C Sp{Ls - Rs) C {A € C : |Im(A)| < n }.
This implies Log(/3) - Log(exp(6$)) = 6s so L * is invariant under Log(/3) = 6g. Hence for each a € A, there is an element a' € A satisfying 6gLa = L a>. For any b € A, we get a'b — L a>b = 6sLab — 6(ab) — a6 (b), so in particular for b — 1 we have a' ~ 6(a) — a<5(l). Thus if we show <5(1) = 0, then k satif fies the derivation equation. Use Runge’s theorem (Theorem 3.3.3) to approximate Log by polynomials {p„}neN on co (S p (a )). Then (5(1) = L o r(r)(1 ) = lim pn( « ) l = lim pn( l ) = L o g (l) = 0. Thus 6 is a derivation, so a = es belongs to {e tf : t € R }. D
610
6: Automatic Continuity
6.4.22
In the above proposition, if a satisfies ||a — /|| < 1, then Log(a) can be defined by the series expansion in powers of a — / as in the proof of
Theorem 3.4.4. Then Runge’s Theorem would no longer be needed. Next we give an analogue of Theorem 3.4.2. 6.4.22 Corollary Let A be a Banach algebra and let G be the topological group of continuous automorphisms of A in the norm topology of B (A ). Then the principal component Gq of G is the subgroup generated by {e 6 : 6 € A b (-4 )} and G /G q is discrete. Proof By the proposition the group generated by {e 6 : 6 € A B(.A)} is an open subgroup. It is obviously connected since t *-» etS is an arc connecting e6 to the identity automorphism. Hence it is the component. □ Examples of Discontinuous Derivations and Homomorphisms
Let A be a commutative Banach algebra and let Af be a Banach A bimodule. Then X is said to be a commutative bimodule if it satisfies ax = xa
Vx£^;ae A
The algebra A ® X defined before Proposition 6.4.3 is commutative if and only if both A and X are commutative. Let 6: A —♦ X be a discontinuous derivation. We will consider the inequivalent algebra norm ||• ||a on A (B X introduced in Proposition 6.4.4. Note that ||a||« is just the norm of A (B X pulled back through tpt- Since the graph of b is just the range of ipt, it is also the graph norm of 6 in A x X where X has the norm ||| • |||. We will denote the completion of A under the norm ||• ||s by A?. Rather than considering the abstract completion, we will take this Banach algebra to be the closure of the graph of 6 in the Banach algebra A d ) X . The next theorem is due to Loy [1974a]. 6.4.23 Theorem Let A be a commutative Banach algebra and let X be a commutative Banach A-bimodule. Let 6: A —* X be a non-zero derivation which is zero on a dense subset of A. Then the Banach algebra A inequivalent complete norms.
has two
Proof The definition of the separating space Xt shows that it is included in the closure <5(.4)~ of the range of 6. We now show that the two spaces are equal. An element x G X is in 6 ( A) ~ if there is a sequence {6n}neN Q A with 6(bn) converging to x. By the density of ker(6) we can choose a sequence {an }n€N Q ker(6) with a„ = bn — c„ converging to zero. But the sequence { a „ } neiN shows that x e Xf. Next we show that the equality of X6 and 6( A) ~ implies the equality of _t —i A ( B ( X s ) and A . The inclusion A = (graph of 6) C A ( B 6 ( A ) ~ = A ® X f is obvious. However, any a ® x € A ® Xf can be written as a ® x =
Derivations
3.4.24
611
c © J(a) + O ffi(i - 6(a )) and both terms belong to the linear space (Graph of 6'~ since x — 6 (a) belongs to 6(A )~ = X&. '" ’i 3 equality of 6(A )~ and A(&X6 shows that 6(A )~ is a Banach algebra under both of the norms ||• ||A and ||• || (restricted from A (B X ). Since 6 is non-zero with dense kernel, it is discontinuous and hence these norms are -equivalent. □ 6.4.24 A Specific Exam ple This example, due to Loy [1974a], satisfies the hypotheses of the above theorem. Let A be the disc algebra introduced in §1.5.2 and discussed further in §3.2.13. Let X be a Banach space and let T £ B (X ) have norm at most one. If a € A has the power series expansion Qnz", then the module action is given by OO
ax — xa — ^T^a„2
~"T"*(x)
V
x £ X.
n=0 This makes X into a commutative Banach .A-bimodule since all a € A and x £ X satisfy ||oz||
=
<
OO
OO
n—0
n—0 l/2 /oo
£ |a„|2-"||x|| < £ |a n|2 -"/22 -"/2||s|| E
w
22-"
n —0
£ /
X 1/2 2“
\ n —0
2tt
<
l- j |a(2-1/2e<e)|2^ v /2||x|| < V2||a||||*||
Note that if we denote the function A ■—►a(A/2) for A € 2D by a, then our module action is ax = a (T )x V a € A\x € X in terms of the analytic functional calculus. To complete this construction, we use results of Dales [1973] discussed briefly in example 3.4.11. Let X be C ([0 ,1 ]) and let T be the indefinite integral operator or Volterra operator:
■ ■
T f (t ) =
J
t
f(s )d s
V / £ X\t £ [0, 1 ].
o In' the reference just cited, a non-zero linear map (3 is defined from the spacs To of functions analytic in a neighborhood of D into C ([0 ,1]) which varMehes on the set of polynomials and satisfies 0 (f g ) = f (T )P (g ) + g (T )p (f )
v /, g e
612
6: Automatic Continuity
6.4.25
Define 6 by 6(a ) = 0 (a ) for all a € A. Then 6: A —* X is a derivation which vanishes on the dense subset of polynomials. Thus we have satisfied all the hypotheses of the theorem and hence have constructed a commutative Banach algebra with two distinct norms. The construction given has other interesting properties which are discussed in Loy [1974a], It is shown that for any particular non-algebraic function / in Tq (such as the exponential function), 0 and a € A may be chosen so as to make f (a ) have different meanings in the functional calculi belonging to the two norms || • || and ||• ||4 on A 6. This implies that the radical of A is infinite-dimensional by Theorem 3.4.12. The paper cited above contains further related results. All bimodule derivations on certain algebras are shown to be continu ous in Ringrose [1972] and Bade and Curtis, Jr. [1974], [1978] and [1984]. Because of point derivations, it is obvious that this can only happen on a commutative unital Banach algebra when M 2 has finite codimension for each maximal ideal M . The [1978] paper shows that in a separable algebra this condition and the absence of prime ideals of infinite codimension are necessary and sufficient. Bade and Dales [1989a], [1989b] give interesting partial results in algebras of power series and weighted convolution algebras. See also Loy [1973], Jewell [1977], Fereidoun Ghahramani [1980], Dales and McClure [1981], Willis [1983a], [1986], Ramesh V. Garimella [1987], [1991], Viet Ngo [1988], J. Vukman [1988], Niels Grpnbaek [1982a] [1989a], Kil Woung Jun, Young-Whan Lee and Dal-Won Park [1990] and Brian Forrest [1988], [1992a], Cusack’s Results on A utom atic Continuity The following discussion is taken from Chapter 2 of Cusack’s thesis [1976]. The basic method used is an extension of the argument used to prove Theorems 6.1.18 and 6.4.20 in the present chapter. In the course of these proofs it is shown that certain separating ideals in a Banach algebra A satisfy the following conditions.
6.4.25
If K } ngN Q A is any sequence, then the decreasing sequence of right ideals (<Ji
6.4.26
Derivations
613
has a non-nilpotent separating ideal. (5) There is a discontinuous homomorphism of a Banach algebra onto a serr.iprime Banach algebra. ;■ ) There is a semiprime Banach algebra which fails to have unique norm topology. (f) There is a derivation of some Banach algebra with a non-nilpotent separating ideal. (6) There is a discontinuous derivation of a semiprime Banach algebra. (7) There is a derivation of a commutative Banach algebra with range not contained in the Jacobson radical. (8) There is a derivation of some Banach algebra which fails to map each primitive ideal into itself. W hether such an algebra exists remains unknown. However, it is shown that ‘ 2) and (3) are equivalent and that (6), (7) or (8) implies (5). 8.4.23 Derivations on radical commutative Banach algebras We give some interesting examples of continuous non-zero derivations on certain radical commutative Banach algebras. Let A be the disc algebra with convolution multiplication which was described in Example 4.8.3. Define 6: A —* A by ( 6f ) ( z ) = z f (z )
V / 6 A.
Any /, g € A and z € D satisfy X
H f * g) (z)
=
zf * g(z) = z
J fWg(z
- w)dw
0 z
=
J
=
W f )* f f )(z ) + (/ * S (g ))(z )
w f(w )g (z — w)dw +
J
z
f {w ){z — w )f (z — w)dw
so thi t 6 is a derivation on A which is obviously bounded with norm one.
Note that the same construction can be applied to the convolution alge bra ?>([0,1]) introduced in Example 3.4.10. In fact, each of these examples is a special case of the general results on derivations of Banach algebras of power series presented in Sandy Grabiner [1974]. This paper determines all t ’’ ; derivations on each Banach algebra in a rather large class of radical Banach algebras of power series. See Example 3.4.14 above for notation. If h is a formal power series with zero constant term, then g >-> g o h (formal composition of power series) defines an endomorphism of C[[z]] and if k is any formal power series then g >-* kg' (formal differentiation followed by formai power series multiplication) is a derivation of C[[z]]. Suppose A is
614
6: Automatic Continuity
6.5.1
a Banach algebra of power series invariant under these operations. Then it is fairly easy to see that the restrictions of these maps are continuous endomorphisms and derivations of A and that all continuous endomorphisms and derivations of A have this form. These simple results are extended in various ways in Grabiner [1974]. Since all the automorphisms and deriva tions of various Banach algebras of power series are known to be continuous (Example 6.1.19 and Loy [1969]), many of the results in the cited reference do not need continuity in their hypotheses.
6.5
Jordan Derivations
We briefly investigate Jordan derivations. W arning: In this section we use the notation a o b to denote the Jordan product a ° b = ab + ba rather than the quasi-product. 6.5.1
D efinition Let A be an algebra. A map 6: A —* A is called a Jordan derivation if it is linear and satisfies 6(a o b ) = 6(a) o b 4- a o 6(b)
V a, b 6 A.
Obviously a derivation is a Jordan derivation. In fact there are not many examples of Jordan derivations which are not derivations. A num ber of results help to explain this scarcity. We will give the elementary algebraic proof due to Cusack [1975], [1976] that every Jordan derivation on a semiprime algebra is a derivation. Although this proof is straightfor ward, it is long and highly computational. Hence before giving it we will prove the much weaker result of Sinclair [1970b] that a continuous Jordan derivation on a semisimple Banach algebra is a derivation. Unlike the proof given here, Sinclair’s original proof depends on Herstein’s theorem [1957] that any Jordan derivation on a prime algebra is a derivation. 6.5.2 P rop osition A bounded Jordan derivation of a semisimple normed algebra is a derivation. Proof Let 6' be a bounded Jordan derivation on a normed algebra A ’. Let A be the completion of A ' and let 6 be the closure of b' in A x A. Then 6 is a bounded Jordan derivation on A. As in the proof of Proposition 6.4.16, we find 6n(an) + V — n!6(a )" + V for any n 6 N, any a € A and any ideal V of A. Hence, as before, ( 6(b) -)- V ) / V is topologically nilpotent in A / V . Next we show that each primitive ideal of A is invariant under 6. Sup pose V is a primitive ideal of A ,b is an element of V and 6 (b) does not belong to V . Let T be an irreducible representation of A on a linear space X with kernel V . Choose x € X so that y = is not zero. Since 6(b ) V and T^fb) have { 0} as spectrum, x and y are linearly independent.
Jordan Derivations
615
Thus Theorem 4.2.13 shows that there is an element a € A so that Tax = x and = x - y. Then T6(b)a+a6{b)x = Tm x + Tay = y + x - y = x, so that T^((,)a+a4(5) is not topologically nilpotent. On the other hand, S(b)o + a6(b) + V = 6(ba + ab) + V shows that Tf(b)a+a6(b) is topologi cally nilpotent. This contradiction proves that 6(b) belongs to V. Hence each 'primitive ideal is ^-invariant. (This paragraph of the proof is valid for any snectral normed algebra A ) /. Jordan derivation satisfies Leibniz’ formula relative to Jordan multi plication, i.e., 6n (a o b) = (jfc)^*(a) ° 6n~ k( b ) . where a ° b = ab + ba. Hence for any ! e R , et6(a o b) — e (a) o etS(b ) holds for all a, b 6 A. Each primitive ideal V of A is invariant under elb so we define a norm continuous one-parameter group of Jordan automorphisms {a t : t € R} of A / V by at(a + P ) = ets(a) + P.
Theorem 6.3.7 shows that each at is either an automorphism or an antiar'jorriorphism of A / V . By continuity, ail at for t sufficiently close to zero must be automorphisms. Now the group property shows that {a t : t 6 R } is a group of automorphisms. Hence its generator is a derivation. Thus 6(ab) — 6(a)b — a6(b)
belongs to each primitive ideal of A . By semisimplicity, 6 is a derivation. Hence S' is also a derivation. □ Note that the second paragraph of the above proof shows that any con tinuous Jordan derivation on a spectral normed algebra leaves the primitive ideals invariant. In order to organize the computations for Cusack’s theorem (which are taken directly from [1976]) we will adopt two notations! conventions: [a, b, c] = abc + cba
V a, b, c 6 A
6(a, b) = 6(ab) — [5(a)6 + a£(6)]
V a,b,c € A
where 6 is a Jordan derivation. Note that 6 is a derivation exactly when 6(a,b) vanishes for all a, b 6 A. Our first lemma does not involve Jordan derivations. From Theorem 4.1.10, recall that if V is a prime ideal, then aAb C V implies either a 6 V or b 6 V. As a slight extension we note that ^AbA C V implies either a 6 V or b 6 V . The proof is the same. P-.2.S Lem m a Let V be a prime ideal in an algebra A. (a) If a, b e A satisfy [a,c]6 6 V
V c 6 A,
i'ie7i i iiher b belongs to V or [a, c] belongs to V for all c € A.
6: Automatic Continuity
616
6.5.4
(b) If a, b € A satisfy [a, c, 6] 6 V
V c € A,
then either a e V or b 6 V. Proof (a): Suppose [a, c] ^ V for some c 6 A. Any d 6 A satisfies [a, c]db = [acd-cda]b + c (d a -a d )b = [a,cd\b-c[a,d\b 6 V. Hence [a, c]A £ V implies b€V. (b): For arbitrary d, e 6 A, we get adaeb+bdaea 6 V . Then adb+bda 6 V and bea + aeb € V imply 2adbea 6 V. This gives aAbAa C V from which we conclude either a € V or b € V . O
The second lemma, due to Herstein [1957], is also computational. 6.5.4 Lem m a If 6 is a Jordan derivation on an algebra A, all a, b, c G A satisfy:
(a) 6 (a2) = a6 (a ) + 6(a)a; (b) 6(aba) = 6(a)6a + a6 (b)a + abS(a); (c ) 6([a ,b,c]) = [6(a), 6, c] + [a,6(6),c] + [a,6, 5(c)]; (d) S (a ,b )[a ,b }= 0; (e) [a,b]6(a,b) = 0; (f) [Cl6]tf(a,6) + M * M ) = 0; (g) [c, a]^(a, 6) + [a, b\6{ c, a) = 0 . Proof (a): All a G A satisfy
26 (a2) — 6(a o a) = a o 6(a ) + 6(a ) o a = 2 (a6 (a) + 6(a)a). (b): Note the identity do (d o e) = do (de + ed) = d2 oe + 2ded
V d, e G A.
From this we obtain 25(oio)
=
£(a o (a o&)) - 6 (a2 o b)
=
a o 6(a o b) + 6 (a ) o (a o b) —a2 o 6(b) —6(a2) o b
=
26(a)ba + 2 a6 (b)a + 2ab6 (a).
(c): Replace a by a + c in (b). This gives 6((a -t- c)b(a + c)) = 6(aba) + [<5(a),6, c] + [a, <5(6), c] + [a, M (c )] + 6(cbe). Then (c) follows by subtracting 6(aba + cbc) from each side. (d): FYom (c) we get b([a, 6,a6]) = [^( u)Auf c] + [a,6(6),a&] + [a,6, 6(a&)]. The identity [a, b,ab] = (ab)2 + at?a, together with (a) and (b) implies 6([a, b, af>]) = ab6 (ab) + 6(ab)ab + 6 (a)t?a + a6(b2)a + ab26(a ) = ab6 (ab) +
Jordan Derivations
617
6 (ab)ab + 6(a)b2a + ab6 (b)a + aS(b)ba + ab26(a). By comparing the two expressions for 6([a, b, aft]), we get 0 = ( S(ab) — 6(a)b —a6 (b))ab — ( 6(ab) — aS(b) — 6(a)b)ba = 6(a, b ) [a, 6], (e): Since 6 is obviously a Jordan derivation on the reverse algebra, this
follows from (d) by reversing the product. (f): FYom (e) we see [a + c,b]6(a + c,b) = 0. Combining this with [a + ',&] = [a, 6] + [c, 6], 6(a + c, 6) = 6(a, b) + 6(c, b), and [a, 6]6(a, b) = 0 = [c, h\i (c, 6), we have [c, b) + [a, 6]<S(c, b) = 0, as required. (g): Interchange a and b in (f). This gives [c, a]6(6,a) + [6, a]6(c, a) = 0. But £(a, b )+ 6(b,a ) = 6(aob)—ao6 (b )—6 (a)ob = 0 implies 6(b,a ) = —S(a,b). Thus [6, a] = [—a, 6] gives the desired conclusion. □ 6.5.5 Lem m a
Let V be a prime ideal of the algebra A , and let 6 be a Jordcn derivation on A. If [a, 6] is in P , then S(a, b) is in P .
Prooj We first consider the case when A / P is noncommutative. We assume [a, 6] = P . Lemma 6.5.4(f) and (g) give [c,6]6(a ,6) € P and [c,a]6(a ,6) € P for all c € A. It follows from Lemma 6.5.3(a) that if there is an element c G A satisfying either [c, a] $ P or [c, 6] ^ P , then 6 (a,b) e P holds. Suppose that [c, a] and [c, 6] are in V for all c in A. By noncommutativity, let c and e be any elements of A satisfying [c,e] ^ V . Then, [e, b] and [a + e, 6] are in P and [c, e] and [c, a -I- e] are not in P . It follows that 6(e, b) and 6(a b) are in P , and that 6 (a, b) 6(a b) — 6 (e, b) is therefore in V. Now suppose that A / P is commutative. Lemma 6.5.4(d) and (e) give [a, 6] ° 6(a, b) = 0 and therefore, since 6 is a Jordan derivation, 0 = <S([a, 6])o 6(a , 6; + [a, b] o S(S(a , b)). Since A / P is commutative 6([a, 6]) o S(a, b), and there'ore 2<S([a, 6])<S(a, b), are in P . Since A / P has no divisors of zero, either 6(a,b, or 6]) is in V . If <$([a,6]) is in P , we get 26(a,b) = 6(ab + ba) + 6(ab~ ba)—2(a6(b)+6(a)b) = 6([a, 6])+ (a o £ (6) —2S(a)b)+(S(a)ob—26(a)b) € P . T ius in either case, 6(a, b) belongs to P . □
+ e,
—
+ e,
<5([a,
The final lemma is concerned with the case in which [a, b] does not bt-ioK j- to a prime ideal P. 6.5.6 Lem m a Let 6 be a Jordan derivation on an algebra A. All a, b, c G A ,<■;' sfy: (a) <5([a, 6]c) = (S(ab) - 66(a) — 6(b)a)c + [a, &]<5(c) + cS(a, b); ( b ) <5(c[a, 6 ] ) = c (6(ab) —bS(a) — S(b)a) + 6(c)[a, 6 ] + S(a, b)c; (c Ha,b],c,S(a,b)] = 0. °m of (a): The identity [o, b]c = [a, b, c\ — c o (6a) and Lemma 6.5.4(c) give i>([a, t\c) = <5([a,6,c]) - 6(c o (6a)) = [6(a), 6, c] + [ a , <5(6), c] + [a,6, 6(c)] c6 (ba) —6(ba)c —6(c)ba —ba6(c) = c(b6 (a ) + 6(b)a —6(ba)) + ( 6 (a )b + a 6 (b) — 5(ba))c — 6(c)ba — ba6 (c ) + a66(c) + 6(c)6a.
618
6.5.7
6: Automatic Continuity
Since 6 is a Jordan derivation, we have 6(a)b + a6 (b) — 6(ba) = 6(ab) — 6(b)a - b6 (a ) and b6 (a ) + 6(b)a - 6(ba) = 6(a,b), and therefore 6([a, 6]c) = c6 (a, b) + ( 6 (ab) - b6 (a ) - 6(b)a)c + [a, 6]<5(c) as required. (b): This may be proved by a similar argument, or by reversing the product. (c): The idea of the proof is to evaluate 6([a, 6]c[a, 6]) in two different ways and then obtain the required result from Lemma 6.5.4(d) and (e). Replacing c by [a, 6]c in (b), we get <5([a, £>]c[a, &]) = 6(a, 6)[a, 6]c + 6([a,6]c)[a,6] + [a,6]c(6(af>) — b6 (a ) - S(b)a) = [a, 6]<5(c)[a,6] + [<5(ab) ~ b6 (a ) — 6 (b)a,c, [a, &]], by (a) and Lemma 6.5.4(d) and (e). We now ob serve 6([a, b]) = ( 6(ab) — bS(a) — S(b)a) + S(a, b). This implies 6( [ a , 6 ] c [ a , 6 j ) = b ([ a ,b \ c \ a , b ] ) - [ A ( « , 6),
by Lemma 6.5.4(b). The result now follows by subtracting <5([a,6]c[a,6]) from both sides. Cl Cusack’s theorem is actually a little stronger than so far noted. 6.5.7 T h eorem A Jordan derivation 6 on any algebra A satisfies 6(ab) — [6(a)6 + a£(6)] 6 A b
V a, b € A
and hence induces a derivation on A / A b ■ In particular every Jordan derivation on a semiprime algebra is a derivation. Proof Let a and b be any elements of A and let V be any prime ideal of A. Lemma 6.5.6 shows [[a, 6],c, 6 (a, &)] = 0, for all c in A. According to Lemma 6.5.3(b), either [a, 6] or 6(a, b) belong to V . But Lemma 6.5.5 shows that [a, 6] 6 V implies <5(a, b) 6 V . Since A b is the intersection of all the prime ideals of A , this completes the proof. □
7
Structure Spaces Let A be an algebra. In this chapter we will study the space ( P a / I I 4 / S a ) of ( prime / primitive / maximal modular ) ideals of A as a topological specy under the hull-kernel or Jacobson topology. For certain classes of a!&3isras A, e.g., completely regular algebras (Section 7.2) and strongly harmonic algebras (Section 7.4), we will show that the subdirect product representation relative to E a (introduced in Definition 1.3.3 and Section 4.6) yields significant information about A. Section 7.3 deals with more detailed questions in ideal theory revolving around primary ideals. We also consider central and weakly central algebras and show that they are completely regular under fairly weak additional hypotheses.
7.1
The Hull-Kernel Topology
”r. Section 3.2 we introduced the hull-kernel topology on the Gelfand space of a commutative Banach algebra A. It is comparatively little used except in the case of completely regular commutative spectral algebras where it is Hausdorff and coincides with the Gelfand topology. In the commutative case, Proposition 3.1.3 shows that the Gelfand space of A can be identified with the set E a of maximal modular ideals of A, and Theorem 4.1.9 shows that the latter set coincides with the set 11* of primitive ideals. In a noncommutative algebra A, the set Pa of prime ideals and its subsets, I U and E a , can each be given the hull-kernel topology. Again, this topology seems to be of comparatively little use unless farther restrictions are placed on the algebra. Generally, these restrictions need to be sufficient to force the topology to be Hausdorff. The most important such restriction is the notion of complete regularity for noncommutative Banach algebras, which was introduced by Alfred B. Willcox in his thesis published as [1956a] and farther elaborated by Charles E. Rickart in his book [I960], Another Is ihe notion of strongly harmonic algebras introduced by Kwangil Koh [1872] building on ideas of Silviu Teleman [1971]. We develop both of these theories in this chapter. G;ven any collection P of ideals in an algebra A, there is a corresponding subdirect product decomposition. This can be thought of as the function ■z 177 ' given by , ~ a (I ) = u + I 6 A / I V I € P
619
7: Structure Spaces
620
7.1.1
which maps A into the full direct product Y\j&p{A/J). Such subdirect product representations ordinarily give little useful information about the algebra even when they are faithful. One main purpose of this chapter is to investigate situations in which the representation in X\m€=a ( A / M ) , and a closely related representation, do become useful. The hull-kemel topology on the space of primitive ideals of a noncom mutative ring was first systematically studied by Nathan Jacobson [1945b]. However, Marshall Harvey Stone [1937] had used a very similar idea in his theory of structure spaces of Boolean rings, and Gelfand and Georgi E. Silov [1941] had explicitly considered the hull kernel topology on the carrier space of a commutative Banach algebra. Most of the ideas of this chapter were first investigated in the commutative case, particularly in Silov [1947a]. For a general algebra A, the set Pa of prime ideals of A seems to be the natural setting for the hull-kemel topology. However since prime ideals are not necessarily closed in a Banach algebra, they play a comparatively small role in the theory of this work. It seems likely that useful classes of closed prime ideals larger than the class of primitive ideals will eventually be found. Partly for this reason, we will introduce the hull-kemel topology for arbitrary subsets of the set Pa of prime ideals of an algebra A. The subsets of chief interest are Pa itself, = { primitive ideals of A } and E.4 = { maximal modular ideals of A }. Theorems 4.1.8 and 4.1.9 show the inclusions Ea C IT* C P^. Some of the following notation was introduced in a preliminary way in Chapter 4. 7.1.1 D efinition Let A be an algebra and let P 4 be a nonempty subset of the set P \ of prime ideals of A. For each subset S of A, the hull of S rrlatii>e to P 4 is the set h (S ) = { P e PA - S c v } .
A subset of Pa which has the form h (S ) for some S C A is called a hull in Pa- When Pa is ( Pa / / Ea ), we write { hp / hn / ha ) instead of h. For each subset B of P a , the kernel of B is the set k (B ) =
n
{V
The next proposition shows that h and km: t up a Galois correspondence oetween the subsets of P a and the subsets of A. 7.1.2 Prop osition Let A be an algebra and let Pa be a nonempty subset of Pa Consider Si Q S? C A and Bi C B 2 C PA . Then k (B\) is an ideal of A and these subsets satisfy H S t)
C
h (S i)
k (B 2 ) C k ( B l )
(1)
The Hull-Kerael Topology
5,
C
hk(Si)
hkh(Si)
=
h (S i)
621
Bi C h k (B t)
(2)
khk(Bi) = k (B i).
(3)
Hence, a subset B of PA is a hull if and only if it satisfies B = hk (B). Any family {<SQ : a 6 A } of subsets of A and any family { B a : a € A } of subse'.s of PA satisfy: h(
U
(J Sa) = h C £ S*) = /»(8P®“ ( ^ a€A a€A
a6/4
= f l h^ a €A
*( u&A 5“) = n&A w a
Any finite set { l i , l 2, ...
a
^
^
of ideals of A satisfies
n
n
Mf| >=1
=
•• Z") * U h^)3=1
(6)
Proof These results are either obvious or are established by the same easy arguments used for Proposition 3.2.6. For (6), we note that the last equation is immediate since each ideal in P is prime. Also, the first set is included in the second by ( 1 ) while it includes the third set from the definition of a hull. □
7.1.3 Th eorem Let A be an algebra and let PA be a nonempty subset of PA . Then the family { P A \ H : H is a hull in PA } of complements of hulls is a topology for PA which satisfies the To separation axiom Proof Let T be the family which is asserted to be a topology. The equations h (A ) - 3 and h ({0 }) = PA show that PA and 0 belong to T. Proposition 7.1.2 t jows that T is closed under arbitrary unions and under (finite) in
ternee. ions. The To separation axiom is the statement that if two points V\ aiwi V 2 are distinct, then at least one is not in the closure of the other. This in obvious since V\ € h k ({P 2} ) is equivalent to V 2 C V\□ 7.1.4 D efinition Let A be an algebra and let PA be a nonempty subset of PA . The topology described above is called the hull-kernel (or Jacobson) topology on PA . Let ( PA / U A / EA ) be the topological spaces of all ( prime ideal;-; / primitive ideals / maximal modular ideals ) under the hull-kernel topo’c 5y. This topological space will be called the ( prime structure space / structure space / strong structure space ) of A.
622
7: Structure Spaces
7.1.5
We have already noted that Theorems 4.1.8 and 4.1.9 show the inclu sions E a C C P a as sets. The definition of the hull-kemel topology shows immediately that the topologies of II.4 and E a agree with their rel ativized topologies as subsets of P a °r IT*. 7.1.5
Proposition For any algebra A, EA satisfies the Tj separation axiom. Furthermore, if each { primitive / prime ) ideal of A is modular or is included in a maximal modular ideal (e.g., if A is unital) and if { I I 4 / PA ) satisfies the Ti separation axiom, then ( II .4 / P a ) equals 5 * . Proof Since V\ € { P 2 }
is equivalent to V\ 5 V 2 , this is obvious.
□
7.1.6 Proposition Let A be an algebra. Let P a be a nonempty subset of P a which includes E a - If 2 is a modular ideal in A , then h (T ) is compact. In particular, P a itself is compact if A is unital. Proof We establish the finite intersection property for h (l ). Let { H a : a 6 j4} be a family of closed subsets of h (l ) satisfying na6/4H a = 0. Each q 6 A satisfies I C k h (l) C k (H a). Hence H = k (H a) includes
the modular ideal J and is therefore modular. Therefore, Theorem 2.4.6(b) shows that 72. is included in some M 6 Ea Q P a unless TL equals A. Equation (4) of Proposition 7.1.2 shows h(^Ta k (H a) ) = nasAhk(Ha) = HaeA^a — 0- We therefore conclude 71 = A . Hence the relative identity e for I can be written in the form e = Oi + •■• + an for some a, 6 k (H aj )• Since any a 6 .4 can be written as a — ( 1 —eJa + ai
/ hs (X ) ) onto ( P a /i / H a /i I ~A/x ). and this fact & included in Theorem ( 4.1.8 / 4.1.10 / 4.5.2 ). If B is any subset of ( hp ( l ) / hn ( I ) / hs (X ) ) and Bf is its image under this map, then k (B ') equals k (B )/T . Hence any V 6 ( P A / U A / Ex ) satisfies P 3 k (B ) if and only if it satisfies V j l 3 k (B '). Therefore the map V >—> V /X is a horneornorphism of h (X) in ( P a I I U / Ea ) onto ( P A / i / n a /x / =-a /t )■ The last sentence of Theorem 4.1.10 asserts that V >—* VCvL is a bijection of P a \ hp (T ) onto P j. In order to prove the second half of the theorem for
*7 1 g
The Hull-Kerael Topology
623
Pa, it only remains to show that this bijection V >-> V fl I is a homeomorphism of Px\hp (X) onto P j. We must show that for any V € P^\/»p ( I ) and any B C P ^ \ /»(!), P 3 fc(B) is equivalent to P f l l 3 fc{P' fl J : P ' 6 B }. The implication "P D k {B ) => V C\ X D k { V n X : V € B ] is obvious and the opposite implication follows since (PIv e a 'P ') H I C V and I g V imply n p 'g s P ' C P by the prime nature of V. Since 11^ and Ea are homeomorphically embedded in Pa, the map P h P f l l restricted to ( II.4 \ hn(J ) / Ha \ h ~ (I ) ) must be a homeomorphism onto its range. It thus remains only to prove that each ( primitive / maximal modular ) ideal of X has the form P H I for some ( primitive / maximal modular ) ideal of A. However, this follows immediately from Theorem 4.1.8 in the case of primitive ideals and from Theorem 4.5.2 in the case of maximal modular ideals. □
7.1.8 C o rollary Let A be an algebra and I the ideal ( A b / A j / Am ). Then ihe map is a homeomorphism of { Pa / TI^ / Ea ) onto { Pa / j / n^/Z / e A/ j ). Proof The choice of I gives ( hp ( I ) — P a I hn ( l ) = n * h r (T ) = Ea )-d 7.1.8 P roposition Let A and Pa be as described in Definition 7.1.1 and also assume 5^ C P * . Any two ideals 1\ and X2 of A satisfy: (a) The sum X\ + X2 equals A if and only if 1\ 4-X2 is modular and h(T\) and fifa ) are disjoint. (b) If H is a hull in P a with k (H ) modular and h (l \ )n H empty, then X\ contains an identity for A modulo k (H ). Proof (a): If A = 1\ + I 2 holds, then 1\ + X2 is trivially modular, and equation (4) of Proposition 7.1.2 shows that h(X 1 ) and h (l 2) are disjoint. Conversely, if h (l i ) and h (l 2) are disjoint, the same result shows that h\li + I 2 ) is empty. If I i + I 2 is modular, Theorem 2.4.6 then shows X\ + I 2 = A. (o): Since k (H )+ 1 1 includes k (H ), it is modular. Since h k (H )n h (Ii ) = H n h(Xi) is empty, (a) shows k (H ) + T\ = A . Let e be a relative identity for k (H ). Write e = c + d with c 6 k (H ) and d € 1\. Then d is a relative identity for k (H ) since d — e = c belongs to k (H ). □
The next result has important consequences in the next chapter. Note that FU could be replaced with any set P a of prime ideals (each one aciosed lor part (b )) satisfying P a 2 n^. 7.1.10 Proposition Let A be an algebra, let X be an ideal of A and let a be ar, element of A . In the following statements, the word “quasi-invertible” can be replaced by “invertible ” if A is unital. (a) The element a + 1 is quasi-invertible in A / 1 if and only if a+k h n (A ) is quasi-invertible in A /k h n ( I ) .
624
7: Structure Spaces
7.1.11
(b) If a is a spectral semi-norm on A, then a + 1 is quasi-invertible in A / 2 if and only if a + 1 is quasi-invertible in A / 2 where 1 is the a-closure of I . Proof Note that khn ( I ) / I is naturally isomorphic to ( A / l ) j by the def inition of the Jacobson radical. Hence (a) follows from Theorem 4.3.6. If a is a spectral semi-norm, its quotient norm a on A / T is also spectral, so Corollary 4.3.10 implies I / I C ( A / X ) j . Hence (b) follows from (a). □
Theorem 7.1.7 shows that if 7 is an ideal in an algebra A , then the map B *-+ B m is a homeomorphism of ( P*\/»p ( I ) / Il^\/»n( I ) / H^\/»=( I ) ) onto ( P i / I I j / E j ). The same results hold for certain subsets other than ideals. The next instance is essentially due to Irving Kaplansky [1950]. Proposition 7.2.14 gives another instance due to Kaplansky [1949a]. We do not know whether the corresponding results hold for prime ideals. 7.1.11 Proposition Let A be an algebra, and let e be an idempotent in A . The map P *-+ P PI eAe is a homeomorphism of ( 11* \ hn (eA e ) / Ea \ hz (eA e) ) onto ( U eJu / EeAe ). Proof If P is a primitive ideal in 11^ \ hn (eA e), let T be an algebraically irreducible representation of A on X satisfying ker(T) = P. Then Te is non zero and each non-zero y 6 TeX satisfies Te_ieV — TcTaV = TeX. Hence b <-> Tblr,x is an irreducible representation o f A on TeX . The kernel of this representation is obviously P n eAe. Hence the map V »-> P fl eAe maps IT* \ hn (e A e ) into II eAe- Let P be a primitive ideal in U eAe and let M be a maximal modular left ideal of e.4e satisfying V = M : eAe (Definition 4.1.7 and Theorem 4.1.8). Define M to be the modular left ideal M = A M +.4(1—e) of A which is proper since it satisfies e M e C M / eAe. Therefore, there is a maximal modular left ideal C of A which includes Ai. Let V be the primitive ideal C : A of A . We wish to show V fl eAe = P . Any b 6 V satisfies bAe = beAe C M C C and 6.4(1 — e) C >1(1 — e) C C, so we conclude V C V n e.4e. Since e (the identity of eAe) does not belong to £, it does not belong to M + ( V fl eA e) C M + V C £. Since M is maximal, this shows M = C fl eAe, V fl eAe C M and therefore P fl eAf- C P . Thin concludes the proof of the identity P fl e A f — V and shows that P P n eAe maps 11^ \ hn(eA e ) onto U eAeNext we show that the map P >-* P fl eAe is injective. If P i , P 2 € IiA \ hn (e A e ) satisfy Pi fl eAe = P 2 fl eAe, then we have (A e )P i (A e ) C A (e P te) C (P, n eAe) C AV? C P 2. Since P 2 is prime and A t D eAe is not included in P 2> we conclude P i Q P 2. Similarly, P 2 C P Y holds. This shows P\ = P 2 as we wished. If P is a maximal modular ideal in "Ea \ h ~(eA e) with relative identity d, then P fl eAe is a modular ideal in eAe with relative identity ede. Fur thermore, this ideal is maximal in eAe since b 6 eAe \ ( P fl eAe) implies
7J
1
The Hull-Kernel Topology
625
eAe = e (A 1 bA l -I- V )e — (e A e )b (e A e ) + V . Conversely, if V C eAe is a maximal modular ideal of eAe with relative identity d 6 eAe, then the ideal V + 1 - e )A + .4(1 — e) + .4(1 —e )A is a modular ideal of A with relative identity d (since a d - a = ( eaed —eae).+ e(ad — a )(l — e) + (1 — e)(ad — a) holdt for all a 6 A ) and any maximal modular ideal M of A including this ideal with relative identity d must satisfy M fl eAe = V . We have shown that the map V »-> V fl eAe sends \ hr (e A e ) bijectively onto EeAe. Finally, it remains to show that the map is a homeomorphism. It is enough to consider the case of primitive ideals. Suppose that F is a closed subset of TIeAe, and let B C IT* \ hn (e A e ) be defined by B = { V e I U \ hn (eA e ) : V fl eAe € F }. In order to show that B is closed in FI.* \ hn (eA e), we must show that any Vo € UA \ hn (e A e ) belongs to B if it includes k (B ). However, this is obvious from the definition of B and the hct that F is closed in II eAe. Conversely, suppose that B is closed in U A \ \n (eA e) and define F C IleAe by F = { V n eAe : V G B }. In order to si' jw that F is closed, we need to show that Vo € II eAe belongs to F if it includes k (F ). We have shown that there is some V\ 6 11^ \ hn (eA e) satisfying Vo = V\ fl eAe. Then V\ satisfies
i k (B )e C k (B ) n eAe = D {V n eAe : V € 5 } = k (F ) C P 0 C p,. Since V\ is a prime ideal satisfying (A e )k (B )(A e ) C V\ and since V\ does not ir. elude Ae, it must include k (B ). Since B is closed, this implies V\ € B and hence Vo 6 F. □ Subdvrect Product Representations
W i will briefly review the construction of a subdirect product represen tation associated with a collection of ideals. (See Definition 1.3.3 and the discussion surrounding Theorem 4.6.1.) Let A be an algebra and let B be a non :mpty collection of ideals of A . (The cases of particular interest are, of course, B = PA , B = I1A or B = EA .) For each a £ A , let a be the cross section in the direct product U- p ^ b ( A / V ) defined by a (V ) = a + V
V VeB.
Lot A be the Hubalgebra {a : a € A ) C \\-p^u (A / V ). Clearly, A satisfies {a (P ) : a 6 >1} = A / V for each V € B so that A is a subdirect product. The ir.ap a n a is a homomorphism of A onto A with kernel DpgsP. Fcr a special case in which B equals ( PA / / Z A ), the kernel of the representation a n d described above is ( A b / A j / A m )■ In this case, the qrotients A / V are ( prime / primitive / unital simple ) algebras and hence are reasonably tractable so that the structure of the direct product UpgB A / V ) is comparatively well understood. However, the subdirect produ't representation will still be of comparatively little use unless useful infoni ation can be found describing which elements of the direct product
626
7: Structure Spaces
7.1.11
belong to A. Situations in which this type of information is available are a main theme of the present chapter. A change in viewpoint may be instructive. Instead of the direct product n V e B {A /V ), consider the disjoint union S = S b (A ) — U-p ^b (A / V ). De fine a function it: S —* B by n(a + V ) = V for all a € A and V 6 B. Then for each V e B, the subset of S is called the fiber (or stalk) over V . Each fiber has the structure of an algebra. The sets B and S are called the base space and total space, respectively, and the map it: S —* B is called the fiber map. A function f : B —* S is called a section (or global section) if jt o / is the identity map on B. The set l '(B , S ) of all sections is obvi ously an algebra under pointwise operations. For each a 6 A, the function a :B —* S defined as before by a (V ) = a + V is a section and a n a i s a ho momorphism of A into the algebra r(fi, S). In terms of this language, the subdirect product representation will become useful if an effective descrip tion can be given of which sections belong to A. The hull-kemel topology would seem to offer some hope for finding such a description. However, in general, it is difficult to find meaningful relations between elements of neighboring fibers, so that the topology of the structure space has only a very weak interaction with the subdirect product. This is in part due to the poor separation properties of the hull-kemel topology. Except on Ea, it is not often even T \, and it is even less often Hausdorff. There are two ways to avoid this difficulty. Any topological space can be made Ti, Hausdorff or completely regular by collapsing certain sets of points into points and giving the quotient space its quotient topology. If this is done to P a , or Ea (with the hull -kemel topology), then the new space may even be iden tified with a certain space of semiprime ideals. (The point in the quotient space obtained by collapsing B C to a point can be identified with the ideal k (B ) of the original algebra.) These Hausdorff or completely regu lar quotients of structure spaces have proved valuable and are exhaustively discussed in John Dauns and Karl Heinrich Hofmann [1968], [1969]. In the first reference, several classes of topological algebras are exhibited as the algebra of continuous sections of a type of bundle or field of algebras over a suitable quotient structure space. This important theory is too technical to be included here. See James Michael G. Fell and Robert S. Doran [1988a], [1988b], Joseph W. Kitchen, Jr. and David A. Robbins [1982], [1983] and [1985], Maurice J. Dupre and Richard M. Gillette [1983], Stephen Allan Selesnick [1979], Anthony Karel Seda [1981] and Gerhard Gierz [1982], We will pursue the second approach. We place more severe restrictions on the algebra A which force its structure space to be well behaved. Propo sition 7.1.5 suggests that the best theory (i.e., the most results with the least restrictive hypotheses) will be obtained by concentrating attention on the strong structure space. This leads to the theory developed in the next section.
7.1.12
627
The Hull-Kernel Topology
Algebras Satisfying Polynomial Identities
This specialized subject will illustrate some of the ideas just introduced. "Ve bagin our discussion with some simple examples. An algebra A is com mutative if and only if any two of its elements are roots of the noncommut ing polynomial S (x ,y ) = xy - yx. Ecre is a second example. Recall that M n is the algebra of all n x n complex matrices. If C € M 2 has trace zero, Theorem (the Jordan decomposition) shows that it can be written as G ~ l D G where G is an invertible matrix and D is either a diagonal matrix with diagonal entries A and —A for some A € C or a strictly lower triangular matrix. In either case, C 2 is a multiple of the identity matrix and is therefore in the center of M 2. However, the commutator A B — B A of any two matrices A and B in M 2 has trace zero. This shows that any three matrices in M 2 are roots of the noncommuting polynomial
2.7.15
W ( x , y, z) = ( xy - y x ) 2z - z( xy - y x ) 2
known as Wagner’s polynomial. We can easily define much more compli cated algebras which satisfy this same polynomial identity. For instance, '.st Q be a compact Hausdorff space and consider the algebra C (fi, Ma) of all continuous functions from Q into Ma- (We have already noted the identification of this algebra with the algebra tensor product C ( f l ) ® Ma.) Clearly any three elements of C(Cl, M 2 ) are roots of Wagner’s polynomial W. Any subalgebra satisfies the same condition. These examples should make clear what we mean by a noncommuting polynomial over the complex field. In order to keep the discussion as ele mentary as possible, we will consider a noncommuting polynomial to be a finite linear combination of monomials where each monomial is just a finite prodv.ct of symbols. The degree of a noncommuting polynomial is the length of its longest monomial. A noncommuting polynomial P of n variables may be emVuated at an ordered n-tuple (a lt 0 2 , . . . , an) of elements from any al gebra A simply by replacing each symbol by its corresponding element in A. Obviously, this gives an element in A. We say that (0 1 , 02,... ,a „) is a root of P if it satisfies P (a t,a2 , ... ,a „) = 0. 7.1.12 D efinition Let A be an algebra and let P — P (x 1 , 3:2,. ••,£„) be a noncommuting polynomial. Then A is said to satisfy the polynomial identity P if any ordered n-tuple (0 1 , 02,... ,an) of elements from A is a root of P. It is said to satisfy a polynomial identity of degree d if it satisfies some noncommuting polynomial of degree d. The standard polynomial in n noncommuting variables is defined by
dn(^hx2, • • • ixn) — ^ ^ s 6 n (ff)**V(l)**'
^
628
7: Structure Spaces
7.1.13
where E „ is the group of all permutations of { 1 , 2 , . . . , n} and sgn(cr) is ±1 depending on whether a is an even or an odd permutation. Obviously the standard polynomial Sn is of degree n and multilinear. (The latter means that for any k = 1,2,..., n, if ai,a 2, .. ., a* , ..., an and at,a2 , ■. ■, bk, ■■ ■,an are roots of Sn, then so is at,a2, . ■., Aa* ■■■,an for any A, n in C.) It is also alternating. (This means that interchanging the order of two elements in an n-tuple substituted into it changes the sign. Hence if the same element occurs twice in an n-tuple, then Sn is zero on the n-tuple.) Multilinearity and the alternating property are useful since they show that an algebra A satisfies the polynomial identity Sn if and only if every set of n distinct elements chosen from a linear space basis for A is a root when arranged in some (hence in any) order. In this subject, the degree of a polynomial is more important than the number of noncommuting variables or the number of terms. Hence the next theorem shows that Wagner’s fifth degree polynomial identity for M 2 with six terms in three noncommuting variables is eclipsed by the 24term, fourth degree polynomial S4 in four noncommuting variables just introduced. We shall omit the proof of the following result of Shimshon Amitsur and Jacob Levitzki [1950] since full proofs can be found in any of the standard references listed at the end of this discussion. 7.1.13 T heorem The algebra M n satisfies the standard polynomial S2n of degree 2n and does not satisfy any polynomial identity of degree less than 2 n.
Perhaps the most fundamental result in the theory is due to Irving Kaplansky [1948e]. We will give the special case for Banach algebras and indicate how that follows from the strictly algebraic statement. 7.1.14 T h eorem A primitive Banach algebra which satisfies a polynomial identity of degree d is isomorphic to a full matrix algebra M n with n < d/2. Proof The version given on page 17 of Nathan Jacobson’s book [1975] states that A is a simple algebra of dimension n2 (where n < d /2 ) over the center A z of A and that A z is a field. However, the Gelfand Mazur theorem (Corollary 2.2.3) shows that this field A z must be just the complex field C. Theorem 8.1.1 shows that any finite-dimensional simple algebra over C is a full matrix algebra. □ 7.1.15 E xam ple In order to have a simple but reasonably representative example, the reader may wish to consider the algebra A defined by
{/: [0, 2]
A/2 : / (0) „ = / (0)22; f ( s ) u = f ( s ) 2l = 0 for j < 1 ; / (2 )ia = 0}
so the matrices f ( t ) are multiples of the identity for t = 0, diagonal for t < 1 and lower triangular for t = 2. The set of primitive ideals is: n
A = {Vu ~P>,k : t €
{0 }u ]l,2 [;
a g]0 ,
1] U { 2 } ;
* € { 1 , 2 } }
629
The Hull-Kernel Topology
'6
where V t — {/ 6 A : f ( t ) = 0} and V,,k — {/ € A : f(s)kk — f ( s ) 21 = 0}. Note that the quotient algebra of A modulo Vo or V ,tk for s G]0,1] U {2 } and k € {1 ,2 } is C while the quotient modulo Vt for t 6]1,2[ is M 2. The hull- kernel topology on IT* is T i since all primitive ideals are maximal. It nsigns the topology of ]1 , 2 [ to {V t '■ t €]1 , 2 [ }, the topology of ]0, 1 ] to {V ,'k '■ s G]0,1] } for k e {1 ,2 }, and for s € {1 ,2 } it puts both V » ti and Vg'2 in the closure of any subset of {V t, - t € ]1 , 2 ( } which has s in t.V losure of its set of indices. Finally for k 6 {1 ,2 }, Vo is in the closure of a subset of { V , k : 3 e]0, 1 ]} if 0 is in the closure of the set of indices. The center of A is A z = {/ € A : f ( t ) is a multiple of the identity for t € {0 } U [1,2] and f ( t ) is a diagonal matrix for t €]0,1[}. We derive some results about semisimple unital algebras satisfying a polynomial identity which are suggested by the main results of Naum Ya. Krupnik [1980a], [1980b], [1981] and [1987] and Krupnik and E. M. Shpigel’ [1982]. These results show that the last example is somewhat typical. 7.1.16 Th eorem
Let A be a semisimple unital Banach algebra which satisfies a polynomial identity of degree d. ( i) Each primitive ideal V of A is maximal with quotient algebra iso morphic to M nv for some n-p < d/2. (b) The center A z of A is the set of elements c such that for each V e n A there is a (necessarily unique) complex number 'y-p(c) satisfying c - ",-p(c) 1 6 V (>;) The map V »-> 7 p is an open surjection of 11^ onto with ker(-'p) = V fl A z - Hence A z is semisimple. (d) There are continuous functions r and 6 from A to B ( U A ) with the former linear satisfying r(ab) = r(ba) r ( c ) ( V ) = np r p (c ) a € Ag
6(ab) = S(ba)
S( c) ( V) = 7 p (c )nr if and only if
V a, b € A
(7)
V c € Az;V € T A
(8)
6(a ) is never zero
Sp(a) = {A € C : 6( A — a) is zero at some V € I I 4 } 6: A g —> B(U.a )g
(9) V a6 A
is a group homomorphism.
(10) (1 1 )
Proof (a): If A satisfies a polynomial identity it is clear that A / V satis fies the same identity. Hence the last theorem shows that this quotient is isomorphic to a full matrix algebra satisfying the indicated bound on its dimeision. Since matrix algebras are simple, V is maximal. f ): If c satisfies the condition, then ac — ca belongs to each primitive ideal. Hence c is in the center of A because A is semisimple. Conversely, if c belongs to the center of A then c + V must be in the center of A / V for
7.1.17
7: Structure Spaces
630
each P. Since this quotient is isomorphic to a full matrix algebra, c + V is a multiple of the identity 1 + P. (c): It is easy to see that 7 p is an element of the Gelfand space of A z with the indicated kernel. The proof of Proposition 7.2.14(c) and (d), which begins by assuming the algebra A is unital as we have supposed here, shows that this is a surjection. Since hulls are always Gelfand closed, it is an open map. (d): For each P , let tp-p be the natural map of A onto A/V and choose an isomorphism 0 p :A / P —* M nv. Define r (a )(P ) and 6(a )(P ) to be Tr(0p o tp-p(a)) and Det(Op o tpv (a)), respectively, for all a 6 A . Since all automorphisms of Mn are inner, the definitions of r and 6 are indepen dent of the choice of Op. Equation (10) (which we are about to prove) shows that f
V tp 6 $ ).
7.1.18 T heorem
The following are equivalent for a unital Banach alge bra A. (a) A has a sufficient family o f n-dimensional representations. (b) A / A j satisfies a polynomial identity of degree 2n. (c) A /A j satisfies the standard polynomial identity of degree 2n. If A / A j has a sufficient family o f n-dimensional representations, then A enjoys the above properties. A Banach algebra A has a sufficient family of n-dimensional represen tations for some n if and only if A/A j satisfies some polynomial identity.
Proof (a) => (c): Let $ be a sufficient family of n-dimensional representa tions. Note that 6 € fl{ker(ip) : tp € implies Det(i^>(Al — 6)) / 0 for all
7.1.19
The Hull-Kernel Topology
631
non-zero A and all
632
7: Structure Spaces
7.2,1
the ideal ®
bk € A&B : Y
t=i
fc=i
l|afc||
||6fc|| <
oo
and
y ^ ( g fc)bfc €
M}.
k=i
For C*-algebras, closely related but more definitive results will be given in Volume II, including those of Laursen [1969a].
7.2
Completely Regular Algebras
The following concept was introduced by Willcox [1956a] and extended by Rickart [I960]. On first sight, the second condition in the next definition appears somewhat artificial. Proposition 7.2.2 explains it: it is not enough to have S^t HausdorfF, S^i must also be Hausdorff.
7.2.1 Definition An algebra A is said to be completely regular if it satisfies: (a) E^ is Hausdorff; and (b) Each M € has a neighborhood V in such that k (V ) is modular. If A is unital, the condition (b) is automatically fulfilled by choosing V = E^. Hence, a unital algebra is completely regular if and only if its strong structure space is Hausdorff. Note also that Proposition 7.1.6 applied to (b) shows that is a locally compact topological space and hence completely regular. The next proposition extends this observation.
7.2.2 Proposition Let A be a nonunital algebra. Define ( 11^ /
)
to be the set ( YIa U {o o } / U {o o } ) with the topology consisting of the open subsets o f { UA / ) and all the sets o f the form ( (11^ \ H ) U {o o } / (E ^ \ 7 /)U {o o } ), where H is a hull (possibly empty) in ( H.4 / ) with k (H ) modular. Then ( ) is a compact ( T 0 / T\) space which is homeomorphic to ( II^i / S^i ) in a natural way. Furthermore, is Hausdorff (equivalently A 1 is completely regular) if and only if A is completely regular. In this case, E^ is the one-point compactification o fS A - Hence, EA is a completely regular topological space when A is completely regular. Proof As usual we regard A as embedded in A 1. Note that the second paragraph is an immediate consequence of the first. (The remarks following Definition 3.2.9 show that E^ is completely regular.) Theorem 7.1.7 shows that the map V >—* V fl A is a homeomorphism of ( \ /»n (.A) / E^i \ h~(A) ) onto ( rU / Ea ). Clearly, A itself is the unique element of ( hn (A ) / h~(A) ). Thus the bijection of ( fl^i / S^i )
7 onto ( n ji /
Completely Regular Algebras
633
) is given by f V ~ V n A, \ A i—►oo.
if V ± A
We need to identify the inverse of this map on ( 11^ / S^t ). If M. is a maximal modular ( left ideal / ideal ) in A with ( right relative / relative ) identity e, define M. C A 1 by M = {A + a € A 1 : Ae + a 6 M } . Then M fi A = M holds and the calculation A 1 /M — (A + M )/ M — A / {M D A ) - A J M shows that M is also a maximal modular ( left ideal / ideal ) of A '. Theorem 4.1.8 shows that we may write any V 6 11 ^ as M : A for some maximal modular left ideal M and then define V to be the primitive ideal M : A 1 o f A 1 which satisfies V D A = {o € A : Ao + ab 6 M. for all A + b e A } = V. (If V is maximal modular, we have now given two definitions of V and these definitions even appear to depend on the choice of the relative identities involved, but since V D A = V holds, Theorem 7.1.7 shows that all these definitions of V must agree.) Thus V >-* V from ( / E^ ) to ( llAi \ hu (A ) / E^i \ h r(A ) ) is the inverse of the above map. To complete the proof, we need only show that the neighborhood sys tem for oo in ( ll\ / E^ ) agrees with the neighborhood system for A in ( Im ' / =U> )• If the nonempty hull H in ( n ^ i / ) does not contain A , t’cin the kernel of its image in {11^ / E^ ) is k(H)C\A which is modular since it satisfies A 1 /k(H) = (A + k (H ))/ A a A/(k(H)C\A). Conversely, if II is i hull in ( 11^ / EA ) with k(H) modular and e is a relative identity for k(H), then the kernel of the image of H in ( II^i / S^i ) contains 1 - e and I ence is not included in A ■ Since this is equivalent to saying that the complement of the image of H is a neighborhood of A in ( II^i / S^i ), this completes the proof. □ 7.2.3 P r o p o s itio n Let A be completely regular and let X be an ideal of A . T'ten X and A/X are completely regular. Prooj Theorem 7.1.7 shows that H i and S A/x are Hausdorff, so it only rema;ns to show that they satisfy condition (b) of the definition. In order to show this for X, choose M € Ex• Theorem 7.1.7 shows that M = M DX for some M £ \ hr(X). Choose a closed neighborhood H of M in disjoint from hr ( I) and with k(H) modular. (This is possible since is completely regular and hr(X) is closed.) Let H be the image of H in Ex under the map described in Theorem 7.1.7 so that H is a neighborhood of M and equals k(H) n l . Proposition 7.1.9(b) shows that I contains a ••elatr'e identity e for k(H) as an ideal of A. Thus, e is a relative identity for H) = k(H) fl J as an ideal of X. Therefore (b) holds in X. In order to establish condition (b) in A f X , let M € E^/i be arbitrary. Theo- em 7.1.7 shows M. = M /X for some M. 6 h(X). Let V be a neighbor
634
7: Structure Spaces
7.2.4
hood of M in E^ with k(V] modular. Then k ( V fl h(T)) is also modular since it includes k(V). Let V be the image of V Pi h(T) in under the map described in Theorem 7.1.7 so that V is a neighborhood of M and k(V) is k(VC\h(T))/l. Then k( V) is modular in A /1 with relative identity e + 2 , where e is a relative identity for k (V f lf t ( J ) ) in A ■ Hence A IT satisfies condition (b) o f the definition. □
7.2.4 C orolla ry An algebra A is completely regular if and only if A j Am is completely regular. Proof The last proposition shows that A / A m is completely regular if A is. Corollary 7.1.8 shows that E^ and ’S.a /a u &Ie homeomorphic under the map M. *-* M ./A m - Hence we need only check that condition (b) holds in A when A / A m is completely regular. Let M G E^ be arbitrary and let V be a neighborhood of M / A m in with k(V) modular. Let V be the cor responding neighborhood of M in so that k(V) equals k(V)/AM- Since A/k(V) is isomorphic to ( A / A m )/(^(V)/A m ) = ( A/A m )/k(V), * (V ) is modular. Hence (b) holds in A j A m q The following result gives a converse to Proposition 7.1.6 in a special case. 7.2.5 P r o p o s itio n Let A be completely regular. modular if and only if hr( J ) is compact.
An ideal I o f A is
Proof Proposition 7.1.6 shows that hr ( I ) is compact if J is modular. As sume that hr(T) is compact. For each M. 6 h~(T), choose a neighborhood Vm of M. with k(Vju) modular. Choose a finite subcover VJ, V2, . . . , V„. The second inclusion of (1) and (5) in Proposition 7.1.2 show k(V) D k(Uj= l Vj) = n "=1 k(Vj). Hence k(V) is modular, since e\ o e2 0 •••° e„ is a relative identity for k(V), where ej is a relative identity for k(Vj), j = 1 , 2 , . ..,n . , □ We will now obtain the detailed structure of the subdirect product rep resentation A C Wm ^ a ( A / M ) for a completely regular A. If A is strongly semisimple as well as completely regular, this will give structural details for A- We begin by obtaining a partition o f the identity for the relative identity of a modular ideal.
7.2.6 Proposition Let A be a completely regular algebra. Let K be a compact subset o f E^ and let V\, V2, . . . , Vn be a finite open cover o f K . Then there exist elements e} € k(E^ \ Vj) fo r j = 1 , 2 such that Ci + e-2 + ----- Hen is a relative identity for k(K) . Proof For j = 1 ,2 ,. . . , n , let T} be the ideal k(EA \ Vj) and let T be T\ + I 2 H----1 - Proposition 7.1.2 shows h ( l ) = n "=1/i(T j) = n "= 1 (E^ \
Completely Regular Algebras
635
v3) = = u \ u "=1 Vj. Hence K and h(T) are disjoint. Since Proposition 7.2.5 shews that k (K ) is modular, Proposition 7.1.9 gives a relative identity e e l .'or k(K). Since J is the sum J j + 12 + •••+ J „, the existence of € Ij sa;;sfying e = ej + e3 H------- h en is immediate. □ 7.;?, 7 D efin ition Let A be an algebra and let £ be a subset of A . An element / € l\Ma'EA{A / M ) is said to belong to £ near an ideal M 6 Ea if there is some neighborhood V of M in EA and some a 6 £ satisfying a = / on V. An element / G YIm£=.a (A/ M ) is said to belong to £ at infinity if there is some open subset V of S _4 with \ V compact and some a G £ satisfying a = / on V. If / belongs to £ at each point of EA and at infinity, t^jen / is said to belong to £ locally. The following lemma shows that we can “paste together" elements of the algebra if their representations in the subdirect product agree on the overlapping parts of certain open sets. 7.2.8 L em m a Let A be a completely regular algebra. Let C be a one-sided ideal o f A . Let H be a dosed subset o f EA which is covered by open sets V o ,V i,...,K , such that Vi, . . . ,Vn have compact closures. If ao, a j , . . . , a„ are elements of £ satisfying a.j — a* on Vj fl VJt for all j, k = 0 ,1 , . . . , n, then there is an element a G £ satisfying a = iij onVjC\H
V j — 0 ,1 , . . . , n.
Proof Let £ be a left ideal. Proposition 7.2.6 gives elements ex, e%,. . . , e„ 6 A satisfying e_, 6 k(EA \ Vj) (i.e., i j vanishes off Vj) for j = 1 ,2 , ..., n and such that e = Y^j=i ej *s a relative identity for k(H\Vo). For each M. G 1st j ( M ) be the set j ( M ) = { j : M G V^}, so that e ( M ) = «i(^ ) holds. Define a G £ by a = (1 — e)ao + eiOj + e202 + ••■+ ena„. For M g H D V* fl Vo, we get ,
n
a(M)
=
( l - e ( A * ) ) o 0(A<) + ] [ ] M A ' l ) a ; ( A < )
=
[(l-e (A < )) +
Y ,
&AM)\ah( M ) = hu(M),
end for M € (H D V*) \ Vo we get a ( M ) = ((T ^ e)d b)(A 4) +
§ ( M ) a i ( M ) = e a i ( M ) = au( M). n
7.2.9 Lemma Let A be a completely regular algebra and let £ be a onejided ideal o f A . If f G J] ( A / M ) belongs to £ near each point o f a MezA
636
7: Structure SpaceH
7.2.10
compact set K C Ea, then there is some b € £ satisfying b = f on an open neighborhood o f K . Proof For each M 6 K , choose an open neighborhood Vj^ of M in and an element a € £ satisfying clm = / on Vm - Since Ea is locally compact, we may assume each Vm has compact closure. By compactness, we can choose a finite subcover Vj, Vi, ■■., Vn with corresponding elements ax,a2, .. . . a , in £ satisfying aj = f on Vj for j = 1 , 2 , . . . , n. Lemma 7.2.8 gives the desired result. □ We can now obtain a useful description of which elements of the full direct product YIm ^ a ( A/ M ) belong to the subdirect product A or to the image in A of a left ideal £ of A.
7.2.10 T heorem Let A be a completely regular algebra and let £ be a (not necessarily proper) left ideal o f A ■ The image £ of £ in the subdirect product representation A C IImsSa ( A/ M ) is the set o f f € I l ^ 63jl (A/M.) which belong to £ locally. If A is unital, A is the set o f f € YIm£Sa(A/M) which belong to A near each point of Ea • Proof If / belongs to £ locally, then it belongs to £ at infinity so there is some compact set K C and some a € £ satisfying a = f on the complement of K . Furthermore, / belongs to £ near each point of and a fortiori near each point of X so there is some b 6 £ satisfying b = f on an open neighborhood of K . Apply Lemma 7.2.8. If A is united, every / 6 ^Me=.A( A / M ) belongs to A at infinity. □ We can use the above description to obtain some remarkably detailed information about the structure of a completely regular algebra. We give an example.
7.2.11 P roposition Let A be a completely regular algebra. For any n € N let H o , H i , . . . , H „ be disjoint closed subsets o f E a with H i, H? , . . . , Hn compact and letao,ai,... ,a„ € A be arbitrary. Then there is some a € A satisfying a, = aj on H j, for j = 0, 1 , . . . , n. Proof Define H to be the set and 8 to be the algebra A/k(H). Then B is completely regular and Eb may be identified with H = U"_0Hj. Regard each Hj as a. subset of Eb and define / 6 ( B/M) by f ( M ) = (aj + k(H)) + M for M € Hj, j = 0 ,1 ,2 ,..., n. Then / belongs locally to B, so Theorem 7.2.10 shows that some b = a + k(H) € B satisfies b = / on Eb = H. Thus a 6 A has the desired property. O Theorem 7.2.10 describes which elements / € H x 6=> ( A/ M ) have the form 6 for some element b in a left ideal £. If the algebra is assumed to
7.2.12
Completely Regular Algebras
637
be strongly semisimple as well as completely regular, so that the subdirect sum representation is faithful, we obtain information on which elements a € ,4 belong to C. We will now apply this result to the case of an ideal, where it can be sharpened by the following lemma.
7.2. 2 Lemma Let A be completely regular and letX be an ideal o f A . For each a e A , a, belongs to X near each point in h({a})° or in the complement o f h( I ) . Proof Since zero belongs to I and 6 = 0 holds on h({a})°, a belongs to I near each point of h({a})°. If M. belongs to s.A \ h(T), we can choose a neighborhood V of M with compact closure disjoint from h(T). Proposition 7.2.5 shows that A;(V) is modular so Proposition 7.1.9 shows that X contains a relative identity e for k(V). Hence (ea)T = a holds on V . Since ea belongs to X, this shows that a belongs to X near each point of E^ \ h(X). □
7.2.13 C orollary Let A be a strongly semisimple completely regular al gebra, and let X be an ideal o f A . (a) An element a 6 A belongs to 1 if and only if a belongs to X near each point o f h(X) \ h({ a}) ° and at infinity. (o) I f V C S^t is an open set containing h(I) and possessing a compact comj lement, then k(V) is included in I . Proo' Theorem 7.2.10 and Lemma 7.2.12 establish (a). Conclusion (b) is an inmediate consequence. □. We now investigate the connection between the structure spaces of an algebra and the Gelfand space of its center. The next result is related to Proposition 7.1.11. For a commutative spectral algebra C, there is a close connection between its Gelfand space I c and its strong structure space Sc- The map which takes 7 6 Tq into its kernel is a bijection of Tc onto Ec = lie. However, we endow the Gelfand space with its Gelfand topology and the structure spaces with their hull-kemel topologies. Hence this map is a homeomorphism if and only if C is completely regular.
7.2.14 P roposition Let A be a spectral algebra with center A z(a) Each primitive ideal in Il^t \ hn ( A z ) w modular with a relative identity in A z ■ (b) The map V ►-» V D A z sends \ hn ( A z ) into TlAz = E^z . (f ) If A z is completely regular, this map is continuous and the images of b:: h n A \ hn ( A z ) and \ h~( Az ) are closed. (d ) If (c) holds and A is ( semisimple / strongly semisimple ) then the imagi Of { n A \ hn ( A z ) / ZA \ hs ( A z ) ) is all of HAz '-‘■Az • Proo} (a), (b): Theorem 4.2.11 shows that for each primitive ideal "P of A the center of A/V is isomorphic to C when it is non-zero. However,
638
7: Structure Spaces
7.2.15
the map c + (V C\A z ) c + V is obviously an injective homomorphism ° f A z/ (V H A z ) into the center of A/V which has non-zero range when V belongs to 11 ^ \ hn ( A z ) Thus, the preimage o f 1 6 C ~ ( A/V) z contains an element e € A z which is a relative identity for V- Furthermore, the isomorphism A z / ( V H A z ) — C shows that V n A z belongs to S ^ 2. Theorem 4.1.9 shows EAz = H^z . (c), (d): At first, we assume that A is unital (so that hn ( A z ) is empty) and we work exclusively with 11a \ hn (Az)If B is any closed set in II^Z then Vo 6 11^ belongs to the closure o f its preimage in 11a if and only if it satisfies Vc D D{ V 6 11A : V D A z £ B}. But this is equivalent to Vq fl A z 2 fV e fiP , which implies Vo H A z G B since B is closed. Since the preimage in 11^ of each closed set is closed, the map V *-* V n A z is continuous from 11a to H^z . Since is compact and nAz is Hausdorff, this shows that the image of 11^ is closed. Finally if A is semisimple, then ripeIjA(V fl A z ) — (Hpgn^'P) D A z = {0 } implies that the image of 11^ is dense. Since this image is also closed, it is all of Ua2 . This establishes the proposition for when A is unital. Suppose A is nonunital. Then ( A 1)z is naturally isomorphic to ( A z ) 1Proposition 7.2.2 identifies n^i and TI^i with 11^ U {o o } and II^Z U {o o }, respectively, under suitable topologies, jfince V >—» V(~\Az carries the point oo (= A in n ^ i) onto oo (= A z in H ^ i), the nonunital case follows from the united case. The proof for the strong structure space is similar. □ Note that the above theorem does not assert that the map V >—>VC\Az is injective on either 11^ \ hn ( A z ) or EA \ h ( A z )• This is not true in general, but the cases in which it is true are of sufficient interest to have been studied. The two parts of the following definition are due to Kaplansky [1949a] and Yosinao Misonou [1952], respectively.
7.2.15 Definition An algebra A is said to be ( central / weakly central ) if ( hn ( A z ) / h ? ( A z ) ) is empty and the map V >-► V fl A z is injective on ( IU / Ea )• A central algebra is obviously weakly central. An algebra which satisfies only hn ( A z ) = 0 is called quasi-central
7.2.16 T h e o re m A central spectral algebra A satisfies HA — SA . Proof Proposition 7.2.14(a) shows that each primitive ideal V of A is modular and hence is included in a maximal modular ideal M (Theorem 2.4.6). This implies V n A z Q M n A z - Proposition 7.2.14(b) shows that V fl A z is maximal and hence equal to M fl A z - Since V >-► V fl A z is injective on 11a, we conclude V = M O Finally we use Proposition 7.2.14 to show that certain weakly central spectral algebras are completely regular. This result is due to Willcox [1956a],
Completely Regular Algebras
7.5 ...7
639
7.3.17 T h e o re m Let A be a weakly central spectral algebra with a com pletely regular center. Then A is completely regular and M •-* M n A z -.3 z homeomorphism o f "S.a onto a closed subset o f E az . If A is either strongly semisimple or central and semisimple, then M h M (1 A z is a homeomorphism of Ea onto Ea z ■ Proof Proposition 7.2.14(c) shows that M >—>M f t A z is a continuous map of E a onto a closed subset of Ea z • Since A is weakly central, this map is a bijection onto its range, and since A z is completely regular, this range is Hausdorff. Hence E a is Hausdorff. Let M o be an arbitrary point in S a - Use the complete regularity of A z to choose a neighborhood V of M o H A z in 5 a z such that k(V) is a modular ideal in A z with relative identity e. Let V be the preimage of V in Ea (i.e., V = { M 6 Ea ■M Pi A z G Vr}). Proposition 7.2.14(a) shows that 3ach M € V has a relative identity e' in Az - Since e + ( M fl A z ) and e! + ( M fl A z ) both equal the identity element of A z / ( M fl A z ) , we conclude e —e' € M fl Az - Hence, e is a relative identity for each M in V. Thus, e is a relative identity for k(V). Therefore, V is a neighborhood of Me, /ith modular kernel. This completes the proof that A is completely reguiar. Lst M o be an arbitrary point in and let AT be a compact neigh borhood of Mo- Then the restriction of the map M >-► M fl A z to K is a. continuous bijection of a compact space onto a Hausdorff space and is therefore a homeomorphism. Thus, the map M >—>MC\Az is a homeomor phism of Ea onto its range. If A is strongly semisimple, Proposition 7.2.14 sh3Y/s that the range of this map is Ea 2 - If A is centred, Theorem 7.2.16 shcrra 11^ = Ea- Therefore, semisimplicity implies strong semisimplicity.□ See also John Francis Rennison [1982], [1987], [1988], [1990] and Sin-ei Tekaiasi [1981], [1983a], [1983b], [1986].
7.3
Primary Ideals and Spectral Synthesis
Identifying the ideals of an algebra A is always one of the most basic and . most important tasks in understanding the structure of the algebra. The :n£-.xi nal modular ideals and the primitive ideals are usually much easier to find than the general ideals. (In many Banach algebras the prime ideals are r.c: easily found.) The ideals which are intersections of maximal modular ideal, or of primitive ideals are therefore tractable. Another way to express th:s property of an ideal is to say that it is the kernel of its hull, taken with respect to Ea or 11^, respectively. Thus, one is led to ask which ideals in an algebra are of this type. When A is a spectral normed algebra, any intersection of maximal modular or of primitive ideals is closed so that one should ask which closed ideals are the intersection of maximal modular or
640
7: Structure Spaces
7.3.1
of primitive ideals. This is often called the problem of spectral synthesis. In order to use the theory of completely regular algebras, we concen trate on the question of which closed ideals are the intersection of maximal modular ideals. The following definitions are essentially due to Georgi Evgenyevic Silov [1947a] and Willcox [1956a].
7.3.1 Definition Let A be an ( algebra / spectral normed algebra ). Then A is called an ( N-algebra / topological N-algebra ) if each of its ( ideals / closed ideals ) with nonempty hull in is equal to the kernel of its hull. It is called ( Tauberian / topologically Tauberian ) if each of its ( ideals / closed ideals ) is included in some maximal modular ideal. An ideal which is included in exactly one maximal modular ideal is said to be primary. If a primary ideal V satisfies h~(V) = {A 4}, then V is said to be M.-primary. We call A a ( weak N-algebra / weak topological Nalgebra ) if each of its ( primary ideals / closed primary ideals ) is maximal modular. Let A be an algebra and let H C 5 ^ be a hull. Let J { H ) be the sum of all ideals of the form k(V) where V is an open set in which includes H, and let J ( H , oo) be the sum of all ideals of the form k(V), where V is an open subset of E^ including H with compact complement. If H = { M } is a singleton, we write J { H ) — J { M ) and J ( H , oo) = J { M , oo) and if H is empty, we write ,7(0, oo) = J ( oo). Note that an ( algebra / spectral normed algebra ) is an ( N-algebra / topological N-algebra ) if and only if each ( homomorphic image / continu ous homomorphic image ) of A is either Brown-McCoy radical or strongly semisimple. (The term N*-algebra used by Silov [1947a], Willcox [1956a], Rickart [1960] and others must be changed since an asterisk in the name of a class of algebras is now universally used to denote a class of algebras with involution.) Our definition of Tauberian algebras differs from that given by Willcox [1956a] and Rickart [I960]. We will show that a strongly semisimple completely regular Banach algebra is topologically Tauberian in our ter minology exactly when it is Tauberian in their terminology. For other algebras, our definition seems more useful. The terminology arises from a close connection with the Tauberian theorems of harmonic analysis. Note that an ( algebra / spectral normed algebra ) is a ( Tauberian N-algebra / topologically Tauberian topological N-algebra ) if and only if each of its ( ideals / closed ideals ) is the kernel of its hull. This happens if and only if each of its ( homomorphic / continuous homomorphic ) images is strongly semisimple. Our definition of primary ideals is a considerable generalization of the classical notion of primary ideals used in commutative algebra (c/., e.g., Os car Zariski and Pierre Samuel [1958]). However, the primary ideals defined
7.3.2
Primary Ideals and Spectral Synthesis
641
here play approximately the same role in the present theory as the classical ones do in commutative algebra. Note that in a spectral normed algebra, the closure of any .M-primary ideal is .M-primary. A maximal modular ideal M is always .M-primary. An .M-primary ideal which is not equal to M is obviously not the kernel of its hull. Hence, any ( N-algebra / topo logical N-algebra ) is a ( weak N-algebra / weak topological N-algebra ). If ar algebra is not a weak N-algebra, we may hope to write some ideals as an intersection of primary Ideals even if they are not an intersection of maximal modular ideals. Corollary 7.2.13 suggested the definition of J ( H ) and J ( H , oo). If Vi and V2 are two open sets in S A both including H, then Vi fl V2 is another such set. Hence, the equation k(V 1 D V2) = fc(Vj) + ^(^ 2 ) shows that J ( H ) can also be described as the union of all ideals of the form k(V) for V an open set including H. The same reasoning shows that J ( H , 00) is the union of all ideals of the form k(V), where V is an open subset of EA including H with compact complement. O f course if A is unital, J ( H ) and J ( H , 00) coincide. I- A is strongly semisimple and we consider it as identified with its subd'rect product representation in YIm £=.a A / M or in Sza (A), then the ideal J ( H ) is the set of elements which vanish near H, and J ( H , 00) is the set of elements which vanish both near H and at infinity. Thus for M G EA , A / J { M ) can be identified with the algebra of germs of functions vanishing near M . This is of course a standard construction in sheaf theory (cf. Roger Godement [1958]). 7.3.?: P r o p o s itio n Let A be a strongly semisimple completely regular algeb~a. For any hull H C J ( H , 00) is minimal among ideals with hull Y. (That is, J { H , 00) satisfies h ( J ( H, 00)) = H and is included in a: y ideal X o f A which satisfies h(X) — H . ) In particular, for each M € E^, J ( M , 00) is minimal among all M -primary ideals. Proo/ First we note that h ( J ( H, 00)) equals H. The inclusion H C h (J ( H, 00)) is obvious. Suppose M G does not belong to H. Since Ea is locally compact, we can choose an open neighborhood V of M in E,4 v ith compact closure disjoint from H. Then U = 3 A \ V~ satisfies k(U) C J ( H , 00). Hence, h( J ( H , 00)) C hk(U) C \ V does not include M . Since M $ H was arbitrary, this shows H = h ( J ( H , 00)). If J is an ideal satisfying h(X) = H, and V is an open set including H with compact complement, then Corollary 7.2.13(b) shows k(V) C I . Hence we conclude J ( H , 00) C l □ 7.3.3 C orolla ry If A is a strongly semisimple completely regular spectral normed algebra, then J ( H , oo)- is minimal among closed ideals with hull H. (That is, J ( H , o o ) ~ is a closed ideal with hull H and is included in
642
7: Structure Spaces
7.3.4
any other closed ideal with hull H .) In particular, fo r each M is minimal among all closed M -prim ary ideals.
6 S^,
Proof As noted above, the closure of an .M-primary ideal in a spectral normed algebra is .M-primary. □ 7.3.4 T h e o re m Let A be a strongly semisimple completely regular ( al gebra / spectral normed algebra ). (a) A is an ( N-algebra / topological N-algebra) if and only if { J ( H , oo) / J ( H , o o ) ~ ) equals k(H) for each nonempty hull H in H^. (b) A is a ( weak / weak topological) N-algebra if and only if { J ( M , oo) / ) equals M for each maximal modular ideal M o f A. (c) A is ( Tauberian / topologically Tauberian ) if and only if { Jf(oo) / J ( og)~ ) equals A. Proof These results are immediate consequences of ( Proposition 7.3.2 / Corollary 7.3.3 ) which shows that any ( ideal / closed ideal) of A satisfies ( J ( h ( I ) , oo) Q I C kh( l ) / J (f c ( I ) ,o o ) - C I C kh( I) ) whether h( I ) is arbitrary nonempty (a), a singleton (b) or empty (c). □ Specializing to the case of a strongly semisiinple completely regular Banach algebra A , we wish to find conditions on the faithful subdirect product representation A —* A C UM f =A ( A/ M) related to the properties of Theorem 7.3.4. Note first that A is a normed subdirect product in the sense that lla lL
=
8UP{II“ ( - ^ ) I I : M
=
sup{||a + M\\ : M € E ^ }
€ H^}
is finite and hence defines an algebra norm ||■||A on A (where, of course, ||a(A<)|| = ||a+ A4|| = inf{||a+6|| : b € M ] is the quotient norm on A / M ). Clearly this norm satisfies N L s IMI
v a€ A
It is not usually complete; in fact, the inverse boundedness theorem shows that it is complete if and only if it is equivalent to ||•||. We say that A C IIM£Ea ( A/ M ) is closed under multiplication by C(Ea ) if for each / € C (E ^) and o € A, there is a b 6 A satisfying b( M) = f ( M ) a ( M )
V M e S A.
The element b is then denoted by fa . 7.3.5 T h e o re m Let A be a strongly semisimple completely regular Ba nach algebra. Let ||•||A be equivalent to || • || and let A be closed under
7.2.6
Primary Ideals and Spectral Synthesis
643
multiplication by C (E ^). Then A is a topological N-algebra if it is a weak topological N-algebra. Proof The last theorem shows that we may assume J ( M , oo)~ = M. h~ "U M e Ea- Hence, we need only show J ( H , oo)~ D k(H) for all rc^siiipty hulls H in Since J ( oo)- includes J ( M , oo)- = M for each' M G S.*, it includes k(H). Hence, for any b € k ( H ) and any e > 0, there is some open set Uq in with compact complement and an element ce £ ‘.{Uo) satisfying ||6 —co|| < £. Let Ko be the compact set H \ <70. The equation J ( M , o o ) ~ = M shows that for each M in Ko there is an open neighborhood Um of M and an element cm € k(U_M) satisfying ||6—cjk|| < e. Since Kq is compact, it is covered by a finite family U\, U2, ■■■, U„ of thes^ neighborhoods Um- For j — 1 , 2 , . . . , n, let Cj € k(Uj) be an element satisfying ||6 —c,-|| < £. Since S .4 is locally compact and Uo U t/j U •••U U„ is aa open set including H with compact complement, there is an open set which includes the complement of Uo U U\ U •••U Un but has compact closure disjoint from H . Since Uo, U\, ■■■, Un+i is a finite cover of S ^, there exists a partition of unity /o , / 1 , •••. fn+i subordinate to this cover. (That is, for j — 0 ,1 , 2 ,. .., n + 1 there exists a function f j 6 C (S ^ ) with range in [0,1] satisfying f j ( M ) = 0 for M Uj and f j ( M ) = 1 for all M € Ea- See Bourbaki [1966] IX.4.3 for a proof.) Define c 6 A by C =
fo C Q
+
flC l
+
•• • +
f n Cn
+
J n + \ b
and U C by U = \ U~+1. Since each cj belongs to k(Uj) and each f j vanishes off Uj for j = 1 ,2 ,..., n, the element c belongs to k(U). Since U is an open neighborhood of H with compact complement, the element c belongs to J ( H , 00). For each A4 € E^, we get ,
Hc(M)-b(M)H
=
n n \\'EfAM)tj ( M ) -' E f A M ) b ( M ) \ \ i=0
<
j=0
;' t f j (M)\\£j ( M ) - b ( M ) \ \ 3=0
n
<
^ 2 fj(M )\ \ C j-b \ \ < £ . i=0
I
*
Hence ||c — 6||A < e holds. Since ||•||A and ||•|| are equivalent and e > 0 iT O arbitrary, we conclude that b belongs to J ( H , o o ) ~ as we wished to show. □ 7.3.3 T h eorem Let A be a strongly semisimple completely regular Banach algebra. Let ||•||A be equivalent to ||•|| and let A be closed under multiplica tion by C (S ^ ). For each a € A , let the function M >—> ||a(.M)|| from to
7: Structure Spaces
644
7.3.7
R+ be continuous at zero and vanish at infinity. Then every closed ideal I o f A satisfies T = kh(T), so that A is a topologically Tauberian topological N-algebra. Proof The previous two theorems show that it is enough to establish ,7(oo)~ = A and J { M , o o ) ' 3 M for each M € EA . We verify both simultaneously. Let M o be ( a maximal modular ideal / all of .A ) and let a be an element of Mo- For any e > 0 there is an open set V with com pact complement satisfying ||a(.M)|| < £ for each M G V and_{ M o € V / no additional condition ). Let U be an open set satisfying U C V and ( M o € (7 / no additional condition ). Urysohn’s lemma (cfi e.g. Rudin [1974]) shows that there is an / 6 C(EA) with range in [0,1] satisfying f ( M ) = 1 for M $ V and f ( M ) — 0 for M € U. Define b to be fa. Then 11° — &IL the supremum of the numbers ||a(.M) - b(M)\\ = | f(M)\\a(M)\\ < e
if M e V .
The equivalence of ||•||A and the complete norm shows ( a € J ( M q , a 6 J { oo ) ~ ).
oo) -
/ □
The abstract commutative Banach algebra formulation of Ditkin’s the orem in Silov [1947a] is generalized to the setting of semisimple completely regular weak topological Banach N-algebras by Willcox [1956a], Corol lary 1.6.1. (In Definition 2.3 of this paper, change “/ ( i ) C J (M o)” to “ /( x ) n X ( M o ) ” .) We now explore the theory of primary ideals in more detail beginning with a beautiful result of Willcox [1956a]. 7.3.7 T h e o re m Let A be a strongly semisimple completely regular algebra. Then each ideal 1 o f A with nonempty hull is the intersection o f the set {1 + J ( M ,o o ) : M e h(I)} o f primary ideals. Proof Proposition 7.3.2 shows that for each M € h(2), the hull of T + J { M , oo) must be { M } so that this ideal is .M-primary. Obviously, the intersection n { I + J ( M , oo) : M € h(X)} includes I . On the other hand, for each element a € I + J ( M , oo), it is clear that d belongs to I near M and at infinity. Hence, each element of the intersection satisfies the criterion of Corollary 7.2.13(a) and thus belongs to J. □ It does not appear to be known whether the analogous topological ver sion of this theorem holds. We would want to show that if a spectral normed algebra or Banach algebra satisfies the hypotheses of the above theorem,
7.3.?
Primary Ideals and Spectral Synthesis
645
then each of its closed ideals J with nonempty hull satisfies J = n { ( I + h ( M , o o ))- : M G h ( l ) } . The set of ideals which are intersected in this equation is obviously the set of ck sed primary ideals which are minimal among closed primary ideals including J, so that this equation holds if J can be written as the intersec tion -.f closed primary ideals. Unfortunately, no proof seems to be known to slnw that the above intersection is included in J = (f){J + h ( M, oo) : M 6 h ( l ) } ) ~ . Some partial results are given in Willcox [1956a]. If A is a strongly semisimple completely regular ( algebra / spectral normsd algebra ), then A has another subdirect product representation be sides the representation A C HM€Sa( A/M) . For each M. € S.*, ( Propo sition 7.3.2 / Corollary 7.3.3 ) asserts that ( ,7(.M ,oo) / ,7 (.M ,oo)- ) is an .M-primary ideal. Hence the intersection of these primary ideals is the zero ideal and for each M € S ^, ( A / J { M , o o ) / A / J { M , oo)"" ) is an ( algebra / spectral normed algebra ) which is local in the sense that it has exactly one maximal modular ideal. For each a G A and M G define a ( M ) to be ( o + J ( M , oo) / a + J { M , oo)- ) ) in this quotient algebra. Then the following result is immediate. 7.3.8 P r o p o s itio n Let A be a strongly semisimple completely regular { algebra / spectral normed algebra }. Then the function a a defined above is a faithful ( representation / continuous representation ) o f A as a ( subc irect product / normed subdirect product) o f local ( algebras / spectral norm :d algebras ).
7.4
Strongly Harmonic Algebras
The hypothesis of strong semisimplicity (i.e., fl^ e = ^ M = {0 } ) seems unnecessarily strong for the above result since the conclusions may hold if ( r\Mz=.AJ { M ,o o ) / r\MeSAJ { M , o o ) ~ ) is zero and each J ( M , o o ) is a primary ideal. We will now introduce a class of algebras which are not all strongly semisimple, but have a very satisfactory subdirect product representa, :on in terms of primary direct factors. As a matter o f fact, for these algebras this subdirect product representation can be replaced by a repre sentation as an algebra o f sections of a suitable sheaf over Sa 33 base space in which the fibers are local algebras. This class of algebras was introduced by Kwangil Koh [1972]. His work is strongly influenced by a series of papers by Silviu Teleman [1969a], [1969b] and [T969c] which are presented in a connected way in Teleman’s survey [19"1] These results are also related to the works by John Dauns and Karl Keinr,: ’ h Hofmann [1966] and [1968].
646
7: Structure Spaces
7.4.1
7.4.1 D efin ition An algebra A is said to be strongly harmonic if for each pair M i , M j of distinct maximal modular ideals of A , there are ideals X\ and satisfying M j £ h(Xj) for j = 1,2 and J 1 J 3 = {0}. These algebras are closely related to completely regular algebras as shown in the next proposition. 7.4.2 P roposition The strong structure space o f a strongly harmonic al gebra is Hausdorff. Hence, a unital strongly harmonic algebra is completely regular. Conversely, a strongly semisimple algebra with a Hausdorff strong structure space is strongly harmonic. Proof Let M\ and M 2 be distinct elements of EA and let X\ and I 2 be the corresponding ideals guaranteed by Definition 7.4.1 satisfying J 1 I 2 = {0 } and M j <$. h(X3) for j = 1,2. Then h(X 1) U h(I2) 2 h{I\X?) = SA shows that EA \ h(I 2) and EA \ h(I\) are disjoint neighborhoods of and J 2 , respectively. Hence, EA is Hausdorff, so if A is united it is completely regular. Suppose A is strongly semisimple and its strong structure space is Haus dorff. Then for any two distinct ideals M i and M 2 in S ^, choose disjoint open neighborhoods Vi and V2 of M\ and M 2 , respectively. Then the ideals I\ and J 2 of Definition 7.4.1 may be chosen as Ji = k(EA \ Vi) and X2 = k(SA \ V2 ) since these ideais satisfy M i £ EA \ Vi = h(Xi), M 2 $■ H^\Va = h{X2) and XxX2 C Z , n l 2 = k(ZA \Vi)nk(SA \V2) = k(SA) = {0 }.
□
For an algebra A which is strongly harmonic rather than strongly semi simple and completely regular, it is not clear that J ( M , 00) will be a pri mary ideal when A f is a maximal modular ideal of A. The next definition gives a suitable substitute for J { M ) . 7.4.3 Definition For each maximal modular ideal M in A, the M component o f the zero ideal is the ideal X ( M ) defined by I ( M ) = {a 6 A : a A lb = {0 } for some b 6 A \ M } Since a maximal modular ideal is prime, it is clear that 1 ( M ) is included in M - (To see this, note that o 6 1 ( M ) implies A * a A lb = {0 } C M for some b in A \ M - Since M is prime, this implies a 6 A 1 a C M - ) We can describe X ( M ) in terms of the left and right annihilators introduced in Definition 1.2.1. X ( M ) = U{ ( A lb)LA b 6 A \ M ) = {a € A : ( oA ^ ra % M ) . Hence, if a belongs to T { M ) for some M € EA , then U - EA \h(( aA1 ) RA) is a neighborhood of M in EA and a belongs to each M ' in U.
7/, /
Strongly Harmonic Algebras
647
We will now show that 1 ( M ) ifl well-behaved in certain strongly harmeric algebras including unital ones. A satisfactory theory depends on melting some hypothesis similar to assuming that the algebra is unital, since T ( M ) is an analogue of J ( M ) rather than J ( M , oo). 7.4.4 P r o p o s itio n The following two conditions are equivalent fo r an algebra A in which each primary ideal is modular. (a) A is strongly harmonic. (b) For each M G T ( M ) is the smallest M.-primary ideal (i.e., T ( M ) satisfies h ~ ( l ( M ) ) = { M } a n d l ( M ) is included in a ny V satisfying fts <~) = {A<})Proof (a)=>(b): Suppose M is a maximal modular ideal of A and V is an yVi-".imary ideal. We first show that any a 6 T ( M ) belongs to V. We have noted that ( uA1) ra is not included in M . If V + ( aA 1)nA were not equal to A, it would be included in some maximal modular ideal M ! / M by Theorem 2.4.6. Hence A = V + ( aA 1)RA holds so that the relative identity s for V can be written a s e = p + b for p e P and b € ( aA l )RA- Therefore, a = a(p + b) — (ae — a) = ap + (ae — a) belongs to V as we wished to show This again shows J ( M ) C M (i.e., M € h ( I ( M ) ) ) and that any //'-prim ary ideal of A includes T( M) . Next we show that no ideal M ' € 5U different from M belongs to h(T( M) ). Given such an M ' , we have ideals T and T o f A satisfying I % M , T % M ! and I ' l = {0 }. Hence we may choose 6 € T \ M satisfying I ' A lb = {0 }. This implies 1' C I ( M ) . Thus T g M ' Implies 1 ( M ) 2 M 33 we wished to show. (b)=»(a): Let M\ and M 2 be distinct ideals in E^. By condition (b), we may choose a € T ( M z) \ M i - Hence, I j = A 1a A 1 £ and I 2 = (a ^ ')flA 2 M 2 satisfy I i l 2 = { 0}. □ For a strongly harmonic algebra in which each primary ideal is modular, ths above proposition and Theorem 7.1.7 show that M / T ( M ) is the unique maximal modular ideal of A/ T( M) and that if Z is any ideal o f A properly included in I ( M ) , then A/T fails to have a unique maximal modular ideal. 7.<1.3 P r o p o s itio n An algebra A satisfies C\m €'S.a T (M ) = {0 } if any of the following conditions hold: (e) A is unital; (b) -Ala = {0 } and A is Tauberian; ■c) -Ala = {0 } and A is a topologically Tauberian spectral normed alC*£ 0 T £ ,.
Proof (a) If a is non-zero, then aA = a A 1 is non-zero so that (oA)/ia is a p-opar ideal. Theorem 2.4.6 shows that there is some maximal (necessarily modular) ideal M which includes ( oA) ra- This implies o ^ 1 ( M ) . The conclusion follows.
648
7: Structure Spaces
7.4.6
(b) and (c): If a is non-zero, either hypothesis shows that there is a maximal modular ideal M which includes (a ^ 1) ^ - Hence, a does not belong to X ( M ) . Since a was an arbitrary non-zero element, C\m£3.a I { M ) is zero. □ We will now give the representation theory for a unital strongly har monic algebra as the algebra of all continuous sections o f a sheaf of algebras. The last result shows that the representation is faithful.
7.4.6 Definition
A shea} o f algebras is a triple (S, 7r, B ) consisting of two topological spaces S and B, a continuous surjection it: S —> B, and, for each b e B, an algebra structure for 7r—1 ( 6), satisfying: (a) Each s e S has an open neighborhood U in S such that the restric tion of 7r to U is a homeomorphism of U onto n(U) which is an open set in B\ and (b) For each A 6 C, the function s —►As is continuous from S to S. The functions ( a , ( ) - * s + ( and (s, t) —» st are continuous from {(s, t) e S x S : jt(s) = ir(t)} C S x S in its relativized product topology to S. For each subset V of B, a function / : V —> S is cfilled a continuous section over V if it is continuous and k o / is the identity map on V . The algebra (under pointwise operations) of continuous sections over V is denoted by r ( V , S ) . The algebra T(B, S) is denoted by T (5) and is called the algebra o f global continuous sections o f S. Finally the sheaf is said to be soft if each continuous section over a compact subset of B is the restriction o f a global continuous section, and is said to be unital if each fiber is a unital algebra and the section which takes the value 1 at each point of the base space is continuous. For a more detailed general discussion of sheaves see Godement [1958]. Other more specialized papers are cited below.
7.4.7 Proposition Let A be an algebra and let S( A) be the disjoint union \j{A/J{M) : M 6 H ^}. Let 7r:S(^ 4) —» be defined by n(a + J ( M ) ) = M . For each a 6 A , let a: —» S(^4) be the global section of it defined by a ( M ) = a + 1 ( M ) . Then {a(U) : a 6 A\U open in S ^ } is the base for a topology on S(A). Under this topology, ( S(A),n, H^) is a sheaf o f algebras and fo r each a € A , a: —* S( A) is a continuous section. Moreover, f : S a —» S( A) is a continuous section if and only if each point in has a neighborhood on which f agrees with some a. Proof First we show that the set described in the proposition is the base for a topology. If b + 2 ( M ) € 5(^4) is a point in d(U)(~\c(V) for some a, c € A and open subsets U, V of E^, then a - b,c — b and hence a — c belong to I ( M ) . By definition of T ( M ) , this implies ( ( a — c ) A 1) r a %. M . Let W be the open subset U n V fl (E^ \ h(((a - c ) ^ 1) ^ ) ) of E^ which includes M .
Strongly Harmonic Algebras
649
Then b + X (M ) is contained in the basic open set a(W ) = c(W ) which is inch ded in a(U) D c(V). Hence we have a base for a topology. Clearly for each a + 1 ( M ) € S(A) and each open neighborhood of M in S^, a(V ) is mapped by it bijectively onto V = ir(a(V)). However, { a(U) ■ U C V is open in H^} is a base for the topology of a(V) since c(W ' Pi a(V') - d ( W D V) holds. Thus ix is also a homeomorphism when restricted to V so that (a) of Definition 7.4.6 is satisfied. Condition (b) of this definition is trivial in this case so that (S ( A ), tt, S^t) is a sheaf of algebras. Prom the definition of the topology, it is obvious that a is a continuous section for each a £ A In order to prove the last sentence, let / : Ey, —►S(A) be an arbitrary section, and let M o G ~ a be arbitrary. Let a(V) be a basic open neighborhood o f f { M o ) . Then f*~(a(V)) = { M 6 V : a ( M ) = f ( M ) } is an open set if and only if the condition is satisfied.□ The above proposition shows that any algebra A has a homomorphic image A which is a subalgebra of T(5(>t)). Clearly, the kernel of this representation is fU ies^ 1 { M ) so that the representation is faithful if this intersection is zero and a fortiori if A is unital or strongly semisimple. However, this is just a reformulation o f the subdirect product representation o f A in llM€s AA / l ( M ) and suffers from all the weaknesses of such a representation unless it can be shown that A is actually all of T(5(>t)). A numier of hypotheses have been explored which force this equality. We ment:on the following papers: Alexander Grothendieck [1960], Dauns and Hoffmann [1966], [1968], Richard S. Pierce [1967], Teleman [1969a], [1969b], [19711, Koh [1971], [1972]. (Several of these papers speak of the base space as Pji, but if PA is Hausdorff in a unital algebra A , then PA equals S^.) We now give the theorem from the last of the above references starting with a result which we would call a lemma except for its independent interest. 7.4.8 P r o p o s itio n Let A be a strongly harmonic algebra and let K C be compact. (a) For any M o $ K , there are ideals I\ and X2 o f A satisfying M o 0 h(I j ) , K n h ( l 2) = 0 and T & = {0 }.
(b) Furthermore, K equals the hull h(C\{2{M) : M G i f } ) . Proof (a): For each M € K , choose ideals Z ^ 1 and satisfying M o £ h(T{y ), M 0 /i(J^*) and = {0 }. Since K is compact, it is covered by a nnite collection of the open sets \ h f t ? 4). Let the corresponding pairs of ideals be (J j, Jo), (2 ? ,2 ? ),. ••, (2T*,2?) and define Jj and J 2 by 2, = and I 2 = Then K is covered by UJ= 1 (S^ \ ii(i*2 ; . = E^\D]J= 1 /i(I2) = E ^ \ /i(J 2) and M o is not an element of h(l\). Also 17,12 = {0 } is obvious. (b}: The obvious inclusion C\MeKl{M) C fijug k M = k (K ) implies K C i (<1m £ k Z { M ) ) . Suppose some M o G h(C\M£Kl{M)) is not in K. Then, (a) gives ideals I i and la satisfying Mq 4 /i(I j), K n h(X3) = 0 and
7.4.9
7: Structure Spaces
650
I i l 2 = {0 }. Hence any M e K satisfies 1 ? % M which implies 1\ C 1 ( M ) . This gives Xi C r\M eK%( M) . Since we assumed M o G h{C\Me KX {M )) , this leads to the contradiction Ji C Mo° 7.4.9 T h e o re m Let A be a unital strongly harmonic algebra. Then the sheaf (S(A), tt, E^) is a unital, soft sheaf o f unital local algebras. Further more, a i-* a is an isomorphism o f A onto the algebra T(5(^4)) o f global continuous sections o f this sheaf. Conversely if (S, 7r, B ) is any unital, soft sheaf of unital local algebras over a compact Hausdorff space B, then T(S) is a unital strongly harmonic algebra and B is naturally homeomorphic to (sjProof Proposition 7.4.5 shows that a a is an isomorphism. Next we show that S( A) is a soft sheaf and that A includes (and hence equals) r(S(./4)) by showing that, for any compact subset Kq of (Ko = E^ is possible since A is unital) and for any / 6 r ( K o , S ( A ) ) , there is an a € A satisfying a|tf0 = / . Since is Hausdorff (Proposition 7.4.2), A'o is closed so Uo = Za \ A'o is open. Since / is continuous, Proposition 7.4.7 shows that for each M € A ’o there is an open set Um in E^ and an element aM e A satisfying (im — f on Um n Kq. Relying on the compactness o f Ko, choose a finite subcover U\, Ui, ■■. ,Un o f the cover {Uj^ '■ M € A'o} and let a i,a 2 , - - - , a „ be the corresponding elements of A . Then Ea equals UJL0Z7*. Define K k and 2* by A * = \ Uk and I fc = f\{J{M) : M € Kk) for k = 0 ,1 , . . . , n. Each K k is compact since it is closed in E which is compact by Proposition 7.1.6. Proposition 7.4.8(b) now shows h\ = h (I k) for k = 0 , 1 , . . . , n. But this gives 0 = n^=cK k = D"k=0h(Ik) = h(Y^k=cZk), which imphes A = ]£fc=o^k- Hence, there are elements ek € Z* satisfying 1 = S k =o e*- For each M G Ko O K k, we have a k{ M) ek( M ) = 0 = f ( M ) e k( M ) and for each M G K 0 D Uk, we have ak( M ) e k( M ) = f ( M ) e k( M ) , so that we get akek = f e k on Ko for all k — 1 ,2 , ..., n. Define a G A by a = eo + $3"=i a*efc- Then for each M G A'o i we find a{M)
=
n e o(M ) + ^ a fc(M )e fc(A<)
=
f ( M ) e 0( M ) +
n f ( M ) e k( M ) k= 1
=
n f ( M ) C £ e k) ( M ) = f ( M ) k=0
since e o ( M ) is zero. Thus we have established that (S(A), sheaf and (taking Ko = S ^) that A equals T(5(>t)).
it, E^)
is a soft
Strongly Harmonic Algebras
651
Let ( 5 , 7r, B) be a unital soft sheaf of unital local algebras over a compact Hft'jsdorff space. The constant section 1 is an identity element for T(5). j o t 3 ’ ch b € B, let Mb be the unique maximal ideal of 7r~1 (b). Since the sheaf is soft, for any b 6 B the set M b = { / € T (5) : f(b) 6 M b ) is a proper subset of T(S) which is obviously a (necessarily modular) ideal of T(S). Since / > —> f(b) is a homomorphism of T (5) onto 7r_1 ( 6) and Mb is maximal in 7r_ 1 (6), Mb is maximal in T (5). Furthermore, since the sheaf is soft, Mb and Mb' are distinct when b and b' are distinct elements of B. rlence, { M b '■b € B } is a subset of Sp(s)- Next we show that this subset equal:; E r ( s ) . If X is any proper ideal of T (5 ), we show that there is some b 6 B satisfying I C Mb- Suppose not. Then for each b € B the ideal {/(b) : f € J } of tt- 1 ( 6) must be tt_ 1 ( 6) itself. Hence for each b we can find fb € X satisfying fb{b) = 1. For each b 6 B, there is an open neighborhood Ub of b satisfying fb(d) = 1 for all d G Ub- Choose a finite subcover {Ubk : k = 1 , 2 , . . . , n } and apply Lemma 7.2.8 to this subcover and the corresponding family of functions {fbk : k = 1 , 2 , . . . , n } . This gives the contradiction: l e i . We conclude that b i-» Mb gives a bijection of B onto =r(S)Next we must show that the map b i-» Mb is a homeomorphism. Since both B and Er(s> are compact and Hausdorff, this will be true if the inverse of this map is continuous. Hence we show that if £/ is an open set in B, then { M b : b € U} is open in Sr(s)- Let 6q G U be arbitrary. Since the sheaf is soft and B is compact, we can find / 6 T(S) satisfying /(b o) = 1 and f(h) = 0 for all b e B \U . Hence, S r(s) \ { M b : b e U } = { M b : b e B \ U } is inc'uded in h ( { f } ) . But then, Mb^ € Er(s) \ h({f}) Q { M b ■ b 6 U) shows that the arbitrary point Mb^ in the image (under b >—> Mb) of U is in t ie interior of this image so that the image of U is open and b >-►M b is a homeomorphism. □ The representation described in the last theorem applies to a number of classes of Banach algebras which are important in analysis. However, since the fibers in a sheaf are necessarily discrete, this representation ignores t })3 topological structure of these algebras. This is at the very least uni&esthetic, and suggests the possibility of a deeper representation theory of topological algebras by continuous sections of some structure more general than a sheaf. Such a theory has been worked out in detail by Dauns and Zio&nann [1968], This beautiful theory is so technically complicated that we will not present even a survey of its results. However, we remark that this theory gives a detailed faithful representation theory for arbitrary C*algebras and in this respect far transcends the theory described here. One reason for the wider applicability of this representation theory is the fact tl'.Ez it is based on the refinement of the subdirect product representation of algebra A over n^t (rather than E^) and hence is faithful for semisimple
652
7: Structure Spaces
7.4.10
rather than strongly semisimple algebras. (All C*-algebras are semisimple.) 7.4.10 E xa m ple There are many examples of completely regular and strongly harmonic algebras. Theorem 3.2.10 shows that a completely reg ular commutative Banach algebra (in the sense of Definition 3.2.9) is a completely regular algebra in the sense of Definition 7.2.1. Examples of commutative completely regular algebras include: ( 1 ) Co(fi) for any lo cally compact Hausdorff space ft (equivalently any commutative Banach algebra satisfying the hypotheses of Theorem 3.2.12, i.e., any commuta tive C*-algebra); (2) for each n € N, the algebra C ^ "'([0,1]) of n-times continuously differentiable functions on [0, 1 ] with the norm
ll/ll = E ^ r ^
V /ec<»>([0,ll);
fc=0
( 3 ) the algebra C BV( [ 0 , 1 ]) of continuous functions of bounded variation on [0 , 1 ] with norm
ll/ll = ll/lloc + II/II b v
V /eC BV([0,l]);
and (4) the classical group algebra L l (G) for any locally compact abelian group. Most of these commutative examples have suitable noncommutative; generalizations. Any algebra A with a discrete strong structure space is obviously completely regular since any M. G has the neighborhood { M } which has A4 as its modular kernel. A fortiori, any algebra with a discrete structure space n^t is completely regular. Theorem 8.4.5 shows that all modular annihilator algebras have discrete structure spaces and hence are completely regular. The next Chapter has many examples in Section 8 .8, the diagram of Theorem 8.7.11 and Example 8.1.2. In particular, L l ( K ) is completely regular when K is a compact group. John R. Liukkonen and Richard D. Mosak ([1974b], Corollary 2.5) and Eberhardt Kaniuth and D«tlef Steiner ([1973], p. 324) independently and by different methods extend this last observation (and a result of Will cox [1956b]) to show that Ll (G) is completely regular for any group G in [FLA]- . A group is said to belong to [FIA]- if it is locally compact and the closure of the group of inner automorphisms in the group of all homeomorphic automorphisms (with its natural topology) is compact. (The abbreviation stands for “finite inner automorphism” and the superscript bar indicates that “finite” should be changed to its topological equivalent “compact” .) Note that this is a common generalization of the result for compact and for abelian groups. Kaniuth and Gunter Schlichting ([1970] Satz 2) and Mosak [1972] showed that the Banach algebra Ll (G) is also strongly semisimple for groups G in [FIA]- . Siegfried Grosser and Martin A. Moskowitz [1971b] obtained the same result for [MAP] groups. The class
Strongly Harmonic Algebras
653
[MAP] was introduced in §3.2.17. The abbreviation stands for “maximally " periodic” and a group G belongs to the class [MAP] if it is locally com pict and has enough finite-dimensional irreducible representations to separate points or, equivalently, if there is some continuous injective ho momorphism of it into a compact group. For many related results see the present author’s survey [1978]. In Volume II, we will show that a C*-algebra A has a Hausdorff structure space Ha if and only if the function V i-> ||o + P|| is continuous on for each a € A. It then follows easily that A is completely regular. In addition, Yosinao Misonou [1952] shows that W*-algebras are weakly central. FYed B. Wright [1954] obtains the same result for AW*-algebras. Theorem 7.2.17 then thows that these algebras are completely regular. Willcox [1956a] notes that the algebra of matrices over C"*([0,1]) is comp.etely regular and that in this case J{to)~ is the set of matrix functions satisfying //j(£o) = /»j(*o) = 0 for all i , j . The results cited in §1.10 shows that certain other tensor products are completely regular. Proposition 7.4.2 shows that strongly semisimple algebras with Haus dorff structure spaces are strongly harmonic. A semisimple commutative algebra with Hausdorff Gelfand space is therefore strongly harmonic. This includes all the commutative examples mentioned above. Similarly we con clude that L l (G) is strongly harmonic if G is an [FIA]— group. If >4 is any AW*-algebra and M i and M 2 are distinct ideals in E^, then there are central idempotents ej £ A4j for j — 1 , 2 , satisfying e\e2 = 0 . Hence AW*-algebras (and a fortiori W*-algebras) are strongly harmonic. These algebras are not strongly semisimple unless they are finite (in the sense defined below). 7.4.11 C en troid s, D o u b le C en tralizer A lg eb ra s and E xtensions Proposition 7.2.14 shows that the structure space of a spectral algebra A and o ' its center A z may be identified if, for instance, A is strongly semi simple and central and A z is completely regular. Theorem 7.2.17 explicitly states the analogous conclusion for strong structure spaces. Quite gener ally there is a connection between the structure spaces of an algebra and its ce ter. However, the center of a nonunital algebra may well reduce to {0 } a aking it useless. The center of A 1 is not a useful substitute in the presp’ t context. For instance, if A is nonunital with a locally compact rlaug' orff structure space, then A 1 may have a structure space which fails to It iausdorff or even T\. These considerations lead us to consider the ce^trc d which is very well behaved. D eii ‘tion The centroid of an algebra A is the center of the double centrailer algebra T>(A) of A. ? •^position 1.2.3 shows that the centroid of A can also be described as
654
7: Structure Spaces
7.4.11
the subalgebra of C(A) consisting of maps T satisfying T(a)b = T(ab) = aT(b)
V o, b e A.
For a Banach algebra with trivial left and right annihilator ideals, Theorem 1.2.4(b) shows that all the maps in the centroid are continuous. The idea of the centroid was independently introduced by Robert C. Busby [1968] and Jacques Dixmier [1968], both working in the context of C*-algebras. A systematic investigation was begun by Dauns and Hofmann [1968], III .6 and [1969]. Under various hypotheses, they determine the structure space of the centroid. This work has been continued by Dauns [1969], [1974] and closely related work is found in Busby [1967], [1971b]. In general, these references deal with various special cases of the problem of finding the structure space of an algebra A in terms of the structure space of an ideal I of A and the structure space of A/1. The best results are obtained in the case of C*-algebras, and some of them will be discussed further in Volume II. Another result on C*-algebras is closely enough related to the present considerations to warrant discussion here. This is the Dauns-Hofmann theorem which asserts that the centroid of a C*-algebra A can be identified with £7( 11 .4 ), so that for each T in the centroid there is an / € C’(IT^) satisfying T(a) + V = f { V ) a + V
V V G II*; o € A.
This was first proved in Dauns and Hofmann [1968]. Other more transpar ent proofs have been given in Dixmier [1968], Theorem 5, Gert K. Pedersen [1972] and George A. Elliott and Dorte Olesen [1974]. The last cited proof is also the easiest and establishes a Banach space generalization.
8
Algebras with Minimal Ideals
Intrc. uction I-: Shis chapter we investigate the role which minimal left ideals play in z'8'i algebras which possess them. ( Minimal here simply means minimal "r e s : inclusion among all non-zero left ideals.) Significant results usually de'Dsr d on assuming that the algebra is semiprime. However, the stronger assumption of semisimplicity, which is common in other areas of our theory, is comparatively seldom needed here. Furthermore, norms behave particu larly 'f/ell on minimal ideals so that a number of results can be proved for normed algebras without assuming completeness. In particular, any norm is spectral on many of the algebras considered in this chapter. Ws will naturally be concerned mainly with algebras which not only have some minimal left ideals, but “enough” minimal left ideals in some sense. Such algebras are quite special, but they have a correspondingly rich theory. Many classes of algebras with “enough” minimal left ideals have been defined, and towards the end of the section we will prove all the indusions which hold between these various classes. (These results are sum marized in a diagram of implications in Theorem 8.8.11.) However, we will cor j x trate attention on the class of modular annihilator algebras which was introduced by Bertram Yood [1958] and has been intensively studied by Ycod [1964] and by Bruce A. Barnes [1964], [1966], [1968a], [1968b], and [1271c■,]. This class seems to give a particularly good compromise between axioms strong enough to give a significant theory and axioms weak enough to be satisfied by most important examples. Furthermore, modular an"ihi!e,tor algebras are defined purely algebraically while many of the other classa; are defined in terms of mixed algebraic and topological criteria. This provides a particularly elegant and straightforward theory. In a semiprime algebra A , every minimal left ideal has the form A e for an k empotent e e A such that eA e is a division algebra. Conversely, A e is s, :r ; ' .-nal left ideal for any such idempotent. These idempotents are called minimal idempotents. If A is a normed or spectral algebra and e and / are minimal idempotents, then eA e and e A f are both one-dimensional (unless the latter is { 0 }). The sum of all minimal left ideals in a semiprime algebra A is the ideal (hot jiist left ideal) generated by the set A m i of all minimal idempotents. This ideal, which was introduced by Jean Dieudonne [1942], is called the •
■!
655
656
8: Algebras with Minimal Ideals
8.0
socle of A and is denoted by A f - It is also the sum of all minimal right ideals. In a semiprime algebra A , the socle and Jacobson radical satisfy A f A j = A j A f = { 0 } and this makes the Jacobson radical tractable. If £ is a minimal left ideal of A, then the restriction of the left regular representation of A to act on £ is an (algebraically) irreducible represen tation of A- Conversely, any irreducible restriction of the left regular rep resentation has this form. These irreducible representations have pleasant properties which are described in Proposition 8.2.6 and in Theorem 8.3.6 for semiprime normed or spectral algebras. In particular, these representations are uniquely determined by their kernels. Any faithful irreducible representation T of an algebra A with a non zero socle maps the socle onto the algebra of those finite-rank operators on the representation spare which belong to T 4 . This allows a very concrete description of any such representation, which we give in Proposition 8.3.3 and Theorems 8.3.4 and 8.3.6. Theorem 8.4.5 contains eight equivalent conditions on a semiprime al gebra which we use to define modular annihilator algebras. The irreducible representations of these algebras are restrictions of the left regular represen tation and hence are uniquely determined by their kernels. Furthermore if A is such an algebra, then the structure space 11 ^ of A is discrete, the hull of A f is empty, every left ideal of A which is not a subset of A j includes a minimal left ideal, and A j is the left and right annihilator of A f - Theorem 8.4.12 shows that all norms on modular annihilator algebras are spectral. Proposition 8.3.3 shows that ideals and semiprime quotients of modu lar annihilator algebras are again of the same type. Furthermore, in any semiprime algebra A, any ideal of k(h( Ap) ) is a modular annihilator al gebra. This allows the introduction in Theorem 8.5.5 of a very detailed Fredholm theory (due to Barnes [1968a]) for elements in k(h(AF)) for any semiprime algebra A. Since any modular annihilator algebra A satisfies A = k( h( AF)), this Fredholm theory holds for all elements in these alge bras. This in turn gives a purely spectral characterization of semisimple modular annihilator algebras among semisimple algebras as those for which no element has a non-zero limit point for its spectrum. This result of Barnes [1968b] is given in Theorem 8.6.4. Section 8.7 introduces many types of algebras which have been shown to have a large socle in some sense. These include dual, annihilator, left anni hilator, completely continuous, left completely continuous, compact, mod ular annihilator, and Duncan modular annihilator algebras. After defining all these terms, we establish their logical relationships. Theorem 8.8.11 presents all these results in the form of a diagram of implications. Enough examples are included to show that no implication beyond those in the the orem are possible between these concepts for semisimple Banach algebras. A number of results are proved for left annihilator and annihilator algebras.
Finite-Dimensional Algebras
8.1.0
657
Finally, we prove the automatic continuity result of Barnes [1967b] which asserts that any homomorphism of a Banach algebra onto a dense subalge bra of a semisimple Duncan modular annihilator is necessarily continuous. Since minimal ideals play so important a role throughout this section it may be well to review the simplest and most concrete representative case before beginning to consider the more abstract results. Let X be a normed linear space and let B( X) be the normed algebra of all continuous linear operators on X . For any x e X and u j 6 X * the symbol x ® u j denotes the eiemei_t of B( X) defined by x ® uj(y) = uj(y)x
V y £ X.
For ea :h fixed non-zero u> £ X* the set X ® u j = { x <3 u j : x £ X ) is obviously a non-zero left ideal of B( X) which contains no smaller non zero left ideal of B(X). Conversely, if £ is any minimal left ideal of B(X), choose a non-zero T £ C and uj' £ X* to satisfy uj = T ' ( uj' ) 0. Then for any x £ X , x ® u j = (x
8.1
Finite-Dimensional Algebras
We begin this chapter by devoting a section to consideration of finitedimenuional algebras. A simple dimension argument shows that any non zero left ideal in a finite dimensional algebra contains a minimal left ideal.
658
8: Algebras with Minimal Ideals
8.1.1
Starting from this observation, we could prove the next two theorems by ar guments similar to those which we use in the rest of this chapter. However, we prefer to give simple proofs based on Theorem 4.2.13. Wedderburn Theorem Our first theorem is due to Elie Cartan [1898]. The theorem was gen eralized to finite-dimensional algebras over arbitrary fields by Joseph H. Maclagan Wedderburn [1907] and to rings satisfying the ascending and de scending chain conditions on left ideals by Emil Artin [1926]. Hence this theorem is often called the Artin-Wedderburn theorem. Charles Hopkins [1939] showed that only the descending chain condition on left ideals is needed for its proof. We could easily prove the theorem with this hypoth esis, but we will not do so. (Of course the generalized form of the theorem without further hypotheses does not give full matrix rings over the com plex numbers, but rather matrix rings over some division ring.) This brief review o f history shows that the theorem deserves to be called the Cartan theorem, but, in part to avoid confusion with other theorems by Cartan, we will follow common usage and name it after Wedderburn. We mention first that every finite-dimensional algebra is a Banach alge bra under some norm and that any two such norms are equivalent. In fact any two linear space norms on a finite-dimensional space are easily seen to be equivalent by a simple basis argument. The same argument shows that any linear operator on a finite-dimensional space is continuous with respect to any norm on the space. To see that any finite-dimensional algebra A can be made into a Banach algebra one can proceed as follows. Choose a basis for A 1. This establishes a linear isomorphism between A 1 and C” for some n. Transfer the Euclidean linear space norm (or any other norm) from Cn to A 1 ■ Now the extended left regular representation is an isomorphism of A into B(A*). Hence the algebra norm o f B ( A l ) can be transferred to A. Since A is finite-dimensional it is complete in this norm. 8.1.1 W ed d erb u rn T h eorem A finite-dimensional prime algebra is isomorphic to a full matrix algebra and hence is simple and unital. A finitedimensional semiprime algebra is isomorphic to a direct sum o f finitely many full matrix algebras and hence is strongly semisimple and unital. Proof Let A be a finite-dimensional prime algebra. Finite dimensionality implies the existence of some minimal left ideal C. (Simply look for a left ideal of smallest dimension.) Let T : A —> C(C) be the restriction of the left regular representation to C, i.e., Tab = ab
V a G A\ b G C.
Then T is irreducible since C is minimal. The kernel of T satisfies ker(T)£ = {0 }. Since A is a prime algebra (i.e., {0} is a prime ideal) and C is non
8.L ?
Finite-Dimensional Algebras
659
zero, this implies ker(T) = {0 }. Hence T is both faithful and irreducible. By ths remarks preceding this theorem A is a Banach algebra. Hence 4.2.13 applies and shows that T is strictly dense. However, it is an immediate consequence of the definition of strict density that a strictly dsras -epresentation on the finite-dimensional linear space C must have all c.' Z(C) as range. Hence T induces an isomorphism of A onto the full matrix, algebra M n where n is the dimension of C. Since M n is simple and united this concludes the proof of the first assertion of the theorem. Now let A be a finite-dimensional semiprime algebra. The last para graph shows that each prime ideal V of A is maximal modular since A/V is a rvll matrix algebra which is simple and unital. A dimension argument sl’ows ;hat there is a finite set Pi, V%, ■■•, Vn of prime ideals in A satisfying = {0 }. We may assume that no proper subset has intersection {0 }. For each a £ A define B(a) by 0(a) = (a + V i , o + V2, . . . , 0 + Vn).
Then 0 is an injective homomorphism of A into the algebraic direct sum (Bj=l J-/Vj. By what we have proved above, each quotient algebra A / V j is isomorphic to a full matrix algebra. Thus it only remains to show that 8, is surjective. This will follow if we show that for any k satisfy ing 1 < k < n and any b e A / V k there is an a G A satisfying 6(a) — ( 0 ,0 , . . . , 0, b, 0 , . . . , 0), where the non-zero entry is in the fcth position. This is equivalent to showing
a=(
n ‘PA+'Pk
i<j<"
for any k satisfying 1 < k < n. However this is an immediate consequence of the fact that Vk is a maximal ideal. □ S .l.S E xam ple We give a simple, completely explicit example of what this decomposition into full matrix algebras looks like in a concrete case. Indeed we have chosen the simplest example which is interesting— namely tb.g s^r-ciimensional algebraic group algebra of the symmetric group on three objects. Since this algebra is not commutative it must contain a full matrix &!ge,b",?. of dimension strictly greater than one, and hence one knows a priori that it Js the direct sum of A/2 and two copies of C. Here are the details. Demote the symmetric group of all permutations of the numbers { 1 , 2 ,3 } by I<. As usual let (12) and (123) denote, respectively, the permutation interc.ianging 1 and 2 and the permutation which replace 1 by 2, 2 by 3, kt.c i by 1 . Using this notation, we denote the six group elements by: e = ics".':ity transformation, a = (12), b = (23), c = (13), d = (123) and / = ' . . 2) . The group multiplication table follows.
660
8: Algebras with Minimal Ideals e
a
b
c
d
f
e
e
a
b
c
d
f
a
a
e
d
f
b
c
b
b
f
e
d
c
a
c
c
d
f
e
a
b
d
d
c
a
b
f
e
f
f
b
c
a
e
d
8.1.2
Let A be the algebraic group algebra (Section 1.10) of £3 which in the case of a finite group such ae £ 3 is just the underlying algebra of the normed £l- or L 1-group algebra. It is natural to identify each group element with the delta function concentrated at it. Then e is an identity for A. Consider the following elements of A:
h = z( a + b + c + d + e + f),
k — ^ ( - o - 6 —c + d + e + / )
o
j=
o
i ( 2e - d - / ) .
These are orthogonal central idempotents with sum e = 1 so that A has the algebra direct sum decomposition A = hA © kA © j A with hA and k A one-dimensional and j A four-dimensional. If we define
X o,X i:£3 -» C by Xo(
1
=
/ \ X' i9)
=
9 G £3
/ +1
9 = d,e o r /
{
g = a,b or c
- 1
we see: gh = hg = Xo(g)h,
gk = kg = \i(9)k
Vg e £3.
This gives the multiplicative structure of the one-dimensional minimal ide als hA = C/i and k A = C k in A. In order to obtain the structure of the four-dimensional minimal ideal j A we let ( be a complex cube root of 1 (e.g., £ = (—1 ± \/3*)/2 and consider the two orthogonal noncentral pro jections m = i(e + £d + £2/ ) and n = j( e + C^d + £/)•) It is easy to check j = m + n and that %: j A —* M 2 is an isomorphism when we define:
0) ’
?)-
8
.
!
Finite-Dimensional Algebras
*(“"*) =(5 o)
***
661
^On) = (o o)'
Thus we have written A as a direct sum of minimal ideals isomorphic to C, C and M 2. The restriction of the left regular representation of A to the ;wo dimensional subspace spanned by m and am yields the group (or algebra) representations tt: E 3 —> M 2 (or n : A M 2) given by:
*(e)-(J J)
»<“> = ( ? J )
'< « = ) = ( “
.< D -(£
£ )
°)
* < « = ( £ 0) A D - ( \
£ )■
Similar explicit calculations have been carried out for many small finite groups. The difficulty increases rapidly with the size of the group. We will show later that a similar analysis is possible for compact groups. Again the details are complicated in all but the simplest cases. The following corollary is taken from Peter G. Dixon and George A. Willis [1993]. The second statement depends on Theorem 8.1.1. Theorem 4.8.9 and Corollary 4.8.10 are closely related. 8.1.0 C orolla ry Let A be a commutative Banach algebra. Then A is unital if and only if it satisfies A 2 = A and it also has a modular closed nilpctent ideal. In particular, a finite-dimensional commutative algebra A is unital if and only if it satisfies A 2 = A. Proof If A is unital, A ? = A and we may choose the ideal { 0 }. Suppose, conversely, that I is a modular closed ideal and n is a positive integer such that all bi, 62, ■■•>bn 6 I satisfy 6162 •■•bn = 0. Let e be a relative identity for T. Then any 0 1 , o2, . . . , o „ € A satisfy
(1 - e)n0 i 02 •• an = (1 - e )o i(l - e)o 2 • • • (!- e)on = 0 . Hence the n-fold quasi-product power of e, which we denote by u, satisfies u aic-2 ■•■an = a i 02 •••o „ for any n-fold product O1 O2 ■■•a„ in A . However, by induction, A? = A implies that every element of A is a sum of n-fold products, so that u is an identity for A. If A is finite-dimensional, Theorem 8.1.1 shows that A / A j is unital, so A j is modular. Any finite-dimensional algebra can be given a norm in which it is a Banach algebra. Furthermore A j is a closed nilpotent ideal, so that the last paragraph shows that A is unital if A 2 — A. □
662
8: Algebras with Minimal Ideals
8.1.4
Feldman Theorem The next theorem is due to Chester Feldman [1951], It is an improve ment of another theorem due to Wedderburn [1907] which deals with finite dimensional algebras. The theorem shows a case in which one can be sure that a nonsemisimple algebra can be suitably decomposed into its Jacobson radical and a subalgebra isomorphic to its semisimple quotient. After the theorem we will give an example, also due to Feldman, of a Banach algebra A in which there is no subalgebra B such that A is the linear space direct sum of A r and B. We will also introduce some standard terminology and additional results. 8.1.4 Feldm an Theorezn Let A be a Banach algebra such that A / A j is finite-dimensional. Let A —* A / A j be the natural map. Then there is a closed subalgebra B C A satisfying: (a ) B + A j = A . (b) B n A / = { 0 }. (c) $ induces a homeomorphic isomorphism of B onto A / A j . Proof The last theorem shows that A / A j is isomorphic to M nk. Choose an isomorphism of Ylk=i M nk onto A / Aj . Let {a^ : 1 < i, j < n*} be the image in A / A j of the matrix units in the fcth matrix under the chosen isomorphism. Proposition 4.3.12(e) shows that we may choose orthogonal idempotents {e^ ■ 1 < k < n ;l < i < n*} in A such that $ (e£ ) = for all i and k. The identities o ^ ofjof, = and show that for all k , i , j satisfying 1 < k < n , l < i < nk and 1 < j < nk we can choose bf, € e^ A e^ and b^ £ ej, A e^ satisfying $ ( 6*',) = a*, and $ ( 6ij) = o fj. Then the equation $ ( 6^ 6*,) = = o f, shows that there is an element c* e A j fl e ^ A e ^ satisfying 6^ 6*, = ef, — cj. Proposition 2.5.3 shows that A j O e ^ A e ^ is a spectral subalgebra of A and Theorem 4.3.6(b) shows that c* has a quasi-inverse, which we will denote by dj, in A and hence in A j H e ^ A e ^ . Define e*, and e\j by e*, = 6*,(ef, —d f) and e = b^j. Then we have
Now we define
by e* = e*, e\} . Clearly these elements satisfy * ( < £ ) = a,*
and
ef c™ = 6km6jPek iq
for all 1 < k, m < n, 1 < i , j < n *, 1 < p,q < n m (where (5jtm is the Kronecker delta and the second equation just reflects the definition of matrix units in the direct sum decomposition M „ k). Hence if we denote the
Finite-Dimensional Algebras
663
linea: span of these elements by B, then $ induces an isomorphism of B onto A f A j . Since B is finite-dimensional, it is complete and hence closed in and the isomorphism from B onto A / A j induced by $ is a homeomorphism. Since $ maps B onto A / A j we conclude, B + A j = A. Finally, 3'” C'; 3 has the form eA e, where e = Y12=i S ? = i e?» *s idempotent, and since 3 is semisimple, Proposition 4.3.12(b) implies B fl A j = {0 }. □ 8.1.5 F eldm an’ s E xam ple The following example is due to Feld man [1951]. However some of the results given here were first obtained by William G. Bade and Philip C. Curtis, Jr. [1960a], Consider the com mutative algebra A 0 of all finite sums n
^ 1Aj Cj 4- nr j= i
V Xj, n €: C
v/kers {ej : j € N} is a family of mutually orthogonal idempotents and r is an element satisfying r 2 = rej — ejr = 0 for all j G N. Norm these elements by || 'jh Ajej + H | = m a x { ( ^ ] |Aj|2) 1/2, |/i - ^
j =i
A ,|}.
j= i
It :s straightforward to check that this is an algebra norm (submultiplicar tiviiy is checked in the first reference above). Let A be the completion of .4 0 :r this norm. It is easy to see that the Jacobson radical of A is the 3 r.3-cimensional space Cr. Furthermore, in the quotient norm we have || Xjej + p r - h Aj\\ = |Aj|2) ,/2 j= l j =l Soi that the map
n A~
£
A^ eJ + •4 j )
v A € *2(N)
j =i
is an isometric isomorphism of -£2(N) with pointwise multiplication onto j . Hence A / A j contains the element YITLi j ~ l (ej + A j ) (convergence L" .its quotient norm since Yl'jLi J -3 ~ T /®) while A does not contain any dement of the form Yl’j Li 3 ~lei because this series diverges in A since
E yOC.= l 3•_ 1 = OO. T ’ie above remarks imply that there is no closed subalgebra B of A .satisfying A = B © A j (see Theorem 6. 1 (b) of our second reference for a prcof). Hence Feldman’s Theorem (8.1.4, above) fails for this Banach
664
8: Algebras with Minimal Ideals
8.1.5
algebra. Since the radical of A is one dimensional and nilpotent this shows that no simple hypothesis on the radical itself will be strong enough to imply the conclusion of Theorem 8.1.4. We follow Theorem 8.6.1(c) of our second reference in constructing a nonclosed subalgebra B of A which satisfies A = B © A j . The map of t 2 onto B which we are about to define is a discontinuous isomorphism. Since l l (N) is a linear subspace of ^ (N ) ~ A / A j we may use a Hamel basis or Zorn’s lemma argument to construct a linear subspace y of l 2(N) satisfying ^ (N ) - /*(N ) © } \ For A € / ! (N) define <^>(A) by OO
‘f i W
=
X3e i 3= 1
which converges absolutely in the norm of A. For any A £ ^ (which must satisfy |A_, |= oo) we define y?(A) as the limit of the Cauchy sequence yj(A)„ = Ylj=i + (E j= i ■\>)r in A . It is easy to show that the nat ural map of A onto A / A j (or equivalently the Gelfand transform) maps ^>(£2(N)) onto ^ (N ) (with suitable interpretation of A / A j or of $.4 , respec tively). It remains to show that tp is a homomorphism and that y?(/2 (N)) is a subalgebra of A. Clearly the restriction of tp to I 1 (N) is a homomorphism and <^>(/! (N)) is a subalgebra. If \,fi are two elements of y , then we have n
n
n
j= 1
j= 1
j=1
n
n
+ C C ^ ) r) = ! C A^ > e> j= l
j= l
and this sequence converges to
8.1.?
Finite-Dimensional Algebras
665
This is a derivation since each term in the derivation equation 6(ab) = 6(a)b + a6(b) is zero. Similarly 6 is also an endomorphism. However, in the original norm of A this map is discontinuous. In particular, for any n € N we have ||£ " = 1 ^ + nr|| = n l/2 but ||<5(E"=i ej + nr)ll = ll5 M /n )+ ™ -)ll = IM I = n. Note that the algebra A has a dense socle in either of the two norms given here. Of course, the socle of any commutative algebra is defined. It is important to note that A is not semiprime since A j = A j A = { 0} holds. However it is obvious from the definition that Cr and Ce< for any i € N are minimal (two-sided) ideals of A so that the socle contains any element of the form Y^ij=i ^iei + lj r - ®y definition, this set of elements is dense in A in its original norm and it is also obviously dense in the norm |||■|||. We introduce some terminology and other examples related to this one.
8.1.8 D efin ition Let A be a Banach algebra and let A j be its Jacobson radical. We say that A has an ( algebraic / topological ) Wedderburn spliti'ng if there is a ( subalgebra / closed subalgebra ) B satisfying A = Aj+B
and
> 4 jn B = {0 }.
Of course, in either case $ : A —* A / A j induces a continuous algebra isomorphism of B onto A / A j which is homeomorphic in the case of a topo logical splitting by the open mapping theorem. In this terminology, Feld man’s theorem (Theorem 8.1.4) assures a topological Wedderburn splitting and his example (Example 8.1.5) has an algebraic but no topological Wed derburn splitting. The following elegant theorem o f Curtis and Richard J. Loy [1989] im proves a similar result of Gregory F. Bachelis and Sadahiro Saeki [1987]. The corollary indicates how this relates to the last example. 8.1.7 T h eorem Let A and B be commutative Banach algebras with A unital. Let tp,i>:A —* B be two continuous homomorphisms satisfying ip(a) —i/j(a) € B j for all a £ A . (a) For a G A g , lim ||o"|| ||o_ "||/n = 0 implies
666
8: Algebras with Minimal Ideals
8.1.8
(b): For any a € A, part (a) shows
or
{a G A : lim ||e"“ ||||e- " “ ||/n = 0.}
Then A does not have a topological Wedderburn splitting. Proof If A had a topological Wedderburn splitting, we could choose ip: A —* A to be the identity map and xl>:A—* A to be the composition of the natural map A —►A / A j and the inverse of the continuous (hence homeomorphic) isomorphism of B onto A / A j. The theorem would imply A j = { 0 }, con
trary to hypothesis.
□
8 .1.9 Non-splitting and Sets o f Non-synthesis The following inter esting examples are due to the combined work of Bachelis and Saeki [1987], Bade [1989], Curtis and Loy [1989] and Curtis [1989], based on the above theorem or variations of it. Sets of non-synthesis were introduced in §3.6.16 where the ideal J { S ) , which is minimal among all closed ideals with hull S, is also described (c/. Definition 7.3.1). (Our notation is essentially dual to that used in some of these references.)
Theorem
Let G be a noncompact locally compact abelian group and let S be a compact set o f non-synthesis in G. Then L1 (G)/J(S) has no algebraic Wedderburn splitting.
T h eorem Let S be a closed set of non-synthesis in R. Then L l (G)/Jr(S) has no topological Wedderburn splitting. A number of hypotheses are also known which force the existence of alge braic or topological Wedderburn splittings. The question of the uniqueness of the subalgebra or closed subalgebra B has also been discussed. See our chapter on cohomology in Volume II and Feldman [1951], Bade and Cur tis [1960a], [1960b], Barry E. Johnson [1968a], [1972b], Nikolai Valeryevich Yakovlev [1989] and the references above. Here are some references to the numerous papers giving criteria for a Banach algebra to be finite-dimensional: Kaplansky [1954], Dixon [1974], Abdullah H. Al-Moajil [1981], [1982], Leoni Dalla, S. Giotopoulos and Nelli Katseli [1989]
Minimal Ideals and the Socle
8.2
667
Minimal Ideals and the Socle
iotents and Ideals :r our discussion of algebras with minimal ideals certain idempotents > y a n important role. For this reason we recall some simple identities v/e have used before. For any idempotent e in any algebra A we have: A e = {o G A : a = ae} = {o e A : a (l — e) = 0}
( 1)
eA = {a € A : a = ea } = {a € A : (1 — e)a = 0}
(2)
eA e = {a € A : a = ea = ae} = {a € A : (1 — c)o = o ( l — e) = 0}.
(3)
All three of these sets are spectral subalgebras and are closed if A is a topological algebra. Furthermore, Proposition 4.3.12(b) shows that the last sat satisfies ( e A e ) j = e A e D A j . Recall also that two idempotents e f are said to be orthogonal if they satisfy e f = f e = 0 . Many of our preliminary results will deal with one-sided ideals. We will sometimes state these results, and even definitions, merely for left ideals. In most cases the result for right ideals follows immediately by considering the reverse algebra. It is easy to see when this is the case. However, the reader shorli bear in mind that a primitive ideal is not necessarily primitive in ' ; - verse algebra. Minimal Ideals S.S.j D efin ition Let A be an algebra. An idempotent e € A is called minimal if e.4e is a division algebra. The set of minimal idempotents in A is denoted by Am i- A left ideal of A is called minimal if it is minimal among jbe S3t of non-zero left ideals ordered by inclusion. If A is a topological algebra, a left ideal of A is called a minimal closed left ideal if it is closed 9xiC is minimal among the set of non-zero closed left ideals of A. Minimal 4qeaL<;, minimal closed ideals, minimal right ideals and minimal closed right ideals are defined similarly. The term “minimal idempotent” as used in the above definition seems to ■have been introduced by Charles E. Rickart [1960] although closely related concepts have a long history. (The related idea of primitive idempotents evp^orcurs in Wedderburn [1907].) It is justified by the next proposition. , .'artial justification can also be given in terms of the order structure of idempotents. If A is an algebra and e and / are idempotents in A , w-e write e < / if f e = e holds. This is equivalent to eA C f A . If A an algebra of linear operators on a linear space X , it is equivalent to ePi C_ f X . (This is the reason we are using principal right ideals rather than principal left ideals here. There is a different, but completely similar, partial order among idempotents based upon left ideals, and it enjoys the
668
8: Algebras with Minimal Ideals
8.2.2
same relationship to minimal idempotents that we are about to establish.) Now suppose / is a minimal idempotent and e is an idempotent satisfying e < / . Then we have e / e / = e e f = e / and f e f = e /. Hence e / = f e f is an idempotent in the division algebra f A f so that either e f = f or e f = 0 holds. Therefore / is minimal in the ordering just introduced if it is minimal according to Definition 8.2.1. The converse is not true. For instance, for any compact connected topological space fl, 1 is an idempotent in the commutative semisimple algebra C (fl) which is minimal in this ordering but does not satisfy Definition 8.2.1 unless fl is a singleton.
8.2.2 Proposition Let £ be a minimal left ideal o f an algebra A . If £ 2 is non-zero, then there is an idempotent e g A which satisfies £ — A e. Any inch idempotent is minimal. Conversely, if A is semiprime and e € A is a minimal idempotent in A , then A e is a minimal left ideal. Analogous results hold for right ideals. Proof Let £ be a minimal left ideal satisfying £ 2 ^ {0 }. Then for some (necessarily non-zero) a € £ ,£ a is a non-zero subset of A which is a left ideal included in £. The minimality o f £ implies £ = Ca. This implies the existence of a (necessarily non-zero) element e e £ satisfying ea = o. Similarly £ (e — 1) is a left ideal included in £. Minimality implies either £ (e — 1) = £ or £ (e — 1 ) = { 0 }. However, the first equation is impossible since it would imply the existence of an element b € £ satisfying 6( e - 1 ) - e which in turn implies the contradiction 0 = ba —ba = bea —ba = b(e —l)a = ea = a. Hence £ (e — 1) = {0 } holds and this implies e(e — 1) = 0 so that e is idempotent. Moreover, the inclusions £ D A e D {e } and minimality imply £ = Ae. Now suppose e is an idempotent satisfying £ = A e. Then ee = e belongs to £ and the subalgebra eA e has e as an identity. Any element eae € eA e \ {0 } satisfies £ 2 A eae D {eeae}. Hence minimality implies £ = Aeae. Thus there is an element b g A satisfying beae = e and hence satisfying (ebe)(eae) = e. Thus each non-zero element of eA e has a right inverse. The proof of Corollary 2.2.3 shows that eA e is a division algebra. Hence e is a minimal idempotent. Finally, in order to establish the converse, we assume that A is semiprime and that e is a minimal idempotent. Let £ be a non-zero ideal included in A e. Since £ 2 C A eC is not zero, e£ is not zero. Hence there is some element ae G £ satisfying eae / 0. Since e.4e is a division algebra there is an element b € A satisfying ebeae = e. Hence e belongs to the left ideal £ so that £ equals A e. Thus A e is a minimal left ideal. □
8.2.3 Corollary Let A be a semiprime algebra. Every minimal left ideal o f .4 has the form A e for some minimal idempotent e. Conversely, if e is a minimal idempotent, then A e is a minimal left ideal.
8.2.4
Minimal Ideals and the Socle
669
If A is also a topological algebra, then every minimal left ideal is closed, so that a left ideal of A is a minimal closed left ideal if and only if it is a minimal left ideal. Consideration of the simple case of a full matrix algebra shows that the idempotent e of the last corollary is not usually uniquely defined. 8.2.4 P r o p o s itio n The following two conditions are equivalent for a pair e and f of idempotents in an algebra A. (a) e f = e and f e = f . (b) A e = A f. These conditions establish an equivalence relation both on the set A i of all idempotents in A and on the set A m i o f all minimal idempotents in A . If A is semiprime the map e i—►A e establishes a bijective correspondence between the set o f equivalence classes of minimal projections and the set of minimal left ideals o f A . P rccf Condition (b) is equivalent to assuming both e € A f and / € Ae. Thuj it is equivalent to (a) by equation (1) above. The conditions ob viously define an equivalence relation on either set mentioned. The last proposition shows that the map mentioned establishes the stated bijective correspondence. □ We now mention another useful equivalence relation on A m i with larger equivalence classes. Let A be a semiprime algebra and let e be a minimal ideir potent of A so that Ae is a minimal left ideal and e^4 is a minimal righ , ideal. Then it is obvious that the restriction of the left regular repre sentation to A e is an irreducible representation. This representation is so useful that we introduce the special notation Le for it. Thus we have L*( 6e) = abe
V o, b e A .
(4)
The right regular representation is an anti-representation of A . An antireprasentation can be thought o f as a representation o f the reverse algebra A r . Using this viewpoint, we carry over all the terminology concerning representations to anti-representations. Then the restriction of the right regular representation to e A is an irreducible anti-representation. Again we give this anti-representation a special notation, /£*. Hence we have R£(eb) = eba
V a, b G A .
(5)
Now we introduce an equivalence relation on A m i by considering when Lf tJid U are equivalent for e, / € A m i • The notation Equiv(Le,L f ) and some remarks about equivalences were introduced following Definition
4.1.:. 8.2.5 P r o p o s itio n Let A be a semiprime algebra and let e and f be minimal idempotents o f A . For each b G e A f , let 14: A e A f be defined
670
8: Algebras with Minimal Ideals
8.2.6
by Vb(ae) = aeb for all ae € A e. Then the map b <-* Vb establishes a linear bijection between e A f \ {0 } and Equiv(L*, L ? ). The following are equivalent: (a) Le is equivalent to iJ . (b) R* is equivalent to R l . (c) e A f is non-zero. (d) f A e is non-zero. Proof Clearly Vj, belongs to Equiv(Le, Z /) for each non-zero b G e A f . Suppose V belongs to Equiv(Le,L,f). Let b = V(e). Then b G A f and eb = eV(e) = L { V ( e ) = V(L*d.) = V(e) = b imply b € e A f . Furthermore V(ae) = V(LZee) = Lle V(e) — aeb for all a e A implies V = Vb. Hence 6 t-» Vj is a linear bijection. This proves the equivalence of (a) and (c). The same argument in A R proves the equivalence of (b) and (d). Since e and / play symmetrical roles in (a) and (b), (c) and (d) are equivalent. □ Properties (a) and (b) in the last proposition are easily seen to be equiv alent when e and / are arbitrary idempotents in an arbitrary algebra. In this more general case they are each equivalent to the existence of elements c G e A f and d G f A e satisfying cd = e and dc = / . However, in our more special setting the easier criteria (c) and (d) will be useful. Note that the conditions just mentioned for pairs of elements of A i define an equivalence relation on A i which generalizes the equivalence relation on Ami defined by the conditions of the last proposition. The next proposition provides an alternative proof of the equivalence of the four conditions in the last proposition. 8.2.6 Proposition Let A be a semiprime algebra and let e be a minimal idempotent o f A . Then L* is an irreducible representation of A and any irreducible representation o f A with the same kernel as L' is equivalent to Lr . Furthermore, if A is a prime algebra, then Le is faithful so that A is in fact primitive. Proof Since Proposition 8.2.2 shows that A e is a minimal left ideal, Le is irreducible. Suppose T is another irreducible representation of A with the same kernel as L‘ . Then Te is non-zero so we may choose a non-zero vector z in TeX T . Define a map V : A e —* X T by V(ae) = Taez ~ Taz. Then we get TaV{be) = TaT^z — Tab^z = V(abe) — VLea(be) for all b € A. Hence V ( Ae) is a non-zero T-invariant subspace of XT. By irreducibility V is surjective. Similarly ker(K) is a left ideal of A e which does not contain e so that V is injective. Hence V is an equivalence. If A is a prime algebra then ker(L').4e = { 0 } and A e ^ { 0 } imply that Le is faithful so A is primitive. □ The following definition is due to Dieudonne [1942],
Minimal Ideals and the Socle
671
S.2.V Definition Let A be an algebra. The left socle of A is the sum of a!) the minimal left ideals of A. The right socle of A is the sum of all ■v: ?.al right ideals of A. If the left and right socle of A coincide then the’r common value is called the socle o f A and is denoted by A fHere, as usual, we follow the common convention that the sum of an err.piy family of one-sided ideals or ideals of A is { 0 }. Curiously, this common convention is not followed by many authors in dealing with left scclss, right socles and socles. Hence they say that a socle exists if the left s nd right socles exist, are non-zero, and coincide. In the same case we would say that A has a non-zero socle. When we say that A has a socle we do not imply that the socle is non-zero. Obviously the left socle is a left ideal and the right socle is a right ideal. We will now prove that in fact they are both ideals. 8.2,8 Proposition Let A be an algebra. The left socle and the right socle of A are both ideals. I f A is semiprime, these ideals coincide so A has a soc'a. In this case the socle equals the left ideal, the right ideal and the ideal geni. ated by the set o f minimal idempotents of A , and therefore A f H A j equois { 0 }. Prco; Let C be a minimal left ideal and let a be an element of A. Then Ca is a left ideal. If V, is a left ideal included in Ca, then {b € C : ba € TZ} is a left ideal included in C and hence is {0 } or C. Therefore Ca is either a minimal left ideal or zero. Hence the left socle © { £ : C is a minimal left ideal} is closed under multiplication from the right. Thus it is an ideal. Similarly, or by considering the reverse algebra, the right socle is an ideal. If A is semiprime, Corollary 8.2.3 shows that the left socle is the left ideal generated by Ami and that the right socle is the right ideal generated by Ami- The result just proved shows that in fact both the left and the right socle are the ideal generated by Ami- Hence they agree, so A has a Spcle. Suppose a non-zero o belongs to A f D A j . Then since o belongs to ^2e.PAMi t^lere 's some e € A m i satisfying ae 0 . By minimality A c is included in A f n A j. This implies e G A j which is impossible by ‘?Vdposition 4.3.12(a). Hence A p n A j is { 0 }. □ T ie first argument in the above proof is useful. It shows that if C is a minimal left ideal of A and a is an element of A, then Ca is either { 0 } or a mkr.mal left ideal. See also Abdullah H. Al-Moajil [1979] and Nelli Katseli (19841 for a new characterization of the socle. Th?. Presocle ‘ Iv(.ost of the theory in this chapter is developed for semiprime algebras and e /en semisimplicity is sometimes required in other treatments. In order
672
8: Algebras with Minimal Ideals
8.2.9
to extend some of the results to a wider context, M. R. F. Smyth [1976] introduced the concept of the presocle in an arbitrary algebra. 8.2.9 D efin ition The presocle A p f of an algebra A is the preimage in A of the socle in the semisimple quotient A / A j o f A, i.e., A p f — {a e A ■a + A j € (A / A j ) f }• 8.2.10 P r o p o s itio n In any algebra, the presocle is an ideal which includes both the left and the right tocle. Proof The presocle is obviously an ideal. Suppose £ is a minimal left ideal in A . If £ 2 is {0 } then it is included in the Jacobson radical. Otherwise, Proposition 8.2.2 shows that it has the form £ = A e for some minimal idempotenet e. Since e + A j is a minimal idempotent in A / A j , £ is again included in the presocle. The argument for the right socle is similar. □ In a semiprime algebra, A f + A j is a subset of the presocle. In fact, this sum is a direct sum by the last proposition. When are the two sets equal? Proposition 4.3.12(d) shows that the preimage of a minimal idemoptent in A / A j can be chosen to be an idempotent in A . When is it minimal? For additional results we refer the reader to the paper already cited and Smyth [1982], Barnes, Murphy, Smyth and West [1982] and Pietro Aiena [1987].
8.3
Algebras of Operators with Minimal Ideals
We will now investigate the socle of a strictly dense algebra of operators on a linear space. Note that such an algebra is primitive, hence semisimple, hence semiprime. Therefore the socle exists. We introduce some concepts and notation which will allow a rather concrete description of the socle in this case. 8.3.1 D efin ition A pairing between two linear spaces V and W is a bilinear map V x W —» C which satisfies ( (v, w) = 0 for all v € V ) ( (v, w) = 0 for all w € W )
=>
w = 0
=>
v = 0.
(1)
If T is an operator in £(V ), then an operator T* € £ (W ) is said to be an adjomt o f T relative to (■.■) if it satisfies (Tv, w) = (v.T^tu)
V v t V;u> G W.
( 2)
The set of operators in £(V ) with adjoints in £ (W ) relative to a pairing (•. •} is denoted by £( >(V).
8.3.2
Algebras of Operators with Minimal Ideals
673
A pairing between V and W is called a normed pairing if V and W are normed linear spaces and the bilinear map is continuous. The set of opera ,ors in B(V) with adjoints in B (W ) relative to a normed pairing (•, •) is denoted by B( )(V). It is easy to check that an adjoint is unique when it exists, and that £( ) (V) and >(V) are subalgebras of £ (V ) and B(V), respectively. We will denote the adjoint o f T 6 (V) by T*. Then the map T *-* T* is an antiisomorphism of C( ) (V) into £ (W ) and of B( ) (V) into B(W). For any subset B Q i ' ( ) (V) we will use to denote the subset B* = (T* : T G B} C £ (W ). If V is a normed linear space and W is a Banach space, the closed graph theorem shows B( )(V) = B(V) D C( )(V ). (A special case of this situation plays a prominent role in Volume II.) Next we will investigate the finite-rank operators in ) (V) and B( ) (V). (The rank of an operator is the dimension of its range.) If V is a linear space or normed linear space, we will denote the set of finite-rank operators in £ (V ) and in B(V) by £ f ( V ) and B f ( V ) , respectively. The results given here are due to Nathan Jacobson [1947]. We need an elementary lemma. 8.3.2 L em m a Let V and W be linear spaces and let {■, ■) be a pairing fo r V and W . For any finite linearly independent set {v i, V2, . . . , vn} in V there exists a (necessarily linearly independent) set {wi,u>2 , . . . , w „ } in W satisfying ( v i , W j ) = 6ij
1
Proof Let X be the linear subspace of V spanned by {t>i,V2, ■■. , v „ } and let ^ be the subspace of linear functionals on X induced by the pairing with elements of W . Since X is finite-dimensional, if y is not the entire space X^ of linear functionals on X there must be an element v G X satisfying oj(v) = 0 for all u> G This is impossible since it would contradict the second implication in ( 1 ). Hence ^ is just Vt and in particular there are elements of W implementing each of the linear functionals v, 6tj. □. Let {•, •) be a pairing between the linear spaces V and W . For any v € V and v) € W define maps v ® w: V —* V and w u: W —►W by v$$w(x) w
v(y)
=
( x,w)v
VxGV
=
(v, y)w
V y G W.
(3)
It is clear that v ® w belongs to £ (V ), w v belongs to £ (W ) and that w®* ./ is the adjoint for v ®w. Furthermore, these operators have rank one when they are non-zero. Hence any operator of the form n (4) j= i
8: Algebras with Minimal Ideals
674
8.3.3
belongs to £ f ( V ) fl £< )(V) and has the operator n
(5) as its adjoint. If is a normed pairing, the operator (4) belongs to B f ( V ) f l B( >(V) since both (4) and (5) are clearly bounded operators. Next we will show that, conversely, any operator in £ f ( V ) f l £ ( >(V) (or in B f (V) n B( )(V) if the pairing is normed) has the form (4). Suppose T is a finite-rank operator in £ ( >(V). Then we may find a basis { v j , V2 , . . . , u „} for its range and we have n Tx = ^ W j(i)t ij
V ie V
where each ui} is a linear functional on V. Use Lemma 8.3.2 to find elements Vi, !/a, ••■, Vn in W satisfying (u ,,y.) = 6{j for 1 < i, j < n. Then we have n
(Tx,yi) =
= W i(i)
for 1 < i < n. Hence if we define w, 6 W by w, = T^yx for 1 < i < n we have T = Y lj-i v}
Q
£ ( )(V);
Bf
=
£ f ( V ) n f i = £ f ( V ) n £ ( ) (V)
(6)
n
= =
Vj <8>Wj : n € N;vj € V ; Wj 6 W for j = 1 , 2 , . . . , n} fl{X : 1 is a non-zero ideal of B}\
B* is strictly dense on W.
(7) (8 )
Algebras of Operators with Minimal Ideals
675
Proof Since B is semiprime (it is even primitive), Corollary 8.2.3 showB that B has a minimal left ideal if and only if it has a minimal idempotent. T 7 / e will show that each minimal idempotent E of B has rank one. If E hts rank greater than one then there are independent vectors vi,v? in EV. Since B is strictly dense there is an operator T e B satisfying Tv j = ?, = v2- Hence E T E is non-zero so there is an operator S e E B E s&tieying S E T E = E. However, this contradicts Ev\ — t»j. Hence E has re "’:, one. 3'nce each minimal projection o f B has rank one, Proposition 8.2.8 shows 3 f — BB m i B C BnCp( V) . In order to establish the opposite inclusion and heac? prove the first equality of (7), we choose an arbitrary T e B nCp (V) . C cn 'd er the natural pairing V v 6 V; w € V*
(v,w) = w(v)
between V and V*. The remarks preceding the statement of this proposition show that we may write n T = j=1
where Vj 6 V, uij E V* and { v Jt v2, . . . , v„ j is linearly independent. The strict density of B shows that for any k = 1 ,2, ... , n there is an operator T V s B satisfying T^Vj = SkjVj. Then BTkT = B(vk
676
8: Algebras with Minimal Ideala
8.3.4
For any T e B the restriction of T* to W is the adjoint o f T relative to this pairing. This proves ( 6 ) which implies BF — £ f ( V ) n B C £ f ( V ) fl £( )(V). However we proved the penultimate equality of (7) before stating this proposition, and our remarks on the set W show that any operator of the form (4) belongs to £ f ( V ) n B . This completes the proof of (7) except for the last equality which we will prove after establishing ( 8). In order to prove ( 8), let {u>i, u>2 >•• •,«>„} and { y lt y2, . . . , yn} be ar bitrary subsets of W with the first linearly independent. Lemma 8.3.2 guarantees a set {jsj, Xa,. . . , i „ } in V satisfying (x3 ,Wk) = bjk- Define T 6 B by T = xj ® Vj- Then T x(wk) = £ " = i Vi x j ( wk) = Vk holds. Therefore B* is strictly dense, proving ( 8 ). To establish the last equality o f (7), it is enough to show that any non zero ideal I of B includes B f . In fact it is enough to show that I contains some rank one operator vo <S>wq since the irreducibility of B and B* then implies that I contains every rank one operator v < 8>w and hence includes Bp. If T € J is non-zero, choose x € V satisfying T x jL 0 and w € W \ {0}. Then Tx ®w = T( x ®w) is a rank one operator in B. Thus (7) is established.
□ We now give a variant of the last proposition which takes into account possible norms on B. For Banach algebras it is due to Rickart [1950]. 8.3.4 T h e o re m Let B be an irreducible spectral algebra of operators on a linear space V and let B have at least one minimal left (or right) ideal. Then there is a norm on V, a normed linear space W and a normed pairing (•, •) between V and W satisfying B C B ( >(V);
(9)
BF = BF( V ) n B = BF( V ) n B { >(V);
( 10)
Bi is strictly dense on W.
(1 1 )
IfB is a Banach algebra under any norm ||■||, then V and W may be taken as Banach spaces and any operator T in B satisfies ||Tl || < ||T||op < ||T||, where ||• denotes the operator norm on W and V. Proof Theorem 4.2.13 shows that B is strictly dense. Choose a spectral semi-norm a on B. Theorem 4.2.8 shows that the choice of any non-zero z € V gives rise to a norm on V by the construction of Theorem 4.2.7. Furthermore, relative to this norm, each operator in B is bounded. We obtain the linear space W as in the proof of the last proposition except that this time the elements of W are all continuous linear functionals in V*. Give W the norm which is the restriction of the norm of V*. Then the pairing between V and W is a normed pairing. For each T 6 B, T* will be the restriction to W o f I 1. Tlu.s proves (9). Hence (10) and (11) follow
8.3.5
Algebras of Operators with Minimal Ideals
677
directiy from the corresponding results in the last proposition. When B is a Banach algebra we use its complete norm in the above construction. Then Theorem 4.2.7 gives all the results stated here. □ Next we introduce some topological considerations on the algebra. 8.3.5 P r o p o s itio n Let A be either a normed or a spectral algebra. Every minimal idempotent e of A satisfies eA e = Ce. If a is either an alge bra norm or a spectral semi-norm fo r A , then there is a continuous linear functional w on (A, cr) satisfying ( 12 )
u(a)e — eae.
Proof Let it be a spectral semi-norm on A ■ Then cr(e) is non-zero since no idempotent is quasi-invertible. Hence under either hypothesis on cr, the restriction of a to eA e is a nontrivial algebra norm. The Gelfand-Mazur theorem (Corollary 2.2.3) shows that eA e consists of complex multiples of its identity element, e. Define w .A —* C by eae = ui(a)e. Clearly u! is a linear functional. It is also continuous since it satisfies |w(a)| = cr(e)~
(,a,b)e
=
ba
L\
C
B( )(Ae);
(13)
( L' ) l
=
J jJeB M )
L\f
=
BF( A e ) n B n (Ae);
(16)
Le and R? are contractive.
(17)
(14) VaeA;
(15)
If A is a Banach algebra, then the norms on eA and A e may be taken as the complete norms which are the restrictions o f the complete norm o f A . Finally, if e and f are two minimal idempotents then either e A f = {0 } or e A f = C c j t { 0 }, where c is any non-zero element in e A f . In the latter case any equivalence between Le and L? is a complex multiple of Vc and hence Ls a topological equivalence. Proof Proposition 8.2.2 shows that A e and eA are minimal left and right ideals, respectively, so that Le and Re are irreducible. Hence Theorem 4.2.13 shows that Le and Re are strictly dense. Let cr be a norm or a
8: Algebras with Minimal Ideals
678
8.3.7
spectral semi-norm on A which is the complete norm of A if A is a Banach algebra. Then a(e) is non-zero since e is not quasi-invertible and hence er(e) = er(e2) < cr(e)2 implies n(e) > 1 . Furthermore, the set {6 € A e : cr(b) = 0} is a left ideal of A which is a proper subset of the minimal ideal A e and hence is zero. Thus the restriction of er to A e is a norm. Similarly a induces a norm on eA. Obviously Le and Re are contractive relative to these norms. Using the notation of Proposition 8.3.5 we define the pairing between A e and eA by (a, b) = ui(ba)
V a e A e; b 6 eA.
Equation (13) is an immediate consequence. It in turn implies {a € A e : (a, b) = 0 for b € e A } = {a 6 A e : ba = 0 for b € eA} . The minimality of A e implies that this last set is {0}. Similarly {6 € e A : (o , 6) = 0 for a 6 A e } equals {0}. Hence (•, •) is a pairing. Equation (13) also gives |(a,6)| < a((a,b)e) — a(fea) < a(b)(r(a) which implies that {•,•) is a normed pairing. Using (13) yet again, we get (L 'a, b)e = bca = (a, Rec b) for any elements a 6 A e, b 6 eA and c e A. This establishes (15) and hence (14). Equation (16) is now a mere restatement of ( 10 ). Proposition 8.2.5 shows that the last paragraph of the present theorem will follow if eA/ is at most one-dimensional. Suppose a, b 6 eA / are linearly independent. Since Le and Re are strictly dense, there are elements c, d € A satisfying ca = b,cb = b,ad = b and bd = 0 . This implies b = cb = cad = bd = 0 which contradicts the independence of a and b. This establishes the last result of the theorem. □ 8.3.7 P r o p o s itio n Let A be a normed algebra. (a) I/e is a minimal idempotent in A , then /e(e) is a minimal idempotent in A ** with respect to either Arens product. (b) 1/A and A ** are also semiprime and A has a minimal ( le/t / right) ideal, then A ** also has a minimal { le/t / right) ideal. Proof By Proposition 8.3.5 eA e is just Ce. Hence K(e)A, , K(e) is just C/e(e) since n(eAe) is weak* dense in it by (15) of Theorem 1.4.2. Suppose A has a minimal left ideal. Corollary 8.2.3 shows that the minim al left ideal has the form A e for some minimal idempotent e. Thus A ” e is a m inim al ideal in A ** by the last portion of Proposition 8.2.2. Q
8.4
Modular Annihilator Algebras
Up to the present point in this chapter we have either assumed directly that the algebras under consideration have minimal one-sided ideals or mini mal idempotents or assumed that they were finite-dimensional. Many large
m
Modular Annihilator Algebras
679
and important classes of infinite-dimensional algebras have minimal one sided ideals. In most cases proof depends either on the spectral properties ; “ v.rnpact operators or on Proposition 8.4.3 below. This proposition shows that if certain left annihilator ideals are non-zero they are minimal left ide als.-1Before stating this important proposition we need to summarize some e.'-Sr-' atary facts about annihilator ideals. 8.4.1 D efin ition Let 5 be a subset of an algebra A- The left annihilator of S n A will be denoted by A l a $ — {a G A : ab = 0 for b e 5 }. The right annihilator of S in A will be denoted by A r a S . If the algebra A is dearly understood, we simply write l a $ and ra S . For the left annihilator in J. of the right annihilator of S in A we write A l r a S or l r a S ■ Similar notation has the analogous meaning. A ( left / right ) ideal of A which is Vie ( left / right) annihilator of some subset of A is called a ( left / right) anriiiiilator ideal. 8.4.2 P r o p o s itio n Let S be a subset o f an algebra A ■ Then l a S is a left idec-l and ra S is a right ideal. Similarly the left annihilator o f a left ideal or the right annihilator of a right ideal is an ideal. Furthermore S C 1 C A implies laS
2
laI
i
5 C RLAS,
ra
S 2
raI
,
(1 )
S C l r a S.
(2)
Jh?m these results we get easily lrlaS
=
l a S,
rlraS
=
(3)
r a S.
An iC'sal T is a { left / right ) annihilator ideal if and only if it satisfies { 1 —■ lra I / 1 = rla ? )■ If A is semiprime and 1 is an ideal, they jcids'V- i -Aj
c
l a (-Af )
•Af
c
la(A j) =
LA? = RA%
= jm M f ) ;
and Z H
r a
(4)
( A j )\
= {0}.
Finally, if A is a topological algebra it is obvious that both are
(5) laS
and
r ^S
Proof Most of these remarks are obvious. The results on semiprime algebras ""lie"' from the last remark in Proposition 8.2.8 and the obvious equations {1 LAi l )2 = { 0 } and ( r a T I )2 = { 0 }. □ V/:' will often use these easy results without comment. The next two results are preparatory for Theorem 8.4.5 which plays a fundamental role
i
680
8: Algebras with Minimal Ideals
8.4.3
throughout the rest of this chapter. The first is due to Yood ([1964], Lemma 3.3, p. 38.) 8.4.3 P r o p o s itio n Let A be a semiprime algebra. The following condi tions are equivalent for a maximal modular left ideal M of A . (a) r a M * { Q } . (b) There is a minimal idempotent e of A satisfying: M = l r a M = >1(1 - e) and r a M = eA . (c) A f % M . The corresponding results hold for right ideals. Proof (a) => (b): If r a M is a quasi-regular right ideal, then Theorem 4.3.6 shows r a M C A j C M . However this implies ( r a M )2 C M r a M = {0 } which contradicts the semiprime character of A. Lemma 4.3.5 gives a right quasi-singular element e € r a M . Then >1(1 - e) is a proper left ideal which includes M { 1 — e) = M . The maximality of M implies l r a M = M = >1(1 — e). Since e belongs to r a M , we conclude (1 - e)e e r a A = {0 }. Hence e is an idempotent and thus h a M = ra ( A ( 1 — e)) = eA. In order to prove that e is a minimal idempotent, Proposition 8.2.2 shows that it is enough to establish the minimality of A e. If £ is a non-zero left ideal included in A e, the maximality of .4(1—e) shows the existence of an element be € C C A e satisfying be + a (l — e) = e. Multiplying on the right by e we get e = be € C and hence C = A e. Therefore e is minimal. (b) => (c): Obviously e 6 A f does not belong to M . (c) => (a): Corollary 8.2.3 shows that there is a minimal idempotent e satisfying A e % M. and hence A e ft M. = {0 } and A e 0 M — A. Let d be a right relative identity for M and let d = b © c be the corresponding direct sum decomposition for d so that b — be € A e and c € M hold. Then we have a(b — 1) = a(d — 1) — oc 6 M . Hence ae(b — 1 ) 6 A e D M = {0 } for all o e A , implies b2 = b and >16 = Ae. So finally, A = A e © M = A b ® M = Ab(B >1(1 - b) and >1(1 - b) C M imply M — >1(1 - b) and r a M = b A * { 0 }. □ The following result is due to Barnes ([1966], Proposition 3.1, p. 567.)
8.4.4 P r o p o s itio n Let A be a semiprime algebra and let e be a minimal idempotent o f A . Then >1(1 — e) and (1 - e)>l are maximal modular left and right ideals o f >1, respectively. Moreover, the set T*
=
l a {A c )
=
r a ( zA )
=
>1(1 - e) : A = {a € A : A a C (1 - e)>l}
is a primitive ideal in both A and A R which includes no other primitive ideal o f A or A R. Every irreducible representation o f A with kernel V e u equivalent to both Le and LA/A^ ~ eK Any prime ideal which does not contain e includes LA(Ae).
Modular Annihilator Algebras
8.4.F
681
Proof Obviously >1(1 — e) is a modular left ideal. Suppose £ is a left ideal which properly includes .4(1 — e). The Peirce decomposition, A = A e ® >1(1 -- e), implies that there is some 6 6 £ satisfying b = ae + c (l —e) with ae ^ 0. Since A e is a minimal left ideal by Proposition 8.2.2, ae 6 £ ft A e implies £ D A e and hence C = A- Thus >1(1 — e) is a maximal modular left ideal of A ■ The same argument applied to A R shows that (1 — e)A is a maximal modular right ideal of A ■ Theorem 4.1.8(a) now shows that >1(1 — e) : A is primitive in >1 and {a € >1: >la C (1 — e)>l} is primitive in
A* The equations l a (A c)
= .4(1 - e ) : A
and
r a ( cA )
= {a € A : Aa C (1 - e).4}
are immediate. Let V e denote the primitive ideal >1(1 — e) : A = l a (A c ). If a belongs to r a {^A) then ea is zero so Ae(ttA{eA)) = {0 } C V e holds. Theo em 4.1.8(b) shows that V e is prime so that either A e C V e or ha (e. 'i) C V e holds. It must be the latter since e e A e does not belong to V e. The same argument in the reverse algebra shows V c — l a ( A c) C {a : Aa C = RA(eA). Hence all four sets are equal as asserted. T ie ideal V e = l a { A c) is the kernel of Le and the ideal V e = >1(1 —e) : >1 is the kernel of so that the last sentence of the proposition follows from Proposition 8.2.6. Suppose V is primitive and satisfies P C P e for some e € A m i - Then V eRAV e = { 0 ( C P implies P e C V C P* or RAV C C P C P * b y Theorem 4.1.8(b). Howeverr a V* = r l a {Ab ) 2 A e ^ {0 } and the semiprime nature of >1 contradicts r a 'P* Q V e. If P is a prime ideal, then e £ V and LA{Ae)Ae = { 0 } C P imply P e = L,t(>le) C P. □ The next theorem combines results of Yood ([1964], Theorem 3.4, p. 38) and Barnes [1966], 8.4.5 T h eorem The following conditions are equivalent for a semiprime algebra A . (a) Each maximal modular left ideal M satisfies r a M ^ {0 }. (b) Each maximal modular right ideal M satisfies l a M ^ {0 }. (c) A /A f is a radical algebra. (d) Every primitive ideal o f A has the form P e (defined in Proposition 8.4.4,) for some e € A m i . (e) Every irreducible representation o f A is equivalent to Le fo r some e g A m i. (f) The hull o f A f is empty. (g) IU is discrete and A satisfies l a { A f ) = r a { A f ) = A j . (h) IU is discrete and every left ideal which is not a subset o f A j includes a minimal left ideal.
678
8: Algebras with Minimal Ideals
8.3.7
spectral semi-norm on A which is the complete norm of A if A is a Banach algebra. Then a{e) is non-zero since e is not quasi-invertible and hence a(e) = tr(e2) < tr(e)2 implies tr(e) > 1. Furthermore, the set {6 6 A e : a(b) = 0 } is a left ideal of A which is a proper subset of the minimal ideal A e and hence is zero. Thus the restriction of a to A e is a norm. Similarly a induces a norm on eA. Obviously Le and Re are contractive relative to these norms. Using the notation of Proposition 8.3.5 we define the pairing between A e and e A by (a, b) = w(fea)
V a e Ae; b 6 eA.
Equation (13) is an immediate consequence. It in turn implies {a 6 A e : (a, b) — 0 for b 6 e A ] = {a 6 A e : ba = 0 for b 6 e A } . The minimality of Ae implies that this last set is {0}. Similarly {6 € e A : (a,b) = 0 for a € A e } equals {0}. Hence (-, •) is a pairing. Equation (13) also gives |(o, b}j < er((a,b)e) = cr(ba) < cr(b)cr(a) which implies that {•,■) is a normed pairing. Using (13) yet again, we get (L*a, 6)e = bca = (a, Rec b) for any elements a 6 A e, b e eA and c G A. This establishes (15) and hence (14). Equation (16) is now a mere restatement of ( 10 ). Proposition 8.2.5 shows that the last paragraph of the present theorem will follow if e A } is at most one-dimensional. Suppose a, b 6 e A } are linearly independent. Since Le and Re are strictly dense, there are elements c, d 6 A satisfying ca = b, cb = b, ad = b and bd = 0. This implies b = db = cad = bd = 0 which contradicts the independence of a and b. This establishes the last result of the theorem. □ 8.3.7 P r o p o s itio n Let A be a normed algebra. (a) I fe is a minimal idempotent in A , then /e(e) is a minimal idempotent in A *‘ with respect to either Arens product. (b) If A and A ** are also semiprime and A has a minimal ( left / right) ideal, then A " also has a minimal ( left / right) ideal. Proof By Proposition 8.3.5 eA e is just Ce. Hence K(e)A**n(e) is just C/e(e) since n(eAe) is weak* dense in it by (15) of Theorem 1.4.2. Suppose A has a minimal left ideal. Corollary 8.2.3 shows that the minimal left ideal has the form A e for some minimal idempotent e. Thus A " e is a minimal ideal in A ** by the last portion of Proposition 8.2.2. □
8.4
Modular Annihilator Algebras
Up to the present point in this chapter we have either assumed directly that the algebras under consideration have minimal one-sided ideals or mini mal idempotents or assumed that they were finite-dimensional. Many large
Modular Annihilator Algebras
679
and important classes of infinite-dimensional algebras have minimal one sided ideals. In most cases proof depends either on the spectral properties :. o . -ipact operators or on Proposition 8.4.3 below. This proposition shows that if certain left annihilator ideals are non-zero they are minimal left ide als.-1Before stating this important proposition we need to summarize some e^r^ntary facts about annihilator ideals. 8 . 4.1 D efin ition Let S be a subset of an algebra A. The left annihilator of S n A will be denoted by A l a $ = {a G A : ab = 0 for b e 5 } . The right annihilator of S in A will be denoted by A r a $ - If the algebra A is dearly understood, we simply write l a $ and ra S . For the left annihilator in >1 of the right annihilator of S in A we write A l r a S or l r a S ■ Similar notation has the analogous meaning. A ( left / right ) ideal of A which is i'.ie { ieft / right ) annihilator of some subset of A is called a ( left / right) annihilator ideal. 8.4.2 P r o p o s itio n Let S be a subset o f an algebra A . Then la<S is a left id?,': and r a $ is a right ideal. Similarly the left annihilator of a left ideal or the right annihilator of a right ideal is an ideal. Furthermore S C l C A implies l a S 2 la I , raS 2 raI , (1) <5 Q
rla S,
S C
lra S.
(2)
5Vior??. these results we get easily l rla S
=
l a S,
rl ra S
=
r a S-
(3)
An i-'.tal T is a ( left / right ) annihilator ideal if and only if it satisfies { I -■ lra I / I — rla ! )■ If A is semiprime and 1 is an ideal, they cciis'y: - . I- -Aj
C
la(Af )
= r a (-4 f);
Af
C
la{Aj)
=
ra %
a*d
la I
=
(4)
r a { A j );
1 C[la % = { Q } -
Finally, if A is a topological algebra it is obvious that both are•closed.
(5) laS
and
raS
Proof Most of these remarks are obvious. The results on semiprime algebras from the last remark in Proposition 8.2.8 and the obvious equations = { 0 } and (RA1 1 ) 2 = { 0 }. □ V /i will often use these easy results without comment. The next two results are preparatory for Theorem 8.4.5 which plays a fundamental role
i
680
8: Algebras with Minimal Ideals
8.4.3
throughout the rest of this chapter. The first is due to Yood ([1964], Lemma 3.3, p. 38.)
8.4.3 Proposition Let A be a semiprime algebra. The following condi tions are equivalent for a maximal modular left ideal M o f A . (a) r a M t {0 }. (b) There is a minimal idempotent e of A satisfying: M = >4(1 - e) and r a M = eA . (c) A f 2 M . The corresponding results hold for right ideals.
lraM
=
Proof (a) => (b): If r a M is a quasi-regular right ideal, then Theorem 4.3.6 shows r a M C A j C M . However this implies ( r a M ) 2 C M r a M = {0 } which contradicts the semiprime character of A. Lemma 4.3.5 gives a right quasi-singular element e € r a M . Then .4(1 - e ) is a proper left ideal which includes _M(1 — e) = M . The maximality of M implies l r a M = M — ^4(1 — e). Since e belongs to r a M , we conclude (1 - e)e € r a A — {0 }. Hence e is an idempotent and thus r a M = j?>i(.A(l — e)) = eA. In order to prove that e is a minimal idempotent, Proposition 8.2.2 shows that it is enough to establish the minimality of A e. If £ is a non-zero left ideal included in A e, the maximality of .4(1—e) shows the existence of an element be e C C A e satisfying be + a (l - e) = e. Multiplying on the right by e we get e = be 6 £ and hence C = A e. Therefore e is minimal. (b) => (c): Obviously e 6 A f does not belong to M . (c) => (a): Corollary 8.2.3 shows that there is a minimal idempotent e satisfying A e M and hence A e D M = {0 } and A e 0 M = A . Let d be a right relative identity for M and let d = b ® c be the corresponding direct sum decomposition for d so that 6 = be 6 A e and c G -M hold. Then we have a(b — 1) = a(d — 1 ) —oc 6 M . Hence ae(b — 1) € A e n M = {0 } for all a € A , implies b2 = 6 and Ab = Ae. So finally, A = A e 0 M = A b ® M = Ab (B A ( l - b) and A (1 - b) C M imply M = >1(1 - b) and r a M = bA * { 0 }. □ The following result is due to Barnes ([1966], Proposition 3.1, p. 567.)
8.4.4 Proposition Let A be a semiprime algebra and let e be a minimal idempotent o f A . Then >1(1 — e) and (1 — e)A are maximal modular left and right ideals o f A , respectively. Moreover, the set V*
=
LA{Ae) =
=
A { 1 - e ) : A = {a € A : Aa C (1 —e ) A }
ra {eA)
is a primitive ideal in both A and A R which includes no other primitive ideal o f A or A R- Every irreducible representation of A with kernel V e is equivalent to both Le and !/*/•'*<1-<). Any prime tdeal which does not contain e includes LA(Ae).
8.4.F
Modular Annihilator Algebras
681
Prooj Obviously .4(1 —e) is a modular left ideal. Suppose £ is a left ideal which properly includes >1(1 — e). The Peirce decomposition, A = A e 0 >1(1 —e), implies that there is some 6 G £ satisfying b = ae + c ( l —e) with ae 0. Since A e is a minimal left ideal by Proposition 8.2.2, ae € £ fl A e implies £ D A e and hence £ = A . Thus >1(1 — e) is a maximal modular left ideal of A . The same argument applied to A R shows that (1 — e)>l is a maximal modular right ideal of A . Theorem 4.1.8(a) now shows that >1(1 — e) : A is primitive in >1 and {a e A : A a C (1 — e)>l} is primitive in .4*. The equations la{A£) = .4(1 — e ) : A
and
ra(cA) = {a € A : A a C (1 —e ) A }
are immediate. Let P* denote the primitive ideal .4(1 — e) : A = LA(Ae). If a belongs to r a { c A ) then ea is zero so Ae(nA{eA)) = {0 } C P e holds. Theo em 4.1.8(b) shows that P* is prime so that either >le C P e or h a (e . 'i) £ P e holds. It must be the latter since e e A e does not belong to P e. The same argument in the reverse algebra shows P® = l a ( A c ) C {a : Aa C (1 — e)>l} = RA(eA). Hence all four sets are equal aa asserted. T ie ideal P e = l a ( A c) is the kernel o f Le and the ideal P e = .4(1 —e ) : >1 is the kernel of Z,^/^(1-e) so that the last sentence of the proposition follows from Proposition 8.2.6. Suppose P is primitive and satisfies P C P e for some e 6 A m i - Then V eRAV e = { 0 } C P implies P ' C P C P ' o r RAV e C P C P e by Theorem 4.1.8(b). Howeverr a P ‘ = r l a ( A c ) D A e ^ {0 } and the semiprime nature of >1 contradicts r a V c C P e. If P is a prime ideal, then e ^ P and LA{ Ae)Ae = { 0 } C p imply P e = LA(Ae) C p . □ The next theorem combines results of Yood ([1964], Theorem 3.4, p. 38) and Barnes [1966]. 8.4.5 T h eorem The following conditions are equivalent for a semiprime algebra A . (a) Each maximal modular left ideal M satisfies r a M ^ {0 }. (b) Each maximal modular right ideal M satisfies ca -M {0}. (c) A /A f is a radical algebra. (d) Every primitive ideal of A has the form P* (defined in Proposition 8.4.4j for some e € A m i . (e) Every irreducible representation of A is equivalent to Le for some e e A mi■ (f) The hull o f A f is empty. (g) IU is discrete and A satisfies l a ( A f ) = r a { A f ) = A j . (h) IU is discrete and every left ideal which is not a subset o f A j includes a minimal left ideal.
682
8: Algebras with Minimal Ideals
8.4.5
Proof (a)<=>(c): The map M —* M / A f is easily seen to send the set of maximal modular left ideal of A including A f onto the set of maximal modular left ideals of A J A f - Hence A j A p has no maximal modular left ideals (i.e., it is a radical algebra) if and only if no maximal modular left ideal of A includes A f . Proposition 8.4.3 shows that this is equivalent to (a). (b)«=>(c): Since (c) holds in A if and only if it holds in A R, and since (b) holds in A if and only if (a) holds in A R, these two conditions are equivalent by what we just proved. (c)<=>(f): Theorem 4.1.6(b) shows that A / A f has an irreducible rep resentation if and only if there is an irreducible representation of A with kernel containing A p . Hence A / A f is radical if and only if H(Af ) is empty. (d)=>(f): This result, which is not needed to close the circuit of impli cations, follows easily from e ^ P e. (f)=»(d): Let V be a primitive ideal of A . Proposition 8.2.8 shows there is some e € A m i satisfying e ^ V, and hence e $ eV C V. Thus the inclusion eV C eA and the minimality of e A imply eV = {0}. Hence V is included in ra ^ A = V e- We conclude that V = V e since V e is minimal among primitive ideals by Proposition 8.4.4. (d )'» (e ): Proposition 8.4.4 shows this. (d)=>(g): Proposition 8.4.4 and (d) imply
In order to show that II* is discrete we will show that the complement A e o f each singleton subset { P e} of TIa is closed. Any “P * in A e is a prime ideal and hence satisfies either V e C or LA"Pe £ V ? ■ Since V ? is a minimal primitive ideal by Proposition 8.4.4 we have l r a ( c A ) = l a ‘P (h): Let £ be a left ideal which is not included in A j = r a A f — <~\{RAA e : e € A m i ] , where the last equality follows from Proposition 8 .2 .8. Let e 6 A mi satisfy £ £ r a A z so that there is some 6 € £ with Aeb ^ {0}. The remark following Proposition 8 .2.8 shows that Aeb is a minimal left ideal included in £. (h) => (f): Let "P be a primitive ideal and let J be the kernel \ { V} ) so that A j = 1 fl V holds. Since is discrete we conclude I £ V and hence J % A j - Hence there is a minimal left ideal £ included in I . However £ is not included in A j — J O V since £ contains a non-zero idempotent by Proposition 8.2.2 and A j contains none by Proposition 4.3.12. Hence £ is not included in V and thus V is not in the hull of A f - Since V 6 11,4 was arbitrary, h( Ap) is empty. □
Modular Annihilator Algebras
683
8.4.0 D efin ition A modular annihilator algebra is a semiprime algebra satisiying the equivalent conditions of Theorem 8.4.5. Yood ([1958], p. 376) defined modular annihilator algebras and their ‘.heory has been considerably advanced by Yood [1964] and by several pa•?3 -c -'Barnes, particularly [1964], [1966], [1968a], [1968b], and [1971a]. 3 e.-r.E3 does not require in his definition that modular annihilator algebras be ssraiprime, although Yood did in [1964]. Most results require that the . algebra be semiprime. John Duncan [1967] used the term modular annihila tor algebra for a much weaker concept introduced in Definition 8.6.1 below. The idea of modular annihilator algebras evolved from the much more re strictive idea of annihilator algebras introduced by Alfred W. Goldie [1954] and in Definition 8.7.1 below. We will develop the theory of modular annihilator algebras very fully here. Theorem 8.4.5 gives a great deal of information on modular annihilator tigeb; as. If A is a modular annihilator algebra then n * is discrete, H(Af ) is e::>oty, A / A f is a radical algebra, the Jacobson radical satisfies A j = l a ( A ? ) = r a { A f ) , every irreducible representation is equivalent to Le for se:E3 ’ 6 A m i , and every left ideal which is not a subset of A j includes a minimal left ideal. Furthermore, we note that A satisfies la
(A f (BAj) =
ra
{A f © A j ) = {0 }
(6 )
so that A f © A j is “most” of A in a certain sense. To see this, note that ss'aiprime character of A gives A f D A j = A f n l a A f = { 0 }, and P.a(/‘-1? ® A j ) = L a { A f ® A j ) = l a ( A f ) n l a { A j ) = A j C \ l a { A j ) — { 0 }. For oi'ier algebras with discrete structure spaces see the references above. j>Tcte that any semiprime Jacobson radical algebra is a modular annihiiator i lgebra by default since it has no maximal modular left ideals at all. This limits the theory somewhat. Characterizations o f algebras with A / A j rcoc v’ar annihilator are given in Theorem 8.6.4. In particular, one of these characterizations is purely in terms of the spectrum of each element. We no\V state a result which gives a reasonably concrete representation of II 4 . 8 .4.? P r o p o s itio n Let A be a modular annihilator algebra. For each ir reducible representation T o f A , let [T\ be the equivalence class o f T relative to equivalence o f representations, and let A be the set o f these equivalence classes. Let ~ be the equivalence relation on A m i defined by ; •' M_
[.Le] = \L*] « . [/I*] = [R! ]
e A f * {0 } o
f A e * {0 }.
(7)
'T'-^er. . .’;e maps [ e ] ~ 7 > e;
[ e ] ~ [ L e];
[Le\~ ker(Le) = P e
( 8)
establish (compatible) bijections of ( A m i / ~ ) onto II 4 ; o f ( A m i / ~ ) onto J. and o f A onto II 4 .
684
8.4.8
8: Algebras with Minimal Ideals
Proof Proposition 8.2.5 establishes (7). Theorem 8.4.5 and Proposition 8.2.6 give the other results. □ The next result shows that if X is an ideal in a semiprime algebra A then A m i is the disjoint union of Im i = Am i n l and A m i n l a 2 = A m i f*l r a 2 . If e and / belong to A m i and satisfy e ~ / (so that e A f and f A e are non-zero), it is clear that e and / either both belong to 2 mi or both belong to A m i H laX . Proposition 8.4.9 shows that if I is an ideal in a modular annihilator algebra then X and A /2 are modular annihilator algebras, provided that J is semiprime in the second case. Furthermore, it is easy to see that e, / 6 2mi satisfy e ~ / relative to X if and only if they satisfy the same relation relative to A . Similarly e, / in A m i n l a I satisfy e ~ / in A if and only if they satisfy e + Z ~ + X in A/2. Hence, when X is a semiprime ideal in a modular annihilator algebra, the construction of the last proposition gives bijections among h(2 ), U^/x, { A m i ^ l a 2 / ~ ), and ((A /T)m i/ ~ ) and among (EU \ h(2)), Fix, and (A m i n X / ~ ) and (2m 1 / ~). 8.4.8 (a) (b) (c) (d) (e)
Proposition An ideal 2 o f semiprime algebra A satisfies: 2m 1 = A m i <^2 . A m i \ 2 mi = A m i n l a 2 = A m i n r a 2 . 2 f — A f n l = A f 2. A f n l a 2 = A f n ra 2. A f = ( A f H I) © n l a 2 ).
Proof (a): Let e be an idempotent in X. Then we have 2 e C A e = (A e)e C Xe which implies Xe = A e and eXe = eA e. This implies (a). (b): The inclusion A m i n r a 2 Q A m i \ 2mi is clear. To prove the opposite inclusion, let e be an element of Am 1 \ 2m i- Then Xe C A e is either {0 } or Ae. Since the latter case implies e € X, it does not hold. Hence e belongs to ra 2 . The second equality follows from (5). (c), (d), (e): Theorem 8.4.5(a) shows that X is semiprime. Hence Propo sition 8.2.8 and the proof of part (a) above show 2 F = Ylee i M, ~ * * and Af =
^2, ^
=
e+
X)
^ e'
Since (b) shows that the second summand is included in expression is a direct sum decomposition.
la 2,
the last □
The first statement in the next proposition is due to Yood [1964] and its proof is due to Barnes [1971a]. The last statement is due to Barnes [1968a]. Remarks following Theorem 8.6.4 will show certain cases when subalgebras o f modular annihilator algebras are again of the same type.
8.4.0
Modular Annihilator Algebras
685
8.4.9 P r o p o s itio n Let A be a modular annihilator algebra and let 1 be an ideal of A . Then 1 is a modular annihilator algebra. If J is a semiprime ideal, then A/1 is also a modular annihilator algebra. Further if A is any semiprime algebra and 1 is an ideal o f fc(/i(«4p)), then 1 is a modular annihilator algebra. Proof In order to show that Z is a modular annihilator algebra it is enough (by Theorem 8.4.5(f)) to show that h (lp ) C IIx is empty. Suppose V is a primitive ideal of 1 which includes I p . Theorem 4.1.8(e) asserts that there is some primitive ideal V of A satisfying V fl 1 = V. However I A f Q A f C V implies A f S V which contradicts Theorem 8.4.5(f). Hence h (lp ) is empty so J is a modular annihilator algebra. Theorem 8.4.5(e) shows that A/1 will be a modular annihilator algebra if and only if every irreducible representation of A j l is equivalent to Le for some minimal idempotent e € (A / 1) m i . Let T be an irreducible repre sentation of A/T on a linear space X . For each a € A denote a + 1 by a. Define T : A —►C {X ) by T a — Ta for each a € A . Then T is an irreducible representation of A so that there is an element e e A m i and a linear bisec tion U: A e —> X satisfying U(bae) = U(L%ae) = TbU(ae) for all a, b 6 A. Suppose ae belongs to Z. Then we get U(ae) = U ((ae)ee) = T aeU(ee) = T-'Uyz) = 0. Hence we can define a linear bijection U: (A / l)e —►X by U(a£) — U(ae). Furthermore U(bae) = U(bae) = T(,{/(oe) = T-bU(ae) = T~bU(ae) holds for all a, ft e A / I. Therefore U is an equivalence between Le and T. This shows that [A / I)e is a minimal ideal so that e belongs to (A/1) m i by Proposition 8.2.2. Thus we have verified condition 8.4.5(e) for A/1. Suppose A is semiprime and 1 = k(h(AF))- Then I is semiprime by Theorem 4.4.8(a) and satisfies I f — A f by Proposition 8.4.8(c). Hence the ideal l / l p = 1 / A f in A /A f is radical by Theorem 4.3.2(a). Thus I is a modular annihilator algebra by Theorem 8.4.5(c). The first statement of this proposition now yields the last statement. □ The last statement in Proposition 8.4.9 shows that any semiprime al gebra A contains a largest ideal k (h (A r )) which is a modular annihilator algebra. Of course it may happen that A /k(h (A p )) has a non-zero socle so that kh((A /k(h(A F)))F) is non-zero. Proceeding in this way, transfinitely if necessary, and then looking back at preimages in A we would eventually arrive at a largest ideal of A which could be analyzed in terms of mod ular annihilator algebras. In this way we could define a radical property which would be something of a curiosity since the radical part would be susceptible of analysis and the radical free part would be intractable. The following result is due to Barnes.
686
8: Algebras with Minimal Ideals
8.4.10
8.4.10 P r o p o s itio n Let A be a modular annihilator algebra. (a) If V is a prime ideal o f A then either: ( 1 ) V does not include A f and both A/V and V are primitive so V has the form V e; or (2) V includes A f and A /V is radical. (b) Let T be a topologically irreducible normed representation o f A on a normed linear space X . Then either ker(T) does not include A f ond there is a dense T-invariant subspace y o f X such that the restriction o f T to y is algebraically irreducible and has the same kernel as T, or ker(T) includes A f and A/ ker(T) is radical. Proof (a): Proposition 8.4.9 shows that A/V is a modular annihilator algebra. If A /V is not radical then it has a primitive ideal which Theorem 8.4.5(d) shows must be of the form la (A / V )c for some e e (A / V )m i■ Since (A/V )e is non-zero and A/V is prime, the primitive ideal LA{A/V)e must be zero. Hence V is primitive. If V does not include A f then Proposition 8.2.8 shows that there is some e 6 A m i which does not belong to V. The minimality of A e implies that A e fl V = {0 }. Hence the restriction of the left regular representation of A/V to (A e + V )/V is irreducible. Hence if A /V is radical V includes A f - If V includes A f then A /V is radical by Theorems 8.4.5(c) and 4.3.2(a). (b): Proposition 4.2.5 asserts that ker(T) is prime. Assume that ker(T) does not include A f Then Theorem 8.2.8 shows the existence of an e g A m i which does not belong to ker(T). Choose z e X satisfying Tez ^ 0. Since T is topologically irreducible the T-invariant subspace y = {Taez : a e A } is dense in X . The surjection V .A e —* y defined by V (ae) = Taez for all a 6 A certainly satisfies VL\ = TvV for all b e A . Hence its kernel is a left ideal properly included in the minimal left ideal A e and therefore equal to zero. Therefore V is an equivalence between L' and the restriction of T to y which must therefore be algebraically irreducible. The argument given in the proof of (a) shows that ker(Z,e) = l a A c equals ker(T'). Q 8.4.11 E xa m ple The following simple example shows that a faithful topologically irreducible representation T of a semisimple modular annihi lator algebra need not be algebraically irreducible. Let X0 be an incomplete normed linear space with completion X and let A be {x 0 w £ B p( X ) : i G Xq,lj 6 ■*"}• Then the identity representation of A on X is topolog ically irreducible but not algebraically irreducible. Its restriction to Xo is algebraically irreducible. The first statement in the next result (for the semisimple modular anni hilator algebras) is due to Yood ([1958], Lemma 2.8, p. 376). The original proof, which we give, evolved from Frank F. Bonsall [1954a],
8 .4 .! 2
Modular Annihilator Algebras
687
S.4.12 T h e o re m Any algebra norm on a modular annihilator algebra is a spectral norm. An algebra semi-norm t on a modular annihilator algebra is a spectra/ semi-norm if and only if the Jacobson radical is r-closed. This occurs if A t is a semiprime ideal included in A j . ~-'vzo ' Let A be a modular annihilator algebra and let ||• || be an algebra r?orr,; on A. Let A be the completion of [A, ||•||). Suppose a is an element of A anr A is a non-zero element of the spectrum of a. Then A_1a is not quasiinvertible in A . Suppose it lacks a left quasi-inverse. Then ,4(A_1a — 1) is a proper modular left ideal since it does not include A- 1 a. Theorem 2.4.6(b) shows that there is a maximal modular left ideal C which includes A(). a - 1 ). Since A is a modular annihilator algebra we may choose a non-zero element b in ra C C r a A {\ ~ 1 o. — 1). Since A~1ab — b belongs tc u/_A and A is semiprime, we conclude X~1ab = b. Now if c 6 A were ?, le* cuasi-inverse for A-1 a in A we would have 0 / 6 = A~lab = (c o /\- I a)J> + c\~1ab —cb = 0 + cb —cb = 0. Hence A_1a is not quasi-invertible in A . If A_1o lacks a right quasi-inverse the argument is similar. Hence PA(a) 5; Pa(0) S lla ll holds. Therefore ||•|| is a spectral norm. Let r be an algebra semi-norm on A. If r is a spectral semi-norm, Corollary 4.3.11 shows that A j is r-closed and A t C A j . Conversely, if A j (which is always semiprime since it is the intersection of prime ideals) is r-closed or if A t = {a £ A : r(a) = 0} C A j is semiprime, then r induces an algebra norm f on A = A / A j or on A = A / A t, respectively. Since A is a modular annihilator algebra by Proposition 8.4.9, f is a spectral norm by ihe argument above. Hence r is a spectral semi-norm since it satisfies r(a) > f(a ) > p(a + A j ) = p{a), where a represents the residue class of a in A and we have used Theorem 4.3.6(d). □ Theorem 8.1.1 asserts that a finite-dimensional semiprime algebra is the direct sum of finitely many full matrix algebras. Hence it is obviously a s^rr-:?;.:"ple unital algebra and Theorem 8.4.5(c) shows that it is a modular : aior algebra since A = A f - The next result, which is essentially due tc Barnes ([1966], Proposition 6.3, p. 573), extends this observation fend provides a converse. The proof is new. First, we provide examples o ' se^iiprime algebras A which are not modular annihilator algebras even though A / A j is a modular annihilator algebra or even finite-dimensional. 8 .4 .1Z E xa m ple Suppose B is one of the semiprime radical Banach alge bras described in Example 8.8.3, 8.8.4 or 8.8.6 and let A be the unitization B 1 of B. Then A j = B is the unique maximal modular left ideal of A. (To see tb-r, note that A + a has spectrum {A} in A.) Hence A / A j is isomorphic to C sr_d is therefore a modular annihilator algebra. However, the equation = { 0 } shows that A is not a modular annihilator algebra although it is p."Tiiprime. (Cf. Theorem 8.6.4 below.)
688
8: Algebras with Minimal Ideals
8.4.14
8.4.14 Proposition Let A be a non-radical modular annihilator alge bra. Then A is a united spectral algebra if and only if A / A j is finite dimensional. In particular a unital, semisimple, modular annihilator, spec tral (or normed) algebra is finite-dimensional. Proof Theorem 8.1.1 and the remarks preceding it show that A / A j is unital and normable if it is finite-dimensional (without even assuming that A is a modular annihilator algebra). Now suppose A is a modular annihilator spectral algebra and A / A j is unital. Then Proposition 8.4.9 shows that we may as well replace A by A / A j and hence assume that A is a unital semisimple modular annihilator algebra. Hence if A f is proper it is included in some maximal modular left ideal and therefore has a non-zero right annihilator. Since Theorem 8.4.5(g) shows r a A f = A j = {0 } we conclude that A = A f - Hence we can write 1 = Ylj=i aj ej f°r some aj € A and e} e A m i - Theorem 8.3.6 therefore shows that any b e A can be written an 6 = 161 = i ai e]^k^k = i where we have used the final remark in Theorem 8.3.6 to write ejAek = Cc^*. Hence {ajCjk : 1 < j , k < n} is a finite set which spans A . 1=1
8.5
Fredholm Theory
We wish to introduce the Fredholm theory of certain elements in semiprime algebras which is due to Barnes [1968a], This requires a number of preliminaries. The major results are stated in Theorem 8.5.5. Let T be a linear operator on a linear space X. For any n € N° we clearly have the inclusions ker(Tn)
C
ker(Tn+1);
and T ' X
3
T n+1 X.
(1)
Furthermore, it is easy to see that if there is some m 6 N° satisfying (ker(T’m) = ker(Tm +l); T’rnA' = Tm+1 X ) then for any n € N we will have (ker(7’m) = ker(Tm+n); TmX — T’m+nA'). We are ready for a definition.
8.5.1 Definition The ascent of T is the smallest integer in N° satisfying ktrr(7’m) = ker(7’m+1) and the descent of T is the smallest integer m in N° satisfying T™X = Tm+1 X . If there is no such integer we will say the ascent or descent is oo. Let A be an algebra and let a be an element of A 1. Let ( La / Ra ) denote the restriction to A of the ( left / right ) regular representation of o on A 1, respectively, i.e., Lab = ab
Rab = ba
a e ^1
b € A-
We will denote the ascent o f ( La / Ra ) by ( a /,(a) / Q «(a) ), and the descent of ( La / Ra ) by ( SL(a) / 6R(a) ).
Fredholm Theory
8.5.5
689
Note that the ascent of T is zero if and only if T is injective and the descent of T is zero if and only if T is surjective. If A is semiprime, then we have the formulas Qi,(a) =
min{m 6 N° : RAA am = RAA am +l}\
6L{a) =
min{m € N° : amA = am+1 A } ;
a R(a) =
min{m £ N° : LAamA = LA
6R(a) =
min{m 6 N° : Aam — A am+1}.
(2 )
8.5.2 L em m a If a linear operator T € £ ( X ) has finite ascent and descent, then they are equal and
X = TmX ® k e r (T m) holds where
m
is
their common value.
Proof Denote the ascent and descent of T by a(T) and S(T), respectively. First assume 6(T) = 0 and suppose a (T) > 0. Then T is surjective but not injective. Choose X\ e ker(T) \ {0 }. Inductively choose X2,X3, ... satisfying T xn+\ = x n. Then for each n , x „ belongs to ker(T") but not to ker(Tn“ 1). This contradicts the finite character of a(T). Hence 6(T) = 0 implies a(T) — 0. Next we will show a (T ) < 6(T). Denote 6(T) by m. Then consider the restriction T' of the operator T to the space T m(X). Since this operator is surjective 6(T') = 0 and hence q (T ') (which is clearly finite and less than or equal m a(T)) must also be zero. Thus T' is invertible. Suppose x belongs to ker(Tm+1). Then T '(T mx) = T m+1 a; = 0 implies T™x = 0. Hence a(T) satisfies a(T) < m = 6(T). Next we will show 6(T) < a(T). In the first paragraph we have covered the case 6(T) = 0 so we may assume m = 6(T) > 0. Let y = T m~l x belong to \ T ^ X . By the choice of m, the restriction of T m to T ^ X is surje: ive so that there is some z € V ' X satisfying T™? = T ^ x . Hence we have T™(z —x) = 0 and T m~1(z —x) = T m~1z —y ^ 0. Thus we conclude a(T) > 6(T) and by our previous result a (T ) = 6(T). Fi: ally, let y be any element of T ^ X fl ker(TTn) where m satisfies m = q (T) = 6(T). Then some x € X satisfies y = T mx so T 2mx = T ^ y = 0. However, this implies y - T "lx — 0 since m = a(T). Hence we have keT(Tm)(~\TmX = {0 }. Now suppose x is any element of X . Since restriction o f Tm to T™X is surjective we may write T ^ x = T ^ y for some y € T m<¥. Hence we have x = y + (x — y) where y e T™X and T m(x — y) = T™ ! — 'I^ y = 0 hold. Hence X is the direct sum of T ^ X and ker(T17*). □ Th ire is a natural concept of dimension or rank in the socle of a semiprime a - g e b ..W e introduce this notion and then prove a simple lemma which explains the concept.
690
8: Algebras with Minimal Ideals
8.5.3
8.5.3 Definition A left ideal has finite-rank if it is included in the sum of finitely many minimal left ideals. In this case we will call the rank of the left ideal the smallest number of minimal left ideals the sum of which contains it. The ideal {0 } will be said to have rank zero. Clearly a similar concept can be defined for right ideals. The next lemma shows that there is no ambiguity in speaking of the rank of a (two-sided) ideal. Note that any principal left ideal A a generated by an element a in the socle has finite-rank since we may write a = $2"= 1 ai with each aj in a minimal left ideal C} . Hence A a is included in 53"=1 In essence the following lemma develops the theory of finite-rank left and right ideals. 8.5.4 L em m a Let A be a semiprime algebra. A left ideal C in A has finite-rank if and only if there is an idempotent e in A f satisfying £ = A e. The rank o f a finite-rank left ideal £ is the number o f orthogonal minimal idempotents in any maximal family included in £ . It is also the number o f terms in any direct sum decomposition o f C as the sum o f minimal left ideals. Any left ideal properly included in a finite-rank left ideal C has finiterank strictly less than the rank o f C. The analogous results hold for right ideals also. Proof By symmetry we only need to deal with left ideals. We use Corollary 8.2.3 repeatedly and without further comment. Let £ be a non-zero left ideal with rank n so that there is a set {e^ : j = 1 , 2 , . . . , n } of minimal idempotents satisfying £ C Ae} . Theorem 8.4.5(h) applied to £ considered as a left ideal of A f shows that £ includes some minimal ideal and hence contains some minimal idempotent. Zorn’s Lemina guarantees a maximal family { / Q : a € A} of orthogonal minimal idempotents in £. Then 'jTiaeAA f a C £ is a direct sum. For each q > € A, / Ql = j a3ej holds for certain a_, € A. There must be some k with a*e* ^ 0 which implies in turn / {0},*4e* = ,4a*e* C A f ai + H j = i '4 e >> and hence £ C A f ai + Aej. Hence if A has more than one element, then for any a 2 € A \ { q i } we have / „ , = aai f a + 6, e,. Since the sum YlaeA A fa is direct, there must be some i / k with ^ 0, so we may write £ C A f a, + A f a. + T '" =i A e } . Proceeding in this way at each step we replace an ideal Ae} by an ideal A f „ k. Hence eventually we exhaust A (since { / „ : a € ^4} is an orthogonal family). We want to show C Q E a€AA f a C £ . Let e be the sum of the finite set { f a : a € A } so that e belongs to £ fl A f - If £ ( 1 — e) is a non-zero left ideal included in £ , it must contain a minimal idempotent f ’,, by the argument given above. Define fp by fp — (1 - e ) f 's e £ ( l - e ) c £. Then / „ = ( 1 - e ) / ' ,f'a = ( 1 - e ) / ' ( 1 - e ) / ' = } } is an idempotent which is non-zero since it satisfies /^ = (fg )2 = / ^ ( l —e)/^ =
8.5.5
Fredholm Theory
691
f'pfp- It is minimal by Proposition 8.2.2 since A fp = Af'p is minimal. Then {/cr : a € A ) U {//)} contradicts the maximality of { f a : a 6 A }. Hence -- e) = 0 implies C = Le C A e C ^2a€AA f a Q £• The definition of rank shows that A has at least n elements, and the construction shows that tit has at most n elements. Hence the rank of a finite-rank ideal is II: s number of orthogonal minimal idempotents contained in any maximal family. The equation C = A e shows that a finite-rank ideal has the form Ac Ter some e e A f . The converse is a special case of the remark that any principal left ideal Aa generated by an element a 6 A f is finite-rank. If we apply the replacement procedure described at the beginning of this . :. r Vo a direct sum C = ©"_ t Aej then linear independence shows that the Process cannot terminate with less than n terms and the proof already given shows that it does terminate with not more than n terms. Hence the number of elements in a maximal family of orthogonal idempotents is exactly n, so any direct sum decomposition contains exactly n terms. Finally, let M be a left ideal included in a finite-rank left ideal C. Clearly M h a s finite-rank. Let { f j : j = 1 , 2 , . . . , m } and {ej : j — 1 ,2 ,..., n} be maximal families of orthogonal minimal idempotents in M and in £ , "espsotively. The argument given earlier in this proof proves m < n. If m = n, the replacement procedure described earlier shows M = C. This proves the inequality of ranks. □ The next theorem contains Barnes’ [1968a] major results on the Fredholin theory o f certain elements in semiprime algebras. Note that for any ideal I in an algebra A the preimage in A of the Jacobson radical in A f T :s "he set k(h( l) ). Hence this set is the same in A and in the reverse algel.'t A r even though it is defined in terms of primitive ideals. Since a semiprime algebra A is a modular annihilator algebra if and only if A f A f is :T'i:cal, we see that A is a modular annihilator algebra if and only if A = b(h(Ap))- Furthermore I = k(h(AF)) is always a modular annihila tor algebra by Proposition 8.4.9. Thus any modular left ideal in I has a r.oE-'soro right annihilator in J and hence in A ■ We will use these facts in the next theorem. S.5.5 T h eorem Any element a in the socle A f o f a semiprime algebra A 'satisfies: (a) Sp(a) is finite. For any non-zero A € Sp(a) and any n € N, l a ((^ ~ a)nA) and r a (A(X - a )") have finite-rank. (c) For any non-zero A e Sp(a), a L(A - a) = 6l (A - a) = a R(A - a) = SR(A - a) ifljw c,U finite.
8: Algebras with Minimal Ideals
692
8.5.5
(d) For any non-zero A € Sp(a) and any n 6 N, there is an idempotent e 6 A f satisfying r a (A(X - a)n) = e A and A ( A — a)n = ,4(1 — e). (e) Any non-zero A € Sp(a) satisfies A =
r a A(X
- a)m © (A - a)mA = LA(X - a)mA © A(X - a)m
where m = a/,(A — a). Any element a in k(h(Ap)) satisfies (b) and if A is a normed, or spectral algebra then a also satisfies (c), (d), (e) and: (a') Sp(a) is either a finite set, or a countably infinite set containing zero which is its only limit point. In particular any element a in a modular annihilator algebra A satisfies (b) and if A is also normed or spectral, then it also satisfies (c), (d), (e), and (a'). Proof (b): We discuss only r a { A ( A—o )n) since this result in A R establishes the other half of (b). Let a belong to A f and let A and n be as stated. The we have (A — a )" = An — ab where b is a polynomial in a. Hence any c 6 r a ( A ( X —a )") satisfies (A" - ab)c 6 r a A — {0 } so c — X~nabc belongs to aA. Lemma 8.5.4 now shows that r a ( A ( X —a)n) C aA has finite-rank. Suppose next that a belongs to k(h(Ap)) and that A, n, and b are as before. Since k(h(Ap))/AF is radical, Theorem 4.3.6(b) shows that there is some c £ k(h( Ap) ) and some d € A f satisfying c o X~nba = d. The equation in ^4*, (1 —c)(X —a)n - > " ( 1 —d) shows A(X —a)n D A ( l —d) and hence r a (A(X — a)n) C ^ ( .4 ( 1 — d)). Since the last ideal has finite-rank by the first part of the proof above, we have established (b). (c): Again the symmetry o f the situation allows us to prove only one' half of the relations. We will first show that a/, (A - a) is finite when a € A f . Let m be the rank of aA. If c*l(A — a) > m we can find an orthogonal set eo, e\, e z ,. . . , em o f idernpotents with eo 6 jm(-4(A —a)) and Ck £ r a ( A ( X - a)*+1) \ r a ( A ( X - a)*) for k = 1 ,2 ,... ,m . However, as in the proof of (b), we conclude that each ek belongs to aA. This contradicts the rank of aA and shows that a ^ ( X — a) < m . Next we assume that a belongs to k(h(Ap)) and that A is a normed or spectral algebra. Again we will show that a/, (A — a) is finite. For each n € N define a right ideal Tin by H n = /^ (^ (A - a)) fl (A - a)n.4. If aj,(A - a) is not finite each 7Zn is non-zero. We will derive a contradiction from assuming this. Since r a ( A ( X - a)) has finite-rank by (b) and the sequence {ftn } of these right ideals is clearly decreasing, this sequence must eventually become constant (and by assumption, unequal to {0 }). Hence there is some e € Ami in n£‘L,7£n. Therefore, for all n > 0, we have e e ^ A -fllln fA -a J M .
(3)
We consider the representation I f described in Theorem 8.3.6. Choose an algebra norm or spectral semi-norm a on A and a corresponding norm
3.5.5
Fredholm Theory
693
on Ae. Proposition 8.4.10 shows that a is a spectral semi-norm in any case. Since k(h(AF))/AF is radical, Theorem 2.2.5 and Corollary 4.3.9 show i m „ _ 00(
A- p( K pyn - D y n)
=
V n - A - ‘ ( A - K ) Y , ( * ~ l K ) ky n - X - pD y n. k=0
p - l
Since the middle term belongs to ker((A — K ) n), we have for any n > m %n
%m — 2In
V
^ ^L)yn ■+■A PDym
with y € ker((A — K ) n). Hence we have \\zn -zm\\
>
||j/n -I/||-||A-p£»y„||-||A-p£»j,m||
>
i - i - U i .
“
2
6
6
6
However, this is impossible since Theorem 8.3.6 shows that J — L% has finite-dimensional range so that each z„ belongs to the compact ball in JA e of radius ||J|| |A|~P. Therefore A — L\ has finite ascent. This contradicts (3). Therefore the original assumption that a i (A —a) was infinite must be rejected. Hence a i (A — o) is finite under either hypothesis. Next we will show that 6r ( A—o) is finite. Let a satisfy either hypothesis. Denote a L ( X — o) by m. We may suppose m > 0, so part (b) shows that there is an idempotent e 6 A f satisfying r a ( A ( X - a)m) = eA. Consider the modular left ideal C — >1((A - o )m - e). We will prove ra£- = {0 }, and then C. — A . Suppose b € r a £ so that (A - a)mb = eb and (A — a)2mb = (A — a)meb = 0. Since m = a i(\ — a) we have (A — a)mb = 0. However this iirplies b € r a ( A ( X - a)m) = eA> so b = eb = (A — a)mb = 0 as we wished to show. Now suppose C / A- Theorem 2.4.6(e) shows that / = A~m((A — o )m + e ) e kh(AF) is the right relative identity for some maxim-il modular left ideal M ^ C D kh(Ap) 2 k h( AF) (l — /)• But Proposition 8.4.9 asserts that kh(Ap) is a modular annihilator algebra, so Theore m 8.4.5(a) gives a non-zero element b in the right annihilator of M sn kh(.-iF). Since kh(AF) is semiprime, we conclude (1 — f ) b = 0. This contra-' icts ra{A{\ - / ) ) = {0}.
8: Algebras with Minimal Ideals
694
8.5.5
Let b be any element of A(X — a)m. Then we may write b = c(A —a)m and c = d((A —a)m - e). Combining these and recalling the definition of e, gives 6 = d(A - a)2m. Hence 6r(A - a) is less than or equal to a/,(A — a). By symmetry a i (A —a), o r (X —a), d^(X - a) and 6r (A —a) are all finite. Lemma 8.5.2 shows that q/, = 6i , a R = 6r and the inequality just obtained together with its translation into the reverse algebra give the rest of the equations in (c). (<■): This is an immediate consequence of (c) and Lemma 8.5.2. (d): Lemma 8.5.4 and conclusion (b) above show the existence of an idempotent e satisfying r a A(X — a)" = eA. Clearly we have A(X — a )" C LRAA{X-a)n = lac A — A ( l - e ) . On the other hand, RA ( A ( X - a ) n+ A e ) = jM*4(A - o )n fl RA(Ae) = eAC\ (1 - e)A = {0 }. Hence ^4(A - a)n + A e = A holds and any b € A ( l - e) may be written as b = (c(A —a )" + de)( 1 —e) = c(A - a )". This gives the opposite inclusion. (a): We have a 6 A f - Suppose { A „ } „ €n is an infinite sequence of distinct non-zero points in Sp(a). For each n 6 N, A ^ a is either left or right quasi-singular. Hence by working in either A or A R, and choosing the subsequence where Xn la is left quasi-singular, we may assume that .4(A„ —a) is a proper left ideal for each n. Hence, for each n e N, conclusion (b) shows that there is some en e Am i satisfying A„e„ = aen. Therefore each en belongs to the finite-rank ideal aA. If for some n > 1 there exist aj € A satisfying e tat + e2
= >t(A - a)m ® where m =
o r (X
- a)”M),
—a). Let C be the left ideal C = { b e A : lim ||fceB||/||e»||=0}. n —* 0 0
Since (A - a ) m e „ = (A - A„)men holds for all n > 0, we have ,4(A - a ) m C C and e,n € (A - a)mA for all n e N. Therefore /,h((A - a)mA ) e n vanishes for all n e N, so that /^ ((A - a)n‘ A) is also included in £. The direct sum decomposition above shows that C = A- However lim ,^ ^ ||oen|j/||e„|| —
8.6.1
Algebras with Countable Spectrum for Elements
695
lim„_>00 ||Ane„||/||en||= |A| shows that this contradicts the assumption that □
A was non-zero. Hence (a1) holds.
8.6
Algebras with Countable Spectrum for Elements
this section we will characterize two kinds of algebras by assuming, in part, that the spectrum of every element is countable. We begin by introducing a dans of algebras larger than the class of modular annihilator algebras. Duncan [1967] defined a class of Banach algebras which he called modular annihilator algebras but his definition differs from that given in this vork which is due to Yood ([1958], p. 376). His modular annihilator algebras were Banach algebras with a family {£<* : a £ ^4} of maximal modular left ideals satisfying n Q€y«£c. = {0 } and
r a Cu
^ 0
V a £ A.
The first equation implies that the algebra is semisimple, hence semiprime. Proposition 8.4.3 shows that the second condition implies the existence of •c. rr:nimal idempotent ea for each a € A satisfying Ca = A ( I —ea ). Hence we have {0 } = n Q€J4£ a = n Q€j4-4(l - ea ) = n Q€J4 LA(eaA ) 2 l a ( A f )Therefore Duncan’s definition is equivalent to requiring that the Banach Rlgsbra A be semisimple and satisfy l a { A f ) = {0 }. In keeping with the spirit'of the development given here we offer the following definition. 6.6.1 D efin ition An algebra A is called a Duncan modular annihilator algebra if it is semiprime and satisfies laA f
=
ra A f
= Aj-
Tbaorem 8.4.5(g) shows that modular annihilator algebras are Duncan modular annihilator algebras. However, if X is any infinite-dimensional Banach space, then A = B( X) is a semisimple (in fact primitive) Duncan modular annihilator algebra which is not a modular annihilator algebra. To see that B ( X) is a Duncan modular annihilator algebra note that T € implies 0 = T (x ® w) = T x ® w for all x £ X and w £ X*. Hence Tx = 0 for all x £ X. The algebra B( X) is not a modular annihilator algebra since it is infinite-dimensional and unital contradicting Proposition g .4 .:4
The following theorem is due to Barnes [1968a], Theorem 2.2. 8.9.? T h eorem If every element of a Banach algebra A has finite or countable spectrum, then A / A j is a Duncan modular annihilator algebra. Proof First we will show that a semisimple commutative Banach algebra A - ’’tS’ Ov.t minimal ideals must contain a point with uncountable spectrum. If
696
8: Algebras with Minimal Ideals
8.6.2
the Gelfand space contains a connected subset which is not a singleton, then there is some a 6 A for which Sp(a) = a (r > ) is a connected subset of C that is not a singleton. Since such sets are uncountable we may assume that TA is totally disconnected. If contained an isolated point then the Silov idempotent theorem (Corollary 3.5.13) would give a minimal idempotent. Hence is totally disconnected without isolated points and each open set is infinite. Choose a nonempty, open, compact subset N in Then choose two disjoint, nonempty, open, compact subsets No and Ni satisfying NqUNi V For k = 0,1 choose disjoint, nonempty, open, compact subsets Nko and Nki satisfying o U Nki = Nk. Proceeding inductively in this fashion, we obtain nonempty, open, compact subsets Nk l k l where ki, k2, ■. ■, kn is any list of n 0’s and l ’s. The Silov idempotent theorem (Corollary 3.5.13) gives corresponding idempotents eklk2...k„ such that eklk2,.k„ is the char acteristic function of For each n € N define an e A by an — ^
•
*„=i
Choose a strictly increasing sequence {m „ },u n so that a = 2~m"a„ converges in A. Given any sequence {/cn} ngN of 0’s and l ’s, the finite intersection prop erty shows that there is a point
£ Sp(a). n= 1 Therefore Sp(a) is uncountable. This completes the proof of our first as sertion. Now suppose A satisfies the hypotheses o f this theorem. Theorem 4.3.6(d) shows that we may assume that A is semisimple by replacing it by A j A j - Denote the closed ideal l a A f by J. If no element o f I has spectrum different from {0 } then Theorem 4.3.6(b) shows I C A j = {0 }. Thus this case is complete. Now suppose that I contains an element b with spectrum different from {0 }. We will show that this implies the existence of a minimal idempotent e € J. This is a contradiction since Proposition 8.4.8(a) shows that e also belongs to A f but the semisimplicity of A gives A f H l a A f = {0 }. Since I is an ideal we have Spj(b) = Sp^(b). Note that there is a non-zero isolated point in the spectrum of b. Hence Proposition 3.4.1 gives a non-zero idempotent in J. Let £ be a maximal family of commuting idempotents in J and let C be a maximal commutative subalgebra of J including E. Since C is a closed spectral subalgebra of A we may apply the first portion of this proof to C and thus obtain a minimal idempotent e in C. We wish to show that e is
8.6.,'
Algebras with Countable Spectrum for Elements
697
minimal in I , or equivalently (by Proposition 8.3.5) that eXe equals Ce. If / is an idempotent in eJe and g is an idempotent in E, then Proposition 8.3.5 gives ge = ege = Xe for some A G C and hence g f = g e f = A e/ = /A e = f g. Therefore / belongs to E by the maximality of E. The last sentence now show3 f = f e — Xe for some A € C. However, Xe = f = f 2 = Aae shows that A is zero or one. Therefore the unital algebra e le contains no non-zero idempotent except its identity e. Hence Proposition 3.4.1 shows that the spectrum of each a G e le is connected. Since e le is a spectral subalgebra of A , each o G eJe has countable spectrum. Thus the spectrum of each a G e le is a singleton. Theorem 4.3.2(a) and Proposition 4.3.12(b) show that eJe is semisimple. We apply an argument of Irving Kaplansky ([1954], Lemma 4, p. 375) to show that the set Af of elements with spectrum {0 } is the Jacobson radical o f e le and hence each element o G e le satisfies o = Ae for A £ Sp(a). The remark following Theorem 4.3.6 shows that it is enough to show that Af is a left ideal. If a € Af, b € A and ba £ Af then Proposition 2.1.8 shows ab ^ Af. Since the spectrum of these elements is a singleton but not {0 }, (ba)~1b and b(ab) ~1 are right and left inverses for a. Hence a € A f implies ba e Af. If o, b G Af and a + b £ Af then a(a + 6)_1 = 1 — 6(o + 6)_1 contradicts our last conclusion that a(a + b)~l and b(a + 6)_1 both belong to Af. Hence o + 6 belongs to Af as we wished to show. □ 8.6.£ E xam ple The converse of this result is far from true. Consider for instance A = £°°(I) (under pointwise algebra operations) where I is the unit interval (or any other uncountable subset of C). Then A f is the set of functions in A with only finitely many non-zero values so that A is clearly a commutative semisimple, Duncan modular annihilator algebra. However, the function f ( t ) = t for all t e / has uncountable spectrum. Consider the set A — c(J) of all / : / —* C for which there is some A/ € C such that the set {t £ / : \f(t) — A/| > e} is finite for all e > 0. Under point vise multiplication this is a unital commutative semisimple Banach algebra where each element has countable spectrum but A is not a modular annihilator algebra by Proposition 8.4.14 or Theorem 8.6.4. If we consider A = ^p(/ ) for 1 < p < oo or A ~ Co(I) = { / : / —* C : the set { t e l : |/(t)| > £} is finite for all e > 0}, each element in these algebras has countable spectrum and they are modular annihilator algebras by Theorem 8.6.4. Example 8.4.13 shows that we cannot replace A / A j in the conclusion of Theorem 8.6.2 by A. P operty (b) of the next theorem shows that a major difference between modrlar annihilator algebras and Duncan modular annihilator algebras is that *he latter may contain idempotents not in their socle. The equivalence of (a) and (c) below is due to Barnes ([1968b], Theorem 4.2, p. 516) but
698
8: Algebras with Minimal Ideals
8.6.4
our proof is much simpler than his. A closely related result is given in Proposition 8.7.8. Example 8.4.13 also shows that this characterization cannot be extended to non-semisimple modular annihilator algebras. Since elements in A and their images in A / A j have exactly the same spectrum, no spectral condition can characterize non-semisimple modular annihilator algebras. 8.6.4 T h e o re m The following are equivalent fo r a Banach algebra A . (a) A / A j is a modular annihilator algebra. (b) Each element of A has finite or countable spectrum and each idempotent o f A / A j belongs to the socle o f A / Aj . (c) No element of A has a non-zero limit point for its spectrum. In fact any modular annihilator algebra has all its idempotents in A f and if it is normed or spectral it satisfies (c). Proof Theorem 8.5.5 shows that if A or A / A j is a modular annihilator algebra then (c) holds. In particular (a) implies (c). If e is any idempotent in a modular annihilator algebra then e.+A p is an idempotent in the radical algebra A/Ap, hence Proposition 4.3.12(a) implies e £ A f For the rest of the argument, Theorem 4.3.6(d) allows us to replace A by A j A j and thus assume that A is semisimple. (c) => (b): Since any uncountable set in C must have more than one limit point we need only prove the statement about idempotents. Suppose e is a non-zero idempotent in A. Then eAe is semisimple by Proposition 2.5.3(d). Hence Theorem 8.4.5(e) shows that e Ae contains minimal idem potents which are obviously minimal idempotents in A since f e A e f = f A f holds for any / £ eAe. Let £ be a maximal family of orthogonal minimal idempotents in eAe. If E is infinite we may choose a sequence { / n}neN of distinct elements in E. Choose a sequence {An} „ e N of distinct numbers converging to zero and satisfying |A„| ||/„|| < oo. Then the element e+ i An/n has / „ as a characteristic vector with characteristic value 1 + A„ for each n € N. Since this contradicts (c), we conclude that E is finite, say E = { f j : j = 1, 2 , . . . , n ). If / = e - £ " = i /> is non-zero then the argument at the beginning of this paragraph shows that f A f contains minimal idempotents. Since this would contradict the maximality of E we conclude that e = $Z"=i f 3 belongs to the socle of A. (b) => (a). (Recall that we are assuming that A is semisimple.) Let M be a maximal modular Soft ideal of A. Let c be a right relative identity for M so that 1 belongs to Sp(c). The spectral condition in (b) ensures that we may apply Proposition 3.4.1 to obtain an idempotent e commuting with c and satisfying c o d = d o c = e f o r some d € A. Hence we have .4(1 - e ) C A(1 - c ) C M . Condition (b) shows that e belongs to A f Hence we have A f 2 M . Thus .A is a modular annihilator algebra by Proposition 8.4.3 and Theorem 8.4.5(a). □
Classes of Algebras with Large Socle
699
Barnes ([1969b], Theorem 3.1, p. 7) deduced the following result as a -crc'a ry of the above theorem and of Theorem 8.4.12. If A is a normed, imple, modular annihilator algebra, then BBj is a modular annihilaior algebra (where B is a subalgebra of A which is a Banach algebra in soire norm) if and only if the norm of A is a spectral norm on B. In par:ic . Li j , a closed subalgebra of a modular annihilator algebra satisfies this ';onc tion. The general result follows since the spectral properties proved in this theorem show that the spectrum and the boundary of the spectrum are equal for any a € A , and hence by Proposition 2.5.10, B is a spectral subaigebra if and only if the norm o f A is a spectral norm on B. In the same paper Barnes also shows that if A is a normed, semisimple, modular annihilator algebra and B is a semisimple subalgebra then B f = A f O B. (A mistake in the proof can be corrected.) We remark that the inclusions 3 f C A p fl B and A m i ("•B C Bmi 816 easy. If e € B is idempotent, then it must belong to A f , since e + A f is an idempotent in the radical algebra A / A f ■ The first inclusion follows. On the other hand if e £ A m i ^ B holds, then the inclusion eBe C eA e = Ce proves its minimality. See also Aupetit [19861.
8.7
Classes o f Algebras with Large Socle
Wo will now discuss the relationship between various conditions on algs’jres or topological algebras which ensure that they have a large socle in son s sense. All these relations will be summarized in a diagram in Theorem 8.7.11. Counterexamples will be provided at the end of the chapter to show thafc 7/e have exhausted all o f the possible implications. Along the way ws will show that many such algebras can be written as topological direct of topologically simple ideals of the same type and that each topolog: simple algebra haa a particularly satisfactory faithful representation ?s compact operators. e begin by defining the remaining classes of algebras which have large socles in some sense and which have been studied in detail in the literature. ' 817.2 D efin ition A topological algebra A is: (a) A dual algebra if it is semiprime and each closed left ideal C satisfies £ = lra C- and each closed right ideal 7L satisfies 7L = r l a R( b ) A left annihilator algebra if it is semiprime and the left annihilator ’ of1’each proper closed right ideal is non-zero. (c) An annihilator algebra if both A and A R are left annihilator algebras. (r ; A left completely continuous algebra if it is normed and La is a operator for each a 6 A. (s' A completely continuous algebra if both A and A R are left completely contiguous.
700
8: Algebras with Minimal Ideals
8.7.1
(f) A compact algebra if it is normed and LaRn is a compact operator for each a 6 ADual algebras, annihilator algebras and left annihilator algebras were defined by Irving Kaplansky [1948a], Prank F. Bonsall and Alfred W. Goldie J1954], and Malcolm F. Smiley [1955], respectively. In their definitions, they did not require that the algebras be semiprime, but the majority of their results were for semisimple algebras. On the other hand, the definitions of the last two classes referred only to Banach algebras. The concept of dual algebras seems a little too strong since it leaves out important examples without providing much better results. (However, Kaplansky [1948a] shows in Theorem 15 that for any comnpact group G, Z^(G) is a dual algebra under convolution.) The existence of an annihilator Banach algebra which is not dual was first established by Johnson [1967b]. His example was semisimple and commutative. The definitions of completely continuous, left completely continuous and compact algebras are due to Irving E. Segal [1940] and [1947a], Definition 1.9, p. 80 (cf. Kaplansky [1948b], p. 698), to A. Olubummo [1957] (whose attribution of the definition to Kaplansky we have been unable to verify), and to Klaus Vala [1967] and James C. Alexander [1968], independently. The last definition grew out of an observation of Vala [1964] (cf. Example 8.8.2). See also Marianne - reundlich [1949], Bonsall [1969b], Alexander [1969], Kari Ylinen [1968], Yood [1982] and Abdullah H. Al-Moajil [1984], Note that a normed algebra A is completely continuous if and only if both La and are compac; operators for each a E A. The implications: dual => annihilator => left annihilator =>• modular annihilator
and
completely continuous => left completely continuous => compact are obvious. For any reflexive Banach space X , Bp ( X) ~ is a topologically simple semisimple annihilator Banach algebra, but a reflexive Banach space X was constructed by Alexander M Davie [1973a] such that Bp( X) ~ is not dual. We will show in Proposition 8.7.4 below that B f ( X ) ~ is a topolog ically simple, semisimple, left annihilator Banach algebra for any Banach space X. Bonsall and Goldie ([1954], Theorem 16, p. 164) showed that B f ( X ) ~ is never an annihilator algebra, unless X is reflexive. Example 8.8.8, which is due to M. R. F. Smyth [1980], shows that a semisimple left completely continuous Banach algebra need not be com pletely continuous. Among the classes of algebras introduced here, annihilator algebras have probably received the most attention. A full presentation of their theory is given in Bonsall and Duncan ([1973b], Section 32). Nevertheless, we believe that modular annihilator algebras and left annihilator algebras (particularly the former) provide more convenient compromises between strong axioms
8.7.2
Classes of Algebras with Large Socle
701
(to give a rich theory) and weak axioms (to give a multitude of examples). Note that, unlike the classes defined here, modular annihilator algebras are defined purely algebraically without reference to any topology. We begin investigating the relationship among these classes of algebras and the size of their socle by considering certain topological algebras A in which the socle or the ideal A f + A j is dense. Result (b) is due to Smiley [1955]. Proposition 8.7.3 contains a partial converse of (b) as well as a structur 3 theorem for topological algebras with dense socle.
8.7.2 P r o p o s itio n Let A be a spectral normed algebra. (a) If A is a left annihilator algebra, then A f + A j is dense. (b) If A is a semisimple left annihilator algebra, then A f w dense. (c) I ' A f + A j is dense in A and A is semiprime, then A is a modular annihilator algebra. Proof (a) Theorems 2.4.7 and 8.4.5(b) show that A is a modular annihilator algebra. Hence equation (6) of Section 8.4 gives l a ( A f + A j ) = {0 }. Since (A f + A j )~ is a closed (right) ideal with zero left annihilator, it must equal A. (b) This follows immediately. (c) Let C be a maximal modular left ideal in A . Then C is closed (by Theorem 2.4.7) and proper, so it does not include A f + A j . However, it does include A j by Theorem 8.3.6(a). Hence it does not include A f - Thus Proposition 8.4.3 shows r a £ ^ { 0}. Hence A is a modular annihilator algebra by Theorem 8.4.5(a). □ The main idea o f the next proposition is due to Kaplansky ([1948b], Theorem 5, p. 692).
8.7.3 P r o p o s itio n Let A be a semiprime topological algebra with dense socle. T'len A is semisimple and equal to the normed direct sum o f its closed topologically simple ideals. Furthermore, the closed topologically simple ide als are exactly those o f the form A eA ~ for some e € A m i Proof Inclusion (4) of Section 8.4 shows A j C l a ( A f )- Clearly anything which ar nihilates A f annihilates its closure, so this inclusion becomes A j C laA ~ 3}. Hence A is semisimple. Let e be any minimal idempotent in A and consider the closed ideal 1C = A eA ~ generated by e. First we will show that K is topologically simple. Theorem 4.3.2(a) shows that K, is semisimple. Let J be a non zero closed ideal of 1C. If Te is {0 } and o and b are arbitrary elements of A tV*n ae e 1C implies la eb = (la e )e b C l e b = {0 }. It follows that A eA — 1^"=1 ajebj : n £ N; bj € A } satisfies T A e A = {0 }. Since tC is the closure of A e A we conclude J 2 C I K = {0 }, which is a contradiction since X, is semiprime by Theorem 8.4.5(a). Hence l e is not zero so there
702
8: Algebras with Minimal Ideals
8.7.4
is some ae € I e \ {0 }. Since /Cae is not zero and ICae is included in the minimal left ideal A e, we have 1 O ACT = Ae. Similarly we find 1 D eA. Hence the closed ideal T includes A e A and hence AC. Thus ACis topologically simple. Conversely, suppose AC is any topologically simple ideal in A- Then K A f is not {0 } since this would imply AC2 C K.A = fC(AF)~ = {0}. Hence ACfl A f is a non-zero ideal in the modular annihilator algebra A f and thus contains a minimal idempotent e by Theorem 8.4.5(h). Thus the inclusion A eA ~ C AC together with the topologically simple nature of AC gives AC = A eA ~ ■ Let A be a subset of A m i so that each ideal of the form A e A " with e 6 A m i can be expressed in the same form for a unique e e A. Clearly (Yie£A( Ae A) ~) ~ includes and hence equals A p — A . Hence A equals the topological sum (i.e., the closure of the sum) of its topologically simple closed ideals. However, this sum must be topologically direct. This can be seen as follows. Suppose { l a : a € ^4} is the set of all distinct topo logically simple ideals in A. Then l al g = l a <11$ is {0 } for each a ^ (3. Suppose there is a subset B C A and an element a 6 A \ B such that Ta n Tfj)~ is not {0 }. Then this intersection must equal Ta , and since l a = W holds by the previous remark, we have 2 ^ = { 0}, contrary to assumption, (Note that we have shown that any sum of distinct topologically simple ideals in a semiprime topological algebra is a topolog ical direct sum.) □ In applying the last proposition to semisimple dual, annihilator, or left annihilator algebras, it is useful to know that the topologically simple direct summands are again of the same type. Any closed ideal in a dual topo logical algebra is again a dual algebra as shown by Kaplansky ([1948b], Theorem 2, p. 690). It is shown in Bonsall and Duncan ([1973b], Theorem 9, p. 163) that a closed ideal I in a semiprime annihilator Banach algebra A is an annihilator algebra if it satisfies {TA)~ = AX~ = X. Since this is obviously true of topologically simple ideals in a semiprime algebra, the result holds here also. For left annihilator algebras the result is included in Smiley [1955]. We remark that Bonsall and Duncan ([1973b], Proposi tion 15, p. 165) also contains a condition under which the quotient of an annihilator Banach algebra is again o f the same type. In Proposition 8.7.4 we will use the approximate B # -condition intro duced in Definition 2.3.9 above to characterize those topologically simple direct summands which occur in Proposition 8.7.3. 8.7.4 P r o p o s itio n The following are equivalent for a left annihilator Banach algebra A . (a) A is primitive. (b) A is topologically simple and semisimple.
a..?.-:
Classes of Algebras with Large Socle
703
(c) There is a faithful, contractive, strictly dense representation T of A on some Banach space X satisfying B p ( X ) C T* C B k ( X ) and such that the ■preimage o f B f ( X ) under T is dense in A. Moreover the norm o f A satisfies the approximate B * condition if and only if T is an isometry.
Proof (b) => (a). Since A is semisimple, it has at least one primitive ideal, which must be { 0} by topological simplicity. .. i(c) => (a). This follows immediately since we have already noted that Bp{<X) is irreducible on X. As noted above A is a modular annihilator algebra. (a) => (b). The socle of A is dense, since otherwise A j = ra A f is non-zero. Theorem 8.4.5(d) shows that there is some minimal idempotent e satisfying {0 } = l a A c. Let I be a non-zero closed ideal of A. Then for eaclj, non-zero b 6 1 there is some ae g A e with bae ^ 0. Hence Te is a .lon-zsro ideal included in the minimal ideal A e. Therefore A e = l e C l inplies e e l Therefore 1 includes A eA ~ which is topologically simple by -reposition 8.7.3. If A eA ~ includes A m i it includes A f and hence includes A and we are done. Otherwise there is some / € A m i satisfying ' f- J.eA~ and hence A f A ~ fl A eA ~ — {0 }. However, taking 1 = A f A ~ previous argument we have the contradiction e £ A f A ~ . (a ‘ => (c) Theorem 8.4.5(e) shows that A has a faithful representation Le for so T ie e 6 A m i ■ Theorem 8.3.6 shows the properties of Le. Proposition 3.7.2(d) shows that A f is dense in A . Since Le is contractive, LeAp C i3 f ( A c ) is dense in LeA . This implies LeA C B f( A c ) ~ C Bn(Ae) . Next we show that all of Bp{Ae) is included in L\. Suppose S is a rank one operator in B(Ae) and suppose la ker(S) is {0 }. This implies succes sively, £,yi(ker(5)v4) = {0 }, (keT(S)A)~ = A, and A e = (ker(<S).4)~e = eAe~ = ker(5)e_ = ker(S). Since this is impossible, we may choose •; 7- ? which satisfies 6 ker(S) = {0 }. Let ce S A e satisfy S(ce) ^ 0. Then we have A e = ker(S) © Cce and hence bee ^ 0 since otherwise we would have L\ — 0. Now since Le is irreducible we can find d € A satisfying Ljbcs — S(ce). Hence S equals Ledb, so includes all rank one operators. Kence it includes all finite-rank operators. T is an isometry then A satisfies the approximate .©^-condition, since have already noted that T* does. Now suppose A satisfies the approx.’~ s i 3 B # -condition. Since a i-* ||Ta|| is an algebra norm on A satisfying r.l" < ||a|j for all a 6 A and a spectral norm by Proposition 8-4.10, Proposivori -3.7.5 shows ||To|| = ||a||
V a e A.
□
'T'se following easy result is essentially due to Kaplansky ([1948b], p. 698).
704
8: Algebras with Minimal Ideals
8.7.5
8.7.5 P r o p o s itio n Let A be a normed algebra satisfying one of the last three properties of Definition 8.7.1. Let B be a closed subalgebra and let I be a closed ideal of A ■ Then B and A/1 satisfy the same property as A . Proof The case of closed subalgebras is obvious. So is the case of quotient algebras after noting the equality ipo La = La+j o y>, where A —* A/T is the natural map. □ The next result is due to Alexander [1968] (Theorem 7.3, p. 15) and Barnes for Banach algebras. 8.7.0 P r o p o s itio n Let A be a aemipnrne spectral normed algebra with dense socle. Then A is a semisimple modular annihilator algebra and if T is any normed topologically irreducible representation of A on a normed linear space X , then there is a dense T-invariant linear subspace y of X such that the restriction o f T to y is algebraically invariant and has the same kernel as T. If A is a Banach algebra, then it is compact. Proof Since the socle is dense, its annihilator is zero and hence A is semi simple by inclusion (4) o f Section 8.4. (Note that the norm need not be spectral in this argument.) Since primitive ideals are closed in a spectral norm, none can contain the dense socle of A. Theorem 8.4.5(f) implies that A is a modular annihilator algebra and Proposition 8.4.10 implies the condition on T since the primitive ideal ker(T) does not include A f Next we will show that if a belongs to the socle o f a semiprime Banach algebra A , then LaRa has finite-dimensional range. (The converse is proved by Alexander in [1968], Theorem 7.2, p. 14.) We can write a = dje_,, where a} € A and e3 € Ami- Hence Theorem 8.3.6 shows that aAa C Yl",k=taj ej A a kek C Ylj,k=iCai cjk for any non-zero cjk 6 ejAek- Thus LaRa has finite-dimensional range so it is compact. The map a >-+ LaRa (although nonlinear) is obviously continuous. Hence if A f is dense in A and LaRa is compact for each a € A f , then LaRa is compact for each a £ A. □ 8.7.7 C orolla ry If A is a semiprime Banach algebra with A f ® A j dense and A j compact then A is compact. Proof If a belongs to A f + A j , we can write o = c + 52"=1 aj eji with c £ A j , aj 6 A, and £ A m i ■ Then for any b £ A, we have LaRa(b) — c6c+5^ "=1 c6o j e j + ^ " _ , a jejb c+ Y lj k=i aj ej^ak^k- Inclusion (4) of Section 8.4 shows that the second and third sums are zero. Hence if A j is compact, the argument in the previous proof shows that A is compact. □ Recall that a bounded operator T on a normed linear space X is called a Riesz operator if its image in B(X)/B k ( X) is topologically nilpotent.
8.7.8
Classes of Algebras with Large Socle
705
(These operators were briefly discussed in Section 2.8.) The next result is due to Barnes [1971a], Note that both the necessity and the sufficiency of the first condition are proved under (different) weaker hypotheses. 8.7.8 P r o p o s itio n A semisimple Banach algebra is a modular annihilator algebra if and only if LaRa is a Riesz operator for each a & A- Hence for any compact Banach algebra A , A / A j is a modular annihilator algebra. Proof If A is a compact Banach algebra, Proposition 8.7.5 shows that A / A j is a compact semisimple Banach algebra. Hence the last statement 3 an immediate consequence of the first. Suppose LaRa is a Riesz operator and C is the commut&nt of o in A. If A 7^ 0 belongs to the boundary of Spc(a2), then Theorem 2.5.7 shows that .here is a sequence { 6 „ } „ e N with ||bn||= 1 for all n and {A_ 1 a26„ — 6 „ } „ e N converging to zero. Hence Sp(LaRa) contains A since {(A — LaRa)bn} neN converges to zero. Also dSp^(a2) equals dSpc(a2) by Proposition 2.5.3. So Sp(a?) is included in Sp(LaRa) which has no limit points except zero as shown in Section 2.8. Thus the same is true o f Sp(a) since Sp(a,2) = Sp(a)2 holds. Theorem 8.6.4 shows that A is a modular annihilator algebra. Conversely, suppose A is a semisimple modular annihilator Banach al gebra. Then A / Ap is a radical algebra by Theorems 8.4.5(c) and 4.3.2(a). Hence Corollary 4.3.8 shows that for any o G A and any n € N there is an element bn £ A f and an e„ > 0 satisfying
| | a " - M 1/n = e „ - 0 . A ny c e A satisfies
||((L0ii 0)n - L ()„iei)Jc||
||{L.*.)" - LbnRbJ l/n
=
| | a "ca "-6„ c M
<
||o"ca" - 6„ca n||+ ||6„c a " - 6„ c 6„||
<
| | a "-6n||(||a"|| + | | M )M
<
e»(||a"|| + ||U)1/n < e „ (2 ||a"|| + O 1/" .
and hence
The last sequence converges to zero. The proof of the last propositon shows that LbnRbn belongs to B f (A) C Bk (A), so LaRa is a Riesz operator. □ The next proposition is due to Barnes ([1966], Theorem 7.2) for semisimple algebras. The last remark is due to Kaplansky ([1948b], p. 698). n.7.9 P r o p o s itio n Let A be a semiprime left completely continuous wormed algebra. Then A is a modular annihilator algebra and all its ir reducible representations are finite-dimensional. Proof Since A is semiprime, L :A -* B k (A) C B(A) is faithful and Proposition 2.5.3 shows Sp^(a) = Spc{A)(La). In order to prove that A is a modular annihilator algebra, Theorem 8.4.5(c) and 4.3.2(a) show
706
8: Algebras with Minimal Ideals
8.7.10
that it is enough to prove that there is no maximal modular left ideal including A f - Suppose M is an exception. Choose c € A satisfying -4(1 — c) C M - Then 1 belongs to SpA {c) = Sp(Lc). Since Lc is com pact, ker(/ — Lc) = jm.4(1 — c) is finite-dimensional and the ascent of I — Lc is finite by the results of Section 2.8. If m denotes this ascent, then Tl = ,m M (1 - c)m) = ker((/ - Lc)m) = k er((/ - Lc)*m) = * * (^ (1 - c)2m) is a finite-dimensional right ideal. We wish to find an idempotent e G A f satisfying e A = TL. Since TL is finite-dimensional, it includes a minimal right ideal o f A which contains a minimal idempotent by Lemma 8.5.4. Let e be the sum of a maximal (nec essarily finite) family of orthogonal minimal idempotents in TL. Obviously e belongs to the socle of A. Suppose e A C TL does not equal TL. Then we may choose a € Tl \ e A and consider 6 = a - ea ^ 0. Since bA C TL is non-zero and finite-dimensional it includes a minimal idempotent / , by the above argument. The minimality of / and the equation e f = 0 (which follows from / € 6.4), show that f € TL is orthogonal to all the idempotents chosen previously with sum e. This contradiction establishes TL = eA. Since we are assuming A f Q M , we have A( 1 —(d + e )) C M , where d e A is defined by 1—d = (1 —c)m. As before, the fact that L xd+e is compact and that 1 belongs to its spectrum implies that r a A (1 —( d+e ) ) = ker(/ —L j +e) is non-zero. Let 6 be a non-zero element in this ideal so that it satisfies (1 —d)b = eb € r a ( A( 1 —c)m). Then we have .4(1 —d)2b = .4(1 —d)eb = 0 which implies 6 £ jm(.4(1 - d)2) = jm M (1 - c)2m) = jm(-4(1 - d)). This implies eb = (1 — d)b = 0. Since 6 was chosen from e A this implies 6 = 0, contrary to its choice. Hence we conclude that no maximal modular left ideal includes A f s o that A is a modular annihilator algebra by Theorem 8.4.5(a) and Proposition 8.4.3. Theorem 8.4.5(d) now shows that any irreducible representation has a kernel of the form V e for some e e A m i - Proposition 8.4.4 shows that V* = {o € A : Aa C (1 — e)^4} is the kernel of the anti-representation Re. Since Le is compact by hypothesis, and its restriction to e A is the identity operator on e A , eA must be finite dimensional. Hence A j V e is also. Thus any irreducible representation is finite-dimensional. CJ
8.7.10 T h eorem Let A be a compact Banach algebra. Every irreducible representation o f A is equivalent to a representation on a Banach space in which each representing operator is compact. Proof Since we are only interested in irreducible representations we may replace A by A / A j and hence assume that A is semisimple. Proposition 8.7.10 asserts that A is a modular annihilator algebra. Theorem 8.4.5(e) shows that each irreducible representation is equivalent to Le for some e 6 A m i - Theorem 8.3.6 gives the structure <>f these representations and we will use the notation defined there.
I it 8.7 I 1
Closes of Algebras with Large Socle
707
Choose and fix an arbitrary e € A m i and an arbitrary a € A (or which : is; :on-zero. We will show that L% is compact. 1 ny 6, c, d 6 A satisfy ||6ec|| ||tie|| > ||6ecde|| = ||6<de, ec)e|| = \{de,ec)\ ||6e||. Hence any b, c € A satisfy ||6ec|J > ||6e||sup{j((fe,ec)| : de € (A e ) i }. Since Le a is non-zero, we may choose d € A satisfying ade ^ 0. Since (•, •) is a pairing we may choose c € A satisfying 0 ^ (ade, ec)e = ecade = (de, em)e. Denote \(de,eca)\l\\de\\ by 6. The inequality above shows that any b £ A satisfies ||6eca|| > £||6e||. Now'iet {6ne}n€N Q {Ae)\ be arbitrary. Since A is compact there is a subsequence {o6n)ieca}fceN of LaRa(bnec) = abneca which converges. The last inequality shows WKbnke - K bn^\\ < <5- 1||ab„heca - abHjeca\\ "/idea in turn shows that a subsequence of {L *6 „e}ngN converges. Hence J'a ~£ compact as we wished to show. □ '.'7 -; now summarize the major results o f this chapter. S.7.Ill T h eorem The diagram on the next page summarizes known im putations between the various classes o f algebras studied in this chapter. Arrv^is indicate implications and letters beside the arrows indicate extra conditions which are required in the proof. Some of these extra conditions may Is unnecessary but no other implications, beyond those shown here, can hold between these classes of algebras for semisimple Banach algebras. Proof Results not proved in Propositions 8.7.2, 8.7.6, 8.7.8, 8.7.9, or The orem 8.7.10 or in the discussions surrounding the various definitions are all trivial. Example 8.8.8 or Barnes ([1966], Theorem 7.1) show that even for Banach algebras with dense socle the implication LCC=>CC fails. When A is a Banach algebra, there are two ways to define the condition KIR given in the diagram. First, any irreducible representation of the algebra A is equivalent to a continuous representation on a Banach space siic-i iliat Ta consists of compact operators. Second, if T is a strongly continuous representation of the algebra A on a Banach space then Ta conwis.s of compact operators. Theorem 4.2.8 shows the equivalence of . -: . ■ -»-o definitions. □
708
8: Algebras with Minimal Ideals FD
8.7.11
C
DMA FD = finite-dimensional C = commutative D = dual CC = completely continuous A = annihilator LCC = left completely continuous LA = left annihilator K - compact DS = semiprime with dense socle DMA = Duncan modular annihilator MA = modular annihilator D(S©R) = semiprime with (socle 0 radical) dense FIR = fill irreducible representations are finite-dimensional KIR = all irreducible representations are equivalent to representations on Banach spaces in which each representing operator is compact. Abbreviations for the conditions are: sp - semiprime; ss = semisimple; S = spectral normed; B = complete normed.
8.7.12
Classes of Algebras with Large Socle
709
Annihilator Algebras We wish to include a few results on the much studied class o f annihilator Banach algebras. Let A be such an algebra. Since it is a left annihilator algebra, Proposition 8.7.2 shows that A f + A j Is dense. Hence the socle A f is dense if A is semisimple. In that case Proposition 8.7.3 shows that A is the normed direct sum of its closed topologically simple ideals, all of which are of the form (AeA) ~ for some minimal idempotent e. Proposition 8.7.4 gives more information on these ideals. None of the above results require more than that the Banach algebra be a left annihilator algebra. The next theorem, which improves on Bonsall and Duncan ([1973b] 32.20), captures the difference between annihilator algebras and left annihilator algebras. We have already noted that B a ( X) is a left annihilator algebra for z ay Banach space X, but this theorem shows that it is an annihilator algebra if and only if X is reflexive. For another approach to this theory using single elements (an abstraction of finite-rank operators), see John A. Erdcs [1971] and S. Giotopoulos, S. [1985], cf. Joachim Puhl [1978]. 8.7.12 T h e o re m Let A b e a semiprime annihilator Banach algebra and let e be a minimal idempotent. Then the pairing of Theorem 8.3.6 establishes homeomorphic linear isomorphisms of e A onto (Ae)* and of A e onto (e A )* when eA and A e have the complete norms they inherit as closed linear subspaces o f A . Hence eA and A e are both reflexive Banach spaces and the linear functional o f Proposition 8.3.5 is autoperiodic. Proof By symmetry it is enough to establish the homeomorphic linear isomorphism of eA onto (Ae)' . Recall from Proposition 8.3.5 and Theorem 8.3.6 that there is a linear functional u> G A* satisfying u (a )e — eae for all a £ A and a pairing between A e and eA satisfying (a, b)e = ba = ui(ba)e for all a € A e and all b G eA . Hence the map b G eA *-» u>(,|Ae is a continuous linear injection of eA into (Ae)*. (As usual, u>t is defined by W|,(o) = u(ba) for all a G 4 .) It is enough to show that this map is surjective, since the inverse boundedness theorem will then establish that it is a homeomorphism. Let r be an arbitrary non-zero element of (Ae)'. Define t G A* by 7(a) = r(ae) for all a G A. In the notation of Definition 4.2.20 we have y A = {b G A : f(ba) = 0 V a G ,4}. We claim ker(r) = A e r rA. (If b = be belongs to the kernel of r then for all a G A , we have r(ba} = r(beae) — r(bui(a)e) = ui(a)r(b) = 0. For the opposite inclusion, note that b G A e implies r(b) = f(b ) = f(b e) which gives A en ^ A C ker(r).) Since r is non-zero, we may find c G A e satisfying r(c) = 1. V/e now switch attention to the topologically simple Banach algebra B = ( AeA)~ and its right ideal 72 = ^A H B. We use Theorem 32.9 of Bonsall and Duncan [1973b] to establish that B is an annihilator algebra. Note that 72 is a proper ideal of B, since it does not contain c G Ae C B. Hence the annihilator algebra property shows that there is a non-zero element d
710
8: Algebras with Minimal Ideals
8.7.13
of B in l aH- However, the proper two sided ideal la A b of B is {0 } by topological simplicity. For any a £ A e, a - r(a )c belongs to ker(r) C V, so we conclude d(a — r(a)c) = 0. Since dAe is not {0 }, dc £ A e is not zero. Using the fact that we have a pairing again, we find b £ eA satisfying bdc = e. Hence any a £ A e satisfies r(a)e = r(a)bdc so r = Ubd\eA ^ we wished to show. Finally, ui is autoperiodic since Theorem 8.3.6 shows that A e and eA are homeomorphically isomorphic to the left and right representation spaces defined by a). □ 8.7.13 Corollary The representation space o f any irreducible representa tion o f a semiprime annihilator Banach algebra can be given the sturcture o f a reflexive Banach space, relative to which the representation is continuous. If the representation is already strongly continuous on a Banach space, the Banach space is reflexive. Proof Theorem 8.4.5 shows that every irreducible representation is equiv alent to the representation Le on the Banach space A e for some minimal idempotent e. Theorem 8.7.12 and Corollary 4.2.4 complete the proof. □ 8.7.14 Corollary A semisimple annihilator Banach algebra is an ideal in its double dual with respect to either Xrens product. Proof By Proposition 1.4.13 we must show that the left and right regular representations La and Ra are weakly compact for each a in a semisimple annihilator algebra A. The theorem shows this for any minimal idempotent a = e since A e and eA are reflexive. However, A is the closed ideal gen erated by its minimal idempotents. Since L and R are continuous and the weakly compact operators form a closed ideal in B(A), we get the desired result. □ Automatic Continuity We close this section with an automatic continuity result too specialized to be given in Chapter 6. The result is due to Barnes [1967b]. The reader may wish to compare this result with Proposition 6.1.13 and Corollary 6.1.14. The set Bv was introduced in Definition 6.1.7. 8.7.15 T h e o re m Let A and B be Banach algebras and let tp:A —►B be a homomorphism with dense range. Then l a ( B f ) includes Bv . Hence if l a { B f ) vanishes (i.e., if B is a semisimple Duncan modular annihilator algebra) then is continuous. In particular a normed representation T of A on a normed linear space X is continuous i f T ^ n B p i X ) is dense in TaProof Proposition 6.1.8(d) shows that Bv is a closed ideal of B. For any minimal left ideal £ of B, BVC C £ equals either £ or {0}. The former case is impossible since £ = BVC C Bv contains a non-zero idempotent by
8.8.:
Examples
711
Corollary 8.2.3, but Bv contains none by Proposition 6.1.10. Hence is included in -- l a ( B f ) equals {0 } then ip is continuous by Proposition 6.1.9(a). The last remark follows from the previous one by noting that X may be com pleted without altering the hypothesis. □ We mention two other results on automatic continuity of homomor phisms which involve minimal ideals. The first result is contained in Bacheiis [1972]. Let A and B be Banach algebras with A semisimple and let tp:A —» B be a homomorphism. Then
V a 6 . 4 F;
b € A F.
Frorr this it is shown that if A F has a bounded approximate identity, then tp is continuous on A f . Note that this result puts no restrictions on B or
•The last result we mention is from Barnes [1967b], Let A be a Banach algebra such that any closed ideal J of A with A/1 finite-dimensional sat isfies X2 = I . (He shows that C*-algebras and dual algebras with bounded approximate identities (such as L l (G) for G a compact group) satisfy this condition.) Let B be a strongly semisimple modular annihilator algebra (e. g,,.a-semisimple completely continuous algebra). Then any homomorphism ip: A —►B is continuous without any restriction on the range of
8.8
Examples
3 .8 .’ E xam ple The argument given before Theorem 8.6.2 shows that B( X) is a semisimple (indeed primitive) Duncan modular annihilator normed a'geb'a or Banach algebra, respectively, whenever X is a normed linear speca or Banach space. It was also shown there that when X is an infinitedimensional Banach space, then B{ X) is not a modular annihilator algebra. 8.8.2 E xam ple Vala [1964] (cf. Alexander [1968], Theorem 3.2) shows that for any compact linear operator T on a Banach space X the map from to B(X) defined by S^TST
S G B(X)
s ' .rnpact linear operator on B(X). This proves that B k ( X) is a comoa"' 3anach algebra for any Banach space X. Hence B k ( X) is a modular annihilator algebra and thus Theorem 8.4.5(e) shows easily that every ir reducible representation of B k ( X) is topologically equivalent to its given
712
8: Algebras with Minimal Ideals
8.8.3
representation on X. Thus every irreducible representation of Bk(X) con sists of compact operators and none of them are finite-dimensional if A' is finite-dimensional. Actually Bk(X) is the largest irreducible subalgebra of B(X) which is a compact Banach algebra with respect to a complete algebra norm majorizing the operator norm (Bonsall and Duncan [1973b], p. 179). Furthermore, Bk(X) is topologically simple when Bk(X) = Bf(X) ~ holds, as it does for many Banach spaces X . That this identity fails for some Ba nach spaces was first shown by Per Enflo [1973]. For such a Banach space, B k ( X ) is a compact Banach algebra in which its socle B f ( X ) is not dense. 8 .8.3 E xa m p le The algebra Ba(X) = Bf(X)~ of approxim ate operators obviously has BF(X) as its dense socle. Since it is a semisimple (in fact, primitive) Banach algebra, it is compact and a modular annihilator algebra by results in this section. It is also easy to see that it is topologically simple. More is true. B a ( X ) is an approximate B * - left annihilator algebra. This is proved in Smiley ([1955], Theorem 2). Furthermore, B a ( X ) is an annihilar tor algebra if and only if X is reflexive. See Antonio Fernandez Lopez and Angel Rodriguez-Palacios [1985]. Hence B a ( X ) , for any nonreflexive Bar nach space X, is a topologically simple, semisimple, left annihilator Banach algebra which is not an annihilator algebra. This too can be generalized. Let B be any algebra of operators which includes B f ( X ) and is complete in som e norm which dominates the operator norm and relative to which Bf (X) is dense in B. Then B is an annihilator algebra if and only if X is reflexive (Rickart [1960], Theorem 2.8.23), and B is a dual algebra if and only if each T e B belongs to TB fl BT (ibid., Theorem 2.8.27). The C p algebras of operators on a Hilbert space and the algebras Bhs(W ) and Bt CH) discussed in §1.7.17 satisfy all the above hypotheses and hence are infinite-dimensional, topologically simple, dual algebras. We remark here that all these examples of semisimple (in fact primitive) dual Banach algebras have natural infinite-dimensional (faithful) irreducible representations and thus are not left completely continuous or completely continuous algebras by Proposition 8.7.9. 8.8.4 E xa m ple Davie [1973a] uses Enflo’s example [1973] of a reflexive Banach space which does not have the approximation property to provide a topologically simple annihilator Banach algebra B a ( X ) which is not dual, thus settling a question which remained open since Bonsall and Goldie’s paper [1954]. 8 .8.5 E xa m ple The algebra of inessential operators on a Banach space X introduced by Kleinecke [1963] is a modular annihilator algebra by Propo sition 8.4.10 above. Similarly one sees that the algebra B w k ( X) o f weakly compact operators on X = C(Si) (f2, a compact Hausdorff space) and on
8.8.6
Examples
713
X - Ll ( M) (M , a measure space) are modular annihilator algebras since Dunford and Schwartz [1958], VI.7.5 and VI.8.13, show that the square of these algebras is included in B k ( X) . By previous remarks these alge bras B w k { X) provide examples o f modular annihilator algebras which are not compact algebras since they properly contain B k ( X) . If Af is a lo cally convex topological linear space then Bk ( X) is shown to be a modular annihilator algebra by Barnes ([1971a], 4.2). 8.8.6 A N orm ed A lg eb ra o f O p era tors w ith a D en se S ocle w hich is n o t a M o d u la r A n n ih ilator A lg eb ra . This example is due to Yood [1964’. The assertions which we do not check here are all verified in the cited reference. Let A1 be an incomplete normed linear space and let X be its completion. Let A = Bf { X) be the algebra of bounded linear operators on X which have finite-rank. Let z belong to X \ X and let U belong to the c ’osure o f A in B ( X ) but satisfy Uz = z where U is the closure of the graph of U in X x X. Let B be the subalgebra of B( X) generated by A and U. Clearly B is a normed algebra with dense socle A. However, the left ideal M — { T € S : Tz = 0} is maximal among closed modular left ideals but is not a maximal modular left ideal. Hence B is not a modular annihilator algebra since it is normed but contains nonclosed maximal left ideals contrary to Theorem 8.4.12. 8.8.7 E xam ple The algebra o f all precompact operators on a normed linear space need not be a modular annihilator algebra or even a Duncan modular annihilator algebra. As pointed out by Barnes [1971a], the example on p. 590 of John Ringrose [1957] contains a precompact operator with the whole plane as spectrum. Hence Theorems 8.6.2 and 8.6.4 show this. 8.8.8 A L eft C o m p le te ly C on tin u ou s Sem isim ple B an ach A lg e b ra w h ich is n ot C o m p le te ly C on tin u ou s Smyth [1980] gave the first such example settling a question which had been long open. Let A be the algebra (under pointwise operations) of complex sequences. Consider the following four rorms: OO
Ni l =
INI2 = 8Up {K I : n € N}
!“ *»! n —l
00
N la =
00
n2la"l
N l 4 = 5 Z n|a»l
n=1
n=1
Let J- ' k) be the subalgebra of elements in A with finite ||•||fc-norm. Define C to be the following algebra under matrix multiplication:
£ = { ( aa
2)
: a * e •*(*);* = l-2 ,3 ,4 }
714
8: Algebras with Minimal Ideals
8.8.9
with norm
/
a2 \
V«3 a4//
k=l It is tedious but straightforward to show that £ is a semisimple Banach algebra. (The ideate Mn = { ( V
: o f c „ = 0 ; * = l , 2, 3, 4} -
are maximal modular since C / M n is isomorphic to M 2. Hence £ is even strongly semisimple. In order to show that £ is left completely continuous, we choose an ar bitrary element b — ^ ^
^ ^ G £ i and show that bC\ is totally bounded.
Let £ > 0 be arbitrary and choose N G N satisfying Y,
<9
E
n=JV
n“ 2 < 9
oo <
|
£
n=N
“’i^
i
<
I
Let T be the finite-dimensional ideal: T = {a
G
£ : akn = 0
n > N;
k = 1 ,2 ,3 ,4 }.
It is easy to verify that the distance from any c G bCi to C t is less than 8e/9. Since T is finite dimensional, T\ can be covered by a finite number of (e/9) - balls. Hence bC\ is totally bounded as we wished to show. Tb see that £ is not completely continuous, we define {a m}meN G -4] by Omn = L ( *oT
o) (o
(the Kronecker delta). Clearly ^ J ) = (S
multiplication by ^
)
^
belongs to C\ but
has no convergent sequence, so that right
o ) ^ n0t a comPact °Perator-
In the cited reference an easy proof is given that a left completely con tinuous semiprime algebra with a dense socle is completely continuous. 8.8.9 E xa m ple Smyth [1980} also gave an example o f a semisimple com pact Banach algebra A and elements b, c G A s o that a *-♦ bac
a &A
is not compact. This contradicts a conjecture of Alexander [1968] who had proved that all such maps are compact if A is a C*-algebra.
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2.6
INDEX
absolutely continuous functions 113 abstract Runge theorem 218 adjoint relative to a pairing 672 Al&oglu’s theorem 79 algebra 3 annihilator 699, 709 l:;euriing 154, 436 central 638 compact 699 . complete matrix 558 completely continuous 699 completely regular 326, 410, 432,632 division 211, 443 caal 699 I uncan modular annihilator 695 ^:>ite-dimensionai 658 j'ourier-Stieltjes 424 Pr6chet 297 :ctionaJ 354 •rjsctive 172, 473 Uft annihilator 699 left completely continuous 699, 713 :ccal (see functional also) 645 ioeally multiplicatively convex 299 modular annihilator 683 nil 481 n >rmed 10 oyrmanently spectral 258, 559 satisfying a poiynomial identity 627 ^emi-normed 10 i-m i-topological 7 rnpie 33, 496 r.octral 233, 309, 497 o jectral normed 212 «;i«ctral seml>normed 212 iiymmetric group 659 tensor product 165, 43, 481, 524 typological 7 topologically nilpotent 515 topologically simple 33, 702 imital 3, 658, 661 weakly central 638 eakly relatively regular 205 .vjighted cross product 511 ' out divisors o f zero 255
linear maps 22 algebraic dual o f a linear map 21 a iinear space 21 algebraic group algebra 128 algebraic multiplicity 274 algebraic tensor product 159 almost commutative 307, 468, 478 almost periodic functions 336 linear functionals 61 almost separably valued 156 alternating noncommuting poiynomial 628 ample subdirect product 497 analytic Banach space valued function 217 at infinity 217 analytic semigroup in A 543 bounded 543 analytic space 549 annihilator 25, 288 left, right 25, 679 in L °°(G ) or L *(G ) 434 anti-homomorphism 5 anti-representation o f an algebra 440 approximate B ^-condition 231, 703 approximate identity 520 bounded 520 central 529 commutative 520 countable 520 left, right, two-sided 520 sequential 520 weak left, right, two-sided 520 approximate point spectrum 286 approximate unit 532 approximately unital 50, 520, 542 approximation property 95, 533 bounded 95, 533 bounded compact, compact 95, 533 metric 95 Arens products 47, 317 examples 55, 79, 102 Arens regular 47, 60, 81, 472 Arens representations 63 Arens-Royden theorem 412 Arens semiregularity 54 Arens’ algebra L* 237 Arens-Calderon theorem 386 Arens-Hoffman extension 369, 608 Aronszajn-Smith theorem 294
of :w?;nal power Heriett 370 ..itid ed double centralizers 26, 51 r vms of analytic functions 349 ;'--ibal continuous sections 648
.
783
784 ascent o f an operator 276, 289, 688 asymptotic ideal 612 automatic continuity 28, 555, 710 automorphism 5 inner 5 autoperiodic linear functional 470 Baer radical 485, 585 balanced convex hull 4 balanced set 4 ball, ckwed, open 14 Banach algebra 10 geometrically injective 473 infective 473 o f power series 370 Banach *-algebra 109 Banach space 10 dual o f a normed linear apace 21 ergodic 86 finitely representable in 82 flat, nonfiat 86 reflexive 79, 470 superreflexive 82 uniformly convex 82 uniformly smooth 82 Banach-Saka property 86 baa« space 626 basis projection 94 basts bounded ly complete 94 ordered 70 Schauder 94 shrinking 94, 533 belong to a set locally 635 near an ideal 635 near infinity 635 Bernoulli, Daniel (1700-1782) 127 Beurling, Arne (1905- ) 11, 128 bilateral ahift 250 bilinear map 161 biznodule 455 Banach 455 biorthogonal system 94, 372 Bochner integral, integrable 157 Bodm er's Theorem 425 Bochner-Schoenberg-Eberlein condition 316 Bohr compactification 337 bound o f an approximate identity 520 bound o f a normed module 455 boundary o f a commutative spectral algebra 319 bounded set in a topological linear apace 298 bounded variation functions 113 Bracounter topology 530 Brown M cCoy radical 490, 585 C*-algebra 109, 186. 329, 376, 532 o f almoat periodic functions 337 algebra norms on 378, 516 C*-condition 109 Calkin algebra, generalized 108
Index Calkin algebra, weak generalized 91 capacity in Banach algebras 231 Cartan, £lie (1869-1951) 6, 72, 504, 658 Cartesian product 38 categorical theory of radicals 498 category 499 cosemiabelian 502 semiabelian 502 of algebras 500 of Banach algebras 500 of normed algebras 500 o f spectral algebras 500 o f spectral semi-normed algebras 500 Cayiey, Arthur (1821-1895) 72 center of an algebra 9, 637 central polynomials 631 centralizer bounded double 26 left, right, double 26 centroid of an algebra 653 Ces&ro mean 122 change of basis 70 character group o f G 419 character o f G 419 character space of an algebra 304 characteristic polynomial of T 273 characteristic value (eigenvalue) 274 characteristic vector (eigenvector) 274 clopen set 67 closed under multiplication by C (S . a ) 642 Cohen’s idempotent theorem 415 cohomology 415 cohomotopy group, first 411 coimage of a morphism in a category 501 cokernel of a morphism in a category 501 commutant 9 commutative bimodule 610 commutative C*-aJgebra 329 commutative subset o f an algebra 9 commutativity, characterizations of 238, 311, 468 compactification of a topological space 335 complete topological linear space 526 completely regular topological space 65, 326 completion of a normed algebra 15 composition in a category 499 conjugacy class 131 connecting homomorphisms 42 consistent norms 228, 559 continuity ideal 571 continuity of homomorphisms from C (O ) 577 continuous spectrum 286 continuum hypothesis 578 contractive map 11 convolution o f linear functionals 136 convolution measure algebra 435 convolution product 114, 129 coordinate function on C 198 coproduct, categorical 38, 165 countable spectrum 695 Cousin data 412 Crabb’s Lemma 266 critical points 436
Index cross Lection 38 croszed product 260 cyclic vector of a representation 440 decomposing subspace 277, 443 decomposition o f representations 443 decomposition property 6 296 degree of '.oncommuting polynomials 627 trigonometric polynomials 113 deriva ion 393 dditive 603 -Jgebra 591 ’.jimodule 591 centralizing 601 iscontinuous examples 610 inner 592 descent o f an operator 289, 688 determinant 71 diagonal matrix 275 D ick sca, Leonard Eugene (1874-1954) 4 differential 1-form 393 dimension o f a representation 441 direct product o f algebras 37 38 direct, product o f two algebras 34 direct sum o f algebras 39 direct sum, algebraic 39 direct sum, Banach algebra 39 direct sum, internal, external 40 DirichJet kernel 128 Dirichlet’s theorem 127 DirichJet, Peter Gustav Lejeune (1805-1859) 6, 127 disc algebra 68, 330 disc algebra under convolution 507 discrete group 130 divisible subspace 364 divisor o f zero 4 joint, two-sided 4 left, right 4 dominate 14 double centralizer algebra 26, 315, 454, 495, 497, 525, 599, 653 t utomorphisms o f 31 examples 30, 53, 66, 104, 148 doubU dual space 22 double dual algebra 47, 316, 527 identities in 49 doubh power bounded 271 dual £ oup o f G 419 dual n ap 21 dual norm o f a linear functional 21 dual o a homomorphism 313 dual o >erator 287 Dunfo d, Nelson (1906-?) 156 Dunfo d-Pettis property 93 Ounfo d-Pettis theorem 170 Eberle n-dmulian theorem 83 Eidelb ‘it’s lemma and theorem 106 Eicle!’ . Jit, Meier (1910-1943) 12, 105, 105, 553 elementary q-form 393
785 elementary tensor 159, 23, 73, 75 epimorphism 501 equivalence o f representations 440 equivalent monomorphisms 500 essential spectrum 295 Euler, Leonard (1707-1783) 127 exponential function in A 201, 411 kernel o f 358 exponential spectrum 203 extension o f A by C 33 split 34 equivalent 33 extension o f homomorphisms 322 extensions o f subalgebras 369 exterior derivative 394 exterior differential calculus 392 factorization o f sequences converging to aero 541 factorization theorems 535-552 [FC] “ -groups 530 feebly spectral subalgebra 254 Fej6r kernel 123 Feldman theorem 662 Feldman’s example 663 [FIA]"-groups 530 fiber 626 fiber map 626 filter 45 Fr6chet 45 proper 45 finite-rank for a left ideal 690 finitely generated normed algebra 383 forcing 578 Fourier algebra 126, 154, 173, 423 Fourier expansion 120 Fourier series 122 Fourier transform 122, 423 Fourier transform, inverse 125 Fourier, Jean Baptiste Joseph (1768-1830) 127 Fourier-Plancherel transform 121, 431 Fourier-Stieltjes transform 424 FVedholm theory in Banach algebras 691 FYedholm, Eric Ivar (1866-1927) 128, 191 free semigroup algebras 509 Freudenthal-Wei! theorem 530 full subcategory of a category 502 functional calculus discontinuous 363 continuous function 376 for spectral algebras 353 analytic (holomorphic) 200, 343, 349 polynomial 199, 342 rational function 199, 342 unique 363 various 341 functional subalgebra 354 fundamental theorem o f spectral semi-norms 228 Gamma function 531 Gantmacher’a theorem 62
786 Gelfand homomorphism 303 Gelfand’s limit a ° ° (a ) 16, 210 Gelfand radical 304 Gelfand space 304, 408 augmented 305 Gelfand theory 309 Gelfand topology 303 Gelfand transform o f a 303 Gelfand, Israel MoiseeviC (1913- ) 11, 156 Gelfaad-Maxur theorem 211 Gelfand-Raikov theorem 429 Gelfand-radical 304 Gelfand aemisimple 304 generalised basis 372 generated by (normed algebra) 318 generator o f one-parameter group 595 geometric category o f Banach algebras 32, 517 Banach spaces 32, 81 normed algebras 32 normed linear spaces 32 geometric multiplicity 274 geometric series 214 germs o f analytic functions 348 Gleason- Kahane-Zelazko theorem 242 global section 626 Goldstine’s theorem 79 Grothendieck topology 96 group algebra o f a locally compact group 143 group o f A 193 groups with invariant neighborhoods o f e 529 groups with small invariant neighborhoods o f e 529 Haar functional 140 Haar integral 140 Haar measure 142 Hamel basis 4 hermit ian element 376 Hilbert space 108 Hilbert space adjoint 109 Hilbert, David (1862-1943) 90, 128, 191 Hilbert-Schmidt norm 73 history of algebras 4 analytic functional calculus 343 Banach algebras 11 Fourier analysis 127 ideals and quotients 6 matrix algebras 72 radicals 503 representations 438 the spectrum 191 homomorphism 5 injective, surjective 5 unital 18 from derivations 593 bomotopic in G 411 bomotopy 411 hull o f a subset o f A 620, 323, 408, 446 hull-kernel topology 621, 326 hyperiavariant subspaces 294
Index ideal (two-sided) 5 left, right, one-sided 5 minimal 667 minimal closed 667 ideal of finite rank operators 23 idealizer 29 left, right 29 ideropotent 4, 67, 479 existence of 357 minimal 667 order among 668 orthogonal 4 identity element, multiplicative 3 image of a morphism in a category 501 implicit function theorem 405 [IN]-group 529 inclusion o f subobjects 501 inductive system o f algebras 42 normed 42 infinite conjugacy class (ICC) groups 131 injective tensor product 99, 175, 473 inner product 108 intertwining operators 443 invariant subset in a group 529 invariant subspace 225, 440 inverse Fourier transform 423 inverse Fourier-Stieltjes transform 424 inverse o f a relation 2 inversion theorem 424 involution 109 irrational rotation algebra 259 isometric 11 isometric Banach spaces 81 iaometry 11 isomorphic Banach spaces 81 isomorphism 5 isomorphism in a category 499 Jacobson radical 225, 474, 585 Jacobson topology 621 Jacobson-radical algebra 225, 474, 507 James space 79 algebra structure of 79 Johnson-Rickart uniqueness o f norm theorem 229, 557 joint spectrum 383 Jordan block 284 canonical form 284 Jordan derivation 614 homomorphism 580 product 579 Kaplansky’s theorem 244 kernel of a linear map 5 morphism in a category 501 set o f prime ideals 620 kernel o f a subset of 323 Koszul complex 416 Kretn-Milman property 85
Index L-homomorphism 436 b'Srs&ce 435 - T^brft 436 ^ ••*■) algebras, functorial properties 145 v*-grouo algebra o f G 130 j**- ,j -module 455 -ormula 592 Lie .^racket 580 Lie homomorphism 591 Lie idsU 592 'y.'t v.:\y ctent 241 Ivo pro iuct 241, 591 Lie subalgebra 241 limit ?T-:ductive or direct 42 normed inductive 42 normed projedtive 44 projective 44 iinear functional 21 linear map, unital 18 linear space normed, semi-normed 10 Bemi-topological 7, 8 topological 7 local spectral theory 296 iocaily bounded topological linear space 298 locally compact group 134 k-calW convex semi-topological linear space 20 logarithms in Banach algebras 359 ;*roup 338 j 'V-c" r onent o f the aero ideal 646 ary 640 14 Iv“r t.vjlb ’evich basis 372 l-febra
e . ar an algebra 74 nc ms on 72 matrix representation o f a linear operator 70 matrix r.nits 69 msociT.f‘ ideal 235 ideal space 304 ^axim ai one-sided ideals 542 ;naximal!y almost periodic group 145, 338 measure algebra o f a group 137 ;rrv;n’rrial polynomial 275 V i’i xvnki functional 20 'n ix ed identity 49 modular function 141 modular ideals 235, 448, 634 rr.od-y.f? 455 1 ' EU.nach 455 ‘ rro’n omcrphism 500 moroh'gr.i has a coinage 501 colcernel 501 ernel 501 lr. ,ge 501 i in a category 499 :».‘. ili-iji:-ar maps 159
787 bounded 166 multilinear noncommuting polynomial 628 multiplicative linear functionals 304 multiplier 28 multivariable functional calculus 378, 388 N-algebra 640 Nalmark, Mark Aronovil (1909-1078) 11 Nagumo, Mltlo (1905-?) 11,301 natural map k : X —* X m* 22 neighborhood 7 Newburgh theorem 222 nil ideal 481 nil radical 483 nil subalgebra, dense 514 nilpotent element 4 nontrivial, proper 4 topologically 18 nilpotent algebra 481 nilpotent ideal 481 non-synthesis, sets of 666 noncommuting polynomial 627 norm complete algebra 215 39 37 minimal 556 minimal spectral 558 regular algebra 19 supremum or uniform 64, 303 weakly spectral 223 norm topology is minimal 558 norm topology is minimum 558 norm-unital 260 normal element 376 normed bimodule 455 normed left .4-module 455 normed representation o f an algebra 456 normed symmetric algebra o f quotients 452 nuclear C*-algebra 186 nuclear norm 98 nullity o f T 274 numerical radius 261, 264 numerical range 261 objects in a category 499 one-parameter group 595 one-sided identity 477 open mapping theorem for analytic spaces 549 operator 22 approximable 88, 712 compact 88, 289, 532, 711 completely continuous 88 decomposable 296 diagonalizable 275 elementary 453 finite-rank 23, 87, 673 FVedholm 108 Hilbert-Schmidt 109 inessential 108, 712 integral 101 nuclear 98
Index
788 operator projection 276 Riess 108, 295, 704 strictly singular 93 trace class 109 transcendental topologically nilpotent 236 weakly compact 88, 91, 712 operator norm 22 orthonormal basis 109 pairing 672 normed 672 Parseval's identity 120, 431 partition o f the identity 634 Peirce, Benjamin (1809-1880) 4, 18 72 Peirce, Charles Saunders (1839-1914) 25, 72 permanently quasi-singular 250 perturbation set 480 Pettis lemma 549 Pl-algebra 631 piecewise sm ooth function 127 Ptancberel theorem 431 point derivation 375 point spectrum 286 polydisc 382 polynomial convex set 318, 381 polynomial identity 627 polynomial polyhedron 381 polynomially convex hull 203, 381 polyradius o f a polydisc 382 Pontryagin dnality theorem 432 positive definite function 424 positive element 532 power bounded 271 presocle 672 primary ideal 640 prime algebra 446, 658 ideal 446, 483 radical 485 structure space 621 primitive ideal 443 principal component o f a topological group
200 product o f sets in a linear space 5 product, categorical 38 projection operator along a linear subspace 276 onto a linear subspace 276 projective Banach algebra tensor product 171, 61 projective system o f algebras 44 normed 44 projective tensor product 99, 166, 340, 631 proper ideal 235 proper vector 274 property 0 296 peeudo-Banach algebra 354 quasi-central algebra 638 quasi-group of A 193 quasi-inverse (left, right two-sided quasi-multiplication 193
193
quasi-nilpotent 18 quasi-product 193 quasi-regular ideal 475 quasi-singular 193 quotient object 501 quotient of C 446 R&ikov, Dmitril Abramovii (1905-?) 11, 189, 417 radical — Jacobson radical 474 radical algebras, examples 613 radical subcategory of a category 502 radical weight function 372 Radon-Nikodym property 85 rank of T 274 rank of a left ideal 690 Ransford’s three circles theorem 227 rational function 198 recalcitrant systems 602 reduction of a representation 441 regular homomorphism 26 regular module action 591 regular unitisation o f algebra norm 20 relative consistency 578 relative identity, left, right, two-sided 235 relative inverse 205 relatively regular algebra 205 element 205 renorming an algebra 15 representation o f an algebra 224, 440, 456 continuous 456 cyclic 440 equivalent 440 extended left regular 25 extended right regular 25 faithful 440 internal models of cyclic 442 irreducible 225, 440 left regular 25 right regular 25 strongly continuous 456 topologically completely irreducible 460 topologically cyclic 456 topologically equivalent 456 topologically irreducible 456 trivial 440 representation o f a group 132 bounded 132 continuous 151 finite-dimensional 338 continuous, bounded 338 normed 132 residual spectrum 286 resolvent function 217 resolvent set o f a 196, 217 restriction of a function 2 representation 441 homomorphism T 441 reverse of an algebra 7 reverse of a function on a group 135 Riemann-Lebesgue Lemma 122 Riesz Lemma 89
Index Riesz representation theorem 137 Riesz, FYigyes (1880-1956) 90 right X~module 455 root of a noncommuting polynomial 627 Rung'* pproximation theorem 347 Schaud^r’s theorem 89, 289 Schmid , Erhard (1876-1959) 128, 192 Scaur’ s ‘emma 443, 463 section 648, 626 semi-norm 10 jebra 10 n ’'-trivial algebra 10 otient 14 equivalent 14 semidir -ct product o f algebras 34 8«migr<. p algebra 155 semipri ne algebra 481, 613, 658 sernipn Tie ideal 481 semisimple (Jacobson) 225, 304, 474, 602 separating function 563 separating spaces o f
789 standard polynomial 627 Stokes’ Theorem 395 Stone, Marshall Harvey (1903-1989) 14 Stone-Cech com pact ification 335 Stone-WeierstraM theorem 66 strict topology 525 strictly dense 464, 674 strong operator topology on B (X ty ) 22 strong radical 490 strong spectral radius 207 strong spectrum 207 strong structure space 621 strongly ^-measurable 156 strongly exposed point 85 strongly harmonic algebra 646 strongly semisimple 490, 567 structure space 304, 621 subaddltlve 10 subalgebra 3 hereditary 6 normal 6 unitaJ 3 subcategory o f a category 502 •-algebra 109 aubdirect product 41, 496, 625 subharmonic functions 230 submultiplicative 10 subobject 500 subrepresentation 441 sufficient family o f n-dimensional representations 630 suitable contour 344 sum o f a family o f ideals 40 sum of sets in a linear space 5 support o f a function 2 Sylvester, James Joseph (1814-1897) 72 symmetric element 265 symmetric subset of a group 134 T-lnvariant subspace 440 tame linear functional 145 Tauberian algebra 640 Tauberian theorem 435 Taylor functional calculus 416 Taylor spectrum 416 tensor map 159 tensor norm 159 admissible 179 dual 180 injective 99, 175 nuclear 180 projective 99, 167 reasonable 180, 524 uniform 180 weak 175 tensor products o f matrices 73 topological category of Banach algebras 32, 517 Banach spaces 32, 81 normed aigebras 31 normed linear spaces 32 topological divisor o f sero 250 joint, two-sided 250 left, right 250
790 topological equivalence o f representations 456 topological group 133 topological N-algebra 640 topological quasi-divisor o f tero 250 joint, two-sided 259 left, right 250 topologically finite inner automorphism group 145 topologically nil semi-normed algebra 477 topologically nilpotent 213 topologically nilpotent algebra 515 topologically Tauberian algebra 640 topology defined by a semi-norm 14 o f coordinatewiae convergence 300 o f uniform convergence on compact sets 96 X on X * 22 X * on X 22 separation properties 8 weak 22 weak* 22 torsion-free group 201 total space 626 trace identity 70 trace o f a projective tensor product 170 trace o o B n ( X ) (X has approximation property) 100 trace o o a matrix algebra 70 normalised 70 translate o f a function on a group 135 translation invariance 112 trigonometric polynomials 113 two-sided continuity ideal 572 ultrafilter 45 countably incomplete 45 fixed, free 45 ultrapower 46 ultraprime algebra 450 ultraprime constant 450 ultraprime ideal 450 ultraproduct 46 uniform algebras 69, 313 uniformly topologically nil algebra 515 unimodular 141, 529 unique Banach algebra topology 553 unique normed algebra topology 553 unitisation o f an algebra 19 norm 20 unordered sum 39 vanish at infinity 65 Varopoulos algebra 173 von Neumann mean 338 weak countable compactness 83 weak N-algehra 640 weak operator topology on B { X ,y ) 22 weak relative inverse 205 weak sequential completeness 83 weak topological N-algebra 640 weakly 5 - measurable 156
Index weakly almost periodic 60, 80, 472 weakly analytic Banach space valued function 217 weakly invertible 207, 492 weakly quasi-invertible 492 weakly quasi-regular 492 weakly relatively regular element 205 weakly sequentially complete 83 Wedderburn splitting, algebraic 665 Wedderburn splitting, topological 665 Wedderburn theorem 658, 662 Wedderburn, Joseph Henry Maclagan (1882-1948) 4, 72, 505, 658 Wedderburn-Artin structure theorems 72 weight function 154, 436 weight sequence 373 basic 373 convex 373 ordinary 373 star-shaped 373 unicelullar 372 weighted shift 372 Weil's formula 146 Wendel’s theorem 148, 149 Weyl spectrum 295 Wiener, Norbert (1894-1964) 11, 128 without order 25 Zermelo-FVankel axioms 578 sero object in a category 499 sero subobject 501
SYMBOL INDEX
A/X (quotient algebra) a t ( a ) , QR(a) (ascent) a (T ) (ascent) 289
M iscella n eou s ( . . . / . . . / . . . ) (choices in text) A, 8, C (algebras) X, y, Z (linear spaces) 0 (empty set) 2 A \ B - [a € A : a £ B ) 2 S + T = { b + t : a € S, t € T } 5 0 + T = {a + 6 : 6 € T } 5 S T = span{ai : a € S, t € T } 4 S.A (idea'iser) 29 (v,tu) (pairing) 672 S ° (interior o f S ) 2 S~ (cio 3ure o f 5 ) 2 d $ (b^vrdary o f 5 ) 2 © c .f/y '.c (direct sum) 39
3
5t 8 688
B ba«S (balanced convex hull) 4 bG (Bohr compactification) 338 B (G ) (Fourier-Stieltjes) 424 £?((*) (Fourier-Stieltjes) 424 B (Q ) (bounded functions) 65 B A (Banach algebra category) 500 II/IIbSE (Bochner-Schoenberg-Eberleln norm) 317 A b (B rown- McCoy radical) 485 B* (dual from pairing) 673 )(V ) (operators with a dual from a pairing) 672 B (X ), B (*,> > ) (operators) 22 B a {X ), B a ( X , y ) (approximable) 88 B f ( X , y ) (finite rank) 23, 87 Br ( X ), Br ( X , y ) (integral) 101 BK ( X ) t BK { X , y ) (com pact) 88 BN ( X ), BN ( X , y ) (nuclear) 98 B c c { X ) t B c c ( X , y ) (completely contin uous) 88 B s s ( X ) , B s s ( X , y ) (strictly singular) 93 B w k (X ), B w k (X , y ) (weakly compact) 88 0Q (Stone-Cech) 335
®a'€.A *>',C (Banach algebra direct sum) 39 1 1 (ur.:Ifcii3Mcative identity) 3 5 (1 — — {6 — 06 : 6 € - 4 } A* (ur: N ation o f .A) 19 A\ (cloced unit ball) 14 20 |] ■\h {unitization o f || ■||) »rl '■ ! ' lion o f a ) 20 ?'l (cloci unit ball) 14 A a o •-- 6 — 06 193 a o ('orr.en product) 579 a {Gelfand transform) 303 [a, 6) (closed interval) 3 [a, 6] (Lie oroduct) 241, 580 ]», oj (open interval) 3 a ?S p(T ) (approximate point) 286 [A] (dr.cs of abelian groups) 529 /1(D ) (Cis.; algebra) 68, 330 ; r ; .ytic on interior) 333 /(c 7 ) (7c rier algebra on G) 423 A (G ) f'!?'o5;.rier algebra on G) 423 A ( T ) (Fonrier algebra on T) 126 AC />?7(T) (absolutely continuous functions) 113 {A C , convolution) 115 A P 'G ' (almost periodic) 337 A (category o f algebras) 500 / / . {^.iij.:!:ilator ideal) 25 -*{ HT' ^ !gtbra o f weight K ) 374
C
c h r (^ ) (characteristic polynomial) 273 coS (convex hull) 4 cS p(T ) (continuous spectrum) 286 C = C (T ) (continuous functions) 115 C (« ) (C, convolution product) 115 C B V = C jB V (T ) (continuous bounded variation) 113 C B V (* ) (C B V , convolution) 115 C (ft) (continuous bounded) 65, 333 C o (0 ) (vanishing at infinity) 65 Coo(ft) (com pact support) 65 C J(T ) (continuous derivative) 68 C ° ° (V ) (infinitely dlfferentiahle) 392 Cv = {u u u -1 : u G G } 131 CH (continuum hypothesis) 579 CHom(.A, 6 ) (continuous homomorphisms) 9 C (complex field) 2 C n (n-tuples from C ) 380
791
792
Symbol Index D
G
[D] (claw o f discrete groups) Det(A f) (determinant o f M ) D f (dotDMtD o f f ) 344
D = {A € C ; |A| < 1)
529 71
G / N (quotient group) G<* (G made discrete)
£* (dual group)
2
T>(A) (double centraliser algebra) 26 Tf*{V) (differential 9-forms) 393 ^ b (-A ) (algebra o f bounded double centralizes) 26 6t (derivation on T>(A)) 599 6tJ (Kronecker delta) 3 ^n(a) (descent) 686 6(T ) (descent) 289 6* (inner derivation) 592 A (modular function) 141 A (-4 ) (derivations o f A ) 591 & (A , X ) (derivations: A to X ) 591 & b (A ) (bounded derivations) 591 A b (.A , •*) (bounded module derivations into X ) 591 A / (-4) (inner derivations) 592 A / (>4, X ) (inner derivations) 592 A (V ) (derivations on V ) 393 tp* (algebraic dual o f ^ ) 313 X* (algebraic dual o f X ) 21 T* (algebraic dual o f T ) 21 to v (rank one from pairing) 673 &A (boundary o f A ) 319 E
€
/.( » ) = /(» u _ I ) 135 •/ ( « ) = 135 fo\ A n»2 A •••A a , 393 f\s (restriction to 5 ) 2 f g y f g fo r f , g € A ” 47 (PC) (class o f groups) 529 [FIA]~ (class o f groups) 145 [F1A]~ (class o f groups) 530 f S ( X ) (free semigroup) 509 F (field: R or C ) 3 T+ (analytic near £>p(a)) 344, £7 (germs o f above) 349, 38'< T = C||ij| 300, 370 T v (analytic on U) 348 ? {X ) = { /( * ) :/ € T, x € X } 2 A r fsocfe) 671
rtl(o)
305
H h (S ) (hull in PA ) 620 h (S ) (hull in T ^ ) 323 hp (S ) (hull in PA ) 620, 446 /in (S ) (hull in n ^ ) 620, 446 * S (-S) (hull in 3 * ) 620, 490 =
/ tt (functional calculus) 349, 388 /J* (functional calculus) 344, 388 [IN] (class of groups) 529 I (ip) (continuity ideal) 572 £l(v>). Za(V’ ) (continuity ideal) 571 A j (idempotents of A ) 4, 667
F (functional calculi for o 199, 342, 349, 353, 388 / • g (convolution) 129 / " (inverse) 2 /(« .) = / ( « - * ) 135 /(« ) = /( « ) ' 135
417
6 (double dual group) 422 G B A , G N A (categories) 500 A c (invertible group o f A ) 193 (Gelfand space) 303 rJi (augmented Gelfand space) To = 419 A r (Gelfand radical) 304
I
e < / (order for idempotenta) 668 c* for a 6 A (exponential) 202 eSp(a) (exponential spectrum) 203 E?3 (matrix units) 70 Equiv(5, T ) (equivalences) 440 E x t(C ,A ) (extensions) 34
/(a )
134 135
A)
■4<“ > (union of .*<">) 384 ||a|r(a) = inf{||a’'||, / " : n 6 N }
16 16, 210
J J (James space) 79 J ( o o ) (vanishing near oo) 640 J ( M ) (vanishing near M ) 640 , o o ) (vanishing near A i 640 J { H ) (vanishing near H ) 640 J (W ,o o ) (vanishing near / / , oo) 640 A j (Jacobson radical) 225, 474-481 A j qz (joint quasi-divisors o f aero) 250 A j z (joint divisors o f zero) 250 K * (B ) (kernel of B C PA ) 620, 323 ker(
Symbol Index
o
L li m a
Aa
itm a
793
(in d u c t iv e lim it )
(projective limit)
44 42
Z,l (M-j.) (semigroup algebra) 531 ^ ( G ) < * 1-group algebra) 142 L x( G r L °°(G ) 418 Ll 113 ' uare summabie) 431 i ? =- L T ) 113 ^ j / (T) 113 L ^ f * ) ( i ao, convolution) 115 Ln (le'*, tgular representation) 25 L * (ext.-. Jed left regular representation) 25 i e (reorientation on w4e) 669 jA .lt ^ ',uction o f L by £ ) 441 , . / '! - / annihilator) 25 Al
0 ( 0 , X ) (analytic functions) u> * r (convolution in X * ) u h 6 i l (G ) ' for h 6 £ ° ° (G ) uja , € A * , o 6 A t u> € A* « / . /w € A m, w 6 .A*, / €
296 136 418 47 47
P p c(K ) (polynomially convex hull) 203 p(a) (functional calculus) 199, 342 p € C ( X l , X 2 , . . . , X n \Z ) (tensor map) 159 p erS pA (a ) (permanent spectrum) 253 p S p (T ) (point spectrum) 286 P a (prime ideals) 621, 446 P (K ) (limits o f polynomials) 384 P(D) (limits o f polynomials) 331 P(T) (limits o f polynomials) 331 P(O) (limits of polynomials) 333, 384 •jr* ( X ) (first cohomotopy group) 4 '1 irm,„ :.A (m) -> .A (n) 385 < 385 n.A (primitive ideals) 621, 4
Q aq (quasi-inverse of a) 193 A 9 q (quasi-group) 193 A g z (quasi-divisors o f zero) 25G •Ag = {P 6 A : W ( p ) C R + } 269 <2(
im il(7j (r uiiity o f T ) 274 N ( n a u '% . numbers) 2 N ° ~ f , 2 , 3 , . . .} 2 Nf ed algebra category) 500 A (x} ~ y e A f : a (x - y) < e } 14 AC >orhoods o f 5 p (o )) 348, 383 A ^ (no ai elements) 376 A nn (* i sdical) 483 A in) (n -tuples from >1) 382 j|a||A (nor:n from (.A /.M )) 642
rank(T) (rank o f T ) 274 rS p (T ) (residual spectrum) 286 r l r a A (annihilator) 679 R € (restriction to A c ) 669 R (0 ) (limits: rational functions) 333 Ra (right regular representation) 25 i?a (extended right regular representation) 25 II ' ||r (regular unitization of || •||) 20 R (real number field) 2 R + = { / e R ;t > o } 2
ll/lloo =
R+ = {t e R;t > 0}
8 Uf { I / ( “ )I : u
e
||(£, fl)|l ( -.orm in V g (A )) j|u;|l (dual norm) 21
64
26
2
R d (discrete real numbers)
134
794
Symbol Index
A t*g (largest completely regular subalgebra) 328 A Pw {r e v e n t algebra) 7 •Aha (right annihilator) 25 A ft+z (right quasi-divisors of zero) 250 A hz (right divisors o f zero) 250 Pa ( o ) — p (o) (spectral radius) 196 S gp+aS (linear span) 4 span(S) (closed spaa) 534 su p p / (support o f f ) 2 5(VV) (spectrum o f W ) 434 £ n (standard polynomial) 627 S p ^ o ) — S p (o) (spectrum) 196, 383 S p ^ (a ) = Sps {* ) (strong spectrum) 207 Sp (S ) (Taylor spectrum) 416 [SIN] (class o f groups) 529 Pj^(a) = p S(a ) (strong spectral radius) 207 A s (symmetric elements) 265 S A (category o f spectral algebras) 500 S 8 A (category o f spectral semi-oormed , algebras) 500 a u (Minkowski functional) 20 (separating function) 562 X 0 = { x € X :
T
9 .G — G 422 T] 0 T j fc* • 0 Tn (tensor product of maps) 176 A*1 ® (V2 ® ® X n (algebraic tensor product) 160 X* ® •••& X n (injectiv« tensor prod uct) 175 X^ •••® X n (projective >ensor prod uct) 166 x ® w (rank-one operator) 23 * l ®*2 (elementary tensor) 159
U |Urn) (unimodular groups)
529
V V (G x G ) (Varopoulos algebra) 173 A y = *4^ -+■ i.4.s 268 W W( a) (numerical range) 261 II •\\w (numerical radius) 261 A*w = {w G A * : w (l) = )M| = 1} 261 A v ij (weakly invertible) 207 W (S ) (weak* closed span) 434 X 3 a (maximal modular ideals) 621, 490 Z * = *r«) 380 2(A) = A for A 6 C 198 * , ( Ai . A 3 ........ Xn) = Xj 380 ZFC (Zermelo-FYanlcel Axioms) 578 Z (all integers) 2 ^ 2 (center of .A) 250
/
