Topological Algebras

and q(x)>/ (5)

146

S o m e Types of Topological Algebras

From (3), (4), (5) we conclude t h a t q ~ pl "~ p, which proves (i). To prove (ii), we observe t h a t if s E S 1, p l ( S X ) -- s u p { p ( t s x )

(6)

S 1} < pl(X)

"t E

(since ts C S1). It follows t h a t q(s)

--

s u p { p l ( s y ) " p l ( y ) <~ 1)~< s u p { p l ( y ) " p l ( y ) <~ 1} (using (6))

~<

1.

(7)

In particular, q(e) ~ 1. But q ( e ) - q(e 2) ~ q(e) 2, whence q(e) >>. 1. Therefore q(e) - 1, as desired. Finally, by (3), Pl is faithful when p is faithful and then, by virtue of 3.5.7, q - I/zll is a p - n o r m . 3.5.10.

(i)

COROLLARY

v ( x ) - Vp(X) - i n f { q ( x ) ' q

is a sm. p - s e m i n o r m with q

p}. (ii) If A has unity e then v ( x ) - inf{q(x) s e m i n o r m with q ~ p, q(e) - 1}.

9 q is a s m .

p-

PROOF. (i) Since u(x)<~ q(x) for all q we get u(x) ~< inf q ( x ) -

t (say).

n --+1 (as Suppose t h a t v ( x ) < t, so t h a t u ( x ) / t < 1. Since n+l n --+ cxD we can choose a sufficiently large n t h a t we have

t

<

n+l

<1.

(1)

1

Set y -- ((n + 1 ) / n t ) ~ x. T h e n q ( y ) - ((n + 1 ) / n t ) q ( x ) ,

so t h a t we have

inf q(y) - n + 1 inf -..a(x) - ~+-----:tn _ q nt q nt

n+l

(2)

Now, by 3.4.5,

- n+l nt

n+ 1 v(x) -

n

t

<

n+l n

.1-

n+l

- inf q(y) q

(a)

w 5. B o u n d e d Linear Transformations on p - s e m i n o r m e d L S ' s 147

where in the last step we have used (2). Again, v(y) -- n + l ~v(x)nt

n + l v(x)

~ . ~ n

Since v ( y ) l-i m p ( y n )

t

<

n + l n

n

n+l

= 1

(using (1)).

~- < 1 we get p(yn) < 1 for all n >/ No,

n

whence it folows that the semigroup S = {y" : n = 1 , 2 , . . - } is p-bounded. By 3.5.9, there is a sm. p-seminorm q0 "~ P and qo(y) ~< 1 < ,~+1 contradicting (2) Therefore we must have n ~ v ( x ) - - t , proving (i). (ii) This can be proved in exactly the same way as (i). 3.5.11. PROPOSITION. ( a ) L e t X be a p - s e m i n o r m e d L S and X* a p * - B a n a c h space. Let Xo be a dense subspace and T :X0 --~ X* a continuous linear tranformation. Then T can be uniquely extended to a continuous linear transformation 2 P - X --+ X* with

TI-

ITI.

(b) Suppose that A is a p - s e m i n o r m e d algebra, Ao a dense subalgebra of A and A m a p - B a n a c h algebra. Then every continuous h o m o m o r p h i s m !p:A0 ~ B can be uniquely extended to a continuous h o m o m o r p h i s m 95" A ~ B with [t5[- ]!PlPROOF. x c X,

(a) Let X = (X,p),

X* = (X*,p*). Suppose that

Xn --+ x (xn E Xo). Then p * ( T x , ~ - T x m ) <~ [T[p(x,~- Xm)~1p

~

0 7

so by completeness of Y, Tx,~ --~ x*. Define T x - x*. By using the uniqueness of limit property of convergent sequences in X* (which is Hausdorff)it is easy to verify that 5b is well-defined (i.e. independent of the particular choice of the sequence xn ---+ x) and linear. Further we have p* (Tx) - lim p* ( T x , ) <. ITlp(x.) ~* p <<.[TIp(x) ~* p, so that ^15b] ~< T I. On the other hand, T being an extension of T, IT[ < IT[. Thus I T [ - IT]. The continuity if T is an immediate

148

S o m e Types of Topological Algebras

consequence of its boundedness. Finally, the uniqueness of 55 is also clear. (b) Extend ~ (as a linear transformation) to ~b on A as in (a); then 95 is continuous linear. We have to show t h a t ~5 is a h o m o m o r p h i s m . If x, y c A, (xn), (y,) E A0, xn ~ x,y,~ ~ y then 9 b ( x y ) - l i m ~ ( x . y , ~ ) - l i m ~ ( x . ) ! p ( y . ) 95(x)gb(y), proving 9b is a h o m o m o r p h i s m and completing the proof.

w6.

Topological

Algebras

with

Inverses

3 . 6 . 1 . DEFINITION. Let A be an algebra (over K) and a E A. Denote by Ia the map x ~ - + a o x ( x E A ) . Similarly r a is the map x ~ xoa. 3 . 6 . 2 . LEMMA. Let A be a W T A . T h e n la,r ao are continuous. I f a is q. invertible then l a0 , r a0 are h o m e o m o r p h i s m s of A.

PROOF. Since la(x ) -- a § x § lax (la " x ~-~ ax) the continuity of l~ follows from the continuity of addition and the continuity of l~. The continuity of r~ follows similarly. If a is q. invertible with a ~ as q.i, so t h a t a ~oa = o - a o a ~, then la,l a - l a ,o oa - - l ~ __ 1A __ lala,. o o 0 Therefore, l~ is invertible with its inverse (/a) -1 - I a, continuous. It follows t h a t I a is a homeomorphism. Similarly, since (ra) -1 -r a,, r a~ is a homeomorphism.

3 . 6 . 3 . LEMMA. I f A is a unital W T A and a E A is invertible then la, ra are linear h o m e o m o r p h i s m s . PROOF. Here we have la I - l a - l , r a 1 - ra-1 and hence the result. 3 . 6 . 4 . DEFINITION. A TA A is called a C algebra or a continuous algebra if the map x ~-~ x ~ of Gq --~ Gq is continuous, where Gq denotes the group of q. invertible elements of A. 3 . 6 . 5 . PROPOSITION. (a) A TA A is a C algebra iff Gq is a

TG.

149

w6. Topological Algebras with Inverses

(b) A unital TA A is a C algebra iff its group Gi of invertible elements is a TG. (c) In a unital C algebra the groups Gq,Gi are TG's and the map 7 -1 " X E G q - - ~ e + x E G i

is a t. isomorphism. PROOF. (a) Since A is a TA the map (x, y ) ~ xy is always continuous. It follows t h a t Gq is a TG iff the map x ~ x ~ is continuous, i.e. iff A is a C algebra. (b) By 1.1.20, y - e+x E Gi iff x E Gq and y-1 _ ( e + x ) - i = e + x ~. It follows t h a t Gi is a T G iff y ~ y-1 is c o n t i n u o u s i f f x ~ x ~ is continuous iff Gq is a TG. (c) The first half of the s t a t e m e n t follows from (a),(b). Also, T- l ' x ~ e + x is an isomorphism of Gq onto Gi Again, r , r -1 being translations in A, are continuous. Hence the second half of the s t a t e m e n t . 9

3.6.6. LEMMA. Every quarter-normed algebra (A,p) with an a.sm. p is a C algebra. In particular, any p - s e m i n o r m e d algebra is a C algebra. PROOF. The first s t a t e m e n t is an immediate consequence of the inequality (.) of 3.3.22. The second s t a t e m e n t folllows from the first since, by 3.4.3, the p - s e m i n o r m p of a p - s e m i n o r m e d algebra is always a.sm.. 3.6.7. DEFINITION. A TA A is called a Q algebra or algebra with q. inverses if there is an open neighbourhood U(0) of 0 consisting of q. invertible elements. A unital TA A is called an I algebra or algebra with inverses if there is an open neighbourhood U(e) of e consisting of invertible elements. 3.6.8. LEMMA (Michael). Let A be a TA. Then: (i) If S is a balanced subset of A comprising q. invertible elements then r(s)<<. 1 for every s in S, where r(s) denotes the spectral radius of s. (ii) Write G = {x E A : r(x) <~ 1}. Then A is a Q algebra iff int G = G~ -~ O.

150

Some Types of Topological Algebras

PROOF. (i) Suppose that r(s) > 1 for some s E S. Then there i s a )~e a(s) with I)~] > 1. By 1.7.8, s 0 - - ) ~ - l s is not q. invertible. But since ])~-11 < 1,s E S and S is balanced we must have so E S, so that so is q. invertible. This contradiction proves (i). (ii) First assume that A is a q algebra with U(0) a nucleus comprising q. invertible elements. Choose a balanced open nucleus V __c U(0). By(i), Y _C G so that G ~ ~ 0. Conversely, 1 assume that in A we have G o r 0. Then U - ~G ~ ~= 0 and open. If x E U then 2x E G, so that by 1.8.11(ii), r(x) ~< ~1 < 1. It follows that - 1 ~ a~(x) whence, by 1.7.8, x is q. invertible. Thus every x C U is q. invertible and A is a Q algebra. 3.6.9. LEMMA. If A is a Q (respy. I) algebra then each q. invertible (respy. invertible) element x has an open neighbourhood. U~ - l;(U(O)) ( respy. U(x) - l,(U(e)) comprising q. invertible (respy. invertible) elements. Hence the set Gq

) of q. i

rtibl (r py i , tibl )

of

Q

spy. I) algebra A is open. PROOF. Since Gq (respy. Gi) is a group we have U~ x o U ( 0 ) C Gi (respy. U ( x ) - xU(e)C_ G , ) F u r t h e r , U~ (respy. U(x)) is open since l~ (respy. l, ) i s , by 3.6.2 (respy.3.6.3) a homeomorphism.

For a TA (respy. unital TA) A to be a Q (respy. I) algebra it is necessary and su~cient that Gq (respy. Gi) is open. In particular, a TA which is a division algebra is an I algebra iff it is Hausdorff. 3.6.10.

PROPOSITION.

PROOF. The "necessity" follows from 3.6.9 and the "sufficiency" is immediate from the definition of a Q (respy. I ) algebra (since 0 C Gq, e E G i ). Since in a division algebra Gi - A\{O} the assertion concerning TA's which are division algebras is clear. 3 . 6 . 1 1 . COROLLARY. Let A be a TA under two topologies 71,r2. If r2 is finer than rl and (A, 71) is a Q algebra (respy. I

w6. Topological Algebras with Inverses

151

algebra) then (A, 72) is also a Q algebra (respy. I algebra) P r o o f . Clear ( Gq r l - o p e n ==~ Gq r2-open). 3 . 6 . 1 2 . LEMMA. Let A be a unital TA with unity e. If there is an open neighbourhood U(O) of 0 consisting of q. invertible

consisting of invertible elements. Conversely, if V(e) is an open n ighbo rhood of con i ting ol i, v tiU th n V(O)= an open n ighbo rhood of 0 con i ting of q. inv rtiU elements. PROOF. By 1.1.20, x E A is q. invertible iff x + e is invertible. Moreover, the translations x ~ x + e, y ~ y - e of A are homeomorphisms. The statements of the lemma are now clear. 3 . 6 . 1 3 . COROLLARY. A unital algebra A is an I algebra iff

it is a Q algebra. 3.6.14.

LEMMA. A Q algebra A is commutative iff Gq is

commutative. commutative.

Also, an I algebra A is commutative iff Gi is

PROOF. Suffices to prove the "if' parts. Suppose t h a t Gq is commutative and x, y E A. Since z_ 0 and Gq is open n ~ n ~_ ~ x there are integers nl, n2 such t h a t ~1' ~y E Ca. Then ~ ~-~ l'b 2 t,~ 1 whence x ~ y, A is commutative. Again, if Gi is commutative so is Gq since Gq isomorphic to Gi (by 1.1.20). This completes the proof. 3.6.15. Remark. The above lemma can fail if the algebra is not a Q algebra. For example, consider the algebra P * = P * ( X , Y ) of all polynomials P = P ( X , Y ) over K in two non-commuting variables X,Y. This algebra can be normed by: IIPII = the m a x i m u m of the absolute values of the coefficients of the monomials of P. It is easy to check this gives a norm t h a t is sm.: IIPQII < IIPIIIIQII. It is clear t h a t the group of invertible elements G{ = K\{0} which is commutative, but P* is not commutative since X Y ~ Y X . Note here that Gi is not open in P*

t

i.e., U ( e ) - {x + e ' x

E U(O)}.

152

Some Types of Topological Algebras

so t h a t P* is not a Q algebra. 3 . 6 . 1 6 . THEOREM ( A r e n s t - B a n a c h ) . A (F) algebra A (A, ]. ]) is a C algebra iff its group Gq i8 a G 5 ~ set of A. PROOF (cf.[31,p.4]). First assume t h a t A is a C algebra, so t h a t the m a p x ~ x ~ is continuous on Gq. (,) For n - 1 , 2 , - . . define W~ - {x c Gq 93 an ~ - ~(x,n) > 0 such t h a t if y , z E 1 Gq, i y - x I < r / , I z - x I < r / then l y ' - z ' I < ~}, where the bar in Gq d e n o t e s closure. It is easy to see, using continuity of the m a p ( , ) , t h a t W,~ is open in Gq. Also it is clear t h a t Wn D Gq, so that WN,~W,~ 2 Gq. Thus, W is a G6 in G and Gq as a

close subset of the metric space A is a G6 of A. It follows t h a t W is a G5 of A. We shall complete the proof of the "only if" p a r t by showing t h a t Gq - W. S u p p o s e t h a t x c W C_ Gq,X n E Gq and x,~ ~ x. T h e n I I x! ~ - X mI l ~ 0 as n , m ~ oo, so t h a t x,~ ~ yEA (since A is c o m p l e t e ) . It follows t h a t x o y - (lim x , ) o (lim x~) - lim(x,~ o x,~) - 0. Similarly y o x - 0. So y - x ~,x E Gq, proves W - Gq. It r e m a i n s to prove the 'if' p a r t and assume therefore t h a t Gq is a G6 of A. Since A is a complete metric space, Gq is topologically complete and consequently there is in Gq an equivalent metric u n d e r which it is complete. By applying p r o p o s i t i o n 2.1.8 to Gq (with this complete metric) we conclude t h a t inversion is c o n t i n u o u s in Gq, c o m p l e t i n g the proof. 3 . 6 . 1 7 . COROLLARY. A unital (F) algebra A is a C algebra iff the group Gi of its invertible elements is a G5 of A. PROOF. It suffices to observe t h a t the m a p x ~ h o m e o m o r p h i s m of A which carries Gq onto Gi. 3.6.18.

e + x is a

COROLLARY. An (F) algebra which is a Q algebra

is also a C algebra.

t Arens proved (essentially) only the "if" part of the theorem and that under the additional assumption that Gq is seperable. tt i.e. Gq is the intersection of a countable family of open sets of A.

w6. Topological Algebras with Inverses

153

PROOF. Since an open set is a G~ this follows from 3.6.16. 3 . 6 . 1 9 . R e m a r k . There exist Q algebras which are not C algebras. For instance, the algebra 9Y~ of germs of meromorphic functions of one complex variable in the neighbourhoods of 0 in C is a commutative division algebra. This can be topologized so as to make it into a complete Hausdorff locally convex TA. It is known that 9.R is not a C algebra (see [29, p.136]). On the other hand since it is a Hausdorff division algebra, by 3.6.10 it is a Q algebra. There are also C algebras which are not Q algebra. For example, if B1 is the closed unit ball in K denote by P the algebra of the restriction of polynomials in one variable to B1 with coefficients from K. P is a normed algebra under the sup norm. By 3.6.6, P is a C algebra. On the other hand since the invertible elements in P are just the non-zero constants, P is clearly not an I (and hence also not a Q ) algebra. 3.6.20. DEFINITION. A Q (respy. I ) algebra is called a CQ (respy. CI) algebra or algebra with continuous q. inverses (respy. continuous inverses) if the map x ~ x' (respy. x ~ x -1 ) from U(0) (respy. U(0) ) into A is continuous, where U(0) (respy. U(e)) as in 3.6.7. 3.6.21. LEMMA. An algebra A is C Q (respy. C I ) a C algebra and a Q (respy. I) algebra.

iff it is

PROOF. The "if" part follows from 3.6.5((a),(b)) For the "only if" part, in view of 3.6.13, it is enough to prove t h a t a C Q algebra is a C algebra. Let A be a C Q algebra and U(0) as in 3.6.7. If y c U(0) then x o y e U~ (as defined in 3.6.9) and (xoy)' = y'o x ~. The m a p x o y ~-~ ( x o y ) ~ is continuous since it is the composite O of the continuous maps x o y ~-~ y - Iz,(x o y), y ~-~ y~, yl ~ yl o x ~. This proves that A is a C algebra. 3 . 6 . 2 2 . LEMMA. Every C I algebra is a C Q algebra. Conversely, every unital C Q algebra is a C I algebra. PROOF. In view of 3.6.21, 3.6.13 it is enough to prove t h a t the m a p x ~ x ' ( x c U(0)) is continuous iff the map ( e + x )

154

Some Types of Topological Algebras

(e -Jr- X) -1 (X E U(0)) is continuous. But this is immediate since (e+

X) - 1 -- e +

X I.

3.6.23. PROPOSITION. (a) A sm. (F) algebra is a CQ, alge-

bra. (b) Every p-Banach algebra is a CQ p-Banach algebra a C I algebra.

algebra and a unital

PROOF. (a) By virtue of 3.3.18, a sm. (F) algebra is a Q algebra and hence by 3.6.18, 3.6.21 it is a CQ, algebra. (b) By the remark in 3.4.4 we may assume that the norm of the p - B a n a c h is sm. and the first half of the required result follows from (a). The second half follows from the first in view of 3.6.13, 3.6.21. 3 . 6 . 2 4 . LEMMA. Let B be a q. inverse-closedt subalgebra (respy. inverse-closed subunitaltt algebra) A. If A is a Q (respy. I) algebra then so is B; if A is CQ (respy. CI) then so is B. PROOF. If Gq(B),Gq(A) denote the groups of q. invertible elements of B, A respy, then we have clearly the relation Gq(B) B 1"7Gq(A) Hence the statements regarding Q and CQ properties of B. The I and C I properties of B follow similarly from the corresponding relation Gi(B) - B ~ G i ( A ) . 3.6.25. PROPOSITION. Let A - (A,p) be a sm. (F) algebra which is a Q algebra. If B is a closed subalgebra of A then B is also a Q algebra. PROOF. Since A is a Q algebra we can choose an open neighbourhood U(0) of 0 in A such t h a t x e U ( 0 ) = ~ x is q. invertible and p ( x ) < 1. Suppose that x e U 0 - B NU(0). By oo

3.3.18, x ~ exists and is given by x I -

~ - ~ ( - 1 ) n x ~. Since x n E B n=l

and B is a closed subspace, x~E B, so that x~E Uo. Thus, Uo is an open

t i.e. if z E B has q.i. x' (respy.inverse x -1) in A then x' (respy. x - 1 ) C B . tt For definition see 1.1.6.

w6. Topological Algebras with Inverses

155

neighbourhood of 0 in B, comprising q. invertible elements and B is a Q algebra. 3 . 6 . 2 6 . L EMMA 9 Let A, A* be T A ' s a n d p ' A - - - . A * a continuous open epimorphism. Then A* is a Q (respy. CQ) algebra provided A is a Q (respy. CQ) algebra. PROOF. If A is a Q algebra then G q is open in A and consequently G* - ~(Gq) is an open neighbourhood of 0 in A*. By 1.1.24, every element y C G* is q. invertible with y~ E G*. Hence A* is a Q algebra. Next suppose that A is a CQ algebra. Then as above G* - ~(Gq) is an neighbourhood of 0 in A*. For Y0 E G*, choose an open neighbourhood V ~ of y~ with V I _c G*. Set U ' - ~ - 1 ( V - 1 ) ~ G q . Then U' is an open set, so t h a t U - {x e A" x I E U ~} is an open neighbourhood of x0, where p(x0) - y0. It follows t h a t V - ~ ( U ) is an open neighbourhood of y such t h a t q#(Y) c_ Y', where q# is the map y ~ y' (y e Gq(A*)). This proves the continuity of q# at Y0 an arbitary point of G*, whence A* is a CQ algebra. 3 . 6 . 2 7 . COROLLARY. In a Q (respy. CQ ) algebra A, if I is a hi-ideal the quotient algebra A # - A / I is a Q (respy. CQ) algebra.

PROOF. This follows from 3.6.26 since the canonical m a p r 9A ~ A # is an open continuous epimorphism. 3.6.28. PROPOSITION. Let A be a TA and A1 its unitization. Then AI is a Q (respy. CQ) algebra iff A is a Q

(respy. CQ) algebra. PROOF. First assume that A is a Q algebra and U an open neighbourhood of 0 comprising q. invertible elements. Choose a balanced open neighbourhood V of 0 with V § V _c U. Then, for 0 ~ [#l ~< 1, we have (I§

c_V§

C_V§

c_ U.

1 Choose ~ such t h a t 0 ~< I#1 < ~. 1 Then I1 § #1 > 1 - I#1 > 2" 1 (1-+-#)-1 Writing A - ~ we get [A[ < 1, so that for x E V, Ax E V.

Some Types of Topological Algebras

156

Since y-

x(1 + # ) - 1 _ 2)~X E 2)~V _C U

it follows t h a t y is q. invertible. Therefore, e l + y is invertible and hence also (1 § # ) ( e 1 Jr- y) z (1 + #)e 1 -[- x.

is invertible. It follows t h a t 1

V1 - {(1 + ~)e I + x" 0 ~ [#[ < ~ , x E V} is an open neighbourhood of el comprising invertible elements, whence A1 is an I algebra and so also a Q algebra. Conversely, assume that A1 is an I algebra; then it is a Q algebra and so, by 3.6.10, the g r o u p Gq1 of q. invertible elements of A1 is open. If Gq denotes the group of q. invertible elements it is clear (see 1.1.21) t h a t Gq - A ~ G q1, so that Gq is open in A. By 3.6.10, A is a Q algebra. It remains to prove that A1 i s a CQ algebraiff A i s a CQ algebra. But for this, by 3.6.21 and the result already proved it is enough to show that A1 is a C algebra iff A is a C algebra. Now by virtue of 1.1.20, Yl -- Ae + x (A E K , x E A) in A1 is invertible (i.e. yl E G~) it~ )~ ~ 0 and A - i x E Gq, and then we have the relation Yl-1 -- A- 1 (e 1 + ()~- 1x)'), w h e r e d a s h d e n o t e s q.i.. Again, if x C Gq then e l + x E G 1 and ( e l + x ) -1 - e l + x ~. It follows from these relations that the map yl ~-~ y~-I in G 1 is continuous, which completes the proof of the proposition. 3.6.29. PROPOSITION. The complexification A of a commu-

tative Q algebra A is a Q algebra. PROOF. Let U be a nucleus ( = a neighbourhood of 0 ) c o m prising q. invertible elements. Then by 2.1.17(iii)(applied twice) and 2.2.14(vi) we can choose nuclei V,W such that

V+V+V+VC_U,

W 2C_V, W c _ v .

If x , y in W then we have u -- 2x + x2 + y 2 E W + W + W 2 + W 2 CC__V --[-V + V --F-V C_ U

157

w6. Topological Algebras with Inverses

so that u is q. invertible in since (x + iy) o ( ~ - iy) - u, It follows that W - W + i W invertible elements, so that A

A and hence also in A. But then N

x + iy is also q._invertible in A.

is a nucleus of A, comprising q. is a Q algebra.

3 . 6 . 3 0 . COROLLARY. The complexification A of a c o m m u tative real I algebra A is an I algebra. PROOF. Since, by 3.6.13, a unital algebra is an I algebra iff it is a Q algebra, the corollary follows from the proposition. 3.6.31. PROPOSITION. The complexification ~i of a c o m m u tative real C I algebra A is a C I algebra. PROOF. In view of 3.6.21, 3.6.30 it is enough to prove t h a t the m a p (.) z ~ z -1 of (~i onto itself is continuous where (~i is the group of invertible elements of A. Suppose that za, z E Gi, za --~ z,z~ - x~+iy~,z - x+iy, where x a , y a , x , y E A. By 1.6.7, z~ - x~ - i y ~ and - 2 - x - iy lie in Gi and hence the products as -- z~z~ -- x 2~ + y 2 a -- z - 2 - x 2 + y2 also lie in Gi. But as, a C A, whence by 1.6.10, an, a E Gi. It follows that N

N

,v

z -~ l __ -~ a ( X 2 +

y2)-I

__

-2 a a -~ l __~ -~ a - 1

_

~-(x 2 _{_ y 2 ) - i

_ z-1

proving the continuity of the map (.). 3.6.32. COROLLARY. The complexification A of a real comalgebra A is a C Q algebra.

mutative CQ

PROOF. By 3.6.28, the unitization A1 of A is a C Q , and hence by 3.6.22, a C I algebra. By 3.6.31, A1 - (A)I is a C I algebra and hence C Q . So, by 3.6.28, A is C Q . N

,v

3 . 6 . 3 3 . WILLIAMSON'S ALGEBRA. The field C ( X ) of rational functions over C can be embedded in the algebra M of example (V) of 3.3.14, by identifying a rational function P ( X ) / Q ( X ) ( P , Q polynomials over C) with the almost everywhere defined function P ( t ) / Q ( t ) in M. The topology thereby inherited from M makes C ( X ) into a Hausdorff topological field (more precisely, a commuttive Hausdorff division algebra in which inversion is continuous). In particular, C ( X ) is a C I algebra as also a p r e - ( F )

158

Some Types of Topological Algebras

algebra. This method of topologizing C(X) is due to Williamson [14' ,p.731]; we call C(X) with this topology as Williamson's algebra and denote it by ~ .

w 7.

Topological

Zero Divisors

3.7.1. DEFINITION. Let A be an algebra. Write GqI (respy. Gq) - the set of 1.q. invertible (respy. r.q. invertible elements) of A. Then Gq - Gqz ~Gqr. Write

S~ - A\Glq, Sq - A\Grq, Sq - S~ U S~. l r The elements of Sq, Sq, Sq are called respectively l.q. singular,

r.q.singular, and q. singular. When A is unital, we have also G~ (respy. G~) - the set of 1. invertible (respy. r. invertible) elements of A. Then Gi - G~ ('1 G'~. Now write

S' -- A\G~,

S r-

A\G~,S b' - S I N St,

S -

SIU

S r.

The elements of S Z, S r S bi, S are called respectively I. singular, r. singular, hi-singular, and singular. 3.7.2. LEMMA. Let A be any algebra. Then x/~ C_ Gq. If A is unital then x/~ c_ S bi. PROOF. The first inclusion follows from 1.2.24 ((a), (b)). Let now A be unital. Then by 1.2.2.4, x / ~ is a bi-ideal with e ~ v/-A. So, no element of x / ~ is 1. or r. invertible, whence __ s 3.7.3. DEFINITION. (Arens). Let A be a TA. An element x in A is called a left (respy. right) topological zero divisor t , or a 1.t.z.d. (respy. r.t.z.d.) if there is a closed set F such t h a t t The notion t.z.d, under the name generalized zero divisor was first introduced by Shilov for commutative Banach algebras [ 7', p.208]; his definition is essentially based on the condition (**) of 3.7.5.

w 7. Topological Zero Divisors

0 ~ F and0ExFrespy.

159

Fx

where bar denotes closure. An element which is both a 1.t.z.d. and a r.t.z.d, is called a hi-topological zero divisor or a bi-t.z.d.. Obviously, when A is c o m m u t a t i v e every 1.t.z.d. or r.t.z.d, is a bi-t.z.d. The same is true (as can be easily seen) when A is anticommutative. We denote by 3 zt, 3 rt, 3 bit respectively the sets of 1.t.z.d's, r.t.z.d.'s and bi-t.z.d.'s of A. 3.7.4. R e m a r k . In any TA whose topology is not indiscrete, 0 is a b i - t . z . d . For, if x E A \ 0 , ~ then 0 ~ ~ - F (say). But OcOF-FO-O. 3.7.5. LEMMA. Let A be a TA. For an element x E A to be a l.t.z.d. (respy. r.t.z.d.) it is necessary and s u ~ c i e n t that

(,)

3 a net (xa) in A w i t h xa 74 0 but xxa

( respy, x~x) -~ O. If A is first countable then (.) can be replaced by"

3 a sequence (xn) in A with xn 74 O, xxn (respy. x,~x) ~ O.

(**)

PROOF. Suppose there is a closed set F such t h a t 0 F and 0 E x F (respy. F x ) . Choose xa E F such t h a t xxa (respy. x~x) -~ 0, then the net (xa) clearly satisfies the condition (.). Conversely, suppose (.) holds. Since xa ~ 0 there is a subnet (xa,) such t h a t none of its subnets converge to 0. This means t h a t if F = {x~,} then F is a c l o s e d such t h a t 0 ~ F, 0 c x F (respy. 0 e F x ) , whence x is a l.t.z.d. (respy. r.t.z.d.). This completes the proof of the first s t a t e m e n t in the lemma. The second s t a t e m e n t is clear. 3.7.6. PROPOSITION. (a) If x E A is a l.t.z.d. (respy. r.t.z.d.) and y C A then yx (respy. xy) is a l.t.z.d. (respy. r.t.z.d.). In particular, if x is a bi-t.z.d, and x ~ y then xy is a bi-t.z.d. t For an element a E A we write ~ for {a}.

Some Types of Topological Algebras

160

(b)

I/xy

(r~py.

i~ a t.t.z.~.

,.t.z.~.)

th~

9 o, y i~ a t.t.z.~.

(respy. r.t.z.d.). In particular, x 2 is a 1.t.z.d. (respy. r.t.z.d.) is a l.t.z.d. (respy. r.t.z.d.).

==~x

PROOF. (a) It suffices to observe t h a t xx~ ~ 0 (respy. x~x 0) implies t h a t yxx~ ~ 0 (respy. x ~ x y ~ 0). (b) Suppose t h a t xyya---+O, ya ~ 0 . If y y a - - ~ O then y is a 1.t.z.d. On the other hand, if yy~ :/-+ 0, writing xa = yy~ we get x~ J-+ O, xx~ ~ 0, whence x is a 1.t.z.. Similarly the proof when xy is a r . t . z . d . . 3 . 7 . 7 . In a unital algebra A a formal sum A + x (A E K, x E A) can be identified with the element Ae + x. But in the case of an algebra w i t h o u t unity A + x can be treated only as a formal sum. The notion of t.z.d, can be extended to formal sums. Thus we call A § x a 1.t.z.d. (respy. r.t.z.d.) if there is a net x~ 74 x with Axa -k xxa = (A + x)xa --~ 0 (respy. xa(A + x) ~ 0).

Let A be a TA and x E A. If 1 + x is a l.t.z.d. (respy. r.t.z.d.) then x is I. (respy. r.) q. singular. If A is 3.7.8.

~ital

a~

LEMMA.

x i~ a t.t.z.~.

(r~py. r.t.z.~.)

th~

9 i~ I. ( , ~ p y .

,.)

singular. PROOF. Suppose t h a t x a - ~ 0 and xa + xxa---+ O. Then x o x~ ~ x. If x~t exists then by premultiplying by x~z on both sides we get x~ --+ 0 - a contradiction. So x is 1.q. singular. Similarly, when 1 + x is r.t.z.d., x is r.q. singular. Next let A be unital with unity e. Then if x in A is a l.t.zd., writing x - e-4-y we see t h a t l + y - e + y - x i s a l . t . z . d . So by above y is 1.q. singular, whence x -- e + y is 1. singular. 3 . 7 . 9 . LEMMA. Let A be a Hausdorff TA. Then" (i) Every l.z.d. (respy. r.z.dt ) is a l.t.z.d. (respy. r.t.z.d.). (ii) When A is finite-dimensional,

a l.t.z.d. nothing but a l.z.d. (respy. r.z.d.).

(respy. r.t.z.d.)

(iii) An element x in A is a l.t.z.d. (respy. r.l.z.d.) spy. rz) is not a topological m o n o m o r p h i s m . t l.z.d. (respy. r.z.d)

-

-

left (respy. right) zero divisor.

is

iff l~ (re-

w 7. Topological Zero Divisors

161

PROOF. (i) If x y = O and x , y r then F = { y } is closed (since A is Hausdorff), 0 ~ F a n d 0 ~ {0} = ~ = z y = x F , so t h a t x is a 1.t.z.d.. Similarly, y is a r . t . z . d . . (ii) Since A is finite-dimensional we may assume t h a t it is a n o r m e d algebra (see 3.4.6. (ii), 2.1.12). If x is a 1.t.z.d., then by 3.7.13 there is a sequence (x,~) with xx,~ ~ O, I1 .11- 1. Since closed unit ball of A is c o m p a c t (A being h o m e o m o r p h i c to Kn), we can choose a subsequence (Yn) of ( x , ) with Yn ---+ Y. T h e n xy,~+O, xy,~xy. So x y = O and x i s a l . z . d . Similarly, if x is r.t.z.d, it is r . z . d . . (iii) If l~ is not a m o n o m o r p h i s m then x is a 1.z.d. and so by (i) a 1.t.z.d. Next suppose t h a t l~ is a m o n o m o r p h i s m which is not topological. T h e n l~-1 is not continuous (since l~ is always continuous) so t h a t there is a net yc~ ---* 0, but xa - l-~lya J-~ O. Therefore x x ~ l , , x ~ - y~--+ O. By 3.7.5, x is a 1.t.z.d. This proves the "if" part. For the "only if" p a r t assume t h a t x is a 1.t.z.d., so t h a t there is a net (x~) such t h a t

(,)

If l~ is not 1 - 1 it is not a m o n o m o r p h i s m , and if it is 1 - 1, the condition (.) implies t h a t l~ 1 is not continuous. Thus, Iz is not a topological m o n o m o r p h i s m in either case, completing the proof. (The proof of the corresponding s t a t e m e n t concerning r.t.z.d. is similar.) o

7.10. Examples

of t.z. divisiors

(i) In the B a n a c h algebra C - C ([0,1] K ) t h e function f o ( t ) t is not a zero-divisor. But it is a t . z . d . . To see this, define if t < n" 1 f,~ E C by 9 f , ~ ( t ) - 0 if t~> ~1 and - 1 - n t Since 0 ~< fn ~< 1 and fn(O) - 1 we have I l f n l l - 1. Also 1 Thus f0 is a t.z.d . . f o f , ~ 0 since Ilfof,~ll <~ ~. (ii) Write 11 - ll (K), the space of all infinite sequences x - (x,~) oo

(Xn E N,) w i t h E lXnl < OO; ll is a B a n a c h space u n d e r the n=l oo

lx.I. Denote by B - B ( l l ) the algebra of a n=l all b o u n d e d linear o p e r a t o r s on /1; B is a B a n a c h algebra (under o p e r a t o r norm). T h e n 1.o's S, T, U defined by S x - (x2, x 3 , ' - - ) , T x = (0, Xl, X2, " " "), U x -- (xx, 0, 0 , . - . )

norm

Ilxll-

162

Some Types of Topological Algebras

belong to B: [tS[t ~< 1, [ITl[- 1, IfUll ~ 1. Further S T - - I (I = identity operator), U T = O. T as a r.z.d, is a r.t.z.d.. On the other hand T being 1. invertible is not a 1.t.z.d.. 3.7.11. DEFINITION. An element x in A is called a symmetric topological zero divisor or s.t.z.d, if there is a net (xa) in A with x ~ - - . O, xx~ and x~x--+ O. Evidently a s.t.z.d, is a bit . z . d . P . G . Dixon has given an example of an element in a Banach algebra which is a bi-t.z.d. (actually a bi.z.d.) but not s.t.z.d. (see [4, p.13, Example 13]). 3.7.12. R e m a r k . It is clear from the definitions t h a t a 1.t.z.d. or a r.t.z.d, x of A continues to be so with respect to any T A B which is an extension t of A. If both A, B are unital then by 3.7.8, x is singular in A as well as B. Following Lorch we call a singularity which continues to be a singularly in any unital extension a p e r m a n e n t singularity. Thus any t.z.d, is a p e r m a n e n t singularity. F. Quigley has shown that every unital Banach algebra A has a unital Banach algebra extension B such t h a t every singular ,element of A is a zero-divisor of B (see [23, pp.25-27]). On the other hand Arens has proved that in commutative unital Banach algebra the permanently singular elements are precisely the t.z.d.'s. For a proof of this theorem (extended to p - B a n a c h algebra) and other interesting connection between permanently singular elements and t.z.d's in TA's see [31, p.32 and pp.112-123]. 3.7.13. element x in A with a s.t.z.d, if

LEMMA. In a p - n o r m e d algebra A = (A, I I II) ~ is a l.t.z.d. (respy. r.t.z.d.) iff there is sequence (xn) [Ixnll-=-1, xxn (respy. xnx)--~ O. In particular, x is there is a sequence (xn) with ]lxnl[ = 1 and xxn --+ O,

XnX --+0.

PROOF. The "if" part of the first statement follows form 3.7.5 (since IIx~ll- 1 ~ x,~ ~ 0). For the, "only if" part assume t h a t x is a 1.t.z.d. Since A is first countable, by (**) of 3.7.5 there is a sequence (xn) with x , 74 O, xxn --~ O. We may assume (after passing to a subsequence if necessary) that IIx,~ll I> r / > 0 for all t i.e.

A is ~ subalgebra of B and the topology of A is the relative

topology from B.

163

w 7. Topological Zero Divisors 1

n and some r/ > O. Set Y n xy,~ --+ 0 since Ilxy, ll - Ilxx, The proof of the "only if" part final assertion of the lemma is

x , / l l x , lf~. Then IlYnll- 11111, and ll/llx, l[ < -lttxx lI, 0. when x is a r.t.z.d, is similar. The clear.

3 . 7 . 1 4 . LEMMA. Let A be a p - n o r m e d algebra and ft its completion. If x c A is a l.t.z.d., r.t.z.d, or s.t.z.d, of ~I then it is accordingly the same of A. PROOF. We shall prove the result only when x is a 1.t.z.d. (the proofs for the other two being similar). By 3.7.13 there is a sequence (~,~)in A with I I ~ , l l - 1, x ~ n - ~ o. Choose x,~ e A such that I I x . - .ll < 1 so that x , ~ - i , - , 0 (as n - + OO). Then

Itx.II II .II- II .- x.II

1

1 2n

1

~> 2'

so that xn 7z~ O. Also, XXn -- X(Xn -- i n ) + Xin --* O. Therefore x is a 1.t.z.d. of A. 3 . 7 . 1 5 . PROPOSITION (Rickart). Let A be a p - B a n a c h algebra and a E A. Then we have" (i) a is a l.t.z.d. (respy. r.t.z.d.) iff

cot(a) -- 0 (respy. wr(a) - 0), where c o l ( a ) - inf Ila~ll/llxll c o t ( a ) - inf IIxall/llxll. zr

~

zr

(ii) Suppose that la (respy. ra) is 1 - 1. Then a is not a l.t.z.d. (respy. r.t.z.d) iff a A (respy. Aa) is closed. (iii) If a is not a l.t.z.d. (respy. r.t.z.d.) then a A (respy. Aa) is closed. PROOF. As usual we shall prove only all results pertaining to 1.t.z.d.. (i) If cot(a)

--0,

(an)in

there is a sequence

A which

1

Iiaanll/lianll -~ o. Writing xn --Ilanll--~an, we get [ [ x n l l 1, Ilax~ll--. O, whence by 3.7.13, a is a 1.t.z.d. Conversely, if Ilxnll - 1, [laxnll -~ 0 then clearly wz(a) - O.

164

S o m e Types of Topological Algebras

(ii) Since la is 1 - 1, la 1 exists and l a l ( a A ) A. If a A is closed then aA, as a closed subspace, is a p - B a n a c h space. By the open m a p p i n g theorem, la I is bounded. But this means t h a t sup Ilxll/llaxll < cr so t h a t wl(a) # O, and so z

by (i), a is not a 1.t.z.d.. Conversely, suppose t h a t a is not a 1.t.z.d. and axn ~ b. Then w z ( a ) > 0, whence 11/21111/wl(a) < cr Since

[[x~ - Xml] <~ [[1~111 IlaXn

-

axml[ ~ o,

as n , m ~ oo, by completeness of A, xn ~ x (say). Then axn --+ ax. By Hausdorff p r o p e r t y of A, a x - b and a A is closed. (iii) This follow from (ii) since if a is not a 1.t.z.d. it is also not a 1.z.d. so t h a t la is 1 - 1. 3 . 7 . 1 6 . LEMMA. Let A be a p - B a n a c h algebra containing an element a satisfying the conditions" (i) There is a sequence (An) in K with A , ~ each Ana is r. (respy. l.) q. invertible.

oo such that

(ii) The element a is not a l.t.z.d. (respy. not a r.t.z.d.). Then A has a I. (respy. r.) unity u. PROOF. Write bn - (Ana)lr. Then Ana + bn + Anabn - O, so that ab.

-

bn

-a

(1)

and hence also bn a ------i- =

IIb.ll

a

IIb.ll

1

bn

1"

(2)

If I]b~lI - ~ c~ then in (2) the R.H.S. + 0 and a would be a 1.t.z.d., contradicting the hypothesis (ii). So we must have Ilbnll <~ C for all n, whence it follows from (1) t h a t abn --~ - a , - a E a A = a A (by (ii) above and 3.7.15 (iii)). It follows t h a t a - au for some u c A. Now for any x E A we have a ( u x - x ) = 0. Since a is

w 7. Topological Zero Divisors

165

not a 1.z.d., we conclude t h a t ux - x, proving u is a 1. u n i t y of A. 3 . 7 . 1 7 . PROPOSITION. Every element of a radical p - B a n a c h algebra A is a bi-t.z.d. . PROOF. S u p p o s e t h a t A contains an element a which is not a b i - t . z . d . . Since every element of ~ A is q. invertible, we can a p p l y 3.7.16 to conclude t h a t A contains a 1. or a r. u n i t y u, and hence a non-zero i d e m p o t e n t u. B u t this c o n t r a d i c t s 1.2.24. Hence the proposition. 3 . 7 . 1 8 . COROLLARY. If A is a p - B a n a c h algebra then every element of v / A is a bi-t.z.d. . P R O O F . If a ~ v/A t h e n by 3.7.17, a is a bi-t.z.d, and so also of A.

of vfA

3.7.19. Let A be a real TA and A its complexification. T h e n the p r o p e r t y of an element x in A being a 1.t.z.d., r.t.z.d., or s.t.z.d, clearly carries over when x is r e g a r d e d as an element of A. F u r t h e r we have 3 . 7 . 2 0 . LEMMA. (i) x + i x is a l.t.z.d., r.t.z.d., or s.t.z.d, of A according as x is a l.t.z.d., r.t.z.d., or s.t.z.d, of A. (ii) If x C A is a 1.t.z.d. (respy. r.t.z.d.) of ~i then it is already a 1.t.z.d. (respy. r.t.z.d.) of A. N

PROOF. (i) Clear. (ii) A s s u m e t h a t x(x~ + iy~) ~ 0, x~ + iya 74 O(xa, ya E A). T h e n either xa 74 0 or y~ 74 0, while xxa --~ O, yya -~ O. ,v T h e r e f o r e x is a 1.t.z.d. of A if it is of A. Similarly x is a r.t.z.d, of A if it is of A.

3.7.21.

Let A (A,]]-II) p- orm d algebra (respy. a unital p - n o r m e d algebra) and (xn) a sequence of q. invertible (respy. invertible) elements g A such that (i) x~ ~ x t t (ii) sup Ilx~ll (respy. s u p I l z n- l l l -- CX:), where x,~ (respy. x n- 1 ) is n

LEMMA.

n

q. inverse (respy. inverse) s.t.z.d. .

of xn.

Then

1 + x

(respy. x)

is a

166

S o m e Types of Topological Algebras

PROOF. First consider the case where x,~ are q. invertible. We assume (as we may) t h a t I1" II is sm. and by passing to a ! subsequence (if n e c e s s a r y ) w e can also assume t h a t IIx, II ~ oo. 1

I I Define Yn - xn/llxnll ~, so t h a t IlYnll- 1. Then, by using the relation xn § x n' § x n x ~ - O, we have

(l+x)yn-yn+xy,~

Xn

- y,~+(x-xn)yn+xnyn-

1 ~ (X-- XI2)yI2 , I

so t h a t

ll(1 + x)y,~l] <~ I]x'~]l + llx- xn]].

IIx'.ll

I ]ix.Itco we get (1 + x)yn---* O. Similarly, yn(1 + x) ~ co. So 1 + x is a s.t.z.d.. Next, consider the case where x , are invertible. Writing zn x,-e, z-x-e we see t h a t z, a r e q . invertible, z n ~ z , sup [Iz~n[[- sup [[xn I - e [ [ - co.

Since x,~--~ x,

By the first part (proved a b o v e ) w e have xyn - (e + z ) y n (1 + z ) y n --~ 0, and similarly y n x -~ O. Also IlYnll- 1. So, by 3.7.13, x is a s.t.z.d.. 3 . 7 . 2 2 . LEMMA. Let A be a unital TA, Gi its group of invertible e l e m e n t s and xa, x c Gi, xa --~ x. Then x~ 1 -+ x -1 iff the net (x~ 1) i8 essentially bounded. PROOF. If x~ 1 --+ x -1 then (x~ 1) being a C - n e t is essentially bounded (by 2.3.6). Conversely, if (x~ 1) is essentially bounded then, by 2.3.8 (iii), m

x a l ( x a - x) ~ O, i.e. x-~lx --+ e.

It follow,~ that xal

__ ( x ~ - l x ) x

1 __~ e X - 1

__ X

1

3.7.23. PROPOSITION. Let A be a unital H a u s d o r f f complete TA whic,h is a C algebra. Suppose that xa E Gi and xa---, x in A. T h e n x c G i iff (x~ 1) is a C - n e t .

PROOF If x E Gi then, A being a C - a l g e b r a , x~ 1 ~ x -1 so t h a t (x~ 1) is a C - n e t . On the other hand, if we suppose t h a t 9

167

w 7. Topological Zero Divisors

(X~ 1) is a C - n e t then by completeness of A, x~ 1 --~ y (say) C A. Since (xa) is convergent it is essentially bounded, and so by 2.3.8 (iii), x ~ ( x ~ 1 - y) ~

0, o r ,

x~y

~

e.

Also, x~y ~ xy, whence by uniqueness of limits (A being Hausdorff) we get xy - e. Similarly we can show that yx = e. Thus x is invertible, x E Gi, completing the proof. 3.7.24. PROPOSITION. Let A be a C I (respy. C Q ) algebra, Gi (respy. Gq) its group of invertible (respy. q. invertible) elements. If (xn) E Gi (respy. Gq) and Xn --~ x E A theft x E G i ( r e s p y . Gq)

i f f Xn I (respy. (Xln)) ~8 bounded.

PROOF. First suppose that xn E Gi, xn --~ x, x E Gi. Then Xni __+ X-I (since A is a C algebra) and hence (x~i) is bounded. Next suppose that x n E G i , X n - + x E A and (x~ 1) is bounded. Then by 2.3.8 (iii), (X n -- X ) X n 1 --+ O, X n l ( X n -- X) --+ 0 i.e. X,Xn 1 --> e, X n l X --~ e.

Therefore, since A is a I algebra, x x nl and x n i x are invertible for sufficiently large n, whence by 1.1.30, x is invertible, x E Gi. This completes the proof of the part relating to inverses. To prove the part relating to q. inverses, consider the unitization A1 of A and observe that (el -Jr-Xn)

--

I

1 _ el jr_ Xn"

It follows that (x~) is bounded iff ((r -1) is bounded. Therefore, by the p a r t proved, el -~-x E Gi iff ((el-Jr-Xn) -1) -! (e I -~- Xn) is bounded iff (x~n) is bounded, completing the proof. 3.7.25. COROLLARY (Gelfand). Let A be a unital p t_ Banach algebra, Gi its group of invertible elements, Xn E Gi and xn --~ x C A. Then x C Gi iff there is a constant C > 0 t Gelfand considered only the case p - 1.

168

Some Types of Topological Algebras

that

of A.

llx II

c

II.II

p- o m

PROOF. Since a unital p - B a n a c h algebra is a CI algebra (by 3.6.23(b)) and the boundedness in A is the same as the boundedness with respect to ]]. ]1 (by 3.2.13), the corollary follows from 3.7.24. 3.7.2,6. COROLLARY. In a unital p - B a n a c h algebra A the limit of a convergent sequence of invertible elements is either an invertible element or is a s.t.z.d. . In particular, every element of OGi t is a s.t.z.d. . PROOF. The first assertion follows by combining 3.7.25 and 3.7.21. The second follows from the first since G~ being open 0Gi is disjoint with Gi. 3.7.2',7. COROLLARY. Every element of the radical x / ~ of a unital p - B a n a c h algebra is a s.t.z.d. (cf. 3.7.17). PROOF. If a E v/A then since na E x/~, na is q. invertible and consequently (e § na) -1 exists. It follows t h a t (~ § a) -1 = n(e § na) -1 exists, whence by 3.7.26, a - l i m , ( ~e + a) is either invertible or a s . t . z . d . . But a cannot be invertible since a E x / ~ , so a is a s.t.z.d, as required. 3 . 7 . 2 8 . PROPOSITION. Let A be a unital p - B a n a c h algebra, x c A and a(x) the spectrum of x. If a(x) 7s tt and )~ c Oa(x) then x - Ae is a s.t.z.d. . PROOF. Since A E On(x) and a(x) is closed, there is a sequence )~n e p ( x ) = K \ a ( x ) s u c h t h a t )~, ~ )~. Then x A~eCG~.. x - ~ e ~ Gi. Since x - ) ~ , ~ e ~ x - ) ~ e it follows t h a t x - Ae E ,~Gi and consequently, by 3.7.26, x - ~e is a s . t . z . d . . 3.7.2!9. PROPOSITION (Rickart). Let A - ( A ,

li" II)

be a

,t For a subset S of a topological space X we denote by OS the frontier of S i.e., OS(- OS') - s - A s ' , where S ' = X \ S and bar denotes closure. tt This condition is satisfied for every element x if A is complex or strictly real. Moreover, whenever cr(x) ~ 0 we have also O~(x) ~ O, since by 6.1.2, a(x) ~ [~, is closed but not open (by connectedness of [K).

169

w 7. Topological Zero Divisors

p - B a n a c h t algebra with I111 ~m. and xn --. x in A with x,~ +-+ X (for all n). Let v - Vll. I. Then"

(i) If A is unital, x,~ invertible a n d V(Xn 1) i8 bounded then x is invertible.

(ii)

If xn are q. invertible q. invertible.

and

u(x~)

bounded

then

x

is

PROOF. (i) Suppose that V(Xn 1) ~< C. Then we have

V(XnlXrt -- XnlX)

V(Xnl)V(Xn -- X)

(using 3.3.7. (iii)) ~<

CII

- -

O.

It follows that for sufficiently large n, v ( e - x n l x ) < 1, whence by 3.3.20 (ii), x ~ l x is invertible, so that x is also invertible. (ii) This can be deduced from (i) by passing to the unitization A1 of A. 3.7.30.

PROPOSITION.

Let A = (A,[[. II) be a unital p n o r m e d algebra with Gi its group of invertible elements. I f xn E G i , x n --+ x in A and x is a t.z.d, then s u p l ] x ; 1 ] l - 0o and x is a s.t.z.d. .

PROOF. Let A be the completion of A. Then x is a t.z.d, of and so not invertible in A, whence by 3.7.26, sup IIx;X]]- oo and so by 3.7.21, x is a s.t.z.d.. 3 . 7 . 3 1 . COROLLARY. A n element x E O G i \ G i iff sup I l x n l ] ] - CX3, where Xn E Gi and xn ~ x.

is a s.t.z.d.

PROOF. This follows from 3.7.21, 3.7.30. 3.7.32.

PROPOSITION.

PROOF.

Since x is singular and A is strictly real we have

In a strictly, real unital p - B a n a c h algebra A every singular element is a bi-t.z.d, and the square of a singular element a s . t . z . d . .

t Rickart considered only Banach algebras (i.e. p - 1 case).

170

Some Types of Topological Algebras

0 C a(x) -- 5(x) __ ~, and consequently we have 0 e 5(x) 2 = 5(x 2) -- a(x2). Since 5(x) _C ~, a(x 2) - 5(x) 2 /> 0. It follows t h a t 0 E 0a(x2), so that by 3.7.28, x ~ is a s.t.z.d.. Hence, by 3.7.6 (b), x is a bi-t.z.d. 3.7.33. COROLLARY. In (a strictly real) A we have S(= s z U s ~)- s hi- s t - s ~ - 3 u - 3 "t- 3 bit. PROOF. It suffices to observe that a bi-t.z.d, is, by 3.7.8, both 1. singular and r. singular. 3 . 7 . 3 4 . R e m a r k s . In the Banach algebra of complex valued continuous functions on the unit interval, with sup norm, every singular element is a t . z . d . . On the other hand, in the Banach algebra A of all complex-valued continuous functions f = f ( z ) on the closed unit Izl ~< 1 which are holomorphic on tzl < 1, the function fo(z) = z (Iz] <~ 1) is singular but not a t.z.d. (see [10, p.70] or [7's p.29]). Further the real algebra A[~] gives an example of a real Banach algebra containing a singular element ([0) which is not a t.z.d.; note that A ~ is not strictly real (since it has complex structure), so that this example does not contradict 3.7.32.

Let B be a complex or strictly real unital p-Banach algebra and A a closed subunital algebra of B. Then, for every x E A, 3.7.3',5. PROPOSITION.

Further, if either (hA(X))~

c

(.)

c

(**)

0 or pB(x) is connected we have =

(, 9 ,)

PROOF. The relation (.) has already been obtained (see 1.7.20). For proving (**) we note that if A E hA(X) then by 3.7.28, x--Ae is a s.t.z.d, of A and hence also of B. In particular x - A e is singular in B and so A e as(x). Thus OaA(X)C_ aS(X). Again, by (**) of 1.7.20, pA(x) C_ ps(x). Therefore

w7. Topological Zero Divisors

-

171

-

which is (**). Assume now that (aA(X)) ~

O. Then

(since aB(X) is closed). Combining this inclusion with the inclusion ( , ) w e get ( , , ,). Next let pB(x) be connected. Since pA(X) C pB(X) we can write pB(x) -- pA(X)O(pB(x)\pA(X)) - p A ( x ) U ( a A ( x ) \ a B ( x ) )

(1)

where 0 denotes disjoint union. Since 0 a A ( X ) C aS(X) every point of aA(X)\aS(X) is an interior point, i.e. aA(X)\aS(X) is open. The connectedness of pB(x) and (1) imply that aA(X) -aB(x), proving (, 9 ,) in this case as well. 3.7.36. COROLLARY. Let A be a closed subunital algebra of a unital complex p-Banach algebra B and x E A. Then

c

o.(x) c

and when these inclusions hold we have actually aA(x) - aB(X). PROOF. Since always fiB(X) _C 6rA(X) it is enough to prove the "if" part of the "iff" assertion. Assume therefore that fiB(X)C_ ~. By (**) of 3.7.35, OaA(X) < OaS(X) C ~, and by 6.1.2, aA(X) is compact. It follows that (aA(X)) ~ = 0, whence by 3.7.35, aA(X) = fiB (X), completing the proof. 3.7.37. PROPOSITION. A closed subalgebra A of a strictly real p-Banach algebra B is strictly real. PROOF. First assume that B is unital with unity e and set A1 - A + ~e (A1 - A if e E A). Consider the complexification B of B and write A1 - A1 + iA1; A1 is a closed subunital algebra

172

Some Types of Topological Algebras

of 1~. Now a [ 4 1 ( x ) - a~l ( x ) , a ' S ( X ) - a[~(x). Since B is strictly a[~(x) !~_ ~. By 3.7.36, a)4x(x) - ax~ (x) - a[~(x) C ~. It follows that A1, and hence A (by 1.9.5) is strictly real. If B has no unity, consider its unitization B1 which again is strictly real. Also A is closed in Bx (since B is closed B1 ). Applying the result just obtained above to the unital Bx we conclude that A is strictly real.

(Zelazko).

3.7.38. DEFINITION. Let A be a TA. A pair ( S , T ) of subsets of A is called a generalized topological divisor of zero [31,p.73] or g.t.z.d, if 0 ~ S, T but 0 E S T , where bar denotes the closure. 3.7.39. LEMMA. A g.t.z.d. ( S , T ) with S {y}) is a l.t.z.d, x (respy. r.t.z.d, y).

{x} (respy. T =

PROOF. Write T - F. Then we have 0 ~- F while 0 C S T =

xT c_ xT - xF, proving x is a 1.t.z.d.. Similarly, y is a r.t.z.d.. 3.7.40. LEMMA. A pair of nets (x~),(y~) such that xa 74 O, y~ ~ 0 and x~y~ ~ 0 determine a g.t.z.d. PROOF. Since xa ~ 0 there exists a nucleus U and a subnet (x~,) of (x~) which lies outside U. It follows that if S - {x~, } then 0 r S. Similarly there is a subnet (yZ,) of (yz) such that 0 @ T, where T - {y~,}. Since (xa, y ~ , ) i s a subnet of (xayz) and x~yr ~ 0 we get 0 C S T , proving ( S , T ) is a g.t.z.d. 3.7.41. PROPOSITION (Zelazko). A commutative unital Hausdorff TA which admits no g.t.z.d, must be a C algebra. PROOF. Suppose that A is not a C algebra. Then we can find a net (x~) in Gi with x~ --. x0 e Gi (1); x~ 1 74 xo 1.

(2)

By 3.7.22, (x~ 1) is not essentially bounded and so by 2.3.6, it is not a C - n e t . It follows that there is a nucleus U and subnets ( x z ) , ( x z , ) of the net (xa) such that x~,1 - x~ 1 ~ U, so that _

§

o.

(3)

w7. Topological Zero Divisors

173

As subnets of (xa) it follows from (1) that X~ --~ X0~ Xt~t --+ X0

whence

x~,z;3 ~ z2o ~ 0

(since x 2 C Gi).

Since A is Hausdorff we have

x~,xz 74 O.

(4)

On the other hand, by using the commutativity of A we get X~tXI3(X-~I 1 -- X~ 1) -- X/~ -- X/~, ~

0.

(5)

From (4), (3), (5) we conclude, by 3.7.40, that A has a g.t.z.d., contradicting the hypothesis. Hence A is a C algebra. 3.7.42. R e m a r k . pp.73-77].

For more information on g.t.z.d, see [31,

3.7.43. R e m a r k . For the class of topological algebras called locally sm. convex (defined in 4.4.11) there is a weaker definition of t.z.d, due to Michael. For results concerning such t.z.d.'s see [20, pp.43-48].

C H A P T E R IV LOCALLY

PSEUDO-CONVEX AND ALGEBRAS

w 1.

SPACES

p-convexity

4.1.1. DEFINITION. Following Landsberg [9~, p.104] a subset S of a LS X (over K) is called p-convex, where 0 < p ~< 1, if

x , y , c S; ~,/~ E ~,~,/~/> 0 and ~P +/~P - 1

(.)

ax + fly E S. It is called absolutely p-convex if x, y e S ; c ~ , / ~ E K

and I~1p+I/~t p~
(**)

ax + fly E S. Evidently when p - 1, p-convex (respy. absolutely p-convex) is just convex (respy. absolutely convex) in the usual sense. Let '~ be a subset of X. An element of the form ~ a j x j J n

~ a j P-- 1 , is called a pj convex linear combination of elements of S. The meaning of an absolutely p-convex linear combination of elements of S is clear. For any subset S of X we denote by Cp(S) (respy. ICpI(S)) the set of all p-convex (respy. absolutely p-convex) linear combination of elements of S; Cp(S) (respy. ICpI(S)) is called the p-convex (respy. absolutely p-convex) hull of S. Obviously, Cp(S) c_ ICp(S)l.

~-~ajxj, with xj C S , j-1

a i />0

and

4.1.2. LEMMA. ( a ) I f S is p-convex (respy. absolutely pc o ~ ~ ) t h ~ s = c~(s) ( ~ p y levi(s)). (b) For any subset S of X, C,(S) (respy. ]Cpl(S)) is the smallest p-convex (respy. absolutely p-convex) subset of X containing S.

w 1. p -convexity

(c)

175

If S is p-convex (respy. absolutely p-convex)so is AS tl

PROOF. (a) Let S be p-convex and x -

~ajxj

with ~ j />

j=l

n l a j P -- 1. We shall show t h a t x E S 0, xj c S and (,) ~ j = by induction on n the length of the combination). If n - 2, x c S by the definition of p-convexity. Assume now t h a t x E S p whenever the length is n - 1. For the a j occuring in (,) write n-1

PZ O~j j=l

n-1

tip (say) , so t h a t we ~ - - ~as xj

E S. Therefore

since

j-1

aj f~P + a~ -- 1, it follows t h a t ~ cux j - ~ -k-xj + c~nx~ E S. j=l \j=l Thus, Co(S ) C_ S whence Cp(S) - S (since always S C_CCo(S)). _ The assertion concerning ICoI(S) is proved similarly. (b) It follows from (a) t h a t if $1 is a p-convex set with $1 _D S then $1 - Co(S1 ) D_Cp(S). We will now show t h a t Co(S ) is pconvex. Suppose t h a t m

rt

x - ~ ~;~;, y - ~ ~y~ (~j, y~ e s) j=l

k=l

m

tI

with O~<cu, /~k; ~-'~lcuI p -

1, ~ l / ~ k l o -

j=l

If O ~ , # ; ~ P + # P - -

1.

k=l

1 then m

n

~x + ~y - ~ ~ j ~ j + ~ . ~ y ~ . j=l

k=l

Since

~--~(Aaj)P+~(#f~k) pj--1

Ap

a

+#P

/~;

- A'+# p - 1

k=l

we conclude t h a t Ax+#y E Cp(S) which proves t h a t Cp(S) is pconvex. The proof of the assertion concerning ICpI(S) is similar.

Locally Pseudo-convex Spaces and Algebras

176

(c) Clear. 4.1.3.

If ~ a i x 3 . , ~~3.x3.

Remark.

3"=1

are absolutely

p-

3.=1

convex linear c o m b i n a t i o n s of elements and I,~1~ + I"1' <-

1, then

n

~-~()~%, + #~i)x3. is also an absolutely p-convex linear combinaj=l

tion of elements. In fact we have

< ~ ( l ~ l I~jl + I~l~jl) ~ J

3"

3"

(since 0 < p <~ 1)

3"

3"

the .other h a n d , the analogous assertion for p-convex linear c o m b i n a t i o n can fail if p < 1. For example, take n - 2; a l = On

_

1

_

1

/32--

4.

)~_/~_

1;

p _

1

Then 1

I

~1 -

(,~al + ~f~i)~

-

12 '

"~2 -

(~c~2 + u Z 2 ) ~

5

-

12

so t h a t "~1 + "/2 - ~/ig+5 12 < 1 showing t h a t ~/2x +-/2x2 is not a p-convex linear combination.

A subset S (of a LS X) is absolutely pconvex iff it is p-convex and balanced. Also, if S is any bal4 . 1 . 4 L EMMA.

a n c ~ ~ub~t th~n C ~ ( S ) i~ ~b~ol~t~ly p-~on,~,, ~o that IC~I(S) c~(s). PROOF. If S is absolutely p-convex it is trivially p-convex. Moreover, it is also balanced since I)~] ~< 1 =~ I,~lp <~ 1. Conversely, s u p p o s e t h a t S is p-convex and balanced. If xj E S, ~--~lai] p ~< 1 J

177

w 1. p -convexity

c~j then ~ l - i f [ ~ where r p~< 1. We assume j 3 that all Icu[ > 0 and set ~j - a j / l a j l . Then ~.xj E S (since S is balanced) and so by p-convexity .

3ix i ~ S ,

i.e. ~ ~ . ~ j x j C S .

3

3

By balanced property of S we get ~ a i x j E S, completing the J proof of the first assertion. For the second we assume that S is balanced. If aj ) 0 , ~ a j O _ l , IA[ ~ 1 and x j E S then J

~ ajxj=~ aj(Axy) E C : ( S ) (since Axj J

E S, S being balanced).

3

Hence Cp(S) is balanced and Co(S ) - I C ] o ( S ) . 4.1.5. LEMMA. If S is p-convex and a,/~ > 0 then 1

1

1

+

If S is convex (i.e. p - l )

c_

then a S + /3S - ( a + fl)S.

PROOF. For x , y E S 1

+

[

1

1 ]

a-; x + /3-;y 1

E

(a + fl); S (by p-convexity).

If p - 1 the above gives a S + flS C_ (a + ~ ) S . On the other hand, since (a + ~)x - ax + fly the reverse inclusion relation also holds. 4.1.6. LEMMA. If U is an open subset of a TLS X Ic~[(u) is open. tI

PROOF. Consider an element x -

~ j-1

a;-aj (aj E U),

then

178

~

Locally Pseudo-convex Spaces and Algebras

# 0, ~

I~jl ~ ~< 1. Since U is open we can choose a balanced 3" open nucleus V such that aj + V _C U ( j 1 , . . - , n ) . Write ~l~;I-

~ ( > o). f o r a C V we h~ve

J x -~- c~a --

oljaj + ~ I~jla - ~ ~j(aj + ~;xl~jla ), J

J

J

Since V is balanced, a ~ ll a j l a E V, aj + a ; l l a j l a E U, so t h a t x + aa c ]Cpl(U),x + a Y cC_ ICpl(U). Since x is an arbitrary element of ICpl(U) we conclude t h a t ICpl(U) is open. 4.1.7. LEMMA. If S is absolutely p-convez (0 < p ~ 1) and 0 < p~ <: p then it is also absolutely p~-convex. In particular, if

S is absolutely convex it is absolutely p-convex for every p with O
P (since ~ ) 1)

pt

p

~<

1~-1.

Therefore, every absolutely p~-convex linear combination is an absolutely p-convex linear combination, whence the lemma. 4.1.8. DEFINITION. Let S be an absorbing subset of a LS X. For x E X , 0 < p ~ < 1, set -

p,~(x) - p o , s ( x ) - p s ( x )

=

i n f { a > O ' x E a-iS}

-

i~f{flP'fl

1

> O,x E flS}.

The non-negative realvalued function p is called the p-gauge (gauge if p - 1). The gauge of S is also known as the Minkowski functional of S. It is clear from the definition of the p-gauge t h a t we have"

Ps2 <~Ps~ if S1 C $2,

(,)

w 1. p -convexity

179

(**)

p~ - 0 , 1 po~ - ~p~

(, 9 ,)

(~ e K \ { o } ) . pt

4.1.9 R e m a r k . p - p;.

If 0 < p~ ~< p < 1 then p;, - p-Z, where

4.1.10. PROPOSITION. Let S be an absorbing subset of X. Then the p-gauge p - ps has the following properties: (i) p(o) - o.

(ii) p ( A x ) - APp(x) if A >10. (iii) p ( A x ) -

[A["p(x) for all A E K, provided S is balanced.

provided S is

(iv) p ( x + y) ~ p(x) + p(y) for all x , y E X p -convex.

In particular, p is a p-seminorm if S is absolutely p-convex (and also absorbing). 1

PROOF. (i) Since S is absorbing, 0 E S and hence 0 E a ~ S for all a > 0, whence p(0) - 0 1

(ii) x E a ~ S

1

iff Ax E (APa)~S(A > 0). Hence p(Ax) -

~p(x) (iii) Assume first that

I A I - 1; then A - 1 S - S 1

since S is

1

1

balanced. It follows that Ax E a ~ S iff x E a ~ A - 1 S - a ~ S . Therefore p(Ax) - p(x). Next for arbitrary A e K, A ~= 0, write A - [A]# where I # l - 1. Then, by (ii) (since I # ] - 1). Finally, the relation clearly holds if A - 0. 1

(iv) Given e > 0, we can choose a, fl > 0 such that x E a ~ S , 1

y C f3-~S ; a < p(x) + e, fl < p(y) + e. Since, by 4.1.5, 1

it follows that

1

1

180

Locally Pseudo-convex Spaces and Algebras

From tile arbitrariness of c we conclude that

;(x + y) < p ( x ) + ;(y) proving the first result of (iv). For the second result we remark t h a t it follows from this and (iii), since an absolutely p-convex set is both p-convex and balanced. 4 . 1 . 1 1 . LEMMA. Let S j ( j - 1 , . . . ,n) be absorbing balanced sets and S - $ 1 ~ " " ~ Sn; then S is absorbing and balanced. If pj, p are p -guages of Sj, S respectively then p - pl V . . . V pn. t PROOF. T h a t S is absorbing and balanced are easy consequences of the definitions (see 2.1.15). Since S c Sj we have pj ~
Again we have, for some en ~ 0, 1

1

x e (pj(~) + ~.)~ sj

C_

(q(x) + cn)~ Sj (since S i is balanced)

=

~sj ( ~ y )

It follows t h a t )~-lx E S, SO that 1

c ~ s - (q(x)+ ~.)~ s. This impities t h a t p(x) <. q(x) + en, whence making en --~ 0 we get p(x):<, q(x) <~p(x). Thus q - p, completing the proof. 4 . 1 . 1 2 . PROPOSITION. Let S be a subset of a LS X (over •), which is absolutely p-convex (0 < p <~ 1) and absorbing. Then: (i) The p-gauage p B1 -

Ps is a p - s e m i n o r m such that if

{x c A'p(x)

<

1},

B1 - {x e A " p(x) < 1}

then B1 C_ S c_ B1; in particular, kerp _c S. (ii) S B1 or B1 p -topology.

according as S is open or closed in the

t See 3;.1.7 (b) for the definition.

w 1. p -convexity

181

(iii) If A is an algebra and S a (multiplicative) subsemigroup of A then p is sm.. PROOF.

(i) By 4.1.10, p is a p-seminorm. If x E B1 then 1

for some e > 0, p(x) < e < 1, x E c ~ S . 1

_1

Then e P x E S, and

1

1

since I c ~ [ - Icl~ < 1 and S is balanced, x -

1

e~(e-~x)e

S, so

1

t h a t B1 C S. Again, if x E S whence S _c B1.

then x E I~S, so that p(x) <~ 1, 1

(ii) Assume first t h a t S is open and x e S. Since ( 1 - ~ 1) ~ x --+ 1

x and S is open, (1 - ~1 ) ; x ~ S for some n/> 2, so t h a t p(x) <~ 1 - ~ 1 < 1, whence B 1 - S. Next assume t h a t S is closed. By 3.2.7, B1 is the closure of B1. Therefore, since by (i), B1 _C S C B1 and S is closed, it follows t h a t S - B1. (iii) By definition of p, since S is balanced, we have for any c > 0 with 0 < e < 1, 1

x c (p(x) + ~)~ s,

1

y e (p(y)+ ~)~ s.

Therefore

xy e [(p(~) + ~)(p(y) + ~)]~s ~ c [p(~)p(y) + c ~ ] ~ s , where C is a constant independent e. By making e ~ conclude t h a t p(xy)<~ p(x)p(y), i.e., p is sm..

0 we

4 . 1 . 1 3 . PROPOSITION. Let p be a p - s e m i n o r m on an algebra A. Then B1, B1 (as defined in 4.1.12) are absolutely pconvex and absorbing. Further, B1 is open, B1 is closed in the p-topology and we have p . , - p-~ - p

(,)

If p is sm. then B1, B1 are subsemigroups. PROOF. Using the identity p ( A x ) - [AIPp(x ) it is straightforward to check t h a t B1, B1 are absolutely p-convex. Next, for any 1

x C A , set ) ~ - 1 if p ( x ) - O and A ~ - ( 2 p ( x ) ) - ~ if p ( x ) ~ 0 . 1 Then p(Axx) - 0 or ~, so t h a t ) ~ x E B1 _c B1, proving t h a t B1, B1 are absorbing. It is further clear from the definition of the

Locally Pseudo-convex Spaces and Algebras

182

p-topology that B1 is open, and B1 is closed, being the closure of B1 (by 3.2.7.). Again it is immediate from the sm. property that BI: B1 are subsemigroups whenever p is sm.. It remains to prove the equalities (.). Since B1 c_ B1 we have (1)

p-~ (x) <~PB~ (x). 1

Again, if A > p(x) then x E A;B1, so that PB~ (x) ~ A, whence

p(x).

(2)

Xw

Finally, p(x)<~ A iff x E A~B1. Hence

(3)

P-B~( x ) - inf{A" A/> p(x)} >/p(x). From (1), (2), (3) we obtain (.).

4.1.14. R e m a r k . If p be as in 4.1.12, and r > 0, then we can show as above (for the r - 1 case) that Br - {x E A " p(x) < r}, Br - {x E A " p(x) <~ r}

are respy, open absolutely p-convex and closed absolutely pconvex. Also, when p is sm., Br, Br are subsemigroups for 0
0 <

! P

p ~ 1, 0 < p' <~ 1. Then p ' - p - ; is a p ' - s e m i n o r m i f f t h e unit ball B1 -- {x 9p(x) < 1} is p'-convex. In particular, p' is a p ' - s e m i n o r m for all p' with 0 < p' < p.

PROOF. Note first that we have also B1 - {x 9 pt(x) < 1}. If p~ is a p~-seminorm then, by 4.1.12, B1 is p~-convex. Conversely, let B1 be p~ convex. Denote by p ~ the pt-gauge 1

determined by B1 9 Then PB~(X) ~ < A

-:

:~ A ~ x

E B1

-:

>

1

p ' ( A - T x ) < 1 iff p'(x) < A. Therefore p' - PB1 and consequently

183

w 1. p -convexity

p~ is a pt_seminorm. Then second assertion in the proposition is an immediate consequence of the first and 4.1.7. 4 . 1 . 1 6 . THEOREM (Rickart t ) . Let A - (A,p) be a real p s e m i n o r m e d algebra, 0 < p <~ 1, and A the complexification of the algebra A. Then fi is a p - s e m i n o r m e d algebra. Further, ffi is unital, p - n o r m e d or p - B a n a c h according as A has the corresponding property. Moreover, we can choose the p - s e m i n o r m of A in such a way that the map x ~ x + i.O of A into A is an N

N

N

~, ~ith p ( ~ ) - 1, ~ d f i ~ I t y ~ i~ ~m. if p i~ ~m.. PROOF (following the method of Bonsall-Duncan [4, p.68]). Write U - {x E A " p(x) < 1}, U - the absolute p-convex hull of U in A. Then U is absorbing in ii.. To see this, suppose that z-x§162 (x, y E A ) and N

#P > m a x { p ( x ) , p ( y ) } .

(1)

Then x / p , y / # E U, so that 1

21/ppz -

1

x

i

21/p ( ; ) +

y

-

2-~/p ( ; ) E U

(2)

N

proving U is absorbing. We next prove that ~r has the property" z-

x§

U ~ x, y E U.

(3)

Assume that z has the representation n

Z

I"/klP ~< 1.

~

k

k=l

Writing "~k -

a k + i/~k

we get Z - - X + iy, w h e r e x -

k

akzk, y -- ~ Z~Yk. k

t Rickart [23, p.8] obtained the results when A is a real normed algebra.

Locally Pseudo-convex Spaces and Algebras

184

Since, lakl, I~1 < I~l it follows, by the absolute p-convexity of U in A, that x,y E U, whence~ (3)holds. Denote by /3 the gauge of U (in A); then /5 is a p-seminorm on A. Writing BI - {z E A " i0(z) < 1}, by 4.1.12(i), we get

/~1 C (/".

(4)

On the other hand, if n

z-

~

Akxk E U (xk E U, Ak E c ) , ~ I A k I p ~< 1,

k--1

k

we can choose e > 0 such that p(xk) < e < 1 for all k. Then 1

1

N

e p xk C U, so that e p z E U, whence /5(z) <~ c < 1. It follows therefore that

U-

B1

(5)

and so in particular U is i6-bounded. From (1),(2)it follows that /5(z) ~< 2 max{p(x), p(y)}.

(6)

Again, if )~ > 0,,~z C U then by (3), ~x,)~y E U, so that

p()~x),p()~y) < 1, whence AP max(p(x),p(y)) < 1.

(7)

For a given e > 0, we can choose the A > 0 such that 1

Ap <

+

(8)

We conclude from (7),(8), since e is arbitrary, that

max{p(x),p(y)} <~l~(z).

(9)

The inequalities (6),(9) imply that /5 induces the topology of A, and also that io is faithful (respy. complete) when p is faithful (respy. complete). If x E A and A E ~ then it is clear from (3) that N

Ax + i0 ~ U iff A x c U, whence p(x) - ~(x). When A is unital we can, using 3.5.9, assume t h a t p ( e ) - 1, so that then we get l o ( e ) - p ( e ) - 1.

w 2. Locally B o u n d e d Algebras

185

Finally, let p be sm.; then U is a subsemigroup of A. We shall show t h a t ~r is a subsemigroup of A. Suppose now t h a t N z, z ~ E U, then they have representations ~--~Akxk, z' -- ~---~Alxz, where xk, xz E U and Z k l

la lo, k

lallo .< 1. Now l zz' - E

and

AkAll xkxl,

k,l

k,l

k

l

Since U is a semigroup, xkx~ E U and consequently the above relations show t h a t z z ~ E ~], so t h a t U is a subsemigroup of A, as desired, completing the proof of the theorem.

w 2.

Locally

Bounded

Algebras

4 . 2 . 1 . L EMMA. Let X be a locally bounded T L S and U a bounded balanced nucleus of X . Then we can f i n d a p,O < p <~ 1 such that 1 U § U C 2-~U. (,) PROOF. Since U is bounded, by 2.1.25, { -1~ U ' n is a basis of nuclei, so that there is a N / > 2 such t h a t U

U

c u.

1, 2, . . . }

(,,)

1

Write p - l o g 2 / l o g N .

X

Then N -

4 . 2 . 2 . L EMMA (K6the). Let S such that

s+sc2

2~ and (,')==~ (,). be a balanced subset of a L S 1

s

(1)

where O < p <~ 1. Then ki

2-7S §

k~

+ 2- o S C_S

(2)

Locally Pseudo-convex Spaces and Algebras

186

lrt

for all integers kj >1 1 such that

2 -kj ~< 1, and further

j-1 1

levi(s) ~ 2~s.

(3)

PROOF. ([18,p.165]). To prove (2) it is clearly sufficient to prove the, special case (2 ~) of (2) where we have rt

2 -kj -- 1.

(*)

j=l

We call k - max kj, the order of the resolution (,). We prove J (2') by induction on k. If k - 1 the relation (2)reduces just to the hypothesis (1) and so (2) holds for this case. Next assume that (2 ~) holds for order k >/ 1. To prove that it holds for order k § 1, observe that each decomposition (,) of order k + 1 is obtained from a decomposition of order k, by replacing one (or more) of the summands 2 -k by 2-(k+l) + 2-(k+l), since it follows from ~ 2 - k J - 1 that the summands 2 -(k+l) occurs an even number of times. Thus, to obtain (2 ~) for order k + 1 starting k from (2 ~) for order k we have to replace the summand 2 - ~ S by k+i

i

k+i

i

2 - - ~ S + 2 - - 7 - S. But this is permissible since 2-~ S + 2-~ S _c S (by (1)). This completes the proof of (2') (by induction). n

It remains to prove (3). Suppose that ~ l a j ] p ~< 1 and xj e j=l

S. Determine integers kj such that 2-kJ 4 [aj] p < 2-kJ + 1 - 2 92-kJ. Then J

l~jl ~ < 2

2-ki ~<2. ~ I~jl~ < 2 , 1 -

2

J

since S is balanced,

-kj+l

ajxj E a i S c Jails c_ 2 ..~ S, so that

kj+l J

J

1(

~) J

1

2~ S (using (2)),

187

w 2. Locally B o u n d e d Algebras

proving (3). 4 . 2 . 3 . PROPOSITION. A TA A (or more generally a T L S ) is p - s e m i n o r m e d (0 < p <~ 1) iff it has a bounded nucleus U such that 1 UtUC2~U. Moreover, when A is s e m i - n o r m e d we can even choose the U so that it is a subsemigroup of A.

PROOF. If A is a p - s e m i n o r m e d algebra (A,p) we may assume t h a t p is sm. (see 3.4.4). By 4.1.13, U - B1 is an open absolutely p-convex nucleus which is a subsemigroup. U is, moreover, b o u n d e d (by 3.2.13). Finally, if x , y E U then by p -convexity, x Yi 1 ~-~EU, 2~ 2~

so t h a t U §

1

Conversely, suppose that U is an open nucleus having the stated properties. Then, by 4.2.2,

v-

Ic l(u) c

1

(,)

and by 4.1.6, V is an open nucleus. The boundedness of U together with the inclusion (,) implies t h a t V is bounded. Since V is absolutely p-convex the gauge p - Pv is a p - s e m i n o r m (by 4.1.10). By 4.1.12(ii), Y - B1 - {x E A " p(x) < 1} and hence (.) gives" 1 - T V ___ U ___V - B1. (**) 2~ Since U , V are bounded nuclei it follows from (**) t h a t the topology of A coincides with the p-topology, whence A is pseminormed. 4 . 2 . 4 . COROLLARY. Every locally boundedt n o r m e d algebra for some p, 0 < p <~ 1.

TA is a p - s e m i -

t A TA is said to be locally bounded if it is locally bounded as a TLS (i.e. it has a bounded nucleus).

188

Locally Pseudo-convex Spaces and Algebras

PROOF. This is an immediate consequence of 4.2.1, 4.2.3. 4 . 2 . 5 . R e m a r k . Rolewicz [12 ~] has shown t h a t to each locally bounded TLS X there is a largest number p0,0 < p0 ~ 1, such t h a t if 0 < p < p0 then the topology of X can be induced by a p-seminorm. 4 . 2 . 6 . COROLLARY. A TA A is a semi-normed algebra iff it has a bounded convex nucleus U. PROOF. If A ---- (A, P) is semi-normed we can take U B1 = {x E A : p(x) < 1}. On the other hand, if A has a bounded convex nucleus U then we have, by 4.1.5, U + U - 1. U + 1 - U - (1 § 1)U - 21U. By 4.2.3, A is semi-normed. 4 . 2 . 7 . COROLLARY (Kolmogorov t ). A Hausdorff TA is a normed algebra iff it has a bounded convex nucleus. PROOF. This follows from 4.2.6 since the topology induced by a semi-norm p is Hausdorff iff p is a norm. 4 . 2 . 8 . PROPOSITION. An infinite direct product of locally bounded ttausdorff T L S ' s - in particular such a product of TA's cannot be locally bounded. PROOF. Suppose X - - I I X a , where {Xa} is an infinite family of locally bounded Hausdorff TLS's. Denote by 7r~ the factor projection: X ~ X~. Then sets of the form U -- 71~1(Uc~1) C I ' ' " N 7t--l(uc~n)c~.

where U~j run through an open basis of X~j, form a basis of nuclei for X. Fixing a U, let /3 be an index with fl 7s a l , ' " , an and x =/= 0 an element of the space Xt~. Then the sequence ( n x ) ( n = 1, 2 , . . . ) of elements clearly belong to U (after suitable

Actually Kolmogorov proved the following result: A Hausdorff TLS is normable iff it has a bounded convex nucleus.

w3. Locally Pseudo-convex Spaces

189

identification?). Since _1. nx - x ~ 0 this sequence does not n converge to 0 and consequently U is not bounded. Since U is an a r b i t r a r y basis nucleus we conclude t h a t X is not locally bounded. 4 . 2 . 9 . COROLLARY. A direct product of p-seminormed algebras is p-seminormed iff it is a 'finite direct product'. PROOF. If A - - A 1 • " " • An, when A j - - ( A j , p j ) are ps e m i n o r m e d algebras then A is p - s e m i n o r m e d under q : q ( x ) m a x j p j ( x ) , where x e A, x - ( x l , ' " , x , ) , x j e Aj (cf. 3.3.11). Conversely, if a direct product A of p - s e m i n o r m e d algebras is p - s e m i n o r m e d then A is locally bounded (by 3.2.14) and hence by 4.2.8, A is a finite direct product.

w 3.

Locally

Pseudo-convex

Spaces

4 . 3 . 1 . DEFINITION. Following Waelbroeck [29,p.4] we call a subset S of a LS X (over ~:) pseudo-convex if it is p-convex for some p (0 < p ~< 1). A TLS X is called locally pseudo-convex if it has a basis {Us} of pseudo-convex nuclei Us; if Us is p~convex we also say t h a t X is locally {p~)-convex. If all the p~ (some) p then X is called locally p-convex, and if p -= 1 then X is called locally convex. A p - s e m i n o r m p is also called a pseudo-seminorm and p is called the homogenity index of the pseudo-seminorm p (cf. 3.2.1). 4 . 3 . 2 . If P is a family of pseudo-seminorms on a LS X, the P - t o p o l o g y (see 3.1.7) makes X a locally pseudo-convex space ( X , P ) ; if P - {p~) and B ~ {x e X 9 p~(x) < r} (r e ~ , r > 0) then the family of all finite intersections of the B~ ( a , r varying) give a basis of pseudo-convex nuclei for ( X , P ) . Conversely, if X is any locally pseudo-convex space with {Us} as a basis of pseudo-convex nuclei t h e m the gauges p~ associated with the U~ determine a family P - - { p ~ } of pseudo-seminorms Pa.

~. The element xt~ E Xt~ is identified with the element x - (x~) in X suchth~t x ~ = 0 if c ~ # p , x ~ - x ~ if ~--fl.

Locally Pseudo-convex Spaces and Algebras

190

4 . 3 . 3 . Let G* denote the set of all pseudo-seminorms on a LS X. We introduce an order -< in G* by writing i

I

p -< p' if p~ < p ;

1

I

(i.e. p(x)-; <<.p ' ( x ) 7 Yx E X ) and p' ~< p,

where p,p~ c ~* are respectively a p - s e m i n o r m and a p~seminorm. Clearly, the relation -< is reflexive and transitive. It is 1 pl ~ pl also anti-symmetric, for, i f p - < p ' , p ' - < p then p ~ 7, p, so t h a t p - p~. Thus -< is a partial order. We also note t h a t if p -< p' then for 0 < e ~< 1, Bep' _c B p (since if p'(x) < e then p

p(x) .< p' 4.3.4,,

< LEMMA.

The poset 0" -

(~*,-<)

is closed for all

finite lattice sums. PROOF. If pj C G* and pj is a p - s e m i n o r m ( j -

1,..-,n)

_R_

, - P 0j ;q(x) - max py(x). , set p - min{py} PY By 3.2.9, py' is a Y ' 1 p - s e m i n o r m and so by 3.2.3, q is a p-seminorm, so t h a t q E G*. It easily follows from the definition of q t h a t q >- py, and t h a t if p t E G * , f f ;>- p y for all j then p' >- q. So q is indeed the lattice sum plV'"Vpn.

4 . 3 . 5 . LEMMA. If pj e and O< c <~ 1 then

(j-

1, 9. . , n ) , q - -

plV'"VPn,

B q C_ B E p ~ N . . . N Bep"-

PROOF. Since py -< q we have B e _ BeP~ (see 4.3.3). The required result is now clear. 4.3.6. then

LEMMA.

1, . . . , n)

If pj,py' C G * ( j !

P l V . . . V P n "" P l V . . .

I

V Pn"

PROOF. By (**) of 3.2.11 we have oy i.~1

I

py <~~jpy

orJ

i I

Py

,pj <. c~py

and PJ "~ PY'

191

w 3. Locally P s e u d o - c o n v e x Spaces

Set C ~ - m a x C~., p - min pj. Then p -7-

Pl V . . . V Pn

=

max pjP1 ~< C t max pjt p1

~< C'(plV...Vp',, 1. Similarly, Pl V . . . V p~ <. C ( p l Hence the lemma.

V

" " " V

Pn), where C - m a x Cj.

4 . 3 . 7 . DEFINITION. A family P of pseudo-seminorms is called saturated if it satisfies the condition Pl,''',Pn

E P ::~ pl V ' ' ' V

pn E P.

(*)

For any subset P of O* we set P -- {q E 6* " q - - p l

V . . . V p n , p j E P}.

The set P clearly satisfies the condition saturated closure of P.

(.)

and is called the

4 . 3 . 8 . LEMMA. (a) P, P induces on X the s a m e topology. (b) If Po is a cofinalt subset of P (with respect to the order in 4.3.3) then Po induces on X the same topology as P. ( c ) / f P is a saturated f a m i l y then the f a m i l y { B p " p E P, e > 0} is a basis of nuclei. PROOF. (a) It is sufficient to observe t h a t p(x~ - x) --~ 0 for each p c P iff q ( x ~ - x ) - - ~ O for each q c P. (b) It is enough to note that i f p - 4 p 0 (PEP,P0EP0) and B-

{x C X'p(x)

< r}, B o -

{x E X ' p o ( x )

< r'}

PO

where r t - r-i-,p o (respy. p) being the homogenity index of p0 (respy p.), then B0 __c_B. (c) This follows from 4 93.5 and the simple fact t h a t B ~ uC B rp2 if 0 < r l < r2.

t i.e. given any p E P

there isa poEPo with p-<po.

Locally Pseudo-convex Spaces and Algebras

192

4.3.9,, Let P be a family of pseudo-seminorms on a LS X. A subset S of X is said to be P -bounded if the numerical functions p(x)(p e 2) are bounded on S (i.e. plS is bounded for each p e P ). The meaning of p - b o u n d e d where p is a pseudo-seminorm is clear.

4.3.11}. LEMMA. Let X - (X, P) be a locally pseudo-convex

space. A .subset S of X is (t.) bounded iff it is P-bounded. PROOF. Let S be bounded. If possible let p~lS be not bounded for some pa in P. Then we can find a sequence (xn) in 1

S with p,~(xn) ) n. Writing Yn - n p~xn we have: P~(Yn) ) 1. It follows t h a t yn ~ 0 in X, contradicting the boundedness of S. Conversely, suppose now S is such t h a t pals is bounded for all Pa. Then, for ( x , ) in S,A,~ ~ 0 we have pot(Anxn) [A,~l;~p~(x~) -+ 0, since the sequence is bounded and pa > 0. Therefore S is bounded. -

-

Let X - (X, P), Y - (Y, g)) be locally pseudo-convex spaces, with P saturated. Then, a linear transformation T " X --~ Y is continuous iff for each q~ E ~_ there is a pa E P and a constant C - C a ~ > 0 such that 4.3.11.

PROPOSITION.

p~

(,)

where pa,p~ are respectively the homogenity indices of pa,p~. PROOF. The condition (.) clearly implies continuity of T at 0 and and hence also everywhere. Conversely, if T is continuous then (by continuity at 0), given q/~ and E > 0 there is a pa and a fi > 0 such that

(**)

p~(x) <~ ~ ~ qz(Tx) <~ e. 1

For an x with p ~ ( x ) ~ O, write X1 - - ( ~ p a ( x ) - I ) - A - ~ X . Then p~(xl) -- ~;, so that by (**) we get q~(Txl) <~ e which reduces o~ to (,) above will C - e~ P~. If p~(x) - O, we can argue exactly

w 3. Locally Pseudo-convex Spaces

193

as in the proof or 3.2.10 and show t h a t q~(Tx) - 0, so t h a t (.) holds trivially in this case. 4 . 3 . 1 2 . LEMMA. Let X be a TLS, Y - (Y,~_) a locally pseudo-convex space, T : X --+ Y a linear transformation. For each qz E D we set p*~(x) - q~(Tx). Then p*~ is a pseudos e m i n o r m on X with the same homogenity index as q~. The transformation T is continuous iff all p*~ are continuous. PROOF. We have

+ y) - qz(Tx + Ty) <.

+

Further, if a is the

homogenity index of qz then: pz(Ax)* - q~(TAx) - q z ( A T x ) -- IA] p *p~(x). To prove the criterion for continuity of T it is enough, in view of 2.1.26, 3.1.3, to consider the continuity of both at 0. Now T is continuous at 0 iff: x~ -~ 0 in X ==~p*~(x~) - q~(Tx~) ~ 0 (for all qt~ ), iff all p} are continuous at 0. This proves the lemma. 4 . 3 . 1 3 . LEMMA. Let X - (X, P) be a locally pseudo-convex space, where P is saturated. Then: (i) For any continuous linear functional f on X there is a p~ E P such that f is p a - c o n t i n u o u s , i.e. f is a continuous functional of the p s e u d o - s e m i n o r m e d linear space

(ii) If p is any continuous p s e u d o - s e m i n o r m on X is a pa such that

then there

p <~ Cp~ ~, where C is some constant > 0 and P, Pa the homogenity indices of p, pa.

PROOF. (i) By 2.1.31, there is a p~ and an e > 0 such t h a t pa(x) < e ==~ [f(x)t < 1.

It follows from 3.5.4 t h a t f " ( X , p ~ ) ~ K is n. bounded, whence by 3.5.5, it is continuous.

Locally Pseudo-convex Spaces and Algebras

194

(ii) The identity m a p : (X, P ) ~ ( X , p ) is continuous by hypothesis on p. So, by 4.3.11, there is a C > 0 and a pa with _2__

as desired. 4 . 3 . 1 4 . DEFINITION. Let X -- ( X , P ) be a locally pseudoconvex TLS - or more generally a TLS with its topology induced by a family P of quarter-norms. A m a p

T : PT(C_ X ) - ~ X where PT is the d o m a i n of T, is called a contraction if we have for each p a E P a c o n s t a n t ea with 0 < c a < 1 such t h a t for all x, y C P T w e h a v e

p~(Tx-

Ty) <<.e ~ p ~ ( x - y).

Note t h a t a contraction m a p is automatically continuous. 4.3.15. THEOREM (Contraction m a p p i n g principle). Let X : (X, P) be a complete (or even sequentially complete) Hausdorff TLS, with P a family of p - s e m i n o r m s (or more generally quarter-norms). Let T " S --~ S be a contraction, where S is a closed subset of X. Then T has a unique fixed point x* E S with x* -- l i m T n x 0 , where xo is an arbitrary point of S. PROOF.

Define X1 ---Txo,xn - - T x , ~ - I : T'~xo. If m < n

then

p~(x,~ - x , ) = p~(Tmxo - T~xo) <~ (e~)mp~(xo - Xn-m) (Ea)m[pa(Xo (Ea)mpa(Xo

---

Xl) -[-''.-[- p a ( X n _ ( m _ l ) Xl)[1 Jr- Ec~-Jr-'''E~ -m]

--

Xm_l) ]

1

< (cc~)mpa(xo -- X l ) l -- c~ It follows t h a t p ~ ( X m - x , ) -~ 0 as m , n ~ c~ (for each a ) . By completeness of X the sequence xn --~ some x* E X; since xn E S and S is closed, x* E S. By continuity of T

w4. Locally Pseudo-convex Algebras we have"

Tx*-

lim T x n .

195

By the construction of T we have

n---+ OO

also

9

lim T x ~ ~t----~ O O

lim X~+l

--

X*

. Therefore, since X is Haus-

~t---~ OO

dorff, Tx* -- x*. Thus x* is a f i x e d point of T. If T y have

y, we

p~(x* - y) - p~(Tx* - T y ) ~ e~pa(x* - y). Since ca < 1 we must p ~ ( x * - y ) - 0 ( X being Hausdorff).

for all p~, whence x * - y

4 . 3 . 1 6 . LEMMA. A locally pseudo-convex space X - (X, P) is locally connected. PROOF. Let P be the s a t u r a t e d closure of P, so t h a t we have also X - ( X , P ) . Since the open balls B~ - {x E X ' p ~ ( x ) < r}(p~ E P) form a base of nuclei the lemma follows since each B~ is connected (see proof of 3.2.8). w4.

Locally

Pseudo-convex

Algebras

4 . 4 . 1 . DEFINITION. A TA A is called locally pseudo-convex if it is locally pseudo-convex as a TLS. This means t h a t a locally pseudo-convex algebra A is of the form A - - ( A , P ) , where P is a family of pseudo-seminorms. If each pa E P is sm. then A is called locally sin. pseudo-convex algebra. Again, if each pa is a semi-norm, A is called a locallyt sin. convex algebra. 4 . 4 . 2 . LEMMA. If A is locally sm. pseudo-convex then it has a basis consisting of pseudo-convex subsemigroup nuclei. Conversely, if A is a TA which admits such a basis then it is locally sm. pseudo-convex. PROOF. If A = ( A , P ) with each pa E P sm. then, by 4.1.13, B~ = {x e A : p~(x) < r } , 0 < r ~< 1, are pseudo-convex subsemigroup nuclei. On the other hand if A has such a bais then the associate gauges pa are sm. pseudo-convex (by 4.1.12 ((i), (iii))). t This is the same as locally m. convex algebra in the terminology of Michael [20].

196

Locally Pseudo-convex Spaces and Algebras

4 . 4 . 3 . L EMMA. If P l , ' " P n are sin. P i - s e m i n o r m s (j -1 , . - . , n ) then q - pl V . . . V pn is a sin. p - s e m i n o r m with p rain{p:-}, where pj is the homogenity index of pj. _L

PROOF.

By 4.3.4,

q(x) -- m a x p i ( x ) PJ, and J seminorm. Further, we have oK_

~

q is a p-

__P__

q ( x y ) - m.~xp;(~y)o; .< m ~ x p ; ( ~ ) o ; p ; ( ~ ) I o ; - q(~)q(y), :

j

whence the lemma. 4 . 4 . 4 . PROPOSITION. Let A be an algebra (over K) P a family of pseudo-seminorms on A, and P the saturated closure of P. Then A is a TA under the P-topology iff for each. pl E P there is a Po C P and a C > O such that I

pl

pl

p (~v) <~cpo(x) Tpo(y)7

(,)

(~, v ~ A)

where pl, po are respectively the homogenity indices of p~,po. PROOF. The proof is somewhat similar to that of 3.4.3. First suppose that (.) holds for each p~ E P. Assume t h a t in A, x~ --~ x, y~ --~ y. Then from the identity

we obtain, using (.) and the subadditivity of pl, the inequality: pl

~, ~, +po(~) ~ p o ( y ~ - y) ;o + p o ( ~

pl

~, ~, } - ~) ;o ;o(y)oo .

It follows t h a t p ~ ( x ~ y p - xy) --+ 0, for every p~, whence x ~ y ~ - ~ x y , and ( A , P ) i s a T A . Next suppose t h a t (A, P) is a TA. Since the m a p (x, y) ~ xy is continuous, given pS E P and e > 0 then is a p0 E - P and a 5 > 0 such that (1)

197

w4. Locally Pseudo-convex Algebras 1

For x , y

~ A with po(x),po(y) ~ O, set

1

Xl

--

1

t~PX/po(X)

P---~

1

Yl -- ~ o(---~y/po(y)~o where P0 is the homogenity index of p0. Then po(xl) - - 5 - - PO(Yl), whence by (1), p ' ( x l y l ) < e which reduces to

Cpo(x)

oa

~

opo(y)oo

(2)

where C - E/5 2p'Ip. It remains to consider the case where po(x) or po(Y) - O. Suppose t h a t po(x) - O. Then po(n2x) - 0 for any integer n ~< 1. For arbitrary y c A we can choose n (large enough) t h a t

, y

p'(y)

n

rt pl

"

Then, by (1), p ' ( n x y ) - p ' ( n 2 x . y / n ) <. e which gives nP'p'(xy) <~ e. Since n is arbitralily large we get p'(xy) - O. Similarly p'(xy)-0 if x is arbitrary and P o ( y ) - O. Thus ( 2 ) h o l d s for all x , y as required. 4.4.5. COROLLARY. Let A -- (A, P) be a locally pseudoconvex algebra. Then for each pl E P we can find a pseudos e m i n o r m p* on A such that pl

pl

(**)

P* "~ Pl V . . . V Pn for some pj E P. 4.4.6. COROLLARY. If P is any family of sm. pseudos e m i n o r m s on an algebra A then under the P-topology A is a locally (sm.) pseudo-convex algebra.

PROOF. By sm. property of p~ E P we have p a ( x y ) p ~ ( x ) p ~ ( y ) and the condition (,) of 4.4.4 evidently holds.

<~

4.4.7. LEMMA. Let A = ( A , P ) be a locally pseudo-convex algebra. If P - {p~} and p o - - i n f p ~ > 0, when p~ is the homogenity index of pa, then A is a locally Po-pseudoconvex algebra. PROOF.

By 3.2.9, qa - p~O/p, is a po-seminorm such t h a t

198

Locally Pseudo-convex Spaces and Algebras

q~,-~p~. Thus A - - (A,Q) where !2 = {q~} i s a f a m i l y of p0seminorms, whence A is a locally p0-pseudoconvex algebra. 4.4.8. PROPOSITION. The locally {pa}-convex algebras A are precisely the TA 's of the form A = (A, P ), where P is a family of p - mi orm o= A ati4yi g (,) of 4.4.1. Moreover, A is Hausdorff iff ker P -- {0}. PROOF. The first statement is essentially a restatement of 4.4.4. The second statement follows from 3.1.21 (iii). 4.4.9. DEFINITION. A family P of pseudo-seminorms on an algebra A is called well-behaved if fi is saturated and further for each p ~ E P there i s a p E P such that pl

pl

7p(y) 7

(**)

where pt,p are respectively a pt-seminorm and a p-seminorm. 4.4.10. PROPOSITION. The locally pseudo-convex algebras are precisely the TA', A of the from A - (A,P), where P is a well-behaved family of pseudo-seminorms. PROOF.

If P is well-behaved then by (**)of4.4.5 and (,) of 4.4.4, A is a TA (under the p-topology) which by 4.3.2, is locally pseudo-convex. Next assume that A is a locally pseudo-convex algebra. Then, by 4.4.4, A - (A,P) with P satisfying (,) of 4.4.4. Let P denote the saturated closure of P and write - { p E G* "p,,,qE--fi} where G* denotes the set of all pseudo-seminorms on A. Then, by 4.3.6, ,P is saturated. Further, by 4.4.5, ,P is well-behaved. 4.4.11. LEMMA. Let A bra such that:

(A,P) be a locally p-convex alge-

(i) P - {p~} is a well-behaved family; (ii) s u p p a ( x ) < oo for each x e A. Pa

w4. Locally Pseudo-convex Algebras

199

Then p ( x ) - sup pc~(x) is a sm. p-seminorm and (A,p) is a pP~

seminormed algebra. PROOF. Since P is well-behaved, for each pa E P there is a p/~ E P such t h a t

pc~(xy) <~p~(x)p~(y) <<.p(x)p(y). Taking the s u p r e m u m of the first term over all a we get

p(xy) <~p(x)p(y). 4 . 4 . 1 2 . PROPOSITION. Let A - (A, P) be a locally pseudoconvex ( respy, locally sm. pseudo-convex) algebra. Then: (i) Any subalgebra Ao of A under the relative topology is locally

pseudo-convex (respy. locally sin. pseudo-convex). (ii) If I is a bi-ideal of a

and A # - A / I , then A #, under the quotient topology, is locally pseudo-convex (respy. locally sin. pseudo-convex).

PROOF. (i) We may assume t h a t P is well-behaved. For p~ C P write P ~ o - P~IAo (restriction to a0) and P 0 - {P~o}. Then it is clear that (Ao, Po) has the required properties. (ii) Let p~ be defined in terms of Pa as in 3.4.15 and write P # - {p~}. Then ( a #, P # ) is the quotient algebra and this has the required properties (the proof t h a t P # induces the quotient topology is similar to that of 3.1.22 (iii)). 4 . 4 . 1 3 . PROPOSITION. The unitization A1 or the complexification A of a locally pseudo-convex or locally sin. pseudo-convex algebra A is again an algebra of the same type. PROOF. If A -- (A, P) then for each p E P denote by Pl the canonical extension of p to A1 as defined in 3.4.11 and by /5 the extension of p to A as defined in the proof of 4.1.16. Write N

Pl -- {Pl " p E P } , Then n I -- (AI, Pl), A -

P -- { p ' p E

P}.

(A, P) have the stated properties.

4 . 4 . 1 4 . LEMMA. Let A - (A,P) be sequentially complete-in particular complete-Hausdorff locally sm. p-convex algebra. If for

200

Locally Pseudo-convex Spaces and Algebras

an x E A we have u~(x) < 1 for all uc~ co

then

x

is q. invertible,

~

( - 1 ) '~xn

converges in

A

and the

n--1

q. inverse co

x'n=l

PROOF. As in the proof of 3.3.19, we obtain for each p~, co

p (x

<

n--1

So

N+r ) N+r P a ( n~N ( -- i ) n x n <~ ~-~ P a ( Xn) --+0, N

as

N , r --+ c~ .

co

This means t h a t

~ - ~ ( - 1 ) n x n converges absolutely and so also n--1

in

A.

Exactly

as in the proof of 3.3.19 we can show t h a t

co

-

x'.

r~--1

4 . 4 . 1 5 . PROPOSITION. A locally sm. pseudo-convez algebra (or more generally a TA whose topology is induced by a family of a. sm. quarter-norms) is a C algebra. PROOF. Let A - ( A , P ) , where P - {pa} and each a. s m . . Then, by 3.3.22, each pa satisfies the inequality (.) therein. It follows t h a t if (each) p a ( x n - x ) --+ 0 ( x ~ , x then p~(x'~ - x I) -+ O, which proves t h a t the m a p x ~-+ x I is continuous, so t h a t A is a C algebra.

A-

4.4.16. (A,P),

p~ is stated e Gq) of Gq

LEMMA. In a locally sm. pseudo-convez algebra where P is saturated, for any continuous pseudo-

w 5. Projective Limit Decompositions

201 1

1

s e m i n o r m p on A there is a p~ E ? such that v~ <~ vi~~, where P, Pn are homogenity induces of p, pn respectively. ~p

PROOF. By 4.3.13 (ii) there is a pa with p <~ Cp~ ~ . Therefore

Up (x) -

lim (p(x'~ ) ) -~-

~--~ O0

lim C ~ ~ [ l i m p ~ ( x ~ ) ~~] _e__ o~

<~

n---+ O0

Ln---~ O0

P

1

1 p~

so t h a t v~ ~< vn , as required. w 5.

Projective

Limit

Decompositions

4 . 5 . 1 . Let {A~ : a E ~ } be a family of TA's with the indexing set .4, a poset directed above by -< . Suppose t h a t for each pair (c~, fl) of elements of .4, with a -< fl, there is given a continuous h o m o m o r p h i s m ~p~Z : A z --. A~, such t h a t (i) p ~ = 1A~ (ii) p ~ o ~ for c~ -< /3 -< ~/. Write A-

the Cartesian p r o d u c t

17I An. nCA

Under coordinate-wise operations A is an algebra. Set

Ao -- { x -

(x~) C A " ~ Z ( x ~ ) -

x~ whenever ~ -4 fl}.

A0 is called the projective limit of the A 0 - - 1 i 4--mAn, x--lim xn (xEA0). l----

An

and we write

The projective limit A0 has the following universal mapping property: for any TA B and continuous h o m o m o r p h i s m s COn " B - + An such t h a t ~n~ o c o Z - con (whenever a - < /3), there is a unique continuous h o m o m o r p h i s m 0 9 B ~ A0 such t h a t ~rn o 0 - con (a E ~q), where r n denotes the n a t u r a l projection A --+ A~. (It is easy to see t h a t 0 is given by 0(b) - (co~(b)), for

boB) 4 . 5 . 2 . LEMMA. Ao is a closed subalgebra of A. PROOF. By 2.2.8 (b), A is a TA, and since ~ Z

are homo-

Locally Pseudo-convex Spaces and Algebras

202

morphisms, A0 is a subalgebra of A. It follows from the continuity of ~ that A0 is closed in A. 4.5.3.

THEOREM (Michael t ). Every complete Hausdorff lo-

cally sm. pseudo-convex algebra A is ( t. isomorphic to) a projective limit, l i m A a of pseudo-Banach algebras tt Aa. In particular, +.._

if A is locally p-convex, then A - l i m A , , Banach algebras. Conversely, a projective limit B -

limBa

with all fia as pof pseudo-Banach

algebras Ba, is a complete Hausdorff locally sm. pseudo-convex algebra. PROOF. Let A - ( A , P ) , where we may assume that P = {p~} is saturated. Set A ~ - A/N~, where N ~ - kerp~. Then A ~ - ( A ~ , p ~ ) i s a p~-normed algebra (p~ being the homogenity index of p~) whose completion A~ is, by 3.4.17, a p~-Banach algebra. Make the indexing set .4 - {a} a directed set (directed above) by defining a -< ~ if pa -< p~, where the latter -< is the order introduced in 4.3.3 (the directedness of the order is a consequence of the family P being saturated). If a -< fl, it is clear that N~ _C N~, where N ~ kerp~, N~ - k e r p . Consequently, the map ~ 9xz - x + NZ ~ xa - x + Na (x C A) is well-defined and is a continuous homomorphism of A~ onto An, which can be extended, by 3.5.11 (b), to a homomorphism ~b~ of A~ into An. It follows that

is a projective system of pseudo-Banach algebras which defines the projective limit t Michael [20, p.17.] established the theorem for the particular case of the locally sm. convex algebras but his method of proof extends to our general case. tt

A pseudo-Banach

algebra is lust a p-Banach

algebra for some

0 < p ~ I. This algebra is not to be confused with the pseudo-Banach introduced

by Allen-Dales-McClure

limit of Banach

algebras.

who

p,

algebra

use the term for a certain inductive

203

w 5. P r o j e c t i v e L i m i t D e c o m p o s i t i o n s

Ao- lim A~ __ 1-I A~" Similarly, {As " 99~f~} gives rise to the projective limit A0 - lim Aa, which is identifiable as a s u b a l g e b r a of Ao. T h e m a p ~p 9A -~ Ao given by ( ~ ( x ) ) ~ - x § N~ - x~ is a h o m o m o r p h i s m , since each of the m a p s ~ 9x ~ x ~ ( a E A) is a h o m o m o r p h i s m . It is 1-1 since x~ - 0 for all a ==~ x e n N~ - {0} (since A is Hausdorff). F u r t h e r , ~p is continuous since each !Pa is continuous. We now show t h a t ~ is "onto". Take any element y - lim ya in Ao. For any finite subset S of A, choose a fl E A such t h a t fl >- a for all a E S. Take an element x s E A such t h a t ( ~ ( x s ) ) z - y~; t h e n ( ~ ( a s ) ) ~ - y~ for all x e S. Also, given finite subsets $1,

$2 with $ 1 ~ $ 2

~ O, we have

( t o ( x s 1 ) ) ~ -- y~ - (99(xs2))~ for all a E S 1

N

This relation implies t h a t XSl - XS2 E N o C_ U~,E - { x E A " p ~ ( a ) < e }

(e>0).

It follows t h a t ( x s ) - S varying a m o n g all the finite subsets of - is a C - n e t . Since A is c o m p l e t e xs ~

(some) x E A.

Since ~ is c o n t i n u o u s we o b t a i n y~ - - ( g 9 ( x s ) ) ~

( S varying)

--, (~p(x))~ -- x~,

i.e. y~ -- x~, so t h a t y - ~p(a), proving t h a t ~p is "onto". We n e x t show t h a t ~ - 1 is continuous. Consider a net (x (~)) in A such t h a t

c A) T h e n by 3.4.16 we have p a ( z (~) - x) -- p#a (x ('x) - x a ) --+ 0 for all a.

Locally Pseudo-convex Spaces and Algebras

204

This means that x (~) -~ x in A, proving the continuity of ~9-1. To complete the proof of the first statement of the theorem it remains to show that A0 - A0. Now any non-empty open set Ifd of A0 contains an open set V of the form

where ( ~ j

N...

N, o,

are open sets in fi.~j and r--1 the natural projection

~rA~-+ A~j ( j - 1 , - - . , n ) choose an (Xrt+l E .~ with an+l >" a j ( j = 1 , . . . ,n). Then we have the homomorphisms

~j,Olnj_l

"

(j= 1,...,n).

l~Olnj_l ~ l~Otj

Set _

,,-1

Gc~,~+l -- ~1,~,~+1 Then

Set

~n,~n+l

~o,j,o,,,+, (G~,,+,) ___r C~rtA-1

n "

(j= 1,...,n).

c~n+l

~

9

Since A~.+I is dense in A~.+~ we can find x E A so that x~.+~ E (~,~+~. It follows that ~(x) E U, so that U n A 0 # 0. Therefore A0 is dense in A0. But A0 is complete, being t. isomorphic to the complete space A and consequently closed. Thus, A0 - fi-0. The second (or converse) statement follows since every pseudoBanach algebra is a complete Hausdorff locally sm. pseudo-convex algebra and these properties carry over to direct products and to closed subalgebras. 4.5.4. For descriptive convenience we call a complete Hausdorff locally sm. pseudo-convex algebra A as a pseudo-Michael algebra. If it is a locally sm. p-convex algebra then it will be referred to as a p-Michael algebra. Finally, if p - 1, A is called

w5. Projective Limit Decompositions

205

a Michael t algebra. If completeness hypothesis is dropped in any of the above definitions then the resulting notions will be indicated by employing pre-Michael instead of Michael in their names 4 . 5 . 5 . COROLLARY. Every pseudo-pre-Michael algebra is a dense subalgebra of a projective limit lim/}~ of pseudo-Banach 4--

algebras [~s. PROOF. of A.

It suffices to apply 4.5.3 to the completion

B -

4 . 5 . 6 . PROPOSITION (Michael). Let A -

(A, P) be a pseudoMichael algebra with projective limit decomposition A--limzi.s (A~ a pseudo-Banach algebra). For x E A, x--limxs (i) x -

(xs C fis) we write x -

(xs). Then:

(xs) is q. invertible in A iff xs E fis is q. invertible

(for ach (ii) A is unital iff all ft~

are unital, and then the unity e (es), es being unity of fis. (iii) u = (u~) is idempotent in A iff for each a, u~ is idempotent in fis. (iv) If A is unital, then an element x - - (xs) is invertible iff for each (~, xa is invertible in fia. PROOF. (i) If x is q. invertible then, by 1.1.24, xs is q. invertible in As and so also in As _~ As. Conversely, if for each a, x s , has q.i. xs, ' then ( x ~ ) C A To see this, note t h a t i f / ~ > a , ~9sZ ..3,Z --~ As the element !psf~(x~) is q.i. of ipsz(xf~) - xs, so that ~f~(x~)-

x~. ' It follows t h a t

~(x~) e limfi,~ - A, whence

there is an element x I E A with (xl)s - x s. Since multiplication and addition are component-wise it is clear t h a t x ~ is q.i. of x. (ii), (iii): These follow at once since multiplication in A is component-wise. t Such algebras h~ve ~lso been considered by Areas.

Locally Pseudo-convex Spaces and Algebras

206

(iv) The proof is similar to that of (i). 4.5.7. COROLLARY. For x -

(x(2) E A we have:

(1 / a~(x) -- Ua2~(x(2); c~

(2) rA(x) -- sup rA,(xa); c~

(3) aA(X) -- U a~i~(x~) whenever A is unital. cg

PROOF. By 1.7.8 A(r 0) E a ~ A ( X ) i f f - - A - i x is not q. invertible, i.e. by 4.5.6 (i), iff for some a, - A - l x a is not q. invertible in A~ iff A E a'^ (xa) This proves (1) and (2)readAs " ily follows from (1). For (3), assume now that A is unital; then each .1i(2 is unital. From (1) and 1.7.21 we obtain (x)U(0)

-

9

If 0 ~ aA(X) then x is invertible and consequently each xa is invertible, so that 0 ~ UaA~(x~). On the other hand, if 0 e c~

aA(X) then x is not invertible and consequently, some x(2 is not invertible so that 0 E aA1 (x(2). Thus, the equality in (3) holds in both cases.

4.5.8. COROLLARY. PROOF. bi-ideal. So

N~I(~/Aa)

_ x/~-- N~9~l(v~a).

(2

(2

By 1.2.17 (i), each ~ I ( ~ A ~ ) o r

(2

(fi)~l(~~a) is a

(2

are bi-ideals of A. By 1.2.26, ~ ( V ~ ) C V~(2. So x / ~ _ I. On the other hand, if x E I then x(2 E V~(2 and so x(2 is q. invertible. Consequently, by 4.5.6 (i), x is q. invertible so that I is q.i. bideal, whence I ___ x/~. Thus x / ~ I. Again, if x E J then

w6. Metrizable Locally Pseudo-convex Algebras

207

x~ E V//i~, x~ is q. invertible and consequently x is q. invertible. Thus, J is a q. invertible ideal whence J c_C_x/~. w6.

Metrizable

Locally

Pseudo-convex

Algebras

4.6.1. PROPOSITION. Every first countable locally pseudoconvex algebra A is semi-metrizable and has the form A - (A, P), where P - { p l , p 2 , ' " } is a countable well-behaved family of pseudo-seminorms such that Pl ~ P2 -< "'" t

(,)

PJ p j ( x y ) ~ Pj+I (X)~176

Pj+I

(j1 , 2 , . . . ) , where pj, indices of Pj,Pj+I.

are

Pi (**)

respectively the homogenity

PROOF. That first countability implies semi-metrizability lows from 2.1.7. Since A is first countable its topology is duced by a countable family Q - {q,} of pseudo-seminorms. P ~ n - ql V . . . V qn and P ' - { p ~ } . Then p~ -< p~ -< . . . . PI --P~. By 4.4.4, there an integer i(1) such that

folinSet Set

Pl (xy) -- pl1 (xy) ~ C l p ~ ( 1 ) ( x ) ~o, pi(1) I (y)-~P'

where Pl, p, are homogenity indices of PI,P~(1) and C1 a constant __E_

which we may clearly suppose /> 1 9 Set p2 - C~~ i'( 1 ) "

Then

P2 ~" P'i(1) and satisfies pl

pl

For P2 we can similarly choose P~(2) ~ with i(2) /> i(1) ~ 2 such that C

!

! 1

1"

t Recall that pj -K Py+l means: pjoy ~ pyPiu

and py+l ~ py.

208

Locally Pseudo-convex Spaces and Algebras

where we can assume t h a t C2 ) C1

9

Set p 3 -

C 2"~2 v"~ i(2)

and we

have

Thus proceeding we get ~3 _ { P l , P 2 , " ' } with the desired properties. It is clear from the way we constructed t h a t P is equivalent to a cofinal subset of P~ ~ whence ~3 ... pt ..~ Q 9 So A - (A, P), completing the proof. 4.6.2. DEFINITION. A first countable Hausdorff locally pseudo-convex algebra is called a pseudo-pre-Frechet algebra or a pseudo-pre-~ algebra. If A is also complete it is called a pseudoFrechet or a pseudo-~ algebra. 4.6.3.

DEFINITION. A p s e u d o - p r e - ~ algebra A is called a locally sm. p s e u d o - p r e - ~ algebra if its topology can be induced by a countable family Q = {qn} of sm. p n - s e m i n o r m s for some p,~ with 0 < p n ~ < 1, n = l , 2 , . . . . 4 . 6 . 4 . LEMMA. Every pseudo-pre-~ algebra A is of the form

A = (A,~_) where ~_ = {qn} is a countable family of pseudoseminorms with ker Q = {0}. If A is locally sm. pre-pseudoalgebra then we can choose ~_ such that each q E ~_ is sin.. PROOF. This follows from 4.6.1, 3.1.21 (iii). 4.6.5. DEFINITION. If in a pseudo-pre- ~ algebra A = ( A , Q ) , all q E Q are p - s e m i n o r m s for some fixed p ( 0 < p ~ < 1) then A is called a p-pre- ~ algebra. If in addition each q is s m . , then A is called a locally sm. p-pre- ~ algebra. The meaning of a p- ~ algebra or a locally sm. p- ~ algebra is clear. Finally, when p = 1, we say simply ~ algebra or locally sm. ~ algebra as the case may be.

The locally sm. pseudo-~ (respy. locally sm. p - ~ ) algebras are precisely projective limits of sequences of pseudo-Banach (respy. p-Banach) algebras. 4.6.6.

PROPOSITION.

PROOF. This follows from 4.6.4, 4.5.3. 4 . 6 . 7 . PROPOSITION. Every pseudo-pre- ~ (respy. pseudo- ~)

209

w 6. Metrizable Locally P s e u d o - c o n v e x Algebras

algebra A is a p r e - ( F )

(respy. ( F )

PROOF. Let A ker ~ - {0}. Then

(A,Q),

Ixl - ~

1

-2~

algebra.

~- -- {qn " n -

l + q,(x)

1,2,---}

and

)

is a (F)norm and A ( A , Q ) is complete.

(n, I.I). Further, I'l is complete whenever

4.6.8. algebras

of metrizable

Examples

locally pseudo-convex

(i) Let (p,) be a sequence of real numbers such t h a t 0 < pn <~ 1. The set C of all ~:-valued continuous functions x - x ( t ) on ~ is an algebra (over K) under pointwise operations. Set

Ilxll.- sup [x(t)lp ;

then

Iris<-

]1" II-

is a sm. p,~-seminorm. It can be seen t h a t : - 1, 2 , . . . } ) is a locally sm. pseudo- ~ pn -- p then C is a locally sin. p - ~ algebra. p : 1, we get a locally sm. ~ algebra which

C = (C, {11" algebra. If all Finally, when we denote by

(ii) Let I p - IP(K) denote the algebra of all sequences (x(n))n~1761 of elements of N such t h a t p ( x ) - ~ - ~ x(n)lP < c~, under rt

coordinate-wise operations. Consider the Cartesian power A-

If y -

(Yl,"',Yn,'")

(IP) w - I p • I p • . . . . Yn E I p then set

E A,

Pn(Y) -

(x)

p(y,~) -

~

ly("~)l p, where y~

-

(Y('~))~=I" Then it is easy

m=l

to check t h a t ( A , P ) is a p - ~ algebra, where P - {pn}. Further, it is locally sm. p- ~ algebra since each Pm is sm..

210

Locally Pseudo-convex Spaces and Algebras

(iii) We have already noted in 3.3.14 (example (iii)) that the algebra ~ of all entire functions (under pointwise operations) is a unital (F) algebra. Now set for f E ~', sup

Ilfll~-

If(z)l.

I~I-<,,

Then I1" II~'s are clearly sm. seminorms* and the family {IJ" [ [ ' n - 1 , 2 , . . . } induces the same topology as the (F) metric I1" I[a since convergence with respect to either is uniform convergence over compacta. Thus, (s {[l" I1.~ 1, 2,...}) is a locally sm. ~-algebra (over C). (iv) For 0 < p ~< 1, let L~ denote all ~:-valued measurable functions x - x(t) (or rather equivalence classes of such functions) on the unit interval [0, 1] such that 1

IlXll(np) - (fO 1 Ix(t) ["Pdt )n

< c~.

Now [[[xy[[(nP)] n

-

fo 1 Ix(t) ]'~Ply(t) ['~Pdt 1

<~

1

~/oI Ix(t)[2"Pdt ]~ [foI ly(t)12"Pdt ~~,

whence we conclude that II~yJl~.~) <

(p)

II~ll~.)lly[[2..

Again, if 0 < r < s, then by applying HSlder's inequality 8 1 = l - - 1 we obtain for x pr and 1, with p - ~,p, r

1

/01[x(t)lPr" ldt <~ (/oI [x(t)lo~'-~dt )~ x (/01lP'dt )~ which gives 1

1

(/oI Ix(t)l~ )~ <<. (/oI [x(t)l~ )~ t Actually they are norms.

w6. Metrizable Locally Pseudo-convex Algebras

211

It follows that we have IIXII1 ~ IIXII2 ~ ' ' "

(i.e.Ilxl[1 -< Ilxl[2 -<...)

so that P - {[[. ] [ n ' n - 1 , 2 , . . . } is a well-behaved (countable) family of p-norms and L~ is a p - ~ algebra. It is not a C algebra. To see this, consider elements of the algebra given by:

gin(t) --

1/(1 + ( m 1

1

1)~)

1

for 0 <~t~<. m~/p for 1/m~ < t ~< 1.

Then

-

gml(t) --

{ l+(m-

1) 1/p

1

(0 ~ t ~ 1/rn l/p) (1/m I/p < t ~ 1).

Since 1

f0

(gin- 1)nPdt-

1

(m--l) p

m (1 + ( m - 1)7)no

--~0

as m --~ c~, for each n, we get gm --~ 1. On the other hand 1

f0

(gin 1 -

m-1

1)nPdt- ~ ( m m

1) "-1 --, cr (if n > 1)

which implies that g ~ l ~ 1. Thus L~ is not a C algebra. Therefore, by 4.4.15, it is not locally sm. ( p - ~ ) algebra. The algebra L ~ - L~ is a Frechet algebra which may be called Aren's algebra since this was introduced and studied first by Arens [1'] who also showed [2'] that it is not a C algebra. (v) Let T be any locally compact, a - c o m p a c t , Hausdorffspace. Then the algebra C - C(T, K), of all the K-valued continuous functions under the topology of uniform convergence over compacta, is a locally sm. ~ algebra. (If T - [.J Kn, K,~ compact, f e C, tlf][n - s u p { I f ( t ) ] "t e gn} then C - (C, {11" [I,~}). If T is compact C is a Banach algebra.

212

Locally Pseudo-convex Spaces and Algebras When T is not compact C may fail to be an I algebra. For example, take T - ~, Kn - [-n,n]. Then [[. [[1 ~< [[" ]12 ~ < " ' . Clearly, if

U,~,~- ( f E c " llfll~ < ~} then {U,~,~ "n/> 1, E > 0} is a basis of nuclei, so that

{Un,e(1)}-{l+Un,~} is a basis of neighbourhoods ~x

2n

u~:

of 1.

Since the function

(z ~ ~), ~x

fn,E(x) - 1 + 2n C Un,E(1) and f,,~ is not invertible since it vanishes at x -

2n -~.

It

follows that C(~) is not an I algebra. (vi) Let C ~ - C~176 1] be the algebra of all K-valued infinitely differentiable functions on the closed interval [0,1], with pointwise operations. For f c C ~ write sup If(t)l ,

Ilflloo-

0~<e~
and set n

llfll~- ~ llflk)Jlo~, /c=O

where f(k) denotes the k th derivative of f ( f ( o ) _ follows from the definitions that we have Ilfll~ ~ Ilfll~§

(n-

f). It

0,1,2,...).

It can be shown by induction, using Leibnitz's theorem, that

Ilfgll~ ~ 2"+lllfll.llgll~. If we

set

Ilfll~- 2~+111fll. then I1" II~ is

sm.

and I1" I1~

I1" rl~. It is easy to see that (C ~r {11" II;~" ~ - 0, 1, 2,...}) is

213

w7. Ample Algebras

a locally sm. Frechet algebra. Further it is an I algebra. To see this consider its group Gi of invertible elements; Gi {f e C ~ 9 f ~ 0 on [0,1]}. We claim that Gi is open. For, given f E Gi we have by compactness of [0, 1], m 0 1 inf0. 0 choose an e with 0 < e < ~m0. If g E C ~ , ] I g - flI0 < e then

g(t)l >1 I f ( t ) ] - I f ( t ) - g(t)] /> m o - e

>

m0

2

> 0,

so that g E Gi. Thus Gi is open and C ~ is an I algebra. Finally, by 4.4.15, it is a C algebra. Thus C cr is a locally sm. ~ algebra which is a C I algebra (cf. Example V). (vii) In 3.6.33 we have seen that the field C(X) of rational functions carries a topology (Williamson topology) under which it is a CI (division) algebra W. There is another topology on C(X) making it a pre-~ algebra which is also an I algebra. This topology as well is due to Williamson (for details, see [14', pp.731-32] or [31, p.83]).

w 7.

Ample

Algebras

4.7.1. DEFINITION. A TLS X is called ample if its (continuous) dual X* separates points of X " given elements Xl ~= x2 in X there is an f E X* with f(Xl) ~= f(x2). The separation condition is clearly equivalent to the simpler condition: given x ~ 0 in X there is an f c X* with f(x) # 0 (it suffices to observe that f ( x l ) # f(x2) iff f(Xl - x2) # 0). A TA A is called ample if it is ample as a TLS. 4.7.2. R e m a r k . Clearly an ample algebra (or TLS) must necessarily be Hausdorff. 4.7.3. PROPOSITION. ( a ) E v e r y subalgebra Ao of an ample algebra A is ample. (b) A direct product or direct sum of ample algebras in ample.

Locally Pseudo-convex Spaces and Algebras

214

PROOF. (a) If y E A0, y # 0 then by ampleness of A there is an f E A* with f(y) #- 0. Write f o - flAo. Then f0 E A~ and fo(y) = f(y) # O, whence A0 is ample. (b) Let A = IF[ A~ be a direct product of ample algebras Aa. If a = ( h a ) # 0 in A then aa o # 0 for some a0. Since Aa o is ample we can choose f~o E A~o with fao(aao) # 0. Define f on A by f ( x ) = f~(rax), where x = ( x a ) a n d r~ is the projection x ~ x~. Clearly f E A* and f(a) = f(aao) # 0 , so that A is ample. Since the direct sum ~ Aa is a subalgebra of A, by (a) it is also ample, completing the proof. 4.7.4. PROPOSITION. ( a ) T h e unitization A1 of an ample algebra A is ample. (b) The complezification A of an ample real algebra A is ample. (c) If an ample real algebra A has complex structure then the resulting complexal gebra /i is ample. PROOF. (a) Let al - he1 + a (a E A) be a non-zero element of A1. We have to show that there is a fl E A[ with f l ( a l ) # 0. For the construction of fl we have to consider two cases. C a s e 1: c z - 0 , a l - a . Since A is ample there is an l E A * with f ( a ) # O. Extend f to fl on A1 by setting for X l - Ael+x ( x E A , ) ~ e K) f l ( X l ) - ) ~ + f ( x ) . Then fl e A~. and f l ( a l ) -

f(a) :/:0. C a s e 2: fl(al)

a-#- 0. Set ] ' l ( ) ~ e l + x ) -- )~. Then fl C A~ and

= (x 76 O. a.,

(b) Suppose that zo e A, zo - xo+ iyo # 0 (xo, Yo C A). Since A is ample, and x0 or Y0 # 0 , there is an f0 E A* with fo(xo) or fo(yo) # O. Setting N

+ N

N

N

(z- z+iv) N

and f(z) - f o ( z ) - i f o (iz), it is easy to check that f is C-linear and f e (A)*. Since ] ' ( z 0 ) - 2(fo(xo)+ifo(Yo)) # O, A is ample. (c) For f e A* set ] ( x ) - f ( x ) - if(ix). Then

w7. Ample Algebras

/ ( ( ~ + iz)~)

215

=

/((,~ + ifl)x) - i / ( i ( , ~ + ifl)~)

=

,~f(x) + fl/(;~)

=

(~ + i~)[/(~) - ;/(;~)] - (~ + ; ~ ) ] ( ~ )

- ;._

fl/(~) - ;,~f(;~)

so that f E (r If x0 E A - A and x0 #- 0 then by ampleness of A there is an f e A* with f(xo) r O. But then ] e (ft.)* and f (xo) - f (xo) - i f (ixo) # O, proving A is ample. 4.7.5. PROPOSITION. For a TA which is a division algebra to be ample it is sufficient that A* 7s {0}.

If a x fa

PROOF. Suppose that A is a division TA with A* ~: {0}. f E A*, f # 0 there is an a0 E A with f(ao) ~ O. For any in A with a ~- 0 define fa(X) - f (xa-lao). Since the maps ~ xa -lao and y ~ f(y) are continuous, fa is continuous, E A*. Since f~(a) - f(ao) # 0, A is ample as required.

4.7.6. PROPOSITION. Every locally convex algebra - i n particular a Banach algebra- is ample. PROOF. This is an immediate consequence of the HahnBanach theorem (see [24, p.59]). 4.7.7. R e m a r k . For 0 < p < 1; there are p-Banach algebras which are ample as well as those which are not ample. For instance the algebra I p = IP(D<) (example (v) of 3.4.6) is an ample p-Banach algebra. To see this define, for each n /> 1, fn(x) - xn, where x - ( X l , X 2 , . . . , X n , . . . ) - - ( X n ) e lP. Then it is easy to see that fn e (lP) *. If a - - (an) e l p, a 7 s then for some n, an ~ 0. Then fn(a) -- an ~ 0, proving I p is ample. Consider next the space L p -- LP[0, 1] consisting of (equivalence classes) of 0<-valued Lebesgue measurable functions f such that

fo If(t)l"dt < c~. With llfll-

fox If(t)] adt,

LP is a p-Banach space. It can be made into a p-Banach algebra by introducing in it trivial multiplication" for x, y E LP, xy - O.

Locally Pseudo-convex Spaces and Algebras

216

L p is not ample since it is known that '0' is the only continuous linear functional on L p (see [24, p.36]).

w8.

Topological

Spectral

Radius

4.8.1. DEFINITION. Let A - (A,p) be a p-seminormed algebra. Then, by 3.4.3, p is a. sm. and so the limit 1

Vp(X)- lim p(xn)-z < cr ll---~O0

exists, by 3.3.6. The non-negative real number Vp(X) is called the topological spectral radius of A, the adjective "topological" being justified by" 4.8.2. Vq

--

LEMMA.

If q is a p-seminorm with q ..~ p then

/2p.

PROOF. By 3.4.1, q is also a. sm.. Further, by 3.2.11 (ii) we have constants Ca,Cp such that p < C q, q <<.c p. Therefore, 1

!

1

vv(x ) -- n -lim p(z'~)-z <. n -lim C~q(x")~ < vq(x), -400 -'-~O0 1

since

aim C q " - 1. Similarly,

n---~ o o

vq(x) <~ Vp(X). Then v p ( x ) -

Vp(y), as desired. 4.8.3. PROPOSITION. The topological spectral radius v(x) vp(x) possesses, besides the properties (i) - (v) of 3.3.7, the fol-

lowing: (vi) v ( A x ) - ]A]Pv(x); (vii) v(x + y) <~v(x) + v(y) provided x ~ y. PROOF. The property (vi) has already been established in 3.4.5. So we have only to prove (vii). For this it is convenient to follow the method of Bonsall-Duncan [4, p.71]. We may assume that p is sm.. Choose e > 0 and set 1 -

+

1 ,

-

217

w 8. Topological Spectral R a d i u s

Since

x ~

y we have also u ~ v. Also, by c o n s t r u c t i o n < 1, so t h a t by 3.3.6, p ( u n ) , p ( v n) < 1 for all sufficiently large n. If follows t h a t the semi-groups { u n 9 n 1 , 2 , - - - } , { v n : n : 1 , 2 , . . . } are p - b o u n d e d . Since u ~ v the semi-group S g e n e r a t e d by u , v is u(u),u(v)

{u k,v y, u k v j 9k , j - 1, 2,---}. Since p is sm. we have It follows from the above t h a t S is p - b o u n d e d . By 3.5.10 we can find a sin. p - s e m i n o r m q such t h a t q ,,, p, q ( u ) , q ( v ) K 1. By 4.8.2 we have Vp - vq - v (say). T h e n 1

q(x)-

q((v(x)+e)-~u)-

(v(x)+e)q(u)

<~ v ( x ) + e

(since q ( u ) <. 1).

Similarly, q ( y ) <. v ( y ) + e. It follows t h a t

,(~ + y) <. q(x + y) .< q(~) + q(y) .< ~(,) + ,(y) + 2~. F r o m the a r b i t r a r i n e s s of e we conclude t h a t u(x + y) ~< u(x) +

-(y). If A is c o m m u t a t i v e t h e n v is a c o n t i n on A ( A , p ) . F u r t h e r , uv - u.

1 . 8 . 4 COROLLARY. uous sm. p-seminorm

PROOF. T h a t v is a sm. p - s e m i n o r m follows from 3.3.7 (iii), 4.8.3 ((vi), (vii)). T h e continuity of v is a consequence of the inequality in 3.3.7 (i). Finally, v~ = v since v ( x n) - v(x)'* (by 3.3.7 (ii b)). 1 2n

4 . 8 . 5 . LEMMA.

lim

-- 2.

n--+ oo

PROOF. Write an -- nn(n!) -1. T h e n a n ~ a n - 1 - ( 1 - 1 ) / ( 1 1)n. Using the well-known result lim (1 + z ) n _ e z t (z e C) we n--+ oo get t We denote the classical exponential function by e z, so that in partic1 1 ularwehave e - l + ~+~+-.-

Locally Pseudo-convex Spaces and Algebras

218

lim

an 1

n--,oo an-

1

=

1 e- 1

lim (1 - ~1) n

~---~ (x)

--e.

It follows that •

lim a , ~ -

lim

n - -,, oo

Therefore

an

(1)

---e.

n - - , oo a n - 1

1

1

2n

~im

lim

n---+ o o

(~)2 1

lira { 2, 1

lim ( 2n' ) ~ n--,c~ (2n)2n -1

--

e

=

2.

1

92- n---+ limO0

( n n ) -~ -~.

92. e, where we have used (1).

4.8.6. THEOREM (Zelazko). Let A be a commutative algebra (over E, p ( r O) a sm. p-seminorm on A, u - up. Then"

(i) Th~ unit b~U B 1 - {* c A ~(~)< 1} i~ ~o~,~,. 1

(ii) ][x[] - u(x)-~ is the gauge of Ul and it is a sin. seminorm. PROOF. (i) By 4.8.4, u is a p-seminorm and consequently, by 3.2.7 (i), B1 is closed in the u-topology. To show that B1 is convex it is enough to show that it is midpoint convex, t Assume first that u ( x ) , u ( y ) < 1. Then l(x§ 1 1 n)-i

~(~

~ ( ~ + Y) - 2-7l i % p((~ + y) ..

(1)

n

Since x ~ y, (x + y)n) _ ~ ( ~ ) x ~ y n-k, so that by subadditivity k=O

of p, we have

p((~ + y~) .< ~

p(~y"-~).

k=O

Choose e > 0 such that u ( x ) , u ( y ) < e < 1. t i.e. x, y C B 1

==~ -~+Y T- EB1

(see [15, p.17])

(2)

w8. Topological Spectral Radius

219

Since u ( x ) , p ( y ) < 1 we can find N1 such that 1 1 p(xn)~,p(yn)-~ < e, or, p(xn),p(y n) < en < 1

for n >~ N1. Therefore, by 3.3.4, p(xn),p(y n) --~ 0 and consequently we can find C > 0 such that p(xn),p(y n) < C, for all n.

Since cn --~ 0 we can choose N2 such that er'C < 1 for all n/> N2. It follows that for n/> N - max(N1, N2) and arbitrary rn

p(xny m) <. p(xn)p(y m) < enC < 1.

(3)

Similarly for arbitrary n and rn ~< N,

p(xny m) < 1.

(4)

From (2), (3), (4) we obtain for n/> N,

p((x + y)2n)~<~-~

p(xky 2n-k) <~~

k:0

<~(2n + 1)

k=O

(5) since (~'~) ~< (2~) for all k. From the definition of u and (15) w e get u(x + y)

:

2

1

1

1 lim p((x + y)2n)~ l i m ( ( 2 n + 1)

)~

2P n ~ c r 1

1

~<

2n

20 n--,oolim(2n + 1 ) ~ ( l i r n

)P

1

1

~<

~-~(lirn

2.

1

)~ (since lim (2n + 1 ) ~ -- 1) n----~ o o

1

~<

2P " 2~ (using 4.8.5)

~<

1,

which proves l ( x + y) E B1. Next, if u(x), u(y) ~< 1, we can choose a sequence An with 0 < An < 1 and An ~ 1. Then u(Anx), u(Any) < 1, so that, by the result above, Zn - 89 (Anx+ A~y) c B~. It follows that ~1 ( x + y ) - l i m z , E B a . Thus B1 is midpoint convex, completing the proof of (i).

Locally Pseudo-convex Spaces and Algebras

220

(ii) Since clearly, B1 - {x E A " II~ll < 1}, it follows from 4.1.13 t h a t I1" II is the gauge of B1. Finally, since p is sm., Si is a subsemigroup and consequently the gauge II. I[ is s m . . 4 . 8 . 7 . COROLLARY. then ]l.]] is a norm.

If the algebra A is a division algebra

PROOF. If x E A, x ~- 0 then x is invertible, and so by 3.3.7

(v),

Ilxll

> o.

4 . 8 . 8 . DEFINITION. An element x of p - s e m i n o r m e d algebra A = (A, p) is called topologically nilpotent or t. nilpotent if v ( x ) = vp(x) = 0. If A = (A, {p~}) is a locally pseudo-convex algebra t h e n an element x E A is called t. nilpotent if each ca(x) --

~ (x)=0 Clearly every nilpotent element is t. nilpotent

(x n -

0

p~(~)~ -o). 4 . 8 . 9 . LEMMA. If x is t. nilpotent then (i) ~ -~ o, (ii) Ax is t. nilpotent for every A E K. PROOF. (i) Take any c with 0 < e < 1. Since va(x) aim p~ (x") 1 - - 0 , we have p~(x n) < en < e for n i> N. It follows t h a t p~(x '~) --~ O, x '~ --~ O. (ii) By 4.8.3 (vii), v ~ ( A x ) - ] A l P v ~ ( x ) - 0. 4.8.10. gebra A ideal of A.

PROPOSITION. In a commutative p - s e m i n o r m e d al( A , p ) the set I of t.nilpotent elements is a closed

PROOF. By 4.8.4, 3.3.7 (iii) we have"

-(~ § y)~< ~(~)§ ~(v), ~ ( ~ ) - I~1'~(~),~(~y)~< ~(~)~(y). Therefore I is an ideal. Moreover, 7" is closed, as follows from the continuity of p (see 4.8.4). 4 . 8 . 1 1 . PROPOSITION. Let A be a p - B a n a c h algebra and x E A. Then: 1 r(x) <~ ~,(x)~. (.)

w 8. Topological Spectral Radius

221

If A is real we have also 1

~(~) < .(~)~.

(**)

PROOF. We may assume the norm !1" II to be sm. and write 1

v-v,,.

If I~1 > v ( x ) ; then

V(--)~-lx)- I~1-,~(~) < 1, so that by 3.3.19, -A-~x is q. invertible. It follows by 1.7.8 that 1

~'(.) c {~ ~ K I~I-< ~(~)a} so that (,) holds. N

If A is real we apply (,) to A and deduce (**). 4.8.12. COROLLARY. In any p - B a n a c h algebra a t. nilpotent element is q. nilpotent. 1

PROOF. Since r(x) ~< v(x)7, hence the result.

v(x) -

0 ::~ r(x) -- 0 and

CHAPTER V SOME

1.

Vector-valued

ANALYSIS

Differentiability

and Analyticity

5.1.1. DEFINITION. Let X be a Hausdorff TLS and X* its (continuous) dual. Let G C_ K be an open set. A function f : G --~ X is called weakly differentiable if there is a function g : G --~ X such t h a t for each x* E X*, the scalar-valued function x ' f (A) - x*(f(~)) has the scalar-valued function x*g(~) as its derivative in the usual sense; we write g - f~(w) and call f~(w) the weak derivative of f. The function f is called strongly differentiable if for each 2 E G, f ( # ) - f(A) lim exists in the topology of X. We denote this limit by /'()~) and call f ' the strong derivative of f. 5.1.2. LEMMA. A strongly differentiable function f is weakly differentiable and continuous, and moreover, the strong derivative fl coincides with the weak derivative fl(w). PROOF. It follows from the definition of the strong derivative that lim ( f ( # ) - f(,~)) - lim (/~ - )~)f'()~) - 0 proving f is continuous. Again, lim

x* f (tt) - x* f (A)

= lim x * ( / ( # ) - / ( ) ~ ) ) ~-~ #-~

x*f'(~),

whence f is weakly differentiable with f l ( w ) _ fl. 5.1.3. DEFINITION. A function f : G - - - , X is called weakly analytic if x*f(A) = x*(f(~)) is an analytic function of ~ on G in the usual sense, for each x* E X. We call f strongly analytic on G

w1. Vector-valued Differentiability and Analyticity

223

if around each point Ao E G _C K, there is a neighbourhood {A E K "l A - ,k0l < r0} in which f has a power series representation t oo

rt--0

where x , E X and the series converges in the topology of X. If X is a complex TLS we will also use the expression weakly holomorphic (respy. strongly holomorphic) for weak analyticity (respy. strong analyticity). 5.1.4.

LEMMA.

A strongly analytic function f is weakly

analytic. PROOF. Suppose t h a t oo

rt--0

Then, for each x* E X*, we have oo

n=O

so t h a t f is weakly analytic. 5.1.5. Remark. If X is a complex, Frechet space (i.e. a complete metrizable locally convex space) then for a function f : G --, X, weak holomorphy of f ==~ strong holomorphy of f (see [24, p.79]). 5 . 1 . 6 . Let X -

(X, I1" tl) be a p - B a n a c h space and oo

rt:0

a series in X. P u t t In the representation we find it convenient to write the scalars on the right side (instead of the customary left side)

Some Analysis

224

1

1

r0 -

1

lim I[x,]l~,R0 - ro 1 , R - Rg - ro ~ (lim - lim sup).

Ft--~ OO

R is called the radius of convergence and {A e K ' I A I -- R} the circle t of convergence of the series (,). Next let X - (X, {p~}) be a complete n a u s d o r f f locally pseudo-convex space. Set 1 n---~ (x)

R

-

1 P~

infR~-infr~

R again is called the radius of convergence of the series ( , ) and {A e K ' I A I - R} its circle of convergence. 5 . 1 . 7 . PROPOSITION. (a) The series (,) converges absolutely

uniformly for IAI <~r, for any r with O < r < R. (b) If IAI > R the terms of the series are unbounded and consequently the series diverges (i.e. does not converge).

(c)

diiT

got ] om (,)

ti t d oo

rt--1

PROOF. We will first treat the case where X is a p - B a n a c h sp~ce ( x , II-II). The proofs are all modelled after those of the corresponding classical results. oo

ll .ll(l l ) - - which is a

(a) The absolute-valued series n-0

numerical series - converges, by the classical Abel's lemma, for IAI p < R0, or equivalently, for IAI < R. Further, this series t In the real case the circle of convergence is to be interpreted as the end-points of an interval.

w 1. V e c t o r - v a l u e d

Differentiability

converges uniformly for IA[p <~ r p, whence formly for IAI ~< r. (b) If IAI > R, choose r so t h a t IAI > r > 1

rP

<

1

Rp

--

1

Ro

..

225

and Analyticity

,,

(,)

converges uni-

R. T h e n ,,

-- llmllXnllK,,,, n --,oo

and consequently there are arbitrarily large n with 1

1

1

tlxnll~ > ~-~, or [[Xn[[ > rp n It follows t h a t ---~ 0 0 .

T h u s the series (,) is not b o u n d e d (for [A] > R ) and so not convergent (by 3.2.15). (c) If we put r ' -

1

lim[l(n+l)xn+l[[~,

then

tl--* oo

1

n-i-1

r' --n-,oolim[{(n + 1) a--4-f}P [[Xn+l II h--4-Y] --K-1

1 _

_

di~rn~ ]]xn+ll['~-I

_ r,

1

where we have used the well-known result " n ~ -~ 1 as n --~ oo. To extend the above proofs to the case where X is locally pseudo-convex, we note t h a t if [A[ < R~ (for each a , ) so t h a t oo

the proof above for (a) shows t h a t

~-[~pa(Xn)([A[P") n < oo, for n--0

each pa, where X ( X , { p ~ } ) . Hence ( a ) f o l l o w s for the locally pseudo-convex case. Further, if [A[ > R t h e n there is an a such t h a t IAI > R~, whence it can be shown, as above, t h a t p~(xnA n) --~ c~, so t h a t the series is not p a - b o u n d e d . Therefore the series is once again divergent, proving (b) for the general case. Finally, if we set r ~'

-

pa((n + n----~ o o lim

1

1)Xn+l)"+'

then as in the proof above for p - B a n a c h spaces we o b t a i n r~ - r ~ot~ whence R ~ - R, where R' is the radius of convergence of the

Some Analysis

226 series (**).

5.1.8. PROPOSITION. The function oo

rt--0

is strongly differentiable inside its circle of convergence and we have

E oo

ft()~) _

Xnn~n-l"

(**)

n=l

In particularly, /(A) is a continuous function of A. PROOF. The proof we give is again modelled after the proof in the classical case (see [25, p.200]). We take x , e X = (X, {p~}) and assume that the radius of convergence of the series (,), R > 0 (if R = 0 there is nothing to prove). Choose a A c N: with IA] < R. Take a real number r with IAI < r < R. Writing O(3

n--1

we have, for ]#1 < r, Oo

#-

A

-g

(1)

--n=l

where ft. - ( ~ " - A " ) / ( # - A ) - n)~ n - 1 . Clearly fll - O , and for n >/2, we get n-1

fin -- ( ~ - )i) E k'~k-lI~n-k-1

(2)

k=l

(as can be easily checked). Since IAI ' I~t] < r, and

I~.1 -< t . - al ( ~ k ) k-1

~ k =n-1 X k ~< n 2 , w e

obtain from (2)

~"-~ -< I.- ~1~=~"-= (- ~>2).

(3)

w 2. Exponential and Logarithmic Functions

227

Therefore oo

co

rt--1

r

n=

1

n:

2

(since fll -- O) oo

n--2

(using (3)) i.e.

(4)

P~ n=l

n--0 oo

For the series

+

", wo h ve

n--O

r~' -

lim p~(x,+2) • (n + 2 ) ~ - -

n - - ~ O<)

lim p~(xn) ~-. - r~.

n---+ OO

It follows that R ~ - R. Since r < R - R ~, the series on the RHS of (4) converges, whence by (1), (4), as tt --+ A we obtain

ft(A) -- g(A) proving (**). The continuity, of f follows from 5.1.2.

w 2.

Exponential

and Logarithmic

Functions

5.2.1. DEFINITION. Let A be a unital Hausdorff TA with unity e. The exponential function in A is defined by c<)

E t(x)-

Exp(x)-

~

~z",

(x ~

e, 0 ! - 1)

(,)

n:0

whenever the series on the right converges. We denote by PE the domain of E and by s its range. Since 0 E P E , e E s both t The Hausdorff assumption ensures that E is well-defined.

Some A nalysis

228

PE, ]~E are non-empty. The inverse function Log is defined on )~E by setting Logy - x if y - E(x). The function Log is in general many-valued. We denote the classical exponential function by )k 2

E(A)-e ~-I+A+2.v +"" (ACE). In particular, we write e--l+l+~+.--

1

5.2.2. PROPOSITION. DE -- A if A is a unital pseudoMichael algebra -inparticular a unital p-Banach-algebra; moreover the series converges absolutely for all x E A. PROOF. Let A Pa

(A, P ) , P -

~

~<

{pa}. Since = un

(say)

and un+X Pa(X)pa --~ 0 , ~ne "" serms " for E(x) converges absolutely un = (n+l) and so also in A for all x E A, by 3.1.24. 5.2.3. PROPOSITION. Let A be a pseudo-Michael algeba- in particular a p-Banach algebra - w i t h unity e. Then: (i) E ( 0 ) - e ;

E(e)-ee.

(ii) E ( x + y) - E ( x ) E ( y ) provided x ~ y. (iii) E ( - x ) -

E(x) -1.

(iv) If A is complex and u an idempotent then E(2~riu) -

0

(i -

PROOF. (i) Clear. oo

(ii)

E ( x + y) -

~(*+u)"=! . Expanding the s u m m a n d s on n-:O

the RHS, using the binomial theorem, and regrouping the terms (which is permissible because of absolute convergence) we find t h a t yn the cofficient of W.t is

229

w2. Exponential and Logarithmic Functions

e+

(n + 1)x n § 1

+

(n § 2)(n + 1)

x2

2!

(n + 2)(n + 1)

III

J

Q

x 2

= e + x + -~. + . . . -

E(x).

Thus, y2

E ( x + y) - E ( x ) ( e + y + ~. + . . . ) - E ( x ) E ( y ) . (iii) This follows from (ii) by taking y = - x

and using (i).

(iv) E ( 2 . i u ) - e + {27ri+ 2! + ' " .}u. The expression in the bracket above - e 2~ri - 1 - 1 - 1 - 0 . So we get E(21riu) - e. 5 . 2 . 4 . COROLLARY. Let T be a bounded l.o. on a p - B a n a c h

space X.

Then T2 E ( T ) - I + T + ~. + . . .

is an invertible bounded l.o. on X. PROOF. Consider the p - B a n a c h algebra B ( X ) (iii). Note t h a t E ( T ) - 1 - E ( - T ) . 5.2.5. ormaliz

and use 5.2.3

If A is a Banach algebra with unity e and I1" It, the.

LEMMA. o,m

]JE(x)]f ~< E(]Jxjj). P R O O F . Clear.

5 . 2 . 6 . If A is a TA w i t h o u t unity we can introduce the quasiexponential function Eq by x2

+...

which makes sense (whenever there is convergence). If A1 is the unitization of A and E the exponential function in A1 then we have the obvious relation

E ( x ) : el + Eq(x)

(x E A),

Some A nalysis

230

where el is the unity of A1. Using this relation and 5.2.3 (ii) we can easily deduce the identity

Eq(x) o Eq(y) - Eq(x + y) whenever x ~-~ y. Since E q ( O ) - 0

we get Eq(-X) - (Eq(x))'.

5.2.7. DEFINITION. Let A be a TA and Gq its group of q. invertible elements. The connected component of Gq containing 0 is called the principal component of Gq and we denote it by Gqo. Similarly, if A has unity e, the connected component of the group Gi (of invertible elements) containing e is denoted by Gie; Gie is called the principal component of Gi. 5.2.8. PROPOSITION. In a C algebra A, Gqo is a closed normal subgroup of Gq. If A has a unity e, Gie is a closed normal subgroup of Gi. PROOF. Since A is a C algebra, by 3.6.5 ((a), ( b ) ) G q and Gi are TG's. The stated results now follow by applying to these groups a standard result [21, p.39] in the theory of topological groups regarding the identity component. 5.2.9. COROLLARY. If A is a p-Banach algebra then Gqo is a clopent normal subgroup; if A is, besides unital then Gie is a clopen normal subgroup of Gi. These results hold, more generally, when A is a pseudo-Michael Q algebra. PROOF. In view of 5.2.8 we have only to prove Gqo , Gie are clopen. By 3.6.23 (b), 3.6.21, 3.6.10 we conclude that Gq,Gi are open when A is a p-Banach algebra. Also they are open in the pseudo-Michael algebra case by virtue of the Q algebra hypothesis we have in this case. Finally, in all the cases A is locally connected (see 3.2.8, 4.3.16). Hence the clopen properties of Gqo, Gie.

Let A -

5.2.10. PROPOSITION. Banach algebra. Then:

(A,[[. ]]) be a unital p-

(x)

(i) The series Log x - - ~

1 ( e - x) n converges absolutely for

n--1

all x E A with v ( e - x) < 1, where u clopen=closed and open

vii.[[.

231

w2. Exponential and Logarithmic Functions (ii) E ( L o g x ) - x.

(iii) For each x e A, the function Ez(A) - E(Ax) is analytic for all A E K. In particular, Ez is entire when K - C. PROOF. Suppose that u ( e - x ) < e < 1, then ]l(e-~)"11 < ~" for n/> (some) N. It follows that 1

En

I1 (e - x) ~ II -- ~ II (e - x)" II < n

< n p

~"-

oo

Since ~

E,~ converges the series for Logx converges absolutely

n-N

for u ( e - x) < 1. The identity (ii) can be checked, as in the classical case, by substituting the series for Log x in each summand of the series oo (Logx)n E(Logx)- ~ n! n=0

and simplifying (the needed steps for the simplication are justified because of absolute convergence). Finally, we have oo

xn ~n

r~--O

n!

Thus radius of convergence for the above power series in A (see 1

5.1.7) by r 0 P, where Xn

I

r o - ,~--,oolimI1~. II~

=

lim Ilxnll 88

.-~oo (~!)~

1 1

P

Now lim ( 1

)-

1 n

- lirn

(1

1)

( n + 1)!/~.

-- lim

n+l

=0.

Some Analysis

232 1

So r0 -- 0, r0 p -- 1 = oo, whence the series Ez(A) converges for 0p

all ,~ and (iii)follows. 5.2.11. R e m a r k . Proposition 5.2.10 can be extended to unital pseudo-Michael algebra A = (A, {p~}) in the following form: For every x in A with c a ( e - x ) < 1, for all a , the series for Log x (defined in 5.2.10) converges absolutely and for all such x the identity (ii) of 5.2.10 holds. Further, the function E , - E,()~) is analytic for all A E K, for each x E A. 5.2.12. PROPOSITION. Let A be a unital p-Banach algebra. The range RE of the expoential function E generates (algebraically) the principal component Gie (i.e. Gie is the smallest subgroup of Gi containing ]~E). PROOF. Since A is connected and E continuous, RE is connected. By 5.2.3 (iii), RE C_ Gi. Since e E RE (by 5.2.3 (i)) and ~ E is connected, ]~E C Gie. Write

U--{xEA']le-x]]

< 1}.

Then, by 5.2.10 ((i), (ii)), U C RE C_ Gie. Since U is an open neighbourhood of e in Gie and Gie is connected, U and hence RE generates Gie (see [20, p.37]). 5.2.13. COROLLARY. ]~E- Gie iff ]~E is a subgroup. In particular, if A is commutative, ]~E- Gie. PROOF. The first statement is immediate. For the second we observe that by 5.2.3 ((i), (iii)), ~E is a subgroup and hence the result.

If y C ]~E then y belongs to a connected abelian subgroup of Gi. Conversely, every connected abelian subgroup H of Gi is contained in ]~E. Hence y E ]~S (i.e. y has a logarithm)iff y belongs to a connected abelian subgroup H of Gi. 5.2.14.

COROLLARY.

(Theorem of Nagumo [11']).

PROOF. If y e ~E then y - E(x), and H - { E ( A x ) ' - o o < A < 0o} is clearly a connected abelian subgroup of Gi. Next, by 1.1.9, there is a commutative subalgebra B0 with H _c B0. Then

w2. Exponential and Logarithmic Functions

233

B - B0 is a c o m m u t a t i v e unital p - B a n a c h algebra with H _c B. By 5.2.13, the range E(B) - Gi~(B) (the principal component in B). Since e E H and H is connected we have

H c G i ~ ( B ) - E(B) c_ E ( A ) -

~E.

The above two results clearly imply the final assertion in the corollary.

Let A be a complex unital p-Banach algeba. A linear functional X on A is a character iff 5.2.15.

THEOREM ( G l e a s o n - K a h a n e - Z e l a z k o t ) .

(i) x ( e ) -

1.

(ii) k e r x comprises non-invertible elements. PROOF. Suppose t h a t X is a character. Then, by 1.3.10, condition (i) holds. Further, if x is invertible, X(x)x(x -1) x ( e ) - 1, so t h a t X ( X ) ~ 0 and (ii) also holds. It remains to prove the "if" part of the theorem. So assume t h a t X is a linear functional satisfying (i), (ii). Write M - ker X. Then X being a linear functional ~= 0, M is a subspace of A o f c o d i m 1, so t h a t we have A C e + M . We shall now show t h a t X is bounded with IIXII ~< 1. If x e M, X ( x ) - O , so t h a t 0 - I X ( X ) I <~ Ilxll. Next consider an element y e A \ M . Then y has the form y -- A t - x

(A E C,A ~: 0, x E M ) .

Since A - i x E M, by hypothesis (ii) it is not invertible and so, by 3.3.20 (ii), l i e - A-lxll tt /> 1. Therefore

Ix(y)l-

Ix(~,~ - ~)1-

I~,1

<

=

I~lll~1

A-lxll ~ -II~,~-

Ilyll ~ (u~ing ~-

~-

~11~

y).

It follows that Ilxll ~< 1, in p,,rti~ul~r that X is ~ontinuou~. To prove t h a t X is a character it is enough by 1.3.19, to show that xEM~x 2EM. Consider now an x E M. We have to show t h a t x 2 E M. We may clearly limit ourselves to the case t h a t I l x l l - 1. Set t These authors have obtained the result for Banach algebras. tt We assume (without loss of generality) that I1" II is sm..

Some A nalysis

234

oo

~

f(A)-

n=O

X(Xn)An

(A C C)

n!

Since

1

(1) "

n

Ix(x")l < IIxll IIx"ll~ < II~ll ~ - 1

(2)

and co

I~1" = E(f~f) n=O

n!

(3)

"

it follows t h a t the series for f(A) is absolutely convergent and hence f(A) is an ordinary entire function. Further, since oo

n!

-

E(IAI),

rt--0

f has order at most 1. By continuity of X we have

I(A) - X

n!

(4)

- x(E(Ax))

where E is the exponential function in A. Since E(Ax) is invertible, f(A) 7- 0, by (4) and (ii). We now apply H a d a m a r d ' s factorization theorem to conclude t h a t f(A)-

e ~+f~

(5)

(~,fl e C)

where e )' denotes the classical exponential function. From (1) we find f ( 0 ) 1 and f ' ( 0 ) X ( X ) - 0 (since x e M). These imply t h a t in (5) a - ~ - 0, whence f(A) - 1. Therefore X(X 2) - f " - 0, whence x 2 E M, completing the proof.

w 3.

Square

Roots

and

Quasi-square

Roots

5.3.1. DEFINITION. Let A be an algebra and x E A. An element y such that y2 _ x is called a square root or sq.r. of x; it may or may not exit. (Since (_y)2 _ y2 _ x a sq.r. even

w 3. Square Roots and Quasi-square Roots

235

w h e n it exists is not unique). Similarly, an element y such t h a t y o y - x is called a quasi-square root or q.sq.r, of x. 5 . 3 . 2 . LEMMA. I f A has a unity e then an e l e m e n t y is a q.sq.r, of an e l e m e n t x iff y + e is a sq.r. of x + e. PROOF. This follows from the identity (y + e) 2 - e + y o y. 5 . 3 . 3 . LEMMA. (cf. [4, p.44, L e m m a 12]). Let A - ( A , P = {pa}) be a p s e u d o - M i c h a e l algebra, x, y E A , x ~ y, and x o x y o y . I f v ~ ( x + y) < 2 p~ (p~ - h o m o g e n i t y index of p~) for all a then x - y. PROOF. Write a - iI ( X + y) , b -- x - y . Then a b - l(x2 y2) _ - b (using x o x yoy). This implies a o b a. Since v ~ ( a ) - l__v2p~~(x + y) < 1, by 4.4.14, a is q. invertible, so t h a t by 1.1.25 b - 0, x - y. 5 . 3 . 4 . PROPOSITION. (cf. [4, p.44, Proposition 13]). In a p s e u d o - M i c h a e l algebra A - ( A , { p a } ) every e l e m e n t a with v ~ ( a ) < 2 p~ - 1 has a unique q.sq.r.b, with ca(b) < 2 p~ - 1. Further, b ~ a and v~(b) <~ v~(a). In particular, every t. nilpotent e l e m e n t a has a unique t. nilpotent q.sq.r.b. PROOF. Following Bonsall-Duncan [4, p.44], we use m e t h o d based on the existence of fixed point for o b t a i n i n g results. For each a, fix an r/a such t h a t c a ( a ) < r/a < 2 p~ T h e n , by 3.5.10(i), we can choose a sm. p a - s e m i n o r m such

the the - 1. that

c a ( a ) ~
-

x2

T x - ----~---(x e S ) .

T h e n we have

t

2~ -

1 -

2.2 ~-~ -

1 -

2~-'

+ (2 ~ - ~ -

1) ~< 2 ~ - ~

(~ince

p < 1).

Some Analysis

236

q~(Tx)

1

-

1

1

~<

1

x 2) ~< 2--~-(r/a + r/2) - 2-7-(1 + r/a)r/a

2P. q . ( a -

92 p" r/~ - r/~,

2p~

so t h a t T is a m a p p i n g S ~ S. Further, if x , y E S, q~(Tx - Ty)

-

x 2 _ y2 1 q a ( ~ 2 ) < 2-;:q~ (~ + y)q~ (~ -- y) 1

2P~

92 r / , ~ q ~ ( x - y) - # ~ q , ( x - y),

where #a - 21-p"Wa < 2 1 - p " ' 2 p"-I - 1. Thus T is a c o n t r a c t i o n m a p p i n g and so by 4.3.15, T has a fixed point b 9 T b - b. It follows t h a t 2b - a - b2 or b o b - a. Since b E S c_ Ba, b +-~ a. Also, va(b) <~ qa(b) ~< r/a < 2 p~ - 1 ~< 2 p~-I.

To prove the uniqueness of b (under the condition v~(b) < 2 p~ 1 ) we suppose t h a t there is a bl with blobl - a, va(bl) < 2 p ~ - 1 ~< 2 p~-I (for all a ) . T h e n abl -- (2bl + b21)b1 - b1(2b1 + b21) - bla,

i.e. a ~-~ bl. Since b E Ba it is the limit of a sequence of polynomials in a. Hence b ,-~ bl. Therefore +

<

< 2

+ 2

2 o.

By 5.3.3, b - bl. Further, the uniqueness assertion implies t h a t b is i n d e p e n d e n t of the p a r t i c u l a r choice of rla satisfying v a ( a ) < ~a < 2 p" - 1 . Hence, since va(b) ~< rl,, we conclude t h a t va(b) ~< va(a). Finally, the inequalities v,(b) ~< va(a) imply t h a t v a ( b ) - 0 w h e n e v e r v a ( a ) - 0. This m e a n s t h a t when a is t. n i l p o t e n t its q.sq.r.b, is also t. nilpotent. 5 . 3 . 5 . COROLLARY. In a unital p s e u d o - M i c h a e l algebra A (A, {p~}) with u n i t y e, every e l e m e n t el with u a ( e - a l ) < 2 p~ 1, for all a, has a unique square root bl with v~(e - bl) < 2 p~ - 1,bl ~ al; then va(e - bl) <~ u ~ ( e - al).

w3. Square Roots and Quasi-square Roots

237

PROOF. Write al - e -- a. Then, by 5.3.4, a has a q.sq.r.b. with b +-~ a , v ( b ) ~ v(a). Writing bi = a-4-b, by 5.3.2, bi is a square root of al. Also, the uniqueness of bl is an i m m e d i a t e consequence of the uniqueness of b. COROLLARY. In a p-Banach algebra A = (A,[]. ]]) with v - v[i.ii , every element a with v(a) < 2 p - 1 is q. invertible and has a unique q.sq.r, b. with v(b) < 2 p - l , b ~ a, v(b) <~ v(a); 5.3.6.

b is also q. invertible. If a is t. nilpotent so is b. PROOF. Since a p - B a n a c h algebra is a Michael algebra the existence of b and all the properties s t a t e d in the corollary except q. invertibility of a, b are i m m e d i a t e consequences of 5.3.4. The q. invertibility of a or b follows from 3.3.19, since v ( a ) , v ( b ) < 2 P - 1 , < 1. 5.3.7. Remark. An element y of an algebra A such t h a t y" = x (x E A) is called an n th root. Similarly if we write

yon = Y o y o . . . o y(n factors ) and yon = x, then y is called a quasi n t h root or q . n t h root of x. If A has a unity e we have the relation e + y~ = (e + y)'~, so t h a t if y is a q . n t h root of x then e + y is n t h root of e § x. Proposition 5.3.4 can be generalized for q.n th roots as follows: Each element a of a pseudo-Michael algebra A with 2n

1)

va(a) < 2 p" - 1/( n

has a unique q. n th root b with b ~ a, va(b) <~ va(a). In particular every t. nilpotent element a has a unique t. nilpotent q. n th root b. The proof is again by the fixed point t h e o r e m m e t h o d . Here we take r/a < 2 p " - 1 / ( ~ - ~ - 1) and define T by

T x - - nl ( a (observe:

(e§

( ~t

) x 2 "'" ( nn - 1

( ~

)

Y§247

) xn-1 _ xn) .

( n n-

1

)yn-l

+

yn

(y E

A).) 5.3.8.

PROPOSITION.

Let A be a complex or strictly real,

Some Analysis

238

commutative unital, p s e u d o - M i c h a e l - in particular p-BanachAlgebra. For every element b E A with a(b) - {0,1} there is a unique element d E x/A such that u - b + d is idempotent. PROOF. Since, by 7.2.21, A is t. spectrally Gelfand we have

{x(b)'x E/ke}.

{0, 1} -- a ( b ) -

It follows that for every X,x(b) x(b), b 2 - b E ker X. Therefore

0 or 1, so that,

c -- b2 - b E ~ f A - x / ~

x(b 2) -

(see 7.2.12).

Now

a ( 2 b - e ) - 2 a ( b ) - 1 - { - 1 , 1} ~ 0, hence (2b e) -1 exists. It follows that - 4 c ( 2 b - e) -2 e x / ~ and so it is a t. nilpotent element (see 7.4.10). This element has, by 5.3.4 a t. nilpotent q.sq.r, f0 E A. By 7.4.10, f0 E x/~. It follows t h a t if f - e + f0 then -

f2 __ e - 4c(2f - e) -2. Set t

d-

2b - e

2b- e

2

2

f.

Then d-

2b- e ----~(f-

e)-

2b- e ------~-fo E x/rA.

Also

b+d-

2

and a simple computation shows that (b + d) 2 - b + d. Thus u - b+d, with d E x/~, is an idempotent. If v - b+dl(dl ~ x/~) is also an idempotent then u - v E v ~ , so that, by 7.2.12, 7.2.21, a ' ( u - v) - {0}. If follows from ( v i i ) o f 7.2.22, and 7.2.23 t h a t u - v, d - dl, proving the uniqueness of the choice of d E x/~.

t For a motivation of the definition of g see [30, p.48].

w4. Complex Vector-valued Line Integrals

239

5.3.9. COROLLARY. If the algebra A is s.s. then every element b e A, with a(b) - {0, 1}, is an idempotent.

PROOF. Since v / A tent.

~4.

Complex

{0}, d - 0 ,

and b -

b + d is idempo-

Vector-valued Line Integrals Cauchy's Theorems

and

5.4.1. Let X be a complex TLS and F a rectifiable (continuous) path in the plane C, represented by z - z(t) (~ < t < ~; ~, ~ e ~)

Let f " F --~ X be a (vector-valued)function on r. If r - { t o a < tl < t2 < " ' " < t~ --/~} be a partition of [a, ~] we set n

s(~)

- s(~

. z;.) - ~

f(g.)(~j

-

~j_,)- ~ f(g.)A~j,

j=l We where zj - - z(tj),z;, _ z(t~) for some t~. with tj-1 ~< tjt ~ < t j . denote m a x l t j - tj-ll by I~1. If l i m S ( ~ ) , as [~[ ~ 0, exists we J

say

that the line integral;

f fdz Jr

exists and write

fr f dz - ~r f (z)dz It is clear that if f r f d z ,

flg)dz (a, fl e C) and we have

lim S ( r ) .

I~l -.o

frg

dz exist so does fr(c~f §

fr(

c~f § flg)dz

5.4.2. LEMMA. Let X , Y

f-

f (z) 9F --~ X

be complex TLS's, F c C a path,

a function such that / f dz exists. Let T " Jr

Some A nalysis

240

X -~ Y be a continuous linear transformation.

Then f T f dz

exists and T fr f d Z - f r T / d z . PROOF.

T fr f dz

T lim ~-" IZxl-~o lim IAl-,o

5.4.3.

~-~(T f(z~))Az k - fr Tfdz.

If ~ fdz exists and x* E X*, t then

COROLLARY.

PROOF. T h i s follows from 5.4.2 by t a k i n g Y - X* and T -

X*. 5 . 4 . 4 . COROLLARY. 5.4.2, and x E A then (i)

If X -

A is a TA, F and f as in

X ( f r f dz ) - fr(xf)dz.

(ii) PROOF. These follow from 5.4.2, by t a k i n g Y - A and T -

lx or rx. Let X - (X, I1"11) be a complex Banach space, F a rectifiable path in C, and f " F -~ X a continuous 5 . 4 . 5 . PROPOSITION. Arm

function. Then / f d z Jr

exists and we have

t X* denotes the (continuous) dual of X.

w4. Complex Vector-valued Line Integrals

241

f

(,)

II Jr f dzJJ < MIFI. where M -

sup{llf(A)l I 9A c F} and IF[ denotes the length of

F. PROOF (following [14, p.62]). Let F have parametric representation z - z(t)(a <.t<~ ~). Since f is continuous on F, the function f * ( t ) - f ( z ( t ) ) i s continuous on I - [a,/~]. Since I is compact, f* is uniformly continuous on I. Hence, given e > 0, there is a 6 > 0 such that c [If*(t')- f , (t")[[ < 2-~ if a ~< t ~ <

tu

~< fl,

It ~ - t"[ < ~, where JF[ denotes the length of F. Let 71"1,~ 2 be two partitions of I with [rl],lr21 < ~. Let ~r3 be a common refinement of r l , Ir2. Then it is easy to see that E

E

[[S(Trj)- S(7r3)11 < 2 - ~ " I F ] - ~

(j-

1,2).

Therefore we have [[S(F1)--S(71-2)II <

E

E

][S(~l)-S(~3)ll+[[s(~3)-s(~2)][ < ~ + ~ -

It follows from the completeness of X that fr f d z -

~.

lim S(~r)

I~1--+0

exists. Further, since clearly IIS(~r)[[ ~< MIF[, for each r, the inequality (,) follows. 5.4.6. The general existence result for line integrals proved above (5.4.5) is no longer valid if X is a complex p-Banach space with 0 < p < 1. However, even for such spaces the line integral exists for a special class of functions (see 5.4.9). 5.4.7. DEFINITION. Let X be a complex p-Banach space and P a rectifiable path in C. A function f 9F --~ X is called p-admissible or p-admissible on F if f has a representation co

f(z) -- ~ xjpj(z) j=l

(1)

Some Analysis

242

where x3. E X, ~3. - !p3.(z) are bounded complex functions on r such that the ordinary line integrals fr 993.dz exists and further

Ilxjllll~.ll ~' < ~

(2)

3. where II~jll - II~jllr - sup I~;(~)1, The condition (2) ensures, by the vector analogue of Weierstrass M-test, that the series in (1) converges uniformly on I'. It follows that the function f will be continuous whenever all the g)f are continuous. oo

5.4.8.

LEMMA.

(a).

If f -

oo

)-~xi~i,

g -

j=l

)-~xj~ i are j=l

p-admissible on r so is f + g. o(

(b) If f -

E x 3 . ~ j is p-admissible and r a continuous func3.=1 oo

tion on r

the,,

~~;(~;r

i~ p - a e m i ~ i b l ~ on r

3"=1

PROOF. (a) Since

we have (3O

j--1

OO

OO

Ilxjll II~j + cjllf, ~ ~ II~jll II~o,llf, + ~ Ilxill IIr j--1

j=l

Hence f + g is p-admissible.

(b) II~jr

- sup I~j(~) Cj(~)lf, ~ II~jllf, ll~llf-,,

so that co

j=l

oo

II~jllll~j~llf, ~ IIr

j=l

IIx,.llll~ojllf,)< ~,

< ~.

w 4. Complex Vector-valued Line Integrals

243

which proves (b). 5 . 4 . 9 . THEOREM.([31, p.57]). If f is p-admissible on F then the line integral fr f dz exists and we have GO

j=l

PROOF. The series in (,) converges absolutely since oo

II jll j--1

j

Irl

j GO

-< Irl

II jllll jllf

< oo,

j=l where IFI denotes the length of the curve F. Therefore, this series converges in X (by 3.1.24). Let z - z(t) (a <~ t <~ fl) be the parametric representation of F. Let 7r - {a - to < tl < " " < tn - fl} be a decomposition of [a, fl], I~-I- max Itj - tj_il , and n

S-

S(Tr) - Zf(ztk)(Zk -- Zk_l) k=l

with z~ - z(t~), zk - z(tk) and tk-1 ~ tlk ~ tk. We have" GO

JJs - ~ , ~j fr ~jdzll = j-1

rt

Z

GO

GO

E Xj~gj(Ztk)(Z k -- Zk_ 1) -- E Xj ~ ~jdz

k=l j = l

j=l

k-1 zj)~j

j--1

3"=1

- lIE1 + E211

(1)

Some Analysis

244

where m

oo

xiAi 3"=1

j=m+l

and

)kj -- E

~J (zlk)(zk -- Zk-1) -

(pjdz.

k=l

Since

If~ ~jdzl

(2)

<. I1~;11 Irl,

where we have written II~;ll for II~;llr, we obtain

I~jl ~ II~jll Irl + I1~,11 Irl- 211~jll Irl. It follows t h a t oo

oo

j~jll 9 < ~

j=m+l

j=m+l

II~lll~;I ~ oo

<2"[rl ' ~

[[~jllll~jll ~

j=m+l co

~<21rl ~ ~

j=m+l

li~jllil~;ll ~,

F r o m (3), by using (2) of 5.4.7, we can find, integer N such t h a t for m / > N we have

(3)

for a given e > 0 an

I1~11 < ~.

(4)

N e x t , for any fixed m / > N we can choose a 5 > 0 such t h a t for a d e c o m p o s i t i o n r w i t h Irl < 5 we have n

I~jl~ - [ f r ~ i d z - ~ ~j(z'~)(zk - Zk_l)l p k=l s

~<

m

(j - 1 , . . . , m ) .

(5)

II~lll ~ ~ llxjll I~jl ~ < ~,

(6)

k=l

It follows t h a t m

j=l

w4. Complex Vector-valued Line Integrals

245

so t h a t by the choice of N we get, using (1),(4),(6),

S- ~Xj

~gjdz - - { 1 ~ 1

+ ~2[I ~< ][~111-+-ll~2l[

< E,

j=l

for any S -

S(Tr) with I~l < ~ Therefore oo

/r

lim S - ~ - ~ X J / r ~ J d z f dz - I~l--'o j=l

completing the proof. 5.4.10. COROLLARY. II fP

fdzll <. E I[xjll II~jll~lr[ ~

PROOF. This follows from (.) of 5.4.9, using the inequalities (2) therein. 5 . 4 . 1 1 . DEFINITION. Let X be a complex p - B a n a c h space, G _c C, an open set, and f 9G --+ X a function. The function f is called p-admissible holomorphic on G if f has a representation oo

f - ~

j-1

xi~J

(*)

with xj C X, Wj an ordinary holomorphic function on G, and further for every compact set K C G, we have oo

j=l

where II~jlIK- ~upl~j(~)l. AEK

The condition ( * * ) i m p l i e s t h a t the series

~ x j w i ( A ) con-

J verges absolutely for every A E K. For any A E G, by taking

K - {~) w~ conclude that the ~eries ~ j ~ j ( ~ )

r

~bso-

3" oo

lutely ~nd we h~ve f ( ~ ) -

~;(~).

~ j=l

5.4.12. be

DEFINITION.

A function f

9G ~

X is said to

locally p-admissible holomorphic at A0 E G if there is an

Some Analysis

246

open set Go such that )~0 E Go _C G and flGo ~ X is padmissible holomorphic. If f is locally p-admissible holomorphic at every point A E G we call f locally p-admissible holomorphic on G. Clearly "p-admissible holomorphic on G" ==~ "locally padmissible holomorphic on G " , but not vice-versa (see 6.2.9). Oo

5.4.13. LEMMA (~) If f -

Oo

~xj~j,

g- ~jCj

j=l

~

tocaUy

j=l

p-admissible (respy. p-admissible) holomorphir on G then so is

f+g, oo

(b) If f

-

~j~j

i~ locaUy p-a~mi~ibl~ (,~py. p-

j-1

admissible) holomorphic on G and r a holomorphic function on G then co

j=l

is locally p-admissible (respy. p-admissible) holomorphic on G. PROOF. Similar to that of 5.4.8. 5.4.14. PROPOSITION. A p-admissible holomorphic function f " G ~ X is strongly holomorphic and its strong derivative S t is p-admissible holomorphic. Moreover, we have oo

s'(~) - ~ j=l

xj~.(~)

(,)

and in general, oo

f(-)(~) - ~ ~j-. j (-) (~)

( ~ - 1,2,...)

(**)

j=l

where f(n) is the nth strong derivative of f and ~.n) the nth ordinary derivative of ~91. PROOF. We have, for )~, ~ + # E G,

247

w4. Complex Vector-valued Line Integrals

f(A + #) - f(A)

~~~(~)il - ~.(,X)}ll

3"

# - ~oj(A)[ p.

j

(1)

#

Let D be a closed disc centre A and radius r such that D c G, and write C - O D {# e c l ~ l r}. Using the classical Cauchy integral representation theorem we have

- ~(~)

# = 2ri

[1{1 ~j(z)

;

z-A-#

1} z-A

, ] - ( z - A ) 2 dz.

r Now choose # such that I#l < ~, so that for z E C ]z-A-#l) Iz-A I- I#l-r-I#[ > 2" Then

.•!

1 1 --II~jllc'~'~'2~r 2

t

we have

211~illc [~1 -

r--------~

From (1),(2)we get

9

(2)

co

II f(~ + ~) - f(~) - ~ ~j~(~)ll #

~ j

j=l

2 2 oo Jlxjll(~)~ll~jll~l~l ~ -- (~)~l~l ~ ~ Ilxjllll~jll~.

(3)

j=l

Since f is p-admissible the last sum ~ in ( 3 ) i s finite. By making # - ~ 0 in ( 3 ) t h e R . H . S - - ~ 0 and we get (,). It remains to prove that f~ is also p-admissible. Let K _c G be a compact set. Since G is Hausdorff locally compact we can find an open set H with K _ H _C K 1 - H _C G and K1 c o m p a c t . Choose an open covering {D~} of K by open discs D~ _C H. Let 5 be the Lebegue number of this covering. Then for any A c K,

248

Some Analysis

we have (the disc) D D(,k, ~)C_ some Da C_ H C_ K1. By Cauchy's estimates we have"

IG'(~)I--<

II~jlID

-<

~/3

311~jlIK~

so t h a t I~;.lIK-< 311~jlIKx ~ . It follows t h a t 3 j--1

cr j-1

whence f~ is p-admissible. Finally, the representation obtained by iteration (using (.)).

(**) is

5 . 4 . 1 5 . We proceed to obtain a version of the Cauchy integral formula for vector-valued functions. First we recall some classical concepts. We call a closed rectifiable path in the plane C a contour. The index of a point A E C \ F with respect to a contour F is the integer I given by the integral

I - I(A, r) -

1 f

az

2 r i Jr z -

A

If F is contained in an open set G _ C, we call F homologous to zero (in G ) , in symbols, F ~ 0, if I()i, F) - 0 for all A 6 C \ G . Again, we call F homotopic to 0 (in G), in symbols, F ~ 0, if F is homotopic to a constant curve (i.e. a point) in G in the usual topological sense. It is known t that" F ~ 0 :~ F,~, O. Let G __ C be an open set and S _ G a s u b s e t . Following [24, p.241] we say t h a t a contour F C_ G surrounds S in G if the following conditions are satisfied: S ~ F - 0,

t See, for example [8, p.93, 6.10].

w4. Complex Vector-valued Line Integrals

1 0

I(I, F)-

249

if A E S if ~ E C\G.

5.4.16. THEOREM (Cauchy integral theorems). Let X be a compl~ p-Ba~ach ~p~c~ (0 < p < 1) a~d f - f(A) a X - v ~ l ~ d (complex) strongly differentiable function on an open set G C C_ C. If 0 < p < 1 we also assume that f is locally p-admissible holomorphic on G. Let F be contour in G with P ... O. Then we have: (a) Integral theorem" fr f ( z ) d z - O. (b) Integral formulae" For every A e G\F with I ( A , F ) - 1,

1 f(A)-

27ri fr

dz

(,)

and more generally,

n! ~r ( z - f(z) d~ A) "+1

f ( ~ ) ( A ) - 2~ri

( ~ - 1,2,...)

(**)

where f('~) denotes the nth strong derivative of f. PROOF. (a)First assume that X is a Banach space ( p - 1). Then by 5.1.2, 5.4.5 fr f ( z ) d z exists. If x* E X then by 5.4.3

x.(/:
/(~)- ~ xj:j(~)(~ e a) Y when ~j are holomorphic. By 5.4.9, fr f (z)dz - ~ xj f r ~pj(z)dz - O, J

Some Analysis

250

since each fr pjdz - 0, by the classical Cauchy theorem. Next consider the case where f is only locally p-admissible holomorphic on G. Since F is compact we find open sets

aj(j-

1,..., n) c__G

such that fj - fIGi ~ X is p-admissible holomorphic on Gj. It is not h a r d t to see that we can find contours F j such that Fj c Gj and F ~ FI+.--+F~. It follows that n

j--1

J

(b) In the case X is a Banach space the integral formula (,) is obtained by applying x* E X*, exactly as in the above proof of (a). Thus

1 /r fz(Z~d z ) - ~1 frX*f(Z)dz-x*f(z ) z- A

*(-f~ri

for every x*, whence (,) follows. The proof of (**) (for this case) is similar. Next consider the case where X is a p-Banach space and f p-admissible holomorphic. Write (3O

f ( z ) - ~ xjpj(z) (~j holomorphic )

(1)

j=l

f(z) ~j(z) g(z)- (z-)~)n+l' Cj(z)- (z-- ,~)n+l" Then oo

g(z) - ~ ~jcj(~). j=l

See [31, pp.62-63 (diagrams)].

(2)

w4. Complex Vector-valued Line Integrals

251

First assume that f is p-admissible on G. Then we see that each Cy is holomorphic on Go - G\{A}. Let K C_ Go, be compact. By compactness of K there is an r > 0 such that Iz - A

>rforallzEK.

Therefore

IlCj

I~j(z)l

IK - zEK sup ] z - ,~In+l ~

II~jll~ r n+l

so that oo

K

IlxjllllCjll~

)-~ IlxJll r(,~+i)p

j=l

J

1

<

oo

r(n+l)p E .i=1 (20.

Ilxjllll~jll~

It follows that g(z) is p-admissible holomorphic on Gj, so that by (,) of 5.4.9 we obtain from (2)

27r~

g(z)dz

=

E xj~ j--1

E J

(z -- )t) n+l dz

~(~)

x y ~ n!

1

n! f('~)(A) (by (**) of 5.4.14).

(3)

Again from (1) we get

1 fr ( z - f(z) dz. g(z)dz - 2~ri ~)n+l

2~i

From ( 3 ) , ( 4 ) w e conclude that

n! /r

2~i

f(z)

dz-

( z - ~)~+1

fr

(4)

252

Some Analysis

The extension of the proof to the case where f is locally pholomorphic on G is carried out as in the proof of (a). 5.4.17. COROLLARY. I f f ( A ) -

EXj~j(A)

then

J ~j

J PROOF. We have

~~ I(")(~)

=

2~;

/(z) (z_a).+i ~z- ~~j~.

(z _ ~).+1 dz

J

J (using the classical Cauchy representation theorem for pj). 5.4.18. COROLLARY(Cauchy's estimates). Suppose that A E G c_ C and F0 a circle centre A, radius r, such that F0 C G. Then: (i) If X is a complex Banach space and f " G ~ X strongly differentiable (strongly holomorphie) then

IIf(~)(~)ll <

Mn! rn

sup l/(z)l. Iz-,Xl--~ (ii) If X is a complex p-Banach space and f " G ~ X missible p-holomorphic, then where M -

n!

oo

IIf(")(~)ll < (~)~ ~ II~Jllll~Jll~o, j=l

PROOF. (i) We have from the integral formula

IIf(")(~)ll <~

n~ f~ (z-~)-+l f (z) I1~. n!

M

dz[i

2r "r '~+1" 2rr( using ( , ) o f 5.4.5) Mn! rn

ad-

(*)

w5. Power Series Operating in TA's

253

(ii) Here we have

n!)o j

.7"=1

using the classical Cauchy estimates. Thus we get (.). 5 . 4 . 1 9 . COROLLARY. A (strongly) entire (respy. p-admissible

entire) function on a complex, Banach space (respy. p-Banach space) is a constant. PROOF. This follows, as in the classical case, from the Cauchy e s t i m a t e for n - 1, by making r ~ oc.

w 5.

Power

Series

Operating

in TA's

co

5.5.1.

Let f(A) -

~ T n A n ('In C K) be a power series. Let n-=0

A be a unital TA (with unity e) and x E A. We say t h a t

f

oo

operates on x if the series ~ ' l n x n (recall x ~ - e) converges in rL:0

A. Even when A has no unity it is meaningful to consider the operation of a power series f with f(0) = "~0 = 0 on an element o@

x" ~--~3'~x". (A power series f with f ( 0 ) i s called a power series n-=l

vanishing at 0.) Let now A = (A, {pa}) be a locally pseudo-convex algebra. T h e n we can speak of f - ~--~Tnx '~ operating absolutely on x n

if ~ p a ( x '~) converges for each pa. If A is complete (or even n

sequentially complete) and f operates absolutely on x then f also operates on x (because of 3.1.24). An entire t function ]" with f(0) - 0 is said to operate on t entire in the rea.1case means real analytic everywhere, i.e. is represented by a power series which converges everywhere in ~ .

Some Analysis

254

an element x of a TA A if its power series expansion ~ / , ) ~ n n

operates on x; if A is unital we can consider the operation of any entire function on x. If a power series f operates on every x E A we say t h a t f operates on A. 5.5.2. Remark. Every entire function f with f ( 0 ) 0 (respy. every entire function f ) clearly operates on any nilpotent element a of a TA A (respy. unital TA A ) . Every such f operates also on an idempotent element u "/nun--~/0e+

~/n

U .

5 . 5 . 3 . PROPOSITION. Let A - ( A , { p a } ) be a complete (or even sequentially complete) locally pseudo-convex TA. Then all entire functions vanishing at 0 operate on an element x iff

u~(x) - limpa(x n) !n

<00~

n

or equivalently ,

1

ua(x ) -- suppa(xn)-~ < oo, n

for every Pa. In particular, entire functions vanishing at 0 operate on any t. nilpotent element.

C-

PROOF. First suppose t h a t ua(x ) < c~ for all a. Choose a Ca > 0 such t h a t u~ ~< C. Then p~(x")<. C n ( n - 1, 2 , . . . ) oo

Let f ( ) ~ ) - ~ T n ) ~ n be an entire function (vanishing at 0). T h e n n-1 1

lim 12~1~ - O. It follows that, given an e with 0 < e < 1, there is n--, oo an integer N Therefore

N~ > 0 such t h a t for n/> N, 17nl !. < (ec_l)o~.x

oo

E n--N

Pa(~'nzn)

--

E n oo

n=N

]"/nlP"Pa (an) ~ E ( E C - 1 ) nCn n

w5. Power Series Operating in TA's

255

T h u s ~-~7~x n converges absolutely and so also in A. T h i s m e a n s n

that

f o p e r a t e s on x, as required . N e x t s u p p o s e t h a t va(x) - cc for some a. Let p - pa be the h o m o g e n i t y index of pa. Since va(x) - co, there is a sequence 0 < n l < n2 < ' " of integers such t h a t 1

pa(x nk)"---k > k (k - 1, 2 , . . . ) . B u t t h e n the entire function oo

~nk

k-1

which vanishes at 0, does not o p e r a t e on x, since pa(x nk)/k nk > 1 xnk (so t h a t ) ~ 74 0 and the series for f ( x ) does not converge (by k~ 2.1.32). For the p a r t i c u l a r case s t a t e m e n t in the p r o p o s i t i o n it is e n o u g h to observe t h a t if x is t. n i l p o t e n t then, by definition, v~(x) = 0 for all a. 5 . 5 . 4 . COROLLARY. All entire functions vanishing at 0 oper-

ate on A iff v~*(x) < c~ for all x and all a. 5 . 5 . 5 . COROLLARY. All entire functions vanishing at 0 operate on any complete locally sm. pseudo-convex algebra A. PROOF.

If A -

(A,{pa})

t h e n since pa

is sm., by 3.3.6,

v~(x)(<~ pa(x)) < co, and c o n s e q u e n t l y v~(x) < co. T h e corollary now follows f r o m 5.5.4. 5.5.6.

PROPOSITION.

algebra with Pl -< P2 -< " " . 1 , 2 , . . . ) such that (i) p j ( x l x 2

""Xn)

< Cj,nPj+l(Xl)"'P.i+l(Xn)

A (ii) sup ~n c . n

Let A ( A , { p j } ) be a p s e u d o - ~ If there are constants Cj,n ( j , n -

-- c i < cc

for X l , ' ' ' , X n

e

256

Some Analysis

then there exists a family {qj " j >1 2} of sin. pj-seminorms with {qj " j >>2} ,-- {pj " j >1 1}, so that A is a locally sm. pseudo-~algebra.

PROOF. Write B j ( r ) (i),(ii) we have

{x E A ' p j ( x )

< r}. By conditions

pj(XlX2"''Xn) <~c~pj(Xl)'''pj(Xn).

(1)

It follows that the Cartesian n th power B~+l(C~ 1) - - ( B j + l ( c ~ l ) ) n C_ Bj(1).

(2)

Set oo

UJ+1 -- Cpj+I ( U B~+l(C21)) (J - 1,2,...)

(3)

n=l

where Cpj+l denotes the p/+l-convex hull. Since pj+l ~ Pj, Bj(1) - which is absolutely p j - c o n v e x - is also, by 4.1.7, absolutely pj+i-convex, so that (2) implies that Uj+I _c Bj(1). Also, from the definition of Uj+I we have Bj+l(cji ) C Yj+ 1. Thus

Bj+I(C; 1) C_ Uj+I C_ Bj+I(1). By 4.1.10, (since Uj+I pj+i-gauges of 4.1.8 and

(4)

Uj+I determines a p/+l-gauge qj+l which is sm. is a subsemigroup as can be easily seen). Taking the of the sets in (4) and using the relations (,), (, 9 ,) Remark 4.1.9 we get

t,j+~ Pj

Pj

~ qj+l ~ Cj-1 Pj+I-

(5)

It follows from (5) that { q n ' n >1 2} ~ { p n ' n >1 1}, completing the proof. 5.5.7. LEMMA. 9 " , x n C A set Xl,

W(,•) k ( xl~

Let A

9,

be a commutative algebra.

x,) -

(x,1 + . . . + x,k)" n

For

257

w5. Power Series Operating in TA's

where the s u m m a t i o n is over all i l , . . . , i k n. Then

with 1 <<.il < i 2 . . . i k

n

X l X 2 . . . X n -- ( - 1n!) '~ ~ ( _ l ) k W ("'k n)

k=l

(Xl'''''Xn)"

(, )

PROOF. Using the multinomial theorem the coefficient of "~. ..xi~~ (1 ~< r ~< k) in th e development the typical term xi~ n, (at!) ( ~ - r )r of W k(n) is (/21!)--.

'

since ( ~ - r )r

of the sub-

sets of { 1 , - - . , n } which have k elements contain i l , ' " , i r and each of these subsets yield the typical term with the coeffin! cient ( n l ! ) - ' . (n~!)" Therefore the coefficient of xi,r t l . . . x ~ f in n

~-~(-1) ktcT(n) "'k , is equal to

k=l n!

k ~r

(n 1!)... (rtr!) _ (--1) k (~

- r _ n! )n-r r ) (nl!)''" (nr!)(1 - 1 -- 0

when r -r n, and is equal to n!(-1) n when r - n This completes the proof. 5.5.8. THEOREM. Let A - (A,{pj}) be a commutative ~ algebra, where we may assume that {pj} satisfy the condition Pl "< P2 4< " ' ' ; pj(xy) ~ Pj+l(X)pj+l(y)

(*) (**)

(see 4.6.1). If there exist constants c~,n such that (i) p j ( x n) <~ c;,, pj+i(x) n ( n , j - 1 , 2 . . . )

(ii) sup, ~ , ~

- cj* < cr

(j-

1, 2, ...)

then A is a locally sin. ~-algebra.

PROOF. Suppose that X l , X 2 , . . . ,x n E A. Assume first that pj+l(Xk) ~ 0 ( k 1 , . . . , n ) . Write yk - xk/pj+l(Xk); then P j + I ( Y k ) - 1 ( k - 1 , . . - , n ) . It follows that

Some A nalysis

258

p j ( X l X 2 " ' ' X n ) -P j + l ( X l ) "" "Pj+I(Xn)pj(YlY2 "'"

Yn)

pj+l(Xl) " ""pj+l(Xn)pj( (-1)n W~ n) n! E (Yl,"',Yn)) k <~

1 -~..Pj+l(Xl) "" "Pj+l(Xn) E P j

(W~n)

(YI,""" ,Yn))

(1)

k where we have used indentity ( , ) of 5.5.7 and the subadditvity of pj 9 Using the definition of W k(n) we obtain the inequality"

w(,~) (y~,--.,y.)) pj(,,~

~

<~

pJ((y,~+... + y,~)")

l ~i l <~i2""~ik
; E c;,.k" ~ E c;.n" (2) From ( 1 ) , ( 2 ) w e get pj(XlX2 "''Xn)

,

<~

) nn)pJ+l(xl) "''pj+l(Xn)

k <

(3)

Cj,nPj+l(Xl)'''p.i+l(Xn)

where _

1 ,

cjo-,cjo(E( k

If for some k , p j ( x k ) - O, then by condition pj(XlX2"''Xn)

--

2nn n ,

n,

(4)

(**) above we have

p j ( X l X 2 " " " Xk-1 Xk+l " " " XnXk) p j + I ( X l X 2 " ' ' x k - 1 Xk+ l ' ' ' x n ) p j ( x k )

-- 0

so that (3) is satisfied trivially. Thus (3) holds w i t h o u t any restriction on xk. This means that condition (i) of 5.5.6 holds. We

w5. Power Series Operating in TA's

259

shall next show condition (ii) of 5.5.6 also holds. Now 2nn

n

2n

~

( n! )~--ff~. so t h a t lim

2n

n

= 2 lim

= 2e

(5)

where we have used for the last equality the result (1) in the proof of 4.8.5. It follows from (4),(5) t h a t lim ( cj,n )~X - 2 e

, ) n~ - ~ - 2 e c j ., lim (ci,

n----~ O0

n---~ O0

The t h e o r e m now follows from 5.5.6 5.5.9. R e m a r k . One does not know whether the result in 5.5.8 is valid for p - ~ - a l g e b r a s (0 < p < 1). The above m e t h o d of proof, however, breaks down in the p- ~ case, since corresponding to the limit in (4) we now have l i m 2 n / ( n ! ) ~ which turns out to be infinite (lim

n

(~!)o

_

__

lim

n

.~oo (,!)

.

~r~ 1 -

p __+ e p . O 0

--

00 ) .

5 . 5 . 1 0 . THEOREM (Zelazko [31, p.140]). Let A (A, {pa}) be a commutative ~-algebra, where we assume that the sequence {p~} ~ati~fy th, c o , d i t i o ~ (,),(**) i~ 5.5.s. If for , v , , y ~ e A and every j we have

~j (~) *

suppj(~")-'

--

n

< 0 0

n

then A is a locally sm. ~ algebra. PROOF. Observe first t h a t v ] ( x ) -

fn(2)

-

l i m n f ~ ( x ) , where

pj(x k) l<<.k<<.n m a x

Since the pj are continuous so are f,~. Hence v ; ( x ) is the pointwise limit of continuous functions of x on a complete metric

Some Analysis

260

space. Therefore u] belongs to the first Baire class of functions and consequently u~ has at least one point of continuity x0 in A. This means that there are constants C, 5 > 0 and an integer k ~ such that

p~,(~- ~o) <~ ~ ~ ~;(x) <~ c. If k = m a x ( j , k I) then we have

; ~ ( ~ - ~o) ~< ~ ~ ~;(~)<- c, since by hypothesis pj <~ Pj+a for every j. So we get from the definition of uj t h a t

pk(x - xo) <~ 6 ==>pi(x n) ~ C n (n = 1 , 2 , . - . ) .

(1)

In particular, we have pj(xr~) ~ C n. Writing y = x - x 0 , we have for any x satisfying the condition in (1),

_ilx_xo,nl-pj_i(

Pj-I(Y n)

k=O

.< 2 - c ~ =

k=O

(2c) ~.

(2)

Let now y be an element of A with Pk(Y) :/: O. Writing yl = 5y/pk(y) we have p k ( y i ) = 5, so that by (2) we obtain

(2C)'~,

Pj-1 (yr~)

~

Pj-l(y n)

<~ (2-~~)npk(y)n.

which gives

(3)

If Pk(Y) = 0 then since k ) j we have

Pj_l(yn) ~ pj(y)pj(yn-1) ~ pk(y)pk(yn-1) _ O. This means t h a t (3) holds in this case as well. The integer k figuring above depends on j, so we shall write

k = k(j). Set n 1 -- 1, rt 2 -- k ( 2 ) , ' - ' ,

rtj-

]r

1 + 1).

w5. Power Series Operating in TA's

261

If we set qj - {Pnj } then {qj} ~ {pj} and by virtue of inequality (3), the conditions (i),(ii) of 5.5.8 are satisfied and the theorem follows from 5.5.8. 5.5.11. COROLLARY (Mitagin-Rolewicz-Zelazko [10' p.295]). A commutative ~-algebra on which all entire functions vanishing at 0 operate is a locally sm. ~-algebra.

PROOF. This follows from 5.5.4, 5.5.10.

C H A P T E R VI SPECTRAL

w 1.

ANALYSIS

Spectral

I N TA s

Properties

6.1.1. PROPOSITION. Let A be a Q algebra (in particular a p-Banach algebra) and x E A. Then the quasi-resolventt set

q a i- p ct, mt

compact

subset of K. PROOF. If A0 E p ' ( x ) t h e n A0 ~ 0 (by 1.7.6)and --~olx is q. invertible. Since A is a Q algebra there is an open neighbourhood U of - A o l x , comprising q. invertible elements. Since the map A ~ - A - i x is continuous, there is an open neighbourhood N of A0 such that for A c N, - A - i x E U, so that - A - i x is q. invertible. Thus N C_ pt(x)and pt(x)is open, its complement at(x) is closed. Next let U(0) be an open nucleus comprising q. invertible elements. Since - A - i x -~ 0 as IAI-~ oo and U(0) is open there is a constant C such that - A - i x E Y(0) for I)~l > c . Hence, for such A, - A - i x is q. invertible and A E p~(x). This implies that if A E a~(x) then IAI ~< C, whence a~(x) is bounded. Being both bounded and closed a t(x) is compact, as required. 6.1.2. C o r o l l a r y . In an I algebra (in particular a unital pBanach algebra) A, p(x) is open, a(x) is compact, and a(x) # K. PROOF. By 3.6.13, A is a Q algebra and so by 6.1.1, a ( x ) U { 0 } = a'(x) is compact. If 0 r a(x) then x is invertible. Since A is an I algebra there is an open neighbourhood U of x, comprising invertible elements. Since x - Ae ~ x as IAI--.0, there is an e > 0 such that for IAI < e , x - ~ e E U and hence invertible, so that A e p(x). It follows that 0 ~ a(x) (the closure of a(x)). On the other hand, since a(x) c a'(x) and

t For definition see 1.7.5.

w 1. Spectral Properties

263

at(x) is closed we have a(x) c_ a'(x) - a(x) [,J{O}. Hence a(x) a(x),a(x) is compact and p ( x ) i s open. Finally, since a ( x ) i s b o u n d e d , a(x) :/: K. If A is a real Q algebra and x E A then the extended quasi-spectrum St(x) is compact, in particular 6.1.3.

PROPOSITION.

PROOF. Let A - a § ifl ~ 0, be a complex number. T h e n x 2 - 2ax

x2 - 2ax

(note lal//IAI 2 ~< & I~l ~ 0) It follows t h a t if U is a nucleus of A, comprising q. invertible elements then there is a C > 0 such t h a t x2 - 2ax

E U,

and hence y q. invertible, if IAI > C. This implies, by 1.8.5, t h a t for IA] > C, A r 5'(x). This means t h a t 5'(x) is bounded. To complete the proof of its compactness it remains to show t h a t 5'(x) is closed, or equivalently, ~'(x) is open. Using 1.8.5 and the fact t h a t x 2 - 2ax x 2 - 2(Re A)x

is a continuous function of A (for A ~= 0), it can be shown, exactly as in the proof in 6.1.1 for p'(x) being open, t h a t ~'(x) is open, completing the proof of the proposition. 6.1.4.

COROLLARY. If A is a u nital Q algebra then 5(x)

is compact. PROOF. It is clearly sufficient to prove t h a t 5(x) is closed in 5'(x) - 5(x)I.J{0} and we carry out the proof as in 6.1.2. We can assume t h a t 0 ~ 5(x), so t h a t 0 E ~(x). Then x, and hence x 2, is invertible. Therefore, A being an I algebra we can choose an open n e i g h b o u r h o o d U of x 2, comprising invertible elements. Since _

_

+

0

264

Spectral Analysis in TA's

we can find an e > 0

such t h a t for ]a),ifl[ < e ,

z~,z -

xa,/~EU. If

9 - (~ + i/~)~

then z~,~,~

- x~,~ e U for [a[, I/~[ < e.

From the choice of U, x~,/~ and hence z~,~, is invertible, whence A - cz + i/~ E ~(x) for tat, I~] < e. This implies t h a t 0 ~ 6(x), whence 6 ( x ) i s desired.

w 2.

The

Resolvent

closed in 6'(x), as

Function

6 . 2 . 1 . DEFINITION. Let A be a unital algebra (over a field F ) , z E A and p(x) the resolvent set of x. We assume t h a t p(x) ~ O. The vector-valued function

R,

~ ~ x~ - x ( A ) -

(~- ~)-1

on p(x) is called the resolvent f u n c t i o n of x; x~ is called a resolvent of x.

6.2.2. PROPOSITION. Let A be a unital algebra and x E A. Then: x~ - x~ - (A - # ) x ~ x t, (A,# E p(x))

(~- ~)-1

_ (~- ~)-1

(Hilbert relation);

_ (~ - ~ ) ( ~ - ~ ) - , ~ ( ~ _

(A - 0 or A-I E p(x), # - 0 or ~ - I E p(x)).

(,)

,~)-1 (**)

PROOF.

(x_~)(x_,~) ~ [(~_~)+(~_~)~] (~_,~)-1_~+(~_~)~. Multiplying both sides of the above equation, on the left, by x~ we get ( x _ ~ e ) - I x~ + (~_ A)x~x~

265

w2. The Resolvent Function

which gives changing signs on both sides we get relation (,). The relation (**) can be obtained by noting that

(~ - ~x)(~ - ~ ) - ~

6.2.3.

-

[(~- ,x)+

(~ - ~)~](~ - , x ) - ~

--

e Jr-(~- ~)x(e-

~x) -1.

COROLLARY. Any two resolvents xa,

xu of x com-

mute" x~ ~-~ xu.

PROOF. Interchanging )~,# in (,) of 6.2.2 we get

,'). Changing the sign in (,') and comparing it with (,) we get

Since we may assume that A ~= # we conclude that xa ~ x~. 6.2.4.

PROPOSITION. Let A be a unital Hausdorff TA and

x c A. Assume that

(i) p(x) is open; (ii)

A ~ xx (A C

p(x))

is continuous.

Then the resolvent function xa is a strongly infinitely differentiable function with dxa = ~ d~

(.)

and in general, d"xa = . ! ~ + 1 d~ ~

(**)

Further, if

(iii) a(x) # 0 and compact and p # ( x ) - (p(x))\{0}) -1 U{0} (p(~)\{o}) -~ - { ~ - ~ - ~

c p(~)\{o}}

266

Spectral Analysis in TA's

then p # (x) is open and on it (e-,,~X) -1 with dn dA n [ ( e - )iX) -1]

i8 ~nfin~tely

differentiable

n ! x n [ ( e - )~X)-I] n ()~ E p # ( X ) ) .

--

* * :4:)

PROOF. If A E p(x) then A + # E p(x) for sufficiently small # since p(x) is open. Using the Hilbert relation we get XA+ # -- X A

xA+#X A .

Making # --~ O, and using condition (iN) above we get (,). The formulae (**) is obtained by induction. Assume thus d n - i xA

dan-1 - - ( n -

Then d

dnx~ dAn

= (n-

1)!x~.

1)! d

~-~(x~). Now

1

dA (x'~)=L,~o-~(x'~+~

-

x~)

-

L.-,o x ~ + g - x ~ ~ #

dx~ n-1 dA 9rLxA

-

-

x~+gx~

O<j+k=n- 1

n x ~ + l ( using (,))

where we have written L for lim. Therefore dd~,~- (n!)x~ +1. (The factorization of x~+~ - x~ used in the above calculation is based on the property x~+u ~-~ xA which is assured by 6.2.3.) It remains to prove the assertions on p#(x). Since p(x) is open, (p(x)/{O}) -1 is also open. To prove that p#(x) is open it is enough to show that 0 is an interior point of p# (x). Since a(x) is compact we have 0 ~ r(x) < co. If r ( x ) - O, then a(x) - {0} and so p # ( x ) - ~ is open. Next assume that 0 < r(x)< co. If 0 < [AI < - ~ then I)~-11 > r(x), so that )~-1 E fl(X),)t E p # ( x ) . Therefore 0 is an interior point of required.

p#(x) and p#(x) is open as

Finally, the relation ( , , , ) i s established by using ( * * ) o f 6.2.2 and induction in exactly the same way as for the proof of

267

w2. The Resolvent Function

(**) above. 6.2.5. COROLLARY. Let A* be the continuous dual of A and x* E A*. Then F~,(A)-x*(x~) is a ~<-valued infinitely differentiable function on p(x) with d~F~,(A) dnx~ dA n =X*( dA n )

(n-

1,2...).

When ~: - C, Fz. is a holomorphic function and xa a weakly holomorphic function on p(x).

PROOF. This follows form 5.1.2. 6.2.6. COROLLARY. The conclusions and formulae ( , ) , ( * * ) , ( , , ,) of 6.2.4 hold in any Hausdorff CI algebra (in particular, a unital p-Banach algebra or more generally a unital sm. (F) algebra). PROOF. It is enough to prove that the hypotheses (i),(ii) of 6.2.4 are satisfied. By 3.6.22, 3.6.23 a unital (F) algebra A is a CI algebra and so, by 6.1.2, hypothesis (i) of 6.2.4 is satisfied. Again the continuity of the maps A ~ x - A e , y~-+ y-1 ( A b e i n g a C algebra) implies that their composite map A ~ x~ is continuous (which is hypotheses (ii)of 6.2.4). 6.2.7. LEMMA. In a C I - in particular a p-Banach - algebra A, x~ --~ 0 as (A PROOF. As IAI-+ c~, II-11 --, 0, and A being a CI algebra we have X,k - -

--)~-l(e

--

A - l x ) -1 --+

6.2.8. PROPOSITION. Let A = (A, algebra and x E A. Then we have

O.

I1"11)

e --

O.

a unital p-Banach

(i) If [AlP > u(x) - limn IIx"[] 88 then A e p(x) and the resolvent x~ is given by oo XA - - ( X -

xn

,'~C)- 1 - - -- E ~ n + l ' n--0

with the series converging absolutely.

(*)

268

Spectral Analysis in TA's

(ii) If A c p(x) and ] A - #1 <

IIx~ll-~

the~ ~ c p(x) a~d

oo

x.-

~(,-

a)"x~§ 1,

(**)

n--0

with the series converging absolutely. PROOF. Write A have, by 3.3.20 (i), XA - - ( X - ) ~ e )

A-1 9 Since v ( A x ) -

-1 -- -A(e-

]A]Pv(x)< 1 we

oo

oo

Xn

--

_

A ~-~1

Ax) -1

which is (,). To prove (**), we first note that we can write - , ~ - (~ - ~ ) k

+ (a - , ) ~ ] .

(1)

Since

[Ix~ll

9 II*~ll-

1,

by 3.3.20, e + (A - #)xa is invertible and so by (1) X#

--

[e -~- ()i -- ~ ) x , ~ ] - l x , ~ oo ( # -- ) ~ ) n x ~ n--O

- - [e -- ( # -- ) i ) x A ] - l x A

oo "X A -- ~(#

-- ) ~ ) n x ~ + l

'

n--0

proving (**). 6.2.9. THEOREM. Let A be a complex unital p-Banach algebra and x E A. Then the resolvent .function x,~ is locally padmissible holomorphic on p(x) U{c~}. PROOF. Let A -

(A, [l" II) where we may assume that []-I1 is

sm.. By 6.2.8(ii), we have for A0 e p(x) and ]A-Aol < IIX~ol]-~, oo

x~ - ~ ( ~ n--O

n

~o/~o

n+l

269

w 2. The Resolvent Function

Write

g~n(A)

-

-

(A-

A0) n , xn

--

and

~o

X n+l

Do-

{A E C

9

1

[ A - Aol < []X~oll-~}. Clearly ~p~ is holomorphic on Do. If K C_ Do and K is compact it is clear t h a t we can encloset K in a 1

closed disc D(,~o, ro)with r0 < tlX~oll- ~ . So

ill-IlK <~ s u p ( [ ~ - ~ol", ~ E Do) ~< r~. It follows t h a t oo

~o Ilronp ~<~llx~ol

IIx, llll~,ll~ ~ ~ l l n--O

xn+l

rg

ro

In + l np

lq,

~ o(3

-- IIx~oll

IIx~oroll" < oo ( since [Ix~oroll < 1)

n--O

proving t h a t x~ is locally p-admissible at Po. Write Gz - {A E 1

c

l~l > ~,(x)~}. By 6.2.8 (i), we have oo X n _ 1

x~ - ( ~ -

~1-~ = - ~

~---~,

(,1

n--1

1

the series on the right converging absolutely for IAI > v(x)~. Writing x,~ - x ~-1, pn(A) - - 1 / A '~, we get for xa the representation x ~ -- E

xn ~n .

rt

Each ~ , is clearly holomorphic on G z. Any compact set in G z can be enclosed in an annulus K-

{A C K ' r l

<~ IAI ~< r2}

1

where v(x)~ < rl < r2. Now we have for A E K, 1" If C - sup{IA-Aol" A E K } , then by compactness of K there is a ,~1 C K with C - I , ~ l - A o I < IIx~0It-~. We can choose for the radius of any ro will C < r o < [[x~01I-~.

Spectral Analysis in TA's

270

1

1

I~.(~) ~< r-~' so t h a t Ll~lIg ~ ~ It follows t h a t oo

oo

EIlIXn-IlIII~gnlIPK <~ Ilxn-lllrl p -- E tlxn-lll < CX), n=

n=l

r~

where for the last inequality we have used the absolute convergence property of the series in (,). This proves the p-admissible holomorphy of x~ on Gx. By interpreting Gz as an open neighb o u r h o o d of c~ we see t h a t what we have just proved a m o u n t s to x~ being locally p-admissible holomorphic at c~. This completes the proof of the theorem. 6 . 2 . 1 0 . PROPOSITION. Let A be a complex unital p - B a n a c h algebra and x c A. Let F be a contour in C surrounding a(x) U{0} (in C). Then:

]r

Amx~dA - - 2 f i x m (m - O, 1 , 2 , . . - ;x ~ - e).

(,)

1

In particular, if r > v(x)~ we have

~

c~ i~ th~ ~i~d~ I~1 = r (~ e C)

/ c AmX~dA - - 2 r i x m.

(**)

r

PROOF. have

Since F

N~(z)

= 0, r

~r A m x ~ d A - -

c p(z)

~oo x" f r n=O

and so by 5.4.9, we

A'~ dA )~n+l

(1) "

We can choose r sufficiently large t h a t Cr C p(x) and Cr ~ F in p(x). Then we have

2ri

A.+ 1 dA - 2~i

r

An+ 1 dA -

01

(2)

w2. The Resolvent Function

271

by the classical results. The formulae (.), (**) now ready follow from (1), (2). 6.2.11. THEOREM (Beurling-Gelfand spectral radius formula). In a complex Banach algebra A we have for any x E A,

r(x)-

u(x)-

If the norm I1" II

1

lim ]]x"}}~.

hav

(*)

also

(**)

r(x) - v(x) <~ Ilxtl.

PROOF. Our proof makes use of the formula (,) of 6.2.10 and is essentially on the lines in the exposition in [24, p.236]. We assume (as we may) that A is unital. By 4.8.11, r(x) ~< v(x) (since here p = 1). So to prove (,) it is enough to show that v(x)<~ r(x). If r > r(x), then by 6.2.10 (**), we have

_xm_

1 / c Amx~dA" 2~ri

(1)

Since x~ is a continuous function of A and C~ is compact we have

M(r)-

sup(llx~]]" IAI - r )

< cr

(2)

using (,) of 5.4.5 we obtain from (1),(2)

1 rmM(r) " 2rr - r m+lM(r). Ilxm]l <~ 2---~ Therefore 1

1

I]xml]-~ <~ r(rM(r))-~ whence

. ( x ) .< r It follows that v(x) <. r(x), completing the proof. 6.2.12.

COROLLARY.

If A is a real Banach algebra then

272

Spectral Analysis in TA's

~(x) -- u ( x ) , where ~(x) is the extended spectral radius. In particular, if Z is strictly real then r ( x ) - u(x). N

PROOF. Apply 6.2.11 to the complexification A. 6.2.13. R e m a r k . We shall obtain later (see 7.4.6) extensions of the above results for p-Banach algebras. 6.2.14. PROPOSITION. Let A - (A, II" II), with a unital p-Banach algebra and x E A. Then:

I1" II

sm., be

(i) a(x) is an upper semi-continuous function of x, i.e., given an open set G C_C_K with a(x) C_ G, there is a ~ > 0 such that a(x + y)C_ G for every y c A with Ilyll < ~. (ii) For ~ c p(x) if d(1) denotes the distance of ~ from a(x) then IIx~ll ~> l/d()~)P; whence I I ~ l l - ~ ~r as d ( ~ ) ~ O. PROOF. By 6.2.6, x~ is a continuous function of )~; by 6.2.7, xa ~ 0 as I)~l ~ c~. It follows that there are numbers C l , r > 0 such that Ilxall < c1 for all )~ with 1)~1 > r, Write D r - {A e K " I)~l 4 r}. Then g - ( K \ G ) N D ~ is compact, so that there is a C2 > 0 such that Ilxal[ < C2 for ~ E K. It follows that if A E K \ G then Ilx~ll < c max{C1,C2}. If y in A with IlY]I < (~ - - C - 1 and )~ E K \ G then

is invertible since )~ ~ p(x) and

II~yll ~< IIx~llllull

<

CC-1-- 1 (

see

3.3.20).

This means that )~ r a(x + y), whence a(x + y) c_ G, proving (i). If A C p(x) and # E a(x) then by 6.2.8 (ii)we have I A - # ] > 1

IIx:~[[-~, so that

II~ll ~ 1/1 ,x -~l p, IIx~ll ~

lld(~) p

proving (ii). 6.2.15. COROLLARY. continuous at x -- O.

PROOF. Take G -

r(x) -+ 0 as x ~ O. Thus r(x) is

{A E N ' [ A I < e}. Then G ___ a ( O ) - O ,

w2. The Resolvent Function

273

whence by u p p e r semicontinuity at 0 we have a(x) c_ G, r(x) ~< e for IIx]] < 5. This shows t h a t r(x) ~ 0 as x -~ O. 6 . 2 . 1 6 . COROLLARY. If A is a p - B a n a c h algebra and x c A then al(x) is upper semi-continuous. PROOF. It suffices to observe t h a t a'(x) - aAl(X) where A1 is the unitization of A. 6 . 2 . 1 7 . PROPOSITION ( R i c k a r t t element of a unital p - B a n a c h algebra either complex or strictly real. Let V of O in K. Then there is a 5 > 0 such IIx - yll < 5 and xy - y z we have"

o(y) c

[23, p.36]). Let x be an A(A,I I 9II) which is be an open neighbourhood that for every y E A with

+ v;

(,)

c o(y)+ v.

(**)

PROOF. Since a(x) + V is an open set containing a(x), the relation (.) follows from 6.2.14(i). To prove (**) we will assume t h a t is is false and show t h a t this leads to a contradiction W i t h o u t loss of generality we can take Y = {A E K : IAI < 2e}. By our a s s u m p t i o n we can find a sequence xn --~ x with xn ~ x, such t h a t a(x) ~ C,~ = a(xn) + V for all n. (1) Choose An C a ( x ) \ G n .

If # e a(xn) then

An - # ~ V, so t h a t IAn - #1/> 2e.

(2)

Since A, e a(x) and a(x) is compact we can, by passing to a subsequence if necessary, assume t h a t

e

(3)

From ( 2 ) , ( 3 ) w e get: IA0 - #I I> 2e > e for every # E a(xn)

(4)

so t h a t A0 r a(x~) for every n. For obtaining the contradiction we have to consider separately the cases A0 - 0, A0 ~ 0. t He obtained the result for complex B~nach ~lgebras.

274

Spectral Analysis in TA's

C a s e 1. ) ~ 0 - 0. Then 0 ~ a(x,) and xn is invertible. Since Igl > c for # e a(x~) we get r ( x ; 1) - s u p ]#-11 ~< e -1. So, by 7.4.6, u ( x ; 1) - r(x;1)P ~< e-p, whence, by 3.7.29(i), x - limx,~ is invertible contradicting 0 - )~0 E a(x). C a s e 2. ,~0 J= 0. Since ,~0 r a(xn), q. invertible for all n. By 1.7.12 a'(Y n) -

1 +a

Yn

-

9a e

-- )io l xn

is

(5)

Since )~ E a(y,) iff -)~,~0 E a(xn), and by (4) I)~0+ )~)~0l > e for - ~ o E a(xn), we get 1

IIx ll

I

1 + )~1 - I)~~ + ,~o,~ I ~<

e

c

~< --e

(6)

for some C > 0 (since the sequence xn being convergent is bounded). From (5),(6) we conclude that r(y~n), and hence by G B formula (see 7.4.6.) u(y~) is bounded. So, by 3.7.29(ii), lim y,~ - -)~o ix is q. invertible, contradicting the fact that )~0 E 6.2.18. COROLLARY. If the algebra A of 6.2.17 is not unital then we have inclusion relations analogous to (,)(**) obtained by replacing a by a I (the quasi-spectrum). PROOF. We have only to apply 6.2.17 to the unitization A1 of A. 6.2.19. COROLLARY.

If A is commutative, r(x) is continu-

ous everywhere. PROOF. This follows from ( , ) , ( * * ) o f 6.2.17. 6.2.20. PROPOSITION. The completion ft of a commutative strictly real p-normed algebra A is strictly real. PROOF. Consider an element x E ti. If possible let a ' i ( x ) contain a complex number )~o - c~o + i~o, with /~o r 0. Write 1 r ] - ~1/~ol > 0. Choose a sequence ( x ~ ) i n A with x~ ~ x. Since

A is strictly real, ~a(x,~) c_ ~. Since A _C Ji we have also Ji _c ti,

275

w3. Pseudo-Resolvent Function and

so

~ ( x . ) c ~(~.) c ~ .

(1)

Write V - {A c C" A - a + i~ with a e R, lilt < ~}. By (**) of 6.2.17, there is a 5 > 0 such that

a(x) c a ( y ) + V whenever I I x - Yll < ~. Choose n sufficiently large t h a t I I x , - ~ll < ~. Then

~5(x) c ~ ( x . ) + It follows t h a t

v c R + v - v. 1 ~1~01, whence / ~ 0 -

A0 e V,l~01 < r l -

0-

a

contradiction. Therefore ~ i ( x ) __ N and A is strictly real.

w 3.

Pseudo-Resolvent

6 . 3 . 1 . DEFINITION. and x E A . Set

Function

Let A be an algebra (over a field F)

pV(x) - {A E F " (Ax)' exists}; pV(x) is called the pseudo-resolvent set of x. Since (0x)' 0 c pP(x) and we have always pV(x) r 0. Write

9'~ - ( ~ ) '

0~ -

0,

(~ c p~(~))

and call x~ a pseudo-resolvent or a p-resolvent of x; A H x~ is called a pseudo-resolvent function, t 6.3.2.

Remark.

The m a p

A ~

_~-1

is a bijection of

p ' ( x ) - p ' ( x ) \ { 0 } onto pP(x)\{O}, where p ' ( x ) d e n o t e s the quasiresolvent set of x. If F - K the map is also a h o m e o m r o h p i s m . If xis q. nilpotent then a'(x) - {0}, p'(x) - F \ { 0 } . Therefore, t This function has been considered implicity by Kaplansky [10 ", p.400, Lemm~ 3.2].

276

Spectral Analysis in TA's

p P ( x ) - - { F \ { 0 } } -1 [..J{0}- {F\{0}} U { 0 } - F. 6.3.3. LEMMA. (a)x~ ~ xu'

(~,~ e p~(~))

I

(b) x . #

4 = ~_ ~ A

(A,~ e p ~ \ { o } )

PROOF. (a) Since Ax ~ #x, by applying 1.1.18 twice we get I Xtt.

(b) We have Ax + (Ax)' + (Ax)(Ax)' - 0

(1)

ux + (u~)' + (u~)(u*)' - 0

(2)

Denoting the expressions in the equations (1), (2) above also by (1), (2)we get u • ( 1 ) - ~ • ( 2 ) - u . 0 - ~. 0 -

0,

which gives ,(~)'

- ~(u~)' - ~ u x [ ( u ~ ) ' - ( ~ ) ' ] .

(3)

Again, (1)x (#x)' and 'Ax" ( ~ x (2) give A tt

x(u~)' + (~)'(u~)' + ~(~)'(u~)' - 0 .

(4)

9( ~ ) , + (~)'(u~)' + ~ ( ~ ) ' (u~), - 0 .

(5)

#

From (4), (5) we obtain by subtraction

1

'(u~)'

and from (3), (6)we get

Ap

A#

(6)

w3. Pseudo-Resolvent Function

277

which reduces, after a change of sign, to the equation in (b). 6.3.4. LEMMA. In a Q algebra A, for each x E A, pC(x) is an open set containing O. PROOF. For A E pP(x), let U be an open neighbourhood of $x comprising q. invertible elements. By continuity of scalar multiplication there is an open neighbourhood N of A such that N x c_ U. If follows that N C_ pP(x) and pC(x)is open. 6.3.5. PROPOSITION. (a) Let A be a Hausdorff C algebra, and x an element of A such that pP(x) is open in ~. Then x~ is a strongly differentiable function of ~ on pP(x), with dx~ dA = - x ( 1 + x~) 2, (,)

dnxk _ (_l)nn!xn(1 + xk) n+l dA,~

(**)

We have also the relation

dn (Xk) d1,~

__ , (_~)n+l

-~-

n.

(A # o).

(b) In a Hausdorff CQ algebra A, the formulae (.), (. **) are valid for every element x in A.

(***) **),

PROOF. (a) From the equation ( 1 ) i n the proof of 6.3.3 we obtain, for ~ E pq(X)\{O}, 9 A

--

A

t = - x ( 1 + xA).

(1)

Making A --+ 0 we obtain

[dx'

(2)

If A, p C pP(x)\{O}, by 6.3.3 (b) we have

!

I

Spectral Analysis in TA's

278

Making #--~ ~ we obtain d,~

(3)

,~2,

which reduces to x~ 1 dx~ ~2 t ~ d~

x~ ~2 '

whence we get dx~ _ x~ (1 + x~) -- - x ( 1 + x~) 2 dA A

(4)

(,)

when )~ ~- 0 . On the other hand (2) shows that the formula in (4) (or (,)) is valid for ~ = 0, completing the proof of (,) for all ~. To prove (**), assume by induction that dn-lx~ _ ( _ l ) n - l ( n 1 ) ! x n - l ( 1 + Xk) n dan_ 1 where we have used

Thus we have proved

Differentiating both sides of the above equation we get dnx~ d/~n

--

( - - 1 ) n - l ( n -- 1)iX n - l " n(1 + xk) n-1 dxk

--

9 d,k (--1)n-ln!xn-l(1 + X~) n - l " - x ( 1 + x~) 2 (using (4))

=

(-1)nn!xn(1 + X~) n+i

which is (**). Finally, by successive differentiation we obtain (, 9 ,) from (3). (b) This follows from ( a ) a n d 6.3.4. 6.3.6. PROPOSITION. Let A be a p-Banach algebra and x E A. Then x~ exists for all ~ such that v(Ax) < 1, or equivalently, 1

lA] < v(x)

p, and we have oo

x~ - ~ (-1)'~(Ax) n

(,)

n=l

with the series on the right converging absolutely. In particular, 1

x~ is analytic for I)~l < v(x)

o.

PROOF. Consider the unitization A1 of A. Then we have (el + ~x) - 1 -

ex + (~x)'

279

w 3. P s e u d o - R e s o l v e n t Function

where el is the unity of A1. By (,) of 3.3.20 we have (3O

~(-1)"(Ax)",

( e I -~- ) ~ X ) - 1 -

rt=0

the series converging absolutely. It follows that we have (x)

x~ - (Ax)' - ~

(-1)n(Ax) n

rt--1

(with the series converging absolutely). 6.3.7.

COROLLARY.

If

A

is a complex p - B a n a c h

then x i is p - a d m i s s i b l e holomorphic on {A e

cl l

algebra I

< ~(~)-~)

PROOF. Set r

(-1)nAn; x . -- x n.

Then (,) of 6.3.6 can be rewritten as oo

n--1 1

For any r0 with 0 ~ r0 < v ( x ) - ~ , set K -- Kr ~ -- {A E C ' l ~ l

r0}.

Then

lie.IlK < So oo

oo

0(3

n= 1

n= 1

n= 1

since v ( r o x ) - r~v(x) < 1. Hence the corollary. 6.3.8. PROPOSITION. Let A be a complex ample C-algebra. I f x e A\{O} is q. nilpotent then {x~ " A e C} is unbounded. PROOF. First note that by 6.3.1 and definition of q. nilpotent, pV(x) - C. Since A is a Hausdorff C-algebra, by 6.3.5 (a),

280

Spectral Analysis in TA 's

z~($z)' is strongly differentiable, and hence for x* E A* F~.($) - x*(x~) is an entire function. If possible let x~ be a bounded function of )~. Then Fzl.()~) is bounded and so by classical Liouville's theorem Fz'.()~ ) is a constant. Since Fz'.(0 ) = 0 we must have F~. (A) - 0

for all A E C.

Putting ) ~ - 1, we get x * ( x ' ) - O. Since A is ample we conclude that x' - 0 which implies x - 0, contradicting the choice of x. Therefore {x~} is unbounded as required. 6.3.9. COROLLARY. (Kaplansky). In a complex normed algebra A, if x ~ 0 is q. nilpotent then {x~} is unbounded. PROOF. Since a normed algebra is an ample C-algebra, the corollary follows from 6.3.8. 6.3.10. LEMMA. Let A be a normed algebra and x E A be q. nilpotent. If ~ . e K are such that the sequence (()~.x)') is unbounded then I~,~l ~ c~. PROOF. First note that since x is q. nilpotent (~x)' exists for all )~ c ~:. Assume now that, to the contrary, I)~n[ ~< C for all n. Since (()~,x)') is unbounded there is, by 2.1.23, a nucleus U and a subsequence ((An,x)') such that U for all n'.

n t

(.)

Since IAn, I ~< C, for all n', we can choose a subsequence (An,,) of (An') with An,,--+ (some) A0. Since A is a C - a l g e b r a we get (An,,x)'--~ (A0x)', whence the sequence ((An-X)') is bounded. But by (*), a II

r u,

contradicting boundedness of ((~n",)'). Hence the Lemma. 6.3.11. PROPOSITION. (Kaplansky). In a complex normed (or more generally, ample p-normed) algebra A every q. nilpotent element x is s.t.z.d.. In particular, every element of the radical

281

w 3. P s e u d o - R e s o l v e n t Function

x/~

is a s . t . z . d . .

PROOF. We may assume that sequence ()~,~) in C with

By 6.3.10,

~.

x r

0. By 6.3.8 there is a

Write

x' /llx ll

1

I

Xn -- ~ n X ~

(so t h a t I]Ynll- 1). Then since !

XnX~ ~

--Xn ~ X n

we get !

1

x~y~ - - x , llxnll-~ - y , ~ . Since x n -

(1)

Anx the equation (1) becomes Xyn

-

- - X [ I X n' l I - ~ -- ~ n l y n

.

(2)

I Since Itx,~ll, lAn[--~ co as n - - . co, and I t y n I [ - 1 we get from (2) xyn --~ 0 (as n --. co). In exactly similar m a n n e r we also get y,~x --+ O. Thus, x is a s.t.z.d., as desired.

6 . 3 . 1 2 . COROLLARY. In a real n o r m e d - or more generally, ample p - n o r m e d - algebra A every ext. q. nilpotent element x is a s.t.z.d.. PROOF. By definition x is a q. nilpotent element of the comN plexification A. By 4.7.4. (b), A is also ample. So, by 6.3.11, x is a s.t.z.d, of ti and hence by 3.7.16 (ii), a s.t.z.d, of A. 6 . 3 . 1 3 . DEFINITION. A TA A is called a topological integral d o m a i n or TID if it has no non-zero t.z.d.. 6 . 3 . 1 4 . PROPOSITION. A complex ample p - n o r m e d algebra A which is a T I D is q.s.s., in particular s.s.. PROOF. This is an immediate consequence of 6.3.11 and the definition of a TID.

Spectral Analysis in TA's

282 w4.

6.4.1.

Spectral

DEFINITION.

Algebras

A unital algebra over a field is called

spectral if for every x in A, a(x) # 0. 6.4.2. LEMMA. Every subunital algebra B of a spectral alge-

bra A is spectral. PROOF. By 1.7.19 (**), aB(X) D_aA(X) ~: 13. 6.4.3. THEOREM. Every complex ample Hausdorff CI algebra

is spectral. PROOF. If possible let a ( x ) = ~, for some x in A, so t h a t p(x) = C. By 6.2.5, F~.(A) = x*(x~) (x* e A*) is holomorphic on p(x) = C. Also, F~* is bounded since by virtue of 6.2.7, x*(x~) 0 as I~] ~ cr By Liouville's theorem, F~. is a constant which must by 0 (since Fz.(A) ~ 0 as I)~1 ~ o0). Thus Fz.($) = 0 for every x* E A*, whence by ampleness of A, x~ = 0. It follows t h a t e = ( x - Ae)x~ = 0 - a c o n t r a d i c t i o n . Thus a(x) # 0, as required. 6 . 4 . 4 . COROLLARY. Every complex Hausdorff locally convex

CI algebra A is spectral. PROOF. By 4.7.6, A is ample and so the result follows from 6.4.3.

Every complex Hausdorff locally sm. convex I algebra A is spectral. In particular, every complex Hausdorff locally sm. convex division algebra D is spectral. 6.4.5.

COROLLARY.

PROOF. By 4.4.15, 3.6.21, A is a CI algebra and hence the first s t a t e m e n t follows from 6.4.4. The second statement follows from the first since D being a division algebra is an I algebra (Gi = D \ { 0 } is open). 6.4.6.

COROLLARY. A complex, ample p-Banach algebra-

in particular, Banach algebra- is spectral. PROOF. This follows from 6.4.3 since every Banach algebra is

w4. Spectral Algebras

283

a CI algebra (by 3.6.23 (b)) and is also Hausdorff. 6.4.7. THEOREM (~;elazko). Every complex unital p-Banach algebra A is spectral. PROOF. This theorem goes beyond 6.4.6 since it covers also those algebras which are not ample. If possible let there be an element x E A with a(x) = O. Then p(x) = C and by 6.2.9, x~ is a locally p-admissible entire function. If F is any circle in C then F ,-~ 0 and hence by 5.14.16

r X~d~ - 0 . On the other hand, by taking m - 0 in the formula (.) of 6.2.10, we obtain

r x~ d)~ -- - 2~ie, where e is the unity of A. This contradiction proves that a(x) 0, for every x E A, and A is spectral. 6.4.8.

COROLLARY. Every strictly real unital p-Banach al-

gebra A is spectral. PROOF. Let A be the complexification of A. Since A is strictly real we have

aA(x) -- aA(x ) -7/:0 (by 6.4.7). 6.4.9. COROLLARY. Every complex or strictly real unital pseudo-Michael algebra A is spectral.

A-

PROOF. By 4.5.3, A has a projective limit decomposition limAa, where each A~ is a unital pa-Banach algebra. Again,

by 4.5.7

(3), OA(X) -

U

x -

(,)

First let A be complex. Then fi.~ is complex and so by 6.4.7, a2~(x~ ) ~ 0. It follows by (,) that ffA(X) r O, and A is spectral.

284

Spectral Analysis in T A ' s

Next let assume that strictly real and (.), A

A be strictly real. In view of 1.7.26, 1.9.11 we may A is commutative. By 1.9.8, As is commutative and so by 6.2.20, Aa is strictly real. Hence by 6.4.8 is spectral.

6.4.10. R e m a r k . A complex unital commutative complete locally convex algebra may fail to be spectral. To overcome this deficiency Waelbroeck [13 t ] has given a modified definition of spect r u m with respect to which these algebras are spectral. His modified spectrum which we shall denote by sp(x) is the complement (in K) of the set of all )~0 e K such that ( x - Xe) -1 exists and is (t). bounded in a neighbourhood of )~0. Using the properties of his spectrum sp(x), Waelbroeck has obtained interesting results concerning locally convex algebras.

w 5.

Gelfand-Mazur

and

Other

Similar

Theorems

6.5.1. LEMMA. If a division algebra A over a field F is spectral then A is of the f o r m A -= Fe, where e is the unity of A. Moreover, if x = ~ze, the map w : x ~-~ Az is an i s o m o r p h i s m of A onto F. PROOF. If x E A and )~z E a ( x ) then x - ~ze is noninvertible and so x - )~,e = 0 since A is a division algebra. Thus, x - - Axe, A - Fe and w is clearly an isomorphism. 6.5.2. COROLLARY. I f a spectral division algebra A over K is a H a u s d o r f f TA then the map w : x ~ ~, is a t. i s o m o r p h i s m . PROOF. This is because of the result that an isomorphism between two finite-dimensional TLS's is automatically a homeomorphism (here dim A - d i r e r - 1) (see 2.1.12). 6 . 5 . 3 . THEOREM. A complex ample CI division algebra A is of the f o r m A = Ce, with the map ~ : x ~ ~ (x = $~e) a t. i s o m o r p h i s m .

PROOF. Since A \ { 0 } - Gi is open, A is Hausdorff. Hence, by 6.4.3, A is spectral. The theorem now follows from 6.5.1, 6.5.2.

w5. Gelfand-Mazur and Other Similar Theorems

285

6.5.4. COROLLARY (Arens). A complex Hausdorff locally sm. convex division algebra A is of the form A Ce (with w a t. isomorphism).

PROOF. Since A is a Hausdorff division algebra, by 3.6.10, A is an I algebra. Further, by 4.4.15, A is a C algebra, and by 4.7.6. it is ample. The required conclusion now follows from 6.5.3. 6 . 5 . 5 . THEOREM (Gelfand-Mazur). A complex normed division algebra A is of the form A - C e and w " x ~ Az is a t. isomorphism. If []e[[- 1 then w is also an isometry. PROOF. W i t h o u t loss of generality we may suppose t h a t the n o r m of A is sm.. Then A is locally sin. convex, so t h a t by 6.5.4, A - Ce and w a t. isomorphism. If I1 11- 1, then [ I x l l - llama I - I A z l and w is an isometry. 6 . 5 . 6 . COROLLARY (Zelazko). A complex p - s e m i n o r m e d division algebra A - (A,p ~ ) has the form A - Ca. Consequently, every locally bounded complex TA B which is a division algebra has the form B - Ca.

PROOF. We may assume t h a t p is sm.. Let a be an element of A and Am a m a x i m a l c o m m u t a t i v e subalgebra containing a (see 1.1.9.); by 1.1.19, Am is a division algebra. For x E Am, 1

write ] ] x [ [ - v~(x)

1

where v ( x ) -

lim p(xn) -~ By 4.8.6 4 8.7

the above defined functional [[-[[ is a norm on Am, so t h a t (Am, ][" ]) is a complex normed division algebra. By 6.5.5, A m - Ca, so t h a t a - Ae (A E C). It follows that A - Ca, proving the first assertion. For the second assertion it suffices to observe t h a t , by 4.2.4, B is p - s e m i n o r m e d (so that it follows from the first). 6.5.7.

PROPOSITION. A strictly real, ample CI division alga-

bra A is of the form A - ~e (with w " x ~ Az a t . isomorphism). In particular, such an algebra is commutative.

t As Mw~ys we ~ssume p :/: 0

286

Spectral Analysis in TA's

PROOF. As in 6.5.6, for any element a r 0 in A, take a maximal commutative subalgebra Am with a E Am; Am is a division algebra and so in particular inverse-closed in A. By 4.7.3 (a), 3.6.27, Am is a commutative ample CI algebra. Further, by 1.9.11, Am is strictly real, and consequently, by 1.9.14, it is also formally real. By 1.6.20, its complexification Am is a division algebra which is moreover ample (see 4.7.4 (b)). By 6.5.3, A,~ Ce, whence A m - Be, so that a - A e (A E ~). Since a is an arbitrary element of A we conclude that A - Re, as desired. 6.5.8. COROLLARY. If A is a strictly real, Hausdorff locally sin. convex division algebra then A - Re. PROOF. The same argument as in the proof of 6.5.4 shows t h a t A is an ample CI algebra. The required result is now an immediate consequence of 6.5.7. 6 . 5 . 9 . PROPOSITION. A commutative real ample CI (in particular, a Hausdorff locally sin. convex) division algebra A is of the form A - B e or Ce ( A ~ R or C) according as A is formally real or not. PROOF. If A is formally real then, by 1.9.14, it is strictly real and so A = ~e (by 6.5.7). It remains to consider the case where A is not formally real. By 1.6.20 (b), A has a complex structure. Since ix = j x , with j E A, multiplication by i is continuous so t h a t ft. is a complex TA which is moreover, ample by 4.7.4 (a). By 6.5.3 (& 6.5.4) we conclude that A - A - Ce. 6 . 5 . 1 0 . THEOREM. Every real ample CI division algebra A is t. isomorphism to R, C, or H (the Hamilton quarternions).

PROOF. In view of 6.5.9, we may assume that A is not commutative. Let Z denote the centre of A. By 1.1.18, Z is a commutative division algebra. For any x r 0 in A the algebra Z(x) generated by Z, x, x -1 is a commutative division algebra which is moreover, ample (by 4.7.3 ( a ) ) a n d CI (by 3.6.24). Applying 6.5.9 to Z(x) we get Z(x) = ~e or Ce. Thus,

w5. Gelfand-Mazur and Other Similar Theorems

287

x - (a + iS)e (a, ~ c R, with 5 - 0 if Z(x) - Re). It follows t h a t (~ -

(~ + iZ)~)(~

- (~ - iZ)~) -

0

or, x 2 - 2 ~ x + (~2 + Z 2 ) ~ _ o,

which implies t h a t A is algebraic of degree 2 at most. By a theorem of Jacobson t A is finite-dimensional, and so by the classical Frobenius theorem, A _~ ~, C or H; the isomorphism is topological since A is finite-dimensional Hausdorff. 6 . 5 . 1 1 . COROLLARY (Arens). Every real Hausdorff locally sm. convex division algebra A is t. isomorphic to ~,C or H. PROOF. As in 6.5.4 we see that A is ample and CI, and the required conclusion follows from 6.5.10. 6 . 5 . 1 2 . T h e o r e m (Zelazko). Every real p - n o r m e d division algebra A (A,p) is t. isomorphic to ~, C or H according as it is formally real, commutative but not formally real, or not commutative. If w denotes the t. isomorphism then 1

Ico(x)l- v ( x ) -

limp(x'~)x tl

where I" I denotes the standard norm tt on ~, C or H. PROOF. Since we do not know t~ priori that A is ample we cannot deduce this theorem from 6.5.10. However, we can prove it by directly considering separately the three cases that arise. C a s e 1. A is formally real. For any non-zero x is A consider a maximal commutative subalgebra Am with x E Am. As in 6.5.7, A,~ is a formally real division algebra, so t h a t Am is t Every algebraic division algebra of bounded degree over a perfect field is finite-dimensional ("Structure theory for algebraic algebras, Annals of Mathematics 46 (1945), p.701" ). tt It I - ( t t ) 8 9 ( t E ~ , C or HI and t the conjugate of t).

Spectral Analysis in TA's

288

now a commutativeN formally real p - n o r m e d division algebra. Its complexification A,~ is a p - n o r m e d division algebra, whence by 6.5.6, Am = Ce, A m = Re, x = At, A = Re. C a s e 2. A is commutative but not formally real. Then A has a complex structure and as in the proof of 6.5.9, we get A --

C a s e 3. A is not commutative. Let Z be the centre of A. For x J= 0 in A we can form, as in the proof of 6.5.10, the algebra Z(x) which is now a real commutative p - n o r m e d division algebra. By the conclusions in cases 1, 2 we have Z(x) = ~e or Ce. This implies, as in 6.5.10, t h a t A is finite-dimensional, and hence by Frobenius, A_~ ~, C or kD. It remain to prove (.). Set l]xII = Iw(x)]. Then I1" ]] is a n o r m on A with ]]xy]l--IIxII []yl]. Since A is finite-dimensional, 11" II ~ P- By 4.8.2

v(x)-

Vv(X ) - v[i.l[(X) - l i m IIx'~I[ 88- l i m I I x l l - IIxll.

6.5.13. Remark. If A is a formally real normed division algebra whose norm I1" II is normalized, then the t. isomorphism w : A --~ ~ is an isometry: if x - Aze, l l x l ] - I A z l - Iw(x)]. 6 . 5 . 1 4 . L EMMA. Let A be a unital p-normed algebra which is a TID. Let A be the completion of A. Then every non-zero element x of A has an inverse in A. PROOF.

Suppose t h a t

x E A has no inverse in r

Then

0 e hA(X), so t h a t aA(x ) 7~ 0. Also, by 6.1.2, aA(x ) is closed and a~(x) r g. It follows t h a t OaA(x ) 7~ O. If A e a2(x ) then, by 3.7.28, x - Ae is a s.t.z.d, of A and so by 3.7.14, it is a t.z.d, of A. By the hypothesis on A, x = Ae and since x is not invertible, X

---

0.

unital p-Banach algebra A is a TID then it is a division algebra. 6.5.15.

COROLLARY.

If a

6.5.16. Remark. (cf. Zelazko [31, p.112]). The result in 6.5.15 does not hold for a locally sm. ~ algebra. For example, consider the algebra E of entire functions (see 4.6.8 (iii)). ~" is

w 5. G e l f a n d - M a z u r and Other Similar Theorems

289

not a division algebra, since for instance, z E $ has no inverse. But ~" is a TID. To see this, suppose t h a t in ~r we have g ~ 0 , and (,) fkg --+ 0 as k ~ oc. Since the zeros of g are isolated we can find a sequence rn of reals with 0 < rn -~ cr and such t h a t inf

Ig(z)] > 0 ( n -

1,2,...).

(**)

If we write iif~ll*=

sup ]f(z)l , i=1=~,,

the family {ll" I1~} of norms is easily seen to be equivalent to the family {ll" II-} defining the topology in ~ (see 4.6.8 (iii)). The condition (,) implies t h a t Ilfkgl~ ~ 0 as k --+ oc, for all n. But this result along with (**) gives: Ilfkll* ~ 0. This means t h a t g is not a t.z.d., proving ~' is a TID. 6 . 5 . 1 7 . THEOREM. (cf. p - n o r m e d algebra which is a A = Ce, A "~ C, if A is and A"~,Cor H if A is a

[31, pp.30-32.]) Let A TID. Then: a complex algebra

be a unital

real algebra.

PROOF. Let .zi be the completion of A. By 6.5.14, every x =/= 0 in A has an inverse x -1 in A. Consider the unital subalgebras A ( x ) of .zi, consisting of all rational functions of x, over ~:, i.e. all elements of the form f ( x ) / g ( x ) -- f ( x ) g ( x ) -1 (g(x) r 0), where f , g are polynomials over K. A ( x ) is a division algebra containing x, which being a subalgebra of A is p - n o r m e d . If A is complex, it follows from 6.5.6 t h a t A ( x ) = Ce, x = Aze, A--Ce. It remains to consider the case where A is real. By 6.5.12, A ( x ) " ~, C or H. It follows t h a t x satisfies a relation of the form a x 2 § ~ x § "/e -- O, where a, ~ , - / c ~, " / ~ O. It follows t h a t x-1 = - - a x - -eft E A, showing t h a t A is a divi.7 -7 sion algebra. Now we apply 6.5.12 to A and conclude t h a t A ___ ~,

Spectral Analysis in TA's

290

C or H. 6.5.18. Remark. Theorem 6.5.17 has been extended by Zelazko to locally sm. convex algebras in the following form" If a unital locally sm. convex algebra A has no nonzero g.t.z.d, then A _ ~ C if A is complex, and _ ~ , C or H if A is real (see [31, pp.l12-14]). 6 . 5 . 1 9 . PROPOSITION. Suppose that the norm of a unital p - n o r m e d algebra A satisfies the condition

cIIxlJ Ilyll-< I1~11

(*)

for some C > 0 and all x, y E A. Then A is t. isomorphic to C if A is complex and to ~, C or H i r A is real. PROOF. In view of 6.5.17 it is sufficient to show t h a t A is a TID. Suppose that x , y , e A, llYn]] = 1, xyn ~ O. Then, by (,) we have

cIl~ll = ctl~ll Ify=ll-< II~y.ll-~ o. This means that Ilxll = 0, x = 0, so that A has no non-zero 1.t.z.d.; similarly it has no non-zero r.t.z.d.. Thus A is a TID, completing the proof. 6 . 5 . 2 0 . COROLLARY (Arens-Shilov). A unital normed algebra A satisfying condition (,) above is t. isomorphic to C if A is complex and to ~, C or H if A is real. 6 . 5 . 2 1 . R e m a r k . A special case of 6.5.20 was proved earlier by Lorch and Mazur (see [14, p . 1 2 7 ] ) i n the form: A complex unital Banach algebra A whose norm satisfies the condition ]]xy]] = I]x]] ]]Y]I for all x , y in A is (t.)isomorphic to C.

w6.

Turpin's

Theorem

on Locally

Convex

Algebras

6.6.1. Let A be a Hausdorff complete locally convex algebra. Denote by Pc the set of all continuous semi-norms on A and by A* (the continuous) dual of A.

w6. Turpin's Theorem on Locally Convex Algebras

291

Set

rl( )

1

sup

limpc~(x'~) ~ ,

p~EPc n~oo

1

sup lim If (xn) l -~, lEA*

n--~oo co

inf{r" (a,)~ ~ an E K, the series ~ a , A " n=l

has radius of convergence > r GO

=> ~ a , ~ x n converges in A}. n=l

We have clearly 0 ~ rj(x) ~ co (j - 1, 2, 3). 6.6.2. LEMMA. r2(x)<~ rl(x). PROOF. If jr E A*, by 4.3.13 (i) there is a pa such that f is p~-continuous, and so p a - b o u n d e d . Therefore we have If(xn)l <. IIflip~(x n) so that lim If(x'~)l ' ~< lim ([Ifll,,p~(x ~

~ - - ~ OO

n - - ~ OO

1- ) ~< 1. lim pa(x'~) -~,~< ~---~ OO

r 1 (X).

By taking the sup over f E A* we get r2(x) <~ rl(x). 6.6.3. LEMMA. If r3(x) < co and I#l > r3(x), then" (i) the series ~

x

converges absolutely;

rt--1

(ii) # c p'(x); (iii) r(x) ~ r3(x), where r(x) denotes the spectral radius of x. PROOF.

By definition of r3(x) there is an r0 with I#l > oo

r0 > r3(x) such that, if ~

~ , ~ n is a series over K with radius

n--1 oo

of convergence /> r0, the series ~ c ~ n x n converges absolutely. rt=l

Spectral Analysis in TA's

292

has radius of convergence

Now the numerical series ~ n--1

- 1/lim Ih!~.188- I'1 > r0, so that by choice of r0, ~

con-

n'--1

verges absolutely. This implies, by 3.1.24, that ~ verges in A, whence, by 2.2.17,

(:)' -

x

con-

n--1

exists, so that # E p~(x).

Therefore, if ,~ E a'(x) then )~ ~ p'(x) a n d s o w e m u s t have I,~] ~< r3(x). It follows that r(x)<<, r3(x), which completes the proof of the lemma. 6.6.4.

PROPOSITION. Let A be a complex Hausdorff com-

plete locally convex algebra which is CQ. Then we have rl(x) -- r 2 ( x ) - r 3 ( x ) - r(x). PROOF. By 6.1.1, a'(x) is compact. Suppose that 1 0 <

(,)

So 0 ~< r(x) < c~.

<

Then r(x) < I~1-1 - I - ~-~l, so t h a t -,X -1 ~ p'(x) a n d cons e q u e n t l y A E pP(x) (see 6.3.2). Since A is a CQ algebra, by 6.3.5, x~ is a strongly holomorphic function with d~x~

d~'~

= ( - 1 ) n n ! x n ( l + x ~ ) n+l

If f E A*, the F ( 2 ) - f ( x ~ ) i s holomorphic on pP(x) and in 1 particular, on I)~I < r--~" Since FC") ( 0 ) - f((-1)nn!x n) - ( _ l ) ' ~ n ! f ( x n) the Taylor expansion for F is given by (x)

F(A)- ~(-1)nf(xn)A n n-----1

(note F ( 0 ) - f ( 0 ) - 0)

w6. Turpin's Theorem on Locally Convex Algebras

293

with the series converging for [AI < 1/r(x). It follows that (-1)~f(x'~)A ~ -

f ( ( - 1 ) n A n x n) ~ O, as n ~ cr

and this is true for every f E A*. This implies that the sequence ( ( - 1 ) " A " x ") is weakly bounded. Since A is locally convex it is also strongly bounded t i.e. we have a constant Ma > 0 (depending on A) such that p~((-1)'~A~x ") - p ~ ( A n x '~) < Ma for all n/> 1. It follows that ..........

lim ~ p ~ ( x '~) <~ rt---~ O0

lim (M~)•- 1 , ~

-I 1-1

rt---~ O0

This being true for every p~ we obtain, by taking sup over pa, rl(x ) ~ IA[-1 (for any A with I)~1-1 > r(x)). By taking the infimum over A we get rl(x) ~< r(x). Combining the above inequality with that in 6.6.2 we get r2(x ) ~ rl(X ) ~ r(x) Next choose r , /

(< oo).

(1)

with r

(z) < r' < r.

By definition of r2, we have for any f c A*. ~//tf(x~)l < r' < r for sufficiently large n. It follows that ~'--~)l < 1

(n/> some N)

Therefore (~-r~-~)isweakly bounded

and consequently strongly bounded. This means that there is an M~ > 0 such that

We have therefore pc~ ?-a- ~< Mc~ t See [24, p.68].

, -r- < 1, so that by the

Spectral Analysis in TA's

294

oe

Weierstrass

X n

~

M-test

rn

converges absolutely in

A.

Let

n=l oo

a~A '~ be a power series with radius convergence > r. Then r~--1

]an] ~< r -~ (for sufficiently large n) so that

p ~ ( a , x ~) - [ a n l p ~ ( x n) <~ P~(xn) <~M~

.

r n oo

It follows that the series ~ a , x

~ converges absolutely in A. This

n-1

means that r ~> r3(x) whence

r (x)

(2)

From (1), (2)we get r3(x) < r 2 ( x ) < r l ( x ) < r(x).

(3)

But by 6.6.3 (iii), r(x) <~ r3(x). This together with (3) gives the relation (.). 6.6.5. THEOREM (Turpin). A commutative complex ~ algebra which is a Q algebra is locally sm.. PROOF. By 3.6.18, 4.6.7, A is CQ. Since A is a locally convex Q algebra we can find an open absolutely convex nucleus U comprising q. invertible elements. Since U is balanced, by 3.6.8 (i), we have r(x)~< 1 for x e U. Set V - U/2. Then V is open and also absolutely convex (see 4.1.2 (c)); since U is balanced, V c_ U. If y c V then y - x / 2 ( x E V ) , so that

r(y) -- r

()

x lr(x) ~ - .11 - - -~ -- ~ 2

1

2

< 1

"

oo

By 6.6.4, r 3 ( y ) -

r(y) < 1. So, by 6.6.3 (i), ~

yn converges

n--1

absolutely in A. O(3

Let ~p(A)- ~ a , ~ A n be an entire function vanishing at 0. n-1

Then

[anAn[ < 1 for n/> n(A).

(,)

w 6. Turpin's Theorem on Locally Convex Algebras

295

If x E A, then using the fact that U as a nucleus is absorbing we find )~o J= 0 such that x - )~oY (Y e g ) . Then, using (,),

for n/> no -- n(A0). Since E

yn converges absolutely we have

rt

oo

n--no

oo

lr/,-~ lq, O

O(3

whence ~ ~ , ~ x n converges absolutely in A. Thus, all entire funcn=l

tions vanishing at 0 operate on A and the theorem follows from 5.5.11.

CHAPTER GELFAND

w 1.

VII

REPRESENTATION THEORY

Ideals of Topological

Algebras

7.1.1. LEMMA. (cf. [20, p.70]). Let A be dense subalgebra of a T A A. Then:

(i)

The closure I in A of a n ( l . o r r . ) i d e a l I of A is an ideal of A of the same type.

(ii) /f the ideal I is regular with a (l. or r.) relative unity u then I is also regular with u as a relative unity. (iii) If I ~ A is a closed regular ideal of A then -[ ~ -A and is a closed regular ideal of A. PROOF. It is enough to prove the results when I is a 1. ideal. Clearly I i s a s u b s p a c e o f A. Further if ~ E A , ~ E I , x ~ E A , az C I and xa --~ 5, az --~ ~ then x~a~ ~ x a. Since each x~a~ C I it follows that ~ a C I, proving I is a l . ideal. Hence M

(i). Next let I be a regular 1. ideal with relative r. unity u. If -~ c A, x~ C A and x ~ 5 then ~u-~-lim(x~u-x~)EI

(sincex~u-x~EI)

and I is regular with u as a relative r. unity, proving (ii). Finally, if u is a relative unity for closed ideal I ~- A, then u ~ I and hence u ~ I (since A ~ I I) and therefore I-7(= A, proving (iii). 7.1.2. COROLLARY. Let A be a TA. Then the closure I of a l. ideal, a r. ideal, or a bi-ideal I of A is an ideal of A of the same type. PROOF. Apply 7.1.1 with A -

A.

w1. Ideals of Topological Algebras

297

7.1.3. COROLLARY. Every maximal ideal M of A is either closed or dense. PROOF. By maximality of M, the closure M -

M or A.

7.1.4. DEFINITION. Following Michael [20] we call a TA A normal if every closed regular 1. (respy. r.) ideal I ~= A is contained in some closed maximal regular 1. (respy. r.)ideal of A. Further, if every such I is contained in some closed hypermaximal ideal then A will be called hypernormal. Trivially, every hypernormal ideal is normal. Finally, we call a TA A hyponormal if every maximal regular 1. or r. ideal is closed. 7.1.5. R e m a r k . Since a radical algebra A has no regular ideal ~ A, such an algebra is vacuously hypernormal and hyponormal. 7.1.6. LEMMA. (a) Every closed maximal regular I. or r. ideal M of a hypernormal TA A is hypermaximal. (b) Every hyponormal TA A is normal. (c) Every hyponormal TA A is functionally t continuous. PROOF. (a) By hypernormality there is a closed hypermaxireal ideal M1 with M C M1. The maximality of M ::~ M - M1. (b) By Krull (1.2.10) if I is a closed regular ideal ~: A there is maximal regular ideal M with I _C M. Since A is hyponormal, M is closed and so A is normal. (c) Since A is hyponormal every hypermaximal ideal is closed and so by 1.3.9, 2.1.30 A is functionally continuous. 7.1.7. We give below examples to show that neither of the properties hypernormality, hyponormality need imply the other. E x a m p l e 1. Let A be any radical algebra (A - x/rA) and A1 its unitization. Then A1 is a T A under the indiscrete topology. The ideal A is hypermaximal in A1. Since A is not closed in A1, A1 is not hyponormal. On the other hand, since no ideal I r A1 is closed it is vacuously hypernormal. Example

2.

The Williamson algebra ~ .

t For definition see 2.2.19.

We have seen in

Gelfand Representation Theory

298

3.6.33 that ~ is a commutative Hausdorff division algebra over C. Since {0} is the only maximal ideal of ~ and it is closed, is hyponormal. But since {0} is not hypermaximal, ~ is not hypernormal.

7.1.8.

PROPOSITION. In a Q algebra A if I # A is a regular l or r. ideal then its closure I ~ A. PROOF. Since A is a Q algebra, Gq is open. If u is a relative (r. or 1.) unity of I the same is true of u + a for any a ~ I (see 1.2.8(a)). Therefore, by 1.2.9(ii), - u - a r Gq, so that ( - u + I)(1Gq = 0 and hence also - u + I N G q = 0 (since Gq is open). It follows that -u+I--u+I#A,

so that I ~ - u + A - A .

7.1.9. COROLLARY. Every maximal regular (l. r. or hi-) ideal M of A is closed. Hence, every Q algebra - i n particular (see 3.6.23) a pseudo-Banach or (more generally) a sm. (F) algebra is hypernormal. 7.1.10. LEMMA. Let A = ( A , I . I) be a unital sin. algebra with unity e. Then for any ideal I # A we have: 1 <. d ( e , I ) * <~ ]el;

in particular when le l - 1, d ( e , I ) -

(F) (,)

1.

PROOF. Since 0 e I, d(e, I) <~ d(e, O) - l e l . If d(e, x) = l e - x I < 1 then, by 3.3.18 (ii), x is invertible and hence x r I (since I # A). It follows that, for x e I, we have d(e,x) /> 1. Hence the inequality (.). 7.1.11. LEMMA. In a TA A, if It (respy. It) is a closed I. (respy. r.)ideal of A then (It : A) (respy. (I~ : A)) is closed. Hence the primitive ideal P associated with a closed maximal regular I. (respy. r.) ideal Mz (respy. Mr) is closed. PROOF. If xa E (It : A) and xa -~ x E A, then, for any y c A, xay c Iz and consequently xy E It (since xay ---+xy and Il is closed). It follows that x E (It: A), proving that (II: A)

w 1. Ideals of Topological Algebras

299

is closed. Similarly, (It" A) is closed. Finally, since P - (Mz" A) (respy. ( M r ' A ) ) , P is closed. 7.1.12. PROPOSITION. A primitive ideal P as well as the radicals x/~, ~c-~, ~ of a hyponormal algebra - i n particular of a Q or of a pseudo-Banach algebra- A, are closed. PROOF. By hyponormality of A and 7.1.11 every primitive ideal is closed. Also, every maximal regular bi-ideal being primitive (by 1.5.10) is also closed. Therefore x/~ t (respy. ~-A) as the intersection of primitive ideals (respy. maximal regular bi-ideals) is closed. Finally, since each hypermaximal ideal of A is closed (see 7.1.6. (c)) their intersection ~/A is also closed. 7.1.13. normal.

A-

PROPOSITION. Every pseudo-Michael algebra A is

PROOF. Let A (A,P), where P is saturated, and lirrl fi.~ the projective limit decomposition, with zi.~ pseudo-

Banach Let (r. orl.) and an with I.

algebras. I -/= A be a closed regular (1. orr.)ideal with relative unity u. Since u ~ I and I is closed there is a p~ E P c > 0 such that B E - { x E A ' p s ( x - u ) < e } is disjoint It follows that p s ( a - u) >>.r ( h E I).

(1)

Using the notations in (the proof of) 4.5.3, 3.4.15 we can write a#-a4-Ns-as;

99s'x~xs-x§

# (Ns-kerps).

Then we have P#a (as -- us) -- p#a (a # -- u #) -- ps(a -- u) >t e(a E I).

(2)

Clearly, I~ - ~ ( I ) is a regular ideal of A~, with relaive unity u~; u~ ~ I~ because of (2). By 7.1.1, J~ - I~ is a closed regular ideal (with relative unity u~) ~ A~. By Krull J~ is t That ~ is closed can also be deduced from the fact that it is the intersection of all maximal regular 1.ideals.

Gelfand Representation Theory

300

contained in some regular maximal ideal M. Since fi.a is pseudoBanach, by 7.1.9, M~ is closed. By continuity of ~p~ and 1.2.17 (iii) M - ~ I ( M ~ ) is a closed regular ideal of A which is maximal, and clearly I C M. This proves that A is normal. 7 .1 .1 4 . COROLLARY. A complex or strictly real commutative pseudo-Michael algebra A is hypernormal.

PROOF. The ideal M~ in the proof above (of 7.1.13) is now closed hypermaximal since A~ is Gelfand (by 7.2.17, 7.2.19)t. Let X~ be the character determined by Ma; X~ is continuous. Then, if we set X - X~~ X ( U ) - X~(U~)- 1, so that X is a continuous character of A. If M - k e r x then M is a closed hypermaximal ideal. Also, if x E I then ~p~(x) E Ja c_ Ma, X(X) - O, so that I ___ker X - M. This proves A is hypernormal. 7.1.15. R e m a r k . There are commutative Michael algebras which are not functionally continuous (and so not hyponormal) (see [20, p.49]). 7.1.16. R e m a r k . There are hypernormal TA's having dense regular maximal ideals. For an example, consider the algebra C = C(~, ~:) of all ~:-valued continuous functions on ~ topologized by the family {lI" Iin} of sm. seminorms, where

llfll, - sup If(t)l (f c c)

(cf. Example of 4.6.8)

Itl<,~

C is a unital commutative locally sm. ~ algebra. Note that C is strictly real when 0 < - ~. By 7.1.14, C is hypernormal. C has, however, dense maximal ideals. For instance, if I-

{f E C" f ( t ) - 0

for t > ( s o m e ) t o ( f ) )

I is clearly an ideal. We claim that I is dense in A. To see this define for an jr E C,

-

I

f(t)

if t < n

f(t)

if n < t <

0

if t > n + e .

+

That A~ r V/fi.~ holds since A~ admits the regular ideal J~.

w 1. Ideals o/Topological Algebras

301

Then f~,~ E I, and IIf,~,~ - / l l ~ - 0 < ~, proving that I is dense in C. By Krull's lemma there is a maximal ideal M _ I and then M is a dense maximal ideal. 7.1.17. PROPOSITION ( R i c k a r t t ) . Let A be a hyponormal algebra-in particular a p-Banach algebra or (more generally) a Q algebra. Let A* be an arbitrary algebra and ~9 9 A ~ A* an epimorphism. Then ~(ker ~) c v/A* where bar denotes closure. In particular, when A* is s.s., ker ~ is closed and x / ~ c k e r ~ .

PROOF. Let Mz* be a maximal regular 1. ideal of A*. By 1.2.17 (iii), Mz - F-I(Mz* ) is a maximal regular 1. ideal of A and ker ~ __ Ml. Since A is hyponormal ML is closed. So her F _ Ml. It follows that ~(ker ~) _C A Mz* - v/A*" When v/A * - {0}, we obtain ~p(ker~) - {0}, so that kertp k e r ~ . Further, By 1.2.26, ~ ( ~ ) _ C v/A * - { 0 } , whence x / ~ _c ker ~p, completing the proof. 7.1.18. DEFINITION. A bi-ideal I of an algebra A is called primary or a primary ideal if it is contained in a unique maximal regular bi-ideal M. An algebra A is called primary if the ideal {0} is primary - which is the same as A having a unique maximal regular ideal. 7.1.19. R e m a r k . Every maximal regular bi-ideal is primary. Moreover, in a Q a l g e b r a - in particular in a pseudo-Banach a l g e b r a - the closure I of a primary ideal I is primary (if I _ M, where M is a maximal regular bi-ideal, then since M is closed, I c M and I is primary). 7.1.20. E x a m p l e s of p r i m a r y ideals (i) Denote by C (1) - C(1)(~,K) the algebra of K-valued continuously differentiable functions on E. C(1) is a locally sm. algebra under the family {ll" II(n1)} of seminorm, where Iftl (1) - s u p ( I f ( t ) l . l t

I < n} + sup{[f(1)(t)] . [t[ < n)

t He has proved this result for Banach algebras [23, p.74].

302

Gelfand Representation Theory

where f(1) denotes the derivative of f. Define in C (1) the ideals" M 0 - { f e C ( 1 ) ' f ( 0 ) - 0 } , Ii-{fEC(1)'f(0)f(1) (0) - 0}. We have I1 c M0, and M0 is maximal (since for f C C (1), f - f(0) C M0). We claim I1 is primary. To see this, suppose that I is an ideal with I D I1, I M0. Then there is an f E I with f(0) - a --fi 0. Then if / 3 - f(1)(0), g ( t ) f(t)-a-[3t, then g e I1 c I. Therefore a + fit E I, whence a t + f i t 2 - ( a + 1 3 t ) t E I. Since clearly f i t 2 E I1 c I we get at E I, t E I. Therefore - (a + fit) - fit C I and I - A. It follows that I1 is not contained in any other maximal ideal, so that I1 is primary. (ii) The algebra C(1)([0,1],0<) is a Banach algebra under the norm [fl ( 1 ) - [[f]l~q-tlf(1)[[~, where [[.[1~ denotes the sup norm. If 11 and M0 are defined analogouly as above then 11 is a primary ideal with M0 as the containing maximal ideal. (iii) Denote by C('~) - C('~) ([0, 1], K) the algebra of all K-valued functions f whose n th derivative exists and is continuous. C ('~) is a Banach algebra with respect to the norm" n 1 Ilfl (")

-

+

9

r----0

Set I~ - ( f c C(~)" f ( 0 ) - f(1)(0) - . . . f ( ~ ) ( 0 ) - 0}. Then it can be shown that I~ (1 <~ r <~ n) are primary ideals with I~ c I~-1 c . . - c I1 c M0. Also, In is the smallest primary ideal c M0 (see [10, p.205]). (iv) In the (F) algebra ~' of entire functions (iii)) if we set

I, - { f E ~'" f ( J ) ( 0 ) - 0

(defined in

(0 ~< j ~< r)}

then Ir are primary ideals such that I n C In_ 1 ( r t -

1,2,...)

t T h e weighted s u m has been taken to m a k e the n o r m s m . .

3.3.14

w2. Gelfand Algebras

303

and I 0 - M 0 - {f E $ 9f ( O ) - O} is maximal regular. Clearly N I ~ - {0}. Since {0} is obviously not primary, there does not exist a smallest primary ideal contained in M0. 7.1.21. R e m a r k . For information regarding primary ideals and other related matters in Banach algebras, see [23, p.92] and references cited therein, as also [22, pp.238-99].

w 2.

Gelfand

Algebras

7.2.1. DEFINITION. A TA A is called Gelfand algebra if (i) A # ~ (ii) every maximal regular 1. or r. ideal is hypermaximal and closed. Note that when A is unital, condition (i) is automatically satisfied (by 1.2.24 (d), e r v/A). 7.2.2 PROPOSITION. Let A be a Gelfand algebra. Then: (i) The radical v/A is the intersection of all hypermaximal ideals of A, i. e . x / ~ -- ~/--A. (ii) Every character of A is continuous, i.e. A is functionally continuous; A - Ac ~ 0. (iii) A is both hypernormal and hyponormal. (iv) If x e A then a'(x) = {x(x) : x e A } U{O}. (v) If A is unital and x E A then a(x) = {X(X) : 2; E A}. PROOF. (i) This follows from 7.2.1 (ii), 1.2.22, 1.2.24 (b). (ii) The ideal k e r x is hypermaximal by 1.3.9, so closed by 7.2.1 (ii) and hence X is continuous by 2.1.30. The condition 7.2.1 (i) ensures the existence of a maximal regular 1. ideal M; M is closed hypermaximal (by 7.2.1 (ii)) and so determines a continuous character, so that A~ :/: 0. (iii) The condition 7.2.1 (ii)implies that A is hyponormal. Also, the same condition together with Krull's lemma (1.2.10) shows that A is hypernormal.

304

Gelfand Representation Theory

(iv) Since always 0 E a'(x) it is enough to consider non-zero elements on both sides of the equation. If A r 0 in a~(x) then - A - i x is not q. invertible and hence, by 1.2.21 (b), 7.2.21 (ii) there is a hypermaximal ideal M - M x for which A-ix is a relative unity. So, by 1.3.10, X ( A - l x ) 1, X ( x ) - A. Thus a'(x)c_ R.H.S. of the equation in (iv). The reverse inclusion follows from 1.7.11, completing the proof. (v) In view of 1.7.24, it is enough to prove that if A E a(x) then A - X(x) for some X- Since x - Ae is not invertible, by 1.2.14, 7.2.1 (ii) there is a hypermaximal ideal M with x - A e E M. If X is the character determined by M then X ( x - Ae) - 0, X(x) - A. 7.2.3. PROPOSITION. The unitization A1 of an algebra A x/rA is a Gelfand algebra iff A is a Gelfand algebra. PROOF. By 1.4.9 (b), there is a bijection between the hypermaximal ideals MI(~-A) of A1 and the hypermaximal ideals M of A, given by" M~ ~ M -- A N M~ (*) M1-/~/-{xlEAlxlueM}-{xleAluxleM}

(**)

where u is the relative unity of M. It follows from (.), (**) that M1 is closed iff M is closed. These results along with those in 1.4.8 (connecting the maximal 1. or r, ideals of A1 and maximal regular 1. or r. ideals of A) prove the proposition. 7.2.4. R e m a r k . The restriction A -~ x / ~ in the statement of 7.2.3 cannot be omitted. For example, any radical TA A is not Gelfand (since A - x/~) though its unitization A1 is Gelfand (by 1.4.11, its sole maximal ideals A is hypermaximal (see 1.1.12) and closed (see 2.2.9)). 7.2.5. DEFINITION. Let A be an algebra without unity. Then A is called spectrally Gelfand if it satisfies the condition - {x(x) ' x e

U{0}

where A denotes the set of characters of A.

e

(,)

305

w2. Gelfand Algebras If A is TA then A is called Gelfand if o-'(~) - {x(x)

t.

(=topologically)spectrally

x9 c/',~} U { o }

(x c A)

(**)

where Ac denotes the set of continuous characters of A. 7.2.6. DEFINITION. A unital algebra A is called spectrally Gelfand if

~(~) = {x(x): x e ~} (~ e A)

(,1)

Similarly, unital TA A is called t. spectrally Gelfand if

~(x) = {x(x): x e ~ } 7.2.7. R e m a r k .

(~ e A)

Since a ' ( x ) = a(x)U{o},

(.1)~

(, 9 1) (.) and

(** 1) ~ (**) 7.2.8. P r o p o s i t i o n . (a) A t. spectrally Gelfand algebra is also spectrally Gelfand; the two concepts coincide when A is functionally continuous- in particular when A is p-Banach or even hyponormal. (b) Every Gelfand algebra is t. spectrally Gelfand. (c) The unitizsation A1 of an algebra (respy. TA) A is spectrally Gelfand (respy. t. spectrally Gelfand). iff A is spectrally Gelfand (respy. t. spectrally Gelfand). PROOF. ( a ) B y 1.7.11 (respy. 1.7.24)

X(X) c a'(x) (respy. X(X) E a(x)) for all X E A. When A is functionally continuous, A - - A c. Also, p-Banach ::~ hyponormal ::~ functionally continuous. Hence (a). (b) This follows from 7.2.2 ((ii), (iv), (v)). (c) If X 1 C A1, Xl -- # e l + x (x E A) then A A t ( X l ) -# § a~(x). Also, A 1 - - A U { X o } , AleA1U{x0}, where A 1 (respy. AI~ denotes the set of characters (respy. continuous characters) of A1, A 1 (respy. A 1) the set of extensions of characters (respy. continuous characters)of A to A1, and X0 the distinguished character of A1 (kerx0 = A). The assertions (c) of the proposition are easy consequences of the above relations. 7.2.9. PROPOSITION. Let A be a spectrally Gelfand algebra and x E A. Then x is q. invertible iff X(X) ~ - 1 for every

Gel/and Representation Theory

306

X E A. If A is a t. spectrally Gelfand algebra and X(X) # - 1 for every X E Ac then x is q. invertible. PROOF. If x is q. invertible then by 1.1.24, X(x) is q. invertible, so t h a t by 1.1.26 (ii), X(X) ~ - 1 . If x is not q. invertible then - 1 E a~(x). Since A is spectrally Gel/and there is a X c A with X(x) = - 1 . This completes the proof of the first assertion. For the second, assume now that A is t. spectrally Gel/and, x E A and X(x)J= 1 for all X E Ac. If possible let x be not q. invertible; then - 1 E a~(x) and so by hypothesis on A there is a X E Ao with X(x) = - 1 , which contradicts our assumption on X- This contradiction shows that x is q. invertible as required. 7.2.10. COROLLARY. Every unital spectrally Gelfand (respy. t. spectrally Gel/and) algebra A has the Wiener property: x c A is invertible iff X(X) ~ 0 for every X e A (respy. every X c At). PROOF. An element x E A is invertible iff x - e is q. invertible iff (by 7.2.9) X ( x - e ) =/= - 1 , i.e. X(X) ~= 1 - 1 = 0, for every X c A (respy. Ac). 7.2.11. R e m a r k . The commutative unital p - B a n a c h algebra WP considered in 3.4.10 has the Wiener property by 7.2.10, since it is spectrally Gel/and by 7.2.17, 7.2.8 (b). 7.2.12.

PROPOSITION.

In a spectrally Gel/and algebra we

have:

- C/At -Aq.tFurther, in a t. spectrally Gel/and algebra we have also

(,)

so that x / ~ is closed. PROOF. By 1.5.12, v~__C_ ~'--A__ ~r-~. Also, by 1.7.15 (iii), t For definitions see 1.5.11, 1.7.13, 1.3.7, 2.2.18.

(1)

307

2. Gelfand Algebras

c A, c r

(2)

If x C ~ it is clear that x c M • g(x)-0:fi-1 for every g e A, whence by 7.2.9, x is q. invertible. Thus ~r~ is a q. invertible ideal, whence by 1.2.24 (b), ~ C_ v/A. This together with (1), (2) proves (,). Now let A be t. spectrally Gelfand. Then, by (,) of 2.2.18, Therefore, to prove (**) it is enough to show that ~/-A C_ x / ~ and this is done exactly as above for ~ (with Ae replacing A). 7.2.13. COROLLARY. A t. spectrally Gelfand algebra- in particular a Gelfand algebra - is s.s. iff it is q.s.s, iff it is h.s.s.. PROOF. This is an immediate consequence of (,) of 7.2.12. 7.2.14. PROPOSITION. A t. spectrally Gelfand p-seminormed s.s. algebra A is p-normed. PROOF. Let us take A = (A,p). Then, if p(x) = 0 we have: x , x , . . . - - ~ O . If X e A e then X ( x ) ~ O , i.e. X ( x ) = O . So ker p C_ ~ -

x/~-

{0},

whence p is faithful and p is a p-norm. 7.2.15. PROPOSITION. Let A ~ V ~ be either a complex or formally real commutative TA. Then A is a Gelfand algebra if it satisfies either of the two conditions: A is p-seminormed and hyponormal. (,) A is locally sm. convex and CQ. (**) PROOF. Let M be a maximal regular ideal with relative unity u. Then the quotient A # - A / M has u # as unity and A # i s a division algebra (see 1.2.5). Let now A satisfy (**). Then M is closed in A (by hyponormality) and so A # is p-normed. By 6.5.6, 6.5.12, A # - K e and M is hypermaximal. Let next A satisfy (,). Then M is closed (by 7.1.9) so that A # is a Hausdorff division algebra. Further, A # is locally sm. convex (by 4.4.12 (ii))and CQ (by 3.6.27). Also, by 1.6.21, A #

308

Gelfand Representation Theory

is formally real whenever A is formally real. It now follows, by 6.5.4, 6.5.8, that A # - ~ : e , M is hypermaximal. Thus we have shown that when either of the conditions (,), (**) is satisfied M is closed hypermaximal, so that A is Gelfand. 7.2.16. COROLLARY. Every complex or formally real unital commutative algebra satisfying either the condition (,) above or the condition A is locally sm. convex and C I ( , , 1) is a Gelfand algebra. PROOF. Since A is unital, A ~ v/A and further, by 3.6.22 the hypothesis "A is C I " is equivalent to "A is CQ ". The corollary now follows from the proposition. 7.2.17. COROLLARY. Every complex or formally real commutative p-Banach algebra which is either unital or at least has A ~ ~/A is a Gelfand algebra. PROOF. By 7.1.9, A is hyponormal. The desired conclusion now follows from 7.2.15 since A satisfies the condition (,) therein. 7.2.18. PROPOSITION. Let A be a strictly real algebra and A its complexification. Then A is a Gelfand algebra iff A is a Gelfand algebra. PROOF. This follows from 1.9.16 (ii), the result (by virtue of 1.9.18) that M § i M is hypermaximal iff M is hypermaximal, and the elementary observation that M § i M is closed in A iff M is closed in A. N

7.2.19. COROLLARY. Every commutative strictly real pBanach algebra A with A ~ x / ~ is a Gelfand algebra. PROOF. By 1.9.17 the complexification A of A satisfies the condition A :/: x/~. Therefore, by 7.2.17, A is Gelfand and so, by 7.2.18, A is Gelfand. 7.2.20. R e m a r k . C is a 2-dimensional real Banach algebra C [e] which is a division algebra. C[~] is not a Gelfand algebra since

w2. Gelfand Algebras

309

the ideal {0} which is (regular) maximal (C [~] being a division algebra) is not hypermaximal (since codim {0} - 2). Note that C [~] is neither formally real (since i 2 + i 2 - 0) nor strictly real (since i 2 - - 1 is not q. invertible). This example therefore shows that a commutative real Banach algebra can fail to be Gelfand if it is neither formally real nor strictly real. The algebra H[~] gives a 4-dim non-commutative real division algebra which again is not Gelfand (here codim {0} - 4). 7.2.21. THEOREM. Let A be a complex or strictly real commutative pseudo-Michael algebra with A ~ x/~. Then" (i) A~ -~ 0 (ii) A is t. spectrally Gelfand. PROOF.

(i) Let A -

lim A~ be the projective limit decom-

position, where A~ are pa-Banach algebras. Since A :/: there is an element x C A which is not q. invertible. By 4.5.6 (i) there is an c~ such that xa c .Aa is not q. invertible. It follows that v/A~ ~ Aa, and by 7.2.17, 7.2.19, Aa is a Gelfand algebra. By 7.2.2 (ii), A~ has a continuous character )~0 (say). Then X~ - f;0 o ~ is a continuous character of A, so that Ac ~ 0. (ii) Suppose that )~ E a~A(X), ~ ~ O. Then, by 4.5.7 (i), A c a3.,l (xa) for some a. But Aa being Gelfand is t. spectrally Gelfand and there is a (continuous)character ~a E / ~ a - A(Aa), with ~ ( x ~ ) - ~. Then X - X~ o ~ E Ac and X ( x ) ~. Further, when A is unital each A~ is also unital (by 4.5.6 (ii)). In this case if 0 c a(x) then x is not invertible, so that some x~ is not invertible in A~, whence 0 E 5~(xa). Since Aa is Gelfand, by 7.2.2 (v) there is a ~ E z ~ with ~ ( x ~ ) 0. Then X -- X ~ C A~ and X(X) - fC~(xa) - O. On the other hand if A e a(x), ~ J= 0 then ~ e a'(x) and so by what has been proved above there is a X E Ac with X(X) - )~. This completes the proof. ^

7.2.22. PROPOSITION. Let A particular Gelfand. Then we have"

be spectrally G e l f a n d - in

(1) For any x, y c A, (i) a'(x + y) _ a ' ( x ) + a'(y)

(ii) a'(xy) c a'(x)a'(y);

310

Gelfand Representation Theory

(iii) a(x + y) c a(x) + a(y)

(iv) a(xy) c a(x)a(y)

(here we assume A is unital);

(v) r(x + y) <~r ( x ) + r(y)

(vi) r(xy)<~ r(x)r(y).

(2) For any two idempotents u , v ( u r v) of A such that u ~ v, (vii) {0} c a ' ( u - v ) = { O , •

r(u-v)=l.

If A is unital and u,v ~ O or e then

(viii) {0} c a ( u - v)_c {0,• PROOF. If ) / 6 A then X(x + y) -- X(x) + X(Y) and X(xy) - X(x)x(y).

From these relations and the definition of spectrally Gelfand the relations (i)- (iv) follow. Also, results (v), (vi) follow from (i), (ii) respectively. It remains to consider (vii), (viii). By 1.7.9 we have" a'(u),a'(v) c_ {0,1}. Therefore, by (i), a'(u - v) C a'(u) + a ' ( - v ) -- a'(u) - a'(v) C_ {0,•

If we now show that a ' ( u - v) # {0} then (vii) will follow. Suppose to the contrary a ' ( u - v ) - {0}. Then -

-

c

c

-

{0}.

Therefore u - uv 6 A qn - vZ-A (see 7.2.12). Since u ~ v, u - uv is easily checked to be an idempotent. Since it is in x/~, by 1.2.24 (d), u - u v - O . Similarly v - u v - O . This, u - v - a contradiction, proving (vii). The result (viii)is proved by similar arguments. 7.2.23.

COROLLARY. (Michaelt).

Let A be a complex or strictly real pseudo-Michael algebra. Then for x, y 6 A with x ~-~ y, results ( i ) - ( v i ) of 7.2.22 hold. Further, results (vii), (viii) of 7.2.22 also hold for A.

PROOF. We may assume that A r x/~ (if A - x/~ the results hold either trivially or vacuously). Let Am be a maximal t He considered and proved the results only for complex Michael algebras.

311

w3. The Gelfand Representation

commutative subalgebra containing x, y. Then we know by 1.7.26 that for a E Am a~m (a)

-- a~4

(a); aAm (a)

--

aA

(a)

(when A is unital). So we may assume that A is commutative. But then, by 7.2.21, A is spectrally Gelfand and so the desired results follow from 7.2.22.

w3.

The Gelfand

Representation

7.3.1 Let A be a TA and A (respy. Ac) the set of all (respy. all continuous) characters of A. As remarked earlier A or Ac may be empty. In the sequel we assume that they are non-empty. We topologize A by equipping it with the weak topology (or topology of simple convergence): thus a net X~ ~ X iff X~ (x) ~ X(X) for every x c A (X~,XE A). Ac a s a s u b s e t of A inherits the relative topology. Let ~ (respy.)~c) denote the set of hypermaximal (respy. closed hypermxaximal) ideals of A. Since there is a bijection between A (respy. A~) and ~ (respy. ~ c ) ( s e e 1.3.9, 2.1.30) the weak topology on A (respy. Ac) can be transferred to (respy.)q~); Aic is a subspace of J~. We refer to A or A c as Gelfand space or spectrum of A; we refer to )~ or ~ as the maximal ideal spectrum of A. If A1 is the unitization of A we denote the corresponding spectra by A1, Alc, ~ 1 , ~ 1 c . 7.3.2. LEMMA. (a) A _~ t l \ { X o } , t c '~ t l c \ { X o } , where Xo is the distinguished character of A1 and "~ denotes homeomorphism. (b) A is homeomorphic to a subspace of the Cartesian product K = I ] K ~ (x c n), where each Kz -- K. In particular, A, Ac are Hausdorff completely regular spaces. PROOF. (a) The homeomorphims are given by X ~-~ X~, where X1 denotes the unique extension of X (see 1.3.17, 1.3.9, 1.4.9). (b) The map

x

(.-.,

312

Gelfand Representation Theory

of A into K is clearly bijective. Moreover, the weak topology of A can be indentified with the relative topology induced by the product topology of K on the image of A under the above map. This means that the above map is a homeomorphism. Since each K~ - ~ or C is Hausdorff completely regular it follows t h a t A has also the same properties (since these properties are preserved by products and subspaces). 7.3.3. PROPOSITION. Let A be a strictly real TA and A its complexification. Then the map N

A ' X C A(A)~-+ ;~ e A(A) (as defined in 1.9.17)

is a homeomorphism. Similarly, if A~ - AIA~ (restriction of A to the subspace Ac) then .

A

(A)

is a homeomorphism. PROOF. We have already seen t h a t in 1.9.18 that bijection. Since

h isa

2(z) - x(x) + ix(y) (z - x + iy, z e A;x, y E A) )~ is continuous iff X is continuous. Finally, it is clear that

So h~ is also a bijection.

Thus, h and hc are homeomorphisms. 7.3.4. Denote by C ( A ) the algebra of K -valued continuous functions on A; these functions can also be regarded as functions on 5t (since ~ can be identified with A). For each x C A, define on A ( = N) by -

- x(M)

(M-

ker X).

It is clear from the definition of the weak topology t h a t ~ E C ( A ) - C ( ~ ) ; ~ is called the Gelfand transform of x. We write - {~ 9 x c A} and call A the transform algebra of A. We

w3. The Gelfand Representation

313

can consider ~ operating on A or ~ according to our contextual needs. We can also clearly make each ~ operate on Ac or ~c. The algebra C(A) is a TA under the weak toplogy. This means that if f ~ , f E C ( A ) then f~--~ f iff f ~ ( X ) ~ f ( x ) for each X c A. Similarly the algebra C(Ac) is a TA under its weak topology. 7.3.5. PROPOSITION. (i) The map ~ : x ~-~ ~ is a homomorphism of A into C(A) and the map ~c : x ~-* ~ is a continuous homomorphism of A into C(A). (ii) ker ~ - ~ (= V ~ if A is sepctrally Gelfand). (iii) ~ is injective iff A is h.s.s. (=s.s. if A is spectrally

C Ifa d). (iv) If A is functionally continuous then ~ (= ,6c) is a continuous homomorphism. PROOF. (i) That ~, ,~ are homomorphisms follow from the fact that each X E A or A~ is a homomorphism of A. Further, if x~--+ x in A and X E A~ then :

Hence ~c is continuous. (ii) ker ~ - N N• E A) - ~ v/A (when A is spectrally Gelfand, see 7.2.12). (iii) This is immediate consequence of (ii). (iv) When A is funtionally continuous we have A _--Ac, so that .~ = ~c is continuous. 7.3.6. COROLLARY. Every real h.s.s. TA (or s.s.t, spectrally Gelfand algebra) A is formally real. PROOF. By 7.3.5 ((i), (iii)), A is (algebraically)isomorphic to A. Since elements of A are real-valued functions, by 1.6.18, /] and hence A is formally real. 7.3.7. COROLLARY. A commutative strictly real s.s., pseudoMichael algebra (in particular, p-Banach algebra) A is formally real. PROOF. By 7.2.21, A is t. spectrally Gelfand and so the result

Gelfand Representation Theory

314 follows from 7.3.6.

7.3.8. PROPOSITION. algebra, with p sm., then

If A -

1

is a p-seminormed

(A,p)

(,)

1

Hence Ilxll ~< 1; if A is unital with p ( e ) - 1, then Ilxll- 1. PROOF. If M k e r x then A x - A / M _~ K. Since X is continuous, M is closed and consequently the identity coset E -u + M (where u is an element of A such that X(u) - 1) is closed. Since E 2 - E we have for any y E E , y n E E (n--1,2,...). Since E is disjoint with M, 0 r E. It follows that y n / - . 0 (since y= c E and E is closed). Therefore, by 3.3.4, we have" p(yn) ~> 1. If x E A and X ( x ) - A # 0, then A - i x e E. So

p(,~-nxn) -- p ( ( , ~ - l x ) n ) ~

1,

which gives

p(~)

~

I~Xl"- Ix(x)l "'~,

so that we have 1

1

Making n ~ c~, we get Ix(~)l-< ~(~)~ -< p(~)~, which is (,). Further, Definition 3.5.1. and (,) imply that IIXII <- 1, ~nd Ilxll1 when A is unital with p ( e ) - 1 7 .3 .9 .

COROLLARY.

1

(since x(e)lp(e)-~- 1).

Ac is an equicontinuous family of K-

valued functions on A. PROOF. For X E Ac we have 1

Ix(~) - x ( y ) l -

Jx(~ - y)l ~< p(~ - y) ~ (~, y c a )

(,)

The equicontinuity of Ar is therefore a consequence of the continuity of p. 7.3.10. COROLLARY. If N - kerp, A # - A / N then At(A) is canonically homeomorphic to A t ( A # ) . PROOF. For X E A(A) define X# E A(A #) by X # ( x + N ) -

315

w3. The Gelfand Representation

X(x). This is well-defined since X vanishes on N (see 3.1.21 (v)). The map X ~-+ X# is clearly a homeomorphism. 7.3.11. PROPOSITION. Let A - (A,p) be a p - s e m i n o r m e d algebra and ]i - (A,~) be its completion. Then each X e A t ( A ) has a unique extension ~ E At(A) and the map X ~ ft is a

hom~omo~phi~m of A~(A) o~to A~(;t). PROOF. The existence of the unique extension )~ of X follows from 3.5.11 (b). Again, if f; e A~(A) then k e r ~ # A and being closed we have A ~ ker~, whence f;IA E A~(A). Thus, the map X ~ :~ ()~ - extension of X) is a bijection. It remains to show t h a t the m a p is bi-continuous. Suppose t h a t X~ --~ X. If ~ E zi, then ~ limx,~, x~ C A . Given e > 0,0 < e < 1, we can find n---~ o o

an n such t h a t

E

/ 3 ( ~ - xn) < ~.

(1)

Since X,~ (xn) --+ X(Xn), there is an a0 such t h a t for a >- a0, E

E x o ( x . ) - x(~.)l < 5.

(2)

Using the inequality (.) of 7.3.9 for /3, we have 1

I ~ ( ~ ) - ~ ( x ~ ) l < ~ ( ~ - ~,)~ <

~

< ~,

E

]~(~.) - ~(~)1 < 5

(3) (4)

From (2), (3), (4) we obtain, for a > a0,

I ~ ( ~ ) - i(~)]

~< i ~ ( ~ ) - ~ ( ~ ) 1 E

~

+ Ixo (~,) - x(~,)l +

E

< 5+5+5-~. This implies t h a t ~ --+ f:. Thus we have shown t h a t X~ ~ X f;~ --~ f:. On the other hand, the reverse implication is trivial since X~ - ~ [ A , X - f;[A. Thus, the map X ~ ~7 is bicontinuous, as we wished to show.

Gelfand Representation Theory

316

7.3.12. PROPOSITION. The spectrum Ac of a p-seminormed algebra A is locally compact Hausdorff. If A is unital, or more generally, if ~ / A t is regular then Ac is compact Hausdorff. PROOF. By 7.3.2, we can identify A~ with a subspace of the cartesian product K - 1-[Kz (x c A). Write Sz - {A e g ~ " 1

IAI <~ p(x)~} and S I-IS~ (x e A). Then S is a compact subset of K. By the inequality (.) of 7.3.8, we have: Ac _ S c K. Denote by Ac the closure of Ac in S(orK). Take a point A - (Az) c A~ and write h(x) - A~. Then A(x) - l i m x ~ ( x ) for some net (X~) in Ac. Since each X~ is a homomophism so is A. Also, since 1

we get 1

[A(x)] ~ p(x)~. Thus, h is bounded and so, by 3.5.5, continuous. By 1.3.3, A is either the zero homomorphism h0 or A E A~. Therefore

A ~ - A~ or A c - Ac U{ho}. Since S is compact and Ac is closed in S, A~ is compact. Thus A~ is either compact or locally compact (with Ac as its l-point compactification). Also, by 7.3.2, A~ is Hausdorff. Finally, if ~/-A is regular, with relative unity u, then x u - x E '~/-A C ker X (X EAc), whence X(x)x(u)

-- X(X), X(U) -- 1.

It follows that if X. -~ A then A(u) - l i m x , ( u ) - 1, so that A C A~ and A~ -- A~ is compact. 7.3.13.

COROLLARY.

Any subset AE of Ac of the form

A ~ - {XE A~ "lX(Xo)l >/ e},

f o r some fixed xo E A

and e > O,

is compact. PROOF. Clearly A~ is closed in A. Let A o o - AU{h0} be the 1- point compactification of A. Since ho(xo) - 0 , while

t For defintion see 2.2.18.

w 3. The Gelfand Representation

317

IX(Xo)l >1 E for X c A~ it follows that h0 is not a limit point of Ae. Therefore AE is closed in Aoo and hence is compact. 7.3.14. L E M M A . L e t a p-seminormed algebra A be a direct sum A = A1 | A2, with A j ( j = 1,2) closed bi-ideals of A. Then the s p e c t r u m Ac = Ac(A) has a decomposition

_ i,o U s o where A~

are (disjoint)closed subspaces such that {xIAa'XEA~

(j-

1,2).

PROOF. The above decomposition into subspaces follows from 1.3.18. It remains to show that they are closed. But this is quite easy. For, if X C A~ then X E A~ iff xIA2 - 0. If x~IA2 - 0 and X~ --~ X then for y E A2 X(Y) = lim X~ (Y) = 0, so that X E Al~

proving A~

is closed. Similarly, A~

is closed.

7.3.15. DEFINITION. A subset S of a TA A which has no unity is said to topologically generate or t. generate A if A is the smallest closed subslgebra containing S; S is called a t. generating set of A. If A is unital, with unity e, then the condition is modified by requiring that A is the smallest closed subalgebra containing S and e. A TA is said to be finitely or countably t. generated according as it has a finite or a countable t. generating set of elements. 7.3.16. PROPOSITION. Let Ac be the spectrum of a pseminormed algebra A = (A,p) and S a t. generating set of A. If (X~), Xo c A~ then for X~ -+ Xo in Ae it is sufficient if X~(Y) -+ Xo(Y) for each y E S.

(,)

PROOF. Suppose that (,) holds. Then given x E A and 0 < < 1, by definition of t. generating set, there is a polynomial P in, say, n (possibly) non-commuting variables and elements Y l , ' " , y, in S such that

318

Gelfand Representation Theory

E

< ~.

p(x - P(Yl,'",Yn))

(1)

Further we have

Fx~(x) - Xo(X)l ~ Ix~(x)- x~(P(yl,... ,yn))} -4--Ix~(P(yi,.. . ,y2)) Now

y.,))- Xo(x)l

- x o ( P ( Y l , "'" , Yn))l 4-]Xo(P(yl,...,

Ix~(~)- xo,(P(yl, ''',y.))l

(2)

~ lx~(x- P(yl,...,yn))l 1

p ( x - P ( Y l , ' " " , Yn)) -~

( b y (,) of 7.3.8) 1

E

< ~( since l i p >1 1).

(3)

Similarly E

IXo(P(yl,...,yn))-

Xo(X)[ < -~.

(4)

By our hypothesis

x(yk) ~ Xo(yk)

(k-

1,...,n)

so that, since P is a polynomial, we have x ~ ( P ( Y l , " ' , Yn)) -- P ( X ~ , ( Y l ) , " ' , X , ~ ( Y n ) ) --* P ( X o ( Y l ) , " " ,Xo(Yn)) - X o ( P ( Y l , " "" ,Yn)).

So we can choose a0 such that for a >- a0 we have

Ix~(P(yl,""

,y,~)) - x o ( P ( y l , " ' , y , ~ ) ) l

From the inequalities (2)-(5) we obtain, for a >~ a0, E

E

E

whence X~(X)--* X(X), completing the proof.

E

< ~.

(5)

w3. The Gelfand Representation

319

The spectrum Ac of a separable pseminormed algebra A is metrizable. 7.3.17.

COROLLARY.

PROOF. Let {xn} be a countable dense subset of A. Set, for XI,X2 E As

d(x,, X2) - ~

1

n=l

- x=(

2n 1 + (X~-(-x~-

n)l

(.)

Then d is a metric. For, if d(X1,X2)= 0 then X,(X•) = X2(X,~), for all xn, so t h a t by density of {xn} and continuity of X~,X2 it follows t h a t X~ = X2. The s y m m e t r y property of d is immediate from the definition of d. Finally, for the triangle inequality property of d it is clearly enough to prove that each of the s u m m a n d s in (.) satisfies the triangle inequality. But this can be established on the same lines as in the proof of 3.1.10. Now it is clear t h a t a net X~ --+ X, under d, iff

xo

x(x,)

= 1,2,...).

(,)

The set {x=}, being a dense subset of A, is clearly a t. generating set of A. So, by 7.3.16, condition (.) is equivalent to the convergence of the net X~ ~ X under the topology of Ac. Thus, the metric topology coalesces with the topology of Ac, completing the proof. 7.3.18. COROLLARY. The spectrum A c of a countably t. generated p-seminormed algebra A is separably metrizable. PROOF. If S is a countable set of t. generators of A then the set S1 of all finite products of elements of S, is countable. Denote by $2 the set of all rational linear combination of elements from S1 (where a rational linear combination in the complex case means the coefficients of the combination have rational real and imaginary parts). $2 is clearly countable and dense in A, whence the required result follows from 7.3.17. 7.3.19. THEOREM. Let A be a pseudo-Michael algebra with

projective limit decomposition A-

limA~ (A~ pseudo-Banach algebras).

Gel]and Representation Theory

320

Let A~,~,~ denote the sets of continuous characters of A, fia respy.. Let Aa denote the subset of Ac comprising characters which are p~ -continuous. $ Then: (i)

A~ - U A c ~ ;

(ii) A~ homeomorphic to }X~; (iii) /f we define r

2z(y~)

-

" }X~ ~

2~(~z(~z)).

}XZ by flZ -

wh~r~

~

~,~

ciated with the projective decomposition:

with

r th~

map~

a~o-

]ca E Aa,kZ E

AZ, ~5~(kZ) - ~ , then { / ~ ; { ~ Z } } is an inductivett system of topological spaces and continuous maps. Write s - ( t h e direct limit ) l i m / ~ . Then there is a bijective co.ti.uous m~p o $ s ~ ~ g i ~ . by (for any a) where f t - limfia.

o(2)(~)

-

xo (x) -

fla(xa)

PROOF. The relation (i) is an immediate consequence of 4.3.13. By 7.3.10, 7.3.11 we have Aa _~ A(A~) _~ / ~ which is (ii). If O(f:l(X)) - X ~ ( x ~ ) - O()~2(x)) - X2~(x~) for all ~ then X1 - X2, so that O is injective. It is also surjective. For, if X C A c , X C A~ for some a and X - )~a o p~ for some ) ~ E A~. It follows that X - O(X,). Finally, the continuity of O is clear from its definition. ^

7 . 3 . 2 0 . COROLLARY. If A is unital, each Aa is compact. PROOF. By 4.5.6(ii) each /~a is unital and consequently, by 7.3.12, / ~ is compact, whence Aa _~/~a is also compact. 7.3.21. DEFINITION. A TA A which is t. generated by a single element is called monogenic. Evidently a monogenic Hausdorff TA is commutative.

i.e. is a continuous character of the algebra (A,p~). tt For definition see [12, pp.184-5]. In general O is not a homeomorphism (see [12,p.161, Remark]).

321

w 3. The Gelfand Representation

7.3.22. E x a m p l e s

of m o n o g e n i c TA~s.

(i) The Banach algebra C[0, 1]. This has fo(t) - t (t e [0, 1]) as t. generator. That f0 is a t. generator is a consequence of the Weirstrass approximation theorem. (ii) (cf. [10, p.33]). The algebra 9.i of complex-valued continuous functions on unit disc [z I ~< 1 which are holomorphic on [z] < 1 is a Banach algebra under the sup norm. This algebra has fo(z) - z as a t. generator. To see this observe that if rE(z) - f ( z / l + e ) (e > 0) then f can be uniformly approximated by f~ in [z I ~< 1. Also, each s being holomorphic on Izl < 1 + c can be uniformly approximated by polynomials in z in Izl ~< 1, whence f0 is a t. generator. (iii) The Banach algebra k I - kl[0, 1] (see 3.4.6 (vi)). If fl denotes the constant function 1 , f l ( t ) 1 (t e [0,1]), then it is easy to see that if f ? - ]'1 * " ' * fl ( n factors), f~(s) - s"-l/(n1). It follows from Weierstrass approximation theorem and the fact that C[0, 1] is dense in k I that fl is a t. generator of [-1. (iv) The subalgebra W~_ of W ~ (defined in 3.4.10) consisting of all elements f -

~ - ~ ] ( n ) e int with ] ( n ) -

0 for n < 0,

nE~

is closed and so a p-Banach algebra. t. generator as can be easily seen.

It has

e~t as a

7.3.23. PROPOSITION. Let A be a monogenic unital p B a n a c h algebra which is either complex, or when real is strictly real or f o r m a l l y real. Let A - Ac be the spectrum of A and a a t. generator of A. Then the map A " X ~ x ( a ) is a h o m e o m o r p h i s m of A onto the spectrum a(a). PROOF. By 7.2.16, 7.2.19,7.2.17 A is Gelfand and consequently, by 7.2.8(5), A is surjective. A is also injective. For, suppose that X, (a) - x2(a). If x e A then x - limn Pn(a) (P,~ e K[x]), so that X.(z)

-

limx.(P.(a)) -limPn(x.(a))n

=

lrl

lim X2 (Pn (a)) - X2 (x). rt

limP.(x2(a)) l,l

Gelfand Representation Theory

322

whence X1 - X2. That A is continuous follows from the definition of the topology of A (the weak topology). By 7.3.12, A is compact, and a(a) is clearly Hausdorff. It follows that A is a homeomorphism, completing the proof. 7.3.24. PROPOSITION (Shilov). If A is a complex monogenic unital p-Banach algebra, with a t. generator a , p ( a ) - C \ a ( a ) is

connected. PROOF. Since a(a) is compact we can enclose it in a closed disc D. Then C \ D is connected unbounded and C \ D _ p(a). If p(a) is not connected it has a bounded component Go which is open (since p(a) is open and A is locally connected). It follows that aG0 C_ a(a). If A0 E Go then by the maximum modulus principle applied to Go we obtain, for any complex polynomial

P, [P(Ao)I

~<

sup IP(A)t ~< sup AEOGo XEa(a)

[P(A)l- sup

JP(x(a))l

xEA 1

--

sup Ix(P(a))] <~ IIP(a)[I ~-

XEA

Let A0 be the subalgebra of A, generated by a; A0 is dense in A. If x e A0 then x - P(a) for some P e C[X]. The map Xo "P(a) ~ P(Ao) is a bounded homomorphism. By 3.5.11 (b) this can be extended to all of A, yielding a (continuous) character Xo of A. Then A0 - Xo(a)e a(a) (by 1.7.24) while by choice of A0, A0 c Go C_ p(a). This contradiction proves the desired result. 7.3.25. R e m a r k . If K is a compact subset of C such that C \ K is connected then it has been shown by Shilov that there is a monogenic unital Banach algebra A such that K is homeomorphic with a(a), where a is a t. generator of A (see [10, p.72]).

7.3.26. PROPOSITION. Let A be a complex monogenic unital p-Banach algebra with a t. generator a. Then the transform algebra ~i can be identified with the algebra of all continuous complex functions on a(a) which are holomorphic on its interior a(a) ~ PROOF(cf.[28, p.412]). Since A is homeomorphic to a(a) we can regard & as a function on a ( a ) " if A E a ( a ) , A - x(a) then

w3. The Gelfand Representation

323

~c(A)- 3:(X)(X C A). Moreover, ~ is continuous. Thus, A can be identified with a subalgebra of the algebra C - C(a(a)) of all complex continuous functions on a(a). If x E A and x lim P,~ (a) then n----~ (x3

I1:~- Pn(a)lloo <~ IIx- P,,(a)ll;

1

so t h a t ~ is a uniform limit on a(a) of polynomials on a(a). Hence - ~(A)is holomorphic on a(a) ~ On the other hand, if f E C is holomorphic on a(a) ~ then since, by 7.3.24, p(a)is connected we can apply a theorem t of Mergelyan to conclude t h a t f is a uniform limit on a(a) of polynomials, so that f E A. This completes the proof. 7.3.27. PROPOSITION. Let A - (A,p) be a p-seminormed algebra, with p.sm., and C0(Ac) the algebra of continuous K-valued functions on the (locally compact) spectrum Ac vanishing at or Then" (i) ~ " x ~ ~: is a continuous homomorphism of A into C0(Ar such that 1 1

11:~11oo~< ,.,(x)~ ~< p(x) ~.

(.)

(ii) If A is also t. spectrally Celfand (in particular, Gelfand) then we have r(x)- II~ll~ < ~ , (**)

which implies in particular that al(x) is bounded. PROOF. We first observe that ~ E C0(Ac) since }(ho) -h0(x) - 0 and Ac U{ho} is the 1-point compactification of Ac. In view of 7.3.5(i), to prove statement (i) we have only to verify the inequality (,). But this is an immediate consequence of (,) of 7.3.8 (since II~ll~ - supx Ix(x)I ). When A is t. spectrally Gelfand we have a'(x) - { X ( X ) ' X E Ac} U{O}, whence I I ~ l l ~ - sup x ( x ) l •

r(x), proving (**).

7.3.28. COROLLARY. If ~c is a t. isomorphism then there is a positive constant C such that

See [25, p.385].

Gelfand Representation Theory

324

p(x) 2 ~ Cp(x 2) for all x c A.

(,)

PROOF. Since ~ - 1 is continuous there is a constant Co such that 1

p(x) F <~c0tl~llo~ Therefore 2

1

p(x)~ <~Co~ll~llL - CoI[~llo~ ~< Cop(x ~) ~, where in obtaining the last inequality we have used (,) of 7.3.27. Thus, we obtain p(x) 2 ~< Cp(x 2) ( C - C~). 7.3.29. PROPOSITION. Let A be a pseudo-Michael algebra with projective limit decomposition A - limti~. Then we have 1

r(x) <~sup uc~(x)Tg.

(*)

PROOF. By 4.5.5 (2), we have

r(x) -- rA(x) -- sup rd_~ (xa).

(1)

c~

Since A]~ is a pseudo-Banach algebra the inequalities (,), (**) of 7.3.27 give 1

r~o(~) .< ~ ( ~ ) ~ o

1

- ~(~)~o.

(2)

The desired inequality (,) now follows from (1),(2). 7.3.30. COROLLARY. A t. nilpotentt element x of a pseudoMichael algebra A is q. nilpotent. PROOF. Since all u~(x) - 0 we conclude from inequality (,) of 7.3.29 that r(x) - 0 and x is q. nilpotent, as required.

w 4.

GB

Algebras

7.4.1. DEFINITION. A p-seminormed algebra A - (A,p) with p sm., is called a Gelfand-Beurling algebra or a GB algebra if it satisfies the condition t For definition see 4.8.8.

w4. GB Algebras

r(x); - u(x) -

325

lira p(x'~)~

(,)

(<. p(x))

n--+oo

for any x c A. A p-Banach algebra A satisfying (.) is called a G B p-Banach algebra. 7.4.2. R e m a r k . By 6.2.11, 6.2.12 every complex or strictly real Banach algebra is a G B algebra. 7.4.3. LEMMA. Let A be a p-seminormed GB algebra. Then we have" (i) For any :,: c A, o'(~) i, bo..U~d. (ii) If x , y ~ A and x +-+ y then 1

r(~ + y) <. (r(x)~ + r(y)~)~

(,)

r(xy) <~ r(x)r(y).

(**)

1

PROOF. (i) Since r(x) - u ( x ) ; < oo, a'(x) is bounded. (ii) By 4.8.3, we have v ( x + y ) <~ u ( x ) + u ( y ) , u(xy) < v ( x ) v ( y ) . It follows t h a t 1

1

1

r(x -+- y)

--

u(x + y)~ <~ (p(x) § u(y)) ~ -- (r(x) p + r(y) p)

r(xy)

--

u(xy) 7 ~< ( u ( x ) u ( y ) ) ~ -- u(x)~ u(y)7 -- r(x)r(y).

1

1

1

1

7.4.4. PROPOSITION. A unital p-seminormed GB algebra A is spectral. PROOF. We have r(x) p = u(x). If u(x) > 0 then r(x) > O, so t h a t there is a A g: 0 in a'(x). It follows t h a t A E a ( z ) and ~(x) :/: ~. If . ( ~ ) = o, then by 3 . 3 . 7 ( v ) , , is n o t invertible. So 0 C a(x) and a(x) 7k O. 7.4.5. PROPOSITION. s e m i n o r m e d algebra we have

In a spectrally Gelfand

GB p-

1

Ilelloo - r(~) - . ( x ) ~

Hence, the concepts "essentially nilpotent", "t. nilpotent" coalesce and we have V~--

(,)

(~ e A ) "q. nilpotent"

{x E A " u(x) - 0} - {x E A " r(x) - 0}.

and

(**)

Gelfand Representation Theory

326

PROOF. The relation (,) above results by combining (**) of 7.3.27 and condition (,) of 7.4.1. The equivalence of "q. nilpotent" and "t. nilpotent" follows readily from (,) above; that of "q. nilpotent" "essentially nilpotent" from (,) of 7.2.12. 7.4.6. THEOREM (Beurling-Gelfand-Zelazko). A complex or strictly real p-Banach algebra A is a GB algebra. PROOF. We have to prove (see (,) of 7.4.1.) that 1

r(x) -- u(x)~

(x E A).

(,)

Since both sides of (,) remain the same when x is considered an element of the unitization of A, we may assume that A is unital. Consider first the case where A is commutative. By 7.2.17,7.2.19, A is a Gelfand algebra. So, by 7.3.27, we have 1

r(x)-

I1~11~ < ~,(x)~.

Write

(1)

1

Ilxll~- ~(~)~. By 4.8.6(ii), II" II, is a sm. semi-norm. Write I A # - A / I is a normed algebra with

(2) ker It" II,. Then

IIx#ll,~ - I I x + Ill,# -Ilxll, (see 3.4.15).

(3)

The completion B of A # is a commutative complex or strictly real (see 6.2.20) Banach algebra. Since B, A are Gelfand algebras, by 7.2.2(v) and (1) (above)we have

r(x#) -

sup I x ( ~ # ) l xEA(B)

sup I x ( : O l - I I ~ l l ~ - r ( x ) , xEA(A)

-

(4)

By 7.4.2, B is a G B algebra, so that we have

r(~#) -

~llll.

(~#)"

Now

~'1.1~(~ # ) =

limoo(llz#-II,#)} - lim (llx"ll,) 1 lim u(x n ) ~ (where we have used (3), (2)) n - - - r OO 1

=

1

lim u(x)-~ - u(x)-~ (using 3.3.7(ii)(6)).

n----~ o o

(5)

327

w4. GB Algebras

Thus

~ll.ll~ (x#)

-

~(~)~.

(6) 1

From (4),(5),(6) we conclude that r(x) - ~ ( x ) ; , completing the proof in the commutative case. When A is not commutative, consider a maximal commutative subalgebra A0 with x E A0. Then A0 is a p-Banach algebra which is again complex or strictly real (see 3.7.32). By virtue of (, 9 ,) of 1.7.25 we have rAo (x) -- rA (x) -- r(x).

Since A0 is commutative, by what we have just proved, rAo ( X )

--

1

~(x) ;. Therefore we obtain 1

r(~) - r~o (x) - ~(~)

completing the proof in the general case. 7.4.7.

COROLLARY. In a p-normed algebra A we have for

x~A u(x) ~

~x P

if A is real

If A is strictly real we have p(x) <~ r(x);.

PROOF. First assume that A is complex and let A] be its completion. Then by applying 7.4.6 to .~ we get ~,(x) - r 2 ( x ) ; <~ rA(x); ( by virtue of 1.7.20).

(,')

Next, if A is real, let A be its complexification. By considering x as an element of A and applying ( , ' ) w e get v(x) ~< ~(x) p. If A is strictly real then 5 ( x ) - a(x) and so we then have .(x)

< ~(x)~-

r(~)~.

7.4.8. THEOREM (Michaelt). Let A be a pseudo-Michael algebra which is either complex or strictly real and A - limA~ 4---

t He considered only the case of complex locally convex algebras.

328

Gelfand Representation Theory

its projective limit decomposition. Then for x e A, x - (xa)(xa C A~) we have the GB formula 1

1

rA(x) -- sup v ~ ( x ) ~ , where v~(x) O~

lim p~(x'~)-~ n----~ (X)

PROOF. By 4.5.7(2)

rA(x) -- sup r/i s (xa). o~

Since by 7.4.6, Aa is a GB algebra we have also

r4~(x~ ) - u ~ ( x a ) ~

- v ~ ( x ) ~ (since p ~ ( x ) - p~(x~)).

Combining these two results we get the above G B formula. 7.4.9. PROPOSITION. Let A be a p-normed algebra. If A is complex, every q. nilpotent element x of A is t. nilpotent; if A is real every ext. q. nilpotent element is t. nilpotent; if A is strictly real every q. nilpotent element is t. nilpotent. The radical v/A of every p-normed algebra A is a topologically nilt (hi-) ideal.

PROOF. The first three assertions are immediate consequences of (.) of 7.4.7 and the fact that when A is strictly real, ~(x) - r(x). For the final assertion we have to show that every element of x / ~ is t. nilpotent. If x c x/~, then by 1.7.15(ii), x is q. nilpotent. Therefore x is t. nilpotent when A is complex (as just seen above). To obtain this conclusion when A is real it is enough to show that x i s e x t , q. nilpotent. IfA=/=0in C t h e n ~ 2zl~_ -~ E~ (c~- ReA) and so this element is q. invertible. It follows from 1.8.5 that A r 5'(x), so that ~ ' ( x ) - {0} and x is ext.q, nilpotent as required. 7.4.10. PROPOSITION. Let A ~ x / ~ be a commutative pseudo-Michael algebra which is either complex or strictly real. Then all the three concepts-essentially nilpotent, q. nilpotent and t. nilpotent-coincide.

t i.e. every element of the ideal is t. nilpotent.

w 4. G B Algebras

329

PROOF. By 7.2.21 (ii), A is t. spectrally Gelfand. So, by 7.2.12, "essentially nilpotent" is the same as "q. n i l p o t e n t ' . Also, by virtue of GB formula of 7.4.8, "q. nilpotent" is the same as "t. nilpotent". Hence the proposition. 7 . 4 . 1 1 . LEMMA. Let A - ( A , p ) be a G B algebra. Then the following two conditions are equivalent: (i) There is a constant C1 > 0 such that C l p ( x ) 2 ~ p(x 2) for all x E A.

(,)

(ii) There is a constant C2 > 0 such that C2p(x) <~ r(x) ~ for all x E A.

(**)

PROOF. Assume t h a t (,) holds and apply the (,) inequality successively to the elements x 2 x 4 --. x n where n is of the form n - 2 k to obtain c r - l p ( x ) n <~ p(xn), from which follows t h a t re--1

C1 n p(x) ~ p(x n)-~ Making, n --, oc, we get C l p ( x ) ~ Vp(X) -- r"(x)

( since A is GB)

which is (**) (with C2 -- C1). Next assume t h a t (**) holds. Then

c p(x)

-

-

which gives (,) (with C 1 --C22). This completes the proof. 7 . 4 . 1 2 . PROPOSITION. Let A (A,I I 9II) be a Gelfand pn o r m e d algebra which is also a G B algebra. Then the map ~ 9 x ~ ~ is a t. i s o m o r p h i s m iff there is a constant C > 0 such that Ilxll 2 <~ Cllx2ll for all x c A.

(,)

Gelfand Representation Theory

330

In particular, the above conclusion regarding ~ holds when A is a complex or strictly real commutative p-Banach algebra with A 7s v~. PROOF. In view of 7.3.28, we have only to prove the "if' part. It follows from (,) that 1

I

1

~ ~ . <. C ( ~ + ~ + ~ ) Ilxll ~< c~-II~ll ~ <~c~c~ltx411 ~."

1

IIz2" II

so that we have IIx]l ~< CII~II~. It follows that ~ - 0 =~ x - 0, which means ~ is 1 - 1. Again it follows from the above inequality t h a t ,~-1 is continuous. Since A is Gelfand, A = A~ and ~ = ~ is continuous (see 7.3.5). Thus, ~ is a t. isomorphism, completing the proof. 7.4.13. COROLLARY.

~ is

IIxtl~ -Itx211

an isometry iff p -

1 and

for all x E A.

(,) 1

PROOF. If ~ is an isometry then IlxlJ- I1~11~ < Ilxll~ (by (,) of 7.3.27). This implies that p - 1 (for, if 0 < p < 1 and 11~11 < 1 1

then IlxlJ~ < lixll since ~1 > 1). Further we have,

Ilxll- Jl~Jl~ -Ilx~JJ~ -I1~1t which is relation (,). Conversely, suppose (,) holds and p C1 - 1) we get (see proof therein)

1. Then by 7.4.11 (with

IIxll ~< r(x) p - r(x) -II~lJ~ ( see (**) of 7.3.27). On the other hand, by (,) of 7.3.27, I1~11~ ~< Ilxll. Thus II~ll~ and ,~ is an isometry.

I1~11-

7.4.14. LEMMA. Let A - (A, II. II) be an ample complex pBanach algebra with II" II sm , and x, y E A. If there is a constant C such that

JlE(Ax)yE(-Ax)I I ~ C for all A ~ C,

w4. GB Algebras then x y -

yx. E(.kx)yE(-Ax). Then we have

PROOF. Write F ( A ) -

F(A)

331

-

[el -~- ,~x -~

(~x) 2 2!

~-...]y[el - )~x Jr ()~x)2

2!

~']

where el denotes the unity element of the unitization A1 of A. By continuity of multiplication in A1 we get F(~)-

[el -~- )~x-~ (~x)2

2!

Y('~x)2 -~--..].

~'" "][y- ~y~ +

2!

The first series on the right above is absolutely convergent (by 5.2.2) and the second series also is absolutely convergent (as can be checked by ratio test). So we can multiply (as in the classical case) the two series term by term to obtain

x2)

F(A) - y + A(xy - yx) + )~2 (x2Y2

xyx + - ~

+....

(1)

Since the series in (1) converges absolutely for all ,~ E C, F()~) is a strongly entire function. If ~ E A*t then ~ o F ( ) ~ ) - p ( F ( ) ~ ) ) i s an ordinary entire function. Further, it is bounded since 1

I~(F(~))I < II~l{llF(~)tl~ < II~IIC, for some constant C. By Liouville's theorem ~9 o F is constant, so that ~(F(A)) = p ( F ( 0 ) ) = !P(Y). Since A is ample, F()~) - y for all )~. Differentiating the series in A in (1) we get 0 = F'(O) = xy-

yx

so that x y - yx = 0, as desired. 7.4.15. THEOREM (La Page-Hirschfeld- Zelazkott). An ample

t A* - continuous dual of A. tt These authors obtained the result for Banach algebras (i.e. p - 1).

Gelfand Representation Theory

332

complex p-Banach algebra A satisfies the condition

(A ,

I1" II), with

I1" II ~m., which

ClllXl] 2 < IIX2I] for all x E A

(,)

is necessarily commutative. PROOF. Let A1 be the unitization of A, with unity el. For x, y C A , A C C s e t

z-

F ( A ) - E(Ax)yE(-Ax).

(1)

Clearly #e 1 -- E ( A x ) # e l E ( - A x )

(2)

(where we have used 5.2.3 (iii)). It follows that

z - #el = E ( A x ) ( y - #el)E(-.~x). The equation (3) shows that z invertible. Therefore we have

o'(z) = ~'(y),

#el is invertible iff y -

(3) #el is

r(~) = r(y)

It follows from (,) and 7.4.11 that there is a constant C2 > 0 with

c~llzlj < r(z)~-~(y)~. Therefore

IIF(A)I I -Ilzll < ~r(y) ~. Applying 7.4.14, we conclude that x y -

2

yx, completing the proof.

7.4.16. L EMMA. A complex or strictly real p-Banach algebra

- (A, ll" II), ~ith II" II ~m, sati4yi~g th~ co~ditio~

c111~112 ~ IIx211for

all x E A

(,)

i8 8.8..

PROOF. The condition (,) together with 7.4.11 and the fact A is G B (see 7.4.6) gives ClllZll ~ l](x) - r ( x ) p.

333

w4. GB Algebras

Therefore, r(x) = 0 ~ x = O, whence A is q.s.s, and so s.s. (see 1.7.18). 7.4.17.

THEOREM (Kaplansky).

Every strictly real s.s. p-

normed algebra A is commutative.

PROOF. First suppose that A is primitive. By 1.9.15, A is a division algebra and so by 6.5.12, A is isomorphic to ~, C, or H. The strict reality of A rules out C or H. Thus A _~ R. Next let A be s.s. and P be a primitive ideal. The quotient A p -- A / P is primitive and by 1.9.8, A p is strictly real and so A p "" ~. Since A is s.s., N P = {0}, so that A is isomorphic to a subalgebra of the direct product I] Ap (of isomorphic copies of ~) and so commutative. 7.4.18.

COROLLARY. Every strictly real primitive p-normed

algebra is isomorphic to ~. Every strictly real p-Banach algebra satisfying condition (,) o/7.4.16 is commutative. 7.4.19.

COROLLARY.

PROOF. By 7.4.16, A is s.s. and so by 7.4.17, A is commutative. 7.4.20. PROPOSITION (Yoodt) Let A (A, II. 11) be a pnormed algebra with II" II sin. Then the following statements are equivalent: (i) A is a Q algebra. (ii) ~(x)P - L,(x) - l i m , ~ o IIx-ll ~

(x e A).

(iii) f(x) p ~< ]lxll (x e A). (Here ~ denotes the spectral radius in A, where A denotes A itself when A is complex and the complexification of A when it is real.)

t Yood proved the proposition for a normed algebra (i.e. p - 1).

Gelfand Representation Theory

334

PROOF. Assume (i). Then there is an rl > 0 such that for any x with Ilxtl < rl, x is q. invertible. Set c - r1-1 if A is complex, and - [ ( l + r l ) 8 9 - 1] -1 if A is real. For a given x e A take a

- a + ifl ~ 0 in C with I~1 p > cllxtl. When A is real we have" II~l-~(x ~ - 2~x)ll ~< I~l-~(llxll ~ + 2~l~l~llxlI)

~< j~l-2~(ll~ll 2 + 21~1~11~11) (since p ~< 1, I~1-< I~1)

-(11~-1~11 + 1) ~ - 1 < (~-1+ 1 ) ~ 1 ~
(bythechoiceofrl).

(1)

W h e n A is complex we have

II-

A-lxtt-

IAi-'~

< ,.

(1')

Using 1.8.5, 1.7.8 we conclude from (1), (1') t h a t A ~ 5(x), whence

~(~)~ ~

,~11~11,

(2)

It follows that

~(x)~ - [ ~ ( ~ . ) ~ ] ! n

<

~ ~11~-ii

<

.(~).

(3)

Now the completion ~ of A is a G B algebra (see 7.4.6). So we have u(x)-

r(x) p ~< ~(x) p

( s i n c e .A _C r).

(4)

From (3),(4) we obtain u(x) = ~(x) p, proving (i) =~ (ii). Again, since 11" II is sm., (ii) =:~ (iii). Finally, to prove (iii) =~ (i), assume to the contrary. Thus, suppose A satisfies (iii) but not (i). Then we can find an x E A such t h a t Ilxll < 1 and x is not q. invertible. It follows from 1.7.8 t h a t - 1 e a'(x), so t h a t I - 11 = 1 ~< ~(x)P ~< IIx]J < 1 a contradiction whence (iii) ~ (i), completing the proof.

w5. Holomorphic Functional Calclus Holomorphic

Functional

Single Algebra

Calculus

Element

335 for a

*

7.5.1. DEFINITION. Let A be a complex unital p-Banach algebra (with unity e) and G C C, a nonempty open set. Write

Aa -- {x E A " a(x) C_ G}. If A E G then a ( $ e ) - { $ } , ) ~ e E A a , so that AG isnonempty. Moreover, by 6.2,14, AG is an open set. Denote by H(G) the set of (ordinary) holomorphic functions on G. 7.5.2. LEMMA. With respect to pointwise operations, H(G) is a commutative unital algebra over C. H(G) is,moreover, a locally sin. Y algebra, the metric convergence in it being the same as uniform convergence on compacta. PROOF. It suffices to prove the second statement. Since G is an open subset of C it is locally compact Hausdorff as well as a - c o m p a c t . It follows Jt that we can find an increasing sequence Kn of compact sets such that (i) U K , - G (ii)if g C G and K is compact then K C s o m e g n . For f c H ( G ) , set

p.(f)

-

sup ,kEKn

oo

1

n=l

2"

Ill-

i

+ pn(].),

Then Pn are sm. seminorms, ]. ] an F-metric, and (H(G),{pn}) a sm. jr algebra. Further, it is clear from the definition of [. [ that metric convergence is the same as uniform convergence over compacta. 7.5.3. DEFINITION. For f E H(G) set l(x) - fr(Ae- x)-lf(A)dA

(x E AG)

(*)

where F is any contour surrounding a(x) in G. If we write }~ - - x ~ - - ( x - Ae) -1 then we can rewrite (,) as } The exposition follows closely that of Rudin [24, pp.240-45]. tt See for example [9, p.241].

336

Gelfand Representation Theory

:/:a f (A)dA.

f(x) -

(**)

7.5.4. LEMMA. The A-valued integral in (**) exists and is independent of the particular contour r surrounding a(x) in G.

PROOF. By 6.2.9, 5.4.13(b) the integrand in (**) is locally p-admissible holomorphic on G \ a ( x ) . The integral exists by 5.4.9 and the independence of the integral on the contour 17 is a consequence of the Cauchy integral theorem (5.4.16 (a)). 7.5.5. LEMMA. Let A be a complex unital p-Banach algebra and suppose that x E A and a C p(x). Let F be a contour in G = C \ { a } such that r surrounds a(x) in a . Then 1

f

/(A-a)n(Ae 2~i Jr

-x)-ldA-

(x-ae) n (n--O,+l

I~ = 2~i

()~ - a)nxa d~.

'

+2 ...). (,) '

PROOF. Set

By Hilbert relation (6.2.2, ( , ) ) w e from which it follows that In = 2:rri

have x), = xc~ + ( A - oz)x;~xa,

(A - a)nxadA + 2ri

lx~xadA"

For n ~= - 1 , the evaluation of the first indefinite integral gives ~+1 x~ so that the corresponding definite integral vanishes since r is closed. On the other hand, for n = - 1 , the first integral, vanishes since the index I ( r , a ) = 0, by hypothesis. Thus we obtain the recurrence relation In+ 1 -- Inx-a 1 -

In(X-

Ole).

(1)

If In denotes the integral in (.) we have /:n - - I n , so that (1) gives i~+1 - I~(x - ae) (n - 0, + 1 , •

w5. Holomorphic Functional Calclus

337

To complete the proof it is clearly enough to show that -T0 - e. By 6.2.10 (,), with m - 0 , we obtain I 0 - - e , so that I 0 - e as required. o

7.5.6. PROPOSITION. Let A be a complex unital p-Banach algebra and suppose that x c A and R - R(A) a rational function over C whose poles do not lie in a(x). Further, let

R(~) - P(~) + ~ ~ ( ~ - ~ ) - ~

(1 ~< ~ ~< m)(1 ~< ~ ~< k~)

r~8

be the representation of R (obtained by division algorithm and partial fraction development) where P is a polynomial (which may be 0), ar the poles of R, and kr the multiplicities of at. Set R(~) - p(x) + ~ c . ( x -

~)-~.

r,8

Then R(x) - R ( x ) -

2ril ~r R(A)(Ae- x)-ldA.

(,)

PROOF. Write RI(A) - ~ c~(A-a~) -~. By (,) of 7.5.5 and ~,~ linearity we obtain

R I ( ~ ) - 2~'i 1 f~ RI(A)(Ae- x)-ldA.

(1)

Again, by (,) of 6.2.10 and linearity we get

P(x) - 2~ril

fr P(A)(Ae - x)-idA.

(2)

Adding (1),(2)we get the representation (,). 7.5.7. LEMMA. Suppose that ~ E G and f E H(G). Then A,

PROOF.

f (c~e)

=

2~i

f(~)(~-~)-i,d~

[2_ fjr f(,~)(,~- ~) -1 e,~]~ 2ri f(c~)e (by classical Cauchy theorem).

Gelfand Representation Theory

338

7.5.8. Set H - [t(Aa) - { f ' f E H(C)}, where each ] is a A-valued function on Aa. Under pointwise operations H is an algebra over C. 7.5.9. THEOREM. H(Aa) is a complex Hausdorff T A under

the weak topology: S%--* / i f f f ~ ( x ) - ~ ](x) for each x E A~. The map A" f ~ f is a continuous isomorphism of H(G) onto H(AG). In particular, H(AG) is commutative. PROOF. It is clear that the map A is linear. If ] - 0 by 7.5.7, f(A)e- f(Ae)- 0 (A e G)

then,

N

so that f - 0 , proving A is 1 - 1 . Since ( A e - x) -1 - - x a is locally p-admissible holomorphic on p(x), it has a local representation ()~e- x ) - I -

~-~ pj(A)xj with pj J 1

ordinary holomorphic for A E G z - {A E C" )~ > u ( x ) ; } proof of 6.2.9). It follows that, for f~, f E H(G) we have

L(x)-

f(x)

-

2~ri 1 f r ( f ~ ( A ) - f ( A ) ) ( A e - x) -1

-

2~i ~

*j

(see

(f~(a) - f ( a ) ) ~ j ( a ) a a

3

so that N

N

(2~)~ ~. Ilxjll I (f~(~)-

]{f~(x) - f(x) ll

f(~))~(~)d~l p

3

1 3

Z

where M -

G-/II~ M,

Iri: (2~)~ ~ II~jlt ll~jll~ < ~ -

It follows that fa --+ f

3 N

in H(G) =~ f ~ - - , f in H(Ac), whence A is continuous.

To

w5. Holomorphic Functional Calclus

339

complete the proof of the theorem it remains to show that A is multiplicative: A(fg) - A(f)A(g). Thus if h - fg we have to show that h(x) - f (x)[t(x) (x e Ac). If f, g are rational functions without poles in a(x), then by 7.5.6,

h(x)- h(x)- fg(x)-

f ( x ) g ( x ) - f(x)[l(x).

In the general case, we make use of Runge's approximation theorem to find rational functions f,~, gn such that f,~ --+ f , g,~ --+ g in

H(G).

Then f,~g,~ ~ f g - h. It follows that

h - fg

=

A(fg) - A(lirn f,~g,~) - lirnoo A(f,g=)

=

lim A ( f n ) h ( g n ) - f~, r~---+ o o

N

completing the proof of the multiplicativity. Finally, the commutativity of H(AG) is an immediate consequence of that of H(G) and the isomorphism. N

7.5.10. COROLLARY. x ~ f(x)

(x E AG).

PROOF. Define j E H(G) by j ( ~ ) - $. Then, since Ac is commutative, j ~ f which clearly implies: j ( x ) - x ~ f(x). 7.5.11.

COROLLARY. (of. [15,p 206]). If f E H(G) has a oo

power-series representation f (A) - E an)~n (an E C) throughout n--O

G then oo

n--O

N

PROOF. Write P N ( A ) - ~

a,~A". Then P N ~

rt=0

So, by construction of map A, we have oo

f(x)-

lim/hN(X ) - l i m P g ( x ) N

~

N n=O

anxn.

f in H(G).

340

Gelfand Representation Theory N

7.5.12. LEMMA. If f E H(G) and x c AG then f (x) is invertible in A iff f has no zero in a(x). PROOF. Suppose that f ~= 0 on a(x). By continuity of f there is an open set G1 such that O(X) C G1 ___G and f ~fi 0 on G1. It follows that g - 1 / f is holomorphic on G1. Since fg - 1 on G1, by applying 7.5.9 (with G1 replacing G ) we obtain f ( x ) O ( x ) - e (using 7.5.7), so that f ( x ) is invertible. Next suppose t h a t f ( a ) - 0 for some a e a(x). Since f is holomorphic we can write f()~) - ( A - a)g()~), with g E H(G). It follows that N

f (x) - ( x - ~e)~(x) - ~(x)(x - ~e). N

Since (x - ae) is not invertible (since a E a ( x ) ) , by 1.1.30, f ( x ) is not invertible. 7.5.13. THEOREM (Spectral Theorem). Suppose that A is a complex unital p-Banach algebra, G C C an open set. If x C A c and f C H(G) then

a(f(x))-- f(a(x)). PROOF. Now o~ E a ( f ( x ) ) iff f ( x ) - (~e is not invertible. By 7.5.12, this is equivalent to f ( A ) - a vanishing in a(x), i.e. c~ c f (a(x)).

for a complex unital pBanach algebra A to have an idempotent u ~ O, e is that there is an element a E A with a(a) disconnected. 7.5.14.

COROLLARY. An n.a.s.ct

PROOF. The condition is necessary since, if u exists, we have by 1.7.9, a ( u ) - {0, 1}, so t h a t a(u) is disconnected. To prove t h a t the condition is sufficient assume t h a t there is t n.a.s.c -- necessary and sufficient condition.

w5. Holomorphic Functional Calclus

341

an element a 6 A with a(a) disconnected. Since a(a) is closed we have a disjoint decomposition

a ( a ) - FI~JF2 where Fj # 0, Fj closed in C ( j - 1,2). It follows t h a t we can find open sets Gj D F j ( j - 1,2) such t h a t G I N G 2 O. Set G - G I U G 2 . Then Gj Aa(a) # O. Define f 6 H(G) by f(A)

_

S

0

/ 1

if )~ 6 G1 if A 6 G 2

Then f is holomorphic and ] . 2 _ f. If u Also,

f(a)

then u 2 -

u.

a(u) - a(f(a)) - f(a(a)) - (0, 1). Since a(O) - O, a(e) - 1, we have u :/: O, e. 7 . 5 . 1 5 . PROPOSITION. Let A be a complex unital p-Banach algebra and A its transform algebra. Holomorphic functions operate on A in the sense that if x 6 A, f - f(A) holomorphic in an open neighbourhood of a(x) there is a y 6 A such that y ~ x and = f o ~ , i.e. Y ( X ) = X ( Y ) = f ( x ( x ) ) (X6A).

The element y is uniquely determined whenever A is h.s.s. (~/A - {0}). PROOF. Set

1 fr()~e - x)-lf()~)d)~ Y - 2ri where P is a contour surrounding a(x). By 7.5.10, y ~-. x. Since X is a continuous h o m o m o r p h i s m we have x(y)

=

2~ri

X ( ( A e - x)-lf(A))dA

f ( x ( x ) ) (by the classical Cauchy theorem).

Gelfand Representation Theory

342

The uniqueness statement for y follows since, when A is h.s.s., X(Yl) - X(Y) for all X E A =~ Yl - Y. 7.5.16. THEOREM (a) (Wiener- Zelazko). If f - f ( t ) e W p t and f ( t ) # O for any t then 1 ~ f E W v. (b) (Wiener-Levi-Zelazko). If f e W p and F is holomorphic in an open set G containing the range of f then F o f E W p. PROOF. (a) Note first that Xtx " f ~ f ( t l ) , t l E ~, is a character of W p. Let next X be any character of W p. Since W p is Gelfand (by 7.2.17), X is continuous. So, if f e WP, f ~ - o o f ( n ) eint then oo

x(f)

- ~ --

/ ( n ) x ( e a ) n. 00

1

Ix(e")l

Now

r(e")

- 1. Similarly, Ix(e-it)l <~ 1, so

tl "ll

that

Ix(e")1-1- Ix(e-")l

1.

Therefore I x ( e i t ) [ - 1, so that x(e it) - e it1 for some t l E ~. It follows that oo

x(f) - E

f(n)x(eit~) n -

f(tl),

so t h a t X - Xt~. The Wiener-Z~lazko theorem is now an immediate consequence of 7.2.10 (since " x ( f ) ~: 0 Vf" r "f(t) r 0, Vt" ). (b) Since the range of f is a ( f ) , the holomorphic function F satisfies the hypothesis (for f ) of 7.5.15. Also l/izP can be identified tt with WP. Hence, by 7.5.15

t tt

See 3.4.10 for D e f i n i t i o n of W p . T h i s is p o s s i b l e s i n c e

0, v t c ~ ~ . f - o ) .

A -

W p is s.s.

(f E ~

=~ x t ( f ) :

f(t) :

w5. Holomorphic Functional Calclus

343

7.5.17. THEOREM (Michael).* Let A be a complex unital pseudo-Michael algebra. Given x E A and a function f holomorphic in an open neighbourhood of a(x), then is a y E A such that y~-~x and [ o

(x e

The element y is uniquely determined if A is h.s.s. (in particular if A is commutative s.s.). PROOF.

Let A - (A, P),,P saturated, and A - limA~ be

the projective limit decomposition of A. For x - (xc~) E A denote by 1-'c~ a contour surrounding a ~ ( x c ~ ) - o~,(xc~). (Note that by 4.5.7(3)), a A ( X ) - L.Ja~(x~).) Since ti~ is a p~-Banach algebra, by 7.5.3, 7.5.4, we can set

ya -- ~

()~eoL- xa)-lf()~)d)~ c~

and the definition of y~ is independent of the particular choice of F~. In the notation of the proof of 4.5.3, ~ap is a continuous homomorphism of Af~ onto A~, when c~-< ft. Then

( ) -

fr

()~e~ -- xfl)-l f()~)d)~

-- JfF ()~ec~-Xo~)-lf()~)d)i:::/F ()~e.-x~)-lf()~)d~ (since, by 1.7.19, a~(x~)c_ follows that y - (ya) E A. X E Aa for some a, and X X(X). By 7.5.15 applied to X ~ ( Y ~ ) - f(x~(x~)). Hence

aZ(xz)), so that ~5~Z(y~)- ya. It If X E Ac -- At(A), then by 7.3.19, determines Xa E Aa with X a ( X a ) Aa there is a ya E Aa such that

X ( Y ) - f ( x ( x ) ) or ~ - f(~:).

t He obtained the result for the case of locally sm. convex algebras.

344

Gelfand Representation Theory's

The uniqueness of y, when A is h.s.s., is clear. Finally, we have y ~ x, since ya ~ xa for each c~. 7.5.18. R e m a r k . The result in 7.5.16(a) can be restated in the following form" if f - f ( t ) is a continuous complex-valued 27r-periodic function on ~ such that (x)

(i)

~

[](n)l p < c~ ( ] ( n ) , n E 7/, denoting the n t h Fourier

nz--OO

coefficient of f ) (ii) f does not vanish anywhere on ~, then 1

g - - ] has the property

OO

9 ~

]t~(n)lp < c o .

For p - 1 the above result reduces to a celebrated lemma of Wiener which he uses for proving his general Tauberian theorem.

w6.

Automorphisms

and

Derivations

7.6.1. An isomorphism of an algebra A onto itself is called an automorphism. If A is unital then every invertible element a in A determines an automorphism Ia -- ~.ara-1; I a ( X ) - - axe -1 (a C G i , x E A). (Note that the identities IaIa-1 = I a - l I a = I guarantee that Ia is bijective; also it is clear t h a t Ia preserves algebraic operations). The automorphism Ia is called an inner automorphism of A. 7.6.2. L EMMA. Ia -- I iff a E Z (the centre of A). In particular, if A is commutative every I a - I. PROOF. Clear. 7.6.3. In a T A we have automorphisms which are continuous as well as automorphisms which are also homeomorphisms, i . e . t , automorphisms. In general, not every automorphism is continuous nor every continuous automorphism a t. automorphisms.

7 .6 .4 . PROPOSITION. Let A be a T A ,

with unity e. Then:

w6. Automorphisms and Derivations

345

(i) Every inner automorphism of A is a t. automorphism. (ii) If A is a (F) a l g e b r a - inparticular a ~ a l g e b r a - then every continuous automorphism is a t. automorphism. (iii) In a semi-simple p - B a n a c h algebra every automorphism is a t. automorphism. PROOF. tinuous, and (ii) This 3.1.15). (iii) This

(i) This follows since I~ -- l a r a - l , l a , r a - 1 are con2ra-1 - 2ra-1. is a consequence of the open mapping theorem (see is proved in chapter 9 (see 9.4.7).

7.6.5. DEFINITION. Let A be an algebra (over F). A linear map D : A -~ A is called a derivation if D ( x y ) = ( D x ) y + x D y for all x, y c A. Let P denote the algebra of polynomials f ( x ) in a single variable x over F. For f ( x ) = "1o + . . . +'1nx n ('1i e F) define f'(x) -- '11 -~- 2'12x + . . . + n'1n xn-1. Then f ~ f~ is a derivation of the algebra P. 7.6.6. R e m a r k s . If /) = P(A) denotes the set of all derivations of A it is clear that it is a linear space (over F ) . Moreover, if we write for D1,D~ E P,[D1,D2] - D : D 2 - D 2 D l , t h e n it is easy to see that [O1,O2] ~ P; [O1,O2] is called the Lie product of D1, D2. P is in fact a Lie algebra i.e. a non-associative algebra (= an algebra without the associative law for multiplication) whose multiplication satisfies: [D, D] = 0, [O1, [02,03]] § IDa, [03, D1]] + [03, [D1, 02]] = O. (D, D1, 02, 03 E D). 7 . 6 . 7 . LEMMA. Let D be a derivation of A and u an idempotent. Then: (i)

-

0

(ii) If u ~ D u - in particular if u lies on the centre of A then D u - O. (iii) If A has a unity e then D e - O , all A E F .

hence also D ( A e ) - 0

for

346

Gelfand Representation Theory's

PROOF. (i) Du - Du 2 -- (Du)u + u D u so u D u - u ( D u ) u + uDu, whence

(1) (2)

u(Du)u-O.

(ii) If u ~ Du then equation (1)gives 2u2Du - 2u(Du)u - O (by (2)).

Du -

2uDu -

(iii) This follows from (ii). 7.6.8. LEMMA (Leibniz rule). n

D'~(xy) - ~ ( n ) D ' ~ - r x ( D r y ) r r=O

(n-

1,2,... ; x , y E A).

PROOF. By induction, making use of the well-known relation (rn) 4- ( r n l )

- (n:l)

9

7.6.9. COROLLARY. Suppose that D 2 x - O. Then: (i) D m x - 0

(m >t 2),

(ii) D ( D x ) '~ - O (m >l 1), (iii) D~(x '~) - n!(Dx) '~ (n >i 1). PROOF. (i) D ' ~ x - - D m - 2 ( D 2 x ) - 0 (m > 2). (ii) For m - 1, we have D ( D x ) - D = x - O. Assume now that n ( n x ) m - i - o ( m ) 2). Then D ( ( D x ) m) -

(1)

D ( D x ( D x ) m - l ) - D 2 x ( D x ) m-1 + D x . D ( D x ) m-1

- O.(Dx) m-1 -4- D x . O - O.

(iii) Assume that Dn-l(x n-l) -(n-

(2)

1)!(Dx) n-1.

Then Dn(x n-l) - D((n-

1)!(Dx) n - l ) - ( n -

where we have used (2). By Leibniz rule

1)!D(Dx) n - l -

0

(3)

w 6. A u t o m o r p h i s m s

D n ( x n)

--

347

and Derivations

Dn(xn-l.x)

D n x n - 1 .x +

D'~-lxn-I.Dx

rt D n _ r x n _ l . D r x

+ r=2

Using (1),(2),(3)the 1)!(Dx)'~-IDz

+ 0 -

r

above RHS reduces to 0 + n . ( n completing the proof (by induc-

n ! ( D x ) '~,

tion). 7.6.10. PROPOSITION. A

Let D

be a d e r i v a t i o n o f an algebra

a n d x an e l e m e n t o f A s u c h t h a t D x ~ x . (i) D x ~ - n x ' ~ - l D x (n >1 1),

(ii)

Dmx n -

xn-mam

for some element

Then: am e A

(1 ~< rn <

n,n>~2),

(iii)

D'~x n - n ! ( D x ) ' ~ + xbn f o r s o m e e l e m e n t b , e A (n >1 1).

PROOF. We prove the results by induction. (i) Assume that D x '~-1 - ( n 1 ) x Z - 2 D x . Then Dx n

-

Dxn-l.x

+ xn-lDx

(n - 1 ) x n - l D x

-

(n - 1 ) x Z - 2 D x . x

+ xn-lDx

(ii) Assume that D m-ix n Dmx n

-

- nxn-lDx.

_ xn-m+lam_l.

--

D(Dm-lx")-D(x"-m+lam_l)

__

Dxn-m+

=

(n - m + 1 ) x n - m D x . a m _ l

-

X'~-mam(

1 . a m _ 1 q-- x n - m +

with

+ xn-lDx

1Dam-

Then 1

+ xn-m+lDam_l

am -- (n - m + 1 ) D x + x D a m - 1 ) .

(iii) Assume that on-ix

n-l-

(n-

1)!(nx)

n-I

q-xbn-i.

(1)

Then Dnx ~

_

D.-~D(x. ) - D.-*(nx.-*Dz)

=

riD"- i x " - t.Dx + n

~l(nr--1

=

n[(n -

1)!(Dx)"-~ +

r

_ nD.-~(x.-XDx )

1)D,-,-1-, x ,-,-1D rDx n - 1 zran+rDr+ 1x.

xbn_~lDx + n r--1

=

n!(Dz)" + zb.,

r

348

Gelfand Representation Theory's

for some element bn, where we have used result (ii) for evaluating each of summands in the second line. 7.6.11. Let A be an algebra (over F ) . For a E A, set Dax-

(~a - ra)X -- a x -

xa.

Then Da is a derivation (see below) called an inner derivation. Further we have clearly the relations D~+b -- D, + Db; D~, - AD, (A E F). 7.6.12. LEMMA. (i) Da is a derivation (ii) D a = O iff a E Z (the centre of A ) ; D a = O for all a if A is commutative. (iii) If A is a TA then Da is continuous. (iv) If A (A,[[. [[) is a p - B a n a c h algebra, with ][. [[sin., then [[Da[[ <~211a][. PROOF. (i)Since Da = ~ a - ra it is linear. Further D~(xy)

-

=

axy - xya - (ax - xa)y + x(ay - ya) Dax.y + xDay.

(ii) Clear. Off) The continuity of D~ follows from that of s (iv) l[Da(x)ll - I [ a x - xa[[ < 211el] [[x[].

ra.

7.6.13. R e m a r k . The inner derivations form an ideal t Pi of the Lie algebra P of all derivations. To see this it is enough to observe t h a t for any derivation D, [D, Da] = Db, where b = Da, (this can be easily checked). n

7.6.14. LEMMA. (D~)n(x) - ~ ( _ l ) r ( n ) a n - r x a

r

r--O

PROOF. The proof is by induction. Assume that

t Since a Lie algebra is anti-commutative ([x,y]- -[y,x]) every ideal is a bi-ideal.

w6. Automorphisms and Derivations

ni

(Da)n_l(x)_ E(_l)r r----0

349

( n - 1) an_l_rxar" r n--1

Then

(D~)"(z)

-

D~(D~)"-~(x)=

D~E.... r---O

~-~(

-

1~~ n - 1

a

,_~

xa

~-

-1

r

r

a

,,-~-1 xa ~+1

r

r--O

a"x+

n-1

r---O

-,

-1)"

[(n-r 1) + (: :)]

an-'za "-(-1)'-lxa"

r--1

E(-1)"

a'~-~xa ~.

r--O

7.6.15. PROPOSITION. Any derivation D of an algebra A can be uniquely extended to its unitization A1. PROOF. Define D l ( A e + x ) - Dx, where x E A, el is the unity of A1 and A E F. It is straightforward to check that D1 is a derivation of A1. The uniqueness assertion follows from the fact that for any derivation D' of A 1 , D ' ( A e ) - 0, by 7.6.7 (iii). 7.6.16. PROPOSITION. Let A be a real algebra, D a derivaN tion of A and A the complezification of A . The (unique) linear extension D of D to A is a derivation of A . PROOF. D ( x + i y ) -- D x + i D y . It is straightforward to verify that D is a derivation of A. 7.6.17. LEMMA. Let A,A* be algebras and 99" A--~ A* an epimorphism. If n is a derivation of A such that D(ker~p) _ k e r ~ then D induces a derivation D* of A*. PROOF. If x* E A*,x* - ~(x) set D ' x * - ~ ( D x ) . Then D* is a well-defined map on A* since if !p(x) - ta(y), ~ ( x y ) - 0 and 9 ~ ( D x ) - g ~ ( D y ) - g ~ ( D ( x - y ) ) 0 (since x - y e ker ~, D(ker p) C ker g~). Further, D* is clearly linear and D* (x'y*) - g)(Dxy) - ~ ( D x . y + x.Dy) - p(Dx)g~(y) + ~(x)g~(xy) -- D*x*.y* + x*.D*y*.

350

Gelfand Representation Theory's

7.6.18. PROPOSITION. Let A be a p-Banach algebra and D a continuous derivation of A. Then E(D) - ExpD is a continuous automorphism of A. PROOF. By 5.2.4, E(D) is an invertible, bounded - hence c o n t i n u o u s - linear operator. So to prove the proposition we have only to show that E(D) preserves multiplication. Now

E(D)(xy)

=

~

~.

(xy)- ~

n=0

~.

n=0

oo n (

1

EE

r r=0

Dn_rx) ( 1 D r )

7'

n=0r=0

(E(D)x)(E(D)y), completing the proof. 7.6.19.

PROPOSITION.

In a unital p-Banach algebra A,

E(Da) -- ~rE(a)(a E A). PROOF. Using 7.6.14 we get oo

E(Da)(x)

--

~

D~x = ~ - ~ n! rt:0

oo

~

EE

n=0

r=0

-1

1

~an-rx(-1) (n-r)'

a r

r_ r!

oo a n oo an E --~.xE(-1)nn---~. n=0

n=0

E ( a ) x E ( - a ) - .rE(a)(x ).

t Note t h a t the expression on the LHS of the equality gives the Cauchy p r o d u c t of the two series on the RHS.

351

w6. Automorphisms and Derivations

7.6.20. THEOREM (Singer-Wermer) t . Let A be commutative p-Banach algebra which is either complex or strictly real. If D is any continuous derivation of A then D(A) C x/~. PROOF (following Sinclair). First assume that A is complex and also that it is unital. For a E A, 27 E A - Ac, write x - a - x(a)e. Then x E k e r x and Dx - De. By 7.6.10 (iii), D'~x '~ - nl(Dx) n + xbn.

Therefore x ( D n x r') - n ! ( x ( D x ) ) r~ (since X ( x ) -

(n!l-!x(D"x'~)-~,,,

1

0). So x ( D x ) -

so that

I x(Dx) l

<

(n!)-& I x(D"x")& (n!)-&llD"(x"ll (by 7.3.8) I

where IID~II denotes the bound of D n (for definition see 3.5.1). The above inequality, gives I x ( D x ) I<~

( !)- llDll (llx"ll ) 1

Making n ~ c~ we conclude (since (hi) - --~ O ) t h a t t x ( D x ) I-- O, Dx e kerx. D x C ~ / A - vIA (since A is Gelfand). If A has no unity, consider its unitization A1. By 7.6.15, D can be extended to D1 on A1. By what we have proved above we get for x c A, Dx - Dlx E ~

N A - ~

(by 1.4.9(c)).

Now we may assume that A ~: x/~ (if A - x/~, trivially Dx E x/~). Then by 7.2.17, 7.2.8, and 7.2.12 ~ V ~ . Thus, Dx E .v

Finally, if A is strictly real consider the complexification A. By 7.6.16 D has an extension /9 to A. Then D A C_ Dfft C_ V ~ . Also D A C A. So t Singer-Wermer proved the result for commutative complex Banach algebras

352

Gelfand Representation Theory's

DA C A ~ V ~ -

~

(by 1.9.17).

7.6.21. COROLLARY. Let D be a continuous derivation of a complex or strictly real, commutative pseudo-Michael algebra A - (A,{pa}) such that each kerpa is invariant under D " D(kerp~) C_ kerpa. Then D(A) C_ x/~. PROOF. Let A - l i m . z i ~

be the projective limit decomposi-

tion. The continuous derivation D induces, by 7.6.17, a derivation D~ on An which can be extended to a derivation /)a on A~. Since Aa is a p-Banach algebra we have by 7.6.20, for any x E A, /)a(x~) _C V/-Aa. It follows that

and hence

Dx

E N ~lV/~a cC_~

(see 4.5.8).

Cg

7.6.22. R e m a r k . It follows from 7.6.20 that in a complex or strictly real commutative s.s. p-Banach algebra, 0 is the only continuous derivation. Johnson has shown that in the case of a complex commutative s.s. Banach algebra, 0 is also the only derivation (continuous or not). Further, Johnson and Sinclair have proved that all derivations on any s.s. Banach algebra are continuous (see [4, pp.93, 95]).

CHAPTER

VIII

COMMUTATIVE t TOPOLOGICAL ALGEBRAS

w1.

F u n c t i o n A l g e b r a s tt

8.1.1. DEFINITION. An algebra A over K is called a function algebra if there is a set S and every element of A is a K-valued funtion f on S and the algebra operations are point-wise; we call A a function algebra over S and write A - ~(S) = ~ = { f : f E A}. If ~ is p - n o r m e d (respy. p - B a n a c h ) we call it a p - n o r m e d (respy. p - B a n a c h ) function algebra. ~ is called a canonically pnormed (respy. canonically p-Banach)funtion algebra iff all f E are bounded and the p - n o r m of A is given by ]]f]l~, where

II/lloo

-- supscs ]f(s)l.

8.1.2. E x a m p l e s . (i) If X is a locally compact Hausdorff space, Co(X) the algebra of K-valued continuous functions on X vanishing at oe, then under the sup norm it is a (canonically) Banach function algebra. (ii) If ~ - ~(S) is a p - s e m i n o r m e d Gelfand algebra then ~ is a canonically normed function algebra. (iii) The Banach algebra A of holomorphic functions (defined in 7.3.22 (iii)) is a canonically Banach function algebra. (iv) The algebra C(~) (see 4.6.8 ( i ) ) i s a locally sm. ~ function algebra. Some of the theory developed in this chapter, especially in w w deals with non-commutative algebras (-algebras not necessarily commutative). tt The treatment here is very limited. For more information on the topic consult the books of A. Browder (Introduction to function algebras, Benjamin, 1969) and T.W. Gamelin (Uniform algebras, Prentice Hall, 1969).

Commutative Topological Algebras

354

8.1.3. R e m a r k . tative.

Every function algebra is evidently commu-

8.1.4. DEFINITION. Let ~-- ~(S) be a funtion algebra. For a subset ~0 of ~ we define ker~0-{seS'f(s)-0,

Vfe~0}.

8.1.5. LEMMA. Let ~ - ~(S) be a function algebra. Then and for an s E S \ k e r ~ , X8 " f ~-~ f(s) is a character of Ms - ker Xs a (hyper) maximal ideal of ~. PROOF. By choice of s, X~ # 0 and so Xs is surjective and hence a character. 8.1.6. DEFINITION. ker x ~ - {f e ~ " f ( s ) ideal.

A maximal ideal of the form M8 ( 0}) is called fixed or a fixed maximal

8.1.7. LEMMA. The p-norm t1"[I of a canonically p-normed algebra ~ = ~(S) satisfies the condition

tif211- Ilfll 2 PROOF. 8.1.8.

ker~-0.

(f e ~).

{if21{- (sups ] f 2 ( s ) l ) P - ((suPs If(s)l)p) 2 - [[f[[2. LEMMA.

Let ~ = ~(S) be

a

function algebra with

Then:

(i) For f e ~, the range ]~(f) C_ at(f). (ii) If ~ is unital, s

__ a(f).

(iii) ~ is h.s.s, and in particular q.s.s, and s.s.. PROOF. (ii) This (iii) If f Thus ~ =

(i) f ( s ) = x~(f) C at(f), by 1.7.11. follows similarly from 1.7.24. c ~ then f(s) = X~(f) = 0 for all s and so f = 0. {0}. Since, by 1.7.15,

{0}, is also q.s.s, and s.s. as required.

Let A - - ~ ( S ) be a p-normed functionally continuous function algebra with its p-norm I1"II sm. and 8.1.9.

PROPOSITION.

w1. Function Algebras Then every f E A is bounded and

kerA-0.

Ilfll~ ~

II/lloo

~

"(f)~ ~ Ilfll ~.

11"

II

of A dominates the canonical

In particular, the p-norm ,o-norm

355

II" II~

00"

PROOF. Since A is functionally continuous and k e r A - 0, for each s c S, X~ ~ A - A ~ , where X~ is given by x ~ ( f ) - f(s) ( l E A ) . By virtue of (,) of 7.3.8 we have

Ilflloo

-sup 8

If(s)l- s u p x . ( f ) ~ II]ttoo ~ ~(f)~ ~ Ilfll ~, 8

completing the proof. 8 . 1 . 1 0 . Let S be any set, K S the set of all K-valued functions on S. K s is, under pointwise operations, a unital algebra over K; the unity element of K s is the constant function 1 9 l(s) - s (V s c S). K s is a TA under the weak topology" a net ]'~ -+ f in K s iff f ~ ( s ) - + f(s) for each s E S. The subset B(S) - B(S,K) of K s comprising all bounded functions is a subalgebra which inherits the relative topology under which it is a TA. B(S) is also a Banach algebra under the sup norm II" Iloo" II]'lloo - s u p If(s)l. Since norm convergence clearly 8

implies weak convergence, weak topology of B(S) is coarser than its norm topology. If S is a topological space we have besides the algebra B(S) the algebra C(S) of all continuous functions on S. Further, BC(S)

-

B(S)NC(S)

is a subalgebra of both B(S) and C(S). Now let S be locally compact Hausdorff. A function f E C(S) is said to vanish at oe if given e > 0 there is a compact set K in S such t h a t IS(~)I < ~ for all s E S \ K . All functions in C(S) which vanish at c~ form a subalgebra C0(S) of C(S). Note t h a t C0(S) = C(S) if S is compact. 8.1.11.

PROPOSITION.

C(S) for compact Hausdorff S and

356

Commutative Topological Algebras

Co(S) for locally compact Hausdorff S are Banach algebras under the sup norm. Moreover, C(S), Co(S) are s.s.. PROOF. The proof of the first statement being straightforward is omitted. For the second we note that for ~ - C(S) or Co(S), ker~ - O, so that { M ~ ' s E S} are all maximal ideals, and clearly N M~ - {0}, whence jr is s.s.. 8.1.12. DEFINITION. Let ~ = ~(S) be a function algebra. A subset ~0 of ~ is said to separate points (of S) if given any pair (Sl, s2) of distinct points there is an f E ~0 with f ( s l ) # f(s2). ~0 is said to strongly separate points if it satisfies in addition the condition ker ~0 = 0. A family ~0 of functions separating (respy. strongly separating) points is also referred to as a separating (respy. strongly separating) family. 8.1.13. R e m a r k . Let S be a locally compact Hausdorff space. Consider the Banach of algebra C0(S). Any strongly separating subalgebra A of C0(S) which is s.a. (i.e. closed for conjugates) is dense by the extended StoneWeirstrass theorem (see [26, pp.166-7, Theorems A,B]). Conversely, we have the result that any dense subalgebra A of Co(S) is strongly separating. To see this, observe first that S being locally compact Hausdorff is completely regular. So for any point so c S we can choose f e C0(S) with f(so) r O. By density of A we can choose g e A with IIg - fll~ < If(s0)l. Then g(so) # O. Again, if Sl ~ s2 are two points of S and fl c C0(S) is such that fl(Sl) # fl(s2), we can choose gl e A 1 with ]lgl-flllcr < ~[fl(Sl)--fl(82)[. Then g l ( s l ) # gl(s2). Thus A is strongly separating. 8.1.14. LEMMA (Rickart). Let ~ - ~(S) be a function algebra which strongly separates points of S. Then, given any finite set of points so, 8 1 , ' " ,Sn (rt >1 1) of S, We can find an f E with f(so) # O, f(si) - 0 (j - 1 , 2 , - . . , n ) . PROOF. We first assume that n - 1 and write so - s, sl - t (s r t). We shall show that there is a f u n c t i o n f~ E with f ~ ( s ) r f~(t)-O.

w 1. Function Algebras

357

By hypothesis there is an f E ~ with f(s) ~: f(t). We have to consider three cases. C a s e 1: f ( t ) = O . We can take f s = f . C a s e 2: f(s) :/: 0, f(t):/: O. By replacing f by a suitable multiple we may assume that f ( t ) - 1 and then f(s) ~ 1. Set f~ _ f

_

f2;

then

f~(t) - O, fs(s) ~- 0.

C a s e 3: f(s) = 0 , f ( t ) = 1. C h o o s e a g e ~ } with g(s) r (this is possible since ker ~} = 0). If g(t) = 0, we can take f8 - g, and if g(t) ~ 0 we take f~ = f - g/g(t). We next assume t h a t n t> 2. By the result for n - 1 just proved above we can choose fj E ~ such t h a t fj(so) ~ 0, fj(sj)--0 (j1 , . . . , n ) . Set f fl""fn (product). T h e n f ( s 0 ) ~ = 0 , while f ( s j ) = 0 (j=l,...n). 8 . 1 . 1 5 . LEMMA. Let S - (S, 7) be a compact Hausdorff space. Then the weak topology t rw on S induced by any separating family ~o of C ( S ) coincides with the initial topology 7.

PROOF. 7w is clearly coarser then 7. Moreover, rw is Hausdorff since ~0 is separating S. It follows t h a t 7 = r,~ (since 7~ c 7, r~ Hausdorff, v compact). 8 . 1 . 1 6 . COROLLARY. If S -- (S, 7) is locally compact Hausdorff and C0(S) the algebra of K-valued continuous functions on S which vanish at oo, then the weak topology 7~ on S induced by any strongly separating family ~o of C(S) coincides with r. PROOF. Let Soo = S U{cc} be the 1-point compactification of S; S ~ is c o m p a c t Hausdorff. E x t e n d the functions f in ~}0 to S ~ by defining f ( c c ) - 0. Then the extended functions form a separating family of C(Soo). The required result now follows from 8.1.15.

Let S be a compact Hausdorff space and A a s.a. tt inverse-closed subunital algebra of C(S). Then 8 . 1 . 1 7 PROPOSITION.

t Anet so-+ s in S under Tw if f(s~) ~ f(s), V f E ~ o . t~ i.e., f E A ==>] E A, where f - f(s), ](s) - f(s), bar denoting complex conjugate.

358

Commutative Topological Algebras

every maximal ideal M of A is fixed, i.e. of the form M-

Ms - { f E A " f (s) - O}, for some s E S.

In particular, every maximal ideal of C ( S ) is fixed. PROOF. Suppose that A has a maximal ideal M which is not fixed. Since 1 c A k e r A - 0, and by 8.1.5, for any s E S , M~ is a (hyper) maximal ideal. So by our supposition M ~= M~. By maximality of M, M ~ Ms, whence there is an fs E M with f~(s) ~ O. By the continuity of ]'8 we have ]'8 r 0 on an open neighbourhood Us of s. Since S is compact there is a finite number of points S l " - , s ~

in S with L J u j -

s, where

j=l

/_/3" -- Us~. Set f-

f l f l + " " f,~s

where f j -

f~j ( j -

1,... ,n).

Then f E M. Since

f - If~l 2 + . . . + If.I 2 > 0, f-1 exists in C(S). Since Z is inverse-closed in C(S), f - 1 E a . Thus, M contains an invertible element f, which is impossible. So we must have M - Ms for some s, completing the proof.

8.1.18. COROLLARY. Let S be a locally compact Hausdorff space and C0(S) the algebra of K-valued continuous functions vanishing at oo. Then every maximal ideal of C0(S) is fixed. PROOF. Consider the 1-point compactification So~ of S. Then C ( S ~ ) i s t the unitization A1 of A - C0(S). The maximal ideals of A1 - C(S~) are, by 8.1.17, all fixed and they are {M~ 9s C Soo}, where M~ {fl e A1 " fl(s) 0}. It follows that the maximal ideals of C0(S) are -

-

-

MINA-M~(sEA), where M~ - { f C A ' f ( s ) -

0}, and so are fixed.

8.1.19. R e m a r k . In the proof of 8.1.17, the hypothesis that t i.e. c~n be identified with.

w 1. Function Algebras

359

A is s.a. was explicitty used. However, it is possible, for a subalgebra A of C(S) which is not s.a. to have the fixity property for the maximal ideals. For instance, the Banach algebra 92 of example (ii) of 7.3.22 is a subalgebra of C(D), which is not s.a. (fo ~ ~t; fo(z) - z). ~t has, nevertheless, the fixity property. In fact, if M is a maximal ideal of 92 and fo(M) = zo then M = Mzo (see [10, p.33]). 8.1.20. PROPOSITION. Let the algebra A - C(S), where S is compact Hausdorff, be a T A under a topology r which is finer than the weak topology rw. Then A is Gelfand and its spectrum A - Ac is homeomorphic to S. PROOF. By 8.1.17, if X C A -- A(A), then X - X8 (s E S). Since x ~ ( f ) - f(s), by definition of Tw,X~ is rw-continuous (f~ --~ f if x ~ ( f ~ ) - f ~ ( s ) ~ f ( s ) - x s ( f ) ) . Since r D_ rw,xs is also r-continuous. It follows that the maximal ideals M M~(s ~ S) are r-closed, hypermaximal and so a (a,r)is Gelfand. Consider now the map A 9s ~ X~ of S to A. By 8.1.17, A is surjective. It is also injective since S is normal if Sl # s2, there is a f ~ C ( S ) w i t h f ( s l ) # f(s2), so that X~x # X~2. Further, A is continuous, since if s~ ~ s and f is continuous, X,~(f)

- f(s~) ~

f(s) -

x,(f)-

Finally, since S is compact and A is Hausdorff (see 7.3.2) it follows that A is a homeomorphism. 8.1.21. COROLLARY. The Banach algebra (C(S),ll.llc~), or more generally, the p-Banach algebra (C(S),I[-I1~) ha~ S its spectrum. PROOF. This follows from 8.1.20 by taking r to be the norm topology. 8.1.22. THEOREM (Gelfand-Kolmogorov-Stone-Banach). If $1, $2 are compact Hausdorff spaces such that the Banach algebras C(S1) and C($2) are isomorphic (as algebras) then S 1 and $2 are homeomorphic. PROOF. Since the spectrum A of the Banach algebra A C(S) depends only on the algebraic structure of A, we have,

Commutative Topological Algebras

360

by 8.1.21, S1 --~ A1 -'~ A2 -~ S2, where _~ stands for homeomorphism. R e m a r k . In the algebra A - C ( U ) - C ( R , K ) , every maximal ideal is fixed. For, let 8.1.23.

I-- {fcA'f(t)-Ofor

not

a l l t > somety}.

Clearly I is an ideal and by Krull's lemma we can find a maximal ideal M containing I. It follows from the definition of I that M cannot be a fixed ideal. 8.1.24. R e m a r k . Let S be a completely regular Tl-space and A - BC(S) the algebra of bounded continuous functions on S. Then A is a unital commutative Banach function algebra. If A is its spectrum then A is compact Hausdorff. It can be shown that A is the Stone-(~ech compactification f l ( S ) o f S (see [26, p.331] or [28, p.415]). 8.1.25. Theorem. Let A - (A, II" II)be an ample, complex or strictly real, p-Banach algebra, with I1" II ~ m T h e n a is t. isomorphic to a canonically Banach function algebra iff II-II satisfies the condition

CIIxll = ~< IIx~ll for all where C is some positive constant (cf.

(,)

x,

[28, P.409, Theorem.

4.8.]). PROOF. Let I1" II satisfy (.). Then, by 7.4.15 or 7.4.19, A is commutative. Also, by 7.4.16, A is s.s.. By 7.4.12, the map ff 9x ~ ~ is a t. isomorphism of A onto A - which is a canonically Banach function algebra. Conversely, suppose that A is t. isomorphic to a canonically Banach function algebra a = (a, [[" II*). Then, by 8.1.7, we have

(liyll*) ~ -Ily~ll *

(y e a).

It follows from 7.4.11 that there is a constant C2 with

c~]]yil*-< r(y)~

( y e a).

w2. Shilov Boundary

361

Let ~9 denote the t. isomorphism between A and ;~. Then we can transfer the norm of a to A by setting for x E A , y--~o(x)

I1~11'- I1~(~)11"- Ilyl{* then I1" II* ~ I}" II, Since r ( y ) - r(x), we get

Since I1" II* ~ I1" tl there is a constant Clllxll*. Therefore we get

C1 >

0 such that Ilxtl

62 flxll ~ r(x)~ C1

whence, by 7.4.11, there is a C > 0 with

cllxll = <. II~=lf as required. 8.1.26. LEMMA. Let A be a p-seminormed algebra. Then the Gelfand transform ~t is always a strongly separating subalgebra

of Co(~X~) PROOF. If X E Ac and a C A \ k e r x then a ( X ) - X ( a ) J:O. Again, if X1,X2 E Ae and X1 -J: X2 there is an element b E A with xl(b) :/: x2(b), and then b(x1) ~: b(x=)- Hence the lemma.

w2.

Shilov Boundary

8.2.1. DEFINITION. Let S be a locally compact Hausdorff space, C0(S) - C0(S,K) the algebra of all continuous K-valued functions on S which vanish at oo, and A ___ C0(S) a subalgebra. A closed set F C_ S is called a pre-boundary for A if for each f c A there is a point s S e F such that I f ( s I ) [ - IlfllooR e m a r k . The set S itself is a pre-boundary for A. For, if f E A , f J=0, f(s0) ~- 0 there is a compact set g such 8.2.2.

362

Commutative Topological Algebras

that (*)lf(s)l < If(s0)l for s E S \ K . Then so e K. Since K is compact there is a S l e K with I f ( s l ) l - s u p i f ( s ) l - suplf(s)] sEK

sES

(by virtue of ( . ) ) . 8.2.3. DEFINITION. A pre-boundary (for A ) is called a Shilov boundary or boundary if it is contained in every pre-boundary (for A). The Shilov boundary, whenever it exists, is clearly unique and is denoted by OAS. A pre-boundary F0 is called a minimal pre-boundary if there is no other pre-boundary contained in F0. Evidently OAS is a minimal pre-boundary. 8 . 2 . 4 . LEMMA. Every pre-boundary F1 contains a minimal pre-boundary Fo.

PROOF. The family 71 of pre-boundaries contained in F1 is clearly a poset with respect to set-inclusion. By Zorn's lemma (or rather by the equivalent Hausdorff maximal principle t ) there is a maximal chain ~0 in ~1 with F1 E ~0. Denote by F0 the intersection of all sets F in ~0; then F0 is closed. We claim F0 is a pre-boundary. To see this, set for f E A,

F s - (~ E s

If(~)l-

II.fllo~).

If S - 0 , F s - S and F0 c_ Ff. If f ~ 0 then Ff is compact (since S vanishes at co ). Since each F c /'0 is a pre-boundary, FS n F =/- 0 and further it is a closed subset of compact Ff. It follows that

FsNF0- Fs N r ) - N (FsNr) F ETo

FETo

Since f is an arbitary function in A the above relation shows that F0 is a pre-boundary. Finally, by its construction F0 is a minimal pre-boundary. 8.2.5. LEMMA. Let F be a minimal pre-boundary for A. Then an element so E S belongs to F iff for every open neighbourhood V of so there exists an f E A such that

1

See

[16, pp.31-36].

w2. Shilov Boundary

sup If(s)l < sup sES\V sEV

If( )l-

363

Ilflloo.

(,)

PROOF. If So C F then F \ V is not a pre-boundary and so there is an f satisfying (.). Conversely, suppose for so E S and an arbitrary open neighbourhood V of so the inequality (.) holds for some f. Then since IIftloo- supif(s)I , F ~ S\V, so sEF that V ~ F ~- 0. It follows that so E F - F. 8.2.6.

LEMMA. If S is infinite and A c_C_C0(S) a strongly

separating subalgebra, then every pre-boundary F (for A) is infinite. PROOF. If possible let F - { 8 1 , " ' , 8 n } and so any point of S\F. By 8.1.14, we can choose f E A with

f(so):O,

f(sj)-O

(j-

1,...,n).

Then IIf]l~ > 0 and f - 0 on F, contradicting that F is a pre-boundary. So F is infinite as required. 8.2.7. PROPOSITION. The Shilov boundary ~A S exists for every strongly separating subalgebra A of C0(S), where S is a locally compact Hausdorff space. PROOF

(cf. [23, pp.133-4]). First assume that S is finite, say, S - { S l , . . . , s n } . Then S being compact and discrete, C o ( S ) C(S) - ~:~. By 8.1.14, there is a f j e A ( j - 1 , . . . , n ) such that

fj(sj) # O, ]'j(sk) - 0 (k -J: j). Since ][fjl[oo is attained only at sj, every pre-boundary F must contain all the sj which means that F - S. Hence OAS exists and OAS -- S. Next assume that S is infinite; by 8.2.6, every pre-boundary F is infinite. By 8.2.4, we can find a minimal pre-boundary Fo. Let F be any pre-boundary. We shall show that Fo C F. Take any point so E Fo and let N be any open neighbourhood of so; since F0 is infinite we can certainly select to E Fo with to ~ so. Set No = N\{to}. Then No _C N and we have 0 r Fo\ No ~ Fo (since to E Fo\No, so E Fo, so r Fo\No ).

Commutative Topological Algebras

364

Since No is an open neighbourhood of so, and the topology of S is the same as the weak topology induced by A (see 8.1.16) we can find functions f j ( j - 1 , . . . , n ) in A and an e > 0 such that

No D_ N1 - {t E S " [fj(t) - fj(so)[ < e, j -

1,...,n}.

(1)

We have also 0 ~ Fo\N1 ~ Fo. The minimality property of F0 implies t h a t Fo\N1 is not a pre-boundary. It follows t h a t we can find an element s l C F o N N 1 a n d a g E A such t h a t I g ( s l ) l - IIgll~ - c > sup{lg(t)l.t c Fo\N1}. Therefore I c - l g ( s l ) l - 1, [c-lg(t)l < 1 for t e Fo\N1. By replacing g by c-lg we may assume c - 1, so that we have I g ( 8 1 ) l - 1,

]g(t)l < 1 for all t E Fo\N1.

(2)

By taking a sufficiently large integer n and setting gl - gn we get

]]gllloo --[gl(Sl)]- 1, Igl(t)l <

E

on Fo\N1.

(3)

2 supj II]'3"I1~ Set hj - f i g 1 - fj(sj)gl. Then we have for t E Fo\N1 (using(3))

I h j ( t ) l - I f j ( t ) - fj(~0)llgi(t)l <

(4)

and for t C N1.

Ihj(t)l < ~llgilloo - ~

(5)

From (4),(5) we conclude, since F0 is a pre-boundary, t h a t

Ilhjlt~ < ~

(6)

On the other hand, since F is also a pre-boundary there is a t l c F such t h a t I g i ( t ~ ) l - Ilglll~ - 1. It follows t h a t

t f j ( t l ) - fj(SO)l- I f j ( t l ) - fj(SO)llgl(tl)l- Ihj(tl)l ~ ]]hjlloo < c,

w2. Shilov Boundary where we have used (6). t l C N1 N F, whence

NNF

365

Therefore, by (1), t l E N1, so that

~ N o r ) F ~ i l ~ ' ) F ~ ~.

Since N is an arbitary neighbourhood of so we conclude that

so E F -

F, Fo -C F.

Hence Fo -- OA S. 8.2.8. COROLLARY. /f A C Co(S) is only separating then also OAS exists. PROOF. Since A is only separating (and not strongly separating) all f ~ A vanish at a point so E S; by separation hypothesis there cannot be another point in S at which all f in A vanish. Set So = S\{so}, Ao = {flSo : f e A}. Then Ao is clearly a strongly separating subalgebra of Co(So). So, by 8.2.7, 8AoSo exists. It is easy to see that

8A~O -- OAS. 8.2.9. REMARK. The above corollary may fail if A is not separating. A counter-example is furnished by the following algebra A. Write S = [-1,1] and denote by C(S) the algebra of continuous K-valued functions on S. Let A be the subalgebra of C ( S ) c o n s i s t i n g of even functions ( f ( s ) = f ( - s ) ) . Then aAS does not exist since [0,1] [-1,0] are both minimal pre-boundaries. Note that A does not separate s and - s (0 < s ~ 1). 8.2.10. PROPOSITION. For every dense subalgebra A of C0(S), S itself is the Shilov boundary" 3 A ( S ) - S.

PROOF. For so C S, choose any compact neighbourhood Vc of so. By local compactness of S there exists a g E C0(S) such that g ( s o ) - 1 and g - 0 on S\Vc. Since A is dense in C0(S) there is a f E A such that

g( )l < 1 for a l l s E S .

(1)

Commutative Topological Algebras

366

Since g ( s o ) so that

1,

If(so)l > Ig(so)l- Ig(so)- f(so)l > 1 1

Ilfltoo > ~ .

2 -- ~,

(2)

On the other hand, if s E S\Vc, g(s) = O, 1

{f(8)l < - . 2

(3)

From (2),(3)we get sup If(s)l < llfll~,

s\v~

Since any arbitrary neighbourhood V of so contains some compact neighbourhood the above inequality implies that the inequality (.) of 8.2.5 holds for V, whence so E Sin, where Sm is a minimal pre-boundary for A. Since so is an arbitary point of S we get S - Sm -- OA(S). Note that in the above proof we have shown directly that every minimal pre-boundary is S, so that we have not made use of a priori the existence of OA(S) (which of course is assured by 8.1.3, 8.2.7).

Let A - (A,p) with p sin., be a p-seminormed algebra having A c ~ 0. Then: 8.2.11.

PROPOSITION.

(i) A is a strongly separating subalgebra of C0(Ae). (ii) The Shilov boundary O~iA~ exists. PROOF. By 7.3.12, Ac is locally compact Hausdorff and by virtue of 7.3.27(i), ft. is a subalgebra. Also, by 8.1.26, A is strongly separating. Finally, (ii) follows from (i) (see 8.2.7). 8.2.12.

COROLLARY.

a~A

exists if A is a p-seminormed

Gelfand algebra. In particular it exists if A ~ V ~ and A is a complex, formally real or strictly real, commutative p-Banach algebra. PROOF. Since A is Gelfand we have A = A c r 0 whence the first assertion follows from 8.2.11. The second statement follows

w 2. Shilov Boundary

367

from the first taking into account the results of 7.2.17, 7.2.19. 8.2.13. DEFINITION. A p-seminormed algebra A with Ac ~: 0 is called self-adjoint or s.a. if to each x E A there is a y with - ~, i.e. Y(X) - x ( x ) for every X c Ac. Note that if A is real it is automatically s.a. (we can take y - x). 8.2.14. LEMMA. s.a. algebra.

The unitization A1 of a s.a. algebra is a

PROOF. If X 1 -- )~el + x (x E A), Choose y E A such that _

_

-- ~ in A. Set Yl -- )~el + y. Consider X1 E A1 -- A ( A 1 ) , X1

X0 (the distinguished character, ker X0 - A). Write X then x E A - - A(A). We have YI (XI)

:

XI ( ~ e i -~ Y) -- ~ nL X ( Y ) - - )k nt-- ~ ' ( X )

--

()~ -~

YI(X0) -- ~ -- x i ( x 0 ) . Thus, Y l - ~:1, p r o v i n g

x(X))

-

-

XllA;

(Xl(X))"

ilso~

A1 is s . a . .

8.2.15. PROPOSITION. The transform algebra A of a s.a. p s e m i n o r m e d algebra A (with p sin.) is dense in C0(Ac). Hence 8 i A ~ - A~. PROOF. By 8.2.11, A is a strongly separating subalgebra of C0(Ac). Also, by s.a. property of A, A is closed for conjugates (i.e. f e A ==~ f e A). By the extended Stone-Weierstrass theorem (see [26, pp.166-7]) A is dense in C0(A~). Finally, by 8.2.10, OAA~ -- A~. ^

8.2.16. S o m e E x a m p l e s of Shilov B o u n d a r y . (i) Let D be the closed unit disc in the complex plane and A - .9(D) the Banach algebra of all continuous complex functions on D which are holomorphic on the interior D ~ of D. By the maximum modulus principle we have OAD -- unit circle $1; thus here the Shilov boundary coincides with the topological boundary 8 D of D. (ii) Consider the bi-cylinder D 2 - {(Zl,Z~) ~ c2- Iz l, lz21 <. 1}. Let C(D 2) denote the Banach algebra of all complex contin-

368

Commutative

Topological Algebras

uous functions on D 2. Let A - P(D) be the uniform closure in C ( D 2) of the subset consisting of restrictions on D 2 of polynomials in zx , z~. We shall now show that C3AD2

-

distinguished b o u n d a r y OoD 2 {(zl,z2) e D 2 " l Z l ] - [ z 2 l -

1}.

Note t h a t 0 0 D 2 is a proper part of the topological b o u n d a r y a D 2 - {(Zl, z2) e D 2 " l Z l l or [ z 2 ] - 1}. It follows from the maxim u m modulus principle t h a t OoD 2 is a pre-boundary, so t h a t we have h A D 2 cC_ OoD 2. Further, for each point p - (e ie~ e ie~) E O0 D2 the function f ( z l , z 2 ) -- (Zl -~- e i01)(Z 2 -Jr-e iOz) o n 0 2. satisfies, as we shall see, the inequality (.) of 8.2.5 with respect to any open neighbourhood of p of the form Yr I -

{(Zl,Z2) C D2 " lzl - e i~ I < ~, Iz2 - e i~ I < ~}.

Since any neighbourhood V of p contains some Vn it means t h a t the inequality (.) is satisfied for all V, and so by 8.2.5. p E OAD 2, proving (~AD ~ - OoD 2. It remains to show t h a t the inequality (.) of 8.2.5 holds for V - Vn. First observe t h a t Ifl attains its m a x i m u m value at Zl - e iel,z2 - e ie2, so t h a t Ilfl[oo - 4. If (Zl,Z2) e D 2 \ V we have Izx

-

or

-

(*)

If [Zll or I z 2 1 - 1 then we must have by (.) above t h a t either a m p Zl ~ amp e ie~ or amp z2 r a m p e i02. It follows t h a t either I z l + e i~ [ < 1 or Iz2 + e ie21 < 2, so t h a t once again the inequality (.) of 8.2.5 holds. (iii) Let C(D) denote the algebra of all continuous K-valued functions on the closed unit disc D in K. Then C(D) is a Banach algebra under sup norm. Its subalgebra A comprising restrictions, of polynomials over K, to D is a normed subalgebra. By StoneWeierstrass, A is a dense subalgebra, so that, by 8.2.10, OAD -D. 8 . 2 . 1 7 . PROPOSITION (Shilov). Let A ~ be either a complex or a strictly real, unital c o m m u t a t i v e p - B a n a c h algebra and A a closed subunital algebra of A I. Let A , A I denote respectively the

w 2. Shilov Boundary

369

spectra of A, A t. Then every character X E O 2 A has an extension to a character X t of A t . In particular, whenever ~t is dense in C(A) every X E A has an extension to a X l E A t . PROOF. Consider the restriction map r] 9 A t --+ A given by rl(X') - restriction X'I A, which is clearly continuous. Since A ~ is compact, rl(A ~) is compact and so closed in A. We shall show t h a t rl(A t) is a pre-boundary for A. If x E A C A t and 1

x ' - x'(x') then IIk'J]~ - u ( x ) ; - t l ~ l l ~ . there is a X~ E A t such t h a t

By compactness of A'

proving t h a t rl(A') is a pre-boundary. But then 0 2 A _ rl(A'), which clearly implies that every X E O~A has an extension X~, proving the first assertion. When A is dense in C(A), by 8.2.10, 0 / i A - A, whence the second assertion. 8 . 2 . 1 8 . COROLLARY. Every maximal ideal M of A, whose associated character X belongs to 0 2 A , is contained in a m a x i m a l ideal M t of A t. PROOF. By 8.2.17, X extends to a character Xt of At; then M - k e r x c k e r x t - M t. 8 . 2 . 1 9 . R e m a r k . In general not every maximal ideal of A is contained in some maximal ideal of A t. For example, if we denote by S 1 the unit circle in the plane C and consider the algebra A t -C(S 1) - C(S 1, C) and its subalgebra A comprising the functions which can be extended to holomorphic functions on the interior of the circle S 1. The function z - ei~ E A C A t. Further z is invertible in A t with z -1 - 2 ' . Since 2 is not holomorphic, 2. ~ A and z is not invertible in A. It follows t h a t there is a maximal ideal M of A with z E M. This ideal M is not contained in any maximal ideal of A ~ since z is invertible in A ~. 8 . 2 . 2 0 . PROPOSITION (Shilov). Let A be a complex unital p - B a n a c h algebra which is monogenic, with t. generator a. Then 9 A A can be identified with (the topological boundary) Oa(a). PROOF. By 7.3.23, we identify A with a(a) and regard A as

Commutative Topological Algebras

370

a function algebra on a(a). By virtue of the maximum modulus principle the topological boundary is a pre-boundary for A, so that O~ia(a) C_ ha(a). For proving the reverse inequality take a point A0 E c3a(a) and an open neighbourhood V of A0 in a(a). Then

V _ {A C C'IA - A0] < e}~a(a), for some e > O. Since A0 C c3a(a) there is a )~1 E p(a) such that ] A 0 AI[ < 3" Set b ( a - Ale) -1. Then b ( A ) 1 / ( A - A1), and

I/~(A0)l >

3 ~,

[b(A)[ ~< 3 ol-

for

A E

a(a)\V

(since

- Xol

Since [[blloo > ~3 > I/~(A)[ (k e a(a)\V ) , by 8.2.5, A0 E c3Aa(a) . This completes the proof.

w3.

Hull-Kernel

Topology

8.3.1. Let A be an algebra (or more generally a ring) with A =/- x/~. Denote by P the set of prime (bi-) ideals P of A with P ~- A, by ~ the set of maximal regular bi-ideals of A, and by II the set of primitive ideals of A. 8.3.2. LEMMA We have E C_ II C p; p, II=fi O. PROOF. The inclusion relations follow from 1.5.10, 1.5.9. Since A ~= x/~, II J= 0. 8.3.3. For any subset S of P we write

k(S)-Is-

r~ P

PES

and call k(S) the kernel of S; k(S) is a bi-ideal of A. We have clearly" For $1 _C $2, k(S2) _c k(Sx). (1) If I is a b i - i d e a l w e set h ( I ) - { P E P the hull of I. We have obviously:

9P_~ I} and call h(I)

For I __ J, h(J)_ h(1).

(2)

w3. Hull-Kernel Topology

371

The following inclusions are clear from the definitions. S _c hk(S).

(3)

I c_ kh(I).

(4)

By virtue of (1),(3) we obtain k(S) - khk(S).

(5)

Similarly, by (2),(4)we get h ( I ) - hkh(I).

(6)

(By applying k to (3)we get, by (1), khk(S) c k(S); on the other hand, khk(S) - kh(k(S)) _ k(S). Combining the above two inclusions we get (5). Similarly, (6) can be proved.) 8.3.4. LEMMA. (i) ~ a h ( I a ) = h(E~ I~); (ii) h(IJ) = h(I) U h(J) = h(I ~ J); (iii) h(A) = 0; (iv) h ( { 0 ) ) = P. PROOF. The properties (i),(iii),(iv) are immediate from the definition of h. For (ii), we first observe that since I J c_ I, J we have h(I),h(J) c h(IJ), so that h(I) Uh(J)_C h(IJ). On the other hand, if P E h(IJ) then P 2 I J and since P is prime, P _~ I or J, so that P E h(I) or h(J), whence h ( I J ) c h(I) U h(J). Combining this with the reverse inclusion relation obtained above we get the first equality in (ii). Again, since h(I) U h(J) _ h(I A J) _c h(IJ), the second equality follows from the first. 8.3.5. PROPOSITION. The set P can be topologized by taking as its closed sets the family {h(I) : I a hi-ideal of A } . The resulting topology is called the hull-kernel or hk topology. We denote P with this topology by Phk" PROOF. It follows from 8.3.4 that the family {h(I)} of subsets of P is closed for arbitrary intersections and finite unions, contains the empty set and the whole space P. Hence the proposition. 8.3.6.

LEMMA. If S is a subset of Phk then S - hk(S),

372

Commutative Topological Algebras

where bar denotes closure in Phk"

PROOF. By definition of hk topology, hk(S) is a closed set and hk(S)D S (by(3)of 8.3.3)Further, if h(I)___ S then h ( I ) hkh(I) D hk(S). Therefore S - hk(S). 8 . 3 . 7 . COROLLARY. For a closed subset S, among the hiideals I of A with h ( I ) - S the largest one is k(S).

PROOF. First observe that k(S) is a bi-ideal and h ( k ( S ) ) hk(S) - S since S is closed. Next, if h ( I ) - S, then I _c k h ( I ) -

k(s). 8.3.8. PROPOSITION. Phk is a To-space. is also a To-space while the subspace E is T1.

The subspace H

PROOF. If P1,P2 6 Phk and P= ~ P1 then P2 r h({P1})= {P1}, proving Phk is To. By a similar argument II is also To. Finally, if M 6 E then {M} = h ( { M } ) = {M} (by maximality of M, whence E is T1.)

A,

8.3.9. DEFINITION. H - HA is called the structure space of ~ -- ~ A the strong structure space of A. 8.3.10. PROPOSITION ([23, p.79]). Let J be a hi-ideal of an

algebra A.

Then:

(i) r ~ A \ h ( J ) i s homeomorphic with Ej

under the map e 9

M---~ J ~ M . (ii) If A # - A / J then the hull h(J) of J in EA is homeomorphic with ~A# under the canonical map 8 # 9M ~-~ M / J .

PROOF. By 1.4.3, the map ~ 9 M ~-, M ~ - J ~ M is a bijection between E A \ h ( J ) and Ej. To prove that 0 is a homeomorphism it is enough to prove that 8 ( h k ( S ) ) - hk(8(S)). Clearly k ( 8 ( S ) ) - J ~ k ( S ) . If M 6 hk(S) then M _ k(S) and ~(M) _ J [-1 k(S), so that 8(M) 6 h(J ["lk(S)). Conversely, suppose that

J F"I M

k(O(s)) - J r'l k(S).

w3. Hull-Kernel Topology

373

Then M D J Ak(S), and since M ~ J, by primality of M, M _~ k(S), so that M E hk(S). This completes the proof of (i). To prove (ii), we observe that if I is a bi-ideal of A which contains J, and its image under the canonical homeomorphism is denoted by I#, then the correspondence I ~-. I # is easily seen to be a bijection preserving the inclusion relation. Therefore maximal bi-ideals correspond to maximal bi-ideals. The required conclusion now readily follows. 8.3.11.

PROPOSITION(cf.

[23, p.79]). If I is a regular hi-

ideal of A then h(I), h(/) N II, h(I) f ] E hk -topology.

are compact under the

PROOF. Let {F~} be a family of closed sets in any one of these spaces such that

VIF -0. Write k ( F s ) - I s

and K -

(,)

t Ia. Since F~ _D h(I) we have S

I c_ hk(I) __ k ( F ~ ) - I~. So

K~Is~I, and K is regular. Suppose that K ~ A. Then by 1.2.10 there is an M C E with K c M. But then M D K D Is, so that M E I'] h ( I s ) -

NFs.

(since h(I~) - hk(F~) - F~). The last conclusion contradicts ( . ) . So we must have K - ~-~'~Is - A. Let u be a relative S

(bi-)unity for I. Since

ucA-K-~Is S

there is a finite subfamily of ideals I 1 , ' . . , I,~ (I i - I~j) such that

uCJ-Ii+...+In. t ZI~

denotes the smallest bi-ideal containing the I~ 's.

374

Commutative Topological Algebras

But since J D Ij D I, J is regular. J - A. This implies that n

n

/=1

/=1

N Fj - N

Therefore, since u E J,

n

hk(Fj)

-

N

h( j)

/=1

-

h(J)-

o.

which proves the desired compactness. 8.3.12. Let A be a T A and ~ (respy. ~ c ) the set hypermaximal (respy. closed hypermaximal) ideals of A. We have clearly the inclusion relations ~c_~__E so that ~4, ~r inherit, by relativization, the hk-topology. This topology on ~ (respy. ~c can be transferred to A (respy. Ac) in view of the bijection. -~ A (respy. 34c-* Ac) (see 7.3.11). If E is a subset of A or Ac we define k(E) -- n ker X

(,k" E E).

If I is an ideal of A, the hull h(I) with respect to A (respy. Ac ) is defined by h(I) - {X e A( respy. A+)" k e r x _D I}. Then a subset E C A (respy. Ac ) is closed iff E 8.3.13. described in Xa --+ X in A and any

hk(E).

LEMMA. The hk-topology on A (or Ac) can be terms of net convergence in the following way: a net the hk-topology if it satisfies the condition: for x E subnet (Xa') of (X~), " X a , ( x ) - 0 for all X ~ , " ~

x(x) - o . PROOF. It suffices to note that if Ma, - ker X~', M - ker X then M D_ N M~, iff ~(M~,) - 0 for all M~,, =~ ~(M) - 0

(x E A).

w3. Hull-Kernel Topology

375

8 . 3 . 1 4 . COROLLARY. If 7w denotes the (weak) topology of the spectrum A (or A c ) and rhk the hk-topology on it, then rhk is coarser than 7w "Thk C_ t 7w.

PROOF. Suppose that a net Xa -+ X in A (or Ac ) under rw. Then X~(x) -+ X(x) for all x e A. In particular, if X~,(x) = 0 for all X~' then X ( X ) = 0, which means that Xa -+ X under rhk. Therefore rhk C 7w, as required. 8.3.15. DEFINITION. Let A be a T A with Ac J= 0. A is called quasi-unital if there is an element u0 E A such that t~0(X) =/= 0, VX c A~. If A is unital (with unity e ) it is also quasiunital since e(x) - 1 J= 0 (X e Ac). 8.3.16. LEMMA. Let A be a s.a. p-seminormed algebra such that Ae is non-empty and compact. Then A is quasi-unital. PROOF. For each X E Ac choose an element x x of A with ~cx(X) - X(Xx) ~ O. By continuity of xx and compactness of Ac we can find, as in the proof of 8.1.17, a finite subset { X 1 , ' " , X,~} of A~, elements x j - Xxi and open neighbourhoods Uj of X1 tI

~i(X) =/= 0 on Uj ( j -

such that

1,...,n)

and

U u j - Ac. j=l Since A is s.a., we can find yj E A withgj - ~j ( J - 1 , . - . , n ) . n

Set u o - ~ ~

xiy i. Then

j--1 n

a0(x)-

n

j(x)yj(x)j=l

I j(x)l > 0 j=l

(since if X e Uj, xo(X) r 0). 8.3.17. PROPOSITION. Let A be a s.a. complex spectrallyGel/and p-Banach algebra with compact spectrum A ( - A c ) . Then v/A is a regular hi-ideal. PROOF. If V ~ - A, v/A is trivially regular. So we may assume that A ~= v/A. Let A1 be the unitization of A and let t i.e. every rbk-open set is rw-open.

376

Commutative Topological Algebras

A1,/~

denote respectively the spectra of A1, A. It follows (see 7.3.2(a)) that we have A1 -- {X1 E t I " x , I A

c a} U{xo},

(,)

where X0 is the distinguished character of A1. Since, by hypothesis A is compact, so is A1. By 8.3.16, there is an u0 ~ A with uo(X) ~ 0 for all X E A. Since A is spectrally Gelfand we have: -

It follows that

O-I(UO ) --

~tO(t)LJ{~o(ttO)},

where al denotes the spectrum with respect to A1. Since A is compact, ~0(A) is also compact and hence closed. It follows from 7.5.14 that there is an idempotent u E A1 with fi(X0) = 0, ~(A) = 1. But then Xo(u) = O, u E A. Further, if z E A and X = x~IA

ux(x)

then

(X1 # XO)

-- fi(X)~c(X) - 1. Sc(X ) - 3c(X ).

This relation holds for every X E A, whence we get u x - x e r'l ker X - ~ x/~ (by 7.2.12), whence v/A is regular.

In a commutative s.a. complex pBanach algebra A with compact spectrum, V ~ is regular. 8.3.18.

COROLLARY.

PROOF. We may assume that A =fi x/~. Then, by 7.2.17, A is Gelfand and so spectrally Gelfand. The required conclusion now follows from 8.3.1 7. 8.3.19.

COROLLARY.

If A is a strictly real commutative

p-Banach algebra with A compact then x / ~ is regular. PROOF. The complexification A of A is s.a." A

=+iy

A

(x) - x ( = ) + i x ( y ) - x ( = ) - i x ( v ) - = - i y

(x).

N

Since, by 7.3.3, A - A ( A ) i s homeomorphic to A - A(A) and A is compact, ,~ is compact.

It follows from 8.3.17 that v ~

is

w4. Completely Regular Algebras

377

regular. By 1.9.17, v/A is self-conjugate and

It follows that

Therefore (see 1.9.16(i)) x / ~ - Rev/-~ is regular. 8.3.20. PROPOSITION. Let A be a commutative s.a., complex or strictly real, p-Banach algebra. Then the following two statements are equivalent: (i) A//V~ is unital . (ii) A = A~ is compact . PROOF. (i) ~ (ii) 9Since A / / ~ is unital, v/A is regular whence ~-A 2 x/~ t is also regular. So, by 7.3.12, (ii) holds. (ii)~ (i)" By 8.3.18 or 8.3.19, x/~ is regular, and so (i) holds. 8.3.21. compact.

C OROLLARY. If A is s.s., then A is unital iff A is

8.3.22. R e m a r k . (cf. [12, p.52, RMK 4.7]). The above corollary may not hold if the hypothesis "A is s.s" is dropped. For example if A1 is commutative unital Banach algebra and A2(=fi {0}) a Banach algebra with trivial multiplication. Then A ( A 1 ) 0 and compact, while A ( A 2 ) # 0. If A = A1 • A2 then by 7.3.14, A(A) --~ A(A1) is compact but A is not unital.

w4.

Completely

Regular

Algebras

8 . 4 . 1 . DEFINITION. Following Willcox an algebra A is called completely regulartt if it satisfies the two conditions" t See (,) of 2.2.18 tt Some authors especially Russian use the term regular for completely regular, following the usage of the term by Shilov who first introduced these algebras in the commutative case. The nomenclature completely regular is due to Rickart.

378

Commutative Topological Algebras

(i) The strong structure space E is Hausdorff. (ii) Each point M E E has an open neighbourhood V such that the kernel k(V) is a regular bi-ideal. When A is unital, condition (ii) above can be dropped since it is automatically satisfied (every ideal then being regular). 8.4.2. E x a m p l e s of C o m p l e t e l y r e g u l a r algebras. (i) The Banach algebra C0(X)

(see 8.4.15).

(ii) The group algebra IX(G) of a locally compact Hausdorff commutative group G (see [22, p.426]). (iii) The group algebra LI(G) of any compact Hausdorff group G (see [23, p.83]). (iv) A von Neumann algebra (see [23, p.290]). (v) The Wiener-Zelazko algebra W p (see Appendix). 8.4.3. PROPOSITION. Let A be a completely regular algebra. Then" (i) E is locally compact Hausdorff. (ii) E is compact Hausdorff if A is unital. PROOF. (i) Since k(V)is a regular bi-ideal (by (ii)of 8.4.1) it follows by 8.3.11 that hk(V) is a compact neighbourhood of M. (ii) When A is unital, every ideal is regular and so in particular, {0} is regular. Therefore, by 8.3.11, h({0})= E is compact. 8.4.4. PROPOSITION. I r A is a completely regular algebra and J a bi-ideal of A then J and A # - A / J are completely regular. PROOF. Let E j denote the strong structure space of J. By 8.3.10 (i), O" M H J ~ M is a homeomorphism of E \ h ( J ) on Ej. By complete regularity of A,E is Hausdorff and hence by the homeomorphism 0, ~J is also Hausdorff. Further, if M E E has V as an open neighbourhood with k(Y) regular then 0(M) has 0(Y)

w4. Completely Regular Algebras

379

as an open neighbourhood and k ( 0 ( V ) ) - J N k(V) is regular (of. proof of 1.4.2 (v)). Therefore J is completely regular. It remains to prove that A # is also completely regular. By 8.3.10 (ii), I],4# is Hausdorff. If g) 9 A --+ A # is the canonical homeomorphism and M - ~ - I ( M # ) ( M # E l ] a # ) then M E l~a. Let V be an open neighbourhood of M 0 - g~-l(M0# ), M0# E EA#, such that k(V) is regular. The intersection I, of all M C EA with M D k(V), M _D J, is also a regular bi-ideal (since I _D k(V)). It follows that i f V # - { M # "M_D k(V)} t h e n V # is an open neighbourhood of M0# with k(V #) - g~(k(V)) regular. This completes the proof. 8.4.5. PROPOSITION. The unitization A1 of an algebra A is completely regular iff A is completely regular. PROOF (cf. [23, p.84]). Assume that A is completely regular. Since A1 is unital, to prove that it is completely regular we have only to show that ~1 -- ~A1 is Hausdorff. Since A is a bi-ideal of Ai, by 8.3.10 (i), E - EA is homeomorphic to EI\{A}. By complete regularity of A, E and consequently E I \ { A } is Hausdorff. To complete the proof that ~1 is Hausdorff it remains to show that the point A can be separated by open sets from any other point M1~ in El. Write M ~ - A N M1~ By complete regularity of A there is an open neighbourhood V of M0 with I - k(V) a regular bi-ideal, with a relative unity u (say). By 1.4.7. (iv), I1 - / ~ - I ~ - A i ( e i - t t ) is a bi-ideal of A1; I1 ~ A since e l - u @ 11. It follows that A c ~1\h1(I1) - U1 (say)were hi denotes the hull relative to A1. For M C V0 consider M1 - M. Then A N M 1 - M. Since M D I - k(V), M1 ~ I1. It follows that V1 - 8-1(V) C_ h1(I1), where 0 is the homeornorphism M1 ~-~ A N M1 - M (see 8.3.10) so that U1/1 V1 - 0. This completes the proof of the "if" part. For the "only if" part assume that ~]1 is Hausdorff. Then E E i \ { A } , is also Hausdorff. If M E E then i - A N M 1 (M1 E El). Since ~1 is Hausdorff we can choose an open neighbourhood V1 of M1 such that A ~ hik (V1) , i.e., k(V1) - /1 ~ A. It follows (see 1.4.7 (ii)) that I - A ('//1 is regular. Since V1 is open we may assume that V1 -- h ~ ( J i ) - ~ i \ h i ( g i ) . ^

Then it is easy to see that V - h~(g), where J - A N gl, is an

380

Commutative Topological Algebras

open neighbourhood of M. Since k ( V ) - ANk(V1) -- A ~ I 1 -- I is regular, the proof of complete regularity of A is finished. 8.4.6. PROPOSITION. Let A be completely regular and F any closed subset of E. Then k ( F ) is regular iff F is compact.

PROOF. If kCF) is regular then by 3.3.11, h ( k ( F ) ) = h k ( F ) = F is compact, which proves the "only if" part. For the "if" part assume that F is compact. Then by using complete regularity of A, we can find a finite open covering {Vj (j = 1 , - . . , n ) } , with each k(Vj) regular. By virtue of 1.2.15, k(Vl[.J...[,JVn) k(V1) ~ . . . k(Vn) = I (say)is regular. Since k(F) is also regular, completing the proof. 8.4.7. THEOREM. Let A be a p-seminormed algebra. If A is completely regular then the hull-kernel topology rhk on Ac coincides with the weak topology rw. Conversely, when A is p-seminormed Gelfand, 7hk - - 7w On A c ( - - A ) implies that A is completely regular.

PROOF. Suppose that A is completely regular. We have always, by 8.3.14, rhk ___ rw. To prove the reverse inclusion, let F be a T~-closed set in A~. Take X0 c A~\F. By complete regularity of A we can choose a rhk-open neighbourhood V of Xo such that k(V) is regular. Let u be a relative (bi-) unity for k(Y). Set Fo - {X E F " X(u) - 1};

F0 is r~-closed. By 7.3.13, the set F1 - {X E AC " IX(u)I >>.1}

is T~-compact. Therefore, F0 as a closed subset of F1 is also T~compact. Since 7hk C Tw, F0 is also 7hk-compact and so 7hkclosed (7hk being Hausdorff by complete regularity of A). Since by choice X0 ~ F0 and Thk is Hausdorff, each point X E F0 has a rhkopen neighbourhood U with X0 ~ U - hk(U). By rhk-compactness there is a finite covering of F by open neighbourhoods U 1 , ' " , Un such that X0 ~ hk(Uj) (j = 1,... ,n). It follows that there is an element aj E k(Uj) with xo(ai) r O. Also, since Ac\Y is ~hk-Closed

w4. Completely Regular Algebras

381

and X0 r A t \ V , there exists a0 6 k(Ac\V) with xo(ao) r O. Set

a - aoal ""an. Then" xo(a) r 0,

(1)

a 6 ker(U1 U " " U UnU Ao\V) _Cker(F0 U At\V)"

(2)

If X 6 V then ker X - k(V) and so ker X has also u has a relative unity. It follows that X(U) - 1 whence V ~ F c F0, so that we have

r c F0 From (2),(3) we conclude that a 6 ker F. Also, by (1), xo(a) # O. Therefore X0 r hk(F), proving F = hk(F), so t h a t F is rhk-Closed and rhk = rw. For the converse part, assume now that A is Gelfand and rw = rhk. Since A is Gelfand, E = A = Ao, so t h a t E is Hausdorff. Further, if X0 6 A we can choose a b 6 A with xo(b) = 2. Setting V = {X 6 Ac: Ix(b)[ > 1}, V is an open neighbourhood of X0 and VC_ { x 6 A o ' I x ( b )

I/> 1}

is compact for Tw - 7hk. Therefore, by 8.4.6, k(V) is a regular ideal and hence the ideal k(Y) _D k ( Y ) i s also regular, completing the proof of the converse (and also of the theorem). 8.4.8. DEFINITION. Let S be a topological space and Y0 a family of continuous K-valued functions on S. Following Naimark (Neumark) :To is called completely regular or a completely regular family if it satisfies the condition: To each closed set F in S and to each point so r F there is an f = f ( s ) i n 70 such that

f (s) - 0 on F and f (so) :/: O.

(,)

70 is called normal if it satisfies the condition: To each pair of disjoint closed sets F1, F2 in S there is an f E ~ro with f(s)-0 8.4.9.

LEMMA.

completely regular.

on F1 and f ( s ) -

1 on F2.

(**)

(a) If S is T1 then every normal family is

Commutative Topological Algebras

382

(b) A space S is completely regular iff the family C(S) is com-

pletely regular. PROOF. Clear. 8 . 4 . 1 0 . Let S -- (S,r) be a completely regular Hausdorff space and C(S) - C(S,~:) the algebra of all continuous K-valued functions on S. Let A be a strongly separating subalgebra of C(S). Denote by N the m a x i m a l ideal s p e c t r u m of A. For s E S, set M~ - M A - {]" C A" f ( s ) - 0}. Since k e r A - 0, each Ms E J~. Further, since A separates points of S, the m a p 12 : s ~-* Ms is injective. Write N0 - {Ms : s E S}. Since

)4o_C N c_ r~ we have hk-topology rhk on J~0 (got by relativization). Moreover, since f~ is a bijection between S and J~0, rhk can be transferred from ~ 0 to S and the transferred topology we denote again by rhk. We also write for an ideal I of C(S)

ho(1)- {M E ~to" M 2 I} - ~to n h(1). 8.4.11.

PROPOSITION (cf. [19, p.57]), rhk C_ r, and rhk -- r

iff A is a completely regular family of functions. PROOF. Let F C S be a rhk-Closed set so t h a t h0k(F*) - F*, where F* - 12(F). If f e A then f e k(F*) iff f - 0 on r . Further, M+ C h0k(F*) iff s e n k e r f, where f ranges in k ( r * ) and k e r f - {s e S " f ( s ) - 0}. It follows t h a t F - n k e r f . Since f is continuous each ker f is r-closed. So F is r-closed, whence rhk C r. Suppose now t h a t A is a completely regular family and F ( C S) is r-closed; then I - k(F*) - { f e A" f - 0 on F}. Clearly M+ c h0(I) - h0k(F*) iff "f - 0 on F ~ f ( s ) - 0". If s ~ F there is, by the complete regularity of the family A, an f C A with f - 0 on F but f ( s ) ~ O. This means t h a t Ms h0k(F*), whence h0k(F*) - F*, so t h a t F* and therefore F is rhk-Closed , proving t h a t r - 7 h k . Conversely, if r - rhk and F C S is closed, then for each point s ~ F - h0k(F) there is an f e A with f - 0 on F and f ( s ) ~ O.

w4. Completely Regular Algebras

383

But this is precisely the condition to be fulfilled for the family A to be completely regular. 8.4.12. COROLLARY. r h k - - 7 on S for the algebra C(S). PROOF. Since S is a completely regular space, by its definition C(S) is a completely regular family and hence the result (by 8.4.11). 8.4.13. PROPOSITION. Let S - (S,r) be compact Hausdorff and A a separating subunital algebra of C(S) such that every maximal ideal of A is fixed. Then rhk -- r (on S) iff A is completely regular algebra. PROOF. If A is completely regular then rhk is Hausdorff. Since, by 8.4.11, rhk _C r, and r is compact Hausdorff (by hypothesis) it follows from a well-known result in topology that rhk = r (on S). Conversely, if rhk = r then rhk on ~0 = E (every maximal ideal of A being fixed) is also Hausdorff. Therefore, A being unital, is completely regular. 8.4.14. COROLLARY. The algebra C(S) is completely regular. PROOF. This follows from 8.4.12, 8.4.13. 8.4.15. COROLLARY. Let S be a locally compact Hausdorff space. Then the Banach algebra C0(S) is completely regular. PROOF. Let S~ be the 1-point compactification of S. Then, by 8.4.14, C(S~) is completely regular and so, by 8.4.5, C0(S) is completely regular since C(Soo) is the unitization of C0(S). 8.4.16. PROPOSITION. Let S - (S,r) be a locally compact Hausdorff space and A be a strongly separating subalgebra of C0(S) - C0(S, I1" I]oo). Then the following two statements are equivalent" (i) A is a completely regular family of functions. (ii) The hull-kernel topology rhk coincides with r. If A is also Gelfand then (ii) (or (i)) is equivalent to

Commutative Topological Algebras

384

(iii) A is completely regular algebra. PROOF. The equivalence of (i) and (ii) has already been demonstrated in 8.4.11. The equivalence of (ii) and (iii) follows from 8.4.7.

8.4.17. PROPOSITION. If A is a completely regular pseminormed algebra then fi is a completely regular family of functions. Conversely, if A is a p-seminormed Gelfand algebra such that fi is a completely regular family then A is a completely regular algebra. PROOF. Assume that A is completely regular. Then, by 8.4.7, r]~k - r~ on A~ - w h i c h is locally compact Hausdorff. By 8.1.25, A is a strongly separating subalgebra of C0(Ac). So by 8.4.16, .3. is a completely regular family. For the converse, assume that A is also Gelfand and that A is a completely regular family. By 8.4.16, 7~ = 7hk on A~(= A -- E) so that by 8.4.7., A is completely regular. 8.4.18. PROPOSITION. Let A be a completely regular pseminormed algebra. Then OAAc -- Ac. PROOF.

Write O,iAc = F; then F is a closed set in A~. If

A ~ \ F =fi 0, take a X0 E Ac\F. Since, by 8.4.17, A is a completely regular family, there is an a e A with a(F) -- 0 and h(X0) =fi 0. Then ]I5]1oo > 0, but suplh(x)] = 0, contradicting that F is the • Shilov boundary. Hence A~ -- F - c3,iAc , as required. 8.4.19. PROPOSITION (cf. [20, p.236], [12, p.54]). Let A be a completely regular p- seminormed algebra. Then: (i) Any closed set F c

(ii) Any open set G C_ Ar

A~

Ae(A) is homeomorphic to

is homeomorphic to A~(k(A~\G)).

PROOF. By complete regularity of A, 7hk -- 7w on Ac (see 8.4.7). Therefore F - hk(F). It follows that X E F iff X - 0 on k(F). The homeomorphism in (i)is now clear.

w4. Completely Regular Algebras

Write A ~ \ G - F. T h e n x E G i f f x r Fiffxr x I k ( F ) c A~(k(F)). Hence the h o m e o m o r p h i s m in (ii). 8.4.20.

385

k(F) iff

PROPOSITION (cf. [12, p.55]). Let A be a completely

regular, complex or strictly real, commutative p-Banach algebra A ( ~ v/A). Let I be an ideal of A, F1 a closed subset of A ( - Ac), F2 a compact subset A such that

(h(z)U F1)N -o. Then there exists an element ao E I N k(F1) such that 50 - 0 F1 and a o - 1 on F2.

(,) on

PROOF. Write I s. -- k(Fj) ( j - 1,2). Then Fj - h(Ij), so t h a t condition (,) above becomes (h(I) U h(I1)) N h(I2) - 0, i.e.,

h(I N 11)Rh(I2)

- 0

(1)

which means t h a t no character X of A can vanish at the same time on both I N I1 and /2. Since /2 is clearly the intersection of all h y p e r m a x i m a l ideals c o n t a i n i n g / 2 and A is Gelfand, it follows t h a t A # - A / I 2 is s.s.. Since by 8.4.19, A ( A #) ~_ F2, and F2 is c o m p a c t we conclude by 8.6.6 t h a t A # has unity u # - u + / 2 (say). Denote by 99 the canonical h o m o m o r p h i s m A --, A # - A/I2. We assert t h a t 99(IN I1) - A #. At any rate J # - ~ p ( I ~ I 1 ) i s an ideal of A #. If J # r A # there is a h y p e r m a x i m a l ideal M # with g)(I n I1) c_ M #. Then M -- p - l ( M # ) is a h y p e r m a x i m a l ideal of A with I N I1,/2 _ M. But this contradicts relation (1). Therefore we must have 99(I N I1) - A, as asserted. It follows in particular t h a t there is an element a0 E I N I1 such t h a t a0 § - to(a0) u #. If X e F1 then X - 0 on k(F1) - I1 so t h a t a0(x) - x(ao) - O. On the other hand if X E F2 then X = 0 on k(F2) = / 2 , so t h a t

ao(x) - x(ao) - x# (ao-4- I 2 ) -

X#(U #) -- 1,

where X # is the character of A # induced by X. This completes the proof. 8 . 4 . 2 1 . COROLLARY. A being as in 8.4.20, if F1 is a closed subset and F2 a compact subset of A ( A ) such that F i e F 2 - O, then there exists an element ao E k(F1) with ~o - 0 on F1 a n d - 1

Commutative Topological Algebras

386

on F2. PROOF. Suffices to take in 8.4.20, I -

A (then h(I) - 0).

8 . 4 . 2 2 . COROLLARY. If I is an ideal of A and F a compact subset of A(A) such that h(I) ~ F - 0, there is an element ao e I such that 5 o - 1 on F. PROOF. Suffices to take in 8.4.20, F1 - 0 (then k(F1) - A) and F 2 - F. 8.4.23.

COROLLARY. If A is a completely regular, complex

or strictly real, commutative unital p-Banach algebra then ~i is a normal family of functions on A. PROOF. This follows from 8.4.21, since now A being compact every closed set in A is compact. 8.4.24. DEFINITION. Let A be a completely regular pseminormed algebra and F a closed subset of J~c (the space of closed h y p e r m a x i m a l ideals). Denote by J = j ( F ) the set of all x c A such t h a t ~ has compact s u p p o r t t disjoint with F. If F is a single point {M} we write j ( M ) for j ( F ) . Further, if F - 0 we write J0 for j(0).

8.4.25. LEMMA. J -- j ( F ) is a bi-ideal of A such that h ( J ) F.

PROOF. Clearly x C j ( F ) iff there is a compact set Cz such t h a t ~ - 0 outside Cz and Cz ~ F - 0. From this result it is easy to see t h a t j ( F ) is a bi-ideal. Further, if X e r then X(x) -- 0 for every x e j ( F ) , whence h ( j ( F ) ) 2 F. Now consider a 2:o e A c \ F . Since A~ is locally compact Hausdorff and F is a closed set there is an open set U in Ac with X0 E U and U compact and disjoint with F. By complete regularity of A there is an x in A with X(Xo) # 0 and ~ - 0 on Ac\U. Then x e j ( F ) and Xo ~ h ( j ( F ) ) , which proves t h a t h ( j ( F ) ) - F. 8 . 4 . 2 6 . PROPOSITION. Let A be a commutative completely t For a function f on a topological space S, by support of f we mean the closure of the largest subset on which it is not zero.

4. Completely Regular Algebras

387

regular s.s., complex or strictly real, p-Banach algebra. Let F be a closed subset of A c - A. Then g - j ( F ) is the smallest of the ideals I such that h(I) - F. PROOF. In view of 8.4.25, it is enough to prove t h a t for any I with h(I) - F we have J C_C_I. If x E J and C is the s u p p o r t of then C is c o m p a c t and C N F - 0. By 8.4.22 there is a y E I with ~) - 1 on C. Since ~ - 0 outside C we have clearly &~ - ~. But A being s.s., we get xy - x, whence x E I, J _c I, completing the proof. 8 . 4 . 2 7 . COROLLARY. ideal I of A with h (I) - F.

The closure J is the smallest closed

PROOF. If M E A then M is a closed ideal, so t h a t M _~ J ==~ M D g. Therefore h(J) - h(J) - r. Further, if I is a closed ideal with h(I) - F then J C_ I and so J C I - I, completing the proof. 8 . 4 . 2 8 . COROLLARY. For each maximal regular ideal M of A , j ( M ) (respy. j ( M ) ) is the smallest primary (respy. closed primary) ideal contained in M. PROOF. Apply 8.4.26, 8.4.27 with F -

(M}.

8 . 4 . 2 9 . THEOREM (Abstract Wiener Tauberian T h e o r e m ) . Let A be a commutative s.s. regular, complex or strictly real, pBanach algebra and Jo the ideal of elements x such that ~ has compact support. If Jo is dense in A then every closed ideal I ( r A) is contained in a maximal regular ideal M. PROOF. If I is not contained in any m a x i m a l regular ideal then h(I) (in ~ ) - 0. Since maximal ideals of A are closed and J0 is dense in A it is clear t h a t h(Jo) - 0. By 8.4.26 we get J0 c I. But then since I is closed and J0 dense we must have I - A - a contradiction. 8.4.30. Remark. For connection between above t h e o r e m and classical Wiener Tauberian t h e o r e m see [19, pp.147-9] or [22, pp.426-7]. See also [23, p.326] and reference cited therein for other related results.

388

Commutative Topological Algebras

w 5. H o l o m o r p h i c Functional Calculus for Several Commutative Algebra Elements 8.5.1. We begin by recalling the notion of polynomial convexity. Let K be a bounded subset of N:n. Set

h(K) - {,,~- ( ~ 1 , ' " , ,,~n) E

K n"

fP(A)I IIPIIK

for each polynomial P over N:n}, where IIPIIK - - s u p { l P ( ~ ) l : g c K}. Then clearly h(K) D_ K and h ( K ) i s called the polynomial convex hull of K. If h(K) = K then K is called polynomially convex or p-convex. If K is the unit circle in C, h(K) is the closed unit disc, as can be seen using the maximum modulus theorem. 8.5.2. LEMMA. ( i ) h ( K ) is compact, in particular a V-convex set K is compact. (ii) K is p-convex iff for each ~o e K'~\g, there is a polynomial Po with IPo(A~ > liPolIK. (iii) If K is polynomially convex and C > O, then for each ~o c ~ n \ K we can find a polynomial Q with

> c,

IIQlIK c.

PROOF. (i) It is clear from its definition that h(K) is closed. Also, if Aj is the polynomial hi(A1,-.. ,A,~) = Aj then by taking P - Aj in the definition of h ( g ) w e get for A e h ( g ) , IAjl ~< sup{l#jl : fi e K} = Cj < c~. This means that h ( g ) i s bounded and consequently it is compact. (ii) Clear from the definition of p-convexity. (iii) Given ,~0, choose first P0 as in (ii) and then set Q(A) -

CPo( )/IJPo]l 8.5.3. R e m a r k . It is known that a compact set K in C is p-convex iff C \ K is connected (see [30, p.37]). Therefore the result 7.3.24 can be restated as: the spectrum a(a) of a t.generator a of a complex monogenic p-Banach algebra is p-convex. In this form the result is generalized in 8.5.15 (ii). 8.5.4.

Recall that a polydisc Pd with poly-radius 5 =

w 5. Holomorphic Functional Calculus

389

(51,-'-,5~) is given by

P ~ - {a ~ ~ " lajl-< ~j, J - 1,... ,~}. A subset II of P g is called a p-polyhedron (in P g) if there are polynomials P 1 , ' " , Pm such that H-

{,~ c [?~." IIPI(,~)I,...,

IPm(X)l ~< 1}.

$.5.5. LEMMA. (a) Pg itself is a p-polyhedron. (b) Every p-polyhedron is p-convex. PROOF. (a) Take m -

n, Pj - 5j-IAj, where A1 is the poly-

__+

nomial h i ( A ) - )~j. 0 (b) If ,~o r 0<~\ii then either some I)~jl > 53. or some IPk(~~

1. In the first case I A j ( ~ ~

>

0 I,kjl > 53./> ]IAjlIH and in the second

case we have IPk(,~~ > 1 i> IIPkllri. Thus in either case ,~o ~ h(H) which means that h(II) - H and II is p-convex 8.5.6. LEMMA. Let K be a p-convex set with K C_ F$ and G an open set with K C_ G C_ K '~. Then there exists a p-polyhedron II with K c II C G. By 8.5.2 (iii), we can choose for each ~0 E R:n\K a polynomial PS0 with P~0(~ 0) > 1, IIPslIg ~ 1. By continuity of PnOOF.

_~

--+

PS ,) we have IPs0(A)I > 1 for A in some (open) neighbourhood A/S0 of ~0. Writing F -- Pd and allowing Ao to range in P\G(_C P\R:) and using the compactness of P\G we can find a finite family of neighbourhoods AfD , corresponding to polynomials PSi, such that

~s,, 9 U" Set II - {X C e "IPD

U-~f,o - P\a.

(X)I .< IIPx; IlK -< 1}, so that

K C II. Suppose

next A r G. If also~ A r F then~of course A r H(since II _ P). On the other hand if A E F then A c P\G and so A E A/~j for some j. _.+

Hence IP~jl > 1, whence A C II. Therefore II C G, as required. 8.5.7. The notion of spectrum of an element can be generalized to that of joint spectrum of a finite set of elements as delineated below.

390

Commutative Topological Algebras

Let A be a commutative algebra with unity e and write d ( a i , ' - ' , a n ) (aj C A). If A - ( A I , ' " , A n ) (Aj C K) then we write I - I ( A ) - I ( A , d ) - the ideal of A generated by a j 1 , . . . , n ) . Clearly we have

Aje ( j -

n

I-

I(A) - ~

A(aj-

(,)

Aje).

j=l

The joint (or simultaneous) spectrum --

a(al,...,an)

o(~)

-

{:~ e ~:". I(:~) # A}.

Not that when n -

8.5.8.

is defined by"

a(d)

1, ~ -

LEMMA.

~ ~nd ~ ( ~ ) -

~(a).

(i) A C K n \ a ( d ) i g there are elements

n

(ii) A C

j=l a(d) iff for every b l , . . . , b ,

E

A

we have

n

(aj - Aje)bj ~ Gi, where Gi denotes the group of invertible el-

j=l ements of A.

(iii) A C a(d) iff there is a maximal ideal M of A, with aj Aje C M (j - 1, . . . , n). PROOF.

(i) The stated condition is clearly equivalent to:

I(A)- A. (ii) The condition is clearly necessary and sufficient for I(A) y6 A. (iii) If A C a(d) then I(X) ~ A and so by Krull, I(X) C_ some maximal ideal M, whence a j - Aje E I(A) C_ M. Conversely, if aj - Aje. (j - 1 , . . . ,n) are contained in some maximal ideal M then I(A) _ M y6 A and ,~ E a(d). 8.5.9. COROLLARY. ,k C a(d) =~ Aj C a(a:i ) ( j so that

(r(~) C o ( a l ) x . . .

x o'(an).

1 , . . . , n),

w 5. Holomorphic Functional Calculus

391

PROOF. ~ C a(d) ::~ aj -- ,,kje E M ::~ ~f E a ( a j ) . 8.5.10.

PROPOSITION.

Let A be a unital c o m m u t a t i v e Gelfand algebra - in particular, a unital, complex or strictly real, c o m m u t a t i v e p - B a n a c h algebra. Then A E a(d) iff there is a character X C A with x ( a j ) -- Aj (j -- 1 , . . . , n), where

X --

(,~1,'",)in).

PROOF. If X E a(d), by 8.5.8 (iii) there is a maximal ideal M with a j - Aje C M . Since A is Gelfand M is hypermaximal and let X be the character determined by it. Then x ( a j ) - Aj, (A

-

(Aj)). Conversely, if X E A satisfies the stated condition then

aj - Aje E M ( M - kerx) and ~ E a(d).

s . 5 . 1 1 . COROLLARY. L~t A - (A, II " If) (ll " ll,m.) b~ ~ ~it~l, complex or strictly real, p - B a n a c h algebra. Then a(d) is n o n e m p t y compact with n

~(~) c I I ~(aj) c p~, j----1 --,

1

PROOF. Since A is Gelfand, A x(aj)Aj then by 8.5.10,

Ac #- 0. If X E A and

- (,kj) c a(d), so that a(d) # O. By 7.3.12,

A

(x(ai),...,X(an))

--

Ae ~ is compact and since the map X ~-* is continuous, a ( d ) i s compact. Finally, if 1

#j ~ a ( a j ) then I#il ~< r(ai) ~ llaj[[~ (see 7.3.27). This completes

the proof. 8.5.12. COROLLARY. Let A be as in 8.5.10, and al, A. Then

(i) For any polynomial P over K, a ( P ( a , , . . . , a,~)) - P ( a ( a l , . . . an)).

,an E

C o m m u t a t i v e Topological Algebras

392

(ii) / f a j E ~ :

(j-l,...,n)

a(alal,...

then

,anan) - {(alAl,"',anAn)"

'~ E a ( a l , . . . , a n ) } .

PROOF. It suffices to observe that each X E A being a homomorphism we have: x ( P ( a I , " " , an)) - P ( x ( a l ) , " " , x ( a n ) ) X(ozjaj) - oLjx(aj).

8 . 5 . 1 3 . R e m a r k . Arens has shown that the conclusion in Proposition 8.5.10 holds also if A is a unital commutative complex locally sin. ~ algebra (see [31, p.105]). 8 . 5 . 1 4 . PROPOSITION. Let A be a unital c o m m u t a t i v e pB a n a c h algebra which is either complex or strictly real, and which is t. generated by a finite set of elements a l , ' " , an. Then the spectrum A(A) is h o m e o m o r p h i c to the j o i n t spectrum a(d) under the map t?:

X ~-~

(X(al), "'" , x ( a n ) ) .

PROOF. By 8.5.10, 0 is surjective. The map 8 is also injective. To see this, suppose that t?(Xt) - t~(X2). Then, for a polynomial P - P ( A I , - - ' , An) over K we have X1 ( P ( a l , " " , an)) - P ( X I ( a l ) , ' ' ' , X1 (a,)) = P(x2(al),...,x2(an))-

x2(P(al,...,an)).

Since the elements of A of the form P ( a l , " " ,an) - where P is a polynomial over K in n variables - form a dense subalgebra A0 of A (since a l , ' - - , a n t. generate A) and X1 = X2 on A0, we conclude from the continuity of X~, X2 t h a t X~ = 2:2 on A, proving injectiveness of 0. Further, the continuity of t~ follows from that of the X'S. By 7.3.12, A = A(A) is compact Hausdorff. Therefore 0 is a homeomorphism. 8 . 5 . 1 5 . PROPOSITION. Let A be a unital commutative, complex or strictly real, p - B a n a c h algebra t. generated by al," " ,an. Then"

393

w 5. Holomorphic Functional Calculus

(i) For any C > O, a E ~( and ft ~ ~ Kn\a(ff) there is a polynomial P (in n variables over K) such that p ( ~ 0 ) _ c~, IIP(g)ll < C where d = ( a l , " " ,an). In particular, there is a polynomial Q with Q ( ~ O ) = 1 > IIO(~)l.

(ii) a(d)

is polynomially convex.

PROOF. If ~o C Kn\a(ff) then n

tl

I(~~ -- E A(aj- ,X~ j=l

E ( a j - ,~~

A,

j=l

where e is the unity of A. It follows that ae E A has a representation n

E (aj - AOe)bj - c~e

(1)

j-1

where bj E A. Since A is t. generated by ~, each bj can be approximated by elements of the form P ( a l , ' " , a,~) where P is a polynomial. It follows from this and equation (1) that there are polynomials Pj (j = 1 , - . . , n ) such that l't

II~e- ~ ( a j - ~~

I < c,

(2)

a;)Pj(2)

(a)

j=l

Write tI

P(a)- . - ~(a; --~

w

0

j-1

Clearly P is a polynomial in )~1,''" ,)in. Further, it is clear from (3), ( 2 ) t h a t we have p ( , ~ o ) _ oe

I]P(~)I] < C.

(4)

If we take C - 1 - ~ in (4) then the corresponding polynomial P which we denote by Q satisfies" Q(~o)-

1 > ]lQ(d)ll.

Commutative Topological Algebras

394

This completes the proof of (i). To prove (ii) take a A e a(d). By 8.5.10 we have

A-

X(d)

-

(X(al),...,x(an))

for some X E A. It follows that the polynomial Q chosen above satisfies:

IQ(X~

-

IQ(x(~))l-

Ix(Q(~))[

1

<~ IIQ(d)ll~ 1.

(since [[xll ~< 1)

(5)

Since K - a(d) is compact, (5)implies that IIQIIK < 1. On the other hand, Q ( ~ 0 ) _ 1 (by construction). Therefore, by 8.5.2 (ii), a(d) is p-convex. 8.5.16. R e m a r k . From 8.5.14, 8.5.15 (ii), 8.5.11, it follows that the spectrum A = A(A), of a finitely t. generated complex commutative p-Banach algebra A, is homeomorphic to a polynomially convex compact subset of C n. Conversely, it can be shown that if K is a polynomially convex compact subset of C n then there is a finitely t.generated (with n t. generators) complex commutative Banach algebra A such that A(A) is homeomorphic with K (see [10, p.45]). 8.5.17. R e m a r k . The joint spectrum or even the spectrum of a single element, of a commutative Banach algebra which is not finitely t. generated, may not be p-convex. For example, in the Banach algebra B of bounded complex-valued functions on the unit interval [0, 1] (under the sup-norm) the function f ( t ) - g 2~it (0 ~< t <~ 1) has for its spectrum its range which is the unit circle. The unit circle, however, is not p-convex (since its p-convex hull is the closed unit disc). To establish the main theorem of this section, which asserts that holomorphic functions of several variables operate on the space of Gelfand transforms, we need a number of preliminary results which we proceed to consider. 8.5.18. LEMMA (Arens-Calderon trick [3', p.205]). Let A be a unital commutative, complex or strictly real, p-Banach algebra. Let

w5. Holomorphic Functional Calculus

395

a l , . . . , a n ~ A and G C_ K '~ an open set with a ( a l , . . . , a n ) C_ G. Then there exists a finitely t. generated closed subunital algebra B of A, with aj C B (j - 1 , . . - , n ) and a B ( a l , . . . , a n ) C_ G, where aB denotes the spectrum with respect to the algebra B.

PROOF. We may assume that the norm I1" II of A is sm. Write

K - {A C ~:" I,~jl < ]lajll~ ( J - 1,... ,n)}. Then K is compact. By 8.5.8 (i), for each ,~ E Kn\a(al, -.. ,an) there are elements bl,-., bn E A such that n

E(aj-

A j e ) b j - e.

(1)

j-1

Denote by B(A) the closed subalgebra of A generated by e, a l , . . . , a , ~ , b l , ' " , b , ~ . By considering the relation ( 1 ) i n B(A), we get, by 8.5.8 (i), that r

a B ( ~ ) ( a l , ' " , an)

--

aB(~) (say).

(2)

Since aB(S) is closed there is an open neighbourhood G(A) of such that

~.(~) N c ( ~ ) - o

(3)

It follows, using the compactness of K \ G and (3) that there are points ~k e K \ G ( k - 1 , . . . , n)such that n

U c(~'~) _DK \ c

(4)

aB(~k ) ~ G(A k) -- 0 (k - 1 , . . . , n ) .

(5)

k=l

and

Since each Bk -- B(A k) is finitely t. generated it is clear that there is a finitely t. generated subunital algebra B of A with all Bk _C B. By 1.7.19 aB(al, " " , an) C aBk ( e l , ' " , an). (6) From (4),(5),(6) we obtain aB(al,...,an)A(K\G

) --0.

396

Commutative Topological Algebras

On the other hand, by 8.5.11,

rtm g .

aB(al, " " , an) C IP[

Therefore,

aB(al,... ,an) C

G, as required.

8.5.19. THEOREM (Oka's Extension Theorem.) Let m, r > 0 be integers, cj > 0 (1 ~<j ~< m + r) be real numbers and

Let P j ( j - 1 , . . . , r) be complex polynomials in rn variables and 0 the Oka map of C m ~ C m+~ given by:

e" X -- ()i,l,''' ,.~m) ~+ (/~1,''',)ira, P I ( X ) , " ' ,

Pr(~)).

Then, for any f holomorphic on an open neighbourhood Of0-1(P) there corresponds an F holomorphic on an open neighbourhood of P such that

F(O(~))- f(X) (,~e O-I(IP)).

PROOF. See [4, p.103]. 8.5.20. PROPOSITION. Let A - (A,I I 9II) be a unital commutative complex p-Banach algebra, f f - ( a l , " ' , an), aj e A ( j 1 , . . . n) and G C_ C n an open neighbourhood of a(d). Then we can find a finite number of elements a n + l , ' " , aN E A such that given any function f holomorphic on G there is a function F holomorphic on the polydisc

{~ E c N " IAjl <~ 1 +

211ajll

(j - 1,..., N ) }

such that

f(~tl(X),'"~tn(X))-

F(al(X),'"

",~tN(X)) (X E A).

PROOF. By 8.5.18, we can find a finitely t. generated subunital algebra B containing a l , " " an with

,a.) c a.

(1)

w5. Holomorphic Functional Calculus

397

We can take as t. generators (~= e) of B to be a l , ' - " ,am, where 1

--,

n ~< m. Fix C > 0 and set a - 2C~ + 2. For any A there is, by 8.5.15(0 , a polynomial P such that

E

Cn\aB(d),

P ( X ) - 26~ + 2, [Ig(~)ll < c, Then we have

IP(X)I- 2c~ + 2 > 1 + 2C~ > 1 + 21tP(~)II ~.

(2)

It follows that there is an open neighbourhood Vp(A) of A such that 1 IP(fi)} > 1 § 211P(d)ll~ (3) for all fi

E

Vp(~). Denote by r the natural projection

(~1,... , ~ ) ~

(~1,.--,~,)

(~ ,< m)

of C "~ onto C n. Write P m - {,~ E C m ' I A j l ~< 1 + 211ail I ( j -

1,...,m)}.

Then " K - Pm\r-l(G) is compact. Also, ~r- I ( G ) contains a(b'), where b ' - ( a l , - "

(4)

(5)

,am) since

r ( a ( b ' ) ) - a ( d ) C G. It follows from (1),(2),(3) and (5) that there are polynomials PI,"',Pr such that Vp~,...,Vp~ cover K. Therefore, for any A c K, we have IIPk(A)II > 1 + 21lPk(b')ll~ (6) for at least one k (1 ~< k <~ r). Set

N-re+r,

am+k - Pk(b) (k - 1 , . . . , r ) ,

FN - { A E C N ' I A j I < ~

l+211ajll~ ( j - - 1 , . . . , N ) } .

Let 0 be the Oka map

0(~1,..., ~ ) -

(~1,-.-, ~m, P~(~),..., P~(~))

398

Commutative

Topological Algebras

where . ~ - (/~1:"",)~m)- Since e(A) E pN => ~ e pm and IPk(A)I ~ 1 +

211Pk(g)ll~

(k-

1,...,r),

using (4),(6)we get

e-l(e N) _c r-l(G). The function f o r is holomorphic on the open neighbourhood 7r-l(G) of o - l ( ? N ) . By 8.5.19, there is a function F holomorphic on an open neighbourhood of 0-1(pN) such that

F(0(A-1)) -- f o ~r(s (~ E For X C A, set

Aj -

x(a.i ) (j -

Then Pk(A) - Pk(A1,'-', A m ) and

e-I(pN)).

(7)

1, . . . , m ) .

-- xPk(b)

-

X(am+~,)

(k -

1,...,r)

so

{)(,~) - - ( X ( a l ) , .

Since

. . ' X(aN)). 1

IAjl- Ix(aj)l and

llajll (J- 1,-..,m)

--,

IPk(~)l

1

-

x(a~+k) < Ila~+kll ~

we conclude that 0(A) c pN, ~ E o-X(pN). Finally we obtain, by

(7) f (x(al),

. . . , X(an))

f o r(A)-

-

F(O(A))- F(X(al),

. . . , X(an)),

which completes the proof. The next lemma is needed for proving the main theorem which follows 8.5.21.

LEMMA. I f fin E

C, limsup

[fin]-

l a n d 0 < p <~ 1,

n--+ oo

t h e n lim sup Ifln Ip

-

IP.

n---+ oo

PROOF. Given e > 0, choose r/ > 0 such that r/p - e. Since limsup[fl= I - I there is a N such that n----~ o o

]fl,~] < l +~T f o r n > N .

(1)

w5. Holomorphic Functional Calculus

399

Also, for any m, there is an n > m with

I~.1 > l - , 7 .

(e)

From (1) we get I/~nl p < ( l §

p~
(n>/N)

(3)

and from(2) we obtain Ifl.I ~ + ~ - t ~ . 1 ~ + ,7 ~ >~ (1~.1 + ,7) ~ > v .

(4)

The inequalities (3),(4) show t h a t l i m s u p l ~ n [ P - I p. 8 . 5 . 2 2 . THEOREM (Shilov-Arens-Caulderon) t. Let A be a unital commutative, complex or strictly real, p-Banach algebra. Let a l , ' " , a n be elements of A, G C Kn , an open set such that a ( a l , . . . , a = ) c_ G and f a function holomorphictt on G. Then there exists an element b E A such that _

b(X) -- X( b) - f ( a l ( X ) , ' " , a n ( x ) )

(X E A).

(.)

The element b is uniquely determined iff A is s.s.. PROOF. First assume t h a t A is complex. Then, by 8.5.20, there are elements a n + l , " " , aN in A and a function F holomorphic on an open neighbourhood G* of the polydisc

_ p i _ {X C cN

I~1 ~ 1 + 211aill~ ( J - 1,-.. ,N)}

such t h a t

F(x(al),'",

X(aN)) -- f ( x ( a l ) , " ' ,

x(an)).

Since P is compact and P _ G* the distance from P to the disjoint closed C~\G * is positive. So we can find an open polydisc P~ with

PcP~ ca*. t These authors considered only the case of complex Banach algebras. tt holomorphic in the real case means real analytic.

400

Commutative

Topological

Algebras

Let the Taylor series representation of F on P~ be F(,~)

--

Og~ '/~lkl " " " ,~N

E

where N - {0, 1, 2,-..}. By Abel's lemma, the (multiple) series for F converges absolutely on P. In particular, the series converges at 1 fi with #j - 211ajll;( j - 1,.-., i). By root test t for convergence we have

~

limsup(l~Zl2k~ttalll~ ... 2k~llaNIl~ ),,I" Ilkll-,o~

1

(,)

where I1~11- ki + . . . + kN. The inequality (,) reduces to k 1

lim

kN

1

1

sup(Io~.t Ilalll-;-""" Ilaull 7 )

~ ~< ~.

II/;ll-,co

By applying 8.5.21, we obtain 1

1

limsup(l~klPllalllkx "''llaNIIk~)~ < 2p

<1.

Therefore, by the root test, the multiple series

laZlPllalll k~''" Ilagll k~ converges, which means that the series E

kX ,

~ fcal

]fN

9 aN

converges absolutely and consequently the series converges to an element b c A, i.e., E

See [2, p.364].

a g a ~lx " " . a ~NN -

b.

w 5. H o l o m o r p h i c

Functional

401

Calculus

If X C A, then X being a continuous homomorphism we have x(b)

-

~-~ OLfcX(al)k,

--

F(X(al)

. . . X(aN)kN

. .. , X(aN))

--

.f(x(al),..

. ,X(aN)),

so t h a t we have

b(x) - J'(al(X),"" ,a,-,(x)) (x ~ A). This proves the theorem when A is complex. Next, let A be strictly real and A its complexification. Here we have ,v

a(al,...,an) C G C •n and f holomorphic on G. Then it is known t that there is a (com7 plex) holomorphic function F on an open set G C C n with G _c G and F I G - f . Since A is strictly real we have N

a(al,'",an)--a(al,-'-,an)

C G C G,

where 5 denotes the joint spectrum with reference to A. Since A is a complex algebra, by applying the theorem for this case (just proved) we can find a b E A such that N

~t(b) -

F(x(al),

N

. . . , x(an))

-

F(X(al),

" " , X(an))

where ~ E A, and X - ;~]A; by 1.9.18, 1.6.14, X is real-valued. Since b C A , b - - b § i c (b, c C A ) . Therefore we have N

N

.~

N

x(b) + ix(c)

-

=

+ ic) -

F(X(al),... , X ( a n ) ) - f(X(al),.., x(an)).

Since the RHS is real we must have

x(b)

-- f (x(al),

. . . , x(an))

which completes the proof of the existence of b for the strictly real case.

1" See [3, p.134].

402

Commutative Topological Algebras

For the uniqueness of b result (in both the cases) we note that if bl is also an element such that (.) holds when b is replaced by bl then clearly we have b - bl E N ker X - ~ x / ~ (since A is a Gelfand algebra). So bl - b iff v / A - {0} iff A is s.s.. 8.5.23. PROPOSITION. Let A be a unital commutative, complex or strictly real, pseudo-Michael algebra such that in its projective limit decomposition A - limA,, each A~ is s.s.. Let Ac be the set of continuous characters of A. For a l , ' " , a n

a*(al,.-.,an)-

{(X(al),-",x(an))'XE

E A write Ac}.

If f is a holomorphic function on an open set G D a * ( a l , . . . , a n ) then there is a unique element b E A such that

b(x)- f(al(X),"',an(x))

(XC Ac).

PROOF. By 7.3.19 we have A: - U A a , A a ~- A~, where / ~ - the space of all characters of .4~ - the space of all continuous characters of A~ (by 7.1.6(c), 7.1.9). Since each Aa is a Gelfand algebra we have (see 8.5.10) the relation a ( a l , ' - - , a n ) --

Ua(ala,...,ano~), Ol

where aja - ~ ( a j ) ,

and ~a the canonical map A -~ Aa. Since

G D a ( a l a , . . . ,a,~a), by 8.5.22 there is an element b~ E Aa such that )ta(ba) - f ( f c a ( a l a ) , " " , fca(ana)) (Xa e h a ) .

Since bjz - ~az(aj~) we have -:

f(

a o o

o -

Since A~ is s.s. we have ~p~e(be) - b~. It follows that lifnb~ - b exists and we have clearly

~'(X) - f(al(X),...,a.,(x))

(x E A:).

To prove the uniqueness of b it is enough to show that ~ r - ~ _ {0}. Suppose that x E A, x =fi 0. Then for some a, xa =/= 0. Since Aa is ^

w6. Shilov Idempotent Theorem

403

s.s. Gelfand there is 2c~ with 2a(xa) r 0. Then Xa -- 2a~ E Ac and x~(x) - 2 ~ ( x ~ ) r o. This shows that ~ {0},t which completes the proof. 8.5.24. R e m a r k . The holomorphic functional calculus embodied in 8.5.22, for t. finitely generated unital commutative complex Banach algebras, using A. Weil's integral formula for holomorphic functions of several variables, was obtained by Shilov. This was extended by Arens and Calderon for arbitrary unital commutative complex Banach algebras. An approach to holomorphic functional calculus based on exterior differential forms had earlier been given by L. Waelbroeck (see [29, pp.41-50], [12']). Waelbroeck's method enabled him to show that the correspondence f ~ b- f(al,"',an)= f(ff). is a h o m o m o r p h i s m - a result not obtained by either Shilov or Arens. Further, Waelbroeck developed the functional calculus for any complex commutative locally convex complete Hausdorff algebra. For an introduction to Waelbroeck's method (for commutative Banach algebras) see [12, pp.41-50]. The above proof of theorem 8.5.22, based on Oka's extension theorem instead of any Cauchy integral formula is modelled after the proof found in Bonsall and Duncan's book [4, w

w6.

Shilov Idempotent

Theorem

8.6.1. L EMMA. Let R be a ring with unity e, which is a direct sum R = I1 + I2 of two I. ideals I 1 # {0} (j = 1,2). Then there are idempotents ej ~ 0 (j = 1,2) in R such that e -- el + e2, el e2 -- e2el -- 0, Ij -- Rej (j = 1, 2). (A similar result for r. ideal decomposition.)

PROOF. Since R - I1 + / 2 , we have for e the decomposition

t It follows from this that the algebra A is s.s. (since x/~ _ c~/-~= {0}).

404

C o m m u t a t i v e Topological Algebras

e - - e 1-q-e2 (el E /1,e2 E /2). Therefore, for x C R, we have x -- xe - xel 4- xe2.

If x C 5 , then x get

xel E 11, xe2 E I2 and s i n c e / 1 ~ / 2 = {0}, we x-

Xel , xe2 - O.

Therefore, I1 - R e l , e~ e~ -- e2, e2el --O.

el, ele2 -

0. Similarly, 12 -

Re2,

8 . 6 . 2 . PROPOSITION. Let A be a unital t. spectrally Gelfand algebra - i n particular a Gelfand algebra. I f A has an i d e m p o t e n t u 7~ O, e then Ac is disconnected. PROOF. By 1.7.9, a ( u ) - {0,1}. Since A is t. spectrally Gelfand there are X1, X2 E Z~c with Xl(U) -- 0, X2(U) -- 1. F u r t h e r , for any X e A~, X(U) - 0 or 1 (since X(U) - X(u 2) - X(U)2). It follows t h a t if we set F, = {X

X(U) = 0};

= {X

= 1}

then the Fj (j = 1,2) are n o n - e m p t y (since Xj e Fj) closed sets in A~ such t h a t

FINE2 - zX~. So Ac is disconnected. 8 . 6 . 3 . COROLLARY. If a unital t. spectrally Gelfand algebra A is a decomposable I. (or r.) A - m o d u l e then Ac is disconnected. PROOF. By 8.6.1, there are i d e m p o t e n t s el =/= 0, e2 ~ 0 with el + e2 = e, when e is the unity of A. Clearly el (or e2) 7~ 0, e. Therefore, by 8.6.2, Ac is disconnected. 8 . 6 . 4 . THEOREM (Shilovt i d e m p o t e n t t h e o r e m ) . Let A be a unital c o m m u t a t i v e , complex or strictly real, p - B a n a c h algebra whose spectrum A = Ac is disconnected, so that we have a t He obtained the result for the case of complex Banach algebras.

405

w6. Shilov Idempotent Theorem

decomposition A - AI(JAo with A1, Ao clopen.t Then there exists a unique tt idempotent u in A such that fi - 1 on A1 and fi - 0 on Ao. PROOF. Since A1 ~ A o - 0, for each Xl E A1 and Xo E Ao there is an element ax~xo E A with Xl(axl,xo) 7s xo(ax~,xo). It follows t h a t we can choose open neighbourhoods V(Xi) of X1 and W(Xo) of X0 in A such that

x,(ax,xo ) 7/= Xto(axl,xo I

) for

I

I

X, E V(X,) , Xo E W(Xo).

The open sets V (X,) • W(x0) cover the compact set A 1 X /X0 and consequently there is a finite subcovering V(X~j) x W(Xoj) (J 1 , . . . , n ) . R e n a m e the associated elements a•215 as a l , ' - ' , a n and denote by 5 the mapping a " )(" ~

(al(X),'",an(x)).

T h e n a(A1)r' a ( A o ) - O, since if )('1 e A1, XO e A 0 then (X1,Xo) b e l o n g s to some V(Xlj) x W(Xoj), so t h a t

Xl(axl,xol) 3s xo(axlixoi). Since A is c o m p a c t Hausdorff there exist disjoint open sets G1, Go in ~n with ~(Aj) _ Gj (j - 1,0). Write G - GILJGo and define f on G by setting f - 1 on G1 and f - 0 on Go. Then f is holomorphic on G, and by 8.5.22, there is a b E A such t h a t

-- f ( ~ t l , ' " a n ) o n Note t h a t b(x) A 1 , A 0 #- 0, a(b) d c ~ such t h a t and ~ - b - 0 on

A.

1 or 0 according as X c A I or Ao. Hence, since - {0, 1}. By 5.3.8, there is a unique element u - b + d is idempotent. Then 6 - b - 1 on A1 A0, completing the proof.

8 . 6 . 5 . PROPOSITION. Let A be a unital commutative, complex or strictly real, pseudo-Michael algebra. If its spectrum Ac is disconnected, with A ~ - A 1 0 A 0 , Aj ( j - 1,0) non-empty clopen then there is a unique idempotent u in A with ~ t - 1 on A1 and ~ t - 0 on A0. t clopen=closed and open. tt The uniqueness follows from the monomorphism property of w proved on pp.397-8.

406

Commutative Topological Algebras

PROOF. By 4.5.1, A has a projective limit decomposition A - limAa, w h e r e / ] a are c o m m u t a t i v e pseudo-Banach algebras. In the notation of 7.3.19, we have _

_

_

where A ~ , / ~ are the spectra of A, f~a respy, and Aa X is p ~ - c o n t i n u o u s }. Set

-{xc/X~"

A,~j -- Aj ~ A,~ (j - 1,0). If for an a we have Aal , Aao ~ ~, then by applying 8.6.4. to A~ we can determine a unique element ua E / i ~ with u ~ ( ; ~ a ) - 1 or 0 ( w h e r e ) ~ - X0~Pa) according as X c A a l or Aa0. If for an a, A~I = 0, then choose u~ = 0~ and if A~0 = 0, choose u~ = ea, where 0a, ea are the zero and unity elements of Aa. Suppose now t h a t a -< fl, Xa E A~ i then X~ o ~a~ ~_ At~j and

X~(~z(uz))-

X~ o ~b~(u~) - 1 or 0

according as X~ c A a l or Aa0. Since ~aZ(ut~ ) is an i d e m p o t e n t the uniqueness of the choice of the ua implies t h a t we have

B u t this means t h a t u-

(u~) E limA~ - A,

and X(u) - 1 or 0 according as X E A1 or A0. Finally, by 4.5.6 (iii), u is an i d e m p o t e n t of A and the proof is complete. 8.6.6. COROLLARY. Let A be a commutative, complex or strictly real, pseudo-Michael (in particular p-Banach algebra) such that its spectrum Ac is compact and non-empty. Then x/~ is a regular ideal. In particular, if A is s.s., then A is unital (cf.

8.3.17,8.3.18). PROOF. Let A1 be the unitization of A. T h e n A1 -- A U{xo } where X0 is the distinguished character of A1. By 7.3.2(c), Ale(--

w 6. Shilov I d e m p o t e n t T h e o r e m

407

(A1)~) and Ac are Hausdorff. Also, by our hypothesis, Ac is compact, whence it follows t h a t Ac is clopen in (A1)c. So, by 8.6.5, there is an i d e m p o t e n t u E A1 such t h a t -- 1 o n A e and

u(x0)-0-

Since u(x0) = 0 it follows t h a t u E A. Further, for x E A and XCAc X(XU - x) = X(x)X(U)

- X(X) = X(X) " 1 - X(X) = 0

(since X(U) -- s - 1). It follows t h a t x u - x E ~ - v/--A, whence x / ~ is regular with u as relative unity. W h e n A is s.s., {0} - Y ~ is regular which means t h a t x u - x - 0 (x E A), i.e. u is unity of A. 8 . 6 . 7 . A partially ordered set or poset S, with ordering relation -<, is called a lattice if any two elements a, b in S has a least u p p e r b o u n d or lattice sum a v b and a greatest lower b o u n d or lattice p r o d u c t a A b. The element a v b of S is characterized by the properties: a, b -< a v b and if a, b -< x E S then a v b -~ x. Similarly a A b has the properties: a A b -< a,b and x -~ a,b ==~ x -< a A b. The least element 0 and the greatest element 1 of S - whenever they exist - are characterized by the properties: 0 -< x -< 1 for every x c S. A lattice L is called distributive if it satisfies the distributive law aA(bVc)

= ( a A b ) V ( a A c ) for every a,b, c e L.

Also, a lattice L with 0,1 is called c o m p l e m e n t e d if each a E L has a c o m p l e m e n t a I satisfying aAaS=O,

aVa~=

1.

A c o m p l e m e n t e d distributive lattice is called a Boolean algebra. 8 . 6 . 8 . PROPOSITION(Stone.) The set Bi of i d e m p o t e n t s of any unital c o m m u t a t i v e ring R is a Boolean algebra. PROOF. Define an order "-< " in Bi by: for u , v E Bi, u -< v if uv = u. It is easily checked t h a t "-< " is a partial order.

408

Commutative Topological Algebras

Moreover, it is straightforward to verify that Bi is a lattice with lattice operations v, A given by" uVv--u§

uAv--uv.

Bi is further complemented with u t - e - u as complement of u, where e is the unity of R. Finally, the distributivity of the ring operations imply the distributivity of the laltice operations. Thus, l~i is a complemented distributive laltice or a Boolean Algebra. 8.6.9. L EMMA. Let X be a topological space. Then the clopen sets in X form a Boolean algebra Z~co(X), with set-union, setintersection and set-complementation as the Boolean operations. PROOF. Clear. 8 . 6 . 1 0 . LEMMA. Let A be a T A and Ac ~ 0. If u E A is an idempotent and Gu - {X c A : X ( U ) - 1},

G u c - {X E A c : X ( U ) - 1}

then Gu, Guc are clopen sets of A, Ae respectively. PROOF. Since u is an idempotent, for any X E A, X ( u ) 2 -- X(U 2) -- X ( u ) ,

so t h a t X ( U ) - - 0 or 1.

Write

a ~ - (x c A

x(u) -

0}.

Then clearly A - GuOG~ By definition of the (weak) topology of A it is clear t h a t G u , G ~ are closed. So Gu is clopen in A. Similarly it can be seen t h a t Guc is clopen in Ac. 8 . 6 . 1 1 . PROPOSITION. Let A be a unital commutative TA. Then the map w " u ~-+ Gu is a homomorphism between the Boolean algebras Bi = ~i(A) and Boo(A). Similarly, wc : u ~ Guc is a homomorphism between Bi and Boo(At). If A is also spectrally

G~lfa~d (r~spy. t ~p~ctraUy C~qa~d) th~ th~ , ~ p ~ ( , ~ p y . ~ ) is a monomorphism. PROOF. Since x(u

v ,,) -

x(u)

+ x(v)

- x(u)x(,,)

-

] if x ( ~ )

or X ( ~ ) -

1,

(t)

409

w6. Shilov I d e m p o t e n t Theorem

we get

Guvv -- Gu U Gv"

Again, since X ( u / ~ v ) obtain

X(U)X(v) -

l i f t X(U) - X(v) -

1, we

-c nc .

Finally, if u' denotes the complement of u then, X(u') - x ( e - u ) 1 X(u) - 1 iff X(u) - O, whence

(2) -

-

Gu,-

A\Gu.

(3)

It follows from (1),(2),(3) that w is a (Boolean algebra) homomorphism. Similarly wc is a homomorphism. Suppose now t h a t A is spectrally Gelfand and Gu~ - Gu2; then X(Ul) - 1 iff X(U2) - 1. Since u l , u 2 are idempotents, X(Ul), X(U2) - 1 or 0. Therefore, X(Ul - u2) - 0, for all X E A, whence e

Since Ul, U2 are idempotents, ul - UlU2 is also an idempotent (as easily checked). But -

U l U 2

-

-

whence by 1.2.24(d), U l - UlU2 -- 0 or ul = UlU2. Similarly, u2 - U2Ul = UlU2. So Ul -- u2. This completes the proof t h a t w is a m o n o m o r p h i s m . Similarly it can be shown t h a t wc is a m o n o m o r p h i s m (when A is t. spectrally Gelfand). 8 . 6 . 1 2 . DEFINITION. A unital commutative T A is called a ShiIov algebra if the m a p u ~-~ G~c of /~i(A) --* /~co(Ac) is an

isomorphism. 8.6.13. THEOREM. A n y unital commutative, complex or strictly real, p s e u d o - M i c h a e l algebra A is a Shilov algebra. In particular, a unital commutative, complex or strictly real, pseudoB a n a c h algebra is a Shilov algebra. PROOF. By 7.2.21, 8.6.11, w~ is a monomorphism. Also, by 8.6.5, w~ is an epimorphism. Thus w~ is an isomorphism and A is a Shilov algebra. 8.6.14.

DEFINITION.

A unital algebra A (over K) is called

Commutative Topological Algebras

410

spectrally connected if the spectrum a(x) of every element x E A is a connected subset of ]4. 8.6.15. THEOREM. A unital commutative, complex or strictly

real, pseudo-Michael (in particular, pseudo-Banach) algebra A is spectrally connected iff its spectrum Ac is connected. PROOF. If Ac is disconnected then by 8.6.4, there is an idempotent u ~ 0, e in A, so that a(u) = {0, 1} is disconnected, whence A is not spectrally connected. On the other hand, if A~ is connected then X ~ x(a) being a surjective continuous map of A~ onto a(a),a(a) is connected for every a e A, so that A is spectrally connected. COROLLARY. A is spectrally disconnected iff A has an idempotent u ~ 0, e. 8.6.16.

PROOF. If u =/= 0, e is an idempotent then by 1.7.9, a(u) {0, 1}, so that a(u)is disconnected, which proves the "it' part. To prove the "only if' part assume that A has an element a with a(a)disconnected. Since X ~ x ( a ) i s a continuous map of Ac onto a(a) it follows that Ac is also disconnected. But then, by 8.6.5, there is an idempotent u r 0, 1 in A, completing the proof.

CHAPTER

NORM

w1.

IX

UNIQUENESS

THEOREMS

Norm- uniqueness Theorem of Gelfand

9.1.1. LEMMA. Let [. Ij ( J - 1,2) norms on a LS X (over K) such that

be two complete

IXll <~ C]zl2 for all x c X and some C > 0.

(F)

(*)

Wh~n I" I1 - I" I~ PROOF. The condition (.) implies that the identity map Z 9(X, I 9I1) --+ (X, [ 912)

is continuous. By the open mapping theorem (3.1.15) for (F) spaces, I is open and consequently I is a homeomorphism and hence ]l'lll"~ I]'1129.1.2.

o~ a LS X

LEMMA. Let I 9Ij ( J -

Th~

1,2) be complete (F) norms I1+J" I~ i~ a compl~t~

I" I1 ~ I" 1~ iff f. I - I "

(F) norm. PROOF. First note that " I (as defined above) is always an (F) norm. If I" t is complete then since ]. Ij ~< I" I, by 9.1.1, 9Ij "~ I" I (J = 1,2). Hence I" ]1 "~ I" ]2. Conversely, assume that I"11 "~ 1"12. Suppose that Ixn-Xml--+ 0. Then Ix,~ - Xmlj --+ 0 (j - 1,2). By completeness of ]. I1 there is an element x with I x n - x ] l - - + O. Since 1"12"~ I ' l l we have also IXn - x12 --+ O. So Ixn - x I - I x n - X]l + ]Xn - - X12 ---+ 0 and 9I is a complete (F) norm.

9.1.3.

COROLLARY.

n o r m s and there is a ( F )

Ill. Ij (J- 1,2) norm

]-Io

with

are two complete

(F)

Iz]o < IXll, Ixlz (x e A)

then I" ]1 ~ ]" ]2.

PROOF. Suppose that I" Io - I " tl -~-I" 12 and Ix. - x m l - + O. Then Ix,~ - Xmlj --+ 0 (j - 1,2). By the completeness of the

412

Norm Uniqueness Theorems

(F) norms I" Ij there are elements x,y E A with I x , - Xll 0, I x n -- Y[2 -+ 0. Since [. ]0 <~1" ]1, [" 12 we obtain Ix, - xl0 0, Ix,~ - y[0 --~ 0. By uniqueness of limits property, we get x Therefore ] x n - x I - [ x , - x ] l + l X , - X [ 2 -~ 0, proving ].] complete. Now we can apply 9.1.2 to conclude that ]. I1 "" ]" completing the proof.

--~ --~ y. is ]2,

9.1.4. THEOREM ( G e l f a n d t ) . Let A be a commutative s.s. algebra which is a p-Banach algebra under either the p - n o r m s

I1 IIj (J-

1,2)

ll " Ill ~ II " lt .

PROOF. First assume that A is complex. We can also assume that the [[. ]]j are sm.. Then, by 7.4.6 (A, 1[" tt3) is a GB algebra, so that we have 1

1

r(x) --/2 l(x) p -- v2(x ) p, where vj - vj(x)

(j - 1, 2)

are topological spectral radii functions with respect to the pnorms II" llj- Writing V ( X ) - Vl(X ) -- V2(X),

by virtue of 4.8.4, 7.4.5 and the hypothesis that A is s.s. we obtain that v is a p-norm. Since v(x)<~ tlxllj (x E A), we can

pply 0.1.3 to

on lude that ll" II1

ll" ll:-

It remains to consider the case where the algebra A is real. If A is the complexification of A, then by 4.1.16, II" 11i can be ,v

extended to [[ "~llj over A such that (A, II "~llj) ( J -

1 , 2 ) a r e comN

mutative complex p-Banach algebras. Furthermore, by 1.6.16, A is s.s.. So applying the result just proved for complex algebra we get II "~tll "" t1" 112, whence (by restriction) [1" II1 "~ I1" ]12-

w 2.

Rickart

Separating

Function

9.2.1. DEFINITION. Let A be a LS over K and I'lj ( J be two (F) norms on A. Set

1,2)

t Gelfand considered only the complex Banach algebra case of the theorem.

w2. Rickart Separating Function

413

A(a) - A12(a) --inf{]x]l -+-la- x12 } (a,x E A). The function A is called the Rickart separating function, or just separating function, for I" Ij. 9.2.2. LEMMA. (i) A 1 2 ( a ) : A21(a) (a E A). (ii) A(a) <~ min{lall , lal2}. (iii) A(0) -= 0. (iv) A ( - a ) = A(a). (v) A ( a § ~
A(Aa) --I)~IPA(a)

(a E A,A E K).

PROOF. (i) This follows by relacing x by a - x in the definition of A12(a). (ii) By taking x = a in the definitions of A12(a), A21(a). (iii) This follows from (ii) since ]011 = 0 (= 1012). (iv) This follows from the definition of A, using the relations I - xlj -- Ix[j (J -- 1,2). (v) Given e > 0, by definition of A, there are Xl,X2 E A with

IX1 ]1 -~- la - X112 IX2[1 -~-In- x2] 2

~< ~<

A(a) + e A(b)-[- E.

Therefore

A(a+b)

~< ~<

Ix1 + x211 -4-]a + b X212 IXlll -~-1X211-[-la- Xl]2 + Ib- x212 Xl

-

From the arbitrariness of e, the inequality (v) follows. (vi) For A - 0 the equality clearly holds (since both sides are zero). Assume next that )~ r 0. Then

~(~a)

~< <~

II~xlli + II~a- ~xll2 ~< J~l ~ {llxlli + tfa- xlJ2} I~IPA(a) ( by taking inf over x).

414

Norm Uniqueness Theorems

On the other hand

IAIPA(a)- IAIPA(A-1Aa) < I~XIPI~X-11p/x(~Xa) - A(Aa). Combining the two inequalities we get the equality (vi). LEMMA. If A is an algebra and I" Ij ( J -

9.2.3. then

A(a) bllal+ ~I~~ (lal~ (+1 12)

A(a,b) ~<

1,2) sin.

(a, b E A).

PROOF.

<~ lab axll + laxl2 <~ [alllb- Xll Jr la121x[2 <~ (lall + fal2)(lb- Xll + Ix12) < (lala+ lal2)zX(b). -

-

Similarly A(ab)

xbll -~- Ixbl2 <~ la - Xlllbll + 1~121bl2

<.

la b

~<

A(a)(Ibll -]-lbt2),

-

-

completing the proof of the lemma. 9.2.4. Let A be the separating function for the (F) norms 9 Ij (J -- 1, 2). If A(a) -- 0, a is called a separating element for

the I ' l j - W r i t e -- {a e A: A(a) --- 0}. 9.2.5. PROPOSITION. (a) a E G iff there is a sequence (Xn) in A with Ix~lx -~ 0, l a - x.12 ~ 0. (b) G is a subspace of A which is closed with respect to both ]" Ij (J = 1,2). (c) If A is an algebra then G is a hi-ideal. PROOF. (a) Clear from the definition of A. (b) That G is a subspace follows from 9.2.2((iii)-(v)), result (a) above, and continuity of scalar multiplication with respect to either (F) norm. That G is closed can be easily shown, using

415

w 2. Rickart Separating Function

(c) This follows from (b) and 9.2.3. 9.2.6. PROPOSITION. Two (F) norms 1" lY ( J - 1 , 2 ) ing a L S X into (F) spaces are equivalent iff G - {0}.

mak-

PROOF. First suppose that ~ - {0}. Then, by 9.2.5.(a), for any sequence (xn) in A, a E A,

But this means that the identity map (A,I. I1) --~ (n, [" [2) is closed and so continuous, by the closed graph theorem (3.1.16) and consequently it is a homeomorphism by the open mapping theorem. Thus, G = {0} implies that ]. [1 "~ l" 12. On the other hand, if I" [1 "~ [" [2 and IXn]l ---+ 0, [ a - x,~12 --~ 0 then also ]x~]2 -~ 0 (since I" [1 " I" 12 )" T h u s , x n ----+ a , 0 under [. [2. By the uniqueness of limit property, a = 0, G = {0}. 9.2.7. Let [ - I j ( J - 1,2) be (F) norms on an algebra A such that (A,I. [ j ) a r e (F) algebras. Let I be a bi-ideal of A closed with respect to both (F) norm topologies and ~r 9A -~ A / I -

A # the canonical homomorphism .

Let I" [~ be the corresponding quotient (F) norms. Let G, G # be the sets of separating elements for I" [~, I" I~ respectively. Then we have 9.2.8. LEMMA. r ( G ) C ~ # .

Hence G # - {0}==~ G _ I.

PROOF. If a E G then [xnll -* O, t x n - h i 2 sequence ( x ~ ) i n

--~ 0 for some

a. Since Ir(x~)l~ ~< IXn]l--~ 0 and [ r ( x n ) -

~-(a)]2# -- ]n(xn - a)]#2 ~ Ixn - at2 --~ 0 it follows that r ( a ) E G #, whence ~(G)_C G #. If G # - { 0 } then r ( G ) {0}, so that GcI. 9.2.9. DEFINITION. An (F) algebra A = (A,I. l ) i s said to have n o r m uniqueness property if for any (F) norm I" I' on A making A into a (F) algebra we have I" I ' " I" I-Similarly, a p-Banach algebra A - - ( A , It" II) is said to have norm uniqueness

N o r m Uniqueness Theorems

416

property if for any p-norm I1" ll' making A into a p-Banach algebra we have I1" I1'~ I1" II. 9.2.10. THEOREM. Let A - (A, I " I1) be a t. spectrally GelSand functionally continuous s.s. (F) algebra. If I" 12 is an (F) norm on A such that (A, ]. 12) is a t. spectrally Gelfand (F) algebra then I" ]l " ]']2. PROOF. Let {M~} be the family of 1.12-closed hypermaximal ideals of A. Since A is s.s., by 7.2.12,

NM~

-

~r-~ _ ~

_ {0}.

Since (A, l" I1) is functionally continuous, each Ma is also I" I1# closed. Since A # - A / M ~ ~_ K it follows that l" t# "~ I" 12" So, by 9.2.6, ~ # - {0}, whence by 9.2.8, G _c M~. Therefore 6 c n

- {o},

so I"

" I"

9 . 2 . 1 1 . COROLLARY. (cf. Michael [20,p59]). Let A -

(A, P) be a commutative functionally continuous locally sm. p s e u d o - 7 algebra which is s.s.. If ~_ - {qj} be a family of locally sm. p s e u d o - s e m i n o r m s on A making it into a locally sm. pseudo-jr algebra, then Q ... P.

PROOF. Assume first that A is a complex algebra. Then by 7.2.21, A is t. spectrally Gelfand. The result for complex algebras now follows from 9.2.10. The result for real algebras A is obtained by considering the complexification A of A and applying the result for complex algebras, and deducing from it the result for A (using the fact that a pseudo-seminorm of A is the restriction of its extension to A). N

,v

w3.

Topological

Modules

9.3.1. D e f i n i t i o n . Let A be an algebra (over K) and X a TLS. X is called a topological A-module or a t. A-module if: (i) X is an A-module;

w 3. Topological Modules (ii) The m a p m # ' ( a , x ) ~ a x

(aCA,

417

xEX)

is continuous.

9 . 3 . 2 . LEMMA. Let A be a p - n o r m e d algebra and X a p - n o r m e d linear space which is an A-module. Then X is a t. A module iff there is a constant C > 0 such that tla~ll* ~< Cllal] Ilxtl* for all a E A, x E X

wher~ II" fl,

(,)

I1" II* are respectively the p - n o r m s of A , X .

PROOF. If (.) is satisfied then clearly the m a p m # is continuous and X is a t. A - m o d u l e . Conversely, suppose t h a t X is a t. A - m o d u l e . T h e n from the continuity of the m a p m # at (0,0) we get for a g i v e n e > 0, a > 0 such t h a t (1)

Ilaxll* ~<~ if Ilall, Ilxll* <~ ~. For any a r 0 in A and x =fi 0 in X, write !

a l _ 6p allall

T h e n r r a l l l - 6, which reduces to

_!

,,

!

Xl _

_!

a0 (llxll,) 0.

I I X l l l - (~ SO t h a t by (1), we get Ilalxlll* <~ c

Ilaxll ~ Cllall tlxll* (c-

c/~2).

(2)

T h e inequality (2) is trivially satisfied if a or x is zero. T h u s (2) holds for all a, x, completing the proof. 9 . 3 . 3 . DEFINITION. If A , X are as in 9.3.2 and condition ( . ) is satisfied then we call X a p-normed A - m o d u l e . If besides, A and X are complete with respect to their p- n o r m s we call X a p-Banach A-module. 9 . 3 . 4 . PROPOSITION. Let A (A, II" II) be a p - B a n a c h algebra (with II" II sm.) and I a closed regular I. ideal of A. Then the quotient module A # - A / I is canonically a cyclic p - B a n a c h module. Further, A # is irreducible iff I is maximal. PROOF. In view of 1.5.16 it is enough to prove t h a t A # is a p- B a n a c h A - m o d u l e . Set

IIx + 111~ -inf{l[x + tll't

E

I}.

418

N o r m Uniqueness Theorems

It is straightforward to check that I1" I1" is a p-norm on A # (11"II is called the canonical p - n o r m induced by II" II on A#). Further, A # is complete with respect to I1.11# (cf. proof of 3.4.14). Finally, we have

Ila(x + z)tl ~

=

Ilax + Ill e <~ inf{llax + t l l . t E I}

~<

inf{liax + atil: t E I} inf{llall IIx + tlI: t E I}

~< < 9.3.5.

~ e x-

LEMMA.

Ilall I1~+111 #.

Let A = (A,I I 9II) be a p - B a n a c h algebra

(x, II" II*) a p-Ba~ach cyclic A-,nodule, with g ~ ~ t o ~

xo. Then I = kerxo is closed in A and ag : a + I ~ t. i s o m o r p h i s m of A # - A / I onto X.

axo is a

PROOF. Since the map a ~ axo is continuous, I - k e r x 0 is closed. Also, by 1.5.14 (i), I is regular. Therefore, by 9.3.4, A # is a p - B a n a c h cyclic A-module and 9 is clearly an isomorphism. Further lie(a-t- I)11* = Iiax0[]* <~ C[[ai[ [[xol[*. If al C a + I then axo = alxo, so that IIr

I)11' = Ilax011* = [falxol[* < CllalI[ [[xoll*.

By taking the minimum over all al E a § I, we get

IIr

+ I)ll* ~ Clla + Ill~ltxol[ *.

It follows that 9 is continuous and consequently by the open mapping theorem t 9 is a t. isomorphism, as required. 9.3.6. PROPOSITION. Let A be a p - B a n a c h algebra, X a cyclic p - B a n a c h A - m o d u l e and Dc the set of all continuous A e n d o m o r p h i s m s of X. Then: (i) Dc is a p - n o r m e d algebra. (ii) D~ is also a division algebra if X is an irreducible module.

t The open mapping theorem is available since p-Banach spaces are (F) -spaces.

w4. Norm-uniqueness Theorem for Non-commutative Algebras 419

PROOF. Let x0 be a generator of X, I = kerx0 and (I) the t. isomorphism A # - A / I ~ X (see 9.3.5). For T E Dc write y -- r

axo - O(a + I).

Then r

+ I) - O - 1 T a x o - a O - l T x o - ay.

It follows that IIO-ITO( a + I ) [ [ * - I[ayl[* -

for any al with a l § 2 4 7 such al we get I1r162

[lalyI[ * ~< Ilal]] []YII*

By taking the infimum over all

a § I)II* ~< inf(llal[[ IiyiI*" a l } -

IIa -+- I]]#[]y[[*"

Thus, T # - (I)-ITcI) is a bounded linear operator of A #. Set I[TII# - I I T # ] I . Then ]]. ]]# is clearly a sm. p - n o r m on Dc and D~ is a normed algebra, proving (i). For proving (ii), consider the algebra D of all Aendomorphisms of X. When X is irreducible, by 1.5.19, D is a division algebra. Clearly, Dc is a subalgebra of D. Further, if T c Dr, T -r 0 then T -1 E D ( D being a division algebra). But, by the open mapping theorem applied to X we have T -1 E D~, proving (ii). 9 . 3 . 7 . COROLLARY. For an irreducible p - B a n a c h A - m o d u l e X , Dc is t. isomorphic to C if A is complex and t. isomorphic to ~ , C or H if A is real.

PROOF. This follows from 9.3.6, 6.5.6, 6.5.12.

w4.

Norm-uniqueness Non-commutative

Theorem for t Algebras

9.4.1 LEMMA (Bonsall-Duncan). Let A be a p - B a n a c h algebra and X an irreducible p - B a n a c h A-module. Then, for any t non-commutative - not necessarily commutative

420

N o r m Uniqueness Theorems

non-zero element xo of X , if M - - k e r x o and I a closed I. ideal with 1 ~ M then we can f i n d an ,7 > 0 such that

~Bizo ~ (IN-B1)xo where B1

denotes the closed unit ball of A.

PROOF. By 1.5.14, M is a maximal regular 1. ideal which by 9.3.5, is closed. Further, by the choice of I and maximality of M, we have" I + M - A. It follows that the map ro " a ~--~ a § M

(a E I)

of the p - B a n a c h space I to the p - B a n a c h space A # - A / M is surjective. By the open mapping theorem, r 0 ( I N B1) t is open, so t h a t we can find an ,7 > 0 such t h a t for every a + M with Ila§ ~< rl p we have a + M E ~ 0 ( I N B 1 ) . This means t h a t there is an element b c I N B 1 with b + M a + M , bxo - axo. If a ~ ~B] then ] ] a + M i ] ~< Ilall ~< ~P, so t h a t a x o - bxo with b c I ~ B1, proving the lemma. 9.4.2. THEOREM (Johnson). Let A be a p r i m i t i v e p - B a n a c h algebra and ~ a faithful irreducible representation of A on a p n o r m e d linear space X such that r ( a ) c B ( X ) for all a in A. T h e n the map

~" A -~ B ( X ) is continuous, where B ( X ) ear operators of X .

denotes the algebra of all bounded fin-

PROOF. If X is finite-dimensional so is B ( X ) and then also A (since ~ is 1 - 1). Thus, in this case ~, being a linear map between two finite-dimensional Hausdorff TLS's, is automatically continuous, tt Assume now that X is infinite-dimensional. Let rz denote the m a p a ~-~ ax - rz(a) (a E A, x E X). Write B1 denotes the open unit ball of A. ~[t See [5,14, Cor. 2]

4. N o r m - u n i q u e n e s s T h e o r e m for N o n - c o m m u t a t i v e Algebras 421

X o - { x E X " r. E B ( A , X ) } ,

where B ( A , X ) denotes the space of all b o u n d e d linear transform a t i o n s of A into X. Then X0 is a submodule of X - (X, lr), since

IIrbz(a)ll - Ilabxll - flr~(ab)J] ~ [[r~llllabll <~ Ilr~lll[a[[l[bll. t Since X is irreducible, X 0 - X or {0}. If X0 - X, and B1 denotes the closed unit ball of X, for x C B1 and a E X we have

IIr~(a)ll- Ilaxll- II~(a)~ll ~ ll~(a)llll~ll < II~(a)li. By Banach-Steinhauss theorem (see 3.2.16) there is a constant C : > 0 with ]]rzl I ~
(x E B1, a E A).

whence I[~(a)[I ~< C(llall), so t h a t by 3.5.5, ~r is continuous. To complete the proof it is enough to rule out the other alternative X0 = {0}. Suppose therefore t h a t X 0 - {0}. Then the definition of X0 implies t h a t for each x ~= 0 in X the operator rz is not bounded, so t h a t (by 3.5.2): the set {llaxll: IiaiI < 1} is not bounded. (1) Now by 9.3.7, Dc is a division algebra over K with dim De ~< 4. Since X is infinite-dimensional over K it is also infinite dimensional over De. It follows t h a t X contains an infinite sequence (x~) of D ~ - i n d e p e n d e n t vectors. By replacing xn if necessary by a suitable scalar multiple we may assume t h a t Ilxnll = 1 for all n. Set M,~ - k e r x n , Ln -

MIN...NM,_I (n- 2,3,.-.).

By density t h e o r e m (1.5.21) we can find an a E A such t h a t ax i -O

(l~<j~n-1),

a~. r 0 (~ ~> 2).

We have assumed that the norm II II of A is sm..

422

N o r m Uniqueness Theorems

Then a E Ln, a q{ Mn, so that Ln ~ Mn. Since by (1), unbounded, by 9.4.1

--Blxn is (2)

(L,~ N B1)2gn is unbounded. We now inductively choose a sequence (an) with an E Ln,

[la~ll ~< 2-".

(3)

and

ItanXnll

an-1)Xnll.

> rt + II(al + . . - - 4 -

(4)

To justify the inductive step, suppose we have chosen a l , - ' - , an-1. Then, by using(2), we can select a bn E Ln ["l B1 such that

lib.~ll

>

2n+1( n -4-]l(al + - "

§ an-1)Xnll).

(5)

Set an - 2 -(n+l)/pbn. Then Ila~ll- 2-(=+l)tlb=ll ~< 2 -(n+l) < 2 -n

( using b. c B1),

and using (5), ]lanXnll - 2-(n+l)llbnxntl > n -+-II(al + . . . + a,-~)x=ll.

Next write

co

c-Ea

co

,

co- E

k=l

k=n+l

These elements are defined since the series representing them converge absolutely (by virtue of the inequality in (3)) and hence also in A. Since ak E Mn (k > n) and M n is closed, we conclude that cn E M,~ and consequently, c n x n - O. It follows that we have CXn --

E

ak § Cn Xn

k=l

ak

Xn,

k:l

so that by inequality (4) we get

Ilcx~ll >~ tla.x~ll- II(ax +"" + a~-i)x~ll ~> n.

w 4. Norm-uniqueness Theorem for Non-commutative Algebras 423 This means that 7r(c) - lc is not a bounded operator contradicting the hypothesis on 7r. Therefore, X0 r {0}, completing the proof. 9.4.3. THEOREM (Johnson). Every primitive p-Banach algebra A has p - n o r m uniqueness property, t PROOF. Let A = (a,I I 9]11) and ]l" {]2 a complete p - n o r m on A such that (A, II" II=) is lso p-B n ch algebra. We h ve to show that ]l" [[2 ~ ]i" [[1. Since A is primitive there is a maximal regular 1. ideal M such that the left regular representation ~ # on A # - A / M is faithful. By 7.1.9, M is closed with respect to either of the norms I]" I[i ( J 1,2). Let [l" I[~ be the canonical p - n o r m on A # induced by [[. IIj. Then (A #, II" [ [ ~ ) a r e p-Banach spaces (see 3.1.22, 3.4.16). It follows that 7r is a faithful representation of the p - B a n a c h algebra A = (A, [I" II1)on the p - B a n a c h space

A # - (A #, []. 112#). By 9.4.2, r is continuous and consequently we have

II (a)ll

CIJaJJ1 (a C A)

for some C > 0. This implies that

ttTr(a)(x § M)l]#2

<<. llr(a)[[#2 iix + M]]#2 #

Cllallx I[

(1)

+Mli2.

Let u be the relative r. unity for M. Then

~r(a)(u + M ) -- a(u + M ) -- a + M, so that #

]]a -+- MII#2 -l[Tr(a)(u + M)II#2 <~ C[lall 1 [lu + M[[2 , where we have used the inequality (i) with x = u. If a + M = b+ M

(2),

then

[ie § Mli#2 -[[b+MIl#2

<~ C[[b[[ll[u + M[[#2 .

(3)

t Johnson obtained the theorem for Banach algebras (i.e. for the case p--l).

424

Norm Uniqueness Theorems

Taking the infimum in (3) over all b satisfying (2) we obtain

Ila + MII2~ ~ ella + Mtll~ Ilu + MII #2"

(4)

The inequality (4) together with the open mapping theorem implies that II" Ill# and I1" 112 # are equivalent. Suppose now that a in A is a separating element with respect to the norms II" [j ( J - 1 , 2 ) , so that there is a sequence (Xn)in A with

tic.ill -~ o, I l a - ~-II~ -~ o Since [Ix + MI] ~ ~< I1~11 it follows that a + M is a separating element of A # with respect to I1" [[3#. Since ll" [[1# "~ I1" 112# we must have a+M-M, a E M , GC_ M, where 6 is the separating set for the I1" llj. Since by 9.2.5(a), G is a bi-ideal we have, by 1.5.3,

G c_ ( M ' A ) -

{0}

( since ~# is faithful ).

So, by 9.2.6, II" 111 "~ ]]" ]]2, as we wished to show. 9.4.4. THEOREM. Every s.s. (real or complex) p-Banach algebra A has p-norm uniqueness property. PROOF. If P is a primitive ideal of A then A / P is a primitive p-Banach algebra and so has, by 9.4.3, p-norm uniqueness property. Let G be the separating ideal for the p-norms I1" IIj on A, where A - (A,[[-][1) and II" [12 is a second p-norm on A making it again into a p-Banach algebra. Then we have S __ P (the proof is similar to that above showing $ _c M ) . It follows that S c_ ('~ P - v / A - {0} ( since A is s.s. ) whence A has p-norm uniqueness property. 9.4.5. R e m a r k . Theorem 9.4.4 admits a strengthening to the following form: If [[.]]j ( j - 1,2) are pj-norms ( 0 < p j ~< 1) on a s.s. algebra A such that (A, [[. [[j) are pj-Banach algebras, then ll" II1 ~ ll-II~

w4. Norm-uniqueness Theorem for Non-commutative Algebras 425

To obtain the above result we first remark that we can assume p_l

that Pl ~< P2 and t1" Ili are sm. If we set I1" I1~ I1" I1~ , then by 3.2.9. II" II~ is a p l - n o r m which from its form is clearly (topologically) equivalent to ]l" 112. Further it is sm." *

,

Pl

P_X_

,

__

P2

,

Ilxylt~ - (llxyWJ~) ~ <<. (llxll211yll2) ~ -IlxllllfyJJ~.

It follows that (A, II" I]~) is a pl Banach algebra. By 9.4.4, I1"111 FJ" lJ~ ~nd so IJ" IJ1 ~ I1" I1~ (sin~e I1" I1~ ~ Jl" 112) ~s claimed. 9.4.6. PROPOSITION. Let A , A * be sm. (F) algebras and 99 9 A --~ A* an epimorphism. If A* is s.s. and has normuniqueness property then 99 is continuous. PROOF. Write I - ker99. By 3.6.23(a), 7.1.17, I is closed and therefore A # - A / I is a sm. (F) algebra with its (F) norm I" I# satisfying 9I# ~< I" I (see 3.1.22, 3.4.15). Since A # is isomorphic to A* we can shift the ( F ) - n o r m I" I# to A*. Then 99" A --, A* - (A*, [. ]g) is continuous. By hypothesis A* has norm-uniqueness property. It follows that I" I* (of A*) --~ I" 1#, whence ~ " A ~ A* - (A*, I" I*) is continuous. 9.4.7. COROLLARY. If A , A * are p - B a n a c h algebras, A* s.s. and 99" A--+ A* an epimorphism then 99 is continuous. PROOF. By 9.4.4, A* has norm-uniqueness property and so the corollary follows from 9.4.6. 9.4.8. COROLLARY. Any automorphism 99 of a s.s. sm. (F) algebra A with norm-uniqueness property is a t. automorphism. In particular, every automorphism of a s.s. p - B a n a c h algebra is a t. automorphism. PROOF. By 9.4.6 applied to p , 99-1 we conclude that they are continuous, so that ~ is a t. automorphism. The second statement follows from the first in view of 9.4.4.

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APPENDIX

LEMMA A. The character space A = A(WP) is h o m e o m o r phic to the unit circle S 1 in C. PROOF. We have already seen in 7.5.16 that the characters X of W P are precisely the maps Xt~ : f ~ f ( t l )

(fEWP,tleR).

Since Xt~(e it) - e it~ determines Xt~ it follows that Xt~ - Xt2 iff eit ~ = eit2 . Thus Xtl ~ eit' -- Xt, (eit)

is a bijection of A o n S 1. Moreover it is continuous (by definition of the (weak)topology of A ) . Since A is compact (by 7.3.12) and S 1 is Hausdorff, the map X ~-+ X(eit) is a homeomorphism. L EMMA B. Let F, C be respectively a closed set and a compact set in an such that F N C = O. Then there exists a C ~ f u n c t i o n g on ~'~ such that g = O on F and g = l on C . PROOF. See [H, p.3]. LEMMA C. Let f = f ( t ) be a C ~ 2 r - p e r i o d i c complex f u n c t i o n on ~ and f(J) denote the j th derivative of f . If

](n), f(i)(n) (n c Z)

th Fo=,i r

of f

and f(J) respectively then we have, for n ~ O,

< I l-Jllf(J)llL , where I I f U ) I I L 1 - ~

IfU)(t)ldt.

PROOF. See [K, p.24, 4.4]. THEOREM. W p is completely regular. PROOF. Since by Lemma A, A can be identified with S 1, for proving the theorem it is enough (in view of 8.14.7) to show that given a closed subset F of S 1 and a point e it~ E S I \ F , there is a fCWPI~P such that f - 0 on F and f ( e it~ # O. Now

Appendix

428

by L e m m a B we can find a C ~ function g - g(x, y) on ~2 with g-0 on F and g(e i t ~ 1 6 2 If we set

f ( t ) - 9(e 't) - g(cos t, sin t) then f is a 2 r - p e r i o d i c C ~ function on R with f ( t ) = 0 if 1 Then e it E F and f(to) # O. Choose integer j such t h a t j > ~. by L e m m a C, o~

I/(~)1 ~ <. r](o)l ~ + 2 ~ nC g

where C proof.

rt=

Ilf(J)lt~l. It follows that

C

i~lj ~ 1

f E W p, completing the

References [HI H. HELGASON" Differential Geometry Spaces, Academic Press, New York, 1962.

and

Symmetric

[K] Y. KATZNELSON" An Introduction to Harmonic Analysis, Douer Publications, New York, 2nd edition, 1976.

TOPOLOGICAL ALGEBRA TYPE CONNECTION CHART

O BanachAlgebra

Locallysm..3 Algebra

\ O L~_ocallysm.convexAlgebra / Locallylt"sm.p-2iI / 0 Algebral ~ Locallyconvex

p-Banach ~ ~

i~

Algebra

Locallysm ~iCgObVaeX ~ sm.(.~)0 Algebra ] ~ X~ [ 0 .Q~algebra ~ 1

0

'Nv

" /

Locally p-convexAlgebra

/

~ o C A l g

Topological Algebra

The arrow indicates that the algebra at the tail of the arrow has the properties of the algebra at its tip.

This Page Intentionally Left Blank

BIBLIOGRAPHY

Books Referred [1] L.V. AHLFORS: Complex Analysis, McGraw-Hill, International Student Edition, 1979. [2] T.M. APOSTOL: Mathematical Analysis, Addision Wesley, World Student Edition, 1963. [3] S. BOCHNER and W.T. MARTIN: Several Complex Variables, Princeton Univ. Press, Princeton, 1948. [4] F.F. BONSALL and J. DUNCAN: Complete Normed Algebras, Springer Verlag, New York, 1973. [5] N. BOURBAKI: Topological Vector Spaces, Chapters 1-5, Springer Verlag, New York, 1987. [6] N. BOURBAKI: Theories Spectrales, Chapters 1-2, Hermann, Paris, 1967. [7] G. CHOQUET: Lectures on Analysis, Vol. 1, W.A. Benjamen, New York, 1969. [8] J.B. CONWAY: Functions of One Complex Variable, Springer International Student Edition, Narosa, New Delhi, 2nd revised edit., 1980. [9] J. DUGUNDJI: Topology, Printice-Hall of India, New Delhi, 1975. [10] I.M. GELFAND, D.A. RAIKOV and G.E. Shilov: Commutative Normed Rings, Chelsea, New York, 1964. [11] L. GILLMAN and M. GERRISON: Rings of Continuous Functions, Springer Verlag, New York, 1976. [12] A. G UICHARDT: Special Topics in Topological Algebras, Gordon and Breach, New York, 1968. [13] N. JACOBSON: Lectures in Abstract Algebras V.I., EastWest Student Edition, Affiliated East-West Press, New Delhi, 1964. [14] E. HILLE and R.S. PHILLIPS" Functional analysis and semigroups, Amer. Math. Soc. Coll. Publ. 31, Providence, R.I., 1957. [15] R.V. KADISON and J.R. RINGROSE: Fundamentals of the Theory of Operator Algebras, V.I., Academic Press, New York, 1983.

Bibliography

432

[16] J.L. KELLEY: General Topology, Van Nostrand, Princeton, 1955. [17] J.L. KELLEY and I. NAMIOKA: Linear Topological Spaces, East-West Student Edition, Affiliated East-West Press, New Delhi, 1968. . .

[18] G. KOTHE" Topologische Lineare Raiime, Springer Verlag, Berlin, 1960. [19] L.H. LOOMIS: An Introduction to Abstract Harmonic Analysis, Van Nostrand, New York, 1953. [20] E.A. MICHAEL: Locally Multiplicatively-Convex Topological Algebras, Memoirs Amer. Math. Soc., No. 11, Providence, R.I., 1952. [21] D. MONTGOMERY and L. ZIPPIN: Topological Transformation Groups, Interscience Publishers, New York, 1955. [22] M.A. NEUMARK: Normierten Algebren, VEB Deutscher Verlag Der Wissenshaftin, Berlin, 1959. [23] G.E. RICKART: General Theory of Banach Algebras, Van Nostrand, 1960, reprinted with corrections by R.E. Kreiger Publishing Co., New York, 1974. [24] W. RUDIN" Functional Analysis, Tata McGraw-Hill, New Delhi, 1974. [25] W. RUDIN" Real and Complex Analysis, Tata McGraw-Hill, Bombay- New Delhi, 1966. [26] G.F. SIMMONS: Topology and Modern Analysis, McGrawHill, New York, 1963. [27] A.M. SINCLAIR: Automatic Continuity of Linear Operators, L.N. Ser. 21, London Math. Soc., CUP, 1976. [28] A.E. TAYLOR and D.C. LAY: Introduction to Functional Analysis, John Wiley, New York, 2nd edit., 1980. [29] L. WAELBROECK: Topological Vector Spaces and Algebras, LNM No. 230, Springer Verlag, New York, 1971. [30] J. WERMER: Banach Algebras and Several complex Variables, GTM. No. 35, Springer Verlag, New York, 1976. [31] W. ZELAZKO" Selected Topics in Topological Algebras, LN Ser.No.31, Mathematisk Institut, Aarhus Universitet, 1971.

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Papers Referred [1 ~] R. ARENS: The space L ~ and convex topological rings, Bull. Amer. Math. Soc. 52 (1946)931- 935. [2 ~] R. ARENS: Linear topological division algebras, Amer. Math. Soc. 52 (1974)623-630.

Bull.

[3 ~] R. ARENS and A.P. CALDRON: Analytic functions of several Banach algebra elements, Ann. of Math (2)62 (1955) 204- 216. [4 ~] G. BERGMAN" A ring primitive on the right but not on the left, Proc. Amer. Math. Soc. 15 (1964)473- 475. [5~] F.T. BIRTELL" Singly-generated Liouville F algebras, Michigen Math. J. 11 (1964) 89-94. [6 ~] V.G. IYER: On the spaces of integral functions I, J. Ind. Math. Soc. 12 (1948) 13-30. [7 ~] I.M. GELFAND, D.A. RAIKOV and G.E. SILOV: Commutative normed rings, Uspehi Matem Nauk I (1946) 48-146; Amer. Math. Soc. Transl. (2)5 (1957) 115-220. [8'] I. KAPLANSKY: Normed algebras, Duke Math.

J. 16

(1949) 399-41s. [9 ~] M. LANDSBERG" Lineare topologische Rafime, die nicht lokalkonvercen sind, Math. Z. 65 (1956) 104-112. [10'] B. MITIAGIN, S. ROLEWICZ, W. ZELAZKO" Entire functions in B0-algebras, Studia Math. 21 (1962) 291 - 306. [11 ~] M. NAGUMO: Einige alalytische Untersuchungen in Linearen metrischen Ringen, Jap. J. Math. 13 (1936) 61-80.

[12 ~ ] S.R. ROLEWICZ: On a certain class of linear metric spaces, Bull. Acad. Polon. Sci., cl.III 5 (1957)471-473. [13 ~] L. WAELBROECK" Le calcul symbolique dans les algebras commutative, J. de Math. P. et App. 33 (1954) 147-186. [14 ~] J.H. WILLIAMSON" On topologizing the field C(t) " Proc. Amer. Math. Soc. 5 (1954)729-734.

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INDEX The bold face figures after the subsection numbers below indicate the corresponding pages in which these numbers occur. Absolutely convergent series 3.1.23,109 Absolutely p-convex 4.1.1, 174 Absorbing set 2.1.15, 78 Almost submultiplicative (a.sm.). function 3.3.3, 118 A-module 1.5.1, 36 Ample algebra 4.7.1,213 Annihilator (left, right) 1.2.1, 12 Arens algebra 4.6.8 (iv), 211 Arens-Calderon trick 8.5.18, 394 Bi-ideal 1.2.1, 12 Binomial theorem 1.1.7, 3 Bi-primitive ideal 1.5.4, 38 Bi-singular 3.7.1,158 Boolean algebra 8.6.7, 407 Bounded or n.(=norm) bounded linear transformation 3.5.1,140 Bound of a linear transformation 3.5.1, 140 C algebra 3.6.4, 148 CI algebra 3.6.20, 3.6.21, 153 CQ algebra 3.6.20, 3.6.21,153 Canonical character 1.4.10, 36 Canonically p-normed function algebra 8.1.1, 353 Cauchy net or C-net 2.3.1, 91 Cauchy's estimates 5.4.18, 252 Cauchy's integral theorem 5.4.16, 249 Character 1.3.1, 22 Circle operation 1.1.13, 6 Closed graph theorem 3.1.16, 105 Closed map 3.1.16, 105 Commutant 1.1.7, 3 Complete p-topology 3.1.4, 101 Complete quarter-norm 3.1.13, 105

436

Index

Complete topological linear space (complete TLS) 2.3.1, 91 Completely regular algeba 8.4.1,377 Completely regular family of functions 8.4.8,381 Complex structure 1.6.1, 43 Contour 5.4.15, 248 Contour surrounding a subset 5.4.15,248 Contraction 4.3.14, 194 Contraction mapping principle 4.3.15, 194 Convergence of series and generalized series 2.1.32, 83 Cyclic module 1.5.13, 41 Derivation 7.6.5, 345 P-independent 1.5.20, 43 Directed set 2.1.1, 73 Discrete 2.1.11, 78 Distinguished character 1.4.10, 36 Distributive lattice 8.6.7,407 Entire functions operating in a topological algebra (TA) 5.5.1,253 Epimorphism 1.1.10, 4 Equivalence of subadditive (sad.) functionals 3.1.4, 101 Essentially bounded net 2.3.5, 93 Essentially nilpotent element 1.2.25, 22 Examples of: Completely regular algebras 8.4.2, 378 (F) algebras 3.3.14, 123 Fixed ideals 8.1.17, 8.1.18, 8.1.23, 354~ 360 Function algebras 8.1.2,353 Hypermaximal ideals 1.3.13, 25 Metrizable locally pseudoconvex algebras 4.6.8,209 Monogenic topological algebras 7.3.22,321 Primary ideals 7.1.20, 301 p-Banach algebras 3.4.6, 133 p-seminormed algebras 3.4.6, 133 Shilov algebra 8.6.13,409 Shilov boundary 8.2.16, 367 Exponential function 5.2.1,227

Index

Extended Extended Extended Extended

quasi-resolvent 1.8.1, 61 quasi-spectrum 1.8.1, 61 spectrum 1.8.1, 61 spectral radius 1.8.10, 64

(F) algebra 3.3.13, 123 (F) norm 3.1.14, 105 Faithful quarter-norm 3.1.14, 105 Faithful representation 1.5.1, 36 Faithful sad.functional 3.1.1,100 Fixed ideal 8.1.17, 357 Formally real algebra 1.6.17, 50 (F) space 3.1.14, 105 Function algebra 8.1.1,353 Functionally continuous 2.2.19, 90 Gelfand algebra 7.2.1,303 Gelfand-Beurling (GB) algebra 7.4.1,324 Gelfand lemma 3.3.6, 119 Gelfand space 7.3.1,311 Gelfand transform 7.3.4, 312 Generalized sum 2.1.32, 83 Hilbert relation 6.2.2, 264 Homogenity index of a pseudo-seminorm 3.2.1,110 Homomorphism 1.1.10, 4 Hull-kernel topology (hk-topology) 8.3.5, 371 Hypermaximal ideal 1.3.6, 24 Hypernormal TA 7.1.4, 297 Hyper-radical 1.3.7, 24 Hyper semi-simple (h.s.s.) 1.3.7, 24 Hyponormal TA 7.1.4, 297 I algebra 3.6.7, 149 Ideal (1., r.) 1.2.1, 12 Index of a point 5.4.15,248 Inverse (1. r.) 1.1.3, 1 Inverse-closed subalgebra 1.7.27, 61 Irreducible module 1.5.2, 1.5.13, 37~ 41

437

438

lndez

Isomorphism 1.1.10, 4 Jacobson density theorem 1.5.21, 43 Joint spectrum 8.5.7, 389 Ker p. 3.1.5,102 Ker P. 3.1.20, 106 Lattice 8.6.7, 407 Leibniz rule 7.6.8,346 Lemma (Bonsall-Duncan) 9.4.1,419 Linear functional 1.3.1, 23 Liouville algebra 1.8.13, 65 Locally bounded algebra 4.2.4 (f.n.), 187 Locally bounded space 2.1.24, 81 Locally pseudo-convex algebra 4.4.1,195 Locally pseudo-convex space 4.3.1, 189 Locally p-admissible holomorphic 5.4.12, 245 Locally sm. pseudo-pre-Frechet (~) algebra 4.6.3, 208 Michael algebra 4.5.4, 204 Minimal pre-boundary 8.2.3, 362 Monogenic 7.3.21,320 Monomorphism 1.1.10, 4 Net 2.1.1, 73 Normal family of functions 8.4.8, 381 Normal TA 7.1.4, 297 Normalized p-seminorm 3.4.13, 136 Norm bounded (n bounded) 3.2.12, 3.5.1,114~ 140 Nucleus 2.1.9, 77 Operator topology 2.2.4, 85 Open mapping theorem 3.1.15, 105 P-bounded 4.3.9, 192 p-bounded 3.2.12, 114 P-complete 3.1.20, 106 Polynomially convex (p. convex) 8.5.1,388 Polynomially convex hull 8.5.1,388

Index

Poset 2.1.1, 73 Power series operating in a TA 5.5.1,253 Pre-boundary 8.2.1,361 Pre- (F) algebra 3.3.13, 123 Primary ideal 7.1.18,301 Prime ideal 1.5.9 (f.n.), 39 (1., r.) Primitive ideal 1.5.4, 38 Principal class 2.3.9, 95 Principal component 5.2.7, 230 Projective limit 4.5.1,201 Pseudo-convex algebra 4.4.1,195 Pseudo-convex space 4.3.1,189 Pseudo Frechet (~) algebra 4.26.2, 208 Pseudo-pre- ~ algebra 4.6.2, 208 Pseudo-resolvent function 6.3.1,275 Pseudo-resolvent set 6.3.1,275 Quarter-norm 3.1.8, 103 Quarter-normed algebra 3.3.1,116 Quasi-inverse, Quasi-invertible 1.1.16, 6~7 Quasi-nilpotent (q. nilpotent) 1.7.13, 56 Quasi-resolvent set 1.7.5, 54 Quasi semi-simple (q.s.s) 1.7.17, 57 Quasi-spectrum 1.7.5, 54 Quasi-square root (q.sq.r.) 5.3.1, 235 Quasi-unital 8.3.15, 375 Radical (Jacobson) 1.2.25, 22 Radical algebra 1.2.27, 22 Radius of convergence 5.1.6, 223 Real character 1.6.14, 49 Regular ideal (1. r.) 1.2.6, 14 Regular bi-ideal 1.2.6, 14 Regular representation 1.5.1, 36 Relative unity (1., r.) 1.2.6, 14 Relative bi-unity 1.2.6, 14 Resolvent set 1.7.1, 52 Rickart separating function 9.2.1,412

439

440

Index

p-admissible function 5.4.7, 241 p-admissible holomorphic 5.4.11,245 p-Banach algebra 3.4.4, 132 p-Banach A-module 9.3.3, 417 p-Banach space 3.2.6, 111 p-convex linear combination 4.1.1, 174 p-guage 4.1.8,178 p-modulus homogenity condition 3.2.1,110 p-normed LS 3.2.6, 111 p-seminormed LS 3.2.6, 111 Saturated closure 4.3.7, 191 Saturated family of pseudo-seminorms 4.3.7, 191 Self-conjugate (subset, ideal) 1.6.4, 1.6.12, 44, 46 Semi-metric space 3.1.2 (f.n.), 100 Semi-simple (s.s.) 1.2.27, 22 Semi-topological group 2.2.6 (f.n.), 86 Separating element 9.2.4, 414 Separating function (Rickart) 9.2.1,412 Separating, strongly separating, family of functions 8.1.12, 356 Shilov algebra 8.6.12,409 Shilov boundary 8.2.3, 362 Simple module 1.5.2 (f.n.), 37 Simultaneous spectrum 8.5.7, 389 Singular (1., r.) 3.7.1, 158 Spectrally connected 8.6.14, 409 Spectrally Gelfand 7.2.5,304 Spectral radius 1.8.10, 64 Spectrum of an algebra 7.3.1,311 Spectrum of an element 1.7.1, 52 Strictly real algebra 1.9.1, 67 Strongly analytic function 5.1.3,222 Strong structure space 8.3.9, 372 Structure space 8.3.9, 372 Submultiplicative (sm.) function 3.3.3,118 sm. pseudo-convex algebra 4.4.1,195 Subnet 2.1.1, 73

Index

Support of a function 8.4.24 (f.n.), 386 Symmetric subset of a group 2.1.3, 74 Symmetric subset of C 1.8.3 (f.n.), 62 T h e o r e m s of: Arens (Proposition) 3.3.2, 116 Arens-Banach 3.6.16, 152 Arens-Shilov (Cor.) 6.5.20, 290 Banach-Steinhauss 3.2.16, 115 Beurling-Gelfand (spectral radius formula) 6.2.11,271 Beurling-Gelfand-Zelazko 7.4.6, 326 Birkhoff-Kakutani 2.1.7, 76 Bonsall-Duncan 3.5.9, 144 Gelfand (norm equivalence theorem) 9.1.4, 412 Gelfand-Kolmogorov-Stone-Banach 8.1.22,359 Gelfand-Mazur 6.5.5, 285 Gleason-Kahane-Zelazko 5.2.15, 233 Jacobson (density theorem) 1.5.21, 43 Johnson 9.4.2, 9.4.3,420~ 423 Kaplansky 1.9.15, 71 La Page-Hirschfield-Zelazko 7.4.15, 331 Michael 4.5.3, 7.4.8,202~ 327 Mitjagin-Rolewicz- Zelazko (Cor) 5.5.11,261 Nagumo 5.2.14,232 Oka (extension theorem) 8.5.19, 396 Rickart 4.1.16, 183 Shilov (idempotent theorem) 8.6.4, 404 Shilov-Arens-Calderon 8.5.22,399 Singer-Wermer 7.6.20, 351 Turpin 6.6.5,294

441

442

Index

Tychnoff 2.1.12, 78 Wiener-Levi-Zelazko 7.5.16, (b) 342 Wiener-Zelazko 7.5.16, (a) 342 Yood (Proposition) 7.4.20, 333 Zelazko 5.5.10, 259 Topological Topological Topological Topological Topological Topological Topological Topological Topological Topological

A-module 9.3.1,416 group (TG) 2.1.3, 74 linear space (TLS) 2.1.9, 77 integral domain (TID) 6.3.13, 281 nilpotent (t. nilpotent) 4.8.8,220 semigroup (TSG) 2.2.5, (f.n.) 86 spectral radius 4.8.1,216 zero divisor (1., r., bi-) 3.7.3, 158 zero divisor (symmetric (s.t.z.d.)) 3.7.11,162 zero divisor (generalized (g.t.z.d.))3.7.38,172

Universal mapping properly 4.5.1,201 Weakly analytic function 5.1.3, 222 Weakly differentiable function 5.1.1,222 Weakly holomorphic function 5.1.3, 223 Weak topology 7.3.1,311 Well-behaved family of pseudo-seminorms 4.4.9, 198 Wiener property 7.2.10, 306 Williamson's algebra 3.6.33,157

List of Special S y m b o l s The numbers after the symbols indicate the subsections in which they are introduced ~/A, 1.2.24-25, ~/A, 1.3.7,

~r-~, 2.218,

ch

~/A, 1.5.11, A qn, 1.7.13 C, 1.6.1 A, 1.3.1, Ac 2.2.18, OAS, 8.23 E(x), 5.2.1; E~(A), 5.2.10; Eq(x), 5.2.6 /~

521; ~ 9

o

~

5 2 1 ; ~, 5 2 1 .

.

.

.

Gs, 3.6.16

Gi, 1.1.4; GI-, 3.7.1; G[, 3.7.1

Gq,

l 1.1.17; Gq, 3.7.1;

r Gq,

3.7.1

Gi~, 5.2.7; GqO, 5.2.7 K, 1.6.1

la, 2.2.2; l~ 3.6.1 vp(x), 3.3.5; u(x) 3.3.7 7rA, 8.3.1; ~A, 8.3.1 ~, 1.6.1 o 3.61 ra, 2.2.2; ra, r(x), 1.8.10; rl(x), 6.6.1; r2(x), 6.6.1; r3(x), 6.6.1 p(x), 1.7.1; p'(x), 1.7.5; pP(x), 6.3.1 1.7.5

S t 371" S ~ 3.7.1- S bi 3 7 1 9

.

~

~

~

~

9

.

x~, 6.2.1; x~, 6.3.1 5 tt 37.3" 5 ~t 3 7 3 ; 5 bit 37.3 9

a ~ b0-

~

~

9

.

~

9

a commutes with b" a b - ba, 1.1.7

empty set

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List of Special A b b r e v i a t i o n s GB (algebra)

Gelfand Beurling

LS

linear space

STG

semi-topological group

TA

topological algebra

TG

topological group

TID

topological integral domain

TLS

topological linear space

TSG

topological group

1. ideal

left ideal

r. ideal

right ideal

bi-ideal

both 1.r. ideal

q.i

quasi-inverse

1.q.i.

left quasi-inverse

r.q.i

right quasi-inverse

q.

quasi in q. invertible

1.q.

left quasi in 1.q. invertible

r.q.

right quasi in r.q. invertible

q. nilpotent

quasi-nilpotent

q.sq.r

quasi-square root

s.q.r.

square root

ext.

extended in ext. q. nilpotent

t.

topological in t. automorphism, t. isomorphism

t.z.d.

topological zero divisor

1.t.z.d.

left topological zero divisor

r.t.z.d.

right topological zero divisor

z.d.

zero divisor

1.z .d.

left zero divisor,

r.z.d.

right zero divisor

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