An Introduction to Abstract Harmonic Analysis

Proof. If
d~J.tp

J

x dp. where p. is a finite positive Baire meas

ure on ;m:, then


=

12 dp. ~ 0,

Jx* dp.

=

(Jx dp.)- =
2 = IJx dp.l 2 ~· I
(JI x 12 dp.) (Jt dp.)

=

II JJ.II
that is,

12

~

k
. • • ~ ki+·. ·+2-n
ki+···+2...,.11
(xx*) 2n

W. . .

98

SOME SPECIAL BANACH ALGEBRAS

Letting n ---+ oo and taking the square root we have I rp(x) I ~ kll xI leo· Since A is dense in e(~), the bounded linear functional I defined on A by I"'(x) = rp(x) can be extended in a unique way toe(~). If/ E: e(~) and/~ 0, then/~ can be uniformly approximated by functions x E: A and hence f can be uniformly approximated by functions I x 12 • Since I"'(l x 12 ) = rp(xx*) ~ 0 and I"'(l 12 ) approximates I"'(/), it follows that I"'(/) ~ 0. That is, I"' is a bounded integral. If JL,p is the related measure, we

x

have the desired identity rp(x) = Jx dJL,p for all x E: A.

26]. The setting for the Plancherel theorem will be a semisimple, self-adjoint commutative Banach algebra A, and a fixed positive functional rp defined on a dense ideal Ao c A. An element p E: Ao will be said to be positive definite if the functional (JP defined on A by 6p(x) = rp(px) is positive and extendable. Then by the Bochner theorem there is a unique bounded integral IP on e(~) (finite positive Baire measure fJ.p on ~) such that

rp(px) = Ip(x) = Jx dp.p. The set of positive definite elements is clearly closed under addition and under multiplication by positive scalars, and we now observe that it contains every element of the form xx*, x E: Ao. In fact, we can see directly that 6.,.,. is not only positive but can also be extended so as to remain positive, for if 6.,.,. (y + ~e) = rp(xx*y + ~xx*), then (J.,.,.((y + ~e)(y + ~e)*) = rp((xy + ~x)(xy + ~x)*) ~ 0. If p and q are positive definite, then I 11 (xp) = rp(pqx) = Ip(xq) for every x E: A and therefore Iq(hp) = Ip(hq) for every h E: e(~). Let S 1 be the support of/, i.e., the closure of the set where f ~ 0. If pq is bounded away from 0 on s,, then h = f/pq E: e(~) and Iq{j jq) = Ip(//p). We now define the functional I on L(~) (the set of functions in e(~) having compact support) by I(/) = IP(f /p) where pis any positive definite element such that p is bounded away from 0 on Stand hence such thatf/p E: L. Such is compact iff E: L and by the Reinea p always exists, for Borel theorem we can find p of the form x 1x 1 * + · · · + XnXn * such that p ~ 0 and p > 0 on S 1 • We saw above that I(/) is

s,

SOME SPECIAL BANACH ALGEBRAS

99

independent of the particular p taken, and using the p just constructed we see that I(/) ~ 0 iff ~ 0. Thus I is an integral. The original identity Ip(hq) = Iq(hp) shows that the integral Ip vanishes on the closed set where p = 0, and this, together with the identity I(/) = Ip(f/p) for all/ which vanish on this closed set, implies that I(gp) = Ip(g) for all g. Therefore I(p) = Ip(t) = II Ip 11. Finally, rp(pq*) = Ip('q) = I(pq) for all positive definite elements p and q, and p - p is therefore a unitary mapping of the subspace of H., generated by positive definite elements onto a subspace of L 2 (I). We have proved the following theorem.

Theorem. Let d be a semi-simple, self-adjoint, commutatiue Banach algebra and let rp be a positiue functional defined on a dense ideal do c d. Then there is a unique Baire measure p. on ffil such that

p E:

V(p.) and rp(px)

= Jxp dp. wheneuer p is positiue definite

with respect to rp. The mapping p - p when confined to the subspace of H., generated by positiue definite elements is therefore a unitary mapping of this subspace onto a subspace of L 2 (p.), and is extendable to the whole of H., if it is known that d 0 2 is Hrp-dense in do. 26K. The last two theorems have assumed, needlessly, that d is a Banach algebra. A more general investigation would start with any complex algebra d having an involution and a positive functional rp on d. The Schwarz inequality for rp implies that rp(xaa*x*) = 0 whenever rp(xx*) = 0 so that the set of x such that rp(xx*) = 0 is a right ideal I, and right multiplication by a becomes a linear operator Ua on the quotient space d/I. The condition that Ua be bounded with respect to the rp scalar product is clearly that there exist a constant ka such that rp(xaa*x*) ~ karp(xx*) for every x E: d, and in this case Ua can be uniquely extended to a bounded operator on the Hilbert space H., which is the completion of d /I with respect to the rp norm. If Ua is bounded for every a, then rp is said to be unitary; the mapping a - U a is then easily seen to be a *-representation of d (see [20]). This is the only new concept needed in the commutative theory to formulate more abstract Bochner and Plancherel thea-

100

SOME SPECIAL BANACH ALGEBRAS

rems. The passage to the space .cl of complex-valued homomorphisms of A is now accomplished by noting that since the mapping a ~ Ua is a *-representation of A, the maximal ideal space gn;rp of the algebra of operators Ua is thus identifiable with a closed subset of .cl. The Bochner and Plancherel theorems are now confined to this subset gn;.,, which consists only of self-adjoint ideals and therefore renders unnecessary the assumption, made above, that x*" = x"- on the whole of .cl. Even when A is a Banach algebra this last point is of some interest; it is easy to prove by characteristic juggling starting with the Schwarz inequality that a continuous functional is automatically unitary, and the hypothesis that x*"' = x~>- can therefore be dropped from both of our theorems. § 27.

H*-ALGEBRAS

The preceding sections of this chapter will find their applications mainly in the theory of locally compact Abelian groups. In this section we give the very special analysis which is possible in the H*-algebras of Ambrose [1], and which includes the theory of the L 2 group algebra of a compact group as a special case. An H*-algebra is a Banach algebra H which is also a Hilbert space under the same norm, and which has an inuolution satisfying (1)-(4) of § 26 and the crucial connecting property

(xy, z) = (y, x*z). It is also assumed that II x* II= II x II and that x*x ~ 0 if x ~ 0. It follows that involution is conjugate unitary ((x*, y*) = (y, x)) and hence that (xy, z) = (x, zy*). The structure which can be determined for such an algebra is as follows. H can be expressed in a unique way as the direct sum of its (mutually orthogonal) minimal closed two-sided ideals. A minimal closed two-sided ideal can be expressed (but not uniquely) as a direct sum of orthogonal minimal left (or right) ideals, each of which contains a generating idempotent. It follows that a minimal closed two-sided ideal is isomorphic to a full matrix algebra (possibly infinite dimensional) over the complex

SOME SPECIAL BANACH ALGEBRAS

101

number field, and operates as such an algebra under left multiplication on any of its minimal left ideals. 27A. We begin with some lemmas about idempotents and left ideals.

Lemma 1. Let x be a self-adjoint element of H whose norm as a left multiplication operator is 1. Then the sequence x 2 n converges to a non-zero self-adjoint idempotent. Proof. Let Illy Ill be the operator norm of y: Illy Ill = lub.ll yz IIIII z II· Since II yz II ~ II Y II II z II, we have IllY Ill ~ IIY II· By hypothesis xis a self-adjoint element such that Ill x Ill = 1. Then lllxnlll = 1 (see llB) and hence llxnll ~ 1 for all n. If m > n and both are even, then (xm, xn) ~ Ill xm-n Ill (xn, xn) = (xn, xn) cxm, xm> ~ Ill xP IWCxn+p, xn+p> = cxm, xn) where 2p = m - n. Thus 1 ~ (xm, xm) ~ (xm, xn) ~ (xn, xn) ~ · · · ~ (x 2, x2) and (xm, xn) has a limit l ~ 1 as m, n ~ oo through even integers. Hence lim II xm- xn l\ 2 = lim (xm- xn, xm- xn) = 0, as is seen on expanding, and xn converges to a self-adjoint element e with ll ell ~ 1. Since x 2 n converges both to e and to e 2, it follows that e is idempotent.

Corollary. idempotent.

Any left ideal I contains a non-zero self-adjoint

Proof. If 0 ~ y E: I, then y*y is a non-zero self-adjoint element of I and the x of the lemma can be taken as a suitable scalar multiple of y *y. Then e = lim x 2 n = lim x 2 nx 2 = ex 2 E:: I, q.e.d. 27B. It is clear, conversely, that the set He of left multiples of an idempotent is a closed left ideal. An idempotent e is said to be reducible if it can be expressed as a sum e = e1 + e 2 of nonzero idempotents which annihilate each other; e1e2 = e2e 1 = 0. If e is self-adjoint, we shall require e1 and e2 also to be self-adjoint, and then the annihilation condition is equivalent to orthogonality: 0 = (eh e2) = (e1 2, e2 2) = (e1e2, e1e2) <=> e1e2 = 0. Since then = II e1 +II e2ll 2 and since the norm of any non-zero II e

W

W

102

SOME SPECIAL BANACH ALGEBRAS

idempotent is at least one, we see that a self-adjoint idempotent can be reduced only a finite number of times, and hence can always be expressed as a finite sum of irreducible self-adjoint idempotents. Lemma. I is a minimal/eft ideal if and only

if it is

of the form

I= He, where e is an irreducible self-adjoint idempotent. Proof. If e E: I and e = e1 + e2 is a reduction of e, then He 1 and He 2 are orthogonal left sub-ideals of I and I is not minimal. Thus if I is minimal, then every self-adjoint idempotent in I is irreducible. Since He is a sub-ideal, it also follows that I = He. Now suppose that I= He where e is irreducible. If I 1 is a proper sub-ideal of I and h is a self-adjoint idempotent in Ill then e1 = eh = ehe is a self-adjoint idempotent in I 1 which commutes with e, and e = e 1 + e2 (e 2 = e - e1) is a reduction of e, a contradiction. (Notice that e 1 ~ 0 since he 1 = heh = h 2 = h ~ 0, and e1 ~ e since He1 c I 1 ~ He.) Therefore, I is minimal. We remark that every minimal left ideal is closed (since it is of the form He). Corollary. His spanned by its minimal (closed) left ideals. Proof. Let M be the subspace spanned by the minimal left ideals of H. If M ~ H, then M is a proper closed left ideal, MJ. is a non-zero closed left ideal, Ml. contains a non-zero irreducible self-adjoint idempotent, and so Ml. includes a minimal left ideal, a contradiction. Remark: If x E: H and e is a self-adjoint idempotent, then xe is the projection of x on He, for (x - xe, he) = ((x - xe)e, h) = (0, h) = 0 so that (x - xe) ..l He. 27C. Theorem. Euery minimal left ideal generates a minimal two-sided ideal. The minimal two-sided ideals are mutually orthogonal, and H is the direct sum of their closures. Proof. Let I = He be self-adjoint idempotent e, erated by I. That is, N is We first observe that N*

a minimal left ideal with generating and let N be the two-sided ideal genthe subspace generated by I H = HeH. = N, for (He H)* = He H. Now sup-

SOME SPECIAL BANACH ALGEBRAS

103

pose that N 1 is an ideal included in N. Since Nd c N 1 n I and I is minimal, we have either that I c Nh in which case N 1 = N, or else Nd = 0. In the latter case N 1(IH) = N 1N = 0 and N 1N 1 * c N 1N = 0, so that N1 = 0. Thus N has no proper subideals and is a minimal ideal. It follows that N2 = N, for N 2 c N and N2 = NN* :;6- 0. Now let N 1 and N 2 be distinct minimal ideals. Then N 1N2 = N2N1 = 0, for N 1N 2 c N 1 n N2 and hence either N1 = N 1 n N 2 = N 2 or N 1N 2 = 0. It follows that, if x,y E: N1 and z E: N2, then (xy, z) = (y, x*z) = 0, so that N 1 = N 1 2 ~ N2. Thus distinct minimal ideals are orthogonal. Since the minimal left ideals span H and every minimal left ideal is included in a minimal two-sided ideal, it follows that H is the direct sum of the closures of its minimal two-sided ideals, i.e., the direct sum of its minimal closed two-sided ideals. Corollary. Every closed two-sided ideal I is the direct sum of the minimal closed two-sided ideals which are included in I. Proof. Every minimal two-sided ideal is either included in I or orthogonal to I, and the corollary follows from the fact that the minimal ideals generate H, and so generate I. Remark: Since the orthogonal complement of an ideal is a closed ideal, and the orthogonal complement of a minimal ideal is a maximal ideal, it follows from the Corollary that every closed two-sided ideal is the intersection of the maximal ideals including it. 27D. We come now to the analysis of a single minimal closed two-sided ideal N.

Lemma 1. If I = He is a minimal (closed) left ideal with generating idempotent e, then eHe is isomorphic to the complex number field. Proof. This is a classical proof from the Wedderburn structure theory. If 0 :;6- x E: He, then 0 :;6- Hx c He, and therefore Hx =He since He is minimal. Hence there exists b E: H such that bx = e. If x E: eHe, so that x = exe, then (ebe)(exe) = ebx = e. Thus eHe is an algebra with an identity in which every non-zero element has a left inverse. Therefore, every such element has an inverse, as in elementary group theory.

104

SOME SPECIAL BANACH ALGEBRAS

Thus eHe is a normed division algebra and hence is isomorphic to the complex number field by 22F. This means, o£ course, that every element of eHe is of the form Xe. Lemma 2. The following conditions are equivalent. (a) {ea.} is a maximal collection of mutually orthogonal irreducible self-adjoint idempotents in N; (b) {Hea.} is a collection of mutually orthogonal minimal left ideals spanning N; (c) {e,.H} is a collection of mutually orthogonal minimal right ideals spanning N. Proof. The equivalence of (a) and (b) follows from the fact that Hea. .1 HefJ if and only if ea. ..l ef3, and 27B. Similarly for (a) and (c). The existence of such a maximal collection is guaranteed by Zorn's lemma. 27E. Given {ea.} and N as above, we choose a fixed e1 and consider any other ea.. Since Hea.H generates N, e1Hea.He 1 ~ 0 and hence e1Hea. ~ 0. If e1a. is one of its non-zero elements, then e1a.e 1a. * is in e1He 1 and e1a. can be adjusted by a scalar multiple so that e1a.e1a. * = e1. Then e1a. * c (e1Hea.) * = ea.Heh and e1a. *e1a. can be checked to be an idempotent in the field ea.Hea. and hence equal to ea.. We define ea. 1 = e1a. *and ea.fJ = ea.lelfJ· The formulas ea.fJeyB = ea.B

=0

(ea.fJ, e7 B)

= 0 unless

(ea.fJ, ea.fJ)

= (eh e1)

if {3 = 'Y if

a

{3~"(

=

'Y

and

{3

=

o

follow immediately from the definitions. Thus (ea.fJ, eyB) = (ea. 1elfJ• e71eu) = (el(Jeu, e1a.e'Y1) = 0 unless {3 = o and a = "(, in which case it equals (eh e1). The set ea.HefJ is a one-dimensional space consisting of the scalar multiples of ea.fJ, for, given any x E:: H, ea.XefJa. E:: ea.Hea. ~ ea.XefJa. = cea. for some c ::::? ea.XefJ = cea.fJ· We know that yefJ is the projection of y on HefJ, and etllJ that of y on ea.H, for any y E:: H. Therefore, if x E:: N we have x =

SOME SPECIAL BANACH ALGEBRAS

105

L:~ xe~ = L:... ~ e,.xe~ = L:,.~ c,.~,.fJ, proving that the set of elements e,.~ is an orthogonal basis for N, and that e,.XefJ is the projection of x on eJ[e~. The Fourier coefficient CatJ can be computed in the usual way, c,.~ = (x, eatJ)/11 eafJ 11 2 • In view of the displayed equations above, it is clear that we have proved the following theorem:

Theorem. The algebra N is isomorphic to the algebra of all com-

plex matrices {c,.fJ} such that L:...~ I cafJ 12 < oo, under the correspondence x +-+ {c,.fJ} where x = La,fJ c,.fJeafJ and c,.fJ = (x, e,.~)/

II

e1

11 2 •

The following theorem is implicit in the above analysis.

Theorem. Two minimal lift ideals of H are (operator) isomorphic if and only if they are included in the same minimal closed twa-sided ideal N. Proof. Let / 1 and / 2 be the two ideals, with self-adjoint generating idempotents e1 and e2 and suppose that / 1 and /2 are both included in N. If / 1 and / 2 are not orthogonal, then the mapping x ---+ xe 1 maps / 2 into a non-zero subideal of /h and since / 1 and / 2 are minimal the mapping is an isomorphism and onto. The fact that such a mapping is an operator isomorphism is due merely to the associative law: y(xe 1 ) = (yx)e 1 • If / 1 and /2 are perpendicular then they can be extended to a maximal collection of orthogonal minimal ideals as in the above theorem and then the mapping x ---+ xe21 furnishes the required isomorphism. Conversely, if / 1 and / 2 are included in orthogonal minimal two-sided ideals N 1 and N 2 , then they cannot be operator isomorphic since N 1(/2) = 0, whereas N 1(/1) = /1.

27F. Theorem. The following statements are equivalent: (a) The minimal closed two-sided ideal N is finite dimensional. (b) N contains an identity, as a subalgebra of H. (c) N contains a non-zero central element.

Proof. If N is finite dimensional, then the finite sum e = L:.. e,. is a self-adjoint idempotent such that, for every x E::: N, x = L:.. xe,. = xe and x = L:.. e,.x = ex. Thus e is an identity for N.

106

SOME SPECIAL BANACH ALGEBRAS

For any x E: H we have xe -= e(xe) = (ex)e = ex, so that e is an element of the center of H. Thus (a) ~ (b) and (c). If e is an identity for N, then ea = eea is the projection of e on Hea and so e = Lea. Since the elements ea all have the same norm, this sum must be finite and N is finite dimensional. Finally, suppose that N contains a non-zero central element x. Since xea = (xea)ea = eaXea, the projection of X on Hea is of the form Co/!a for some constant ca, and x = L Col!a· Then C[J(f18 = Xef18 = ep&X = caepa, so that cp = ca and the coefficients Ca are all equal. Thus again the sum must be finite and N is finite dimensional. Also we have seen that any central element of N is of the form ce where c is the common value of the coefficients Ca. Now let {Na} be the set of minimal closed two-sided ideals of H. Then every x E::: H has a unique expansion x = L Xa, where Xa is the projection of x on Na. It is clear that (xy )a = XaJ = XaYa, and it follows in particular that Xa is central if x is central: XaJ = (xy)a = (yx)a = yxa. Therefore, given a, either Xa = 0 for every central element x or else Na has an identity ea. We thus have the following corollary. Corollary. If H 0 is the subspace of H which is the direct sum of its finite dimensional minimal ideals, then euery central element x is in H 0 and has an expansion of the form x = L Caea, where Ca is a scalar and ea is the identity of Na. Ho is the closed two-sided ideal generated by the central elements of H. 27G. If H is commutative, then every element is central and the minimal ideals Na are all one-dimensional, each consisting of the scalar multiples of its identity ea. The minimal ideals Na are the orthogonal complements of the maximal ideals Ma, and the homomorphism X ~ x(Ma) is given by x(Ma) = (x, ea)lii ea 11 2 • That is, x(Ma) is simply the coefficient of X in its expansion with respect to the orthogonal basis {ea}. This follows from the fact that the isomorphism between H /Ma and the complex number field is given by H / Ma :=::: Na = { cea}, and cea +--+ c. The continuous function CIJ thus has the value 1 at Mp and is otherwise zero, so that the space mt of maximal ideals has the discrete topology.

Chapter 171 THE HAAR INTEGRAL

In this chapter we shall prove that there exists on any locally compact group a unique left invariant integral (or measure) called the Haar integral. For the additive group R of real numbers, the group R/I of the reals modulo 1 and the Cartesian product R .. (Euclidean n-space) this is the ordinary Lebesgue integral. For any discrete group, e.g., the group I of the integers, the Haar measure attaches to each point the measure 1. In the case of an infinite Cartesian product of unit intervals (groups R/I) it is the so-called toroidal measure of Jessen. In each of these cases Haar measure is the obvious measure already associated with the given space in terms of its known structure. However, in the case of matrix groups it is not so clear what the measure is in terms of structural considerations. And in more complicated groups the reverse situation has held-the invariant measure has been used as an aid in discovering structure, rather than the structure giving rise to the measure. A general non-structural proof of the existence of an invariant measure is therefore of the utmost importance. The first such proof, and the model for all later proofs, was given by Haar in 1933 [22]. The first three sections center around the Haar integral, § 28 developing the necessary elementary topological properties of groups, § 29 giving the existence and uniqueness proofs, and § 30 treating the modular function. The group algebra and some elementary theorems on representations are discussed in § 31 and § 32, and the chapter closes with the theory ofinvariant measures on quotient spaces in § 33. 107

108

THE HAAR INTEGRAL

§ 28.

THE TOPOLOGY OF LOCALLY COMPACT GROUPS

A topological group is a group G together with a topology on G under which the group operations are continuous. Any group is a topological group under the discrete topology, and in fact is locally compact. The special groups R, Rjl, Rn, (R/f)N mentioned in the introduction above are all locally compact Abelian topological groups; the second and fourth are compact. A simple example of a non-Abelian locally compact group is the group of all linear transformations y = ax + b of the straight line into itself such that a > 0, the topology being the ordinary topology of the Cartesian half-plane of number pairs (a, b) such that a > 0. More generally any group of n X n matrices which is closed as a subset of Euclidean n 2-dimensional space is a locally compact group which is usually not Abelian. Groups such as these are interesting to us because, being locally compact, they carry invariant measures by the general theorem of § 29 and therefore form natural domains for general harmonic analysis. Of course, there are large classes of topological groups which are not locally compact and which are of great importance in other fields of mathematics. In this section we develop the minimum amount of topological material necessary for integration theory. 28A. We start with some simple immediate consequences of the definition of a topological group. 1) If a is fixed, the mapping x ~ ax is a homeomorphism of G onto itself, taking the part of G around the identity e into the part around a. If Vis any neighborhood of e, then aVis a neighborhood of a, and if U is any neighborhood of a, then a- 1 u is a neighborhood of e. Thus G is "homogeneous" in the sense that its topology around any point is the same as around any other paint. The mapping x ~ xa is another homeomorphism taking e into a. The inverse mapping x ~ x- 1 is also a homeomorphism of G onto itself. 2) Every neighborhood U of e includes a symmetric neighborhood W of e, that is, one such that W = w- 1 • In fact W = U n u- 1 has this property. Similarly if j is continuous and vanishes off W, then g(x) = j(x) + j(x- 1 ) is symmetric (g(x) =

109

THE HAAR INTEGRAL

+

j(x- 1) is Herg(x- 1 )) and vanishes off W, while h(x) = j(x) mitian symmetric (h(x) = h(x- 1)). 3) Every neighborhood U of e includes a neighborhood V of e such that V2 = v. V c U, for the inverse image of the open set U under the continuous function xy is an open set in G X G containing (e, e), and hence must include a set of the form V X V. Similarly, by taking V symmetric, or by considering the continuous functions xy - 1 and x- 1y, we can find V such that vv- 1 c U or such that v-l v c u. We notice that v- 1 V c U if and only if aV c U whenever

e E: aV. 4) The product of two compact sets is compact. For if A and B are compact, then A X B is compact in G X G, and with this set as domain, the range of the continuous function xy is AB, which is therefore compact (see 2H). AV, the intersection 5) If A is any subset of G, then A= being taken over all neighborhoods V of the identity e. For if y E: A and Vis given, then y v- 1 is an open set containing y and so contains a point x E: A. Thus y E: xV c AV, proving that A c AV for every such V, and hence that A c nvAV. ConAV, then yV- 1 intersects A for every V and versely, if y c AV =A. y E: A. Thus 6) If v is a symmetric com pact neighborhood of e' then vn is compact for every n and the subgroup j7rl> = limn_.., vn is thus IT-compact (a countable union of compact sets). It is clearly an open set, since y E: j7rl> implies yV c j?ri>V = V, and since j?rl>- c j7rl> V = V it is also closed. The left cosets of j7rl> are all open-closed sets, so that G is a (perhaps uncountable) union of disjoint IT-compact open-closed sets. This remark is important because it ensures that no pathology can arise in the measure theory of a locally compact group (see lSD).

nv

nv nv

28B. Theorem. If j E: L (the algebra of continuous junctions with compact support), then f is left (and right) uniformly continuous. That is, given E, there exists a neighborhood V of the identity e such that s E: V implies that I f(sx) - j(x) I < E for all x, or, equivalently, such that xy- 1 E: V implies that I j(x) - f(y) I < E {or all x andy.

110

THE HAAR INTEGRAL

Proof. Choose a compact set C such that f E: Lc (i.e., such that f = 0 off C) and a symmetric compact neighborhood U of the identity. By SF the set W of points s such that I /(sx) /(x) I < E for every x E: UC is open, and it clearly contains the identity s = e. If s E: U, then f(sx) and /(x) both vanish outside of uc. Therefore if s E: v = w n u, then I /(sx) - /(x) I < E for every x, q.e.d. Given a function f on G we define the functions f, and/" by the equations/.(x) = f(sx),j"(x) = J(xs- 1). They can be thought of as left and right translates off. The choice of s in one de6 nition and s- 1 in the other is necessary if we desire the associative laws f., = (/,) 1, /" 1 = (/") 1• (For other reasons/. is often defined by J,(x) = f(.r- 1 x).) Corollary. Iff E: L, 1 ~ p ~ oo and I is an integral on L, then J, and j", as elements of LP(I), are continuous junctions of s. Proof. The case p = oo is simply the uniform continuity of j, for we saw above that 11/.- f 11 .. < E whenever s E: V. Using this same V and choosing C so that/ E: Lc, let B be a bound for I (see 16C) on the compact set PC. Then II /. -fliP = I(l f, - f IP) 11P~ B 11PII J, - f 11 .. < B 11PE if s E: V, proving the continuity of J, in LP. Similarly for j•. 28C. Quotient spaces. If S is any class and II is a partition of S into a family of disjoint subclasses (equivalence classes), then the quotient spaceS /II, or the fibering of S by II, is obtained by considering each equivalence class as a single point. If S is a topological space, there are two natural requirements that can be laid down for a related topology on S /II. The first is that the natural mapping a of S onto S/II, in which each point of S maps into the equivalence class containing it, be continuous. Thus a subset A of S/II should not be taken as an open set unless a- 1 (A) is open in S. The second requirement is that, iff is any continuous function on S which is constant on each equivalence class of II, then the related function F on S /IT defined by F(a(x)) = /(x) s.hould be continuous on S/II. In order to realize this second requirement the topology for S/II is made as strong (large) as possible subject to the first requirement: a subset A of S/Il is defined to be open if and only if a- 1 (A) is an

THE HAAR INTEGRAL

111

open subset of S. Then the mapping a is continuous, by definition. Moreover, if/ is continuous on S and constant on the sets of II, then a- 1 (F- 1 (U)) = j- 1 (U) is open whenever U is open. Therefore, F- 1 ( U) is open whenever U is open and the related function F is continuous. Clearly a subset B of S/II is closed if and only if a- 1 (B) is closed in S. In particular, if each equivalence class of II is a closed subset of S, then each point of S/JI is a closed set in S/II, and S/II is what is called a T 1-space. These considerations apply in particular to the quotient space of left cosets of a subgroup H in a topological group G. Moreover, such fiberings G/ H have further properties which do not hold in general. Thus:

Theorem. If H is a subgroup of a topological group G, then the natural mapping a of G onto the left coset space G/H is an open mapping, that is, a(A) is open if A is open. Proof. Suppose that A is open. We have a- 1 (aA) = AH = U {Ax: x E:: H}, which is a union of open sets and therefore open. But then aA is open by definition of the topology in G/H, q.e.d. } Corollary 1. I/G is locally compact, then G/H is locally compact, for any subgroup H. Proof. If Cis a compact subset of G, then a(C) is a compact subset of G/H (since a is a continuous mapping-see 2H). If x E:: interior (C), then the above theorem implies that a(x) E:: interior (a(C)). Thus compact neighborhoods map into compact neighborhoods, proving the corollary. Corollary 2. If G is locally compact and B is a compact closed subset of G/H, then there exists a compact set A c G such that B = a(A). Proof. Let C be a compact neighborhood of the identity in G and choose points XIJ · · · , x .. such that B c U1"a(x;C) = aCU1"x,C). Then a- 1 (B) n CU 1"x,C) is a compact subset of G, and its image under a is B since it is simply the set of points in Ut "x,C which map into B.

112

THE HAAR INTEGRAL

28D. We now show that for the purposes of integration theory a topological group can be assumed to be a Hausdorff space. Theorem. A topological group G has a minimal closed normal subgroup, and hence a maximal quotient group which is a T 1-space. Proof. Let H be the smallest closed set containing the identity. We show that H is a subgroup. For if y E: H, then e E: Hy- 1 and soH c Hy- 1 (H being the smallest closed set containing e). Thus if x, y E: H, then x E: Hy- 1 and xy E: H. Since H- 1 is closed, it follows similarly that H c H-1 • Thus H is closed under multiplication and taking inverses; that is, H is a subgroup. Moreover, His normal, for H c xHx- 1 for every x and consequently x- 1Hx c H for every x, so that H = xHx- 1 for every x. Since His the smallest subgroup of G which is closed, G/H is the largest quotient group of G which has a T 1 topology. Lemma. A topological group which is a T 1 -space is a Hausdo'ljf space.

Proof. If x -:;6. y, let U be the open set which is the complement of the point xy- 1 and choose a symmetric neighborhood V of the identity so that V2 c U. Then V does not intersect Vxy - 1 and Vy and Vx are therefore disjoint open sets containing y and x respectively. Every continuous function on a topological group G is constant on all the cosets of the minimal closed subgroup H. It follows that the same is true for all Baire functions. Thus the continuous functions and Baire functions on G are, essentially, just those on G/H, so that from the point of view of integration theory we may replace G by G/H, or, equivalently, we may assume that G is a Hausdorff space. For simplicity this property will be assumed for all groups from now on, although many results (for example, those of the next three numbers) will not depend on it. § 29.

THE HAAR INTEGRAL

We shall follow Weil [48] in our proof of the existence of Haar measure and then give a new proof of its uniqueness. This procedure suffers from the defect that the axiom of choice is invoked

THE HAAR INTEGRAL

113

in the existence proof, here in the guise of choosing a point in a Cartesian product space, and then is demonstrated to be theoretically unnecessary when the chosen functional is found to be unique. There are proofs which avoid this difficulty by simultaneously demonstrating existence and uniqueness (see [7] and [33]), but they are more complicated and less intuitive, and we have chosen to sacrifice the greater elegance of such a proof for the simplicity of the traditional method. 29A. Iff and g are non-zero functions of L +, then there exist positive constants c, and points si, i = 1, · · ·, n, such that f(x) ~ L~ cig(s,x).

For example, if m1 and m 11 are the maximum values off and g respectively, then the c, can all be taken equal to any number greater than mJfm 11 • The Haar coueringfunction (/;g) is defined as the greatest lower bound of the set of all sums D c, of coefficients of such linear combinations of translates of g. The number (/;g) is evidently a rough measure of the size of f relative tog, and the properties listed below, which depend directly upon its definition, show that if g is fixed it behaves somewhat like an invariant integral. (/.;g) = (/;g).

(1)

+ /2; g)

(2)

(/1

(3)

(cf; g)

(4)

(5) (6)

~

(/1; g)

+ (/2; g).

= c(f; g) where c > 0.

f1 ~ /2 => (/1; g) ~ (/;h)

~

(/;g)

(/2; g).

(/; g)(g; h). ~

mJfm 11 •

In order to see (6) we choose x so thatf(x) = m" and then have m, ~ L c;g(s,x) ~ (L c,)m,, so that (L c,) ~ mJim,. (5) follows from the remark that, if f(x) ~ c;g(s,x) and g(x) ;:;;; E d;h(t;x), then f(x) ~ Ei.i c;,d;h(t;s;x). Thus (/; h) ;:;;; glb E cid; = (glb E ci)(glb Ed;) = (/; g)(g; h). The other properties are obvious. 29B. In order to make (/;g) a more accurate estimate of relative size, it is necessary to take g with smaller and smaller sup-

L

114

THE HAAR INTEGRAL

port, and then, in order that the result be an absolute estimate of the size off, we must take a ratio. We accordingly fixf0 E: L + ( /0 .= 0) for once and for all, and define I.,(/) as (/; rp)/(/0 ; rp). It follows from (5) that

(7) so that I.,f is bounded above and below independently of rp. Also, it follows from (1), (2) and (3) that I., is left inuariant, subadditiue and homogeneous. We now show that, for small rp, I., is nearly additive.

Lemma. Giuen f 1 and f 2 in L + and borhood V of the identity such that

Irpft

(8)

+ lrpj2

~

I.,(fl

E

> 0, there

+h)+

exists a neigh-

E

for all rp E: L +v.

Proof. Choose j' E: L + such that f' = 1 on the set where f 1 + h > 0. Let 5 and E' for the moment be arbitrary and set f = f 1 + h + 5f', hi = fdf, i = 1, 2. It is understood that hi is defined to be zero where f = 0 and it is clear that h, E: L +. Choose V such that I h,(x) - h .. (y) I < E' wheneverx- 1y E: V (by 28B). If rp E: L +v and f(x) ~ 2: c;rp(s;x), then rp(s;x) .= 0 ~ I h.. (x) - h.. (s;- 1) I < E1, and

L; c;rp(s;x)h,(x) ~ L; c;rp(s;x)[h,(s;- + E'] (/.. ; rp) ~ L; c;[h .. (s;- + (11; rp) + (/2; rp) ~ 2: c;[1 + 2E']

f,(x) = f(x)h,(x) ~

1)

1)

E1]

2: c; approximates (/; rp), we have + I.,/2 ~ I.,f[1 + 2E'] ~ [I.,(/1 + /2) +

Since

I.,fl

U.,(/')][1

+ 2E'].

The lemma follows if 5 and l are initially chosen so that 2E'(/t + /2;f0) + 5(1 + 2E')(/';f0) < E. 29C. We have seen above that I., becomes more nearly an invariant integral the smaller rp is taken, and it remains only to take some kind of generalized limit with respect to rp to get the desired invariant integral.

THE HAAR INTEGRAL

Theorem.

115

There exists a non-triuial non-negatiue left inuariant

integral on L.

Proof. For every non-zero/ E: L +letS 1 be the closed interval [1/(/0 ; / ) , (/; / 0 )], and let S be the compact Hausdorff space

Il1 S 1 (the Cartesian

product of the spaces S 1). For each nonzero tp E: L + the functional I"' is a point in S, its/ coordinate (its projection on S1) being Irpf· For each neighborhood V of the identity in G, let Cv be the closure in S of the set {I"': tp E: Lv +}. The compact sets Cv have the finite intersection property since Cv, n ··· n Cv,. = Ccv,n···nv,.J· Let I be any point in the intersection of all the Cv. That is, given any Vand given/h · · ·,/.. and E > 0, there exists tp E: Lv + such that I I(j,) - I"'(j,) I < E, i = 1, · · ·, n. It follows from this approximation and 29B that I is non-negative, left invariant, additive and homogeneous, and that 1/(/0 ; / ) ~ I(/) ~ (/; / 0 ). The extension of I from L + to L, obtained by defining I(/1 - /2) as l(/1) - I(/2), is therefore a non-trivial left invariant integral, as desired . . The integral I is extended to the class of non-negative Baire functions as in Chapter 3 and the whole machinery of Lebesgue theory is available, as set forth in that chapter. We emphasize only one fact: that a locally compact group is a union of disjoint open-closed cr-compact subsets (28A, 6) and therefore the difficulties which can plague non cr-finite measures do not arise here. For instance the proof that (D)* = L«> as sketched in 15D is available. (See also 13E.)

29D. Theorem. The aboue integral is unique to within a multiplicatiue constant.

Proof. Let I and J be two left invariant non-negative integrals over L and let f E: L +. We choose a compact set C such that f c Lc+, an open set U with compact closure such that C c U and a function/' E: L + such that/' = 1 on U. Given E, we furthermore choose a symmetric neighborhood V of the identity such that II / 11 - fe 11,., < E if y, z E: V, and such that .(CV U VC) cU. The latter condition guarantees that/(xy) = j(xy)f'(x) and f(yx) = f(yx)f'(x) if y E: V, and together with the former implies that I j(xy) - f(yx) I < Ef(x) for all x. Let

116

THE HAAR INTEGRAL

h be any non-zero symmetric function in Lv +. The above conditions, together with the Fubini theorem and the left invariance of I and ], imply that

I(h)](J) =

I11]~h(y)j(x)

=

I11]~h(y)J(yx)

](h)I(J) = I,J~h(x)j(y) = I,J~h(y- 1 x)j(y)

= ]J 11h(x- 1y)j(y) =

I I(h)](j)

- ](h)I(j)

J~I11h(y)j(xy)

I ~ I11]~(h(y)l f(yx) ~ d11]~h(y)f'(x)

- f(xy)

I)

= d(h)j(f').

Similarly, if g E: L' and h is symmetric and suitably restricted, then I I(h)f(g) - J(h)I(g) I ~ EI(h)f(g'), where g' is fixed, and has the same relation tog that f' has to f. Thus JU). - J(g) ' s I(j) I(g) -

' JU') E

I(j)

+ J(g') I(g)

j

' and, since E is arbitrary, the left member is zero and the ratios are equal. Thus if g0 is fixed and c = j(g0 )/I(g0 ), then ](f) = ci(f) for all f E: L +, completing the proof of the theorem. If I is known to be right invariant as well as left invariant, as is the case, for instance, if G is commutative, then the uniqueness proof is trivial. For iff and h E: L + and h*(x) = h(x-1 ), then ](h)I(f)

=

]11I~h(y)j(x)

=

JJ~h(y)f(xy)

=

I~]11 h(y)J(xy)

= I~]11 h(x- 1y)j(y) = ]11I~h*(y- 1 x)J(y)

=

]11 I~h*(x)f(y)

= I(h*)j(J),

so that ](f) = ci(f) where c = j(h)/I(h*).

29E. Theorem. G is compact if and only if p.( G) constant junctions are summable).

<

CXJ

(the

Proof. If G is compact, then 1 E: L, and JJ.(G) = I(l) < CXJ, Conversely, if G is not compact and if Vis a neighborhood of the identity with compact closure, then G cannot be covered by a finite set of translates of V. Therefore we can choose a sequence

117

THE HAAR INTEGRAL

{Pn} of points in G such that Pn fl. Ul'- 1 PiV. Now let U be a symmetric neighborhood such that U 2 c V. Then the open sets PnU are all disjoint, for if m > n and PnU n PmU ~ {ZJ, then Pm E: PnU2 c PnV, a contradiction. Since the common measure of the open sets Pn U is positive, it follows that the measure of G is infinite, and the constant functions are not summable, q.e.d. If G is compact, Haar measure is customarily normalized so that 1-1(G) = 1 (J1 d1-1 = /(1) = 1).

§ 30.

THE MODULAR FUNCTION

30A. Left invariant Haar measure need not be also right invariant-in general /(f) ~ I(j). However, if tis fixed, I(f) is a left invariant integral, I(fs') = I(f,) = I(f), and because of the uniqueness of Haar measure there exists a positive constant ~(t) such that I(f) tJ.(t)I(J).

=

=

The function tJ.(t) is called the modular function of G; if tJ.(t) 1, so that Haar measure is both left and right invariant, G is said to be unimodular.

Lemma. lfG is Abelian or compact, then G is unimodular. Proof. The Abelian case is trivial. If G is compact and if then tJ.(s) = tJ.(s)I(j) = J(j•) = /(1) = 1, so that the compact case is also practically trivial. We have already observed earlier that, iff E: L, then /(f) is a continuous function of t. Thus tJ.(t) is continuous. Also tJ.(st)I(j) = J(JBt) = I( (j•) 1) = tJ.(t)J(j•) = tJ.(s) tJ.(t)/(j). Thus tJ.(t) is a continuous homomorphism of G into the positive real numbers. JOB. As might be expected, Haar measure is not ordinarily inverse invariant, and the modular function again plays its adjust-

f

= 1,

ing· role.

We use the customary notation

J

f(x) dx instead of

I(f) below because of its greater flexibility in exhibiting the variable.

118

THE HAAR INTEGRAL

Theorem.

f j(x-

1)

a(x- 1) dx =

f j(x) dx.

Before starting the proof we define the function!* by

j*(x)

=

j(x-1 ) a(x- 1).

The complex conjugate is taken so that, at the appropriate moment, the mapping j ~ f* will be seen to be an involution. The formulas !*" = a(s)J,*, P* = a(s)J*. follow directly from the definition. Proof. In proof of the theorem we first observe that the integral above, ](f) = l(j*-), is left invariant: ](f.) = I(f. *-) = a(s- 1)1(/*-•) = I(j*-) =](f). Therefore ](f) = cl(j). But now, given E, we can choose a symmetric neighborhood V of e on which 11 - a(s) I < E and then choose a symmetric/ E: Lv+ such that I(j) = 1, giving

11 - c I =

= Since

E

is arbitrary, c

I (1 I 1((1

I = I I
c)I
= 1, q.e.d.

Corollary 1. j* E: V if and only iff E: L\ and II/* ll1 = This follows from the theorem and the fact that L is dense in D.

II f lk

Corollary 2. Haar measure is inuerse inuariant G is unimodular.

if

and only

if

30C. The following theorem does not really belong here, but there seems to be no better spot for it.

Theorem. If g E: LP (1 ~ p < oo), then g. and g•, as elements of LP, are continuous functions of s. Proof. Given

II g, -/.liP < E/3 < E/3 if s E:

V.

we choose f E: L so that II g - f liP = and by 28B we choose V so that II f- /.liP Then E,

II g- g. liP~ II g- flip+ II f- /.liP+ II/.- g. liP< if s E: V, showing that g. is a continuous function of s at s

E

= e.

THE HAAR INTEGRAL

119

Since II P - g' liP = ..1(s) 11 PII f- g lip, the above inequality remains valid when g, andf, are replaced by g• andj" if the choice of Vis now modified so that II f- j" liP < E/6 and ..1(s) 11P < 2 when s E:: V. The equations II fu: - /z liP = II f, -fliP and II pz - P liP = ..1(x) 11PII f' - f liP show that both functions are continuous everywhere, and, in fact, that j, is left uniformly continuous. 30D. The classes of groups treated in detail in this book, Abelian and compact groups, are unimodular. However, many important groups are not unimodular, and it will be worth while to exhibit a class of such groups. Let G and H be locally compact unimodular groups and suppose that each u E:: G defines an automorphism of H (that is, G is mapped homomorphically into the automorphism group of H), the result of applying u to an element x E:: H being designated u(x). The reader can check that the Cartesian product G X H becomes a group if multiplication is defined by (uh Xt)(u2, x2) = (u1u2, u2Cx1)x2). Furthermore, if u(x) is continuous in the two variables u and x simultaneously, then this group is locally compact in the ordinary Cartesian product topology. Such a group is called a semidirect product of H by G. It contains Has a normal subgroup and each coset of the quotient group contains exactly one element of G. If A and Bare compact subsets of G and H respectively, then (u 0 , x 0 )(A X B) is the set of all pairs (u 0 u, u(x 0 )x) such that u E:: A and x E:: B. If we compute the Cartesian product measure of this set by the Fubini theorem, integrating first with respect to x, we get ,.,.(A),(B), where ,.,. and " are the Haar measures in G and H respectively. This is also the product measure of A X B. Thus the Cartesian product measure is the left invariant Haar measure for the semi-direct product group. To compute the modular function ..1 we consider (A X B)(u 0 , x 0 ) = (Au 0 ) X (u 0 (B)x 0 ), whose product measure is ,.,.(A)P(u0 (B)). If the automorphism u 0 multiplies the Haar measure in H by a factor Cl(u0 ), then the above measure is Cl(u0 ),.,.(A),(B). The modular function ..1((u, x)) therefore has the value Cl(u), and the semidirect product is unimodular only if Cl 1.

=

120

THE HAAR INTEGRAL

As a simple example we take H = R, the additive group of the real numbers, and G as the multiplicative group of the positive reals, with u(x) = u·x. Thus (a, x)(b, y) = (ab, bx + y) and Ll((a, x)) = a. This group is, therefore, not unimodular. 30E. It is clear that, if G and Hare not unimodular, but have modular functions .6. 1 and .6. 2 respectively, then the above calculation will give the modular function Ll in terms of .6. 1 and .6. 2 • Cartesian product measure is seen to be the left invariant Haar measure exactly as above, but now the product measure of (A X B)(uo, xo) = (Auo) X (uo(B)xo) is 5(uo)Ll1(uo)Ll 2 (xo) [~-&(A)v(B)] so that Ll( (u, x)) = 5(u)Ll 1(u)Ll 2 (x).

If in this situation we take Has the additive group of the real numbers (so that .6. 2 = 1) and u(x) = x[Ll 1(u)]- 1, then 5(u) = Ll 1(u)- 1 and Ll((u, x)) = 1. Thus any locally compact group G can be enlarged slightly to a unimodular group which is a semidirect product of the reals by G. This was pointed out by Gleason [18].

§ 31.

THE GROUP ALGEBRA

In this section we prove the existence of the convolution operation and show that with it as multiplication V(G) becomes a Banach algebra. 31C-G then establish some of the special properties of this algebra which follow more or less immediately from its definition. 31A. We define the convolution of j with g, denoted j * g, by [f * g](x)

=

J

f(xy)g(y- 1) dy

=

J

j(y)g(y- 1x) dy,

the equality of the integrals being assured by the left invariance of the Haar integral. The ordinary convolution integral on the real line, i:J(x - y)g(y) dy, is obviously a special case of this definition. It also agrees with and generalizes the classical notion used in the theory of finite groups. A finite group G is compact if it is given the discrete topology, and the points of G, being congruent non-void open sets, must have equal positive Haar measures. If the Haar measure of G is normalized by giving

121

THE HAAR INTEGRAL

each point the measure 1, and iff and g are any two complexvalued functions on G, then

f * g(x)

=

J

f(xy)g(y- 1 ) dy = Lllf(xy)g(y- 1 ) = Luv=:z,J(u)g(v)

which is the classical formula for convolution. The above definition is purely formal and we must start by showing that f * g exists.

Lemma. Iff E: LA. and g E: LB, then J * g E: LAB· Proof. The integrandf(y)g(y- 1x) is continuous as a function of y for every x and is zero unless y E: A and y- 1x E: B. Thus f * g(x) is defined for every x and is zero unless x E: AB. Also

I f * g(x1)

- f

* g(x2) I ~ II /:1:1 - /,.~ Jlao ·

Jl

g(y- 1)

I dy

and since

j,. is by 28B continuous in the uniform norm as a function of x it follows that f * g is continuous. Theorem. If j, g E: CB +, then f(y)g(y- 1x) E: CB +(G X G) and

II

f•glJP ~II f 11~·11 giJP, where 1 ~ p ~co, Proof. Let/ og be the functionf(y)g(y- 1x). If/ E: L +,then the family of functions g E: CB+ such that fog E: CB +(G X G) is L-monotone and includes L +, and is therefore equal to CB +. Thus if g is an L-bounded function of CB +, then the family of functions f E: CB + such that fog E: CB + includes L + and is Lmonotone, and therefore equals CB +. In particular fog E: CB + whenever f and g are L-bounded functions of CB +. The general assertion then follows from the fact that any non-negative Baire function is the limit of an increasing sequence of L-bounded Baire functions. The Fubini theorem implies thatf(y)g(y-1x) is

integrable in y for every x, that (/ * g)(x) = integrable,

II /IJ1·1l Kill·

and

that

II f * g ll1

=

JJ

J

f(y)g(y- 1x) dy is

f(y)g(y- 1x) dx dy =

Ifh E: CB+then also/(y)g(y- 1x)h(x) E: CB+(G X G)

(see 13C) and, as above, (f.g, h)= J!(y) [Jg(y- 1x)h(x) dx] dy

by the general HOlder inequality 14C.

~II f 11~11 g[JPII h II~~

122

THE HAAR INTEGRAL

It follows, using lSC, that 11/*giiP~ IIJII1IIgiiP, and, in particular, that f * g E:: LP if f E:: V and g E:: LP. The case p = oo is obvious. Corollary. Iff E:: V and g E:: LP, thenf(y)g(y- 1x) is summable iny for almost all x, (f * g)(x) E:: LP and II f * g liP ~ II f lhll g liP· Proof. The sixteen convolutions obtained by separating each off and g into its four non-negative parts belong to LP by the

above theorem, and their sum, which is equal to

J

f(y)g(y- 1x) dx

wherever all sixteen are finite, is thus an element of LP (with the usual ambiguity about its definition at points where the summands assume opposite infinities as values). Moreover, II f * g liP ~ Ill f I* I g I liP ~ II f ll1ll g lip, by the above theorem. 31B. Theorem. Under convolution as multiplication V(G) forms a Banach algebra having a natural continuous involution

f

~J*.

Proof. We have seen earlier that the mapping/~ j* of U onto itself is norm preserving, and we now complete the proof that it is an involution. It is clearly additive and conjugate linear. Also U * g)*(x) = f(x- 1y) g(y-1).1(x-1) dy

J

=

J

g(y-1).1(y-1)j((y -1x) -1).1((y -1x) -1) dy

= (g* *f*)(x),

so that (f *g)* = g* *f*. The associativity of convolution depends on the left invariance of the Haar integral. If/, g, h E:: CB+, we have ((f *g)

* h)(x)

=

= = =

jlU * g)(xy)]h(y- 1) dy

JJ JJ J

f(xyz)g(z- 1)h(y - 1) dy dz f(xz)g(z- 1y)h(y- 1) dy dz

f(xz)[(g

* h)(z-1)] dz =

(f * (g

* h))(x).

123

THE HAAR INTEGRAL

The assoctatlve law is therefore valid for any combination of functions from the LP classes for which the convolutions involved are all defined. The convolution f * g is also clearly linear in f and g. Since V is closed under convolution and II f * g 1!1 ~ II f Ik II g ll1 by 31A, it follows that V is a Banach algebra with convolution as multiplication. It is called the group algebra (or group ring) of G, and generalizes the notion of group algebra used in the classical theory of finite groups. We prove below a number of useful elementary properties of the group algebra V(G).

31C. Theorem. V(G) is commutative if and only if G is commutative. Proof. If G is commutative, then G is also unimodular and f

*g =

J

f(y)g(y- 1 x) dy =

J

g(xy)j(y- 1) dy = g

•f.

Now suppose that V(G) is commutative and that/, g E: L. Then 0 = f * g- g •! = j[f(xy)g(y- 1) - g(y)f(y- 1 x)] dy

= Jg(y)[j(xy-I)Ll.(y-1) - f(y-Ix)] dy. Since this holds for every g in L, it follows that j(xu) Ll.(u) - j(ux)

= 0.

Taking x = e we have Ll.(u) = 1, so that /(xu) - f(ux) = 0 for every f in L. Since L separates points, it follows that xu - ux = 0 and G is commutative.

31D. Theorem. V(G) has an identity if and only if G is discrete. Proof. If G is discrete, the points of G are congruent open sets having equal positive Haar measures which may be taken as 1. Thenf(x) is summable if and only ifj(x) = 0 except on a countable set {Xn} and I: I f(xn) I < oo. The function e(x) which is 1 at x = e and zero elsewhere is an identity: f

* e(x) =

J

f(y)e(y- 1x) dy

= I:,.J(y)e(y- 1x)

=

f(x).

124

THE HAAR INTEGRAL

Conversely suppose that u(x) is an identity for L. We show that there is a positive lower bound to the measures of non-void open (Baire) sets. Otherwise, given any E there exists an open neighborhood V of the identity e whose measure is less than E, and hence one such that fvi u(x) [ dx

<

Choose a symmetric

E.

U so that U 2 c V and let/ be its characteristic function. Then

j(x)

=

(u *f) (x)

=

f

u(y)j(y- 1 x) dy

=

iu

u(y) dy

~ fvi

u

I<

E

=

for almost all x in U, contradicting /(x) 1 in U. Therefore, there is a number a > 0 such that the measure of every non-void open Baire set is at least a. From this it follows at once that every open set whose closure is compact, and which therefore has finite measure, contains only a finite set of points, since otherwise its measure is seen to be !5::; na for every n by choosing n disjoint non-void open subsets. Therefore, every point is an open set, and the topology is discrete. 31E. In any case V(G) has an "approximate identity." Theorem. Giuen f E::: LP (1 ~ p < oo) and E > 0, there exists a neighborhood V of the identity e such that II/* u -/liP < E and II u */- f liP < E whenever u is any function of L 1 + such that

u = 0 outside of V and fu = 1. Proof. If h E::: L'l the Fubini theorem and Holder inequality imply that

I (u */-/,h) I =

iff u(y)(j(y- 1x) - j(x))h(x) dy dx I

~ II h II a fll !11-l - I

llpu (y) dy.

II u *f- fliP ~ fll!yilpu(y) dy. If Vis chosen (by 30C) so that II / fliP< whenever y E::: V and if u E::: (V)v+, then II u *f - / II P < e u = E. Thus

1 -/

11- L -

f

E

f * u is similar but somewhat complicated by modularity. If m = u(x- 1 ) dx, then The proof for

J

125

THE HAAR INTEGRAL

I (f * u - f, h) I = IJJC!(xy - j(x)/m)u(y- 1 )h(x) dy dx I

~

II h llq

j11 mf"-

1

-

Jllp(u(y- 1)/m) dy

andll/*u-JIIP~JIIm/"-1 -/llpu(y- 1 )/mdy. as V decreases.

~

1

(For m = J[u(x- 1 ) a(x- 1)] a(x) dx, where

J

u(x- 1) a(x- 1 ) dx =

= 1.)

Nowm

J

u(x) = 1, a(x) is continuous and a(e)

Thus there exists V such that II m.f'- 1 -fliP < E if y- 1 E:

f

Vand u E: (V)v+, giving II

* u- fliP< EJ[u(y- 1)/m] dy = E.

The neighborhoods of the identity V form a directed system under inclusion, and if uv is a non-negative function vanishing off V and satisfying

J

uv

1, then uv *f and f * uv converge

=

tof in the pth norm for any f E: LP by the above theorem. The directed system of functions uv can be used in place of an identity for many arguments and is called an approximate identity.

A closed subset of L 1 is a left (right) ideal and only if it is a left (right) invariant subspace.

31F. Theorem.

if

Proof. Let I be a closed left ideal and let u run through an approximate identity. Iff E: /, then Uz *f E: /. But Uz *f = (u *f)z ~ /z, since u *f ~ j, and thereforefz E: /. Thus every closed left ideal is a left invariant subspace. Now let I be a closed left invariant subspace of V. If I l. is the set of all g E: L ~ such that (/,g) = 0 for every f E: /, then we know (see 8C) that I = (fl.)\ i.e., that, if/ E: V, then/ E: I if and only if(/, g) = 0 for every g E: P. But if hE: V,j E: I and g E:

I\

then (h *j, g)

=

JJh(y)j(y- 1x)g(x) dy dx = Jh(y)

J[J(y - 1x)g(x) dx] dy = 0 (since/11

1_

g for every y), which proves

by the above remark that h *f E: /. That is, I is a left ideal. The same proof, with only trivial modifications, works for right ideals.

126

THE HAAR INTEGRAL

31G. We conclude this section with some remarks about positive definite functions. For this discussion we restrict ourselves exclusively to unimodular groups. It will be remembered that the notion of positivity was introduced in connection with an abstract Banach algebra having a continuous involution. The algebra which we have before us now is V(G), and we start with the ideal L 0 consisting of the uniformly continuous functions of V, and the function rp on L 0 defined by rp(j) = f(e). Then

rp(f */*)

=

J

ll2 2~ 0. Thus rp is positive and II g *f ll2 ~ II g ll1·ll f ll2, the operators U,

f(x)j(x) dx

=

II f

H"' is L 2(G). Since defined by UJ = g *fare bounded and the mapping g ~ U, is a *-representation of Ll(G). It is called the left regular representation and will be discussed further in 32D. The functional rp is thus unitary, but is clearly not continuous (in the V norm). At the other extreme are the positive functionals which are continuous in the L 1 norm. Lemma. A positive junctional on V(G) is continuous only if it is extendable.

if

and

Proof. We have seen earlier (261) that any extendable positive functional is automatically continuous. The converse implication follows here because of the existence of an approximate identity. Given a continuous positive functional P, we have

I P(f)

12

=

lim I P(f * u)

12

~ P(f *f*) lim P(u* * u)

~ I! p IIPU */*),

P(j*) = lim P(f* * u) = lim P(u* *f) = P(j), proving the lemma. Now every continuous functional P on L 1 is given by a function p E:: La:J, and, if P is positive, p is called positive definite. Notice that then p = p*, for (f, p) = P(f) = P(J*) = (j*, p) = (p,j*) = (J,p*) for every f E:: L 1 • Thisnotionofpositivedefiniteness is formally the same as that introduced in 26], the fixed unitary functional rp being that considered in the first paragraph above: P(j) = (f, p) = (f, p*) = (p,j*) = [p */](e) = rp(p *f).

127

THE HAAR INTEGRAL

The condition that a function p E: be written

L~

be positive definite can

for every f E: L 1•

§ 32.

REPRESENTATIONS

32A. A representation T of G is a strongly continuous homomorphism of G onto a group of linear transformations on a complex vector space X. That is, if T, is the transformation associated with the group element s, then T. 1 = T,Tt and T,(x) is a continuous function of s for every x E: X. If X is finite dimensional, this requirement of continuity can be expressed as the continuity of the coefficients in the matrix for T. when a fixed basis has been chosen for X. When X is infinite dimensional some kind of topology for X must be specified. Usually X is a Banach space and here we shall confine ourselves to reflexive Banach spaces and Hilbert space. T will be said to be bounded if there is a uniform bound to the norms II T, II, s E: G. T is a unitary representation if X is a Hilbert space and the transformations T. are all unitary.

of G on a reflexive Banach f(x)T~ dxfor euery f E: £1, then

32B. If T is a bounded representation space X and if T 1 is defined as

J

the mapping f ~ T 1 is a bounded representation of V(G). If T is unitary, then the integrated representation is a •-representation of L 1 (G).

Proof.

The function F(j, x,y) =

J

f(s)(T.x,y) ds =

J

f(s)y(T.x) ds

is trilinear and satisfies I F(j, x, y) I ;:;;; II f II1BII x II IIY 11. It follows exactly as in 26F that there is a uniquely determined linear transformation T 1 such that (T1x, y)

II T, II over,

;:; ; Bll f II h

and the mapping

f

J

= f(s)(T.x, y) ds,

~ T, is linear.

More-

128

THE HAAR INTEGRAL

(T, qX, y)

JJ = JJ =

f(t)g(t- 1s)(T.x, y) dt ds f(t)g(s)(Tt.x,y) ds dt

= Jg(s)(T,T.x,y) dsdt = Jg(s)(T.x, (T1)*y) ds = f(T,x, (T,)*y) = (T1T,x,y). Thus T 1 ,., = T 1 T, and the mapping f ~ T 1 is a homomorphism. If X is a Hilbert space and T is a unitary representation, then (T1•x,y)

= jJ(s- 1)A(s- 1 )(T.x,y) ds = (jf(s)(T,y,x) ds)= (T1y, x) = (x, T 1y).

Thus (T1)* = T,. and Tis a *-representation. Remark: No non-zero element of X is annihilated by every T 1 .

For, given 0 ~ x E: X andy E: X* such that (x, y) ~ 0 we can find a neighborhood V of the identity in G such that (T.x, y) differs only slightly from (x,y) if s E: V. If/is the characteristic function of V, then (T1x, y) = fv(T.x, y) ds differs only slightly from

p. ( V)

(x, y), and hence T 1x is not zero.

32C. Theorem. Conversely, a bounded representation T of V(G) over a Banach space X arises in the above way if the union of the ranges of the operators T,,j E: V, is dense in X. If X is a Hilbert space and Tis a •-representation, then Tis unitary. Proof. Let Xr be the union of the ranges of all the operators T1 (j E: V); Xr is dense in X by hypothesis. Let u run through an approximate identity of V. We know that Ua *f = (u */)a ~ fa, and therefore

II T u..Tt -

Tt.ll ~ 0.

Thus T,.. converges strongly (that is, pointwise) on Xr to an operator Ua satisfying UaTt = T,., and since II T u..ll ~ Bll Ua ll1 = B for every u, it follows that II Ua II ~ B and that U a is uniquely defined on the whole of X (see 7F). Since UabTt = 1/.b = T(f.lb = UbTf. = UbUaTt (i.e., Uab = UbUa on Xr), we have Uab = UbUa and

129

THE HAAR INTEGRAL

the mapping a ~ Ua. is therefore an anti-homomorphism. Finally, Ia. is a continuous function of a as an element of Lh and therefore Ua.Tt = T1o is a continuous function of a. Thus Ua. is strongly continuous on Xr, and hence on X. U is an anti-homomorphism, and we set To. = Uo.-1 to get a direct homomorphism.

It remains to be shown that (T1x, y)

=

J

f(s)(T.x, y) ds for

every x E:: X, y E:: X* and f E:: L 1 • We may clearly restrict x to any dense subset of X and f to L. Thus, it is sufficient to show that (T1 .,z,y)

=

jf(s)(T.T,z,y) ds for all/,g E:: L. Now

if x andy are fixed, the linear functional ](f) = (T1x, y) satisfies I ](f) I ~ Bll x II II Y II II f II h and is therefore a complex-valued integral (the sum of its four variations). We therefore have from the Fubini theorem that

(T,. ,x, v) = ]t =

[J

f(s)g(s- 11) ds

l =J

f(s)U,g(s- 1t)J ds

jf(s)(T.T,x,y)ds

as desired. If X is a Hilbert space and T is a *-representation, we want to show that T. is unitary for every s, i.e., that (T.) * = (T.) - 1 = T,-1. But (T.)* = T,-1 c::::> (T.)*T, = T.-1T1 = T1,forall/E:: £1 c::::> (u,-1) * *f ~ j. as u runs through an approximate identity c::::> f* * u,-1 ~ j. * c::::> g * u,-1 ~ g*. *. Now by direct computation g*. * = ~(s- 1 )g• and g * u,-1 = ~(s- 1 )g" * u. Since we know that g' * u ~ g', the result follows. 32D. If we take X as the Banach space LP(G) for some fixed p.in [1, oo] and define the operator T, on X by T,g =f*K for every f E: £1, g E:: LP, then we can check directly that the mapping/ ~ T, is a bounded representation of the algebra L 1 (G). For instance, T, Teh = j * (g *h) = (f *g) * h = T1 .,h, giving T1 . , = T,T,. Moreover the inequality II f * g liP ~ II f II til g liP implies that II T, II ~ II j II h so that the representation has the bound 1. There is also an obvious group representation s ~ T. defined by T.j = j,-1, or [T.j](t) = f(s- 11). The transformations T. are all isometries by the left in variance of Haar measure.

130

THE HAAR INTEGRAL

This pair of representations is interconnected in exactly the same way as in the above two theorems. Thus if g E: LP and h E: U (1/p + 1/q = 1), then

J

f(s)(T.g, h) ds

=

JJ

j(s)g(s- 1t)h(t) ds dt

= (/ * g, h).

If p = 2, the transformations T. are unitary by the left invariance of Haar measure: (T.j, T sg) =

J

f(t)g(t) dt = (/, g).

J

f(s- 1t)g(s- 1t) dt =

It follows from our general theory that

the representation of L 1 is a *-representation. Of course, this fact can also be checked directly-the equality (/ * g, h) = (g, f* * h) can be computed directly from the Fubini theorem and the left invariance of the Haar integral. These representations are called the (left) regular representations of G and V(G) respectively. They are both faithful representations (one-to-one mappings), for if s ~ e we can choose g to1 vanish outside a small giving in particular enough neighborhood of e so that gg. = T.g ~gin V(G), and if/~ 0 then we can choose u E: L 1 n L 2 similarly so that II/* u - / ll2 < Ell/ ll2, giving in particular that T,u = / * u ~ 0 in L 2 (G).

0,

§ 33.

QUOTIENT MEASURES

In this section we consider invariant measure on quotient groups, and, more generally, quotient spaces, and prove the Fubini-like theorem relating such quotient measures to the invariant measures on the whole group and kernel subgroup. This material is not needed for most of the subsequent discussion, and since it gets fairly technical the reader may wish to omit it. The method of procedure is taken quite directly from Weil [48]. JJA. Let H be a fixed closed subgroup of G, and let I and J be respectively the left invariant Haar integrals of G and H. It is understood that ](/) is the integral over H of the restriction of/to H. If/ E: L(G), it follows from the uniform continuity off that the function/' defined by f'(x) = J(/z) = Jd(xt) is continuous. Also f'(xs) = f'(x) for every s E: H by the left invariance of Jon H, so that/' is constant on the left' cosets of H.

131

THE HAAR INTEGRAL

If/ E:

Lc, then the related function Fan G/H defined by F(a(x)) f'(x) = ](/z) = ]t/(xt) belongs to La.
Theorem. Iff E: ffi+(G), then /z E: ffi+(H) for every x and ](/z) = ]t/(xt) E: ffi +(K). If I,] and K are suitably normalized, then Kz]t/(xt) = I(j) for every f E: ffi +(G). Proof. We have seen above that the family of non-negative functions for which the theorem is true includes L +,with a suitable normalization for I. If the theorem holds for a sequence {fn} of L-bounded functions of ffi+ which converges monotonically to j, then I(j) = lim I(/n) = lim Kz]t/n(xt) = Kz(lim ]t/n(xt) = Kz]t(lim/n(xt) = Kz]rf(xt). Thus the family is L-monotone and hence equal to ffi +(G), q.e.d. Corollary. Iff E: V(G), then /z E: V(H) for almost all x, ]t/(xt) E: V(K) and Kz]t/(xt) = I(/). 33B. We return now to the case where H is closed but not necessarily normal. The mapping f --+ F of L(G) into L(G/ H) defined above is clearly linear, and we now show that it is onto. Lemma. IfF E: L +(G/H), then there exists that F(a(x)) = ](/z). .

f

E: L +(G) such

Proof. Let B be a compact subset of G/H such that FE: LB +, let A be a compact subset of G such that a(A) = Band choose h E: L +(G) such that h > 0 on A. Notice that ](hz) > 0 if the coset containing x intersects A, i.e., if x E: AH = a- 1 (B), and

132

THE HAAR INTEGRAL

that F(a.(x)) = 0 in (All)' = a.- 1 (B'), an open set. Therefore, g(x) = F(a.(x))/](hz) if ](hz) > 0 and g(x) = 0 if ](hz) = 0 defines an everywhere continuous function. Also g(x) is constant on the cosets of H. Therefore f = gh E: L +(G) and Jt/(xt) = g(x)]th(xt) = F(a.(x)), q.e.d. 33C. If H is not normal, it is still true that every element y E: G defines a homeomorphism of G/H onto itself: a.(x) --+ a.(yx), or xH --+ yxH. If an integral K is invariant under all these homeomorphisms, then the theorem of 33A goes through unchanged. More generally K may be relatively invariant in the sense that K is multiplied by a constant D(y) under the homeomorphism induced by y: K(F,;) = D(y)K(F), where, of course, F,;(a.(x)) = F(a.(yx)). The function D(y) is called the modulus, or the modular function, of K, and, like .r:l, is seen to be continuous and multiplicative. Theorem. If K is relatively invariant, with modulus D, then the functional M(j) = Kz]tD(xt)f(xt) is (essentially) the Haar integral I(j), and all the conclusions of 33Afollow. Proof. M(f,;) = KJtD(xt)f(yxt) = D(y- 1 )Kz[JtD(xt)f(xt)] 11 = Kz]tD(xt)f(xt) = M(j). Thus M(j) is left invariant on L(G) and equals I(j). The extensions tom+ and to D are the same as in 33A. 33D. Any continuous homomorphism of G into the multiplicative group of the positive real numbers will be called a real character. Examples are the modular functions .r:l and D. Theorem. In order that a real character D be the modular function for a relatively invariant measure K on the quotient space G/ H, where H is a closed subgroup of G, with modular function o, it is necessary and sufficient that D(s) = .r:l(s)/o(s) for all s E: H. Proof. If D is the modular function of K, then M(j) = I(j) as above and we have, for s E: H, fl(s)M(j)

= M(j") = Kz]tD(xt)f(xts- 1 ) = D(s)Kz]tD(xts- 1 )f(xts- 1 ) = D(s) o(s)Kz]tD(xt)f(xt) = D(s) o(s)M(j).

Thus .r:l(s) = D(s) o(s) for every s E: H.

THE HAAR INTEGRAL

133

Now suppose, conversely, that D(x) is a real character such that Ll(s) = D(s) 5(s) for every s €: H, and define the functional K on L(G/H) by Kr:(](f.)) = l(D-y). It follows from 33B that this functional is defined on the whole of L(G/H) once we have shown that the definition is unique, i.e., that, if ](f:r:) = 0, then I(D-y) = 0. But if ]t/(xt) = 0 for all x and if g €: L +(G), then

0 = I:r:]tg(x)D- 1 (x)f(xt) = ]d:r: 5(t-1 )g(xr 1 )D- 1 (x)f(x)

= I:r:{D- 1 (x)j(x)]t[5(t- 1 )g(xt- 1 )]} = I:r:D- 1 (x)j(x)]tg(xt). In order to prove that l(D-y) = 0 it is therefore sufficient to choose g €: L +(G) so that ](g:r:) = ]tg(xt) = 1 for every x for which j(x) ~ 0. But iff €: LA and if a function is chosen from L +(G/H) which has the value 1 on the compact set a(d), then the required function g is constructed by the lemma of 33B. It remains to check that the functional K thus defined on L +(G/H) is relatively invariant with D as its modular function: K:r:](/11 :r:)

= l(D-lj11) = D(y)l((D-lj) 11 ) = D(y)l(D-y) = D(y)K:r:](/:r:).

This completes the proof of the theorem. 33E. Returning to the case where H is normal we now see, since G/H has an invariant integral, with modular function D == 1, that Ll(s) = 5(s) for all s €: H. Taking H to be the normal subgroup on which Ll(t) = 1, it follows that 5 == 1 and H is unimodular. The real character Ll is itself a homomorphism having this group as kernel, and the quotient group is therefore isomorphic to a multiplicative subgroup of the positive real numbers. In a sense, therefore, every locally compact group is almost unimodular. Another sense in which this is true has been discussed in JOE.

Chapter VII LOCALLY COMPACT ABELIAN GROUPS

The V group algebra of a locally compact Abelian group is a commutative Banach algebra with an involution, and much of the general theory of Chapter V is directly applicable. Thus we can take over the Bochner and Plancherel theorems whenever it seems desirable, and we can investigate questions of ideal theory such as the Wiener Tauberian theorem. We shall be especially concerned with reorienting this theory with respect to the character function (the kernel function of the Fourier transform), and much of our discussion will center around topological questions. The character function is discussed and the character group defined in § 34 and various standard examples are given in § 35. § 36 is devoted to positivity, and includes the Bochner, Plancherel, and Stone theorems. A more miscellaneous set of theorems is gathered together in § 37, such as the Pontriagin duality theorem, the Wiener Tauberian theorem, and a simple form of the Poisson summation formula. The chapter concludes in § 38 with a brief discussion of compact Abelian groups; the general theory of compact groups is given in the next chapter. § 34.

THE CHARACTER GROUP

We know that each regular maximal ideal Min V is the kernel of a homomorphism f --+ j(M) of L 1 onto the complex numbers (see 23A, B) and that this homomorphism, as an element of the conjugate space (V) * of V, is uniquely represented by a func134

LOCALLY COMPACT ABELIAN GROUPS

tion a.M E:: Lrzl, }(M) =

J

f(x)a.M(x) dx (see 15C, D).

135 We shall

see below that a.M is (equivalent to) a continuous function, I a.M(x) I 1 and a.M(xy) = a.M(x)a.M(y). Thus a.M is a continuous homomorphism of G into the multiplicative group of complex numbers of absolute value one; such a function is called a character of G. Moreover, every character arises in the above way. Thus the space mt of regular maximal ideals of V is now identified with the set G of all the characters of G. Through this identification we know that G, in the weak topology of L
=

34A. Theorem. Giuen a fixed M E: mt, let f E:: L 1 be such that j(M) ~ 0. Then the function a.M defined by a.M(x) = }.,(M)/}(M) is a character. It is independent of j, uniformly continuous, and continuous in the two uariables x and M if the parameter M is also uaried. If u runs through an approximate identity, then u.,(M) conuerges uniformly to a.M(x). Proof. That a.M is uniformly continuous follows from the inequality I j.,(M) - } 11(M) I ~ II f., - /11 U1 and the fact that/., as an element of L 1 is a uniformly continuous function of x (30C). The inequality I j.,(M)- }.,.(Mo) I~ II /:r:- /.,. ll1 +I J.,.(M)/ •• (M0 ) Ishows that}.,(M) is continuous in the two variables x and M together, and the same holds for a.M(x) upon dividing by j(M). The remaining properties of a.M are contained in the equation

j(M)g.,(M) = j.,(M)g(M)

(which comes from f *g., =f., *g). Thus}.,/} = g.,/g, proving that a. is independent off. Taking g = / 11 and dividing by (}) 2 , we see that a.M(yx) = a.M(y)a.M(x). Thus a.M is a continuous homomorphism of G into the non-zero complex numbers. If I a.M(xo) I > 1, then I a.M(Xon) I = I a.M(Xo) In ---7 oo. But a.M is

136

LOCALLY COMPACT ABELIAN GROUPS

bounded; therefore II a.M II"" ~ 1. Then I a.M - 1 (x) I = I a.M(X-1 ) I ~ 1, so that I a.M(x) I == 1. Thus a.M is a character. Finally, taking g = u, we get u.,(M) = ka.M(x), where k = u(M), and since u(M) --+ 1 as u runs through an approximate identity
34B. Theorem. The mapping M --+ a.M is a one-to-one mapping of mr onto the set of all characters of G, and

=

j(M)

J

f(x)a.M(x) dx.

Proof. We first derive the formula. Given M, the homomorphism f --+ }(M) is in particular a linear functional on £1, and therefore there exists a. E:: L'" such that j(M) = Then

J

f(x)a.M(x) dx = limu

= limu

Jf(x)u.,-~(M) JJ

J

f(x)a.(x) dx.

dx

f(x)u(x- 1y)a.(y) dx dy

= limu Ju * u)(y)a.(y) dy =

Jf(y)~(y)

dy = }(M),

q.e.d. In particular, M is determined by the character a.M, so that the mapping M --+ a.M is one-to-one.

If a. is any character, then the linear functional

J

f(x)a.(x) dx is

multiplicative:

JJ = JJ

Ju * g)(x)a.(x) dx =

=

J

f(xy)g(y- 1 )a.(xy)a.(y- 1 ) dx dy f(x)a.(x)g(y)a.(y) dx dy

f(x)a.(x) dxJg(y)a.(y) dy.

137

LOCALLY COMPACT ABELIAN GROUPS

It is therefore a homomorphism, and if M is its kernel the equation

J

f(x)a(x) dx

J

= f(x)aM(x) dx shows

that a

= aM.

This theorem permits us to identify the set G of all the characters on G with the space ffil. of regular maximal ideals of V(G) (which has already been identified with, and used interchangeably with, the subset of (£1)* consisting of non-trivial homomorphisms). Generally from now on the domain of j will be taken to be G. Also we shall use the notation (x, a) for the value of the character a at the group element x, so that the Fourier transform formula is now written:

l(a) =

J

f(x)(x, a) dx.

34C. The topology of G is that of uniform convergence on compact sets of G. That is, if Cis a compact subset of G, E > 0 and a 0 E: G, then the set of characters U(C, E, a 0 ) = {a: I (x, a) - (x, ao) I < E for all x E: C} is open, and the family of all such open sets is a basis for the topology of G. Proof. That U(C, E, a 0 ) is open follows at once from SF. Moreover the intersection of two such neighborhoods of a 0 includes a third: U(Cl u c2, min (ElJ E2), ao) c U(Ch f ! ) ao) n U(C2, E2, ao). It is therefore sufficient to show that any neighborhood N of a 0 belonging to the usual sub-basis for the weak topology includes one of the above kind. Now N is of the form N = {a: I j(a) - j(ao) I < ~} for some/ E: V and o > 0. If C

r If I < 6/4, E = 6/211 fIll and I (x, a) - (x, Jc• + a 0) I < E on C, then I j(a) - j(ao) I ~ Jl f(a - iio) I ~ ~ II f [lt(6/2ll f ll1) + 2(6/4) = o. That is, U(C, E, ao) c N,

is chosen so that

L L,

q.e.d.

34D. Theorem. The pointwise product of two characters is a character, and with this definition of multiplication G is a locally compact Abelian group. Proof. It is obvious that the product of two characters is a character. The constant 1 is a character and is the identity under

138

LOCALLY COMPACT ABELIAN GROUPS

multiplication. For any character a, the function a- 1 is a character, and is the group inverse of a. These facts are evident, and there remains only the proof that multiplication is continuous. But if Cis a compact subset of G, and E > 0, then

I a{3

- aof3o

I ~ Ia

- ao

1·1

f3o

I + I a 1·1 f3

- f3o

I<

E

on C if I a - ao I < E/2 on C and I {3 - {3 0 I < E/2 on C. It follows from 34C that multiplication is continuous. 34E. Remark: It may seem to the reader that it would have been more natural to define aM so that }(M) = instead of }(M) =

J

J

f(x)aM(x) dx

f(x)aM(x) dx. The only justification for our

choice which can be given now is that it stays closer to the formalism of the scalar product and convolution; we have defined }(M) as (/, aM) = [f * aM](e). Later we shall see that it leads to the conventional Fourier transform on the additive group of the reals, whereas the other choice leads to the conventional inverse Fourier transform. Notice that the character which we assign to a regular maximal ideal M is the inverse of the character assigned by the other formula.

§ 35.

EXAMPLES

In calculating the character group for specific simple groups we have two possible methods of procedure. First, a character can be determined from the homomorphism of the group algebra associated with it, and this method was used in working out the examples of § 23. The second, and more natural, method in the present context, is to proceed directly from the definition of a character as a continuous homomorphism of G on to T. The direct verification of the correct topology for C takes perhaps a little longer because the theorem of SG on the equivalence of a weak topology with a given topology does not now intervene, but this is a minor objection at worst.

35A. Theorem. The character group of a direct product G1 X G2 of two locally compact Abelian groups is (isomorphic and homeomorphic to) the direct product C1 X C2 of their character groups.

LOCALLY COMPACT ABELIAN GROUPS

139

Proof.

The algebraic isomorphism is almost trivial. Any character a on G1 X G2 is the product of its restrictions to G1 and G2, a((xh x 2)) = a((xh e2))a((eh x2)), and these restrictions are characters on C1 and C2 respectively. Conversely the function a((xh x 2)) = a 1(x1)a 2(x 2) defined as the product of two characters on G1 and G2 respectively is evidently a character on G1 X G 2. The one-to-one correspondence a ~ (ah a2) thus established between (G 1 X G2 ) « and C1 X C2 is clearly an isomorphism. Let C be a compact subset of G containing e = (eh e2) and of the form c = c1 X c2. If I Q! - a 0 I < E on c, then (taking x2 = e2) I Q!l - 0!1° I < Eon cl and, similarly, I 0!2 - 0!2° I < Eon c2. Conversely if I O!I - 0!1° I < Eon cl and I 0!2 - 0!2° I < Eon c2, then I Q! - a 0 I = I 0!10!2 - a1°a2° I < 2E on c. It follows from 34C that the topology of (G1 X G2)" is the Cartesian product topology of X C2. This fact also follows directly from the next theorem.

c1

35B. Theorem. If His a closed subgroup of a locally compact Abelian group G, then the character group of the quotient group G/H is (isomorphic and homeomorphic to) the subgroup of C consisting of those characters on G which are constant on Hand its cosets. Proof. If (3 is a character on G/H, then a(x) = (3(Hx) is continuous on G and a(x1x2) = (3(Hx1x2) = (3(Hx1Hx2) = (3(Hx1){1(Hx2) = a(x 1)a(x 2), so that a(x) is a character on G. Conversely, if a(x) is a character on G which is constant on H, and therefore on each coset of H, the function {1 defined on G/ H by (3(Hx) = a(x) is a character. We still have to show that the topology induced in this subgroup is the correct topology for the character group of G/H. We know that C = {Hx: x E:: F} is compact in G/H ifF is a compact subset of G, and that every compact subset of G/H arises in this way (see 28C). Then I (3(u) - (3 0 (u) I < E for all u in C if and only if I (3(Hx) - (3 0 (Hx) I < E for all x in F, and the result follows from 34C.

35C. Theorem. Every character a(x) on the additive group R of the real numbers is of the form a(x) = e'""', and!<. is isomorphic and homeomorphic toR under the correspondence e'""' ~ y.

140

LOCALLY COMPACT ABELIAN GROUPS

+

Proof. Every continuous solution of the equation a(x y) = = ea"', and, since a is bounded, a must be pure imaginary. Thus a(x) = e'""', for some real y.

a(x)a(y) is of the form a(x)

Conversely, every real number y defines the character e'""', and this one-to-one correspondence between R and R is an isomorphism by the law of exponents. The set of characters a such that I a(x) - 1 I < E on [- n, n] is a neighborhood of the identity character, and the set of such neighborhoods is a neighborhood basis around the identity. But I e'""' - 1 I < E on [- n, n] if and only if y is in the open interval (- o, o), where o = (2/n) arc sin (E/2), so that the mapping R ~ R is continuous both ways at the origin, and therefore everywhere.

35D. Theorem. Each character on R/ I is of the form e2 ..'""', and (R/I)~ is isomorphic and homeomorphic to I under the correspondence e2.. '""' ~ n. Proof. This follows at once from 35B and C; the character e'""' has the constant value 1 on the integers if and only if y = 21rm for some integer m, and the discrete subgroup {21rm} of R is, of course, isomorphic (and trivially homeomorphic) to I.

35E. Theorem. Each character on I is of the form e2.. '"'", where 0 ~ x < 1, and 1 is isomorphic and homeomorphic to R/I under the correspondence e 2,.iz" ~ x. Proof. If a(n) is a character and a (1) = e2..'", then a(n) = e2..'""·

It is clear that every suchy defines a character, and that the mapping y - e2 .. '"" is a homomorphism of R onto 1, with kernel I. Thus 1 is isomorphic to R/I. The set {y: 0 ~ y < 1 and I e 2,.'"" - 1 I < E, n = 1, · · ·, N} is a basic neighborhood of e in I, and is easily seen to be an open interval about 0 in R/I. Conversely, any open interval about 0 in R/I is such a set (with N = 1 and E to be determined), and the two topologies are identical. 35F. Each of the three groups R, R/ I, and I is thus (isomorphic and homeomorphic to) its own second character group. These results are special cases of the Pontriagin duality theorem, which asserts that every locally compact Abelian group is its own second character group, and which we shall prove later in § 37.

LOCALLY COMPACT ABELIAN GROUPS

141

The Fourier transform formula becomes for these groups the well-known formulas }(y) = [

j(x)e-i11 "' dx -co

}(n)

=

1

Cn = J/(x)e-2rinz dx

}(x) = £f(n)e-2rizn dn =

L::: . . Cne-2rinz.

The fact that the Fourier transform is initially a mapping of V(G) into e(C) reduces here to the fact that each of the above integrals is absolutely convergent if j E:: V (which, in the last case, means I::: . . I Cn I < oo), and} is continuous and vanishes at infinity (in the first two cases). The corresponding formulas for multiple Fourier series, multiple Fourier transforms, and mixtures of these, can be written down by virtue of 3SA. The duality theorem can also be directly verified for these product groups.

§ 36.

THE BOCHNER AND PLANCHEREL THEOREMS

36A. We shall now apply to the group algebra V(G) the general commutative Banach algebra theory of § 25 and § 26, keeping in mind the extra information that the space C of characters (maximal ideal space of V(G), space of homomorphisms) is itself a locally compact group, with its own Haar measure and its own group algebra. We start with positivity. We have already seen in 31G that a positive linear functional on L 1 (G) is extendable if and only if it is continuous. The Bochner theorem of 261 can thus be taken to refei· to positive definite functions p E.: L 00 • Moreover, the hypothesis in that theorem that/*" = p- is now automatically

met, forf* ... (a) = J!cx- 1 )(x, a) dx = J!(x)(x- 1 , a) dx =

(Jj(x)

Finally, if IJ.p is the measure on

C asso-

(x, a) dx)-

=

(}(a))-.

ciated with the positive definite function p E:: L 00(G), we have

J

f(x)p(x) dx

J

J

= }(a) dp.p(a) = j(x) [Jcx, a) dp.p(a)] dx

142

LOCALLY COMPACT ABELIAN GROUPS

for all/ E.: D, implying that p(x)

=

Jcx, a) dp,p(a) for almost all

x. Any function p defined this way is positive definite, for it clearly belongs to L, and

The Bochner theorem can therefore be reformulated in the present con text as follows: Theorem.

The formula p(x) = Jcx, a) dp,(a)

sets up a norm preserving isomorphism between the convex set of all finite positive Baire measures p, on C and the convex set of all positive definite junctions p E.: Lrr>(G). Corollary. Every positive definite junction p E.: Lrr>(G) is (essentially) uniformly continuous. Proof. This is because the measure /J-p is mostly confined to a compact set C and the character function (x, a) is continuous in x uniformly over all a E.: C. Given Ewe choose a compact subset C of C such that p,p(C') < E/4 (C' being the complement of C), and then find a neighborhood V of the identity in G such that I (xh a) - (x2, a) I < E/2p,p(C) if x 1x 2- 1 E.: V and a E.: C (by SF again). Then

ljlcxh a) when x1x 2-

1

-

Cx2, a)J dp,p(a) 1 ~ f + f ~ ~ + ~ Jc Jc, 2 2

E.: V. Thus Jcx, a) dp,p(a) is uniformly continuous.

36B. Let

is dense in D, and if/ E.: L 1 n Lrr> and u runs through an approximate identity (also chosen from L 1 n Lrr>), then j * u is at once an approximation to j and a linear combination of four positive definite functions like (j u) * (j u) *. By exactly the same argument [D n
+

+

143

LOCALLY COMPACT ABELIAN GROUPS

The Bochner theorem above and the abstract Plancherel-like theorem of26J now lead us to the following D-in version theorem.

Theorem.

If j E: [D

n


f(x) = J(x, a)j(a) da for almost all x, where da is the Haar measure of malized.

G suitably

nor-

Proof. Let tp be the positive functional defined on the ideal L of uniformly continuous functions of D by tp(j) = f(e). We have observed earlier (31G) that a function p E: D n

that p E: D(m) and (j, p) =

f 1P dm whenever p E: D

n

1P and

f E: D. The formula p(x) = J(x, a)p(a) dm(a) is the same as

that of the Bochner theorem in 36A. We have left only to prove that m is the Haar measure of G. But if p E: D n IP, then a direct check shows that p(x)(x, a 0 ) E: D n IP for any character ao, and we have Jp(a) dm(a) = p(e) = p(e)(e, ao) = Jp(a 0 a) dm(a),

where we have used the directly verifiable fact that Pao is the Fourier transform of pa 0 • Since the algebra generated by L 1 n 1P is dense in L I, and the transforms p are therefore dense in e( G), the above equation shows m to be translation invariant and therefore the Haar measure of G. Remark: If Tis the Fourier transformation f ~ 1 from L 1 (G) into e(G), then the mapping p. ~ p defined by p(x) = J(x, a) dp.(a) from the space of all bounded complex-valued Baire measures on into L 00 (G) ( = L 1 (G) *) is the adjoint mapping T*. It must be remembered, however, that the identification of Loo with (L 1 ) * is taken to be a conjugate-linear rna pping in order that the

G (e(G) *)

144

LOCALLY COMPACT ABELIAN GROUPS

ordinary scalar product formula can be used: P(/)

=

J

f(x)p(x) dx,

where P and p are corresponding elements of (£1)* and£«~. The same remark holds for the identification of e(C)* with the space of bounded Baire measures. It is this twist which is responsible for the factor (x, o:) in the integrand ofT* whereas (x, o:) occurs in that ofT. The inversion theorem above shows that T* = T- 1 when Tis restricted to [L 1 n
}(y) =

[c-:.'12c-;ll"'

dx

we differentiate with respect to y and then integrate by parts we find that u = j(y) satisfies the differential equation du = -uy dy, so that}(y) = Cc- 11' 12• In order to determine C we observe that}(O) = C =

f_. . c-'"'

12

dx.

C 2 = J~[c-c"''+ 11'JI2 dxdy =

Then

.r:.,.£«~c-•'12rdrd0 =

2'11",

and C = (2·nV~. Therefore if we take Haar measure on R to be dx/(27r)Y: instead of dx the function c-'"'12 transforms into itself.

LOCALLY COMPACT ABELIAN GROUPS

145

Now we must choose Haar measure C dy on~= R so that the inverse formula

holds. Since this equation is real it is the same as its complex conjugate above and C must be 1/(211-)3'~. Thus ifj(x) E: [V n P], then j(y) E: [V n P], where f(y)

and f(x)

. dx = -1- J"" f (x )e-•y:JJ (211-)3'~

-QO

= -1- J"" J(y)e•Y:JJ dy· A

(211v~

_..,

,

'

both integrals being absolutely convergent. This, of course, is the classical choice for the Haar measures on R and~= R. It is easy to see that all other pairs of associated measures are of the form (X dx/(27r)v., dy/X(27r)Y.). The proper pairing of measures for compact and discrete groups such as R/I and I will follow from 38B.

36D. The Planchere1 theorem. The Fourier transformation f --+ Tf = j preserves scalar products when confined to [L 1 n
146

LOCALLY COMPACT ABELIAN GROUPS

Now if FE: V n L 2 (G), then j = T*(F) E: L 2 (G). For j is bounded and uniformly continuous, and if g E: U n L 2 (G), then I (g,j) I = I (l, F) I ~ II F ll21i g 1!2 = II F 1!2 ·II g 1!2, proving that n L 2(G), then f E: L 2(G) and II f 112 ~ II F 112· If also H E: hE: L 2 (G) and lif = T*(H *F) E: [V n (V]. Thus H * F = T(hj) by the inversion theorem, and since such convolutions H *Fare dense in L 2 (G) we are finished. We have also, as a corollary of the above method of proof, the following:

v

Corollary.

U(G) is semi-simple and regular.

Proof. Iff E: U and g E: L 2, then f * g E: L 2 and T(j *g) = }g. Iff~ 0, the convolution operator U1 defined on L 2 (G) by f is not zero (32C) and therefore the operator on L 2 (G) defined by multiplication by J is not zero. That is, iff~ 0 in U(G), then/~ 0 in e(G), and V(G) is semi-simple. The reader is reminded that V(G) is regular if for every closed set F c G ~nd every point a t{_ F there exists f E: U such that J = 0 on F and/(a) ~ 0. We prove directly the stronger assertion about local identities in L 1 (25C), namely, that if F and U are subsets of G such that F is compact, U is open and F c U, then there exists f E: U (G) such that j = 1 on F and f = 0 on U'. To see this we choose a symmetric Baire neighborhood V of the identity in G such that V 3F c U and let g and h be respectively the characteristic functions of V and a Baire open set between VF and V 2F. Then the convolution g * h is identically equal to m(V) on F and to zero outside of U. Since g * h = T(gh), the desired function is/= gh/m(V). 36E. We conclude this section by translating the representation theorem of 26F to the group setting. If T (s ~ T.) is any unitary representation of an Abelian group by unitary transformations on a Hilbert space H, the considerations of § 32 tell us that Tis completely equivalent to a *-representation of the group algebra V (G). Then, as in 26F, T can be "transferred" to the algebra of transforms J and then extended to a *-representation of the algebra of all bounded Baire functions on G. The trilinear functional I(j, x, y) = (T1x, y), for fixed x andy, is a bounded complex-valued integral on e(G), and satisfies I(Fg, x, y) =

LOCALLY COMPACT ABELIAN GROUPS

147

I(F, T,x,y) for FE: e(G) and g E: V(G). Thus (T.T,x,y) = (T,.-~x,y) = l((s-t, a)g(a), x,y) = l(J, T,x,y). The operator T.,E corresponding to the characteristic function f{JE of the set E is a projection, designated PE, and (PEx,y) is the complex-valued measure corresponding to the above integral. (The correspondence E ---+ P E is called a projection-valued measure.) The above equation can therefore be written (T,u,y) =f(s, a) d(Pau,y), where we have set u = T ,x. Since such elements are dense in H, this restriction on u can be dropped. We can summarize this result, using the symbolic integral of spectral theory, as follows:

Stone's theorem. If T is a unitary representation of the locally compact Abelian group G by unitary transformations on a Hilbert space H, then there exists a projection-valued measure PE on G such that The projections PE form the so-called spectral family of the commutative family of operators T.. Each such projection defines a reduction of the representation T into the direct sum of TPE operating on the range of PE and T(l - PE) = TPE' operating on its orthogonal complement in H. The further analysis of the reducibility of T depends directly on the multiplicity theory of the family of projections P E and is beyond the reach of this book.

§ 37.

MISCELLANEOUS THEOREMS

This section contains the Tauberian theorem and its generalizations, the Pontriagin duality theorem, and a simple case of the Poisson summation formula. 37 A. The Tauberian theorem, as proved in 25D, applies to any regular commutative Banach algebra, and hence (see 36D, corollary) to the V-group algebra of an Abelian locally compact group. The extra condition of the theorem, that the elements whose Fourier transforms vanish off compact sets are dense in the algebra, is always satisfied in this case. For l is dense in

148

LOCALLY COMPACT ABELIAN GROUPS

L 2 " and hence the elements of L 2 whose transforms lie in 1. are dense in L 2 • Since every function of U is a product of functions in L 2 , it follows via the Schwarz inequality that the functions of L 1 whose transforms lie in 1. are dense in L 1• We can now restate the Tauberian theorem.

Theorem. If G is a locally compact Abelian group, then euery proper closed ideal of U(G) is included in a regular maximal ideal. Corollary 1. Iff~ U is such that translates off generate L 1•

j neuer uanishes, then the

Proof. By hypothesis/ lies in no regular maximal ideal. The closed subspace generated by the translates off is (by 31F) a closed ideal, and therefore by the theorem is the whole of L 1 • Corollary 2. (The generalized Wiener Tauberian theorem.) Let G be any locally compact Abelian group which is not compact. Let k ~ U be such that k neuer uanishes and let g ~ L~ be such that the continuous function k * g uanishes at infinity. Then f * g vanishes at infinity for euery f ~ U. Proof. The set I of functions f ~ U such that f * g vanishes at infinity is clearly a linear subspace, and I is invariant since f, * g = (/*g)., vanishes at infinity iff* g does. I is also closed. For iff~ 1 and E is given, we can choose h ~I such that II f- h ll1 < E/(21\ g 1\oo), and we can choose a compact set C such that \ (h *g) (x) I < E/2 outside of C. Since I (/ * g)(x) - (h * g)(x) I ~ II f - h ll1ll g \loo < E/2, it follows that I (f*g)(x) I< E outside of C. Thereforef*g vanishes at infinity and/ ~ I. Therefore (by 31F again) I is a closed ideal, and since it contains k it is included in no regular maximal ideal. It follows from the theorem that I= U. Corollary 3. (The Wiener Tauberian theorem.) If k ~ L 1 (R) is such that k neuer uanishes and g ~ L~ is such that (k * g)(x) --+ 0 as x --+ +oo, then (/*g) (x) --+ 0 as x --+ +oo for euery f ~ U. This is not quite a special case of the above corollary since it concerns being zero at infinity only in one direction. However the proof is the same.

LOCALLY COMPACT ABELIAN GROUPS

149

37B. The Tauberian theorem in the general form of the theorem of 37 A asserts that a closed ideal is the kernel of its hull if its hull is empty, and is thus a positive solution to the general problem of determining under what conditions a closed ideal in a group algebra V(G) is equal to the kernel of its hull. That the answer can be negative was shown by Schwartz (C. R. Acad. Sci. Paris 227, 424-426 (1948)), who gave a counter example in the case where G is the additive group of Euclidean 3-space. The positive theorem also holds for one-point hulls. This was proved for the real line by Segal [44] and extended, by means of structure theory, to general locally compact Abelian groups by Kaplansky [27]. Helson has given a proof of the generalized theorem which is independent of structural considerations [24]. In the proof below we use a function which has been discussed both by Helson and by Reiter (Reiter's paper is unpublished at the time of writing) to give a direct proof that the condition of Ditkin (25F) is valid in the L 1 algebra of any locally compact Abelian group. This immediately implies (by 25F) the strongest known positive theorem. Lemma. If a regular, semi-simple Banach algebra A has an approximate identity and has the property (of Wiener's theorem) that elements x such that x has compact support are dense in A, then d satisfies the condition D at infinity. Proof. This lemma has content only if :m is not compact. We must show that for every x and E there exists y such that j has compact support and II xy - x II < E. Because A has an approximate identity there exists u such that II xu - x II < E/2, and then, by the Wiener condition, there exists y such that j has compact support and II y - u II < E/211 x II· Then II xy - xu II < E/2 and II xy - x II < E, q.e.d. Corollary. It then follows that, if I is a closed ideal with a compact hull, then I contains every element x such that hull (I) c int (hull (x)). Proof. It follows from the theorem that there exists y such that j has compact support and II xy - x II < E. But then xy E: I by 25D, and since I is closed we have x E: I.

150

LOCALLY COMPACT ABELIAN GROUPS

37C. In order to establish the condition D at finite points we introduce the function mentioned above. Let U be any symmetric Baire neighborhood of the identity e in G such that m(U) < oo, where m is the Haar measure of G. Let V be any second such neighborhood whose closure is compact and included in U, and taken large enough in U so that m(U)/m(V) < 2. Let u and v be the characteristic functions of U and V respectively, and let u and u be their inverse Fourier transforms in L 2(G). Then the function j = uu/m(V) belongs to V(G), and IU ll1 ~ II u II2JI u ll2/m(V) = II u ll2ll vll2/m(V) = [m(U)/m(V)]l-
=

1 if {3 E:::

w.

Lemma. Giuen any compact set C c G and any E > 0 there exists a function j E::: V such that j = 1 in some neighborhood of the identity ein G, II j lb < 2, and IU- II; Ill< Ejor euery x E::: C. Proof. We define j as above, taking for U any symmetric Baire open subset of the open set {a: 11 - (x, a) I < E/4 for all x E::: C} (see SF). Since j -II; = [u(u - u.,) + u.,(u - u.,)]/m(V) and since, with the above choice of u, II u - u., ll2 2 = fl u(a) - (x, a)) 12 da ~ m(U) lub..E:u 11 - (x, a) 12 < m(U)(E/4) 2 if x E::: C (and similarly for II u- u., ll2), we have IU- j., \11 ~ 2[m(U)m(V)]~E/4m(V) < E for every x E: C, q.e.d.

o

Corollary. Iff E::: V (G) and }(e) = 0 then f * j ~ 0 as U decreases through the symmetric Baire neighborhoods of 2. Proof. Given 6 we choose C in the above lemma to be symmetric and satisfying f * j(y)

r If I < Jc,

6/8, and set E = 6/211 fIll· Then

= J!(x)j(x- 1y) dx = J!(x)(j(x- 1y) - j(y)) dx, and

L J:.

+ < II/ II1E + II/* j \l1 ~ fiJ(x) I 1\j.,-t - j ll1 dx = (6/8)4 = 6. This corollary leads at once to the condition D at the identity e; it then follows for other points upon translating.

LOCALLY COMPACT ABELIAN GROUPS

151

Theorem. There exists a uniformly bounded directed set of functions v E: U(G) such that v = 0 in a neighborhood of ~ and such thatf * v --+ f for every f E: V(G) such thatj(~) = 0. Proof. Let u run through an approximate identity and set v = u - j * u. Then II v Ill ~ 3 and v = u - uj = 0 in the neighborhood aU wherej = 1. Also II/- f * v ll1 ~ II/- f * u ll1 + ll/•jll1ll u ll1--+ O,q.e.d. It follows that Silov's theorem (25F) is valid for the group algebras of locally compact Abelian groups. We restate the theorem slightly. Theorem. Let I be a closed ideal in L 1 (G) and f a function of U(G) such that j(a) = 0 whenever a is a character at which every function of I vanishes; that is, hull (I) c hull (j), or f E: k(h(l)). Suppose furthermore that the part of the boundary of hull (f) which is included in hull (I) includes no non-zero perfect set. Then/ E: I.

Corollary. If I is a closed ideal in V(G) whose hull is discrete (i.e., consists entirely of isolated points) then I= k(h(l)).

37D. We have seen that the character function (x, a), x E: G, a E: C, is continuous in x and a together. Also that (x, a1a2) = (x, a 1 )(x, a 2 ). Therefore, every element x in G defines a character on C, and the mapping of G into C thus defined is clearly an algebraic homomorphism. The Pontriagin duality theorem asserts that this mapping is an isomorphism and a homeomorphism onto C, so that G can be identified with C. By using the considerable analytic apparatus available to us we can deduce a short proof of this theorem. The space [V n tY] corresponds exactly to the space [l 1 n
152

LOCALLY COMPACT ABELIAN GROUPS

tions, it follows that G is imbedded inC with the relative topology it acquires from C, and that G is dense in C. (Otherwise we could construct a non-zero convolution of two functions of (V n L 2 )(C) which vanishes identically on G, contradicting the isomorphic extension.) Since the subsets of G and C on which I j I > E > 0 have compact closures in both topologies and one is dense in the other, we can conclude that their closures are identical. Therefore G = C, finishing the proof of the duality theorem. 37E. The Poisson summation formula. Let j be a function on R = ( -oo, oo) and let F be its Fourier transform. Let a be any positive number and let {3 = 21rja. Then under suitable conditions onf it can be shown that

Va E: . . j(na)

=

v'(3 E: .. F(n{3),

which is the Poisson formula. The formal proof proceeds as follows. The function g(x) = E: .. j(x + na) is periodic and hence can be considered to be defined on the reals mod a. The characters of this quotient group are the functions eimz(2T/a) = eimPz, and its character group, as a subgroup of~ = R (see 35B), is the discrete group {m{3}: "'' Then G(m{3)

! E:= _,. [e-imP"'f(x + na) dx a o 1 i(n+l)ae-•mP"'j(x) . 1 fao . = - E:=-oo dx = e-•mP"'f(x) dx a na a

=

!

rae-imPzg(x) dx aJo

=

-oo

v2; F(m{3)

= -

a

=

J~- F(m{3). a

But by the inversion theorem g(x) fore

E: .. j(na)

= g(O) =

=

E: .. G(m{3)

E: .. eimP"'G(mf3). =

There-

~ E: .. F(m{3),

giving the Poisson formula. We now present a proof in the general situation, with the function f severely enough restricted so that the steps of the formal proof are obviously valid. More could be proved by careful

153

LOCALLY COMPACT ABELIAN GROUPS

arguing, but our purpose here is only to display the group setting of the theorem. Theorem. Let G be a locally compact Abelian group and let H be a closed subgroup. Let the Haar measures on G, Hand G/H be

adjusted so that such that g(y) y.

Then

f= f

Ja

f, and letf be a function of [L 1 n
Ja;HJH

= Lf(xy) dx is a continuous function (on G/H) of

rf(x) dx = Ja;H r- }(a) da.

JH

Proof. The character group of G/ H is by 35B the set of characters a E:: C such that (xy, a) = (y, a) for every x E:: H. Then

r

g(a) =L (y, a)g(y) dy =L (y, a) [ f(xy) dx] dy G/H G/H JH

Lr G/HJH

(xy, a)f(xy) dxdy =Lex, a)f(x) dx =}(a). G

But by the inversion theorem g(y)

=

f.--..

Ja!H

(x, a)g(a) da.

There-

fore

f f(x) dx = g(O) = (~J(a) da = f.--.. }(a) da, ~ ~H ~H which is the desired formula. § 38.

COMPACT ABELIAN GROUPS AND GENERALIZED FOURIER SERIES

If G is compact and Abelian, its Fourier transform theory is contained in the analysis of H*-algebras carried out in § 27 and its application in Chapter VIII. However, a simple direct discussion will be given here.

if and only if Cis discrete. Proof. Reversing the roles of G and C we remark, first, that, if G is discrete, then V(G) has an identity (31D) and C is com38A. Theorem. G is compact

pact as the maximal ideal space of a commutative Banach algebra with an identity (19B). Second, since V(G) is semi-simple

154

LOCALLY COMPACT ABELIAN GROUPS

and self-adjoint, it follows from 26B that, if C is compact, then an identity can actually be constructed for V(G) by applying the analytic function 1 to an everywhere positive function j in the transform algebra, and (by 31D again) G is discrete. The theorem as stated now follows from the duality theorem. The second part of the argument can be proved directly as follows. If G is compact, then the set of characters a such that II a - 1lloo
value 0 elsewhere. Since convolutions on C correspond to ordinary pointwise products on G, is the transform of u = 1. Thus

u(a)

=

= Jcx, a) dx =

u 1 if a = e and

1, this implies that JJ.(G)

= 0 otherwise. Since (x, e)

J

= 1 dx = 1, so that the measures

match as stated in the theorem. The above identity can also be written Jcx, a 1 )(x, a 2 ) dx =

Jcx,

a 2a 1

1)

Corollary.

dx = 1 if a 2 =

a1

and = 0 if a 2

=r!= a1.

Thus:

The characters on G form an orthonormal set.

38C. The characters form a complete orthonormal set in L 2 (G) and the (Fourier series) development f(x) = Li (f, an)an of a function/ E: L 2 (G) is the inverse Fourier transform. Proof.

If j(x) E: L 2 , then j E:

1.2,

so that j(a) = 0 except on

a countable set {an} and, if Cn = iean) =

U,

an), then

J

f(x)(x, an) dx =

LOCALLY COMPACT ABELIAN GROUPS

155

which is Parseval's equation. The continuous function /n = Ei ciai is by its definition the in verse Fourier transform of the finite-valued function }n equal to ci at ai if i ~ n and equal to 0 elsewhere. Since}n clearly converges (L 2 ) to}, it follows that/n converges (V) to/ and that the formulaf(x) = Ei cn(x, an) =

J

}(a) (x, a) dais the inverse Fourier transform.

38D. Euery continuous function on G can be uniformly approximated by finite linear combinations of characters. This follows from the Stone-Weierstrass theorem, for the characters include the constant 1 and separate points. Since the product of two characters is a character, the set of finite linear combinations of characters forms an algebra having all the properties required. A direct proof can be given. First/ is approximated uniformly by f * u where u is an element of an approximate identity in V(G). Then,asabove,jisapproximatedV byfn = E!ci(x,ai) and u by Un = Eidi(x, ai), where the sequence {an} includes all the characters at which either} or ~ 0. Then/* u is approximated uniformly by /n * Un = Ei ci di(X, ai). The last equation can be verified by writing out the convolution or by remembering that/n * Un = T*(}ngn). The partial sums of the Fourier expansion off will in general not give uniform approximations.

u

Chapter f7l/I COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

If G is compact L 2 (G) is an H*-algebra, and the whole theory of § 27 can be taken over. Thus L 2 (G) is the direct sum of its (mutually orthogonal) minimal two-sided ideals, and it turns out that these are all finite dimensional and consist entirely of continuous functions; their identities (generating idempotents) are the characters of G. These facts, which are in direct generalization of the classical theory of the group algebra of a finite group, are presented in § 39, and in § 40 they are used to obtain the complete structure of unitary representations of G. § 41 presents the theory of almost periodic functions on a general group, following a new approach. § 39.

THE GROUP ALGEBRA OF A COMPACT GROUP

39A. The Haar measure of a compact group is finite and generally normalized to be 1. Then !! f ll1 ;:; ; II f liz ;:; ; II f lloo· Moreover, on any unimodular group the Schwarz inequality implies that II f * g lloo ;:;;; II f !Ia[[ g lla· Together, these two inequalities show that L 00 and L 2 are both Banach algebras. The latter is in fact an H* -algebra; it is already a Hilbert space, and the further requirements that (/ * g, h) = (g,j* *h), II/* lla = II f II a and f */* ~ 0 iff ~ 0 are all directly verifiable. Thus f */* is continuous and (/ *f*)(e) = !If [[a 2 > 0 iff~ 0, and the other two conditions follow from the inverse invariance of the Haar measure of G. The fact that L 2 (G) is an H*-algebra 156

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

157

means that its algebraic structure is known in great detail from the analysis of § 27. Moreover, further properties can be derived from the group situation; for instance we can now show that all minimal ideals are finite dimensional. Because of 27F this will follow if we can show that every closed ideal contains a non-zero central element, and we give this proof now before restating the general results of§ 27. Lemma 1. A continuous function h(x) is in the center of the h(yx). algebra L 2 (G) if and only if h(xy)

=

Proof.

h E: center (L 2 ) if and only if J[h(xy)j(y- 1 )

-

j(y)h(y- 1x)] dy = 0.

Replacing y by y - 1 in the second term of the integrand and remembering that f is arbitrary, the result follows. This proof works just as well for the V algebra of a unimodular locally compact group. Lemma 2. Every non-zero closed ideal I c L 2 contains a nonzero element of the center of L 2 •

Proof. (After Segal [44].) Choose any non-zero g E: I and let = g * g*; f is continuous since j(x) = (g.,, g) and since g., is a continuous function of x as an element of L 2 • Thenf(axa- 1) is a continuous function of a if x is fixed, and, as an element of L~ is a f

continuous function of x (see 28B). Thus h(x) =

J

j(axa- 1) da

is continuous, and h(e) = f(e) = I! g 11 2 2 > 0, so that h -;e. 0. Finally, if a is replaced by ay- 1 in the integral for h(yx), it follows from the right invariance of Haar measure that h(yx) = h(xy), and hence that his central. Moreover it is easy to see that h E: I, for/ E: I and/.." E: I by 31F, and it then follows from the Fubini theorem as in 31F that h E: Iu =I, q.e.d. It now follows from 27F that every minimal two-sided ideal N of L 2 (G) is finite-dimensional and contains an identity e. It then follows further that every function f E: N is continuous, for f = f * e, or f(x) = (f.,, e), and f., as an element of L 2 is known to be a continuous function of x.

158

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

We now state the global structure theorem from § 27 as modified above.

Theorem. L 2 (G) is the direct sum in the Hilbert space sense of its (mutually orthogonal) minimal two-sided ideals N". Each minimal two-sided ideal Na is a finite-dimensional subspace of continuous functions. It has an identity ea, and the projection on Na of any function f E: L 2 (G) is fa = f * ea = ea *f. The regular maximal ideals of L 2 are the orthogonal complements of the minimal ideals, and euery closed ideal in L 2 is at once the direct sum of the minimal ideals which it includes and the intersection of the maximal ideals which include it. The only central functions in Na are the scalar multiples of the identity ea, and euery central function f E: L 2 has a Fourier series expansion/= L Xaea, where "Aaea = f *ea. If G is Abelian as well as compact, then each Na is one-dimensional and the identities ea are the characters of G. 39B. The fine structure of a single minimal two-sided ideal N is also taken over from § 27, with additions. This time we state the theorem first.

Theorem. If It, ···,I,. are orthogonal minimal/eft ideals whose direct sum is a minimal two-sided ideal N, then I 1 *, · · ·, I.,.* are minimal right ideals with the same property and Ii* n I; = I,** I; is one-dimensional. Therefore there exist elements eii E: Ii* n I;, uniquely determined if i = j and uniquely determined except for a scalar multiple of absolute ualue 1 in any case, such that eii * ekz = e,z if j = k and = 0 if j :;e. k, e,; = e;i* and e = Lie.... The functions {ei;} form an orthogonal basis for N and the correspondence f +-+ (ci;), where f = L ci;ei;, is an algebraic isomorphism between Nand the full n X n matrix algebra, involution mapping into conjugate transposition. Furthermore II e,; ll2 2 = n and neik(xy) = Li ei;(x)eil,(y). The left ideals Ii are isomorphic to each other (under the correspondence Iieii = I;) and to euery other minimal left ideal in N.

Proof. The only new facts concern the equation ne,k(xy) = Li ei;(x)e;k(y). The left ideal Ik has the basis elk, · · ·, e.,.k, and is invariant under left translation. Therefore, given x and i

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

159

there exist constants Ci;(x) such that eik(xy) = Li Ci;(x)e;k(y). Then ei;(x) = [ei1 * e1;](x) = Jeil(xy)eli(y- 1) dy =

Lm Cim(x) Jem1(y)eil(Y) dy

= c;;(x)

=

c;;(x)

II e;1 1!2 2

II eu 1!2 2,

which gives the desired result if II eu ll2 2 = n. But II eu ll2 4 = II eu ll2 2eu(e) = LJ=1 eli(x)e;1(x- 1) = L:i I eli(x) 12, and integrating we get II eu ll2 4 = I:i II eli ll2 2 = nil eu ll2 2, or II eu ll2 2 = n as desired. We have used the symbol e ambiguously for both the identity of the group G and of the ideal N. 39C. The identities ea are called characters in the compact theory as well as in the Abelian theory; this again is terminology taken over from the classical theory of finite groups. It is not surprising that there is a functional equation for characters here as well as in the Abelian case. Theorem. If j E: L~(G), then there exists a scalar X such that j /X is a character if and only if there exists a scalar 'Y such that J j(xsx- 1t) dx

=

j(s~(t) ·

Proof.

Given a character (or identity) ea in a minimal ideal Na, we have Jea(xsx- 1t) dx = Jea(sx- 1tx) dx because ea is central, and then we observe as in 39A, Lemma 2, that because of the left in variance of Haar measure the integral is a central function oft for every fixed s. Since this central function belongs to Na, it is a scalar multiple of ea, Jea(xsx- 1t) dx = k(s)ea(t).

We evaluate k(s) by taking t = e, gtvmg ea(s) = k(s)n and Jea(xsx- 1t) dx = ea(s)ea(t)/n,

where n = ea(e) =

II ea ll2 2·

If

j = Xea, then clearly J J(xsx- 1t) dx = j(s)j(t)/'Y, where 'Y = n>...

160

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

Now suppose conversely that/ satisfies this identity. As above

J

f(xsx- 1t) dx is a central function in s for every fixed t, and it

follows from the assumed identity that/ is central. If Na is some minimal ideal on which f has a non-zero projection f * ea = Xea, then

JJ JJ

f(xsx- 1t)ea(t- 1 ) dx dt

=

f(xsx- 1t)ea(t- 1 ) dx dt =

XJea(xsx- 1 ) dx

j~)

=

Xea(s)

J

f(t)ea(t- 1 ) dt = j(s)

(J~ea) ·

Since (f, ea) = [J * ea](e) = Xea(e) = Xn, we see that f(s) = -rea(s)/n, which is the desired result. Notice also that 'Y = Xn as before. 39D. A function/ E:: L 2 is defined to be almost invariant if its mixed translatesfab generate a finite-dimensional subspace of L 2 • This subspace is then a two-sided ideal by 31F, and is therefore a finite sum of minimal ideals. Thus/ is almost invariant if and only iff lies in a finite sum of minimal two-sided ideals. The functions of a minimal ideal can be called minimal almost invariant functions, and the expansion/= Lfa can be viewed as the unique decomposition off into minimal almost invariant functions. The set of all almost invariant functions is a two-sided ideal, being, in fact, the algebraic sum La Na of all the minimal ideals. The ordinary pointwise product jg of two almost invariant functions/ and g is also almost invariant. For if {j.t} is a set of mixed translates off spanning the invariant subspace generated by f and similarly for {g;}, then the finite set of functions {fig;} spans the set of all translates of fg. Thus the set of almost in.:. variant functions is a subalgebra of e(G). Theorem. Every continuous function on G can be uniformly approximated by almost invariant functions. Proof. If u is an approximate identity, then u •f approximates f uniformly (for if u = 0 off V, then I u •f(x) - f(x) I =

I Ju(y)(j(y- 1x)- f(x))

dy

I ~ max11-lcv II / 11 - !I lao,

etc.), and

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

161

it is sufficient to show that u *f can be uniformly approximated by almost invariant functions. Let the sequence an include all the indices for which /a :;C 0 or Ua :;C 0, and choose n so that II f- Lifa; liz < E and II U - Li Ua; liz < E. Then II U* j Li Ua• *fai lloo = II (u - Li ua) * ( j - Lila) Iloa < Ez, q.e.d. Another proof follows from the above remark that the vector space of almost in variant functions is closed under ordinary pointwise multiplication upon invoking the Stone-Weierstrass theorem. 39E. We conclude this section by showing that the ideal theory of L 1 (G) is identical with that of V(G). Since II f lb ;:; ; II f liz the intersection with L 2 of any closed ideal I in D is a closed ideal in V. If I :;t. 0, then I n L 2 :;e. 0. In fact, I n L 2 is dense (V) in I, for if 0 :;t. j E: I and u E: V n L2 belongs to an approximate identity, then u *f belongs to I n L 2 and approximates f in D. Since the L 1-closure of a closed ideal in L 2 is obviously a closed ideal in D, the mapping I ~ I n L 2 is a oneto-one correspondence between the closed ideals of L 1 and L 2 • The minimal ideals of L 2, being finite-dimensional, are identical with their D-closures and any L 1-closed ideal I is the direct sum in the D sense of the minimal ideals it includes. In order to see this we approximate f E: I by g E: I L 2 and g by a finite sum of its components. This approximation improves in passing from the L 2 norm to the D norm so that f is approximated L 1 by a finite sum of minimal almost invariant functions in I. The other LP algebras can be treated in a similar way, and a unified approach to the general kind of algebra of which these are examples has been given by Kaplansky [26], who calls such an algebra a dual algebra.

n

§ 40.

REPRESENTATION THEORY

40A. Let T be any bounded representation of G. That is, T is a strongly continuous homomorphism x ~ T., of G onto a group of uniformly bounded linear transformations {T.,} of a normed linear space H into itself.

162

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

Theorem.

If H is a Hilbert space, with scalar product (u, u),

then [u, u] = j(T.,u, T.,u) dx is an equivalent scalar product under which the transformations T., are all unitary.

Proof. It is clear that [u, u] has all the algebraic properties of a scalar product except possibly that u o:;e 0 implies [u, u] o:;e 0. This follows from the fact that (T.,u, T.,u) is a continuous function of x which is positive at x = e. Also [T11 u, T 11u] = j(T.,T11u, T.,T 11u) dx = j(T., 11 u, T., 11u) dx = j(T.,u, T.,u) dx = [u, u], so that

the transformations T., are all unitary in this scalar product. This completes the proof if H is finite-dimensional. If B is a bound for the norms {II T., II: x E: G} and if I u I = (u, u)~, Ill u Ill= [u, u]l'', we have at once that Ill u 111 ~Ell u II· Since u = T.,-au if u = T.,u, we have II u II ~ Ell T.,u and therefore II u II ~ Bill u Ill- Therefore the norms II u II and Ill u Ill are equivalent and the proof of the theorem is complete. 40B. We shall assume from now on that T is always a representation of G by unitary operators on a Hilbert space H. We know (see 32B, C) that Tis equivalent to a norm-decreasing, involution-preserving representation of V by bounded operators on H. Since £2 c V and II f ll2 ~ II f ll1 if G is compact and ,.,.(G) = 1, we thus have a norm-decreasing representation of L 2 (G), which we shall denote by the same letter T. Our next arguments are valid for the *-representations of any H*-algebra.

Theorem. Euery *-representation of an H*-algebra A is uniquely expressible as a direct sum of (faithful) representations of certain of its minimal two-sided ideals. Proof. Let N 1 and N 2 be distinct (and hence orthogonal) minimal two-sided ideals of A and let H be the representation space. Iff E:: Nh g E: N 2 and u, u E: H, then (T,u, Tgu) = (u, T 1..1 u) = 0 since f*g = 0. If we normalize H by throwing away the intersection of the nullspaces of all the operators T1 and for every index a set Ha. equal to the closure of the union of the ranges of the operators T, such that f E:: Na., and set T"

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

163

equal to the restriction of T to N.,., then it is clear that the subspaces H.,. are orthogonal and add to H and that T is the direct sum of the representations T"' of the (simple) algebras N.,.. Since each N.,. is a minimal closed ideal, it follows that T"' is either 0 or faithful, and the kernel of T is exactly the sum of the ideals N.,. for which T"' = 0. 40C. The above representations T"' are not necessarily irreducible and we can now break them down further, although no longer in a unique manner. In our further analysis we shall focus attention on a single minimal ideal N and a faithful *-representation T of N by operators on a Hilbert space H, it being understood that the union of the ranges of the operators T 1 is dense in H. Moreover, we shall assume that N is finite-dimensional; our proof that the irreducible constituents are all equivalent is not valid if N is infinite-dimensional. Given a non-zero u1 E: H, there exists a minimal left ideal I c N, with generating idempotent e, such that T.u 1 -;e. 0. Then H 1 = {T1 u1 :f E: I} is a finite-dimensional subspace of H, invariant under the operators Th f E: N, and operator isomorphic to I under the correspondence f ~ T1 u1 • That the mapping is one-to-one follows in the usual way from the observation that, if it were not, then the kernel would be a non-zero proper subideal of I. That it is an operator isomorphism follows from the equation Te(T1 u1 ) = (TeT1 )u 1 = Te1 u., which establishes the desired relation oetween left multiplication by g on I and operating by T e on H 1 • Finally H 1 is irreducible, for if u = T1 u1 E: H1, then there exists g E: N such that gf = e, so that Teu = T 8 u1 and the subspace generated by u is the whole of H 1 • If H 1 is a proper subspace of H, we take u2 E: H/- and notice that then (Teu 2 , T 1 u1 ) = (u 2 , Te"'f, u1 ) = 0, so that any minimal subspace derived from u2 in the above manner is automatically orthogonal to H 1 • Continuing in this way we break up H into a direct sum of finite-dimensional irreducible parts H.,.. If T"' is the irreducible representation derived from T by confining it to H.,., then the preceding paragraph and 27E imply that the T"' are all equivalent. We have proved: Er:ery faithful representation of a minimal closed ideal is a direct sum of equivalent irreducible represent ali ons.

164

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

40D. We can now restrict ourselves to the case where T is an irreducible representation on a finite-dimensional Hilbert space V. Let N be a minimal ideal on which T is an isomorphism and let (ei;) be the matrix as described in 39B. We shall use the notation T(j) instead of T 1 . T(e,.,.) is a projection since e,.,. 2 = e,.,.; let V,. be its range. If v1 is any non-zero vector of Vh then Vj = T(e;1)vl is a vector in vb for T(e;;)v; = T(e;;e;1)v1 = T(e; 1 )v 1 = v;, and by a similar argument the vectors v; have the property that v; = T(e;i)vi. Since T(ei;)vn = 0 if j o:;e n, the set v; is linearly independent, and if V' is the subspace it spans, the equation v; = T(e;i)Vi shows that the coefficient matrix in the expansion of any f E::: N with respect to (ei;) is identical with the matrix of T(j) with respect to the basis {v;} in V'. Thus V' is invariant under T, and is operator isomorphic to any minimal left ideal in N. Also, since Tis irreducible, V' = V. If we now use the hypothesis that T is a *-representation, i.e., that T(j*) = T(j)*, then II v; W= (T(e;1)v1, T(e;1)v1) = (vt, T(e1;e;1)v1) = (vh T(e11)v1) = (vt, v1) = II V1 11 2 , and (vi, v;) = (T(ei 1)vt, T(e; 1)v1) = (vh T(elie;1)v1 = 0 if i o:;e j. Therefore, the basis {vi} is orthonormal if II v1 II = 1. We now again assume that T arises from a representation T., of a compact group. Since U.,f = e"' *f on N, and e"'(y) = e(yx- 1) = L: eii(yx- 1) = Li,i ei;(y)e;i(x- 1 )/n = L: [ei;(x)/n]ei;(y), it follows that the matrix for T., with respect to the orthonormal basis {v;} of Vis ei;(x)/n. Any other orthonormal basis for V arises from the one considered here by a change of basis transformation, and under this transformation the functions eii give rise to a new set of similar functions related to the new basis. We have proved:

Theorem. Let T be an irreducible representation of G by unitary transformations on the finite-dimensional Hilbert space V. Let T be extended to L 2 (G) as usual, and let N be the minimal ideal on which T is an isomorphism, the kernel of T being the regular maximal ideal M = Nl. If ci;(x) is the matrix ofT., with respect to any orthonormal basis for V, then the functions ei;(x) = nci;(x) are matrix generators for N as in 39B, and every set of such matrix generators arises in this way.

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

§ 41.

165

ALMOST PERIODIC FUNCTIONS

In this section we offer the reader an approach to almost periodic functions which as far as we know is new. We start off with the simple observation that under the uniform norm the left almost periodic functions on a topological group G form a commutative C*-algebra (LAP) and therefore can be considered as the algebra e(:m) of all complex-valued continuous functions on a compact space :m, the space of maximal ideals of LAP. The points of G define some of these ideals, a dense set in fact, and :m can be thought of as coming from G by first identifying points that are not separated by any f E: LAP, then filling in new points corresponding to the remaining ideals, and finally weakening the topology to the weak topology defined by LAP. The critical lemma states that when G is considered as a subset of :m the operations of G are uniformly continuous in the topology of :m and hence uniquely extensible to the whole of :m. We thus end up with a continuous homomorphism a of G onto a dense subgroup of a compact group :m, with the property that the functions on G of the form/( a(s)), f E: e( :m), are exactly the almost periodic functions on G (i.e., that the adjoint mapping a* is an isometric isomorphism of e(:m) onto LAP). 41A. A left almost periodic function on a topological group G is a bounded continuous complex-valued function f whose set of left translates S, = {/., s E: G} is totally bounded under the uniform norm. The reader is reminded that a metric space S is totally bounded if and only if, for every E > 0, S can be covered by a finite number of spheres of radius E. Also, a metric space is compact if and only if it is totally bounded and complete. Since the closures, of S 1 in the uniform norm is automatically complete, we see that f is left almost periodic if and only if S1 is compact. Let LAP denote the set of all left almost periodic functions on G. If g is a second function in LAP, the Cartesian product space X Sg is compact and the mapping (h, k) ~ h k is continuous. Therefore the range of this mapping, the set of all sums of a function in S1 with a function in S g, is compact, and the subset St+B is therefore totally bounded. ·we have proved that

s,

+

166

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

LAP is closed under addition, and it is clear that the same proof works for multiplication. Now suppose thatr E:: LAP andr converges uniformly tof. Since II f., II..,= II !II.., the sequence/n, converges uniformly to j.,, uniformly over all x E:: G. If, given E, we choose fn so that II fn - f II.., < E/3 and choose the points x, so that the set {fn .,.} is E/3-dense in S /", then the usual combination of inequalities shows that {/.,J is E-dense in S 1. That is,/ E:: LAP, proving that LAP is complete. Iff is left almost periodic, then so are its real and imaginary parts and its complex conjugate. Thus LAP is closed under the involution f - ]. We have proved:

r.,-

r-

Theorem. Under the uniform norm the set of all left almost periodic functions on a topological group G is a commutatiue C*algebra with an identity. 41B. It follows from the above theorem (and 26A) that LAP is isomorphic and isometric to the space e(:m:) of all complexvalued continuous functions on the compact Hausdorff space :m: of its maximal ideals. Each points E:: G defines a homomorphism f - f(s) of LAP onto the complex numbers and there is thus a natural mapping a of G into :m: (where a(s) is the kernel of the homomorphism corresponding to s). The subsets of :m: of the form {M: I}(M) - k I < E} form a sub-basis for the topology of :m:. Since the in verse image under a of such a set is the open set {s: I /(s) - k I < E}, the mapping a is continuous. a(G) is dense in :m: since otherwise there exists a non-zero continuous function on :m: whose correspondent on G is identically zero, a contradiction. The mapping a is, of course, in general many-to-one and it is not a priori clear that the families of unseparated points in S are the cosets of a closed normal subgroup and that a therefore defines a group structure in the image a(G) c :m:. Supposing this question to be satisfactorily settled there still remains the problem of extending this group structure to the whole of :m:. An analysis of these questions shows the need of a basic combinatorial lemma, which we present in the form of a direct elementary proof that a left almost periodic function is also right almost periodic.

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

167

The key device in this proof is the same as in the proof that the isometries of a compact metric space form a totally bounded set in the uniform norm.

Lemma. Iff E:. LAP and E > 0, there exists a finite set of points {b.:} such that given any b one of the points b.: can be chosen so that I f(xby) - f(xb.:y) I < Ejor every x andy. Proof. Let the finite set {/aJ be E/4-dense among the lefttranslates of f. Given b E:. G and i, la,b is within a distance E/4 of /a; for some j, and letting i vary we obtain an integer-valued function j(i) such that II /a,b - /aj(i) Ilao < E/4 for every i. For each such integer-valued function j(i) let us choose one such b (if there is any) and let {b;} be the finite set so obtained. Then, by the very definition of {b;}, for every b there exists one of the points bk such that II !a,b - !a,bk llao < E/2 for all i. Since for any x E:. G we can find a.: so that II /1) -/a; II < E/4, we have II /xb -

/:r.b" II,., ~ II /xb - /a;b llao

+ II /a;b

- /a;bk "'"'

+

I /a;bk - /,.bk ll:c

<

E/4 + E/2 + E/4 = E. That is, for every b E:. G there exists one of the points bk such that I f(xby) - f(xbky) I < E for every x and y, q.e.d.

Corollary 1. The set {/.:;} defined by /.:;(x) = f(b.:xb;) is 2Edense in the set of all mixed translates {Ia b}. Corollary 2. If x andy are such that I f(b.:xb;) - /(b.:yb;) I < for all i and j, then I f(uxv) - f(uyv) I < 5Ejor all u and v E:. G.

E

Proof. Given u and v we choose i and j so that II f(uxv) f(b.:xb;) llao < 2E. This inequality combines with the assumed inequality to yield I f(uxv) - f(uyv) I < 5E, as desired. 41C. We now show that multiplication can be uniquely extended from G to the whole of ~ by showing that, given Mh M 2 E:. ~ and setting = a(x), if x 1 - M 1 and x 2 ~ M2, then ~x 2 converges. What we already know is that, given f and E, there exist weak neighborhoods N 1 and N 2 of the points M1 and M 2 respectively such that, if xh :9 1 E:. N 1 and x2, Y2 E:. N2, then I j(x1x2) - f(YlY2) I < Ej this follows from the second corollary of the paragraph above upon writing/(x 1x2) - j(y1y2) = j(x1x2) - j(x1y2) + j(x1y2) - j(x2y2). The family of sets {X';X2: X1 E:. N1

x

168

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

.

and x 2 E: N 2 } obviously has the finite intersection property as N 1 and N 2 close down on M 1 and M 2 respectively, and, mt being compact, the intersection of their closures is non-empty. If M and M' are two of its points, then I}(M) - }(M') I < E for every E > 0 and every f E: LAP by the second sentence above, so that M = M'. We conclude that, given M 1 and M 2, there is a unique point M such that, for every weak neighborhood N(M), there exist weak neighborhoods N 1(M1) and N 2 (M2 ) such that X1 E: N1 and x2 E: N 2 imply that ~'X2 E: N. This shows first that the product M 1M 2 is uniquely defined, and second, letting X1 and 2 converge to other points in N 1 and N 2, that the product M1M2 is continuous in both factors. The reader who is familiar with the theory of uniform structures will realize that the above argument is a rather clumsy way of saying that ~'X 2 is a uniformly continuous function of and in the uniform topology of mt and hence is uniquely extensible to the whole of mt. We still have to show the existence and continuity of inverses, i.e., loosely, that, if ---+ M, then x- 1 converges. But by Corollary 2 of 41B 1 given/ E: LAP and E > 0 there exists a weak neighborhood N(M) such that, if x,y E: N, then I j(uxu) - f(uyu) I < E for all u and u. Taking u = x- 1 and v = y - 1 this becomes I f(y- 1) - /(x- 1) I < E and proves exactly in the manner of the above paragraph the existence of a unique, continuous homeomorphism M ---+ M-1 of M onto itself such that x- 1 = (x- 1) 6 for all X E: G. Thus MM- 1 = lim = and M- is the inverse of M. We have our desired theorem.

x

xl

x2

x

(xx-1) e

1

Theorem. If G is any topological group, there is a compact group mt and a continuous homomorphism a of G onto a dense subgroup of mt such that a function/ on G is left almost periodic if and g(a(s)) only if there is a continuous/unction g on mt such thatf(s) (i.e.,/= a*g).

=

There is a further remark which we ought to make in connection with the above theorem, namely, that the group mt is unique to within isomorphism. For if mz:' is any second such group, with homomorphism {3, then {3*- 1a* is an algebraic isomorphism of e(mz:) onto e(mz:') and defines an associated homeomorphism 'Y of M onto M'. Since 'Y is obviously an extension of {3a- 1, which is

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

169

an isomorphism on the dense subgroup a(G), "Y itself is an isomorphism. That is any two such groups M and M' are continuously isomorphic. 41D. Because of the above theorem the set of (left) almost periodic functions on a group G acquires all the algebraic structure of the group algebra of a compact group. Thus any function of LAP can be uniformly approximated by almost invariant functions, that is, by almost periodic functions whose translates generate a finite-dimensional subspace of Lrz>. The more precise £2 theorem according to which any almost periodic function has a Fourier series expansion into a series of uniquely determined minimal almost invariant functions is also available, but it has an apparent defect in that the Haar integral on the associated compact group seems to be necessary to define the scalar product. However, von Neumann [39] showed that an intrinsic definition can be given for the mean M(j) of an almost periodic function by proving the' following theorem.

Theorem. The uniformly closed conuex set of functions generated by the left translates of an almost periodic function f contains exactly one constant function. The value of this constant function will, of course, be designated M(j). Von Neumann proved the existence and properties of M(j) by completely elementary methods, using only the definition of almost periodicity. It will be shorter for us in view of the above theory to demonstrate the existence of M(j) by using the Haar integral on ;m:, Since it will turn out that M(j) =

Jj

dp,, we will also have proved that the expansion theory based

on the von Neumann mean is identical with that taken over from the associated compact group. Proof of the theorem. The convex set generated by S 1 is the set of finite sums .L: c.J(a .. x) such that ci > 0 and .L: ci = 1. We have to show that M(j), and no other constant, can be approximated uniformly by such sums. In view of the isometric imbedding of S 1 into e(;m:) and the fact that a(G) is dense in ;m:, it will be sufficient to prove the theorem on ;m:, Given E, choose V so that I f(x) - f(y) I < E when xy- 1 E:: V and choose h E:: Lv +,

170

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

Since grr; is compact, there exists a finite set of points ah · · ·, a,. such that H(x) = L: h(xai- 1) > 0 on mL The functions g.; = ha;; H thus have the property that g.; = 0 outside of Va.; and that

J

L: g.;(x) = 1. If c.; = g.;, then ljf(x) dx - L: c.;f(a.;y) I = I L: since

I f(xy

- f(a.;y)

I<

E

J

g.;(x)[j(xy) - f(a._y)] dx

I<

Thus M(j) =

J

when g.;(x) =;e. 0.

E

f is

approximated by the element L: c.;fa; of the convex set determined by S,. Conversely, if k is any number which can be so approximated,

II k -

L

c.;f(a.;y) [loo

<

integrating, so that k =

E,

then I k -

Jf =

J

j(y) dy I

<

E

follows upon

M(f), q.e.d.

Corollary. Using M(j) as an integral on the class of almost periodic functions, euery almost periodic function has a unique L 2 expansion as a sum of minimal almost invariant junctions. 41E. If G is Abelian, then its associated compact group Gc is both compact and Abelian, and can be readily identified. The minimal almost invariant functions on Gc are, of course, its characters, and it follows that the minimal almost invariant functions on G are the characters of G, for f(x) = h(a(x)) is clearly a character on G if h is a character on Gc. The adjoint of a thus determines an algebraic isomorphism between the character groups of G and G.. Since Cc is discrete, it is completely identified as being the abstract group of characters on G with the discrete topology. Thus Gc is the character group of G0 , where G0 is the character group C, but in the discrete topology. On an Abelian group the approximation and expansion theorem states that every almost periodic function can be uniformly approximated by linear combinations of characters and has a unique L 2 expansion as an infinite series of characters. In particular a function is almost periodic on the additive group of real numbers if and only if it can be uniformly approximated by linear combinations of exponentials, L:;oc,.f'~n"'. This result will become

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

171

the classical result when we have shown that the von Neumann definitions of almost periodicity and mean values are equivalent to the classical definitions. This we do in the next two theorems.

41F. Theorem. A continuous/unction! on R is almost periodic

if and

only if, for euery E, there exists T such that in euery interval of length T there is at least one point a such that II f - /a.lloo < E.

Proof. We first rephrase the above condition as follows: for every E there exists T such that for every a there exists b €: [- T, T] such that II fa. - /b lloo < E. Iff is assumed to be almost periodic and if the set {/aJ is E-dense in St. then we have only to take [- T, T] as the smallest interval containing the points {a;! to meet the rephrased condition. Conversely, suppose that T can always be found as in the rephrased condition. We first show that/ is uniformly continuous. To do this we choose o such that I f(x + t) - f(x) I < E if x €: [- T, T] and I t I < o. Then, given any x, we can choose b €: [ -T, T] so that II f,- /b lloo < E and have

I J(x + t)

I ~ I J(x + t) + I f(b + t) Thus II f- /t II < 3E if It I < o. - J(x)

+ t) I f(b) I + I f(b)

J(b -

- f(x)

I < 3E.

Now let {at, · · ·, a,.) be o-dense on [- T, T]. Given a, we first choose b €: [- T, T] so that II /a. - fb II < E and then a; such that I ai - b I < o, and therefore such that II fb -fa, lloo = II fb-a.;- Jlloo < 3E as above. Thus II /a.- !a, II ~ II fa.- /b II+ II /b -/a, II < 4E, proving that the set {/a 1, • • · , fa..l is 4E-dense in S 1 and therefore that f is almost periodic.

41G. Theorem.

M(f)

=

1 lim 2 T

T--+"'

fT f(x) dx. -T

Proof. Given E, we choose the finite sets {cd and Iai} such that II M(f) - "'£.. c;J(x - ai) lloo < E. It follows upon integrating that I M(f) -

("'£.. ci [TTJ(x

- ai) dx)/2T

I < E.

If we then

172

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

choose T

>

T.

=

(II f

JT j(x- ai) dx 2T -T

-1

where

lloo·max I ai

1 2T

=-

iT-a; f(y)dy -T-a;

I 6,1 < I ai I II f 1!/T < E. I M(/)

[)/E, it follows that

- -1

2T

1 2T

=-

JT f(y) dy + 6i -T

Therefore

JT f(y) dy I < 2E -T

whenever T > T., q.e.d. The alternate characterizations of almost periodicity and M(/) given in the above two theorems are the classical ones for the real line. Thus the basic classical results now follow from our general theory. 41H. We ought not to close this section without some indication of the usual definition of the compact group Gc associated with G by its almost periodic functions. We have defined a left almost periodic function f to be such that the uniform closure S, of the set S 1 of its left translates is a compact metric space. Let U/ be the restriction of the left translation operator Ua ( Uaf = /a-1) to S,. U / is an isometry on S1 and an elementary (but not trivial) argument shows that the homomorphism a ~ U/ of G into the group of isometries on S, is continuous, the uniform norm being used in the latter group. Now, it can be shown by an argument similar to that of 41B that the isometries of a compact metric space form a compact group under the uniform norm. Thus the uniform closure G1 of the group { U/: a E: G} is a compact metric group, and the Cartesian product IT 1 G1 is compact. Let G be the group of isometries T of LAP onto itself such that for every f the restriction T 1 of T to S1 belongs to G1 • Each isometry in G determines a point of G, and it is easy to see that the subset of ITt G, thus defined is closed and hence compact. That is, G itself is compact if it is given the following topology: a sub-basic neighborhood of an isometry T 0 E: G is specified in terms of a function f E: LAP and a positive number E as the. set of isometries T E: G such that II T 1 - To' II < E. Also, under this topology the homomorphism a of G into G defined by a(s) = U, is continuous.

rr,

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

173

If/ E: LAP, then IJ(x) - f(y) I~ llf,- f~~[[oo ~ II U/- U/ II· That is, the function F defined by F( U/) = f(x) is uniformly continuous on a dense subset of G1 and hence is uniquely extensible to the whole of GI· Since the projection of Gonto GJ( T ~ T') is continuous, we may then consider F to be defined and continuous on the whole of G. Moreover F(a(s)) = f(s) by definition. Thus for each f E: LAP there exists F E: e(G) such that f(s) = F(a(s)). Conversely any continuous F on the compact group G is almost periodic as we have observed earlier, and F(a(s)) is therefore almost periodic on G. Thus G is (isomorphic to) the compact group associated with G. This method of constructing the compact group is essentially that given by Weil [48].

Chapter IX SOME FURTHER DEVELOPMENTS

In this chapter we shall give a brief resume of results in areas of this general field which are still only partially explored. Our aim is not so much to expound as to indicate a few of the directions in which the reader may wish to explore the literature further. It is suggested in this connection that the reader also consult the address of Mackey [35] in which many of these topics are discussed more thoroughly.

§ 42.

NON-COMMUTATIVE THEORY

42A. The Plancherel theorem. We start with the problem of the generalization of the Plancherel theorem. It may have occurred to the reader that there is a great deal of similarity between the Fourier transform on an Abelian group and the expansion of a function into a sum of minimal almost invariant functions on a compact group. The following properties are shared by the two situations. There exists a locally compact topological space C, and with each point a E:: C there is associated a minimal translation-invariant vector space V.. of bounded continuous functions on G. For each f E:: V there is uniquely determined a component function /a E:: V.. such that, for each fixed x,j.. (x) is a continuous function of a vanishing at infinity. There is a measure p. on G such that, for each f E:: V n CP(G), j .. (x) E:: Ll(p.) for each x andj(x) =

J

j .. (x) dp.(a).

V.. is a (finite-dimen-

sional) Hilbert space in a natural way, and, iff E:: V 174

n

L2(G),

SOME FURTHER DEVELOPMENTS

then

II f ll2 2= jll f., 11 2diJ(a)

(Parseval's theorem).

175 Thus the

mapping f ----+ f., can be extended to the whole of L 2 (G) preserving this equation, except that now f., is determined modulo sets of IJ-measure 0. This is the Plancherel theorem. In each subspace V., there is a uniquely determined function e., characterized as being a central, positive definite, minimal almost invariant function. The component f., in V., of a function f E: L 1 is obtained as f., = f *e., = U1e.,, where U is the left regular representation. The subspaces V., are invariant under U, and, if U':z is the restriction of U to V.,, then the function e., is also given by e.,(x) =trace (U"'.,). The component/., has a similar alternate characterization, f.,(x) = trace (U"' .,U"'! ). The Abelian case is simplified by the fact that the subspace V., is one-dimensional, consisting of the scalar multiples of a single character a(x), but is the more complicated case in that a function f is expressed in terms of its component functions f., by an integration process rather than a summation process. The compact group case is simpler in the latter respect; f is an ordinary sum of its components f.,. But the subspaces V., are not one-dimensional. It is, of course, impossible without some experience to select from the above rich collection of properties common to the compact groups and the Abelian locally compact groups those which could be expected to hold, and constitute a Plancherel theorem and its associated expansion theorem, on a locally compact group which is neither compact nor Abelian. However, the success of the above composite theory is in some sense due to the presence of sufficiently many central functions, and the extent to which it is a general phenomenon must be connected with the extent to which the general group has or does not have central functions, or at least central behavior of some kind. What success has been achieved in the general situation has been based on von Neumann's reduction theory for rings of operators [40] which we now proceed to outline. We suppose given a measure space S, with each point a E: S an associated Hilbert space H.,, and a vector space of functions f on S such that/(a) E: H., for every a E: Sand II /(a) II E: L 2 (S).

176

SOME FURTHER DEVELOPMENTS

The ordinary L 2-completion of this vector space is then a Hilbert space H of functions f of thti: above type, and we write

H =

JHa

dp.(a), where p. is the given measure on S.

If we are given a separable Hilbert space H and a complete Boolean algebra
JHa

dp.(a), such

that the projections of
JHa

dp.(a) and if T is a bounded operator on H, then

T is defined to be a direct integral if and only if there exists for each a a bounded operator Ta on Ha such that, if x =

JXa

dp.(a)

E.: H, then T(x) = J(Taxa) dp.(a). One of von Neumann's basic theorems is that, if T commutes with all the projections Ec associated with the measurable subsets of the index space S, then T is a direct integral. An algebra of bounded operators on a Hilbert space H is said to be weakly closed if it is a closed subset of the algebra of all bounded operators in the weak topology generated by the functions J.,, 11 (T) = (Tx, y) and if it is closed under the involution T -----) T*. For any set a of operators, a' is defined to be the set of all operators commuting with every operator in a. Then it is a fundamental fact that a *-closed set a of operators is a weaklyclosed algebra if and only if a = a". If a is a weakly-closed algebra, then a n a' is clearly the center of a The center of a consists only of scalar multiples of the identity if and only if the union a U a' generates the algebra B(H) of all bounded operators on H; in this case a is said to be a factor. Von Neumann and Murray's dimensional analysis of factors, with the resulting classification of all factors into types I,., I~, I It, II~ and III, is one of the basic results of the theory, but will not be gone into here.

177

SOME FURTHER DEVELOPMENTS

Now let a be any weakly-closed algebra of bounded operators on H and let H _=

J

Ha d,.,.(C!I.) be the expression of Has a direct

integral with respect to the center of a. That is, the projections Ec associated with the measurable subsets C of the index space are exactly the projections in the center of a. Let T n be a sequence of operators in a which generates a, let Tn =

J

Tn,a d,.,.(C!I.)

be the direct integral expression ofT n and let aa be the weaklyclosed algebra of bounded operators on Ha generated by the sequence T n,a• Then a is the direct integral of the algebras aa in the sense that a bounded operator T on H belongs to a if and only if it is a direct integral

J

Ta d,.,.(C!I.) where Ta E: aa.

Another of von Neumann's fundamental theorems is that in this direct integral decomposition of a almost all the algebras aa are factors, and that a' is the direct integral of the commuting factors aa'· Godement, Mautner and Segal ([21], [37], [38], [45]; Godement's major work 'm th'ts d'trection is unpublished at the time of writing) have considered V(G) on a separable unimodular locally compact group G as both a right and left operator algebra on the Hilbert space L 2 (G) and have observed the fundamental fact that, if .c and CR are the weak closures of these operator algebras, then .c' = (R and (R 1 = .c. Thus the intersection .c n (R is the center of each algebra, and von Neumann's reduction theory implies that H = L 2 (G) can be expressed as the direct integral of Hilbert spaces Ha in such a way that each operator A E: .c is the direct integral of operators Aa acting on Ha, and the weaklyclosed algebra £a generated over Ha by the operators Aa is, for almost all a, a factor. This analysis of .c into factors is what corresponds to the analysis of V (G) into minimal parts in the simpler theories, and a Parseval formula is valid; if/ E::

v

n L2,

then

II f lb 2

= Jtrace

U,"U1 .a dCJI.. However, these oversimplified statements conceal a multitude of complications and the reader is referred to the literature cited above for details. Godement has so framed his work that it applies to other two-sided representations in addi-

178

SOME FURTHER DEVELOPMENTS

tion to the two-sided regular representation mentioned above, and he has introduced a topology into his index space, starting with the observation that the common center .c n
SOME FURTHER DEVELOPMENTS

179

commutative self-adjoint Banach algebra, and the other is the theorem of Gelfand and Neumark [13] to the effect that every C* -algebra is isometric and isomorphic to an algebra of bounded operators on a Hilbert space. These are, of course, very restricted conclusions, but little is known of general representation theory, even for such a special case as that of a Banach algebra with an involution (see [28]).

§ 43.

COMMUTATIVE THEORY

43A. The Laplace transform. Mackey [34] has observed that the kernel e(z +iYl • of the classical Laplace transform can be given a characterization valid on a general locally compact Abelian group in much the same way as Fourier transform theory is shifted to this setting by replacing the kernel e;11 " by the general character function a(s). The following description is an oversimplification but will illustrate the kind of step to be taken. We iy)s, where x andy are fixed, by replace the mappings --+ (x a continuous homomorphism u of an Abelian topological group G into the additive group of the complex numbers. The set of all such homomorphisms is clearly a complex vector space V and an element u E:: V will be called simply a vector. The kernel e("' +iyl 8 is then replaced by the kernel ev(s), u E:: V, s E: G, and the generalized Laplace transform is the mapping f --+ F where

+

F(u) = Jev(•lj(s) ds. A complex-valued function defined on a suitable subset of V will be said to be analytic at u if for every u E: V the limit F(u, u) of Xu) - F(u)/X exists as the complex variable}.. approaches of F(u 0. Leaving aside questions of existence, we see that formally the Laplace transform F of a function/ on G is analytic and

+

F(u, u) = Jev(alu(s)f(s) ds. Mackey has shown that in a suitably restricted L 2 sense a function F defined on Vis analytic if and only if it is a Laplace transform. Many other theorems of Laplace transform theory have

180

SOME FURTHER DEVELOPMENTS

natural extensions, and the abstract setting of the theory seems to lead to a fruitful point of view. In particular, it is expected that some of the theory of functions of several complex variables will be illuminated by this general formulation. 43B. Beurling's algebras. Another problem which leads to this same generalization of the Laplace transform, and which has been discussed in a special case by Beurling [4] is the theory of subalgebras of D(G) which are themselves £1 spaces with respect to measures "larger than" Haar measure. The example considered in the second part of 23D is a case in point. More generally, suppose that w is a positive weight function on an arbitrary locally compact Abelian group such that w(xy) ;:;;; w(x)w(y) for all x, y E: G. Then the set Lw of measurable functions/ such that

Jl f !w <

oo is the space D(fJ.) where fJ.(A)

= Lw(s)

ds, and the

inequality on w assumed above implies that Lw is closed under convolution and is commutative Banach algebra. A simple condition guaranteeing that Lw is a subset of D(G) is that w be bounded below by 1. Then every regular maximal ideal of D (G) defines one for Lw (see 24B) and the maximal ideal space of L 1 (G) is identifiable with a (closed) subset of that of Lw, generally a proper subset. The homomorphisms of Lw onto the complex numbers are given by functions 9M E: LrxJ(G),j(M) =

J

j(s)9M(s)w(s) ds,

and a simple argument as in 23D shows that the product 9M(s)w(s) is a generalized character, that is, a continuous homomorphism of G into the multiplicative group of the non-zero complex numbers, and therefore that its logarithm, if it can be taken, is what we have called in the above paragraph a vector. The transforms j of the elements f E: Lw are thus generally to be considered as bilateral Laplace transforms, and the theory of such algebras Lw merges with the general Laplace transform theory. 43C. Tauberian theorems. In case w is such that the convolution algebra Lw = {/:

Jl f !w <

ro} has the same maximal ideal

space as the larger algebra D(G) it is natural to consider whether ideal theoretic theorems such as Wiener's Tauberian theorem remain valid. Questions of this nature were taken up by Beurling in

SOME FURTHER DEVELOPMENTS

181

1938 [4] for the special case of the real line. Beurling's basic theorem m this direction is roughly that, if

[

I log w(x) I d. 1 +x2 X < oo, ---«~

then L"' is regular and the Tauberian theorem holds: every closed proper ideal lies in a regular maximal ideal. It should be remarked that Beurling proved and leaned heavily on the formula: IIJ [loo = lim II j"' W'"'· His proof uses the properties of the real numbers in an essential way and does not generalize; Gelfand's later proof of the same formula for a general Banach algebra is based on an entirely different approach. Returning to our main theme, we notice that the Tauberian theorem on a locally compact Abelian group can be rephrased as the assertion that any proper closed invariant subspace of V(G) is included in a maximal one, and then can be rephrased again, via orthogonality, as the statement that every non-void translation-invariant, weakly-closed subspace of L 00 includes a minimal one (the scalar multiples of a character). In another paper [5] Beurling has demonstrated this form of the Tauberian theorem for a different space, the space of bounded continuous functions on the real line, under the topology characterized by uniform convergence on compact sets together with the convergence of the uniform norm itself. In addition he showed that unless the closed invariant subspace is one-dimensional it contains more than one exponential. This is clearly the analogue of the SegalKaplansky-Helson theorem, and suggests that we leave the general Tauberian problem in the following form. It is required to find conditions which will imply that a closed ideal in a commutative Banach algebra A shall be the kernel of its hull, or, dually, that a weakly-closed invariant subspace of A* shall be spanned by the homomorphisms which it contains. The condition can be either on the algebra or on the subspace, and in the case of groups it can be on the measure defining the convolution algebra or on the topology used on the conjugate space. 43D. Primary ideals. Although a closed ideal in a commutative Banach algebra is not, in general, the intersection of maximal ideals (i.e., not the kernel of its hull) there are two spectacular

182

SOME FURTHER DEVELOPMENTS

cases in which it is known that every closed ideal is the intersection of primary ideals, a primary ideal being defined as an ideal which is included in only one maximal ideal. The first (due to Whitney [49]) is the algebra of functions of class cr on a region of Euclidean n-space, its topology taking into account all derivatives up to and including order r. The second (due to Schwartz [43]) is the algebra of all bounded complex-valued Baire measures on the real line which are supported by compact sets, with the topology which it acquires as the conjugate space of the space of all (not necessarily bounded) continuous functions under the topology defined by uniform convergence on compact sets. The continuous homomorphisms of this algebra into the complex numbers are given by exponentials

eat (Jl(a) =feat dp.(t)) and the regular maximal ideal space is thus identified with the complex plane, the transforms J1 forming an algebra of entire functions. Every primary ideal of this algebra is specified by a complex number a 0 and an integer n, and consists of the measures p. such that J1 and its first n derivatives vanish at a 0 • And every closed ideal is an intersection of primary ideals. In both of these examples the presence and importance of primary ideals are connected with differentiation. 43E. We conclude with a brief mention of a remarkable analogy which has been studied by Levitan, partly in collaboration with Powsner, in a series of papers of which we have listed only [31]. Levitan observed that the eigen-function expansions of SturmLiouville differential equation theory can be regarded as a generalization of the Fourier transform theory of group algebras as discussed in Chapter VII. Specifically, let us consider the differential equation y" - (p(x) - X)y = 0 on the positive real axis, with a boundary condition at x = 0. With this ordinary differential equation we associate the partial differential equation Uu - u 1111 (p(x) - p(y))u = 0 with initial conditions u(x, 0) = f(x), u 11 (x, 0) = 0, where p and f are even functions. The solution u(x, y) = [T11 j](x) is regarded as a generalization of the translation of f(x) through y. If generalized convolution is defined in the obvious way an L 1 convolution algebra is obtained

SOME FURTHER DEVELOPMENTS

183

which has very far-reaching analogies with the L 1 group algebras. The following topics can be developed: generalized characters (eigen-functions), positive definite functions, Bochner's theorem, the L 1-inversion theorem and the unique measure on the transform space, and the Plancherel theorem. The theory is valid not only when the spectrum is discrete but also in the limit point case when the spectrum is continuous, although it is not yet fully worked out in the latter case. The reader is referred to the literature for further details.

BIBLIOGRAPHY

The following bibliography contains only those items explicitly referred to in the text. For a more complete bibliography see the book of Wei! ([48] below) for the years preceding 1938, and the address of Mackey ([35] below) for subsequent years.

1. W. AMBROSE, Structure theorems for a Jptcial clau of Banach algebra;, Trans. Amer. Math. Soc. 57, 364-386 (1945). 2. R. ARENS, On a theorem of Geijand and Neumark, Proc. Nat. Acad. Sci. U.S.A. 32, 237-239 (1946). 3. S. BANACH, TMorie deJ oplrationJ lineaireJ, Warsaw, 1932. 4. A. BEURLING, Sur leJ intlgraleJ de Fourier abJo/ummt convergenteJ et leur une transformation fonctionnelle, Congres des Mathematiques a application Helsingfors, 1938. 5. A. BEURLlNG, Un tMor~me JUr leJ fonctiom bornleJ et uniformement continueJ Jur /'axe riel, Acta Math. 77, 127-136 (1945). 6. H. BoHNENBLUST and A. SoBCZYK, ExtemionJ of functiona!J on complex linear;paces, Bull. Amer. Math. Soc. 44, 91-93 (1938). 7. H. CARTAN, Sur Ia meJure de Haar, C. R. Acad. Sci. Paris 211 1 759-762 (1940). 8. H. CAR TAN and R. GoDEMENT, TMorie de Ia dualite et analyu harmonique danJ /e; groupeJ abelians localemmt compactJ, Ann. Sci. Ecole Norm, Sup. (3) 64, 79-99 (1947). 9. P. DANIELL, .d genera/form of integral, Ann. of Math. 191 279-294 (1917). 10. V. DlTKlN, On the Jtructure of idea/J in certain normed ringJ, Ucenye Zapiski Moskov. Gos. Univ. Mat. 30, 83-130 (1939). 11. J. DlXMlER, Sur un tMor~me de Banach, Duke Math. J. 151 1057-1071 (1948). 12. I. GELFAND, Normierte Ringe, Rec. Math. (Mat. Sbornik) N.S. 9, 3-24 (1941). 13. I. GELFAND and M. NEUMARK, On the imbedding of normed ringJ into the ring of operatorJ in Hilbert Jpace, Rec. Math. N.S. 12 (54), 197-213 (1943). 14. I. GELFAND and M. NEUMARK, Unitary repreuntationJ of the Lorentz group, lzvestiya Akademii Nauk SSSR. Ser. Mat. 11 1 411-504 (1947). 15. l GELFAND and D. RAlKov, On the theory of character of commutatiue topological groupJ, C. R. (Doklady) Acad. Sci. URSS 28 1 195-198 (1940). 16. L GELFAND and D. RAlKov, Irreducible unitary represmtationJ of locally bicompact groupJ, Rec. Math. (Mat. Sbornik) N.S. 13 1 301-316 (1943). 17. I. GELFAND, D. RAlKov, and G. S1wv, Commutatiue normed ringJ, Uspehi Matematiceskih Nauk. N.S. 2 1 48-146 (1946). 185

a

186

BIBLIOGRAPHY

..r8. A. GLEASON, .d note on locally compact groupJ, Bull. Amer. Math. Soc. 55, 744-745 (1949). 19. R. GooEMENT, Lu fonctionJ de type poJitif ella tMorie du groupu, Trans. Amer. Math. Soc. 63, 1-84 (1948). 20. R. GooEMENT, Sur Ia tMorie du repr!JentationJ unitairu, Ann. of Math. (2) 53, 68-124 (1951). 21. R. GooEMENT, Memoire Jur Ia tMorie deJ caractereJ danJ lu groupu localement compactJ unimodulairu, Journal de Math. Pur~e e Appliqu~ 301 1-110 (1951). 22. A. HAAR, Der Maubegriff in der Theorie der kontinuierlichen Gruppen, Ann. of Math. 341 147-169 (1933). 23. P. HALMos, MeaJure Theory, Van Nostrand, New York, 1950. 24. H. HELSON, Spectral JynthuiJ of bounded functionJ, Ark. Mat. 1, 497-

502 (1951). 25. N. JAcoBsoN, The radical and umi-Jimplicityforarbitrary ringJ, Amer. J. Math. 671 300-320 (1945). 26. I. KAPLANSKY, Dual ringJ, Ann. of Math. (2) 491 689-701 (1948). 27. I. KAPLANSKY, Primary idealJ in group algebraJ, Proc. Nat. A cad. Sci. U.S.A. 35, 133-136 (1949). 28. I. KAPLANSKY, The Jlructure of certain operator algebraJ, Trans. Amer. Math. Soc. 701 219-255 (1951). 29. M. KREIN, Sur une generaliJation du tMoreme de Plancherel au caJ deJ integralu de Fourier Jur lu groupu topologiquu commutatijJ, C. R. (Doklady) Acad. Sci. URSS (N.S.) 30, 484-488 (1941). 30. M. KREIN and D. MILMAN, On extreme pointJ of regular convex utJ, Studia Mathematica 91 133-138 (1940). 31. B. M. LEVITAN, The application of generalized diJplacement operatorJ to linear differential equationJ of the ucond order, Uspehi Matern. Nauk (N.S.) 4, no. 1 (29), 3-112 ( 1949). 32. D. E. LITTLEWOOD, The Theory of Group CharacterJ and Matrix Repruentatiom ofGroupJ, Oxford Univ. Press, New York, 1940. ~. L. H. LooMis, Haar meaJure in uniform Jtructuru, Duke Math. J. 16, 193-208 (1949). v.M. G. MACKEY, The Laplace tramform for locally compact Abelian groupJ, Proc. Nat. Acad. Sci. U.S.A. 341 156-162 (1948). 35. G. MACKEY, FunctionJ on locally compact groupJ, Bull. Amer. Math. Soc. 56, 385-412 (1950). 36. G. MACKEY, On induced repreuntationJ of groupJ, Amer. J. Math. 73, 576-592 (1951). 37. F. I. MAUTNER, Unitary repreuntatiom of locally compact groupJ I, Ann. of Math. 51 1 1-25 (1950). 38. F. I. MAUTNER, Unitary repruentatiom of locally compact groups I I, Ann. of Math., to be published. 39. J. voN NEUMANN, .dlmoJI periodic functiom in a group, Trans. Amer. Math. Soc. 361 445-492 (1934). 40. J. voN NEUMANN, On ringJ of operatorJ. Reduction theory. Ann. of Math.

so,

401-485 (1949).

41. D. A. RAIKov, Generalized duality theorem for commutatiue groupJ with an inuariant meaJure, C. R. (Doklady) Acad. Sci. URSS (N.S.) 301 589-591

(1941).

BIBLIOGRAPHY

187

42. D. A. RAIK.OV, Harmonic ana/yJiJ on commutatiue groupJ with the H aar meaJure and the theory of ch.aracterJ, Trav. lnst. Math. Stekloff 14 (1945). 43. I.. ScHWARTZ, Th.!orie glnlrale dn jonctionJ moyenneplriodiqun, Ann. of Math. 48, 857-929 (1947). 44. I. E. SEGAL, The group algebra of a locally compact group, Trans. Amer. Math. Soc. 61, 69-105 (1947). 45. I. E. SEGAL, .dn extenJion of Plancherei'J formula to Jeparahle unimodular groupJ, Ann. of Math. (2) 52, 272-292 (1950). 46. M. H. STONE, The generalized WeierJtrau approximation theorem, Math. Mag. 21, 167-184, 237-254 (1948). 47. M H. STONE, Notn on integration: II, Proc. Nat. Acad. Sci. U.S.A. 34, 447-455 (1948). 48. A. WEIL, L'intlgration danJ leJ groupeJ topologiqueJ et JeJ applicationJ, Paris, 1938. 49. H. WHITNEY, On idea/J of di.fferentiahle junftiom, Amer. J. Math. 70, 635-658 (1948).

INDEX The references in the index are, principally, either to the definitions of terms or to the statements of theorems.

Absolute continuity, 40 Adjoint, 21 Adverse, 64 Algebra, 8 Banach algebra, 16, 48 function algebra, 53 group algebra, 123 (see Banach algebra) Almost invariant function, 160 Almost periodic function, 165 (§ 41) Analytic function theorem, 78 Annihilator, 20 Approximate identity, 124 Axiom of choice, 2 Baire function, 32 Banach algebra, 16, 48 C* algebra, 28, 88, 91 H* algebra, 100 regular Banach algebra, 82 self-adjoint Banach algebra, 88 semi-simple Banach algebra, 76 Banach space, 13 reflexive Banach space, 20 Basis, 3 Bessel's inequality, 26 Bochner theorem, 97, 142 Boundary, 80 C* algebra, 28 commutative C* algebra, 88, 91 Carrier, 84 Cartesian product, 10

Central element (function), 105, 157, 175 Character, generalized, 180 of a compact group, 159 of an Abelian group, 135 of an algebra, 71 real, 132 Character group, 137 Closed graph theorem, 17, 18 Closed set (closure), 4 Compactification, of a completely regular space, 55, 56 one-point, 7 Compact set, 5 Completely regular space, 55 Complex LP-space, 46 Conjugate space, 18 Continuity, 4 Convolution, 120-122

Daniell integral, 29 (§ 12) Directed set, 1, 2 Direct integral, 176 Direct sum, 16 Ditkin, condition of, 86 Dual algebra, 161 Equivalent functions, 36, 37 Extendable linear functional, 96 Factor, 176 Finite intersection property, 5 Fourier series, 153 (§ 38) Fourier transform, 71, 135 188

INDEX

Fubini theorem, 44 Function algebra, 53 inverse closed, 54 regular, 57 self-adjoint, 55

Gelfand representation, 53, 70 Group algebra (ring), 123 (§ 31) Haar integral, 115 (§ 29) normalized, 144, 154 on quotient spaces, 130 (§ 33) Hahn-Banach theorem, 19, 20 H* algebra, 100 (§ 27) Hausdorff space, 5 Heine-Borel property, 4 Herglotz-Bochner-Weil-Raikov theorem, 97, 142 Hilbert space, 23, 24 (§ 10) Holder inequality, 37, 46 Homeomorphism, 4 Hull, 56 Hull-kernel topology, 56, 60

Idempotent, 101 reducible idempotent, 101 Identity, added to an algebra, 59 modulo an ideal, 58 Integrable function (and set), 34 Integral, 30 bounded, 34 on Cartesian products, 44 the Haar integral, 115 (§ 29) Interior, 4 Invariant subspace, 125 Inversion theorem, 143 Involution, 28, 87 Irreducible idempotent, 102 Irreducible representation, 163

189

Linear functional (Cont.) extendable, 96 positive, 96 unitary, 99 Linear order, 2 Linear transformation, bounded,15 self-adjoint, 95 unitary, 127-129 L-monotone family, 33 Locally compact space, 7 Local membership in an ideal, 85 LP-norm, 37, 46 LP-space, 37 (§ 14) complex LP-space, 46

Maximal ideal space, 54, 69 (§ 23) Mean value of von Neumann, 169 Measurable set (and function), 36 Measure, 34 on a Cartesian product, 45 Minimal left ideal, 102 Minimal two-sided ideal, 102 Minkowski inequality, 38, 46 Modular function, 117, 132 Monotone family of functions, 32 L-monotone family, 33

Kernel, 56

Neighborhood, 4 symmetric neighborhood, 108 Norm, 13 Banach algebra norm, 16, 48 LP-norm, 37, 46 of a bounded transformation, 15 uniform norm, 9, 14, 39 Normal space, 6 Normed algebra, 16 N ormed field, 68 N ormed linear space, 13 Null function, 35 Null set, 35 Null space, 16

Laplace transform, 74, 179 L-bounded function, 33 Linear functional, bounded, 18

One point compactification, 7 Open set, 3 Orthogonal complement, 25

190

INDEX

Parallelogram law, 23 Parseval's equation, 155, 175 Partial order, 1 Plancherel theorem, 99, 145, 174 Poisson summation formula, 152 Pon triagin duality theorem, 140, 151 Positive definite element, 98 Positive definite function, 126, 142 Positive linear functional, 96 Primary ideal, 181 Projection, onto a coordinate space, 11 in Hilbert space, 25, 26 Pseudo-norm, 52 Quotient space, 14, 110 Radical, 76 Radon-Nikodym theorem, 41 Real character, 132 Reduction theory, 175 Reflexive Banach space, 20 Regular Banach algebra, 82 (§ 25) Regular function algebra, 57 Regular ideal, 58 Relative topology, 4 Representation, irreducible, 163 of a group, 127 of an algebra, 92 of an H* algebra, 163 regular, 130

Spectral theorem, 94, 95 Spectrum, 64 Stone's theorem, 147 Stone-Weierstrass theorem, 9 Sturm-Liouville differential equations, 182 Sub-basis, 3 Summable function, 31, 33 Summable set, 34 Symmetric neighborhood, 108

Tauberian theorem, 85, 148, 180 Topological group, 108 Topology, 3 compact, 5 completely regular, 55 Hausdorff, 5 hull-kernel, 56, 60 locally compact, 7 normal, 6 relative, 4 Th 111 weak, 10 Triangle inequality, 13, 24 T1-space, 111 Tychonoff theorem, 11 Uniform norm, 14 Unimodular group, 117 Unitary functional, 99 Urysohn's lemma, 6 Variations of a furtctional, 40

Scalar product, 24 Schwarz inequality, 24 Self-adjoint element, 90 Semi-direct product, 119 Semi-simple ring (algebra), 62, 76 Silov's theorem, 151 Spectral norm, 75

Weak topology, 10 on a conjugate space, 22 Weight function, 74, 180 Wiener Tauberian theorem, 85, 148, 180 Zorn's lemma, 2