An Introduction to Abstract Harmonic Analysis

An Introduction to Abstract Harmonic Analysis by

LYNN H. LOOMI S

Associate Professor of M"athematics Harvard University

D. VAN NOSTRAND COMPANY, INC.

TORONTO

PRINCETON, NEW JERSEY MELBOURNE LONDON

VAN N OSTRAND R EGIONAL O FFICES: New York, Chicago, San Francisco D.

VAN N OSTRAND C OMPANY, LTD., London

D.

VAN N OSTRAND C OMPANY, (C anad a) , LTD., Toronto

D.

VAN N OSTRAND

AUSTRALIA

PTY. LTD., Melbourne

Copyright © 1953, by D. VAN NOSTRAND COMPANY, INC.

Published simultaneously in Canada by D. VAN NOSTRAND COMPANY (Canada),

LTD.

No reproduction in any form of this book, in whole or in part (exceptfor brief quotation in critical articles or reviews), may be made without written authorization from the publisher.

PRINTED IN TH E UNITED STATES OF AMERICA

PREFACE

This book is an outcome of a course given at Harvard first by G. W. Mackey and later by the author. The original course was modeled on Weil's book [48] and covered essentially the material of that book with modifications. As Gelfand's theory of Banach algebras and its applicability to harmonic analysis on groups became better known, the methods and con ten t of the course in­ evitably shifted in this direction, and the present volume concerns itself almost exclusively with this point of view. Thus our devel­ opment of the subject centers around Chapters IV and V, in which the elementary theory of Banach algebras is worked out, and groups are relegated to the supporting role of being the prin­ cipal application. A typical result of this shift in emphasis and method is our treatment of the Plancherel theorem . This theorem was first formulated and proved as a theorem on a general locally compact Abelian group by Weil. His proof involved the structure theory of groups and was difficult. Then Krein [29] discovered, apparently wi thou t knowledge of Weil' s theorem, that the Plancherel theorem was a natutal consequence of the application of Gelfand's theory to the L l algebra of the group. This led quite naturally to the form­ ulation of the theorem in the setting of an abstract commutative (Banach) algebra with an involution, and we follow Godement [20] in taking this as our basic Plancherel theorem. (An alternate proof of the Plancherel theorem on groups, based on the Krein­ Milman theorem, was given by Cartan and Godement [8].) The choice of Banach algebras as principal theme implies the neglect of certain other tools which are nevertheless important in the investigation of general harmonic analysis, such as the Krein­ Milman theorem and von Neumann's direct integral theory, each of which is susceptible of systematic application. It is believed, however, that the elementary theory of Banach algebras and its v

VI

PREFACE

applications constitute a portion of the subject which in i tself is of general interest and usefulti.ess, which can serve as an introduc­ tion to present-day research in the whole field, and whose treat­ ment has already acquired an elegance promising some measure of permanence. The specific prerequisites for the reader include a knowledge of the concepts of elementary modern algebra and of metric space topology. In addition, Liouville's theorem from the elementary theory of analytic functions is used twice. In practice it will probably be found that the requirements go further than this, for wi thout some acquaintance with measure theory, general topology, and Banach space theory the reader may find the pre­ liminary material to be such heavy going as to overshadow the main portions of the book. This preliminary material, in Chapters I-III, is therefore presented in a condensed form under the as­ sumption that the reader will not be unfamiliar with the ideas being discussed. Moreover, the topics treated are limited to those which are necessary for later chapters ; no effort has been made to wri te small textbooks of these subjects. With these restrictions in mind, it is nevertheless hoped that the reader will find these chapters self-contained and sufficient for all later purposes. As we have mentioned above, Chapter IV is the central chapter of the book, containing an exposi tion of the elemen ts of the theory of Banach algebras in, it is hoped, a more leisurely and systematic manner than found in the first three chapters. Chapter V treats certain special Banach algebras and introduces some of the notions and theorems of harmonic analysis in their abstract forms. Chap­ ter VI is devoted to proving the existence and uniqueness of the Haar integral on an arbitrary locally compact group, and in Chap­ ters VII and VIII the theory of Banach algebras is applied to deduce the standard theory of harmonic analysis on locally com­ pact Abelian groups and compact groups respectively. Topics covered in Chapter VII include positive definite functions and the generalized Bochner theorem, the Fourier transform and Plancherel theorem, the Wiener Tauberian theorem and the Pontriagin duality theorem. Chapter VIII is concerned with

PREFACE

Vll

representation theory and the theory of almost periodic functions. We conclude in Chapter IX with a few pages of introduction to the problems and literature of some of the areas in which results are incomplete and interest remains high.

CONTENTS

CHAPTER I: TOPOLOGY

SECTION

1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Sets

.

.

.

.

.

.

.

.

.

. . .

Topology .

.

.

.

.

.

.

.

.

PAGE

1 3 5 8 10

.

Separation axioms and theorems The Stone-Weierstrass theorem Cartesian products and weak topology CHAPTER II: BANACH SPACES

N ormed linear spaces

.

.

.

13 15 18 22 23 27

.

Bounded linear transformations Linear functionals .

.

.

The weak topology for X*

Hilbert space

.

.

.

.

Involution on ffi(H) CHAPTER III: INTEGRATION

The Daniell integral

.

.

.

.

29 34 37 40 43 46

.

Equivalence and measurability The real LP-spaces .

.

.

.

.

.

The conj ugate space of LP Integration on locally compact Hausdorff spaces The complex LP-spaces .

.

.

.

.

.

.

.

.

.

.

CHAPTER IV: BANACH ALGEBRAS

Definition and examples Function algebras Maximal ideals Spectrum ; adverse Banach algebras; elementary theory The maximal ideal space of a commutative Banach algebra Some basic general theorems

ix

.

.

.

.

.

.

.

.

.

.

.

.

48 50 58 63 66 69 75

x

CONTENTS

SECTION

25. 26. 27. 28. 29. 30. 31. 32. 33.

CHAPTER V: SOME SPECIAL BANACH ALGEBRAS

Regular commutative Banach algebras Banach algebras with involutions H*-algebras.

.

.

.

.

.

.

.

. .

CHAPTER VI: THE HAAR INTEGRAL

The topology of locally compact groups The Haar integral

.

The modular function

.

The group algebra Representations . Quotient measures CHAPTER VII: LOCALLY COMPACT ABELIAN GROUPS

34. 35. 36. 37. 38.

The character group . Examples . . . . . .

. . . . . . . . . . . . . .

The Bochner and Plancherel theorems Miscellaneous theorems. .

. . . . .

Compact Abelian groups and generalized Fourier series

PAGE

82 87 100

108 112 117 120 127 130 134 138 141 147 153

CHAPTER VIII: COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

39. 40. 41.

The group algebra of a compact group Representation theory

.

Almost periodic functions . CHAPTER IX: SOME FURTHER DEVELOPMENTS

42. 43.

Non-commutative theory Commutative theory Bibliography Index

156 161 165

174 179 185 188

TOPOLOGY

§

1.

S ETS

IA. The reader is assumed to be familiar with the algebra of sets, and the present paragraph is confined to notation. The curly bracket notation is used to name specific sets. Thus { a, b } i s the set whose elements are a and b, and { a l ) .. . , a n } i s the set whose elements are al) ..., a n . However, sets cannot generally be named by listing their elements, and more often are defi n ed by properties, the notation being that { x : ( ) } is the set of all x such that ( ) . We write " a � A" for " a is a member of A" and " B c A" for " B is a subset of A" ; thus A { x : x � A} . The ordinary " cup" and " cap" notation will be used for com­ binations of sets : A U B for the union of A and B, and A n B for their intersection ; U:= l An for the union of the sets of the countable family { An } , and n:= l An for their intersection. More generally, if g: is any family of sets, then U AE:::f A or U { A : A � g:} is the union of the sets in g:, and n { A: A � g: } i s their intersection. The complement o f A, designated A', is understood to be taken with respect to the " space" in question, that is, with respect to whatever class is the domain of the given discussion. The null class is denoted by 0. The only syn1bols from logic which will be used are " ==}" for " if ... then" and for " if and only if." lB. A binary relation "<" between elements of a class A is called a partial ordering of A (in the weak sense) if it is transitive (a < b and b < c ==} a < c), reflexive (a < a for every a � A) and if a < b and b < a ==} a = b. A is called a directed set =

"{::}"

2

TOPOLOGY

under " < " if i t is partially ordered by " < " and if for every a and b E: A there exists c E::: A such that c < a and c < b. More properly, A in this case should be said to be directed downward; there is an obvious dual definition for being directed upward. A partially ordered set A is linearly ordered if either a < b or b < a for every distinct pair a and b E: A. A is partially ordered in the strong sense by " < " if " < " is transitive and irrejlexive (a
3

TOPOLOGY

there exists a function f w ith domain D such that f(x) every x €: D.

€:

F(x) for

Proof. We consider a function to be a class of ordered couples

(a graph) in the usual way. Let 5' be the family of functions g such that domain (g) c D and such that g(x) €: F(x) for every x €: domain (g) . 5' is non-empty, for if Xo is any fixed element of D and Y o is any fixed element of F(xo), then the ordered couple (xo , Y o) is in 5'. (More properly, it is the unit class { (xo, Y o)} which belongs to 5'.) 5' is partially ordered by inclusion, and the union of all the functions (sets of ordered couples) in any linearly ordered sub­ family 5'0 of 5' is easily seen to belong to 5' and to be an upper bound of 5'0. Therefore 5' contains a maximal element j. Then domain (f) = D, since otherwise we can choose a point Xo €: D domain (f) and an element Y o €: F (xo), so that f U {(xo, Y o ) } i s a function o f 5' which i s properly larger than f, a contradiction. This function f, therefore, satisfies the conditions of the theorem. -

§ 2.

TOPO LOGY

2A. A family 3 of subsets of a space (set) S is called a topology

for S if and only if: (a) 0 and S are in 3 ; (b) if3 1 c 3, then U { A : A E: 3 d E: 3 ; that is, the union o f the sets of any subfamily of 3 is a member of 3 ; (c) the intersection o f any finite number o f sets o f 3 i s a set of 3. If 3 1 and 32 are two topologies for S, then 3 1 is said to be weaker than 32 if and only if 3 1 C 32 . ' 2B. If a is any family of subsets of S, then the topology gen­ erated by a, 3( a), is the smallest topology for S which includes a; if 3 = 3( a), then a is called a sub-basis for 3. I t follows readily that A E: 3( a) if and only if A is 0 or S, or if A is a union (per­ haps uncountable) of finite intersections of sets in a. If every set in 3 = 3( a) is a union of sets in a, then a is called a basis for 3. 2e. If a topology 3 is given for S, then S is called a topological space and the sets of 3 are the open subsets of S. If A is any sub-

4

TOPOLOGY

set of S, then the union of all the open subsets of A is called the interior of A and is denoted int (A) ; evidently int (A) is the largest open subset of A, and A is open if and only if A = int (A) . If P E: int (A) , then A is said to be a neighborhood of p. Neigh­ borhoods are generally, but not always, taken to be open sets. A set of neighborhoods of p is called a neighborhood basis for p if every open set which contains p includes a neighborhood of the set. 2D. A subset of S is closed (with respect to the topology 3) if its complement is open. It follows that [25 and S are closed, that the intersection of any number of closed sets is closed, and that the union of any finite number of closed sets is closed. If A is any subset of S, the intersection of all the closed sets which include A is called the closure of A and is commonly denoted A; evidently A is the smallest closed set including A, and A is closed if and only if A = A. Also, p E: A if and only if p fL int (A'), that is, if and only if every open set which contains p also contains at least one point of A. 2E. If So is a subset of a topological space S, then a topology can be induced in S o by taking as open subsets of So the inter­ section of So with the open subsets of S. This is called the rela­ tive topology induced in So by the topology of S. 2F. A function I whose domain D and range R are topologi­ cal spaces is said to be continuous at Po E: D if l-l (U) = { p : /(p) E: U} is a neighborhood of Po whenever U is a neigh­ borhood ofI(P o) . If I i s continuous a t each point o f its domain, it is said to be continuous (on D). It follows thatl is continuous if and only if l- l(U) is an open subset of D whenever U is an open subset of R, and also if and only if 1-1 (C) is closed whenever C is closed. If I is one-to-one and both I and I- I are continuous, then I is said to be a homeomorphism between D and R. Evidently a homeomorphism defines a one-to-one correspondence between the topology 3D for D and the topology 3R for R. 2G. The following conditions on a subset A of a topological space S are obviously equivalent (by way of complementation) : (a) Every family of open sets which covers A includes a finite subfamily which covers A. (Heine-Borel property.)

5

TOPOLOGY

(b) Every family of relatively closed subsets of A whose in­ tersection is JZf includes a finite subfamily whose intersection is JZf. (c) I f a family of relatively closed subsets of A has the finite intersection property (that every finite subfamily has non-void intersection), then the family itself has a non-void intersection. A subset A which has any, and so all, of the above three prop­ erties is said to be compact. It follows immediately from (b) or (c) that a closed subset of a compact set is compact. 2H. Theorem. A continuous function with a compact domain has a compact range. Proof. If {Ua} is an open covering of the range R of f, then ff- 1 (Ua) } is an open covering of the compact domain off, and hence includes a finite subcovering {f l ( Ua,)} whose image {Ua.} is therefore a finite covering of R. Thus R has the Heine­ Borel property and is compact. 21. An indexed set of points {Pal is said to be directed (down­ ward) if the indices form a (downward) directed system. A di­ rected set of points {Pal converges to p if for every neighborhood U of p there exists an index {J such that pa E=: U for all a < {J. All the notions of topology can be characterized in terms of con­ vergence. For example, a point p is called a limit point of a di­ rected set of points {Pal if for every neighborhood U of p and for every index {J there exists a < {J such that Pa E=: U. Then i t can easily b e proved that a space i s compact if and only if every directed set of points has at least one limit point. Li ttle use will be made of these notions in this book and no further discussion will be given. -

§ 3.

S E PARATI O N AXIOMS A N D TH E O R EMS

3A. A Hausdorff space is a topological space in which every two distinct points have disjoint neighborhoods. Lemma. If C is a compact subset of a Hausdor:ff space and p tl C, then there exist disjoint open sets U and V such that p E=: U and C c V. Proof.

point

q

Since S is a Hausdorff space, there exists for every

E=: C a pair of disjoint open sets Aq and B q such that

6

TOPOLOGY

p E: Aq and q E: Bq. Since C is compact, the open covering {Bq} includes a finite subcovering {BqJ , and the sets V = UBq" U = n Aq, are as required. Corollary 1. A compact subset of a Hausdorff space is closed. Proof.

q.e.d.

By the lemma, if p

fl

C, then p

fl

C, so that C

c

C,

If J I is a Hausdorff topology for S, J 2 a compact topology for S and J I CJ2 , then J I = J2 . Corollary 2.

Every J 2-compact set C is J I -compact since every J I covering of C is also a J 2-covering and so can be reduced. But then C is J I -closed since J I is a Hausdorff topology. Thus every J 2-closed set is J I -closed and J 2 = J I . 3B. A topological space is said to be normal if it is a Haus­ dorff space and if for every pair of disjoint closed sets FI and F2 there exist disjoint open sets UI and U2 such that Fi C Ui, i = 1 , 2. Proof.

Theorem.

A

compact Hausdorff space is normal.

Proof. The proof is the same as that of the lemma in 3A. For each p E: FI there exists, by 3A, a pair of disjoint open sets Up , Vp such that p E: Up and F2 C Vp . Since FI is compact, the open covering { Up } includes a finite subcovering { U� } and the sets UI = U Up" U2 = n VPi are as required. 3C. Urysohn's lemma. If Fo and FI are disjoint closed sets in a normal space S, then there exists a continuous real-valued func­ tion on S such thatf = 0 on Fo, f = 1 on FI and 0 �f � 1 . Proof. Let VI = FI '. The normality of S implies the exist­ ence of an open set V� such that Fo c V72' V72 C VI . Again, there exist open sets V� and V� such that Fo c V�, V74 C V72' 1772 C V�, V� C VI . Continuing this process we define an open set Vr for every proper fraction of the form m/2n, 0 �m � 2n, such that Fo c Vr, Vr C VI and Vr c Vs if r < s . We now define the function f as follows : f(P) = 1 i f p is in none of the sets Vr, andf(p) = glb {r p E: Vr } otherwise. Then :

TOPOLOGY

7

f = 1 on Fh f = O on Fo, and range (f) c [0, 1]. If O < b � 1 , then f(P) < b if and only if p E:: Vr for some r < b, so that { p :f(P) < b} = Ur a if and only i f p fl. 17r for some r > a, and {p : f(p) > a } = U r 17,,', also an open set. Since the in tervals [0, b), (a, 1 ] and their intersections form a basis for the topology of [0, 1], i t follows that the inverse image of every open set is open, i.e., that f is continuous. 3D. A topological space is locally compact if every point has a closed compact neighborhood. A locally compact space S can be made compact by the addi­ tion of a single point. In fact, if SOC) = S U { POC) } ' where POC) is any point not in S, then SOC) is compact if it is topologized by tak­ ing as open sets all the open subsets of S, together with all the sets of the form 0 U {POC) } , where 0 is an open subset of S whose complement is compact. For if { Oa } is an open covering of SOC)' then at least one set Oao contains POC)' and i ts complement is there­ fore a compact subset C of S. The sets Oa n S are open in both topologies and cover C. Therefore, a finite subfamily covers C, and, together with Oao, this gives a finite subfamily of { Oa } cover­ ing SOC). Thus SOC) has the Heine-Borel property and is compact. I t is clear that the original topology for S is i ts relative topology as a subset of SOC). SOC) is called the one point compactification of S. If S is a Hausdorff space, then so is SOC)' for any pair of points distinct from POC) is separated by the same pair of neighborhoods as before, while POC) is separated from another point p by taking an open set 0 containing p and having compact closure, so that (5' is an open subset of SOC) containing POC). >a

3E. Theorem. If S is a locally compact Hausdorff space and if C and U are respectively compact and open sets such that C c U, then there exists a real-valued continuous function on S such that f = 1 on C, f = ° on U' and ° �f � 1 .

Proof. The proof of 3C could be modified to yield a proof of 3E. However, it is sufficient to remark that the one point com­

pactification of S is a Hausdorff space in which C and U' are dis­ joint compact sets, and 3C can be directly applied.

8

TOPOLOGY

§ 4.

TH E

STO N E-W E I E R STRA S S TH E O R EM

4A. If S is compact, we shall designate by e (S) the set of all complex-valued continuous functions on S. If S is locally com­ pact but not compact, e(S) will designate the set of all complex­ valued continuous functions on S which " vanish at infinity," in the sense that {p : , f(p) , � } is compact for every positive If S is compactified by adding POCJ (as in 3D) , then e(S) becomes the subset of e(SOCJ) consisting of those functions which vanish at POCJ; hence the phrase " vanishing at infinity." In each of these two cases eR(S) will designate the corresponding set of all real­ valued continuous functions. 4B. An algebra A over a field F is a vector space over F which is also a ring and in which the mixed associative law relates scalar multiplication to ring multiplication : ( �x)y x( �y) �(xy) . If multiplication is commutative, then A is called a commutative algebra. Under the usual pointwise definition of the sum and product of two functions i t is evident that e(S) is a commutative algebra over the complex number field and that eR(S) is a commutative algebra over the real number field. If S is compact, the constant functions belong to these algebras, and they each therefore have a multiplicative identity. If S is locally compact but not com­ pact, then e(S) and eR(S) do not have identities. We remark for later use that eR(S) is also closed under the lattice operations : f U g = max (f, g),f n g min (f, g) . E

E.

=

=

=

4C. Lemma. Let A be a set of real-valued continuous functions on a compact space S which is closed under the lattice operations f U g and f n g. Then the uniform closure of A contains every continuous function on S which can be approximated at every pair of points by a function of A. Proof. Letf be a continuous function which can be so approxi­ mated, and, given let fp,q be a function in A such that ' f(P) - fp,q(p) , < and ' f(q) - fp,q(q) , < Let Up,q = {r:fp,q(r) < fer) + } and Vp,q = {r: fp,q(r) > fer) } Fixing q and varying p, the open sets Up,q cover S, and therefore a finite sub­ family of them covers S. Taking the minimum of the correE,

E

E

E.

-

E

.

9

TOPOLOGY

sponding functions jp , q, we obtain a continuous function jq in A such that jq < j + e on S and jq > j - e on the open set Vq which is the int�rsection of the corresponding sets Vp , q . Now varying q and in a similar manner taking the maximum of a finite number of the functions jq, we obtain a function jE in A such that j - < jE < j + e on S, q.e.d. 4D. Lemma. A uniformly closed algebra A oj bounded real­ valuedjunctions on a set S is also closed under the lattice operations. Proof. Since j U g = max (j, g) = (j + g + I j - g j) /2, it is sufficient to show that I j I E:: A if j E:: A. We may suppose that I I j I I = maxpE:S I j(p) I � 1 . The Taylor's series for (t + e 2) � about t = ! converges uni­ formly in 0 �t � 1 . Therefore, setting t = x2 , there is a poly­ nomial P(x2) in x2 such that I P(x 2) - (x2 + e2 ) � I < e on [ - 1 , 1]. Then I P(O) I < 2e, and I Q(x 2) - (x2 + e 2) � I < 3e, where Q = P - P(O) . But (x2 + e2) � - I x I � so that I Q(x2 ) - I x I I < 4e on [ - 1 , 1]. Since Q contains no constant term, Q(j2) E:: A and I I Q(j2) - I ji ll < 4e. Since A is uni­ formly closed, I j I E:: A, q.e.d. 4E. The Stone-Weierstrass Theorem (see [46]) . Let S be a compact space and let A be an algebra oj real-valued continuous junctions on S which separates points. That is, if PI rf P 2 , there exists j E:: A such that j(PI) rf j(P2 ) . Then the uniform closure A oj A is either the algebra eR(S) oj all continuous real-valued junc­ tions on S, or else the algebra oj all continuous real...,valuedjunctions which vanish at a single point Poo in S. Proof. Suppose first that for every p E:: S there exists j E:: A such that j(P) rf o. Then, if PI rf P 2 , there exists j E:: A such that 0 rf j(PI) rf j(P2 ) rf O. But then, given any two real num­ bers a and b, there exists g E:: A such that g(Pt) = a and g(P2 ) = b. (For example, a suitable linear combination of the above j and j2 will suffice.) Since A is closed under the lattice operations ( 4D), i t follows from 4C that A contains every continuous real-valued function and A = eR(S) . There remains the possibility that at some point Poo every j in A vanishes. We wish to show that, if g is any continuous func­ tion vanishing at POO) then g E:: A. But if the constant functions e

E,

10

TOPOLOGY

are added to A, the first situation is obtained, and g can be ap­ proximated by a function of the form f + c, I I g - (f + c) I I < E/2, where f � A and c is constant. Evaluating at POCl we get I c I < E/2, so that I I f - g I I < E. Therefore, g E::: A, q .e.d. § 5.

CARTESIAN

PRODU CTS A N D WEAK TO POLOGY

SA. The Cartesian product SIX S of two sets S 1 and S is defined as the set of all ordered pairs (p, q) such that P � S 1 and q E::: S SIX S = {(P, q) : P E::: S 1 and q � S } Thus the Cartesian plane of analytical geometry is the Cartesian product of the real line by itself. The definition obviously ex­ tends to any finite number of factors : SIX S2 X· . .X Sn is the set of ordered n-ads (Ph . . . , P n) such that P i E::: Si for i = 1 , . . . , n. In order to see how to extend the defini tion further we reformulate the definition just given. We have an index set, the integers from 1 to n, and, for every index i, a space Si . The ordered n-ad (P h . . . , P n) is simply a function whose domain is the index set, with the restriction that P i E::: S i for every index i. In general, we consider a non-void index set A and, for every index a E::: A, we suppose given a non-void space Sa. Then the Cartesian product IIa(:A Sa is defined as the set of all functions P with domain A such that p e ) = Pa � Sa for every a E::: A. The axiom of choice ( ID) asserts that naSa is non-empty. SB. Let {fa } be a collection of functions whose ranges Sa are topological spaces and which have a common domain S. If S is to be topologized so that all of these functions are continuous, then fa - 1 ( Ua) must be open for every index and every open subset Ua of Sa. The topology generated in S by the totality of all such sets as a sub-basis is thus the weakest topology for S in which all the functions fa are continuous, and is called the weak topology generated in S by the functions {fa } . In defining a sub­ basis for S it is sufficient to use, for each a, only sets Ua in a sub­ basis for Sa. I t is clear that : If SI C S and fa' is the restriction of fa to SI, then the weak to­ pology generated in S 1 by the functions {fa'} is the relativization to S 1 of the weak topology generated in S by the functions {fa } . 2,

2

2

2

2

a

a

•

11

TOPOLOGY

se. The function fa which maps every point p in a Cartesian product S = ITa Sa onto its coordinate Pol in the ath coordinate space is called the projection of S onto Sa. If the spaces Sa are all topological spaces, then we shall certainly require of any to­ pology in S that all the projections be continuous, and S is con­ ventionally given the weakest possible such topology, that is, the weak topology generated by the projections.

If M is any subset of ITa Sa, then the relative topology induced in by the above topology in ITa Sa is the weakest topology in which the projectionsfa, confined to M, are all continuous. A-f

SD. Theorem (Tychonofi).

of compact spaces is compact.

The Cartesian product of a family

Proof, after Bourbaki. Letff be any family of closed sets in S = ITa Sa which has the finite intersection property (that the intersection of every finite subfamily offf is non-void) . We have to show that the intersection offf is non-void. First we invoke Zorn's lemma to extendff to a familyffo of (not necessarily closed) subsets of S which is maximal with re­ spect to the finite intersection property. The projections of the sets offfo on the coordinate space Sa form a familyffoa of sets in that space having the finite intersection property, and, since Sa is assumed to be compact, there is a point Pol which is in the clo­ sure of every set offfoa . Let p be the point in S whose ath coordi­ nate is Pol for each a. We shall show that p is in the closure of every set offfo, and therefore is in every set of ff, which will finish the proof. Accordingly, let U be any open set in S containing p. Then there exists (by the definition of the topology in S) a finite set of indices ah . . . , an, and open sets Uai C SOli' i = 1 , , n, such that .

.

.

wherefa is the projection of S onto Sa. This implies in particu­ lar that Pol, E:: UOl., and hence that Uai intersects every set in But thenfOli- I ( UOl,) intersects every set offfo, and so be­ ffOOli. longs toffo (sinceffo is maximal with respect to the finite inter­ section property) . But then n�fOli- I ( UOl) E::ffo, for the same reason, and so U E::ffo. Thus U intersects every set offfo) and

12

TOPOLOGY

since U was an arbitrary open set of S containing p it follows that is in the closure of every set of 5"0, q .e.d.

p

SE. The Cartesian product of a family of Hausdorff spaces is a Hausdorff space.

Proof. If p � q, then pa � qa for at least one coordinate and, since Sa is a Hausdorff space, there exist disjoint open sets Aa and Ba in Sa such that Pa E:: Aa and qa E:: Ba. But then fa -I (Aa ) and fa -I (Ba) are disjoint open sets in ITa Sa contain­ ing p and q respectively, q.e.d. SF. The following lemma will be useful in Chapter VI. a,

Lemma. Let f(P, q) be a continuous function on the Cartesian product S I X S2 of two Hausdorff spaces S I and S2 , and let C and o be respectively a compact subset of S and an open set in range (f) . Then the set W = {q :f(P, q) E:: 0 for all p in C} is open in S2 . 1

Proof. This proof is a third application of the device used to prove 3A and 3B. If qo E:: W is fixed and p E:: C, there is an open set UX V containing (p, qo) on which f(P, q) E:: O. But C, being compact, can be covered by a finite family Ub . . . , Un of such sets U, and if V = n i Vi is the in tersection of the corre­ sponding V sets, we have f(P, q) E:: 0 for q E:: V and E:: C. Thus if qo E:: W, then there exists an open set V such that qo E:: V c W, proving that W is open.

p

If 5" is a family of complex-valued continuous functions vanishing at infinity on a locally compact space S, sepa­ rating the points of S and not all vanishing at any point of S, then the weak topology induced on S by 5". is identical with the given to­ pology of S. SG. Theorem.

Proof. Let SYJ be the one point compactification of S and J 2 its compact topology. The functions of 5" extend to J2-continu­ ous functions on Soo which vanish at Pooo If J l is the weak topology defined on Soo by the family of extended functions, then J l c J 2 by definition. Also J1 is a Hausdorff topology, for the extended functions clearly separate the points of Soo . Therefore, J l = J 2 by 3A, Corollary 2, and the theorem follows upon relativizing to S.

Chapter II BANACH SPACES

§ 6.

N O RM E D

L I N EA R S PA C E S

normed linear space is a vector space over the real num­ bers or over the complex numbers on which is defined a non­ negative real-valued function called the norm (the norm of x be­ 6A. A

ing designated I I x I I ) such tha t I l x l l = 0 {:::} x = 0 I I x + y I I � I I x I I + I I y I I (triangle inequality) I I AX I I = I A 1 · 1 1 x I I (homogeneit y) .

A normed linear space is generally understood t o b e over the complex number field, the real case being explicitly labeled as a real normed linear space. A normed linear space becomes a metric space if the distance p(x, y) is defined as I I x - y I I, and it is called a if it is complete in this metric, i.e., if � 00, then there exists an whenever I I Xn - Xm I I � 0 as element x such that I I X n - x I I � 0 as � 00. The reader is reminded that the (metric space) topology in question has for a basis the family of all open spheres, where the sphere about Xo of radius S(xo, is the set { x : I I x - Xo I I < } The open spheres about Xo form a neighborhood basis for Xo. It follows directly from the triangle inequality that I I I x I I I I y I I I � I I x - y I I, so that I I x I I is a continuou s function of x. Euclidean n-dimensional space is a real Banach space with the norm of a poin t ( vector) x taken to be its ordinary length ( L x i2) J1,

Banach space

r.

n, m

r,

n

r) ,

13

14

BANACH SPACES

where X I , . . . , xn are the coordinates (or components) of x. I t remains a Banach space i f the norm o f X i s changed t o I I x l i p = p l (L I Xi I ) / p for any fixed � 1, though the triangle inequality is harder to prove if � 2. These Banach spaces are simple examples of the LP spaces of measure theory, whose basic theory will be developed completely (though concisely) in Chapter III. We mention now only that LP is the space of all " integrable" real-valued functions on a fixed measure space such that the p integral I I p is finite, with I I l i p = . Another I Ip

p

Jf

p

f

f

(J f )"

example of a Banach space is the set of all bounded continuous functions on a topological space S, with I I 1 100 = lubxE:s I f(x) I · This norm is called the norm because n � i n this norm (i.e., I I n � 0) if and only if the functions f n con­ II verge to The fact that this normed linear space is complete is equivalent to the theorem that the limit of a uni­ formly convergent sequence of bounded continuous functions is itself a bounded continuous function. 6B. The following theorem exhibits one of the most important ways in which new Banach spaces are generated from given ones.

f -f uniformly f.

uniform

f

f f

Theorem. If M is a closed subspace of a normed linear space vector space X/M becomes a normed linear X, then the quotient space if the norm of a coset Y is defined as its distance from the ori­ gin: l I y I I glb { I I x I I : x E: Y } · If X is complete, then X/M is complete. =

Proof. First, I I Y I I = 0 if and only if there exists a sequence Xn E: Y such that I I Xn I I � o. Since Y is closed, this will occur if and only if 0 E: y, so that I I Y I I = 0 <=> Y = Next I I Y I + Y 2 1 1 = glb { I I X l + X 2 1 1 : X l E: Y I , X2 E: Y 2 } � glb { I I X l I I + I I x2 1 1 } = glb { I I X l I I : X l E: y d + glb { I I x2 1 1 : X2 E: Y 2 } = is thus I I Y I I I + I I Y 2 1 1 · Simi larly I I AY I I = I A 1 · 1 1 Y I I , and a normed linear space. we can suppose, by If {Yn } is a Cauchy sequence in passing to a subsequence if necessary, that I I Yn +1 - Yn I I < 2 -n . Then we can inductively select elements Xn E: Yn such that n n I I Xn + l - Xn I I < 2 - , for P(Xn, Yn + l ) = P(Yn, Yn + l ) < 2 - o If is complete, the Cauchy sequence { Xn } has a limit Xo, and if

M.

X/M

X/M,

X

BANACH SPACES

15

Yo is the coset containing xo, then I I Yn - Yo I I � I I Xn - Xo I I so that {Yn} has the limit Yo . That the original sequence converges to Yo then follows from the general metric space lemma that, if a Cauchy sequence has a convergent subsequence, then i t itself is convergent. Thus XIM is complete if X is complete. § 7.

B O U N D E D L I N EA R T RA N S FORMATI O N S

7A. Theorem. If T is a linear transformation (mapping) of a normed linear space X into a normed linear space Y, then the fol­ lowing conditions are equivalent: 1) T is continuous. 2) T is continuous at one point. 3) T is bounded. That is, there exists a positive constant C such that I I T(x) I I �C I I x I I for all x E: X. Proof. If T is continuous at Xo, then there is a positive con­ stant B such that I I T(x - xo) I I = I I T(x) - T(xo) I I � 1 when II x - Xo I I �B. Thus I I T(h) I I � 1 whenever I I h I I �B, and, for any non-zero y, I I T(y) I I = (l l y I I IB) 1 I T(y (Bll l y I I ) ) I I � II y I I IB, which is 3) with C = I IB. But then I I T(x) - T(X l ) I I = I I T(x - Xl) I I � C I I x - Xl I I < e whenever I I x - X l I I < e/C, and T is continuous at every point X l 7B. The norm" T II of a continuous (bounded) linear trans­ formation is defined as the smallest such bound C. Thus I I T I I = lub x�o I I T(x) 1 1 11 1 X I I , and it follows at once that :

The set of all bounded linear transformations of X into Y forms itself a normed linear space. If Y is complete (i.e., a Banach space), then so is this space of mappings. For example, if { Tn } is a Cauchy sequence with respect to the above-defined norm, then { Tn(x) } is a Cauchy sequence in Y for every X E: X, and, if T(x) is i ts limit, then it is easy to see that T is a bounded linear transformation and that I I Tn - T I I � o.

7C. The set CE-(X) of all bounded linear transformations of X into i tself is not only a normed linear space but is also an alge­ bra, with the product TI T2 defined in the usual way : TI T2 (X) = Tl ( T2 (X)) . Moreover, I I TI T2 1 1 � I I Tl 1 1·1 1 T2 1 1 , for I I TI T21 1

16

BANACH SPACES

lub I I TI ( T2 (x)) 1 1 / 1 1 x i i � lub ( I I TI 1 1 · 1 1 T2 1 1 · 1 1 x 1 1 ) / 1 1 x I I I I TI 1 1 · 1 1 T2 1 1 · Any algebra over the complex numbers which has a norm under which it is a normed linear space and in which the above product inequality holds is called a A complete normed algebra is called a Thus
=

normed algebra. Banach algebra.

(Xl

=

7D. Theorem. I N is the nullspace of a bounded linear trans­ formation T, then I IfT I I remains unchanged when T is considered as a transformation with domain XIN.

T can be considered to be defined on XIN since, if X l and X2 belong to the same coset y, then X l - X2 � N and T(X I ) = T(X2 ) ' Let I I I T I I I b e the new norm o f T. Then I I T(x) I I I I T(y) I I lub lub I I I T i l l = lub II X II FO xf:.y FO II y II I I T(x) I I = II T il. = lub x tlN r l X I I 7E. A normed linear space X is said to be the of subspaces M and N if X is algebraically the direct sum of M and N, and if the projections of X onto M and N are both continu­ ous (i.e., the topology in X is the Cartesian product topology of M X N). If X is the direct sum of and N, then the algebraic isomorphism between M and XIN (which assigns to an element of the coset of XIN containing it) is also a homeomorphism. For the mapping of M onto XIN is norm-decreasing by the defi­ nition of the norm in XIN, and the mapping of XIN onto M has the same norm as the projection on M by 7D. 7F. We remark for later use that a transformation T which is linear and bounded on a dense subspace M of a normed linear space X, and whose range is a subset of a Banach space, has a unique extension to a bounded linear transformation on the whole Proof.

Y

=

Y

direct sum

M

M

BANACH SPACES

17

X.

of This i s a special case o f the theorem that a uniformly con­ tinuous function defined on a dense subset of a metric space has a unique continuous extension to the whole space. An easy di­ rect proof can be given, as follows. If T is defined on the se­ quence { Xn } and Xn � Xo, then the inequality I I T(xn) - T(xm) I I � I I T 1 1 · 1 1 Xn - Xm I I shows that the sequence { T(xn) } is Cauchy and hence convergent. If i ts limit is defined to be T(xo), it is easy to verify that this definition is unique (and consistent in case T is already defined at xo) and serves to extend T to the whole of This section is devoted to the important closed graph theorem. Here, for the first time, it is essential that the spaces in question be Banach spaces.

X.

7G.

Lemma 1. Let T be a bounded linear transformation of a Banach space into a Banach space Y. If the image under T of the unit sphere S 1 = S(O, 1 ) in is dense in some sphere Ur = S(O, r) about the origin of Y, then it includes Ur.

X

X

Proof. The set A = Ur n T(S 1 ) is dense in Ur by hypothesis. Let y be any point of Ur. Given any 0 > ° and taking Y o = 0, we choose inductively a sequence Yn E:: Y such that Yn+1 - Yn n+1 r for all n � 0. There exists, n E:: o A and I I Yn+ 1 - Y I I < O therefore, a sequence { Xn } such that T(Xn+1 ) = Yn+1 - Yn and n I I Xn+1 I I < o . If X = 2:; Xn, then I I x I I < 1 /(1 - 0) and T(x) = 2:; (Y n - Yn - 1 ) = y; that is, the image of the sphere of radius 1 /(1 - 0) covers Ur. Thus Ur (1 -li) C T(S 1 ) for every 0, and hence Ur C T(S 1 ) , q.e.d. Lemma 2 . If the image of S 1 under T is dense in no sphere of Y, then the range of T includes no sphere of Y. Proof. If T(S 1 ) is dense in no sphere of Y, then T(S n) = {T(x) : I I x II < n } = n T(S 1 ) has the same property. Given any sphere S c Y, there exists therefore a closed sphere S(y I , r1 ) c S disjoint from T(S 1 ) , and then, by induction, a sequence of closed spheres S(Y n , rn) c S(Yn - h rn 1 ) such that S(Yn, rn) is disjoint from T(Sn) . We can also require that rn � 0, and then the sequence { Y n } is Cauchy. I ts limit Y lies in all the spheres S(Yn , rn) and hence in none of the sets T(Sn) . Since U n T(Sn) -

18

BANACH SPACES

= T(X), we have proved that T(X) does not include any sphere

S

c

Y, q.e.d.

Theorem.

If T is a one-to-one bounded linear transformation

of a Banach space X onto a Banach space Y, then T - 1 is bounded.

Proof. Lemma 2 implies that T(S l ) is dense in some sphere in Y, and therefore, by translation, that T(S2 ) is dense in a sphere Ur. But then Ur C T(S2 ) by Lemma 1 , T- 1 ( Ur) C S2 and I I T- 1 I I � 2/r, q.e.d. Corollary.

If T is a linear transformation of a Banach space

X into a Banach space Y such that the graph of T is closed (as a subset of the Cartesian product X X y), then T is bounded.

Proof. The graph of T( = { (x, Tx) : x €:: X} ) is by assumption a Banach space under the norm I I (x, Tx) I I = I I x I I + I I Tx I I · Since the transformation (x, Tx) � x is norm-decreasing and onto X, it follows from the theorem that the inverse transforma­ tion x � (x, Tx) is bounded and in particular that T is bounded, q.e.d. This is the closed graph theorem.

§ 8.

L I N EAR FU NCTI O NA L S

SA. If Y is the complex number field (with I I y I I = I y / ) , the space of continuous linear mappings of a normed linear space X into Y is called the conjugate space of X, denoted X*, and the indi vidual mappings are called linear functionals. Since the com­ plex number field is complete, it follows from 7B that

X* is always a Banach space.

If F €:: X* and N is the nullspace of F, then XjN is one-dimen­ sional, since F becomes one-to-one when transferred to X/N (see 7D) . Otherwise stated, the closed subspace N has deficiency 1 in X. SB. The well-known Hahn-Banach extension theorem implies the existence of plenty of functionals. We shall use the notation [A] for the linear subspace generated by a subset A of a vector space.

19

BANACH SPACES

M

Theorem (Hahn-Banach). If is a linear subspace of the normed linear space X, if F is a bounded linear functional on M, and Xo is a point of X not in then F can be extended to + [xo] without changing its norm.

M

M,

if

Proof. We first give the proof when X is a real normed linear space. The problem is to determine a suitable value for a F(xo) ; after that the definition F(x + XXo) F(x) + Xa for every and every real X clearly extends F linearly to x E::: + [xo]. Assuming that / I F / I = 1, the requirement on a is that I F(x) + and every real X � O. Xa I � I I x + XXo I I for every x E::: After dividing out X, this inequality can be rewritten =

=

M

M

M

- F(X I ) - I I X l + Xo I I �a � - F(X2 ) + I I X2 + Xo I I

M.

But F(X2 ) - F(X I ) F(X2 - X l ) .:s for all X l , X2 E::: I I X2 - X l II � I I X2 + Xo I I + II X l + Xo I I, so that the least up­ per bound of the left member of the displayed inequality is less than or equal to the greatest lower bound of the right member, and a can be taken as any number in between. We now deduce the complex case from the real case, following [6] . We first remark that a complex normed linear space becomes a real normed linear space if scalar multiplication is restricted to real numbers, and that the real and imaginary parts, G and H, of a complex linear functional F are each real linear functionals. Also G(ix) + iH(ix) = F(ix) iF(x) - H(x) + iG(x), so that H(x) = - G(ix) and F(x) = G(x) - iG(ix) . If II F II 1 and, by the above proof, G can be then I I G I I � 1 on on extended to the real linear space M + [xo] in such a way that I I G I I � 1. If we similarly add ixo, we obtain the complex sub­ space generated by and Xo and the real linear functional G defined on it. We now set F(x) = G(x) - iG(ix) on this sub­ F is space; we have already observed that this is correct on obviously a real linear functional on the extended space, and in order to prove that it is complex linear it is sufficient to observe iF(x) . that F(ix) = G(ix) - iG( - x) = i[G(x) - iG(ix)] Finally, if X is given we choose ei8 so that e1,8F(x) is real and non­ G(ei8x) � I I e1,8x I I negative, and have I F(x) I I F(e i8x) I 1, q.e.d. so that F I � X II II, II I =

=

=

=

M

M,

M

M.

=

=

=

=

20

BANACH SPACES

SC. I t follows from Zorn's lemma that a functional F such as above can be extended to the whole space X without increasing its norm. For the extensions of F are partially ordered by inclusion, and the union of any linearly ordered subfamily is clearly an extension which includes all the members of the subfamily and hence is an upper bound of the subfamily. Therefore, there ex­ ists a maximal extension by Zorn's lemma. I ts domain must be X, since otherwise it could be extended further by SB. In particular, if Xo is a non-zero element of X, then there exists a linear functional F E::: X* such that F(xo) = II Xo II and II F II = 1 . For the functional F(Axo) = A l l Xo II is defined and of norm one on the one-dimensional subspace generated by xo, and F can be extended to X as above. Also, if M is a closed subspace of X and Xo f[ M, then there ex­ ists a functional F E::: X* such that I I F II = 1 , F = 0 on M and F(xo) = d, where d is the distance from Xo to A1. For if we define F on [xol + M by F(Axo - x) = Ad, then F is linear and II F II = lub I Ad I 1I I AXo - x II = lubxcM di l l Xo - x II = dlglbxcM II Xo - x I I = did = 1 . We then extend F as above. Let Ml. be the set of all F E::: X* such that F(x) = 0 for every x E::: M. Ml. is called the annihilator of M. The above para­ graph implies that x E::: M if and only if F(x) = 0 for every F E::: Ml.. This fact might be expressed : (Ml.)l. = M.

SD. Theorem. There is a natural norm-preserving isomorphism (imbedding) x � x** of a normed linear space into its second con­ jugate space X** defined by x**(F) = F(x) for every F E::: X*.

If x is fixed and F varies through X*, then x**(F) = F(x) is clearly linear on X*. Since I x**(F) I = I F(x) I � II F 11· 1 1 x II, we see that x** is bounded, with II x** I I �II x II· Since by SC we can find F such that F(x) = I I x II and II F II = 1 , it follows that II x** I I = lubF I F(x) I II I F II � II x II. Therefore, I I x** II = I I x II, q.e.d. Generally X is a proper subspace of X**; if X = X** we say that X is reflexive. SE. Let T be a bounded linear mapping of a normed linear space X in to a normed linear space Y. For every G E::: Y*, the functional F defined by F(x) = G(T(x» is clearly an element of Proof.

BANACH SPACES

21

X*. The mapping T* thus defined from y* to X* is evidently linear; it is said to be the adjoint of T. If X is imbedded in X* *, then T* * is an extension of T, for if x E:: X and G E:: Y*, then (T**x* *) G = x**(T*G) = (T*G) x = G( Tx) = (Tx) **G, showing that T* *(x**) = (Tx) * *. Since I F(x) I = I G(T(x)) I � I I G 1 1 · 1 1 T 1 1 · 1 1 x I I , we see that I I T*G I I = I I F I I � I I G 1 1 · 1 1 T I I, and hence T* is bounded with I I T* I I � I I T I I · Since T** i s an extension of T, we have con­ versely that " T" � " T* * I I � I I T* I I · Therefore, I I T* I I = I I Til· SF. We shall need the following theorem in Chapter 4:

Theorem. If {Fn} is a sequence of bounded linear functionals on a Banach space X such that the set of numbers {I Fn(x) I } is bounded for every x E:: X, then the set of norms {I I Fn I I } is bounded.

Proof. I t is sufficient to show that the sequence of functionals is bounded in some closed sphere. For if I Fn(x) I � B whenever x E:: S(xo, r) = {x: I I x - Xo I I < r} , then I Fn(Y) I � I Fn( y - xo) I + I Fn(xo) I � 2B if I I y I I < r, and the norms I I Fn I I have the common bound 2B/r. But if {Fn} is unbounded in every sphere, we can find induc­ tively a nested sequence of closed spheres {8m} with radii con­ verging to 0 and a subsequence {FnJ such that I Fnm(x) I > m throughout Sm. We do this by first choosing a point Pm + l in­ terior to Sm and a functional Fnm+l such that I Fnm+l(Pm +l) I > m + 1 , and then this inequality holds throughout a sphere about Pm + l by the continuity of Fnm+1• Since X is complete, there is a point Xo in niSm, and {I Fnm(XO) I } is bounded by hypothesis, contradicting I Fnm(XO) I > m. Therefore, {Fn} cannot be un­ bounded in every sphere, and the proof is finished.

Corollary. If {xn} is a sequence of elements of a normed linear space X such that {I F(xn) I } is bounded for each F E:: X*, then the set of norms {I I Xn I I } is bounded. For X* is a Banach space and Xn E:: X** by SD ; the above theorem can therefore be applied.

22

BANACH SPACES

§ 9. TH E WEAK TO POLOGY FO R X* 9A. The conjugate space of a normed linear space X is a set of complex-valued functions on X and, therefore, is a subset of the Cartesian product space IIxE:x Cx, where the elements x E: X serve as indices and the coordinate space Cx is the complex plane. The relative topology thus induced in X* is called the weak to­ pology for X*. We know (see SC) that it is the weakest topology for X* in which the functions x** (the projections of X* on the coordinate spaces) are all continuous. The sets of the form { F: I F(x) - Ao I < e } , depending on x, Ao, and e, form a sub­ basis for the weak topology of X*. 9B. Theorem. The strongly closed unit sphere in X* is com­ pact in the weak topology. Proof. The strongly closed unit sphere is, of course, the set S = { F: I I F I I � 1 } . The values assumed by these functionals at a point x E: X are complex numbers in the closed circle Sx of radius I I x I I . S is thus a subset of the Cartesian product IIxE:x Sx , which, by the Tychonoff theorem ( SD), is compact. It is there­ fore sufficient to show that S is weakly closed in IIxE:x Sx . Ac­ cordingly, let G be a function in its closure. Given x, y, and e, the set { F: I F(x) - G(x) I < e } n { F: I F(y) - G(y) I < e } n { F: I F(x + y) - G(x + y) I < e } is open in II x Sx and contains G(x), and therefore contains an element F E: S. But F(x + y) = F(x) + F(y ), so that I G(x) + G(y) - G(x + y) I < 3e. Since e is arbitrary, i t follows that G is additive. Similarly i t follows that G(Ax) = AG(x) and that I G(x) I � I I x I I . Therefore, G E: S, proving that S is weakly closed in IIx Sx, q.e.d. 9C. There are a number of interesting and important theo­ rems about the weak topology in X* which will not be needed in this book but which perhaps ought to be sampled in the inter­ ests of general education. We confine ourselves to one su'ch situ­ ation. It was seen in BC that a closed subspace M c X is the annihilator of its annihilator eM = M.l.. .l.. ) . A similar characteri­ zation can be given to subspaces of X* which are closed in the weak topology.

23

BANACH SPACES

Theorem. Let M be a subspace of X*, let M.L be the intersection of the nullspaces of the functionals in M and let (M.L) .L be the set of all functionals in X* which vanish on M.L. Then M = (M.L) .L if and only if M is weakly closed. Proof. For every x E:: X the functional x**(F) is by definition continuous with respect to the weak topology on X*, and there­ fore its nullspace is weakly closed. I t follows that, if A is any subset of X, then the set of functionals which vanish on A is an intersection of such nullspaces and hence weakly closed. In par­ ticular (M.L) .L is weakly closed. Since M c (M.L) .L, there re­ mains to be shown only that, if Fo is not in the weak closure of M, then Fo is not in (M.L) .L, that is, that there exists an Xo E:: M.L such that Fo(xo) � O. But if Fo is not in the weak closure of M, then there exists a weak neighborhood of Fo not intersecting M, i.e., there exists E and xI, " ' , Xn such that no G E:: M satisfies I G(Xi) - FO(Xi) I < E for every i = 1 , " , n. The elements Xi map M onto a subspace of complex Euclidean n-space G � (G(Xl), " ' , G(xn) , and no point of this subspace is in an E-cube about the point (FO(Xl) " ' , Fo(xn) . In particular the subspace does not contain this point and is of din1ension at most n 1, so that there exist constants C I, " ' , C n such that L CiG(Xi) = 0 for every G E:: M and L CiFO(Xi) = 1 . Taking Xo = L1 CiXi, these conditions become G(xo) = 0 for every G E:: M and Fo(xo) = 1 , proving that Fo is not in (M.L).L. Remark: I t is possible, by refining this type of argument to show that, if M is such that its intersection with every strongly­ closed sphere is weakly closed, then M = (M.L) .L. (See [1 1].) '

-

§ 10.

H I L B ERT S PA C E

lOA. A Hilbert space H is a Banach space in which the norm satisfies an extra requirement which permits the introduction of the notion of perpendicularity. Perhaps the neatest formulation of this extra property is the parallelogram law :

(1)

II x + y 1 1 2 + I I x

-

y 1 1 2 = 2[ 1 1 X 1 1 2 + /I y 1 1 2] .

24

BANACH SPACES

A scalar product (x, y) can now be defined,

(2) 4(x, y) = I I x + y 1 1 2 - I I x - y 1 1 2 + i l l x + iy 1 1 2 - i l l x - iy 1 1 2 , and rather tedious elementary calculations yield the laws : (3)

(4)

(5)

(6)

( Xl + X 2 , y) = ( Xl) y) + ( X2 ' y) ;

(Xx, y) = X(x, y) ; (x, y) = (y, x) ;

(x, x) > 0 if x � O.

The additivity condition (3) can be proved by treating the real and imaginary parts separately, and applying the parallelogram law to four sums of the type I I Xl + X2 + y 1 1 2 + I I Xl - X2 + y 1 1 2 to obtain the real part. Repeated applications of (3) lead to « m/2 n )x, y) = (m/2 n ) (x, y), and hence, by continuity, to (ax, y) = a(x, y) for positive a. On the other hand, i t follows from di­ rect inspection of (2) that ( - x, y) = - (x, y) and ( ix, y) = i (x, y), completing the proof of (4) . The remaining two laws also follow upon direct inspection. lOB. In the applications, however, the scalar product is gen­ erally more basic than the norm, and the usual development of the theory of Hilbert space starts off with (3)-(6) as axioms, the parallelogram law now following at once from the definition of I I x i i as (x, x) Y2. We also obtain the Schwarz inequality, I (x, y) I � I I x 1 1 l l y I I , (7) · with equality only if x and y are proportional, from the expansion o � (x - Xy, x - Xy) = (x, x) - X(y, x) - �(x, y) + I X 1 2 (y , y), by setting X = (x, x)/(y, x) . If (y, x) = 0, the inequality is trivial. Again, substituting from the Schwarz inequality in the expansion of I I x + y 1 1 2 = (x + y, x + y), we obtain the triangle inequality : I I x + y I I �I I x I I + I I I , (8) I Y with equality only if x and y are proportional, with a non-nega­ tive constant of proportionality. Thus H is a normed linear

BANACH SPACES

25

space, and now, rather inelegantly, the final assumption is made that

H is complete in the norm I I x I I = (x, x) �. Two elements x and y are said to be orthogonal if (x, y) = o. An immediate consequence of orthogonality is the Pythagorean property : I I x + y 1 1 2 = I I X 1 1 2 + I I y 1 1 2 . lOCo Theorem. A closed convex set C contains a unique ele­ ment of smallest norm. (9)

Proof. Let d = glb { I I x I I : x E: C} and choose X n E: C such that I I X n I I 1 d. Then (xn + xm) /2 E: C since C is convex, and so I I X n + Xm I I � 2d. Since I I X n - Xm 1 1 2 = 2 ( 1 1 Xn 1 1 2 + I I Xm 1 1 2 ) - I I X n + Xm 1 1 2 by the parallelogram law, it follows that I I X n - Xm I I � 0 as n, m � 00. If Xo = lim X n, then I I Xo I I = lim I I Xn I I = d. If x were any other element of C such that I I x I I = d, then (x + xo) /2 would be an element of C with norm less than d by (8) above. Therefore, Xo is unique.

lODe Theorem. If M is a closed subspace of H, then any ele­ ment x E: H can be uniquely expressed as a sum x = X l + X2 of an element X l € M and an element X2 -.l M. The element X l is the best approximation to x by elements of M. Proof. If x fl M, the coset { x - y : y E: M} might be called the hyperplane through x parallel to M. I t is clearly convex and closed, and hence contains a unique element x - X l closest to the origin ( IOC) . Setting X2 = x - X I , we have I I X2 - Xy 1 1 2 � I I X2 1 1 2 for every y E: M and every complex X. Setting X = (X2 ' y)/(y, y) and expanding, we obtain - I (y, X2 ) I � 0, and hence (y, X2 ) = o. Thus X2 -.l M. This argument can be run backward to deduce the minimum nature of I I X2 I I from the fact that X2 -.l M, which proves the uniqueness of the decomposition. Of course, this can easily be proved directly. IOE. The set of elements orthogonal to M forms a closed sub­ space Mi., and the content of the above theorem is that H is the direct sum of M and M.L. The element X l is called the projec­ tion of x on M, and the transformation E defined by E(x) = X l

26

BANACH SPACES

is naturally called the projection on M. It is clearly a bounded linear transformation of norm 1 which is idempotent (E2 = E) . Its range is M and its nullspace is M.l.. Lemma. If A is any linear transformation on H such that, in the above notation, A(M) c M and A(M.l.) C M\ then AE = EA. Proof. If X l � M, then AX I � M, so that AEx l = AX I = EAx l ' If X2 � M\ then AX2 � M\ so that AEx2 = 0 = EAx 2 . Since any X can be written X l + X2 , it follows that AEx = EAx for all x, q.e.d. If MI and M are orthogonal closed subspaces, the Pythagorean property I I X l +2 x2 1 1 2 = I I Xl 1 1 2 + I I x2 1 1 2 for a pair of elements X l � Ml and X2 � M2 leads easily to the conclusion that the algebraic sum MI + M2 is complete, hence closed. Now let MI , . . " Mn be a finite collection of mutually orthogonal closed sub­ spaces and let Mo be the orthogonal complement of their (closed) algebraic sum. Then H is the direct sum of the subspaces ]vlo, . " Mn, the component in Mi of a vector X is its orthogonal proj ection Xi on Mi, and Pythagorean property generalizes to the formula I I X 1 1 2 = I I 2:0 Xi 1 1 2 = 2:0 I I Xi 1 1 2 . As a corollary of this remark we obtain Bessel's inequality : if Ml ) " ' , Mn are orthogonal closed subspaces and Xi is the projection of X on Mi, then 2:1 I I Xi 1 1 2 � I I X 1 1 2 . .

lOF. Theorem. Let { Ma } be a family, possibly uncountable, of

pairwise orthogonal closed subspaces of H, and let M be the closure of their algebraic sum. If Xa is the projection on Ma of an element X � H, then Xa = 0 except for a countable set of indices a n, 2:i Xan is convergent, and its sum is the projection of X on M. Proof. Let ah " ' , a n be any finite set of indices. Since 2: I I Xai 1 1 2 � I I X 1 1 2, it follows that Xa = 0 except for an at most countable set of indices a n and that 2: i I I Xai 1 1 2 � I I X 1 1 2 . Set­ ting Yn = 2:1 Xai) we have I I Yn - Y m 1 1 2 = 2:� I I Xai 1 1 2 ---7 0 as n, m ---7 00. Hence Yn converges to an element Y � M, and, since the projection of Y on Ma is clearly Xa, it follows that X - Y is orthogonal to Ma for every a and so to M, q.e.d. lOG. Theorem. For each linear functional F � H* there is a unique element Y � H such that F(x) = (x, y).

27

BANACH SPACES

Proof. If F = 0, we take y = O. Otherwise, let M be the nullspace of F and let z be a non-zero element orthogonal to M. Let the scalar c be determined so that the element y = cz satisfies F(y ) = (y, y) (i.e., c = F(z) /(z, z)) . Since HIM i s one-dimen­ sional, every x E: H has a unique representation of the form x = m + Xy, where m E: M. Therefore, F(x) = F(m + Xy) = XF(y) = X(y, y) = (m + Xy, y) = (x, y), q.e.d.

§ 11.

I N V O L UTIO N

ON

ill (H)

I IA. We have already observed that the bounded linear trans­ formations of a Banach space into itself form a Banach algebra In a Hilbert space H this normed algebra ill (H) admits another important operation, taking the adjoint (see BE) . For under the identification of H with H* (lOG) the adjoint T* of a transformation T E: ill (H) is likewise in ill (H) . In terms of the scalar product the definition of T* becomes :

(7C) .

( Tx, y) = (x, T*y) (1) for all x, y E: H. We note the following lemma for later use. Lemma. The nul/space of a bounded linear transformation A is the orthogonal complement of the range of its adjoint A* . For Ax = 0 if and only if (Ax, y) = 0 for all y E: H, and since (Ax, y) = (x, A*y), this is clearly equivalent to x being orthogonal to the range of A*.

l IB. Theorem. The involution operation T � T* has the following properties: 1) T* * = T 2) (S + T) * = S* + T* 3) (XT) * = X T* 4) (ST) * = T*S* 5) I I T*T I I = I I T 1 1 2 6 ) (1 + T*T) - 1 E: ill (H), where 1 is the identity transformation. Proof. Properties 1 ) to 4) are more or less obvious from ( 1 ) . For instance 4 ) follows from (x, (ST) *y) = (STx, y) = ( Tx, S*y) = (x, T*S*y) . For 5), we already know that I I T* T I I � / I T* / I . I I T i l = I I T 1 1 2 (using BE) , and i t remains to be shown that

28

BANACH SPACES

I I T2 1 1 � I I T* T I I . But I I T2 1 1 lub ( Tx, Tx)/ 1 1 x 1 1 2 = lub (X, T* Tx) / 1 1 X 1 1 2 � I I T* T I I by Schwarz's inequality. To prove 6) we notice that «/ + T* T)x, x) � I I (/ + T* T)x 1 1 · 1 1 x I I , I I X 1 1 2 + I I Tx 1 1 2 =

=

so that I I (I + T* T)x I I � I I x I I . In particular I + T* T is one­ to-one, and, since it is clearly self-adjoint, its range is dense in H by I IA. If Yo is any element H, we can therefore find a se­ quence X n such that (I + T* T)xn � Yo, and since (I + T* T) - 1 is norm-decreasing i t follows that X n is a Cauchy sequence and converges to some Xo E:: H. Thus (I + T* T)xo = Yo, proving that the range .of (I + T* T) is the whole of H. Therefore, (I + T* T) - 1 is a bounded linear transformation on H with norm at most one, q .e.d. I IC. If A is any algebra over the complex numbers, a *-opera­ tion on A satisfying the above properties (1) to (4) is said to be an involution. A Banach algebra with an identity and an invo­ lution satisfying all the properties (1) to (7) is said to be a C*­ algebra. Gelfand and Neumark [13] have shown that every C*­ algebra is isometric and isomorphic to an algebra of bounded transformations on a suitable Hilbert space. Thus a subalgebra of (B(H) is the most general C*-algebra. We shall see in the fifth chapter as an easy corollary of the Gelfand theory that, if a C *­ algebra is commutative, then it is isometric and isomorphic to the algebra of all continuous complex-valued functions on a suit­ able compact Hausdorff space. An application of this result yields a very elegant proof of the spectral theorem. l ID. The following lemma holds for any pair of linear map­ pings A and B of a vector space in to itself:

Lemma. If AB = BA and if N and R are the nullspace and range of A respectively, then B(N) c N and B(R) c R.

If x E:: N, then 0 = BAx = A(Bx), so that Bx R then x = Ay for somey, so that Bx = BAy = ABy

Proof.

If x

E::

E:: E::

N. R.

Chapter III INTEGRATION

The theory of Haar measure on locally compact groups is con­ veniently derived from an elementary integral (linear functional) on the continuous functions which vanish outside of compact sets (see Chapter VI) . We shall accordingly begin this chapter by presenting Daniell's general extension of an elementary inte­ gral to a Lebesgue integral [9] . The rest of the chapter centers around the LP spaces and the Fubini theorem. § 1 2.

TH E DANI E L L I NTEGRAL

12A. We suppose given a vector space L of bounded real­ valued functions on a set S and we assume that L is also closed under the lattice operations / U g = max (/, g) and / n g = min (/, g) . We can take absolute values in L since 1 / 1 = / U 0 - / n o. Second, we suppose defined on L a non-negative linear functional 1 which is continuous under monotone limits. These conditions on 1 are explicitly as follows :

(1) (2)

(3) (3') (4)

1(/ + g) = 1(/) + l(g) l(cj) = cl(/) / � 0 ==} 1(/) � 0 / � g ==} 1(/) � l(g) /n 1 0 ==} 1(/n) 1 0

Here "/n 1 " means that the sequence /n is pointwise monotone · 29

30

INTEGRATION

p,

decreasing, i.e., thatfn + l (P) � fn(P) for all n and and "fn l f" means that fn decreases pointwise to the limit f. Such a functional will be called an integral, to avoid confusion wi th the bounded linear functionals of normed linear space theory. Our task is to extend I to a larger class of functions having all the properties of L, and which is in addition closed under certain countable operations. As an example, L can be taken to be the class of continuous functions on [0, 1] and I to be the ordinary Riemann integral. Properties (1 ) to (3) are well known, and (4) follows from the fact that on a compact space pointwise monotone convergence implies uniform convergence (see 1 6A) . The extension of L is then the class of Lebesgue summable functions, and the extended I is the ordinary Lebesgue integral. 12B. Every increasing sequence of real-valued functions is con­ vergent if + 00 is allowed as a possible value of the limit function. Let U be the class of limits of monotone increasing sequences of functions of L. U includes L, since the sequence fn = f is trivi­ ally increasing to f as a limit. It is clear that U is closed under addition, multiplication by non-negative constants, and the lat­ tice operations. We now want to extend I to U, and we make the obvious defi­ nition : I(f) = lim I(fn ), where fn i f and fn E: L,/and where + 00 is allowed as a possible value of I. I t will be shown in 12C that this number is independent of the particular sequence { fn } converging tof. Assuming this, it is evident that the new defini­ tion agrees with the old in casef E: L (for then we can setfn = f) and that the extended I satisfies ( 1 ) and (2) with c � 0. I t will also follow from 12C that I satisfies (3') .

12C. Lemma. If { fn } and { gm } are increasing sequences 0/ functions in L and lim gm � lim fn, then lim I(gm) � lim I(fn) . Proof. We first notice that, if k E: L and limfn � k, then lim I(fn) � I(k) , for fn � fn n k and fn n k i k, so that lim I(fn) � lim I(fn n k) = I(k), by (3') and (4) . Taking k = gm and passing to the limit as m � 00, i t follows that lim I(fn) � lim I(gm), proving 12C. If limfn = lim gm, we

INTEGRATION

31

also have the reverse inequality, proving that I i s uniquely de­ fined on U. 12D. Lemma.

i I(f) ·

If fn E:: U and fn i f, then f E:: U and I(fn)

n

Proof. Choose gn m E:: L so that gnm i mJn' and set hn = gi U . . . U gn n . Then hn E:: L and { hn} is an increasing sequence.

Also gin � h n � fn if i � n. If we pass to the limit first with re­ spect to n and then with respect to i, we see thatf � lim hn � f. Thus h n i f and f E:: U. Doing the same with the inequality I(gin ) � I(h n) � I(fn) , we get lim I(fi) � I(f) � lim I(fn ), proving that I(fn) i I(f) , q.e.d. 12E. Let U be the class of negatives of functions in U: U = { f: -f E:: U} . If f E:: - U, we make the obvious defi­ nition I(f) = - I( -f) . I f f is also in U, this definition agrees wi th the old, for f + ( ....f) ... = 0 and I is addi ti ve on U. The class U obviously has properties exactly analogous to those of U. Thus, - U is closed under monotone decreasing limits, the lattice operations, addition and multiplication by non-negative constants, and I ' has on U the properties (1), (2) and (3') . It is important to remark that, if g E:: - U, h E:: U, and g � h, then h - g E:: U and I(h) - I(g) = I(h - g) � o. A function f is defined to be summable (better, I-summable) if and only if for every E > 0 there exist g E:: - U and h E:: U such that g � f � h, I(g) and I(h) are finite, and I(h) - I(g) < E. Varying g and h, it follows that glb I(h) = lub I(g), and I(f) is defined to be this common value. The class of summable func­ tions is designated L l (or L l (I)) ; it is the desired extension of L. We note immediately that, if f E:: U and I(f) < 00, then f E:: Ll and the new definition of I(f) agrees with the old. For then there exist fn E:: L such that fn i f, and we can take h = f and g = fn for suitably large n . 12F. Theorem. L l and I have all the properties of L and I. -

-

-

-

Proof. Givenfl andf2 E:: L l and given E > 0, we choose g l and

g2 E:: - U, and h I > h 2 E:: U, such that gi � fi � hi and I(hi) I(gi) < E/2, i = 1 , 2. Letting 0 be any of the operations + , n , U , we' have g l 0 g2 � fl Of2 � h i 0 h 2 and h i 0 h 2 - g l 0 g2 �

32

INTEGRATION

(hi - g l ) + (h 2 - g2 ) , and therefore I(h l 0 h 2 ) - I(gl 0 g2) < E. I t follows at once thatfl + f2 , fl u f2 and fl n f2 are in L l , and, since I is additive on U and - U, that I I(fl + f2 ) - I(fl ) I(f2 ) I < 2E. Since E is arbitrary, (1 ) follows. That cf E:::: L l and I(cf) = cI(f) if f E:::: L l is clear; we remark only that the roles of the approximating functions are interchanged if c < 0. If f !?; 0, then h !?; ° and I(f) = glb I(h) !?; 0, proving (3) . (4) follows from the more general theorem below.

12G. Theorem.

If fn E:::: L l (n = 0, 1 , . . . ) , fn i f and lim I(fn ) < 00, then f E:::: L l and I(fn) i I(f) . Proof. We may suppose, by subtracting off fo if necessary, that/o = 0 . We choose h n E:::: U (n = 1 , . . . ) so that (fn - fn - l ) � h n and I(h n ) < I(fn - fn - l ) + E/2 n . Then fn � L:1 h i and L:1 I(hi) < I(/n ) + E. Setting h = L:r h i, we have h E:::: U and I(h) = L:r I(hi) by Moreover, f � h and I(h) � lim I(/n ) + E. Thus if m is taken large enough, we can find g E:::: - U so that g � /m � f � h and I(h) - I(g) < 2E. It follows that f E:::: L l and that I(f) = lim I(fm) . A family of real-valued functions i s said to b e monotone if it is closed under the operations of taking monotone in­ creasing and monotone decreasing limits. The smallest mono­ tone family including L will be designated (B and its members will be called Baire functions. If h � k, then any monotone family mr which contains (g U h) n k for every g E:::: L also con tains (/ U h) n k for every f E:::: (B, for the functions f such that (f U h) n k E:::: mr form a monotone family which includes L and therefore includes (B. In particular the smallest monotone family including L + is (B + (where, for any class of functions C, C+ is the class of non-negative functions in C) .

12D.

12H.

Theorem.

(B

is closed under the algebraic and lattice operations.

(From Halmos [23].) For any function f E:::: (B let mr(/) be the set of functions g E:::: (B such that f + g, f U g and / n g E:::: (B. mr(f) is clearly a monotone family, and, iff E:::: L, then 'SIT(/) includes L and therefore is the whole of (B. But g E:::: 'SIT(/) if and only if / E:::: .'SIT(g) ; therefore, 'SIT (g) includes L Proof.

INTEGRATION

33

for any g E::
Proof. The family of functions I E::
Proof. Let 5= be this smallest family. For any fixed g E:: L + the functions I E::
including L+ and therefore equal to
31A

33A,

Proof. The necessity is trivial. In proving the sufficiency we may suppose that I � o. Then the family of functions h E::
34

INTEGRATION

12G)

and includes L +, such that h n g E: L 1 is monotone (by and is therefore equal to ill +. Thus f = f n g E: L 1 , q.e.d. We now extend I to any function in ill + by putting I(f) = 00 if f is not summable. A function f E: ill will be said to be integrable if either its positive part f+ = f U 0 or i ts negative part f- = (f n 0) is summable. Then I(f) = I(f+) - I(f- ) is unambiguously defined, although i t may have either + 00 or - 00 as its value, and f is summable if and only if i t is integrable and have the following and I I(f ) I < 00. Theorems theorem as an immediate corollary.

12K.

-

12F

12G

Theorem. Iff and g are integrable, then f + g is integrable and I(f + g) = I(f) + I(g), provided that I(f) and I(g) are not oppo­ sitely infinite. If fn is integrable, I(f1 ) > 00 and fn i f, then f is integrable and I(fn) i I(f) · -

§ 13.

13A. If A

EQUIVA L E N C E A N D M EAS U RA B I LITY

c

S, then " CPA " conventionally designates the char­ acteristic function of A: CPA (P) = 1 if p E: A and CPA (P) = 0 if p E: A'. If CPA E: ill , we shall say that the set A is integrable and that, if A define its measure p.(A) as I(CPA) . I t follows from and B are integrable, then so are A U B, A n B, and A B ; that, if { An } i s a disjoint sequence o f integrable and from sets, then U � An is integrable and p.( U � An) = L� p.(An) . If A is integrable and p.(A) < 00 , then A will be said to be summable. The integral I will be said to be bounded if S is summable : i t is clear that I is then a bounded linear functional with respect to the uniform norm. We shall now further restrict L by adding a hypothesis used by Stone [47], namely, that

12F

-

12G

13B.

f

E:

L

=>

fn 1

E:

L.

Then also f U ( - 1 ) E: L, and these properties are preserved through the extension, so thatf E: ill => f n 1 E: ill . This axiom is added to enable us to prove the following theorem. Theorem. If f E: ill and a > 0, then A = { p : f(P) > a } integrable set. If f E: L 1 , then A is summable.

tS

an

35

INTEGRATION

Proof. The function fn = [n (f - f n a)] n 1 belongs to (B and it is easy to see that fn i
Theorem. If f � 0 and A = { p : f(p) > a } is integrable for every positive a, then f E::: (B. m m Proof. Given 0 > 1 , let Am5 = { p : o < f(p ) � o + l } , - co < m < co . Let 0, then fa E::: (B +. Proof. fa > b if and only iff > b 1 / a, and the corollary follows from the theorem and 13B. Corollary 2. If f and g E::: (B +, then fg E::: (B +. Proof. fg = [(f + g) 2 - (f - g) 2] /4.

i(

Corollary 3. If f E::: (B +, then I(f) =

taken in the customary sense.

ff dp.., where the integral

Proof. The function f5 in the proof of the theorem was defined

with this corollary in mind. We have f5 � f � 0 (f5 ) and I(f5) = L o mI(
f

ff5 dp.. � ff � off5 dp.., we see that, if either of the numbers I(f) and ff dp.. is finite, then they both are, and that I I(f) - ff dp.. 1 � ( 0 1 )I(f5) � ( 0 - l )I(f) . Since 0 is any oI(f5 ) and

-

number greater than one, the corollary follows. 13D. A function f is said to be a null function if f E::: (B and 1(\ f I ) = O. A set is a null set if its characteristic function is a null function, that is, if it is integrable and its measure is zero. Any integrable subset of a null set is null, and a countable union

36

INTEGRATION

of null sets is a null set. Any function of ill dominated by a null function is a null function, �nd a countable sum of null functions is null. Two functions J and g are said to be equivalent if (J g) is a null function. -

Theorem. A Junction J E: ill is null if and only if {p :J(P) -:;e o }

is a null set.

Proof. Let A = {p : J(p) -:;e o} . If A is a null set, then ncpA is a null function and ncpA i 00CPA, so that the function equal to +00 on A and zero elsewhere (OOcpA) is a null function. But o � 1 J 1 � OOCPA so that J is null. Conversely, if J is null, then so is (n l J I ) n 1, and, since (nl J I ) n 1 i CPA, it follows that CPA is a null function and A is a null set. We are now in a position to remark on an am bigui ty remaining namely, that, since the values ±oo are allowed for a sum­ in mabIe function, the sum J1 + J2 of two summable functions is not defined on the set where J1 = ±oo and J2 = =Foo. However, i t is easy to see that, if J is summable, then the set A where 1 J 1 = 00 is null, and we can therefore eliminate ± oo as possible values if the convergence theorems are restricted to almost every­ where convergence (pointwise convergence except on a null set) . The other method of handling this ambiguity is to lump together as a single element all functions equivalen t to a given function and then define addition for two such equivalence classes by choosing representatives that do not assume the values ± oo . 13E. Very often the space S is not itself an integrable set, a fact which is the source of many difficulties in the more technical aspects of measure theory. I t may happen, however, and does in the case of the Haar measure on a locally compact group, that S is the union of a (perhaps uncountable) disjoint family { Sa } of integrable sets with the property that every integrable set is in­ cluded in an at most countable union of the sets Sa . Now the above-mentioned difficulties do not appear. A set A is defined to be measurable if the intersections A n Sa are all integrable, and a function is measurable if its restriction to Sa is integrable for every a. It follows that the intersection of a measurable set with any integrable set is integrable, and that the restriction of a measurable function to any integrable set is integrable. The

12F,

INTEGRATION

37

notion of a null function can be extended to cover functions whose restrictions to the sets Sa are all null, and the same for null sets. Two functions are equivalent if their difference is null in the above sense. The families of measurable functions and of measurable sets have much the same elementary combinatorial properties as the corresponding integrable families and there seems little need for further discussion on this point. The important property of a measurable space of the above kind is the possibility of build­ ing up a me�surable function by defining it on each of the sets Sa. In a general measure space S there is no way of defining a measurable function by a set of such local definitions. Since every integrable set is a countable union of summable sets, the sets Sa may be assumed to be summable if so desired.

14A. If

§ 14. THE REAL LP-SPACES

� I, the set of functions f � CB such that I f I p is summable is designated LP. The next few paragraphs will be devoted to showing that LP is essentially a Banach space under the norm I I f l i p = [1 ( 1 f I p )] 1 /p · Since I I / l i p = 0 if/ is null, it is, strictly speaking, the quotient space LP /'J"{, ('J"{, = the sub­ space of null functions) that is a Banach space. However, this logical distinction is generally slurred over, and LP is spoken of as a' Banach space of functions. If fg � Ll, it is convenient to use the scalar product notation (/, g) = 1 (/g) . The Holder inequality, proved below, shows that L2 is a real Hilbert space, and it follows similarly that the com­ plex L2 space discussed later is a (complex) Hilbert space. These spaces are among the most important realizations of Hilhert space. (Holder's inequality.) If f � LP, g � L q and ( l /p) + ( l /q) = 1 , then /g � Ll and I (/, g) I � I I / I I p l l g I l q · Proof. From the ordinary mean value theorem of the calculus it follows that, if x � 1 and p � 1 , then Xl /p � x/p + 1 /q . I f x = a lb this becomes a1 /Pb1 / q � � + � . Iff � LP, g � L q , and p q

14B.

,

P

38

INTEGRATION

1 1 1 l i p � ° � I I g I l q, then we can take a = 1 1 I p/ l 1 1 I l pp, b = I g I q/ l l g I l q q in this inequality. I t follows from 13C and 12J that the left member is summable, and integrating we get 1 ( 1 19 I I ( 1 1 1 l i p ' I I g I l q)) � 1 . Holder's inequality now follows from I l(h) I � 1( 1 h I ) · If 1 1 1 l ip = ° or I I g I l q = 0, then 19 is null and the inequality is trivial.

m +, then (1, g) � I I 1 I l p l l g I I q . 14C. (Minkowski's inequality.) 111 and g €. LP (p � 1 ) , then 1 + g €. LP and I I 1 + g l lp � 1 1 1 l i p + I I g l lp· Proof. The case p = 1 follows directly from 1 1 + g I � 1 1 1 + I g I · If P > 1 and 1, g €. LP, the inequality 1 1 + g I p � [2 max ( 1 1 I , I g D] p � 2P ( 1 1 I p + I g I p) shows that1 + g €. LP. Then I I 1 + g I l pp � 1( 1 1 + g I p - 1 1 1 I) + 1( 1 1 + g I p -1 1 g I ) Corollary.

11 1, g

€.

� 1 1 1 + g I lpp -1 1 1 1 l ip + 1 1 1 + g I l pp -1 1 1 g l ip , where the second ineq uali ty follows from the Holder inequal­ i ty. Minkowski's inequality is obtained upon dividing by 1 1 1 + g I l pp - l . If 1 1 1 + g l ip = 0, the inequality is trivial. Corollary. 111, g €. m +, then 1 1 1 + g l ip � 1 1 1 l ip + I I g l ip·

14D. Since the homogen eity property I I A1 l i p = I A I I I 1 l ip is

obvious from the definition of I I 1 l ip , Minkowski's inequality shows that LP (actually LPInull functions) is a normed linear space. It remains to be shown that LP is complete. Theorem. LP is complete if p � 1 .

Proof. We first remark that, if1n €. LP ,1n � ° and L: r I I 1n l ip < 00, then 1 = L:r1n €. LP and 1 1 1 l ip � L: r I I 1n l ip· For gn =

L:11i €. LP and I I gn l i p � L:1 1 1 1i l i p by the Minkowski in­ equality, and since gn i 1 it follows that 1 €. LP and 1 1 1 l ip = lim I I gn l ip � L:r I I 1n l ip , by 12G. Now let { 1n } by any Cauchy sequence in LP . We may sup­ pose for the moment, by passing to a subsequence if necessary, that I I 1n + l - 1n l ip < 2 - n . If gn = 1n - L:: I 1i l - 1i I and +

39

INTEGRATION

h n = fn + 2:: I fi + l - fi I , it follows from our first remark that gn and h n E: LP and that I I h n - gn l i p < 2 -n +2 . Moreover, the sequences gn and hn are increasing and decreasing respectively, and if we take f as, say, lim gn , then f E: LP and I I f - fn l i p � I I h n - gn l i p < 2 - n +2 . Our original sequence is thus a Cauchy sequence of which a subsequence converges tof, and which there­ fore converges to f itself, q.e.d. We say that a measurable function f is essentially bounded above if it is equivalent to a function which is bounded above, and its essential least upper bound is the smallest number which is an upper bound for some equivalent function. The essential least upper bound of I f I is denoted I I f 1 1 00 ; this is the same sym­ bol used elsewhere for the uniform norm, and I I f 1 1 00 is evidently the smallest number which is the uniform norm of a function equivalent to j. If LOO is the set of essentially bounded measur­ able functions, equivalent functions being identified, it is clear that LOO is a Banach space (in fact, a Banach algebra) with re­ spect to the norm I I f I I The following theorem can be taken as motivation for the use of the subscript " 00" in I I f 1 1 00 . Theorem. Iff E: LP for some p > 0, then lim q 00 I I f I I q exists and equals I I f 1 1 00, where 00 is allowed as a possible value of the limit. Proof. If I I f 1 1 00 = 0, then I I f I l q = ° for every q > ° and the theorem is trivial. Otherwise, let a be any positive number less than I I f 1 1 00 and let Aa = { x : I f(x) I > a} . Then I I f I I q � a [IL ( Aa) P / q . Moreover, ° < IL (Aa) < 00, the first inequality hold­ ing because a < I I f 1 1 00 and the second because f E: LP. There­ fore lim q 00 I I f I l q � a . Since a is any number less than I I f 1 1 00, lim q =-= I I f I l q � I I f 1 1 00 . I I f I I q � I I f 1 1 00 is obvious if The dual inequality lim q I l f l l oo = 00 . Supposing, then, that I l f l l oo < 00, we have I f l q � I f I p ( 1 1 f 1 1 (0) q -p and there fore I I f I l q � (I I f I I p ) p /q ( 1 1 f 1 1 (0 ) 1 -p /q,

14E.

<Xl"

14F.

-+

-+

-+ 00

00 I I f I l q � I I f 1 1 <Xl" Theorem. L is a dense subset of LP, 1 � P < 00.

giving the desired inequali ty lim q

14G.

-+

40

INTEGRATION

Proof. We first observe that L is dense in L l by the very defi­

nition of L l (via U) . Now let J be any non-negative function in LP. Let A = { x : l /n < J(x) < n } and let g = J
I I f - h l i p � I I f - g l ip + I I g - h l ip < e, § 15.

TH E C O NJ UGATE

S PA C E O F

LP

15A. The variations of a bounded functional. We now con­ sider the space L of 12A as a real normed linear space under the uniform norm I I J 1 I c
Theorem. Every bounded linear Junctional on L is expressible as the difference oj two bounded integrals (non-negative linear Junc­ tionals) .

Proof. Let P be the given bounded functional, and, if J � 0, let P+(J) = lub { P(g) : 0 � g �J} . Then P+(J) � 0 and I P+ (f) I � I I P 1 1 · 1 1 J I I · It is also obvious that P + (cf) cP+(J) if c > O. Consider now a pair of non-negative functions /1 and f2 in L. If 0 �gl �Jl and 0 � g2 � J2 ' then 0 �gl + g2 �J1 + J2 and P+(fl + J2 ) � lub P(gl + g2 ) = lub P(gl) + lub P(g2) = P+(fl ) + P+(f2 ) ' Conversely, if 0 � g � J1 + J2 ' then 0 �J1 n g �J1 and 0 �g - J1 n g � J2 ' so that P+(J1 + J2 ) = lub peg) � lub P(f1 n g) + lub P(g - J1 n g) �F+(Jl ) + P+(f2 ) ' Thus P+ is additive on non-negative functions. But now P+ can be extended to a linear functional on L by the usual definition : P(J1 - J2 ) = P(J1 ) - P(J2 ), where Jl and J2 are non­ negative. And P+ is bounded since I P+(f) I �P+( I f I ) � II P II · ll J II· Now let P-(J) = P+(J) - P(f) · Since P+(f) � P(J) if f � 0, we see that P- is also a bounded non-negative linear functional, and P = P+ - P-, proving the theorem. An integral I is said to be absolutely continuous with re­ spect to an integral I if every I-null set is I-null. ==

15B.

41

INTEGRATION

The Radon-Nikodym Theorem. If the bounded integral J is absolutely continuous with respect to the bounded integral I, then there exists a unique I-summable function fo such that ffo is I-sum­ mabIe and J(f) = I (ffo) for every f � L 1 (J) .

Proof. (From [23].) We consider the bounded integral K = I + J and the real Hilbert space L2 (K) . If f � L2 (K), then f = f · 1 � L 1 (K) and

I J(f) I � J( I f I ) � K ( I f I ) � I I f 1 1 2 · 1 1 1 1 1 2

by the Schwarz (or HOlder) inequality. Thus J is a bounded linear functional over L2 (K), and there exists by a unique g � L2 (K) such that

lOG

J(f) = (f, g) = K(fg) ·

Evidently g is non-negative (except on a K-null set) . The ex­ panSlOn J(f) = K(fg) = I (fg) + J (fg) = I (fg) + I (fg2) + J (fg2) = . . = I (f 2:1 gi) + J (fgn ) .

shows first that the set where g � 1 is I-null (by taking f as its characteristic function) and so J-null. Thus fgn 1 0 almost everywhere if f � 0, and since f � L 1 (J), then J(fgn) 1 0 by Second, the same expansion shows that, if fo = 2:� gi, then ffo � L 1 (I) and J(f) = I (ffo), again by Taking f = 1 , it follows in particular that fo � L 1 (I) . The I-unique­ ness of fo follows from the K-uniqueness of g and the relation g = fo/( 1 + fo) . Since the integrals J(f) and I(f/o) are identical on L2 (K) and, in particular, on L, they are iden tical on L 1 (J) . This proof is also valid for the more general situation in which J is not necessarily absolutely continuous with respect to I. We simply separate off the set N where g � 1 , which is I-null as above but not now necessarily J-null, and restrict / to the com­ plemen tary set S N.

12G.

12G.

-

15C.

Theorem. If I is a bounded integral and F is a bounded linear functional on LP(I), 1 � p < 00, then ihere exists a unique

42

INTEGRATION

function fo €: Lq (where q = p/(p 1 ) if p > 1 and q = 00 if p = 1) such that I I fo I I q = I I F I I and F(g) = (g, fo) = I(gfo) for every g €: LP. That is, (LP) * = Lq . Proof. The variations of F are integrals on L since fn 1 ° implies I I fn l i p 1 ° and since F + and F - are bounded by I I F I I . Since I I f li p = 0, and therefore F+(f) = F- (f) = 0, when f is I-null, the variations of F are absolutely continuous with respect to I. Hence there exists by the Radon-Nikodym theorem a sum­ mabIe function fo such that F(g) = I(g/o) -

for every function g which is summable with respect to both F+ and F-, and in particular for every g €: LP. If g is bounded, ° � g � I fo I and p > 1 , then I(gq) � I(gq - l sgnfo -fo) � I I F 1 1 · 1 1 gq - l li p = I I F I I · [I(gq)] 1 IP

so that I I g I I q � I I F I I . Since we can choose such functions k so that gn i I fo I , it follows from 12G that fo €: L q and I I fo I I q � II F II· The case p = 1 is best treated directly. Suppose that I I fo 1 1 00 � I I F I I + E ( E > 0) and let g be the characteristic function of A = { p : I fo(p) I � I I F I I + E/2 } . Then ( I I F I I + E/2)p,(A) � 1(1 gfo \ ) = F(g sgnfo) � I I F 1 1 · 1 1 g i l l = I I F 1 I p,(A), a contradic­ tion. Therefore l i fo 1 1 00 � I I F I I · Conversely, we have I F(g) I � 1 ( 1 gfo I ) � I I g I Ip ll fo I I q (by the Holder inequality if p > 1 ) , so that I I F I I � I I fo I I q . To­ gether with the above two paragraphs this establishes the equality I I F I I = I I fo I I q . I t can The uniqueness of fo can be made to follow from be as easily checked directly, by observing that, iffo is not (equiv­ alent to) zero, then F is not the zero functional, so that non­ equivalent functions fo define distinct functionals F. The requirement that I be bounded can be dropped from the preceding theorem if p > 1 , but if p = 1 i t must be replaced by some condition such as that of 13E. The discussion follows. n

ISB.

ISD.

INTEGRATION

43

If the domain of F is restricted to the subspace of LP consisting of the functions which vanish off a given summable set S h then the norm of F is decreased if anything. The preceding theorem furnishes a function /1 defined on S 1 . I f S 2 and /2 are a second such pair, then the uniqueness part of 15C shows that /1 and /2 are equivalent on S I n S2 . Suppose now that p > 1 . Let b = lub { I I / I l q : all such/} and let S n be a nested increasing sequence such that I I in I l q i b. Then { in } is a Cauchy sequence in L q and its limit /o is confined to So = U i S n, with I I /0 I I q = b. Be­ cause of this maximal property of /0, there can be no non-null /' whose domain S' is disjoint from So. Now if g E: LP and if the set on which g � ° is broken up into a countable union of disjoint summable sets Am, then the restriction /m of/0 to Am is the function associated by 15C with Am, and F(g) = Li F(gm) = Li I(gm/m ) = I(g/o) . As before, I I F I I � I I /0 I J q , and since I I /0 I I q = b � I I F I I the equality follows. Now if p = 1 this maximizing process cannot be used to fur­ nish a single function /0 E: LOO and there is in general no way of piecing together the functions/a associated by 15C with summable sets Sa to form a function /0 defi n ed on the whole of S. How­ ever, if there is a basic disjoint family { Sa } with the property discussed in 13E, then the function /0 defined to be equal to /a on Sa is measurable and is easily seen (as in the case p > 1 con­ sidered above) to satisfy F(g) = I(gfo) for all g E: L l . § 1 6. INTEGRATION ON LOCALLY COMPACT HAUSDORFF S PACES

16A. We specialize the considerations of § 12 now to the case where S is a locally compact Hausdorff space and L is the algebra of all real-valued continuous functions which vanish off compact sets. We write L + for the set of non-negative functions in L, and LA and LA. + for the sets of functions in L and L + respec­ tively which vanish outside of A. Lemma 1 . 1/ /n E: L and /n 1 0 , then /n 1 ° uniformly.

Proof. Given E, let en

=

{ P : /n (P) � E } . Then en is compact

and n n en = 0, so that en = 0 for some n. That is, I I /n 1 1 00 � E for some, and so for all later, n, q.e.d.

44

INTEGRATION

Lemma 2. A non-negative linear functional is bounded on Lc whenever C is compact. Proof. We can choose g E: L + such that g � 1 on C. Then f E: Lc implies I f I � I I f l loog and I I(f) I � I(g) · 1 1 f 1 100, so that I I I I I � I(g) on Lc. Theorem. Every non-negative linear functional on L is an in­ tegral. Proof. If fn 1 0, then I I fn 1 100 1 ° by Lemma 1 . I ffl E: Lc, then fn E: Lc for all n, and, if B is a bound for I on C (Lemma 2), then I I(fn) I � B I I fn 1 100 and I(fn) 1 0. Therefore, I is an integral. 16B. Let 81 and 82 be locally compact Hausdorif spaces and let I and ] be non-negative linear functionals on L(S 1 ) and L(82) re­ spectively. Then Ix (]yf(x, y)) = ]y (Ixf(x, y)) for every f in L(81 X 82), and this common value is an integral on L(81 X 82) . Proof. Given f(x, y) E: L(81 X 82), let C1 and C2 be compact sets in 81 and 82 respectively such that f vanishes off C1 X C2, and let Bl and B2 be bounds for the integrals I and ] on C1 and C2 respectively. Given E, we can find a function of the special form k(x, y) = Ll gi(x)hi(y), gi E:: Lcl, hi E:: Lc2, such that I I f - k 1 1 00 < E, for by the Stone-Weierstrass theorem the al­ gebra of such functions k is dense in LCl x c2. I t follows, first, that I ]yf(x, y) - L ](hi)gi(X) I = I ]y (f - k) I < EB2, so that ]yf is a uniform limit of continuous functions of x and is itself therefore continuous, and, second, that I Ix]yf - L I(gi)](hi) I < EB1B2• Together with the same inequality for ]yIxf, this yields I Ix]yf - ]ylxf I < 2EBIB2 for every so that Ix]yf = ]yIxf. It is clear that this functional is linear and non-negative, and therefore an in tegral. 16C. (Fubini Theorem.) If K is the functional on L(S 1 X 82) defined above and f(x, y) E:: (B + (8 1 X 82), then f(x, y) E:: (B + (8 1 ) as a function of x for every y, Ixf(x, y) E:: (B + (82) and Kf = ]y (Ixf(x, y)) . Proof. We have seen in 16B that the set if of functions of + B (8 1 X 82) for which the theorem is true includes L + (81 X 82) . E,

if

1.�1N n::r:::;:s� E� 6 '" � , . / '1 .{� ..

' J-

.

45

INTEGRATION

It is also L-monotone. For suppose that { in } is a sequence of L-bounded functions of g: and that in i i or in l i. Then K(f) = lim K(in) = lim Jy(Ixin ) = Jy(lim Ixfn) = Jy(Ix lim in) = Jylxf, by repeated application of 12G or 12K (or their nega­ tives) . Thus g: is L-monotone and includes L +, and is therefore equal to CB + by 121. The Fubini theorem is, of course, valid in the absence of to­ pology, but the proof is more technical, and the above case is all tha t is needed in this book. 16D. The more usual approach to measure and integration in Cartesian product spaces starts off with an elementary measure }.t(A X B) = }.t 1 (A)}.t2 (B) on " rectangles" A X B (A c Sh B c S2) . This is equivalent to starting with an elementary integral

II:, Ci

C(JAi X Bi

=

I:, Ci }.t 1 (Ai)}.t2 (Bi)

over linear combinations of characteristic functions of rectangles. The axioms of 12A for an elementary integral are readily verified. We first remark that for a function f(x, y) of the above sort the

If is actually in terms of the iterated integral : fi f(fi dX) dy f(fi dy) dx. If now i. ex, y) is a se­ definition of =

=

quence of elementary functions and fn l 0, then in 1 ° as a function of x for every y and therefore

Iin dx 1 ° for every y

by the property 12G of the integral in the first measure space S 1 . But then

f(fin dX) dy

1 0 by the same property of the inte-

gral in the second space. Hence

Iin d}.t 1 0, proving (4)

of

12A. The other axioms are more easily checked and we can therefore assume the whole theory of measure and integration as developed in this chapter. In case S 1 and S2 are locally compact spaces and }.t 1 and }.t2 arise from elementary integrals on continuous functions, we now have two methods for in troducing in tegra tion in S 1 X S2 and it must be shown that they give the same result. If CB and CB ' are the two families of Baire functions thereby generated (CB by con-

46

INTEGRATION

tinuous elementary functions and CB ' by the above step functions) , i t must be shown that CB = CB ' and that the two integrals agree on any generating set for this family. Now it is easy to see that ({JA xB � CB if A and B are integrable sets in 8 1 and 82 , for ({JA xB = ((JA (X)({JB (Y) lies in the monotone family generated by functions j(x)g(y), where j and g are elementary continuous functions on 8 1 and 82 . Thus CB ' C CB. Conversely, if j � L(8 1 ) and g � L(82 ) , then j(x)g(y) can be uniformly approximated by a step function of the form (I: ({JAi(X) (I: ((JBj(Y» , and, therefore, by the Stone-Weierstrass theorem any continuous function j(x, y) which vanishes off a compact set can be similarly approximated. Thus CB C CB ' , proving that CB = CB ' . By the same argument the two integrals agree for functionsj(x, y) of the formj(x)g(y) and there­ fore on the whole of CB = CB ' . § 17. THE COMPLEX LP-SPACES

17A. We make the obvious definition that a complex-valued function j = u + iv is measurable or summable if and only if its real and imaginary parts u and v are measurable or summable. Evidently a measurable function j is summable if and only if I j I = (u 2 + v 2) Y2 IS summable, and we can prove the critical inequality : I I (j) I � I ( I j I ) · Proof. We observe that h = u 2 j(U 2 + v 2 ) Y2 � L 1 . The Schwarz inequality can therefore be applied to I u I = h Y2(u 2 + v 2) � giving l I(u) 1 2 � I(h)I( (u 2 + v 2) Y2) . Writing down the corresponding inequality for v and adding, we obtain as was desired. 17B. The definition of LP is extended to include all complex­ valued measurable functions j such that I j I p is summable with I I j l i p = [1( 1 j I p) P l p as before, and we see at once that j � LP if and only if u, v � LP. The Holder and Minkowski inequalities remain unchanged by virtue of the inequality in 17A. The ex­ tended LP is complete, for a sequence jn = U n + i Vn is Cauchy if

INTEGRATION

47

and only if its real and imaginary parts U n and Vn are real Cauchy sequences. 17C. A bounded linear functional F on LP, considered on the subset of real-valued functions, has real and imaginary parts each of which is a real-valued bounded linear functional. Taken together these determine by 15C a complex-valued measurable function /0 E:: L q such that F(g) = l(gJo) for every real function g E:: LP, and hence, because F is complex-homogeneous, for every g E:: LP. The proof that I I F I I = I I /0 I I q holds as well in the complex case as in the real case if we replace /o by]o throughout 15C and remember that sgn/o = ei arg/o. Since this time we already know that /o E:: L q a slight simplifi­ cation can be made, as follows. That I I F I I � I I /0 I I q follows, as before, from the Holder inequality. If p > 1, we can take g = I /0 I q - 1 ei arg/o and get F(g) = 1(1 /0 I q) = I I /0 I I q q, I I g l i p = ( I I /0 I I q ) q /p and I F(g) 1 /1 1 g l i p = I I /o l l q · Thus I I F I I � 1 1 /0 I I q , and so I I F I I = I I /0 I I q '

Chapter IV BANACH ALGEBRAS

This chapter is devoted to an exposition of part of the theory of Banach algebras, with emphasis on the commutative theory stemming from the original work of Gelfand [12]. This theory, together with its offshoots, is having a marked unifying influence on large sections of mathematics, and in particular we shall find that it provides a basis for much of the general theory of harmonic analysis . We begin, in section 1 9, with a motivating discussion of the special Banach algebra e(S) of all continuous complex­ valued functions on a compact Hausdorff space, and then take up the notions of maximal ideal, spectrum and adverse (or quasi­ inverse) thus introduced. The principal elementary theorems of the general theory are gathered together in § 24. § 1 8 . DEFINITION AND EXAMPLES Definition. A Banach algebra is an algebra over the complex numbers, together with a norm under which it is a"Banach space and which is related to multiplication by the inequality : I I xy I I � I I x I I I I y I I · If a Banach algebra A has an identity e, then I I e I I = I I ee I I � I I e 1 1 2, so that I I e I I � 1 . Moreover, it is always possible to re­ norm A with an equivalent smaller norm I I I x I I I so that I I I e I I I = 1 ; we merely give an element y the norm of the bounded linear transformation which left-multiplication by y defines on A ( I I I y I I I = lub I I yx 1 1 I1 1 x I I ) and observe that then I I y I 1 I 1 I e I I � 48

BANACH ALGEBRAS

49

I I I y I I I � I I y I I · We shall always suppose, therefore, that I I e I I = 1. I f A does not have a n identity, i t will b e shown i n 20C that the set of ordered pairs { <x, X) : X is a complex number and x � A} is an extension of A to an algebra with an identity, and a routine calculation will show the reader that the norm I I <x, X) I I = I I x I I + I X I makes the extended algebra into a Banach algebra. We have already pointed out several Banach algebras in our preliminary chapters. These were :

1 a. The bounded continuous complex-valued functions on a to­ pological space S, with the uniform norm I I f 1 100 = lubx�s I f(x) I (see 4B) . 1 b . The essentially-bounded measurable complex-valued func­ tions on a measure space S, with the essential uniform norm (see 14E) . 2a. The bounded linear transformations (operators) on a Banach space (see 7C) . 2b. The bounded operators on a Hilbert space ( § 1 1 ) .

Of these algebras, 1 a and 2 b will be of continued interest to us. Still more important will be the group algebras of locally com­ pact groups, of which the following are examples : 3a. The sequences of complex numbers a = { a n } with I I a I I L :' oo I a n I < 00 and with multiplication a * b defined by (a * b) n = L :: � : a n -mbm .

Here we have the group G of integers as a measure space, the measure of each point being 1 . The norm is simply the L l norm, I I a I I = I a I, and the algebra is L l (G) . I ts multiplication as

fa

defined above is called convolution. We shall omit checking the algebraic properties of convolution, but it should be noticed that the norm inequality I I a * b I I � I I a I 1 I I b I I arises from the re­ versal of a double summation, which is a special case of the Fu­ bini theorem : I I a * b I I = L n I L m a n -mbm I � Lm (L n I a n -mbm I ) = Lm I b m I I I a I I = I I a 1 1 1 1 b I I ·

50

BANACH ALGEBRAS

It is dear that the sequence e defined by eo = 1 and en n � 0 is an iden ti ty for this algebra. 3b. L l ( - 00 , 00 ) with convolution as multiplication : [h g] (x)

=

=

0 if

f00l(x - y)g(y) dy

Here the group is the additive group of the real numbers, with ordi­ nary Lebesgue measure. The inequality 1 1 1 * g 1 1 1 � 1 1 1 1 1 1 1 1 g i l l is again due to the Fubini theorem : 111* g 1\1 =

r I f00l(x - y)g(y) dy I dx � f r00I l(x - y)g(v) I dx dy -ao

-ao

= 1 1 1 1 1 11 1 g i l l' In each of 3a and 3 b the measure used is the so-called Haar measure of the group. The followmg are examples of important types of Banach al­ gebras with which we shall be less concerned. 4. The complex-valued functions continuous on the closed unit circle I I z 1 \ � 1 and analytic on its interior I I z I I < 1, under the uniform norm. 5 . The complex-valued functions on [0, 1] having continuous first derivatives, with I I 1 I I = I I 1 1 100 + I I l' 1 100 ' § 1 9.

F U N CTION A LGE B RA S

The imbedding of a Banach space X in its second conjugate space X* *, that is, the representation of X as a space of linear functionals over the Banach space X*, is an important device in the general theory of Banach spaces. In Gelfand's theory of a commutative Banach algebra A the corresponding representation is all-important. We consider now the space Ll of all continuous homomorphisms of A onto the complex numbers, for every x E: A the function x on Ll defined by x(h) = h ex) for all h E: Ll, and the algebraic homomorphism of A onto the algebra A of all such func­ tions x. It is clear that Ll is a subset of the conjugate space A* of A considered as a Banach space, and the function x is obtained

BANACH ALGEBRAS

51

by taking the functional x**, with domain A*, and restricting its domain to d. This change of domain greatly changes the nature of the theory. For example, d is a weakly closed subset of the closed unit sphere in A* and is therefore weakly compact. More­ over, the linear character of x** becomes obliterated, since d has no algebraic properties, and x is merely a continuous function on a compact space. The more nearly analogous representation theorem for Banach spaces would be the fact that, if X O is the restriction of x** to the strongly closed unit sphere in A*, then the mapping x � XO is a linear isometry of X with a Banach space (under the uniform norm) of continuous complex-valued functions on a compact Hausdorff space. But, whereas this rep­ resentation theorem for a Banach space is devoid of implication and is of the nature of a curio, the corresponding represen ta tion x � x of a commutative Banach algebra by an algebra of func­ tions is of the utmost significance. Thus the representation al­ gebra A is the vehicle for the development of much of the theory of Banach algebras, and the transformation x � x is a generali­ zation of the Fourier transform. If our sole concern were the commutative theory we would take the simple concept of a function algebra as central and let the theory grow around it. We shall not adopt this procedure because we want to include a certain amount of general non­ commutative theory. However, we shall start out in this sec­ tion with some simple observations about function algebras, partly as motivation for later theory and partly as first steps in the theory itself. 19A. A homomorphism h of one complex algebra onto another is, of course, a ring homomorphism which also preserves scalars : h(Ax) = Ah(x) . The statements in the following lemma are obvious. Lemma. Let x � x be a homomorphism of an algebra A onto an algebra A of complex-valuedfunctions on a set S. If P is a point of S, then the mapping x � x(p) is a homomorphism hp of A into the complex numbers whose kernel is either a maximal ideal Mp of deficiency 1 in A or is the whole of A. The mapping p � hp im­ beds S (possibly in a many-to-one way) in the set H of all homo-

52

BANACH ALGEBRAS

morphisms of A into the complex numbers. Conversely, if S is any subset of H and if the function x is defined on S by x(h) = h ex) for every h � S, then the mapping x � x is a homomorphism of A onto the algebra A of all such functions. A pseudo-norm is a non-negative functional which satisfies all the requirements of a norm except that there may exist non-zero x such that I I x II O. A pseudo-normed algebra has the obvious definition. The inequalities I I x + y I I � I I x I I + I I y I I and I I xy I I � I I x l i i i y I I show that the set I of all x such that I I x I I = 0 is an ideal in I. Moreover I I x I I is constant on each coset of the quotient algebra A/I and defines a proper norm there. Any linear mapping with domain A which is bounded with respect to the pseudo-norm is zero on I and therefore by 7D is transferable to the domain A/I without changing its norm. In particular the conjugate space £1* is thus identified with (£1/1) *. =

Lemma. If the algebra A in the above lemma consists of bounded functions, then A defines a pseudo norm in A, I I x I I I I x 1 1 00, and the mapping p � hp identifies S with a subset of the unit sphere in the conjugate space £1* of A. =

Proof.

Obvious.

19B. The following theorem is very important for our later work. It is our replacement for the weak compactness of the unit sphere in £1*.

Theorem. If A is a normed algebra and if A is the set of all continuous homomorphisms of A onto the complex numbers, then A is locally compact in the weak topology defined by the functions of A. If A has an identity, then A is compact. If A is not compact, then the functions of A all vanish at infinity. Proof. If h � A, then its nullspace Mh is a closed ideal and the quotien t space is a normed linear space isomorphic to the complex number field. If E is the identity coset, then h ex) = X if and only if x � }..E. Now E cannot contain elements of norm less than 1 since otherwise, being closed under multiplication, it would contain elements of arbitrarily small norm. In general, if x � XE then x/X � E, II xl}.. II � 1 and I I x I I � I X I = I h ex) I ·

53

BANACH ALGEBRAS

Thus I I h I I � 1 for every h E:: d, and d is a subset of the unit sphere 8 1 of ..1*. We know 8 1 to be weakly compact (9B) and therefore the closure .1 of d in 8 1 is compact. Now the same argument as in 9B shows that, if F E:: .1 , then F is a homomorphism. In fact, if x, y and E are given, there ex­ ists by the hypothesis that F E:: .1 and the definition of the weak topology an element h E:: d such that I F(x) - hex) I < E, Since h(xy ) = I F(y) - h(y) I < E and I F(xy) - h(xy) I < h(x)h(y), it follows that I F(xy) - F(x)F(y) I < 3 E for every E, and therefore that F(xy) = F(x)F(y) . Thus either F is a non­ zero homomorphism and F E:: d, or F = o. We thus have two possibilities : either .1 = d and d is compact, which occurs if the zero homomorphism ho is not in the closure of d, or else .1 = d U { ho } and d is a compact space minus one point, and hence locally compact. In the second case every function x E:: A vanishes at infi n ity, for x is continuous on the one-point compactification .1 = d U r ho } and x(oo) ho(x) = o. If A has an identity e, then e(h) = h (e) = 1 for all h E:: d and it follows that e(F) = F(e) = 1 for any F E:: .1 . This rules out the possibility that ho E:: .1, so that the fi rst case occurs and d is compact. This completes the proof of the theorem. A by-product of the above proof is the fact that I I x 1 1 00 � I I x I I for every x E:: A. We thus have a standard norm-decreasing rep­ resentation of a normed algebra A by an algebra A of continuous complex-valued functions on the locally compact Hausdorff space d of all continuous homomorphisms of A onto the complex num­ bers. We shall call this the Gelfand representation of A. In the simple case of a normed linear space which we sketched earlier the mapping x � xO was an isometry roughly because there exist lots of continuous linear functionals (BC) . Now it is obviously harder for a functional to be a homomorphism than to be merely linear; in fact, there may not exist any non-zero homo­ morphisms at all. In any case the set d is sparse com pared to the unit sphere of ..1* and we must expect that the Gelfand rep­ resentation is in general norm decreasing and that there may be x such that x o. I f the mapping is one-to-one (although prob­ ably norm decreasing), A will be called a function algebra, and will generally be replaced by the isomorphic algebra of functions A. E.

=

==

54

BANACH ALGEBRAS

19C. The correspondence h ----; M between a homomorphism and its kernel allows us to replace Ll by the set mt of all closed maximal ideals of A which have deficiency 1 as subspaces. In general there will exist many other maximal ideals, some not closed and some not of deficiency 1, which therefore do not corre­ spond to points of Ll. It is very important for the successful ap­ plication of the methods of algebra that this cannot happen if A is a commutative Banach algebra with an identity; now every maximal ideal is closed and is the kernel of a homomorphism of A onto the complex numbers. This will be shown later for the general case; we prove it here for the special algebra which in the light of the above lemma must be considered as the simplest kind of function algebra, namely, the algebra e(S) of all continuous complex-valued functions on a compact Hausdorff space S. h

Theorem. If S is a compact Hausdorff space and if I is a proper ideal of e(S) , then there exists a point p E:: S such that I c Mp . If I is maximal, then I = Mp . The correspondence thus established between maximal ideals of e(S) and points of S is ane­ ta-one.

Proof. Let I be a proper ideal of e(S) and suppose that for every p E:: S there exists fp E:: I such that fp (P) � O. Then \ fp \ 2 = fpJp E:: I, \ fp \ 2 � 0 and \ fp \ 2 > 0 on an open set con­ taining p. By the Heine-Borel theorem there exists a function f E:: I, a finite sum of such functions \ fp \ 2, which is positive on S. Then f- 1 E:: e(S), 1 = f}- 1 E:: I and e(S) = I, a contradic­ tion. Therefore, I c Mp for some p, and, if I is maximal, then I = Mp . Finally, if p � q, then there exists f E:: e(S) such that f(P) = 0 and f(q) � 0 (see 3C) so that Mp � Mq and the map­ ping p Mp is one-to-one. 19D. We proved above not only that all the maximal ideals of e(S) are closed and of deficiency 1 but also that they are all given by points of S. In the general theory of function algebras these two properties can be separated. Thus it follows from simple lemmas proved later (2 1D-F) that, if A is an algebra of bounded functions, then every maximal ideal of A is closed and of deficiency 1 provided A is inverse-closed, that is, provided A has the property that whenever f E:: A and glb I f I > 0, then ----;

BANACH ALGEBRAS

55

l /f � A. We can then separately raise the question as to when an inverse-closed algebra A of bounded functions on a set S is such that S = � . A necessary condition (supposing that A has an identity) is clearly that S be compact under the weak topology defined by A, so that in view of 5G we may as well suppose that we are given a compact Hausdorff space S and a separating, in­ verse-closed algebra of continuous functions on S. This is not sufficient, however; we still may have S � � . If we add one further property to A, the situation suddenly becomes again very sim pIe. We define a function algebra A to be self-adjoint if f � A =} ] � A.

Lemma. If A is a separating, self-adjoint, inverse-closed algebra of continuous complex-valued functions on a compact space S, then every maximal ideal of A is of the form M = Mp for some p � S. In particular � = S. Proof.

Identical with that of the above theorem in 19C.

Corollary. If A is a separating, self-adjoint, inverse-closed al­ gebra of bounded complex-valued functions on a set S, then S is dense in � . Proof. If S is the closure of S in � , then S is compact and the above lemma applies to the extensions to S of the functions of A, proving that S = � as required. A topology ::i on a space S which is completely determined by its continuous complex-valued functions is said to be completely regular. By this condition we mean that ::i is identical with the weak topology ::i w defined on S by the algebra e (S) of all bounded complex-valued ::i-continuous functions. Since the inclusion ::i w C ::i always holds, complete regularity is equivalent to the in­ clusion ::i C ::i w, and i t is easy to see that a necessary and sufficient condition for this to hold is that, given any closed set C and any point p ([.. C, there exists a continuous real-valued function f such that f 0 on C and f(p) = 1 . I f S is completely regular, then the above corollary applied to e (S) gives an imbedding of S as a dense set of the compact Haus­ dorff space � such that the given topology of S is i ts relative to==

56

BANACH ALGEBRAS

pology as a subset of .1. This is a standard compactification of a completely regular space. 19E. The fact that the property proved in 19C for the alge­ bra e(S) of all continuous complex-valued functions on a com­ pact Hausdorff space S is now seen (in 19D) to hold for a much more general class of function algebras suggests the presence of further undisclosed properties of e(S) . We consider two here, starting with a relatively weak property which e(S) also shares with many other function algebras. In order to present these properties in a form most suitable for later comparison with other algebras we make the following general definitions.

In any ring with an identity the kernel of a set of maximal ideals is the ideal which is their in tersection. The hull of an ideal I is the set of all maximal ideals which include I. D E FINITION.

The reader is reminded of the other conventional use of the word "kernel," as in 19A. Theorem. Proof.

If B

c

S, then B = hull (kernel (B) ) .

Let IB be the kernel of B. IB = { f: f � e(S)

and f = O on B } .

If C = B, then IB = Ie, for a continuous function vanishes on B if and only if it vanishes on B = C. If P is not in C, there exists by 3C a continuous functionf which vanishes on C (1 � Ie) but not at p (f tl. Ip) . Thus Ie c Ip if and only if p � C, and hull (Ie) = C. Altogether we have B = C = hull (Ie) = hull (IB), q.e.d. 19F. We saw above that a subset of a compact Hausdorff space S is closed if and only if it is equal to the hull of its kernel, when it is considered as a set of maximal ideals in the algebra e(S) . Now this hull-kernel definition of closure can b e used to introduce a topology in the space of maximal ideals of any alge­ bra with an identity. This will be discussed further in 20E ; our present concern is the comparison of this hull-kernel topology 3hk, defined on a set S by an algebra A of complex-valued func­ tions on S, with the weak topology 3 w defined by A, or with any

BANACH ALGEBRAS

57

other (stronger) topology 3 in which the functions of A are all continuous. We prove the following theorem.

If A is an algebra of continuous complex-valued functions on a space S with topology 3, then 3 = 3hk if and only if for every closed set C c S and every point p not in C there exists f E: A such thatf 0 on C andf(p) � O. Theorem.

==

Proof. We first remark that since there may be maximal ideals of A other than those given by points of S it is the inter­ section with S of the actual hull that is referred to here. Let C be 3M-closed; that is, C = hull (I) where I = kernel (C) . Now the set of maximal ideals containing a given element f E: A is simply the nullspace of the function f and is therefore closed. The hull of I, being the intersection of these nullspaces for all f E: I, is therefore closed. Thus every hull-kernel closed set C is also closed and we have the general inclusion 3hk C 3. The theorem therefore reduces to finding the condition that a closed set C be the hull of its kernel. But the kernel of C is the ideal of elements f E: A such that f = 0 on C, and the hull of this ideal will be exactly C if and only if for every p t[ C there exists f in the ideal such that f(P) � O. This is the condition of the theorem. If A = e(S) in the above theorem, then the condition of the theorem is exactly the one characterizing completely regular spaces. Thus for a completely regular space 3 = 3w = 3hk. A function algebra A which satisfi e s the above condition is called a regular function algebra. 19G. The following result is much less susceptible of generali­ zation than that in 19E. The extent to which it holds or fails to hold in such regular algebras as the algebra of L I-Fourier transforms over ( - 00, (0 ) is related to such theorems as the Wiener Tauberian theorem.

Theorem. If S is a compact Hausdorif space and I is a uni­ formly closed ideal in e(S), then I kernel (hull (/) ) . =

Let C = hull (I), C = { p : f(p) = 0 for all f E: I} . Since each f E: I is continuous the null set (hull) of f is closed, and C, the intersection of these null sets, is therefore closed. Let Proof.

58

BANACH ALGEBRAS

S be the space S - c. Then S is locally compact and the func­ tions of Ie, confined to S I > form the algebra of all continuous functions which vanish at infinity (see 3D) . The functions of I form a uniformly closed subalgebra of Ie. Let P I and P 2 be dis­ tinct points of S I and let ] be a function of e(S) such that ] = 0 on C, ](P 2 ) = 0 and ](Pt) = 1 . Such a function exists by 3C. Let g be a function of I such that g(Pl) � o. Then gf E: I, gf(P t ) � 0 and gf(P 2) = o. Therefore by the Stone-Weierstrass theorem, 4E, we can conclude that I = Ie, q.e.d. 1

1

§ 20.

MAXIMAL I D E A L S

We begin the general theory with some results on maximal ideals in arbitrary rings and algebras. The following simple theorem is perhaps the basic device in the abstract development of harmonic analysis which we are pursuing. I ts proof depends explicitly on Zorn's lemma. 20A. Theorem. In a ring with an identity every proper (right) ideal can be extended to a maximal proper (right) ideal. Proof. We consider the family g: of all proper (right) ideals in the ring R which includes the given ideal I. This family is par­ tially ordered by inclusion. The union of the ideals in any lin­ early ordered subfamily is an ideal, and is proper since it excludes the identity. Therefore, every linearly ordered subfamily has an upper bound in g:, and g: contains a maximal element by Zorn's lemma, q .e.d. 20B. If R does not have an identity, the above proof fails be­ cause the union of a linearly ordered subfamily of g: cannot be shown to be a proper ideal. However, the proof can be rescued if R has a (left) identity modulo I, that is, an element u such that ux - x E: I for every x E: R, for then u has the necessary prop­ erty of being excluded from every proper (right) ideal ] including I (u E: ] and ux - x E: I :::::} x E: ] for every x E: R, contra­ dicting the assumption that ] is proper) , and the same proof goes through using u instead of e. A (right) ideal I modulo which R has a (left) identity is said to be regular. We have proved : Theorem. Every proper regular (right) ideal can be extended to a regular maximal (right) ideal.

59

BANACH ALGEBRAS

20C. We shall find throughout that the presence of an iden­ tity in an algebra makes the theory simpler and more intuitive than is possible in i ts absence. I t is therefore important to ob­ serve, as we do in the theorem below, that we can always enlarge an algebra deficient in this respect to one having an identity, and we shall use this device wherever it seems best to do so. However, in many important contexts this extension seems un­ natural and undesirable, and we shall therefore carry along a dual development of the theory so as to avoid i t wherever feasible.

If d is an algebra without an identity, then d can be imbedded as a maximal ideal of deficiency one in an algebra Ae having an identity in such a way that the mapping Ie I= d n Ie is a one-to-one correspondence between the family of all (right) ideals Ie in Ae which are not included in A and the family of all regular (right) ideals I of A. Theorem.

--7

Proof. The elements of de are the ordered pairs (x, X), where x E: d and X is a complex number. Considering (x, X) as x + Xe, the definition of multiplication is obviously (x, X)(y, j.l) = (xy + Xy + j.lx, Xj.l) ; we omit the routine check that the enlarged system Ae is an algebra. It is clear that (0, 1 ) is an identity for Ae and that the correspondence x (x, 0) imbeds A in Ae as a maximal ideal with deficiency 1 . Now let Ie be any (right) ideal of de not included in d and let I = Ie n A. Ie must contain an element v of the form (x, - 1 ). Then the element u = v + e = (x, 0) E: A is a left identity for Ie in Ae (uy - y = (u - e)y = vy E::: Ie for all y E: de) and hence automatically for I in A. Thus I is regular in d. Moreover, since uy - y E: Ie and uy E::: A for all y, we see that y E::: Ie if and only if uy E: I. Conversely, if I is a regular (right) ideal in A and u = (x, 0) is a left identity for I in d, we define Ie as {y : uy E: I} . Direct consideration of the definition of multiplication in de shows that I is a (right) ideal in de ; hence Ie is a (right) ideal in de . It is, not included in A since u (x, - 1 ) = u (u - e) = u 2 - u E: I and therefore u - e = (x, - 1 ) E: Ie. Moreover, the fact that uy - y E: I for every y E: A shows that y E::: I if and only if uy E: I and y E: d, i.e., I = Ie n d. --7

.

60

BANACH ALGEBRAS

We have thus established a one-to-one inclusion preserving correspondence between the family of all regular (right) ideals in A and the family of all (right) ideals of Ae not included in A. In particular the regular maximal (right) ideals of A are the in­ tersections with A of the maximal (right) ideals of Ae different from A.

Corollary. If S is a locally compact but not compact Hausdorff space and e( S) is the algebra of continuous complex-valued func­ tions vanishing at infinity, then the regular maximal ideals of e(S) are given by the points of S in the manner of 19C. Proof.

The extended algebra is isomorphic to the algebra

C(Soo) of all continuous complex-valued functions on the one point compactification of S, under the correspondence (f, A) �

f + A. In the latter algebra e(S) is the maximal ideal corre­ sponding to the point at 00, and its other maximal ideals, corre­ sponding to the points of S, give the regular maximal ideals of e (S) by the above theorem. 20D. If M is a regular ideal in a commutative ring R, then M is maximal if and only if RjM is a field.

Proof. If Rj M has a proper ideal j, then the union of the cosets in j is a proper ideal of R properly including M. Thus M is maximal in R if and only if RjM has no proper ideals. There­ fore, given X E::: RjM and not zero, the ideal { XY: Y E::: RjM} is the whole of RjM. In particular, XY = E for some Y, where E is the identity of RjM. Thus every element X has an inverse and RjM is a field. The con verse follows from the fact that a field has no non-trivial ideals. If R is an algebra over the complex numbers, then the above field is a field over the complex numbers. If R is a Banach al­ gebra, we shall see in § 22 that this field is the complex number field itself, and we shall therefore be able to proceed with our represen ta tion program. 20E. We conclude this section with some simple but important properties of the hull-kernel topology, taken largely from Segal [44] . If grr is the set of regular maximal (two-sided) ideals of a ring R and B c grr, we have defined B as h ( k ( B) ) = hull (kernel

61

BANACH ALGEBRAS

(B)) . In our earlier discussion i t was unnecessary to know that the operation B � B was a proper closure operation. Actually, the properties A c B =} A c B and B c B = E follow at once from the obvious monotone properties A c B =} k(B) c k(A), and I e ] =} h e]) c h (I) . However, the law A U B = A U B is more sophisticated, and we present a formal proof. Lemma.

closed.

If A and B are closed subsets of

ffil,

then A

U

B is

Suppose that A U B is not closed. Then there exists such that k (A U B) c M and M fl A U B. Since A is closed, k (A) ¢ M and so k (A) + M = R. If e is an identity modulo M, it follows that there exist a E::: k (A) and mi E::: M such that e = a + m I ' Similarly, there exist b E::: k (B) and m 2 E::: M such that e = b + m 2 ' Multiplying we get e 2 - ab E::: M, and since e2 - e E::: M we see that ab is an identity modulo M. But ab E::: M (since ab E::: k(A) n k(B) = k (A U B) e M) , and this is a contradiction. Remarks: The reader should notice that this proof is not valid in the space of regular maximal right ideals, since we then can conclude neither that e2 - ab E::: M nor that ab E::: M. The no­ tion of hull-kernel closure is still available, but we cannot con­ clude that it defines a topology. We also call explicitly to the reader's attention the fact that every hull is closed. This, like the elementary topological prop­ erties listed before the above lemma, is due to the inclusion re­ versing nature of the hull and kernel mappings. Thus if C = hull I, then I c k (C) and h (k(C)) c h (I) = C, or E c C, as re­ quired. Similarly any kernel is the kernel of its hull. 20F. We saw above in 19G that I = k(h (I)) if I is a closed ideal in the algebra e(S) of all continuous complex-valued func­ tions o.n a compact Hausdorff space S. This result does not hold for ideals in general rings, even when they are closed under sui ta­ ble topologies. The best general result of this nature is the fol­ lowing theorem, various versions of which were discovered inde­ pendently by Godement, Segal, and Silov. I t is the algebraic basis of that part of harmonic analysis exemplified by the Wiener Proof.

M E:::

ffil

62

BANACH ALGEBRAS

Tauberian theorem. We define a ring R to be semi-simple if the intersection of its regular maximal ideals is zero. Lemma. If I and j are ideals with disjoint hulls and j is regu­

lar, then I contains an identity modulo j.

Proof. An identity e modulo j is also an identity modulo I + j, and I + j is therefore regular. But by hypothesis I + j is included in no regular maximal ideal; therefore I + j = R, and in particular there exist i E::: I and j E::: j such that i + j = e. The element i = e - j is clearly an identity modulo j, q.e.d.

Theorem. Let R be a semi-simple ring, I an ideal in R and U

an open set of regular maximal ideals such that h(I) k( U') is regular. Then k( U) c 1.

c

U and

Proof. The hypothesis that U' is closed is equivalent to U' =

h(j), where j = k( U') . Thus I and j have disjoint hulls and by the lemma I contains an identity modulo j, say i. If x E::: k( U) i t follows that ix - x belongs to every regular maximal ideal, and since R is semi-simple this means that ix - x = o. Thus x = ix E::: I, q .e.d. We include the following variant largely for comparison with the later theory of regular Banach algebras.

Let R be a commutative semi-simple ring and let I be an ideal in R. Then I contains every element x such that h (I) c int h (x) and such that x = ex for some e E::: R. Theorem.

Proof. If C is the complement of h (x) and j = k (C), then the

first hypothesis says that I and j have disjoint hulls. The sec­ ond hypothesis implies that e is an identity modulo j, though this is by no means immediately obvious. I t depends on the following lemma. Lemma. Two subsets of a semi-simple commutative ring R an­ nihilate each other if and only if the union of their hulls is R. Proof. If x and y are elements such that h (x) U h(y) = R, then xy belongs to every regular maximal ideal and so xy = 0, by the semi-simplicity of R. Conversely let /l and B be subsets

BANACH ALGEBRAS

63

of R such that AB = 0. If h (A) U h (B) =;t. R, then there exist a regular maximal ideal M and elemen ts a €:: A and b €:: B such that neither a nor b belongs to M. Thus neither a nor b is zero modulo M whereas ab = 0, contradicting the fact that R/M is a field. Returning now to the theorem, the fact that x(ey - y) = (xe - x)y = ° implies by the lemma that ey - y belongs to every regular maximal ideal which does not con tain x and hence be­ longs to J. Thus e is an identity modulo J, and x €:: I exactly as before. We conclude this section with the theorem on the per­ sistence of the hull-kernel topology under homomorphisms.

ZOG.

If I is a proper ideal in a ring R, then a subset of R/I is an ideal in R/I if and only if it is of the form J/I where J is an ideal of R including I. J/l is regular and/or maximal in R/I if and only J is regular and/or maximal in R. The space of regular maximal ideals in R/I thus corresponds to the hull of I in the space of regular maximal ideals of R, and the correspondence is a homeomorphism with respect to the hull-kernel topologies. Theorem.

if

Proof.

Direct verification. § 21.

S P E CTRUM ; ADVE R S E

In this section we shall add to our repertoire the two very im­ portant notions of spectrum and adverse. These concepts arise naturally out of the question as to what can be meant in general by the statement : the element x assumes the value A. This should have the same meaning as the statement that x - Ae as­ sumes the value 0, and should reduce to the ordinary meaning in the case of our model algebra e(S) of all continuous functions on a compact Hausdorff space. In this case there are two ob­ vious algebraic formulations of the statement : first, that x - A belongs to some maximal ideal, and second, that ex - A) - 1 fails to exist. We show below that these two properties are equiva­ lent in any algebra, so that either can be taken as the desired generalization. \Ve set y = x - Ae, and prove the equivalence for y in any ring with an identity.

64

BANACH ALGEBRAS

2 1A. Theorem. If R is it ring with an identity, then an element y has a right inverse if and only if y lies in no maximal right ideal. If y is in the center of R, then y - 1 exists if and only if y lies in no maximal (two-sided) ideal.

If y has a right inverse z and lies in a right ideal I, then yz = e L I and I = R. Thus y can lie in no proper right ideal. Conversely, if y does not have a right inverse, then the set {yz : z L R} is a proper right ideal containing y, and can be extended to a maximal right ideal containing y by 20A. Thus the first statement of the theorem is valid. The same proof holds for the second statement, needing only the additional remark that since y now is assumed to commute with every element of R the set {yz : z L R} is now a two-sided ideal. Proof.

2 1B. If x is an element of an algebra A with an identity, then

the set of all X such that (x - Xe) - 1 does not exist, corresponding to the range off in the case of the algebra e (S), is called the spec­ trum of x. If A does not have an identity, we define the spectrum of an element x to be that set of complex numbers which becomes the spectrum of x in the above sense when A is enlarged by adding an iden ti ty, as in 20C. In this case X = ° must always be in the spectrum of an ele­ ment x, for an element x L A cannot have an inverse in the ex­ tended algebra Ae (x(y + Xe) = e => e = xy + Xx L A) . More­ over the non-zero spectrum of x can still be determined within A by rephrasing the discussion of (x - Xe) - 1 so that e does not occur. In fact, after setting x = Xy and factoring out X, we can write (y - e) - 1 , if it exists, in the form u - e, and the equation (y - e) (u - e) = e becomes y + u - yu = 0, the desired condi­ tion. Such an element u must belong to A, for u = yu - y L A. The following definition is clearly indicated. In any ring R, if x + y - xy = 0, then y is said to be a right adverse of x, and x is a left adverse of y. We shall see below that, if x has both right and left adverses, then they are equal and unique, and this uniquely determined element is called the adverse of x. The conclusions we reached above now take the following form.

65

BANACH ALGEBRAS

Theorem. If A is an algebra without an identity, then 0 is in the spectrum of every element, and a non-zero X is in the spectrum of x if and only if xIX does not have an adverse in A.

2 IC. We return now to the proof that right and left adverses are equal.

Theorem. If an element x has both right and left adverses, then the two are equal and unique.

Proof. If u and v are left and right adverses of x, then the proof of their equality and uniqueness is equivalent to the usual proof that e - u and e - v, the left and right inverses of e - x, are equal and unique, and can be derived from that proof by cancelling out e. A better procedure, which can be systematically exploited, is to notice that the mapping e - x � x takes multiplication into a new operation x 0 y = x + y - xy and takes e into O. I t fol­ lows at once that x 0 y is associative and that 0 0 x = x 0 0 = x. The desired proof now takes the following form :

u = u

0

0= u

0

(x 0 v) = (u

0

x)

0

v = 0 0 v = v.

2 ID. If we carry out the proof of 2 IA for an element x - e and then cancel out e, we get the following replacement theorem.

In any ring R an element x has a right adverse if if there exists no regular maximal right ideal modulo which and only x is a left identity. Theorem.

Proof. If x has no right adverse, then the set { xy - y : y € A} is a right ideal not containing x modulo which x is clearly a left identity (xy = y mod I since xy - y € I) , and this ideal can be extended to a regular maximal ideal with the same property by 20B. Conversely, if x has a right adverse x' and if x is a left iden­ tity for a right ideal I, then x xx' - x' € I. Then y = xy (xy - y) € I for every y € R and I = R. Thus x cannot be a left identity modulo any proper right ideal. If x € center (R), then the above theorem has the expected counterpart involving adverse and regular two-sided ideals. I n particular w e have the corollary : =

66

BANACH ALGEBRAS

Corollary. If x is an element of the center of an algebra A and

if A � 0, then A is in the spectrum of x if and only if xIA is an iden­ tity modulo some regular maximal ideal. 2 1E. Theorem. If P is a polynomial without constant term, then the spectrum of P(x) is exactly P(spectrum (x) ) .

Proof. We assume a n identity, enlarging the algebra i f neces­ sary. Given Ao, let JLIT (x - JL n e) be the factorization of P(x) Aoe into linear factors. Then (P(x) - Aoe) - 1 fails to exist if and only if (x - JL ne) - 1 fails to exist for at least one value of n. Since P(JLn) - Ao ° by the definition of the JL n , we have shown that Ao E: spectrum P(x) if and only if there exists JLo E: spec­ trum (x) such that Ao = P(JLo), q.e.d. =

§ 22.

B A NACH A LG E B RAS ;

E L EM ENTARY TH EORY

Except for minor modifications the results of § 22-25 are due to Gelfand [12]. The principal deviations are the provisions, such as the use of the adverse, regular ideals, etc., made to take care of the lack of an identity element. The existence of the identity will be assumed only in those contexts in which it, or an inverse, is explicitly mentioned. In this section it is shown that the set of elements which have inverses (adverses) is open, from which it follows that a maximal ideal M is closed, and hence that the quotient AIM is a normed field. An elementary application of analytic function theory leads to the conclusion that there exists only one normed field, the complex number field, and this completes the first step in our representation program. 22A. The ordinary geometric series can be used in the ordi­ nary way to prove the existence of (e - x) - 1 in a Banach algebra. Theorem. If I I x I I < 1 , then x has an adverse and e - x has an inverse, given respectively by x' = - Li x n and (e - x) - 1 = e - x' = e + Li x n , and both are continuous functions of x.

If Yn - L1 Xi, then I I Y m - Yn I I I I L� + 1 Xi I I � L�+ 1 1 I x W < I l x l l m + 1 /(l - l I x l l ) � O as n � 00. Thus the sequence {Yn} is Cauchy, and its limit y is given by the infinite Proof.

=

=

67

BANACH ALGEBRAS

series - Li Xi in the usual sense. Then x + y - xy lim (x + Yn - XYn) = lim xn = 0, and y is a right adverse of x. Similarly, or because y commutes with x, y is a left adverse and therefore the unique adverse of x. If e exists, then, of course, e - y = (e - x) - 1 . Since the series is clearly uniformly con­ vergen t in the closed sphere I I x I I � r < 1 , it follows that the adverse (and inverse) is a continuous function of x in the open sphere I I x I I < 1 . The above geometric series for the inverse (e - x) -1 is the classical Neuman series in case x is an integral operator. Remark: I I x' I I � I I x 1 1 / ( 1 - I I x i I ) · 22B. Theorem. If y has an inverse, then so does y + x when­ ever I I x I I < a = 1 /1 1 y - l I I and I I (x + y) -1 - y - l I I � I I x I I I (a - I I x I I )a. Thus the set of elements having inverses is open and the inverse function is continuous on this set. Proof. If l l x / I < I l y - 1 1 1 - 1 , then I l y - i x / I � I l y - l / I / I x I I < 1 andy + x = y (e + (y- l X» has an inverse by 22A. Also (y + X) -1 - y - l = ( (e + y - lX) -1 - e)y - l = - ( _y - lX) ,y -1, so that by the remark at the end of 22A / I (y + x) - 1 - Y - 1 / I � /I x I I I I y - l 1 1 2 / ( 1 - / I x / 1 / 1 y -l I I ) = I I x 1 1 / (a - / I x I l )a, where a l il l y -I I I · 22C. Theorem. Ify has an adverse, then so does y + x whenever I I x I I < a = ( 1 + I l y' 1 1 ) - 1 , and / l (y + x) ' - y' / I � / l x l l l (a - I I x I l )a. Thus ihe set of elements having adverses is open and the mapping y � y' is continuous on this set. Proof. I f / I x / I < a = ( 1 + / I y' I I ) - 1 , then / I x - xy' / I � x I I 1 1 (1 + I I y' I I ) < 1 and u = x - xy' has an adverse by 22A. But (y + x) 0 y' = x - xy' = u, so that y + x has y' 0 u' as a right adverse. Similarly y + x has a left adverse, and so a unique adverse equal to y' 0 u'. Finally, (y + x) ' - y' = (y' 0 u') - y ' = u' - y'u', so that / I (y + x) ' - y' / I � ( 1 + / I y ' 1 1 ) / 1 u' I I � ( 1 + / I y ' 1 1 ) / 1 u 1 1 / ( 1 - / I u / I ) � / I x 1 1 ( 1 + I I y' 1 1 ) 2 / ( 1 I I x 1 1 ( 1 + I I y' I I ) = / I x 1 1 / (a - I I x I l )a. - Ay'. Remark: I f we take x = AY, then u A(y - yy') Therefore [(y + AY) ' - y '] /A [( - AY')' - y' ( -Ay')'] /A = Li ( _ l ) nA n -l[(y,) n - (y,) n + l] � (y') 2 - y' as A � 0. This =

=

=

=

=

68

BANACH ALGEBRAS

proves the analyticity of (AY) ' as a function of A, and will be of use later. 22D. If u is a relative identity for a proper regular ideal I, then p(I, u) � 1 . For if there exists an element x E: I such that I I u - x I I < 1, then u - x has an adverse a : (u - x)a - a (u - x) = o. Since x, xa and ua - a are all in I, it follows that u E: I, a contradiction (see 20B) . It follows that :

Lemma. If I is a proper regular ideal, then so is its closure 1. In particular, a regular maximal ideal is closed.

Proof. 1 is clearly an ideal, and p(I, u) � 1 implies that p(I, u) � 1 so that 1 is proper.

22E. Theorem. If I is a closed ideal in a Banach algebra A, then A/I is a Banach algebra.

6B

Proof. We know already from that A/l is a Banach space. If X and Y are two of its cosets, then I I XY I I = glb { I I xy I I : x E: X and y E: Y} � glb { I I x 1 1 · 1 1 y I I } = glb { I I x I I : x E: X} . glb { I I y I I : y E: Y} = I I X 1 1 · 1 1 YI I . If I is regular and u is a rela­ ti ve iden ti ty, then the coset E containing u is the iden ti ty of A/I and I I E I I = glb { I I x I I : x E: E} = gl b { I I u - y I I : y E: I} = p(I, u) � 1 by 22D. If A has an identity e, then e E: E and 1 and we are I I E I I � I I e I I = 1 so that in this case I I E I I done. If A does not have an identity, then it may happen that I I E l l > 1 . In this case, as we saw in § 1 8, we can renorm A/I with a smaller equivalent norm so that I I E I I = 1 . =

Corollary. If I is a regular maximal ideal and if A is commu­ tative, then it follows from 20D and the above theorem that A/l is a normed field.

22F. Theorem. Every normed field is (isometrically isomorphic to) the field of complex numbers. Proof. We have to show that for any element x of the field there is a complex number X such that x = Ae. We proceed by contradiction, supposing that x - Ae is never zero and there­ fore that (x - Ae) - 1 exists for every X. But if F is any linear

69

BANACH ALGEBRAS

functional over the field considered as a Banach space, then F((x - Ae) - 1 ) , as a function of A, is seen by direct calculation as in 22B to have the derivative F((x - Ae) -2 ) , and is consequently analytic over the whole plane. Also (x - Ae) - 1 � 0 as A � 00, for (x - Ae) - 1 = A - 1 (X/A - e) - I , and (x/A - e) - 1 � - e as A � 00 by 22A. Thus F( (x - Ae) - 1 ) � 0 as A � 00 and F( (x - Ae) - 1 ) 0 by Liouville's theorem. It follows from Be that (x - Ae) - 1 = 0, a contradiction. Remark: The above proof has not made use of the fact that multiplication is commutative, except, of course, for polynomials in a single element x and i ts inverse. Thus it actually has been shown that the complex number field is the only normed division algebra. =

§ 23.

TH E

MAXIMA L

IDEAL

S PA C E

OF

A

COMMUTATIVE

B ANACH

A LG E B RA

23A. We have proved in the preceding section all the ingredi­ ents of the following theorem, which is the basic theorem in Gel­ fand's theory.

Theorem. If A is a commutative Banach algebra, then every regular maximal ideal of A is closed and of deficiency 1 , every homo­ morphism of A onto the complex numbers is continuous with norm � 1 , and the correspondence h � Mh between such a (continuous) homomorphism and its kernel thus identifies the space .1 with the set mr of all regular maximal ideals. Proof. 22D-F say explicitly that every regular maximal ideal

is the kernel of a homomorphism of A onto the complex numbers. Conversely, the kernel of any such homomorphism is clearly a regular maximal ideal. The theorem then follows if we know that every such homomorphism is bounded by 1 (which is equiv­ alent, in fact, to i ts kernel being closed) . The most direct proof is to suppose that I hex) I > I I x I I for some x and notice that, if A = hex), then I I x/A II < 1, (x/A) ' exists and so hex/A) � 1, con­ tradicting the definition of A. In clarification of the roles played by the various properties asserted in the above theorem, we prove the following lemma.

70

BANACH ALGEBRAS

Lemma. If A is a commutative normed algebra, then every regu­ lar maximal ideal of A is closed if and only if A has the property that the adverse of x exists whenever I I x I I < 1 .

Proof. We remark first that, i f M is a closed regular maximal ideal with relative identity u, then p(M, u) � 1 , for if there ex­

ists x E: M such that I I x - u I I = 0 < 1 , then y = u - (u - x) n is a sum of positive powers of x and so belongs to M, and I I u - y I I = I I (u - x) n I I � on, proving that u E: M = M, a contradiction. Suppose then that every regular maximal ideal is closed and that I I x I I < 1 . Then x' must exist, for otherwise x is an identity relative to some regular maximal ideal, by 2 ID, and 1 > I I x I I = I I x - 0 I I � p (x, M), contradicting the above remark. Conversely, the existence of x' whenever I I x I I < 1 implies that every regular maximal ideal is closed, exactly as in 22D. If a commutative normed algebra has either, and hence both, of these equivalent properties then 22F shows that every regular maximal ideal is of deficiency 1 and the rest of the above theo­ rem then follows. In particular, the theorem holds for an al­ gebra A of bounded functions which is inverse-closed (see I9D) . 23B. Let us review the facts of the Gelfand representation, re­ placing the underlying space A by the space mr of all regular maximal ideals. Let FM be the homomorphism of A whose ker­ nel is the regular maxim'al ideal M. The number FM (x) is ex­ plicitly determined as follows : if eM is the identity of the field AIM and if x is the coset of AIM which contains x, then FM (X) is that complex number A such that x = AeM. I t also follows from 22E that I FM (X) I � I I x I I · I f x is held fixed and M is varied, then FM (x) defines a complex­ valued function x (x(M) = FM (x)) on the set mr of all regular maximal ideals of A. The mapping x � x is then a norm-de­ creasing homomorphism of A onto an algebra Ji of complex­ valued functions on mr, the uniform norm I I x I I � being used in Ji. mr is given the weak topology which makes the functions x E: Ji all continuous, and it was shown in I9B that mr is then either compact or locally compact.

71

BANACH ALGEBRAS

The function algebra d is not generally the whole of e(�) , nor even a uniformly closed subalgebra ; it may or may not be dense in e(�) . I t always shares with e(�) the property that its regular maximal ideals correspond exactly to the points of �. And d is always closed under the application of analytic func­ tions : that is, if x E: A and f is analytic on the closure of the range of X, then there exists y €: A such that y (M) f(x(M» . These properties are all very important, and will be discussed in some detail later on, mostly in § 24. The function x will be called the Fourier transform of the ele­ ment x, and the homomorphism x � x(M) = FM (X) associated with the regular maximal ideal M a character of the algebra A. Actually the situation should be somewhat further restricted be­ fore these terms are used ; for instance, in some contexts it is more proper to call x the Laplace transform of x. This section will be largely devoted to well-known examples illuminating this terminology. We start, however, with a simple preliminary theorem relating the spectrum of x to the values assumed by x. ==

Theorem. The range of x is either identical to the spectrum of x or to the spectrum of x with the value 0 omitted. If x never assumes the value 1 , then x/ (1 - x) €: d. If A has an identity and x is never 0, then 1 /x €: d.

Proof. If A � 0, then A is in the spectrum of x if and only if X/A is a relative identity for some maximal ideal M, i.e., if and only if FM (x/A) = 1, or x(M) = FM (x) = A, proving the first statement. (If FM (X) = 0, then x €: M and it follows that 0 €: spectrum (x) .) If the spectrum of x does not contain 1 , then x has an adverse y by 2 ID, and from x + y - xy = 0 it follows that y = x/(x - 1) €: do The last statement is similarly a trans­ lation of 2 1A. Both of these facts will also follow from the gen­ eral theorem on the application of analytic functions, proved in 24D. 23C. As a first example let A be the algebra of sequences a = { a n } of complex numbers such that L�: I a n I < 00, with I I a I I defined as this sum and multiplication defined as convolu­ tion, (a * b) n = L; = - oo an -mbm. This was example 3a of § 1 8 .

72

BANACH ALGEBRAS

The fact that A is a Banach algebra under these definitions will follow from later general theory (of group algebras) and can also be easily verified directly. A has an identity e where eo = 1 and e n = 0 if n � O. Let g be the element of A such that gl = 1 and gn = 0 otherwise. Then g has an inverse (gn - 1 0 unless n = - 1 and = 1 if n = - 1 ) and A is simply an algebra of infinite series a = L : a ngn in powers of g, under the ordinary, formal multiplication of series. Let M be any maximal ideal of A and let A = FM(g) . Then I A I � I I g i l = 1 . But FM(g- l ) = A - I and similarly I A- I I � 1 . Thus I A I = 1 and A = eifJM for some OM c: ( ] Then FM(gn ) = einfJM and FM (a) L : oo a neinfJM for any a c: A. Conversely any 0 c: ( ] defines a homo­ morphism F of A onto the complex numbers, F(a) = L : oo a n ei n8, and the kernel of F is, of course, a maximal ideal of A. Thus the space M of maximal ideals can be identified with the interval ] and the transforms a are simply the continuous functions ( on ( ] having absolutely convergent Fourier series. If such a function a is never zero, then, by the above theorem, its recip­ rocal 1 /a is also a function with an absolutely convergent Fourier series. This is a well-known result of Wiener. Notice that the weak topology on ( ] considered as a set of maximal ideals is identical to the usual topology (with and identified), for g(O) = ei8 is continuous in the usual topology and separates points so that 5G can be applied. Now let Ao be the subset of A consisting of those sequences a whose terms a n are all zero for negative n . Ao is easily seen to be closed under convolution and forms a closed sub algebra of A. Ao contains the generator g but not its inverse g- l , and the above argument shows that the homomorphisms of Ao onto the com­ plex numbers (and hence the maximal ideals of Ao) are defined by the complex numbers in the closed unit circle I z I � 1 , Fz(a) = L f a nz n . The maximal ideal space of Ao is thus identified with the closed unit circle (the spectrum of the generator g) and the function algebra Ao is simply the algebra of analytic functions on I z I < 1 whose Taylor's series converge absolutely on I z I � 1 . Since g(z) z separates points, i t follows from 5 G as i n the above example that the weak topology on I z I � 1 is identical wi th the usual topology. =

00

- 7r,

7r .

=

- 7r ,

7r

- 7r, 7r

- 7r, 7r

- 7r,

7r

7r

- 7r

==

73

BANACH ALGEBRAS

The theorem corresponding to the Wiener theorem men tioned above is :

Theorem. If fez) is an analytic function on I z I < 1 whose Taylor's series converges absolutely on I z I � 1 , and iff is not zero on I z I � 1 , then the Taylor's series of l /f is absolutely convergent on I z I � 1 . 23D. As a third example let A L 1 ( - 00, 00 ) with I I f 1 and multiplication again as convolution (f * g)

= = (x)" = "f " [""""f(x - y)g(,'V) dy (Ex. 3b of § 1 8 ) . I f F is any homomorphism of A onto the complex numbers such that I F(f) I � " f " then, since F is in particular a linear functional in (L l ) *, there exists a LOO with I I a 1 100 � 1 such that F(f) = [� f(x) a(x) dx. b

€

Then

f:f:f(x)g(Y) a(x + y) dx dy = [J:f(x - y)g:yOa (x) dx dy = F(f g) F(f) · F(g) = f�f(x)a(x) dx[g(y)a(y) dy = [J:f(x)g(Y)a(x) a(y) dx dy Thus a(x + y) = a(x) a(y) almost everywhere, and if we accept for now the fact (which will follow from later theory) that a can be assumed to be continuous, then this equation holds for all x *

=

and y. But the only continuous solution of this functional equa­ tion is of the form eax, and since I eax I � 1, a is of the form iy eiy x • The above argument can be reversed to show and n that every fu ction e iyx defines a homomorphism . Thus the regular maximal ideals of A are in one-to-one correspondence with the points y € - 00, 00 ) and} is the ordinary Fourier trans-

a(x) =

a (x) = ( form, }(y ) = [:f(x) e -iYX dx.

These functions are easily seen to

be continuous in the ordinary topology of ( - 00, 00) and it fol-

74

BANACH ALGEBRAS

lows as usual from SG that the weak topology and ordinary to­ pology coincide. Now let w (x) be a non-negative weight function on ( 00 00 ) such that w (x + y) � w (x)w (y) for all x and y, and let A be the subset of functions of L I ( - 00, 00 ) such that I f(x) I w(x) dx -

,

i�

< 00 , with this integral as I I f \ I and multiplication taken to be

convolution. The closure of A under convolution is guaranteed by the inequality w (x + y) � w (x)w(y) , as the reader can easily check. If F is a norm-decreasing homomorphism of A on to the complex numbers, then, as above, there exists a E: LOO such that

i�

f(x)a (x)w (x) dx, leading this time to I I a 1 1 00 � 1 and F(f) = the functional equation a(x + y)w(x + y) = a (x)w(x)a ( y)w(y). Therefore, a (x)w (x) = e -sxei tx for some s such that e-SX � w ( x) . Conversely, any complex number s + it such that e -SX � w (x) for all x defines a regular maximal ideal. As an example, suppose that w(x) = ea l x I for some a > o. Then the regular maximal ideals of A are in one-to-one correspondence with the strip in the complex plane defined by I s I � a . The transform function f(x)e - (8 +i t) X dx is a bilateral Laplace transform f(s + it) =

i�

and is analytic interior to this strip. That the usual topology is the correct maximal ideal topology follows, as in the above ex­ amples, from 5G. 23E. We add an example which is a generalization of that in 23C. Let A be any commutative Banach algebra with an iden­ tity and a single generator g. I t may or may not happen that g - l exists. In any case we assert that the maximal ideal space � is in a natural one-to-one correspondence with the spectrum S of g. For M � g eM) is a natural mapping of � onto S, and since g generates A the equality g (MI ) = g (M2 ) implies X(Ml ) = X(M2 ) for every x E: A and hence Ml = M2 , so that the mapping is one-to-one. As before g becomes identified with the function z and the weak topology induced by g is the natural topology of S in the complex plane. A is identified with an algebra of con­ tinuous complex-valued functions on S. If the interior of S is not empty (under this identification), then x is analytic on int (S)

75

BANACH ALGEBRAS

for every x E::: A, for g is identified with the complex variable and x is therefore a uniform limit of polynomials in z. § 24.

SOME

z

B A S I C G E N E RA L TH EO REMS

We gather together here the basic tool theorems of the com­ mutative theory. First comes the formula for computing I I x 1 100 in terms of I I x I I , and then the theorem that conversely, in the semi-simple case, the norm topology in A is determined by the function algebra .d. The third main theorem is the earlier men­ tioned theorem that .d is closed under the application of analytic functions. Finally we prove the existence of and discuss the boundary of the maximal ideal space mL. 24A. The formula for the computation of I I x 1 100 can be de­ veloped in a general non-commutative form if I I x 1 100 is replaced by the spectral norm of x, I I x 1 1 8P' which is defined for x in any complex algebra as lub { I A I : A E::: spectrum x} . The commuta­ tive formula follows from the equality I I x 1 100 I I X l i sp proved in 23B. Theorem. In any Banach algebra I I x l i sp = lim n _ I I xn I I l / n . =

CWJ

Proof. We observe first that p. E::: spectrum y =:} I p. 1 � I I y I I , for, if I p. 1 > I I y I I , then I I y ip. I I < 1 and y ip. has an adverse by 22A. Thus I I y 1 1 8P � I I y I I . Also, we know (2 IE) that A E::: spectrum x =:} An E::: spectrum xn, so that I I x 1 1 8P � ( I I x n I I sp) l / n . Combining these inequalities we get I I x 1 1 8P � I I xn I I l / n for every n, and so I I x l i sp �limn _ I I xn I I l / n . I t remains to be proved that I I x l i sp � lim n _ I I xn I I l / n . By the definition of spectrum and spectral norm, (AX) ' exists for I A I < 1 1 1 1 X l i sp. If F is any functional of A*, it follows from 22C that f(A) = F((AX)') is an analytic function of A in this circle, and its Taylor's series therefore converges there. The co­ efficients of this series can be identified by remembering that, for small A, (AX) ' = L� (AX) n, giving CWJ

CWJ

-

f(A) = F( (AX) ') = L:= l F(X n)An. 0 as I t follows in particular that I F(Xn)A n I = I F(Anxn) I This holds for any functional n -? 00 if I A I < 1 I I I X l i sp . -

-?

76

BANACH ALGEBRAS

F E: A*, and it follows from a basic theorem of Banach space theory proved earlier (SF) that there exists a bound B for the sequence of norms I I 'Anx n I I . Thus I I x n I l l / n � B l / nl l 'A I and lim I I x n I l l / n � I I I 'A I . Since 'A was any number satisfying I 'A I < 1 / 1 1 X l i sp we have lim I I xn I l l / n � I I X l i sp, as desired. Corollary. In any commutative Banach algebra I I x 1 1 «) = lim n _ I I x n I l l / n . 24B. The radical of a commutative algebra is the intersection of its regular maximal ideals ; if the radical is zero, the algebra is said to be semi-simple. If A is a commutative Banach algebra, then x E: radical (A) if and only if x(M) = 0 for every M, that is, I I x 1 1 «) = lim I I x n I l l / n = O. Thus a necessary and sufficient condition that A be semi-simple is that x 0 ==} x = 0 ; hence that the mapping x � x of A onto A is an algebraic isomorphism. There then arises the natural question as to whether the topology of A is determined by the function algebra A, or, equivalently, by the algebraic properties of A. I t is not obvious that this is so, as it was for the algebra e(S), for now I I x 1 1 «) is in general less than I I x I I and the inverse mapping x � x is not in general norm continuous. The answer is nevertheless in the affirmative and depends directly on the closed graph theorem. We prove first a more general result. <Xl

==

Theorem. Let T be an algebraic homomorphism of a commuta­ tive Banach algebra A l onto a dense subset of a commutative Banach algebra A2 . Then: ( 1 ) The adjoint transformation T* defines a homeomorphism of the maximal ideal space mI2 of A2 onto a closed subset of the maximal ideal space mI l of A I ; (2) If A2 is semi-simple T is continuous. Proof. Since we have not assumed T to be continuous, the adjoint T* cannot be assumed to exist in the ordinary sense. However, if a is a homomorphism of A2 onto the complex num­ bers, then a( T(x» is a homomorphism of A l into the ccmplex numbers, and since a is automatically continuous (23A) and T(A I ) is dense in A2 it follows that this homomorphism is onto. We naturally designate it T*a. It is clear that T*a l � T*a2 if

77

BANACH ALGEBRAS

al a2, so that T* is a one-to-one mapping of Ll2 onto a subset of Ll l . Since T(Al) is dense in A2 , the topology of Ll2 is the weak topology defined by the algebra of functions (T(X A. But [T(x)] "(a) a(Tx) [T*a](x) x(T*a) , and since ))the func­ tions x define the topology of Ll l the mapping T* is a homeo­ morphism. Now let f30 be any homomorphism of Ll l in the closure of exists a €: Ll 2 T*(Ll2) . That is, given and Xl > , xn, there such that I f30(Xi) - a(Txi) I < i = 1 , . . . , n. This implies first that, if T(X l) T(X2) , then f30(Xl ) f30(X2) , so that the functional ao defined by ao(T(x)) = f3o(x) is single-valued on T(Al) , and second that I ao(Y) I � I I y I I . Thus ao is a bounded homomorphism of T(Al) onto the complex numbers and can be uniquely extended to the whole of A2 • We have proved that, if f30 belongs to the closure of T*(Ll2) , then there exists ao Ll2 such that f3o(x) ao(Tx) , i.e., f30 = T*ao. Thus T*(Ll2) is closed in Llh completing the proof of ( 1 ) . If Xn � x and T(xn) � y , then X n � x uniformly and Y uniformly, and since z(T*(a)) = (Tz) A(a) for all (T(Xn)),A it�follows that x(T*(a)) yea) , i.e., that (Tx) � = y. If zA isAsemi-simple, I Tx, and the graph of T is therefore then y 2 closed. The closed graph theorem (7G, Corollary) then implies that T is continuous, proving (2) . Corollary. Let A be a commutative complex algebra such that the homomorphisms of A into the complex numbers do not all vanish at any element of A. Then there is at most one norm (to within equivalence) with respect to which A is a Banach algebra. Proof. If there are two such norms, then A is semi-simple ;:e

=

=

=

E

•

•

.

E,

=

=

€:

=

€:

=

=

with respect to each and the identity mapping of A into itself is, therefore, by (2) of the theorem, continuous from either norm into the other. That is, the two norms are equivalent. 24C. This is the natural point at which to ask under what circumstances the algebra A is uniformly closed, i . e., is a Banach space under its own norm. If A is semi-simple, the answer is simple.

A necessary and sufficient condition that A be semi­ simple and A be uniformly closed is that there exist a positive conTheorem.

78

BANACH ALGEBRAS

stant such that 1 1 2 � KI I x2 I I for every x E: A. The map­ ping xK� x is thenI aI xhomeomorphism between the algebras A and A. Proof. If A is semi-simple, the mapping x � x is an algebraic and, if its range A is uniformly closed, then the in­ isomorphism verse mapping is continuous by the closed graph theorem (7G) . Thus there exists a constant K such that I I I I � K I I x 1 1 00 ' Then I I x 1 1 2 � K2 1 1 X 1 100 2 = K2 1 1 (x 2) 1 100 � K2 1 1 x 2 1 1 . Thus the condition is necessary. If, conversely, such a K exists, then

x

6

II

x I I � K �I I x2 1 1 � � K �+�I I x4 1 1 � � . . . � K � + " ' + 2 -n 1 1 x2n 1 1 2 -n.

x

Thus I I I I � K lim I I xn 1 1 1 1 n = KI I X 1 100 ' I t follows that A is semi-simple and that A is complete under the uniform norm, so that the condition is sufficient.

Corollary. A necessary and sufficient condition that A be iso­ to A is that I I x 1 1 2 I I x2 I I for every x E: A. metric =

The case K = 1 above. 24D. Our third theorem states that the algebra of Fourier transforms A is closed under the application of analytic functions. Proof.

Theorem. Let the element x A be given and let F z) be analytic in a region R of the complex plane which includes the (spectrum of x the range of the function x, plus 0 if mr is not compact). Let be (any rectifiable simple closed curve in R enclosing the spectrum of .":. Then the element y A defined by y 27r1 i Jrr (AeF(-A) x) is such that F(A) = F(x(M)) 1 r y (M) 27ri Jr A x(M) or every regular maximal ideal M. Thus the function algebra A is fclosed under the application of analytic functions. Proof. For the moment we are assuming that A has an iden­ tity e. Since contains no point of the spectrum of x, the ele­ ment (Ae - x) - 1 exists and is a continuous function of A by E:

r

E:

=

=

r

_

dA

_

dA

A

79

BANACH ALGEBRAS

22B. The integrand F(X) (Xe - X) -l is thus continuous, and hence uniformly continuous, on the compact set r. The classical existence proof for the Riemann-Stieltjes integral shows, without any modification, that the element YA = L FCAi) (Xie - x) I IlX converges in the norm of A, and the integral Y is defined as its limit. Since the mapping x X is norm decreasing, the func­ tion YA (M) L F(Xi) (Xie - X(M)) -l llXi converges at least as rapidly in the uniform norm to Y (M) . But the limit on the right, for each M, is the ordinary complex-valued Riemann-Stieltjes in­ tegral, and the theorem is proved. If A is taken as the algebra of functions with absolutely con­ vergent Fourier series, then the present theorem is due to Wiener, and generalizes the result of Wiener on the existence of recipro­ cals mentioned at the end of 23C. We now rewrite the above formula in a form using the adverse. First, (Xe x) - 1 = X -lee - x/X) - 1 = X -lee - (x/X) ') = X - Ie X - I (x/X) '. Thus -

i

-?

=

-

Y =

(X) (X) (:)'dA. � [� r F dX ] e rF Jr Jr X 7rt X X 7rt 2 2 _

If A does not have an identity, we cannot write the first term, and we therefore define y in the general case by :

The existence proof for y is the same as in the above case, except that the inverse is replaced by the adverse. Remembering that x '� (M) = x(M)/(x(M) - 1) = 1 + l /(x(M) - 1 ) , we see that y (M)

=

_

(X) (X) 1 r F dX + r F dX . 2 7ri Jr X 2 7ri Jr X - x (M) 1

_

__

The first integral will drop out, giving the desired formula y (M) = F(x(M)), if either (a) F(O) = 0, or (b) r does not enclose z = O. The latter cannot happen if mz is not compact, for then the spectrum of x automatically contains 0 (x being zero at in­ finity) . If however mz is compact and if there exists an element x such that x never vanishes, then (b) can be applied. If we

80

BANACH ALGEBRAS

take F 1 in this case we get y 1 , proving that A has an identity, and therefore, if A is semi-simple, that A has an identity. ==

==

A is semi-simple and is compact, and if there f I exists an element x such that x does not vanish, then A has an identity . Remark: The curve may be composed of several Jordan curves in case the spectrum of x is disconnected. We must there­ fore be more precise in the specification of R: it shall be taken to mL

Corollary.

r

be a region whose complement is connected, and whose compo­ nents are therefore simply connected. 24E. Our last theorem, on the notion of boundary, is due to Silov [1 7] .

Theorem. Let A be an algebra of continuous complex-valued at infinity on a locally compact space S, and unctions fsuppose thatvanishing A separates the points of S. Then there exists a uniquely determined closed subset F S, called the boundar o S with re­ spect to A, characterized as being the smallest closed ysetfon which all the functions 1 f I , f A, assume their maxima. c

E:

The assumption that A separates points is intended to include the fact that each point is separated from infinity, i.e., that the functions of A do not all vanish at any point of S. We have earlier proved in these circumstances that the weak topology induced in S by the functions of A is iden tical to the gi ven topology We come now to the defini tion of the boundary F. Let 5= be the family of closed subsets F c S such that each function 1 f I , E:= A, assumes its maximum on F. Let 5=0 be a maximal linearly ordered subfamily of 5= and let Fo = n { F: F E: 5=0 } . Then Fo E: 5=, for, given E:= A and not identically zero, the set where 1 f 1 assumes its maximum is a compact set intersecting every F E:= 5=0 and therefore intersecting Fo. Fo is thus a lower bound for 5=0 and therefore a minimal element of 5=. We show that Fo is unique by showing that any other minimal element FI is a subset of Fo. Suppose otherwise. Then there exists a point not E: FI - Fo and a neighborhood N of I < meeting Fo. N is defined as the set Proof.

(5G) .

f

f

PI

P { p: I fi (P) -fi(P I ) I

E,

81

BANACH ALGEBRAS

i 1 , . . . , n } for some > 0 and some finite set fh . . . , fn o f ele­ ments of A. We can suppose that max I fi - Ci I � 1 , where Ci fi(P I ) . Since FI is minimal, there exists fo (: A such that I fo I does not attain its maximum on FI - N. We can suppose that max I fo I = 1 and that I fo I < on FI - N (replacing fo by a sufficiently high power fon) . Then I fifo - cifo I < on the whole of FI and hence everywhere. Therefore if Po E: Fo is chosen so thatfo (p o) = 1 , it follows that I fi(Po) - Ci I < i = 1 , . . . , n , and s o Po E: N. Thus Po E: Fo n N, contradicting the fact that N n Fo = 0. Therefore FI c Fo and so FI = Fo ; that is, Fo is the only minimal element of 5=. Remark: The reason for calling this minimal set the boundary of S can be seen by considering the second example in 23C. The function algebra Ao was the set of all analytic functions in I z I < 1 whose Taylor's series converge absolutely in I z I � 1 and the maximal ideal space mr is the closed circle I z I � 1 . The ordinary maximum principle of function theory implies that the boundary of mr with respect to Ao is the ordinary boundary I z I l. Corollary. If A is a regular function algebra, then F = S. =

E

=

E

E

E,

=

Proof. If F � S and P E: S - F, then there exists by the con­ dition of regularity ( 19F) a function f E: A such that f = 0 on F andf(p) � o. This contradicts the definition of the boundary, so that F must be the whole of S. Corollary. If A is a self-adjoint function algebra, then F = S.

The proof is given later, in 26B.

Chapter F SOME SPECIAL BANACH ALGEBRAS

In this chapter we shall develop the theory of certain classes of Banach algebras which we meet in the study of the group al­ gebras of locally compact Abelian groups and compact groups. § 25 treats regular commutative Banach algebras, which are the natural setting for the Wiener Tauberian theorem and i ts gen­ eralizations. In § 26 we study Banach algebras with an involu­ tion, which are the proper domain for the study of positive defi­ niteness. Finally, in § 27 we develop the theory of the H*-alge­ bras of Ambrose, which include as a special case the L2-group algebras of compact groups. § 25.

R EG U LA R COMMUTATIVE B A NACH A LG E B RAS

This section is devoted to a partial discussion of the ideal theory of a commutative Banach algebra associated with the Wiener Tauberian theorem and its generalizations. The gen­ eral problem which is posed is this : given a commutative Banach algebra A with the property that every (weakly) closed set of regular maximal ideals is the hull of its kernel, when is it true that a closed ideal in A is the kernel of i ts hull ? The Wiener Tauberian theorem says that it is true for ideals with zero hull, provided the algebra satisfies a certain auxiliary condition. Questions of this kind are difficult and go deep, and the general situation is only incompletely understood. A Banach algebra A is said to be regular if it is commu­ tative and i ts Gelfand representation A is a regular function al-

25A.

82

83

SOME SPECIAL BANACH ALGEBRAS

gebra. This means by definition that the weak topology for mr defined by A is the same as its hull-kernel topology, and (by 19F) is equivalent to the existence, for every (weakly) closed set C c mr and for every point Mo tl C, of a function f E:= A such that f ° on C and f(Mo) � 0. We show in the lemma below that the representation algebra of a regular Banach algebra pos­ sesses " local identities," and the discussion in the rest of the sec­ tion holds for any regular, adverse-closed function algebra which has this property of possessing " local identities." ==

Lemma. If A is a regular Banach algebra and Mo is a regular

maximal ideal of A, then there exists x some neighborhood of Mo.

E:=

A such that x

=

1 tn

Proof. We choose x E:= A such that x(Mo) � 0, and a com­ pact neighborhood C of Mo on which x never vanishes. Then C is hull-kernel closed by the hypothesis that A is regular, and we know that the functions of A confined to C form the representa­ tion algebra of A/k(C) . Since this algebra contains a function (x) bounded away from 0, it follows from the analytic function theorem 24D that it contains the constant function 1 . That is, A contains a function Xo which is identically 1 on C, q.e.d. 25B. From now on A will be any adverse-closed algebra of continuous functions vanishing at infinity on a locally compact Hausdorff space S, such that S = .1, and A is regular and has the property of the above lemma. Lemma. If C is a compact subset of S, then there exists f such thatf is identically 1 on C.

E:=

A

I f f1 = 1 on B l and f2 = 1 on B2 , then obviously f1 + f2 - f1 f2 = 1 on B 1 U B2 . This step is analogous to the well-known algebraic device for enlarging idempotents. By the compactness of C and the previous lemma there exist a finite number of open sets Bi covering C and functions fi such that fi = 1 on Bi. These combine to give a function f = 1 on U Bi by repeating the above step a finite number of times. Proof.

Corollary. If A is a semi-simple regular Banach algebra and the maximal ideal space of A is compact, then A has an identity.

84

SOME SPECIAL BANACH ALGEBRAS

Since mL is compact, the lemma implies that the con­ stant 1 belongs to the representation algebra A; thus A has an identity. The semi-simplicity of A means by definition that the natural homomorphism x � x of A onto A is an isomorphism ; therefore A has an identity. Proof.

25C. Lemma. If F is a closed subset of S and C is a compact set disjoint from F, then there exists f C A such that f = 0 on F and f = 1 on C. In fact, any ideal whose hull is F contains such an f· Proof. By virtue of the preceding lemma this is an exact rep­ lica of the first lemma of 20F. However, in view of the impor­ tance of the result we shall present here a slightly different proof. Let Ae be the function algebra consisting of the functions of A confined to C; Ae is (isomorphic to) the quotient algebra A/k(C) . Ae has an identity by the preceding lemma, and since C is hull­ kernel closed the maximal ideals of Ae correspond exactly to the points of C. Now let I be any ideal whose hull is F and let Ie be the ideal in Ae consisting of the functions of I confined to C. The functions of I do not all vanish at any point of C, for any such point would belong to the hull of I, which is F. Thus Ie is not included in any maximal ideal of Ae, and therefore Ie = Ae. In particular Ie contains the identity of Ae; that is, I con­ tains a function which is identically 1 on C.

25D. Theorem. If F is any closed subset of S, then the func­ tions of A with compact carriers disjoint from F form an ideal j(F) whose hull is F and which is included in every other ideal whose hull is F.

Proof. The carrier of a function f is the closure of the set where f � o. The set of functions with compact carriers dis­ joint from F clearly form an ideal j(F) whose hull includes F. If P tl F, then p has a neighborhood N whose closure is compact and disjoint from F. By the fundamental condition for regularity there exists f C A such that f(p) � 0 and f = 0 on N'. Thus f has a compact carrier ( eN) disjoint from F and f C j(F) , prov­ ing that p tl h (j(F)) . Thus p tl F implies that p tl h (j(F)) and h (j(F)) = F.

SOME SPECIAL BANACH ALGEBRAS

85

Now if I is any ideal whose hull is F and / is any function of }(F), with compact carrier C disjoint from F, then by the pre­ ceding lemma I contains a function e identically one on C, so that / = /e E::: I. Thus }(F) c I, proving the theorem. As a corollary of this theorem we can deduce the Wiener Tauberian theorem, but in a disguise which the reader may find perfect. I ts relation to the ordinary form of the Wiener theorem will be discussed in 37A.

Corollary. Let A be a regular semi-simple Banach algebra with the property that the set 0/ elements x such that x has compact sup­ port is dense in A. Then every proper closed ideal is included in a regular maximal ideal.

Proof. Let I be a closed ideal and suppose that I is included in no regular maximal ideal. We must show that I = A. But the hull of I is empty and therefore I includes the ideal of all elements x E::: A such that x has compact support. Since the latter ideal is dense in A by hypothesis, we have I = A as de­ sired. 25E. A function / is said to belong locally to an ideal I at a point p if there exists g E::: I such that g = / in a neighborhood of p. If P is the point at infinity, this means that g = / outside of some compact set. Theorem. 1/ / belongs locally to an ideal I at all points 0/ S and at the point at infinity, then / E::: I. Proof. In view of the assumption on the point at infinity, we

may as well suppose that S is compact and that A includes the constant functions. Then there exists a fi n ite family of open sets Ui covering S and functions /i E::: I such that / = /i on Ui• We can find open sets Vi covering S such that Pi C Ui, and the theorem then follows from the lemma below. Lemma. 1/ /i E::: I and / = /i on Ui, i = 1 , 2, and if C is a compact subset 0/ U2 , then there exists g E::: I such that / = g on U1 U C. Proof. Let e E::: A be such that e = 1 on C and e = 0 on U2 '. I f g = /2 e + /1 (1 - e) , then g = /2 = / on C, g = /1 = / on

86

SOME SPECIAL BANACH ALGEBRAS

U1 - U2 and g = fe + f(1 - e) = f on U1 n U2 • These equa­ tions add to the fact that g = f on U1 U C, as asserted. In applying this theorem the following lemma is useful.

Lemma. An element f always belongs locally to an ideal 1 at every point not in hull (1) and at every point in the interior of hull (f) · Proof. If p ([. hull (1) then there exists by 25C a function e E: l such that e = 1 in a neighborhood N(P) . Then ef E: J andf = ef in N(P), so thatf belongs locally to 1 at p. The other assertion of the lemma follows from the fact that 1 contains o. 25F. The above theorem leads to the strongest known theorem guaranteeing that an element x belong to an ideal 1 in a commu­ tative Banach algebra A. We say that the algebra A satisfies the condition D (a modification of a condition given by Ditkin) if, given x E: M E: mt, there exists a sequence Xn E: A such that xn = 0 in a neighborhood Vn of M and XXn � x. If mt is not compact, the condition must also be satisfied for the point at infinity.

Theorem. Let A be a regular semi-simple Banach algebra sat­ isfying the condition D and let 1 be a closed ideal of A. Then 1 contains every element x in k(h(J)) such that the intersection of the boundary of hull (x) with hull (1) includes no non-zero perfect set.

Proof. We prove that the set of points at which x does not belong locally to 1 is perfect (in the one point compactification of mt) . It is clearly closed. Suppose that Mo is an isolated point and that U is a neighborhood of Mo such that x belongs locally to 1 at every point of U except Mo. There exists by the condition D a sequence Y n such that Y nX � x and such that each function Y n is zero in some neighborhood of Mo. Let e be such that e = 0 in U' and e = 1 in a smaller neighborhood V of Mo. Then Y nxe belongs locally to 1 at every point of mtoo and therefore is an ele­ ment of 1 by the preceding theorem. Since 1 is closed and Y nX � x, it follows that xe E: 1 and hence that x belongs to 1 at Mo (since xe = x in V) . Thus the set of points at which x does not belong locally to 1 is perfect. Since it is included in both hull (1) and the boundary of h ull (x) by the lemma in 2 5E

SOME SPECIAL BANACH ALGEBRAS

87

and the assumption that h(J) c h ex), it must be zero by hypoth­ esis. Thus x belongs locally to 1 at all points, and x €: J by 25E. We shall see in 37C that the group algebra of a locally compact Abelian group satisfies condition D, and this gives us the strong­ est known theorem of Tauberian type in the general group setting. § 26. BANACH A LG E B RA S WITH I NVO LUTI O N S We remind the reader that a mapping x x* defined on an algebra A is an involution if it has at least the first four of the following properties : x** = x (1) (x + y) * = x* + y * (2) -)0

(Ax) * = �x*

(3)

(4)

(5) (6)

(xy) * = y *x * / I xx* / I = I I X 1 1 2 - xx * has an adverse (e + xx* has an inverse) for every x.

Many important Banach algebras have involutions. For in­ stance all the examples of § 18 possess natural involutions except for 2a and 4. In the algebras of functions 1 and 5 the involu­ tion is defined by f* = J and the properties (1) to (6) above can all be iminediately verified. In the algebra 2b of bounded opera­ tors on a Hilbert space, A* is the adjoint of A, and we have al­ ready seen in that properties ( 1 ) to (6) hold. Group algebras will be discussed in great detail later. The existence of an involution is indispensable for much of the standard theory of harmonic analysis, including the whole theory of positive definiteness. We begin this section by investigating the elementary implications of the presence of an involution, and then prove a representation theorem for self-adjoint Banach al­ gebras, the spectral theorem for a bounded self-adjoint operator, the Bochner theorem on positive definite functionals, and a gen­ eral Plancherel theorem. 26A. Our systematic discussion in the early letters of this sec­ tion gets perhaps a trifle technical and we shall try to ease mat-

lIB

88

SOME SPECIAL BANACH ALGEBRAS

ters somewhat by proving ahead of time and out of context one of the simplest and most important theorems. The reader will then be able to omit 26B to 26E if he wishes and go on immedi­ ately to the numbers having more classical content.

If A is a commutative Banach algebra with an iden­ tity and with an involution satisfying (1 )-(5), then its transform algebra A is the algebra e(�) of all continuous complex-valued functions on its maximal ideal space � and the mapping x � x* is an isometry of A onto e(�) . Theorem.

Proof. We first prove that, if x i s self-adjoint (x = x*) , then x is real-valued. Otherwise x assumes a complex value a + bi (b � 0) , and, if y = x + iBe, then y assumes the value a + i (b + B) . Remembering that y * = x - iBe we see that a 2 + b 2 + 2bB + B 2 � l I y 1 100 2 � I l y 1 1 2 = I l yy * I I = I I x 2 + B 2e I I

� I I x2 1 1 + B2 which is a contradiction if B is chosen so that 2bB > I I x 1 1 2 . This argument is due to Arens [2] . For any x the elements x + x* and i(x - x*) are self-adjoint and 2x = (x + x*) - i[i(x - x*)]. Since the functions x sepa­ rate points in � the real-valued functions of A form a real alge­ bra separating the points of � and hence, by the Stone-Weier­ strass theorem, dense in eR (�) . Therefore A is dense in e(mz) . The above expression for x in terms of self-adjoint elements also proves that (x*) � = (x) -. Finally we prove that A is isometric to A, hence complete in the uniform norm, hence identical with e(mz) . I f y i s self-adjoint, we have exactly as in 24C the inequality I I y I I � I I y 2 I I 72 � � I I y 2n 1 1 2 ->1, and therefore I I y I I = I I y 1 100 . In general I I x I I = I I xx* 1 1 72 = I I X(x*) " 1 100 72 = I I I x 1 2 1 100 72 = I I x 1 100, as asser ted. 26B. A commutative Banach algebra is said to be self-adjoint if for every x E.: A there exists y E.: A such that y = x - (a being the complex conjugate of a) . We note several simple consequences of this definition. •

•

•

If A is a self-adjoint commutative Banach algebra, then A is dense in e(�) . Lemma 1.

SOME SPECIAL BANACH ALGEBRAS

89

Proof. If x and y are related as above, then (x + y)/2 is the real part of x and (x - y)/2 is the imaginary part. It follows that the real-valued functions of A form a real algebra which separates points of mr and therefore, by the Stone-Weierstrass theorem, is dense in CR (mr) . Therefore A itself is dense in e(mr) . Corollary. If A is self-adjoint, then mr is its own boundary.

Proof. Otherwise let Mo be a point of mr not in the boundary, let f be a continuous function equal to 1 at Mo and equal to 0 on the boundary, and let x be any function of A such that I I f - x l lao < !. Then I x(Mo) I > ! and I x I < ! on the boundary, con­ tradicting the fact that x must assume its maximum on the bound­ ary. Thus no such Mo can exist and mr equals its boundary.

Lemma 2. If A is self-adjoint and C is a compact subset of mr, then there exists x E: A such that x � 0 and x > 0 on C.

Proof. For each M E: C there exists x E: A such that x (M) � O. Then I x 1 2 E: A by the definition of self-adjointness and 1 x 1 2 > 0 on an open set containing p. It follows from the Heine-Borel theorem that a finite sum of such functions is posi­ tive on C, q.e.d. This is the same argument that was used in 19C.

If A is semi-simple and self-adjoint and mr is com­ pact, then A has an identity . Corollary.

Proof. There exists x E: A such that x > 0 on mr and it fol­ lows from the corollary of 24D that A has an identity. 26C. If A is self-adjoint and semi-simple, then, given x, there exists a unique y such that y = x -. If this y be denoted x *, then the mapping x � x * is clearly an involution on A satisfying ( 1 ) - (4) . I t also satisfies (6), for - I x 1 2 never assumes the value 1 , and - xx * is therefore never an identity for a regular maximal ideal, so that - xx * has an adverse by 2 ID. The converse is also true, without the hypothesis of semi-simplicity.

If A is a commutative Banach algebra with an in­ volution satisfying (1) - (4) and (6) , then A is self-adjoint and x * '" x for every x E: A. Theorem. =

-

90

SOME SPECIAL BANACH ALGEBRAS

Proof. An element x such that x = x * is called self-adjoint. In proving that x * � = x -, it is sufficient to prove that, if x is self-adjoint, then x is real-valued, for in any case x + x * and i(x - x*) are self-adjoint elements, and, if they are known to have real transforms, then the conclusion about x follows from its expression as [(x + x*) - i(i(x - x*))]!2, together with (3) . Accordingly, let x b e self-adjoint and suppose that x i s not real, x(M) = a + bi for some M, with b � o. Then some linear combination of x(M) and (X 2) A (M) has pure imaginary part, for the number pairs (a, b) and (a 2 - b 2, 2ab) are linearly independ­ en t. The actual com bina tion is y = [( b 2 - a 2) x + ax 2]!b( a2 + b 2), giving y (M) = i. Then ( _y 2) A (M) = 1 and _y 2 = -yy * can­ not have an adverse. This contradicts (6), and proves that x is real-valued if x is self-adjoint, q.e.d.

26D. Theorem. If a commutative Banach algebra is semi­ simple, then it follows from properties (1) - (4) alone that an involu­ tion is continuous.

Proof. 24B can be applied almost directly. Actually, the proof of 24B must be modified slightly to allow a mapping T of A l onto a dense subalgebra of A2 which departs slightly from being a homomorphism (to the extent that involution fails to be an isomorphism) . The details will be obvious to the reader. 26E. We now raise the question, suggested by 26B, as to when a self-adjoint algebra has the property that A = e(mr) . Some of the argument is the same as in 24C, but will nevertheless be repeated. If .d = e(mr) and A is semi-simple, the continuous one-to-one linear mapping x � x of A onto A = e(mr) must have a continuous inverse by the closed graph theorem. Thus there exists a constant K such that I I x I I � K I I x 1 100 for every x, and, in particular, I I x 1 1 2 � K2 1 1 x 1 100 2 = K2 1 1 xx - l loo � K2 1 1 xx * I I . Conversely, this condition is sufficient to prove that .d = e(mr) even without the assumption of self-adjointness. Theorem. If a commutative Banach algebra A has an involu­ tion satisfying (1) - (4) and the inequality I I x 1 1 2 � K I I xx * I I , then I I x I I � KI I x 1 100 for all x E: A, A is semi-simple and self-adjoint, and A = e(mr) .

91

SOME SPECIAL BANACH ALGEBRAS

Proof. If y is self-adjoint, then the assumed inequality be­ comes I I y 1 1 2 � KI I y2 1 1 · Applying this inductively to the powers y 2", we get I I y I I � Kyzl l y2 1 1 yz � KYZK�I I y4 I I U � • • • � KYZKu . . . K2-1 1 y2" 1 1 2-. Therefore I I y I I � K lim I I ym 1 I 1 1m = K I I Y 1 100' The equation x*"(M*) = x(M) implies that I I X* A 1 100 = I I x 1 100' Thus I I x 1 1 2 � KI I xx* I I � K21 1 XX*A 1 100 � K21 1 X 1 1002, and I I x I I � KI I X 1 100, as asserted in the theorem. One consequence of this in­ equality is obviously the semi-simplicity of A. Another is that the one-to-one mapping x -7 X is bicontinuous and that the al­ gebra A is therefore uniformly closed. It will follow from that A = e(m) if A is self-adjoint. The equation x*A (M*) = x (M) shows that, if F is a minimal closed set on which every function I x I assumes i ts maximum, then so is F*. But the only such minimal closed set is the bound­ ary Fo, so that Fo = Fo * . We now show that M = M* if M is a poin t of the boundary Fo. Otherwise we can find a neighborhood U of M such that U n U* = 0, and a function x E: A which takes its maximum absolute value in U and nowhere else on the boundary Fo . We can suppose that max I x I = 1 and that I x I < on Fo U (replacing x by (x) n for n sufficiently large) . Then I (x*) A I < on Fo U* and I xx*" I < on the whole boundary Fo and hence everywhere. Then 1 � I I X 1 1 2 � K21 I XX*A 1 100 < K2 e, a contradiction if < 1 / K2 . Thus (X*) A = x - on the boundary. Since A is uniformly closed, we see from that the restriction of A to the boundary Fo is the algebra e(Fo) of all continuous complex-valued functions vanishing at in­ finity on Fo. We know (19C and 20C) that Fo is the set of all regular maximal ideals of this algebra, and since the restriction of A is, by the definition of the boundary, isomorphic (and iso­ metric) to A itself, it follows that Fo is the set of all regular maxi­ mal ideals of A, i.e., Fo = m. Thus X*A = xA - everywhere and A is self-adjoint. Corollary. If A is a commutative Banach algebra with an in­ volution satisfying ( 1 ) -(5), then A is isometric and isomorphic to e(m) = A. Proof. Now K = I , and the inequality I I x I I � KI I x 1 100 im­ plies that I I x I I = I I x 1 100 '

26B

E

-

E

e

-

E

26B

92

SOME SPECIAL BANACH ALGEBRAS

Corollary. If A is a commutative algebra of bounded operators on a Hilbert space H, closed under the norm topology and under the adjoint operation, then A is isometric and isomorphic to the algebra e(�) of all continuous complex-valued functions vanishing at infinity on a certain locally compact Hausdor:ff space mt.

The above corollary and l iB. It should perhaps be pointed out that the theorem can be proved without using the notion of boundary by adding an iden­ tity and applying 26A. We start, as in the first paragraph of the given proof, by showing that the norm on A is equivalent to the spectral norm. The following lemma is then the crucial step. Proof.

Lemma. If A is a commutative Banach algebra with an in­ volution satisfying I I x 1 1 2 � K I I xx * I I , then I I x I I sp 2 = I I xx * l i sp .

I I (xx*) n I I I / n = I I x nx* n I I I / n � I I Xn I I I / n i l X* n I I I / n . Taking limits we get I I xx* l i sp � I I x I I spl l x* l i sp = I I X I I sp2 . Conversely, I I xn 1 1 2 � K I I Xn (Xn) * I I = K I I (XX *) n I I , I I X n 1 1 2 / n � KI / n l l (XX*) n I I I / n and, letting n � 00, I I X I I sp2 � I I xx * l i sp . Supposing, then, that the original algebra has been renormed with its spectral norm, we now add an identity and observe that the norm I I x + Xe I I = I I x I I + I X I satisfies the K inequality wi th K = 1 6. For if II x I I < 3 1 X I , then I I x + Xe 1 1 2 = ( I I x I I + I X 1 ) 2 < 1 6 1 X 1 2 � 1 6\ 1 (x + Xe) (x * + Xe) I I , while, if II x I I � 3 1 X I , then I I xx* + Xx* + Xx I I � I I xx * I I I I Xx* + Xx I I � I I x \ 1 2 /3 and obviously ( I I x \ I + I X 1 ) 2 � 1 6 ( 1 1 x 1 1 2 /3 + I X 1 2 ) . The lemma therefore implies that \ I y I I sp 2 = I I yy * 1 1 8P i n the extended algebra and the theorem is now a direct corollary of 26A. 26F. A representation T of an algebra A is a homomorphism (x � Tx) of A onto an algebra of linear transformations over a vector space X. If S has an involution, then X is generally taken to be a Hilbert space H and T is required to take adjoints into adjoints : Tx * = ( Tx) * I t is then called a *-representation (star represen ta tion) . Proof.

-

.

Lemma. If A is a Banach algebra with a continuous involution,

then every *-representation is necessarily continuous.

SOME SPECIAL BANACH ALGEBRAS

93

Proof. Since T is a homomorphism, Ty has an adverse when­

ever y does and therefore I I Tx l i sp � I I x l i sp for every x. Now Tx*x is a self-adjoint operator, and for it we know (see 26A) that I I Tx 1 1 2 = I I ( Tx) *( Tx) I I = I I Tx*:c I I = I I Tx*:c l i sp. Thus I I Tx 1 1 2 = I I Tx*x l i sp � I I x*x l i sp � I I x*x I I � I I x* I I I I x I I � B I I x 1 1 2, where B is a bound for the involution transformation. That is, T is bounded with bound B �. If the involution is an isometry ( I I x* I I = I I x I I ), then B = 1 and I I T I I � 1 in the above argument. The theorem of Gelfand and Neumark quoted earlier ( § 1 1 ) says simply that every C*­ algebra has an isometric *-representation. In this paragraph we are principally concerned with *-represen­ tations of commutative, self-adjoint algebras. I t follows from the above argument that in this case I I Tx I I � I I I x 1 2 1 1 00 � = I I x 1 1 00 so that any such representation can be transferred to a norm­ decreasing representation of the function algebra .d, and, since .d is dense in e(mt), the representation can then be extended to a norm-decreasing representation of e (mt) . We now show that T has a unique extension to the bounded Baire functions on mt.

Theorem. Let T be a bounded representation of the algebra e(mt) of all continuous complex-valued (or real-valued) functions vanish­ ing at infinity on a locally compact Hausdor.f1 space mt by operators on a reflexive Banach space X. Then T can be extended to a repre­ sentation of the algebra m (mt) of all bounded Baire functions which vanish at infinity on mt, and the extension is unique, subject to the condition that Lx , y (f) = ( Tfx, y) is a complex-valued bounded in­ tegral for every x E: X, y E: X*. If S is a bounded operator on X which commutes with Tf for every f E: e( mt) , then S commutes with Tf for every f E: m (mt) . If X is a Hilbert space H and T is a *-representation, then the extended representation is a *-representa­ tion.

The function F(f, x, y) = ( Tfx, y) defined for f E: e(mt), x E: X, y E: X* is trilinear and I F(f, x, y) I � I I T I I I I f 1 100 II x l i i i y I I · If x and y are fixed it is a bounded integral on e(mt) and hence uniquely extensible to m(mt) with the same inequality holding. The extended functional is, moreover, linear in f, x and y. If x andf are fixed, it belongs to X** = X; that is, there

94

SOME SPECIAL BANACH ALGEBRAS

exists an element of X which we designate T,x such that F(/, x, y) = ( T,x, y) for all y. From the linearity of F in its three argu­ ments and the inequality I F(/, x, y) I � I I T 1 1 1 1 / 1 1 001 1 x l i i i y I I we see that T, is linear and bounded by I I T 1 1 1 1 / 1 1 00 and that the mapping / � T, is linear and bounded by I I T I I . Now for /, g � e (mr) we have (a)

or

(b)

F(/g, x, y)

=

F(/, Tgx, y)

=

F(/, x, ( Tg) *y) .

Keeping g fixed this identity between three integrals in ! persists when the domain is extended from e ( mr) to CB ( mr) , proving (a) for g � e ( mr) and / � CB ( mr) . Since (a) is symmetric in / and g, we have (b) for g � CB ( mr) and / � e ( mr) . Extending once more we have (b) and hence (a) for all /, g � CB ( mr) . Thus the ex­ tended mapping / � T, is a representation of CB(mr) . I f S commutes with T, for every / � e ( mr) , we have ( T,Sx, y) = (ST,x, y) = ( T,x, S *y), i.e., F(/, Sx, y) = F(/, x, S *y) . This identity persists through the extension, as above, and then trans­ lates back into the fact that S commutes with T, for every / � CB(mr) .

Finally, i f X i s a Hilbert space H the assumption that T i s a *-representation is equivalent to the identity F(], y, x) = F(/, x, y), which again persists through the extension and trans­ lates back into the fact that the extended T is a *-representation. This completes the proof of the theorem. 26G. The essential content of the spectral theorem is that a bounded self-adjoint operator on a Hilbert space can be approxi­ mated in the operator norm by linear combinations of projec­ tions. This is similar to the fact that a bounded continuous function on a topological space can be approximated in the uni­ form norm by step functions (i.e., by linear combinations of characteristic functions) . The Gelfand theory reveals these ap­ parently unconnected statements to be, in fact, equivalent asser­ tions, and this fact gives rise to an elegant and easy proof of the spectral theorem.

95

SOME SPECIAL BANACH ALGEBRAS

Let a be a commutative algebra of bounded operators on a Hilbert space H, closed under the operation of taking the ad­ joint, and topologically closed under the operator norm. We may as well suppose that a contains the identity, for in any case it can be added. Then a is isometric and isomorphic to the al­ gebra e (mr) of all continuous functions on its compact maximal ideal space (26A and and the inverse mapping can be uniquely extended (26F) to the algebra CB(mr) of all bounded Baire functions on mr. Let A be a fixed self-adjoint operator from a and A its image function on mr. (This conflicts, momen­ tarily, with our earlier use of the symbol A.) We suppose that - 1 � A � 1, and, given we choose a subdivision - 1 = Ao < Let Ex be Al < . . . < An = 1 such that max (Ai - Ai - I) < the characteristic function of the compact set where A � A, and choose A / from the interval [Ai - b A J . Then

lIB) E,

E.

and hence

/ / A - 2:1 A/ (EAi - EAi_I) / /00

<

" A - 2:1 A / ( EAi - EAi_I) " <

E

E

where Ex is the bounded self-adjoint operator determined by the Baire function Ex . Ex is idempotent (since (EX) 2 = Ex) and hence a projection. The approximation above could be written in the form of a Riemann-Stieltjes integral,

A

=

f

A dEx,

which is the integral form of the spectral theorem. All the standard facts connected with the spectral theorem follow from the above approach. For instance, simple real­ variable approximation arguments (one of which can be based on the Stone-Weierstrass theorem) show that, for fixed A, there ex­ ists a sequence of polynomials P n such that P neAl 1 Ex . Re-

membering that (P n(A) x, x)

=

fPneAl dJ.Lx,x

it follows that

Pn(A) converges monotonically to E in the usual weak sense : (P n(A) x, x) 1 ( Ex x, x) for every x E:= H. 26H. We conclude § 26 with a study of positive-defi n iteness culminating in a general Plancherel theorem. Most of the ele-

96

SOME SPECIAL BANACH ALGEBRAS

mentary facts appeared in the early Russian literature, but the Plancherel theorem itself is a modification of that given by Gode­ ment [20] . If A is a complex algebra with an involution, a linear functional ep over A is said to be positive if ep(xx*) � ° for all x. The signifi­ cance of positivity is that the form [x, y] = ep(xy *) then has all the properties of a scalar product, except that [x, x] may be zero without x being zero. The linearity of [x, y] in x and its conju­ gate linearity in y are obvious, and the only remaining property to be checked is that [x, y] = [y, x] . This follows upon expanding the left member of the inequality ep((x + AY) (X + AY ) *) � 0, showing that Aep(yX*) + �ep(xy *) is real-valued for every com­ plex number A, from which it follows by an elementary argument that ep(yx *) = ep(xy *) as required. It then follows (see l OB) that the Schwarz inequality is valid :

I ep(xy *) I � ep(xx*) %ep(yy *) � .

If A has an identity, we can take y = e in the Schwarz inequality and in the equation ep(xy *) = ep(yx*) and get the conditions

I ep(x) 1 2 � kep(xx*), ep(x*) = ep(x)

where k = epee) . In any case, a positive functional satisfying these extra conditions will be called extendable, for reasons which the following lemma will make clear.

Lemma 1 . A necessary and sufficient condition that ep can be extended so as to remain positive when an identity is added to A is that ep be extendable in the above sense.

The necessity is obvious from the remarks already made. Supposing then that ep satisfies the above conditions and taking epee) = k, we have ep((x + Ae) (x + Ae) *) = ep(xx*) + 2 1 A I k %ep(xx*) % + I A 1 2 k = 2
-

Lemma 2. If A is a Banach algebra with an identity and a continuous involution, then every positive functional on A is con­ tinuous.

SOME SPECIAL BANACH ALGEBRAS

Proof.

If

A

97

is a Banach algebra with an identity and if

" x " < I, then e - x has a square root which can be computed by the ordinary series expansion for VI t about the origin. If A has a continuous involution and x is self-adjoint, then so is the series value for y = �. Thus ep(e - x) = ep(yy*) � ° and ep(x) � epee). Similarly ep( -x) � epee), and we have the con­ clusion that, if x is self-adjoin t and II x II < 1 , then I ep(x) I � ep( e) . For a general x we have the usual expression x = (x + x*) j2 i[i(x - x *) j2] where x + x* and i(x - x*) are self-adjoint. If B is a bound for the continuous involution, i t follows that I ep(x) I � V2ep(e) whenever I I x I I < 2j(B + 1), proving that ep is con­ -

tinuous with (B + l)ep(e)jV2 as a bound.

261. Theorem. (Herglotz-Bochner-Weil-Raikov.) If A is a semi-simple, self-adjoint, commutative Banach algebra, then a linear functional on A is positive and extendable if and only if there ex-

ep

ists a jinite positive Baire measure j.Ltp on ffir such that

ep(x) = fx dj.Ltp

x E::: A. Proof. If ep(x) = fx dj.L where j.L is a finite positive Baire meas

for every

ure on ffir, then

ep(xx*) = II X j 2 dj.L � 0,

� (x*) = J� * dl' = (J� dl'r = �(x),

� (x) [ 2 = [J� dl' [ 2 ;;;:. (I[ � [ 2 dl') (It dl') = I I I' [[ � (xx*) ; that is, ep is positive and extendable. Conversely, if ep is positive and extendable, then ep is continuous (by Lemma 2 above) and I ep(x) 12 � kep (xx*) � kl+�ep((XX *) 2) � � � k 1 + ... + 2 -nep((Xx*) 2n) 2 -n � k 1 + ... + 2 -n 11 ep 112 -nj I (XX *) 2n 112 -n• I

.

.

.

98

SOME SPECIAL BANACH ALGEBRAS

Letting n � 00 and taking the square root we have I cp(x) I � kl l x 1 100 ' Since A is dense in e (mr) , the bounded linear functional I defined on A by Itp(x) = cp(x) can be extended in a unique way to e (mr) . If/ E: e(mr) and / � 0, then /72 can be uniformly ap­ proximated by functions x E: A and hence / can be uniformly approximated by functions I x 1 2 . Since Itp( 1 x 1 2) = cp(xx *) � 0 and Itp( 1 X 1 2) approximates Itp(/), it follows that Itp(/) � O. That is, Itp is a bounded integral. If }J.tp is the related measure, we

f

have the desired identity cp(x) = X d}J.tp for all x

E:

A.

26 J. The setting for the Plancherel theorem will be a semi­ simple, self-adjoint commutative Banach algebra A, and a fixed positive functional cp defined on a dense ideal Ao C A. An element p E: Ao will be said to be positive definite if the functional (}p defined on A by (}p (x) = cp(px) is positive and extendable. Then by the Bochner theorem there is a unique bounded integral Ip on e ( mr) (finite positive Baire measure }J.p on mr) such that

f

cp(px) = Ip (x) = x d}J.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 {}xx* is not only positive but can also be extended so as to remain positive, for if {}xx* (y + Xe) = cp(xx*y + Xxx *), then (}xx* ( (y + Ae) (y + Xe) *) = cp« xy + Xx) (xy + AX) *) � O. If p and q are positive definite, then Iq (xp) = cp(pqx) = Ip (xq) for every x E: A and therefore Iq(hp) = Ip (hq) for every h E: e(mr) . Let Sf be the support off, i.e., the closure of the set where / � O. If pq is bounded away from 0 on S" then h = /Ipq E: e ( mr) and Iq(/Iq) = Ip (/Ip) . We now define the functional I on L ( mr) (the set of functions in e ( mr) having compact support) by 1(/) = Ip (/Ip) where p is any positive definite element such that p is bounded away from 0 on Sf and hence such that/lp E: L. Such a p always exists, for Sf is compact if/ E: L and by the Heine­ Borel theorem we can find p of the form X I X 1 * + . . . + XnXn * such that p � 0 and p > 0 on Sf. We saw above that 1(/) is

SOME SPECIAL BANACH ALGEBRAS

99

independent of the particular p taken, and using the p just con­ structed we see that I(f) � 0 if f � o. 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 wi th the identity I(f) = Ip (flp) for all f which vanish on this closed set, implies that I(gp) = Ip (g) for all g. Therefore I(p) = Ip (l ) = " Ip I I . Finally, cp(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 Hep generated by positive definite elements onto a subspace of L2 (I) . We have proved the following theorem. --7

Let A be a semi-simple, self-adjoint, commutative Banach algebra and let cp be a positive functional defined on a dense ideal A c A. Then there is a unique Baire measure f.L on ffi"l: such that p €: L l (f.L) and cp(px) = xp df.L whenever p is positive definite Theorem. 0

f

with respect to cpo The mapping p P when confined to the sub­ space of Hep generated by positive definite elements is therefore a unitary mapping of this subspace onto a subspace of L 2 (f.L), and is extendable to the whole of Hep if it is known that A0 2 is Hep-dense in Ao. --7

26K. The last two theorems have assumed, needlessly, that A is a Banach algebra. A more general investigation would start with any complex algebra A having an involution and a positive functional cp on A. The Schwarz inequality for cp implies that cp(xaa *x*) = 0 whenever cp(xx*) = 0 so that the set of x such that cp(xx*) = 0 is a right ideal I, and right multiplication by a becomes a linear operator Ua on the quotient space AII. The condition that Ua be bounded with respect to the cp scalar prod­ uct is clearly that there exist a constant k a such that cp(xaa*x*) � kacp(xx*) for every x €: A, and in this case Ua can be uniquely extended to a bounded operator on the Hilbert space Hep which is the completion of AII with respect to the cp norm. If Ua is bounded for every a, then cp is said to be unitary; the mapping a --7 Ua is then easily seen to be a *-representation of A (see [20]) . This is the only new concept needed in the commutative theory to formulate more abstract Bochner and Plancherel theo-

100

SOME SPECIAL BANACH ALGEBRAS

rems. The passage to the space A of complex-valued homo­ morphisms of A is now accomplished by noting that since the mapping a � Ua is a *-representation of A, the maximal ideal space �II' of the algebra of operators Ua is thus identifiable with a closed subset of A. The Bochner and Plancherel theorems are now confined to this subset �II" which consists only of self-ad­ joint ideals and therefore renders unnecessary the assumption, made above, that x*� = xl!- - on the whole of A. Even when A is a Banach algebra this last point is of some interest ; it is easy to prove by characteristic j uggling starting with the Schwarz in­ equality that a continuous functional is automatically unitary, and the hypothesis that x*1!- = x � - can therefore be dropped from both of our theorems. § 27. H *-AL G E B RA S

The preceding sections of this chapter will find their applica­ tions 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 involution satisfying (1) - (4) of § 26 and the crucial connecting property

(xy, z) = (y, x *z) .

It is also assum ed that I I x* I I = I I x I I and that x *x � 0 if x � O. 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. I t fol­ lows 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 multi­ plication on any of its minimal left ideals. We begin with some lemmas about idempotents and left ideals.

27A.

1.

Let x be a self-adjoint element of H whose norm as a left multiplication operator is 1 . Then the sequence x2n converges to a non-zero self-adjoint idempotent. Lemma

Proof. Let I I I y I I I be the operator norm of y : I I I y I I I = lub z I I yz 1 1 11 1 z I I . Since I I yz I I � l I y 1 1 1 1 z I I , we have I I I y I I I � I I y I I · By hypothesis x is a self-adjoint element such that I I I x I I I = 1 . Then I I I xn I I I = 1 (see and hence I I xn I I � 1 for all n. I f m > n and both are even, then

lIB)

(xm, xn) � I I I xm -n 1 1 1 (xn, xn) = (xn, xn) (xm, xm) � I I I xP 1 1 1 2 (xn +p, xn +p) = (xm, xn)

where 2p = m - n. Thus 1 � (xm, xm) � (xm, xn) � (xn, xn) � . . . � (x2, x2) and (xm, xn) has a limit I � 1 as m, n � 00 through even integers. Hence lim I I xm - Xn 1 1 2 = lim (xm - xn, xm - xn) = 0, as is seen on expanding, and xn converges to a self-adjoin t element e with I I e I I � 1 . Since x2n converges both to e and to e 2, i t follows that e is idempotent. Corollary.

idempotent.

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

If 0 � y E::: I, then y *y is a non-zero self-adjoin t ele­ men t of I and the x of the lemma can be taken as a sui table scalar multiple of y *y. Then e = lim x2n = lim x2nx2 = ex2 E::: I, q.e.d. 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 = e l + e2 of non­ zero idempotents which annihilate each other; e l e2 = e2 el = o. If e is self-adjoint, we shall require e l and e2 also to be self-adjoint, and then the annihilation condition is equivalent to orthogonality : o = (eb e ) = (e 1 2 , e 2 2 ) = (e l e2 , e l e 2 ) ele2 = o. Since then 2 2 2 2 e and since the norm of any non-zero = e e + II 11 II l 11 II 2 11 Proof.

27B.

{=}

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 left 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 = e l + e 2 is a reduction of e, then He l and He2 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 II is a proper sub-ideal of I and h is a self-adjoint idempotent in It , then e l = eh = ehe is a self-adjoint idempotent in II which com­ mutes with e, and e = e l + e2 (e2 = e - e l ) is a reduction of e, a contradiction. (Notice that e l � ° since he l = heh = h 2 = h � 0, and e l � e since He l c II � He.) Therefore, I is mini­ mal. We remark that every minimal left ideal is closed (since it is of the form He) . Corollary. H is 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, MJ.. contains a non-zero irre­ ducible self-adjoin t idempotent, and so MJ.. includes a minimal left ideal, a con tradiction. 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) = ° so that (x - xe) ..1 He.

27C. Theorem. Every minimal left ideal generates a minimal two-sided ideal. The minimal two-sided ideals are mutually or­ thogonal, and H is the direct sum of their closures. Proof. Let I = He be a minimal left ideal with generating self-adjoint idempotent e, and let N be the two-sided ideal gen­ erated by I. That is, N is the subspace generated by IH = HeH. We first observe that N* = N, for (HeH) * = HeH. Now sup-

SOME SPECIAL BANACH ALGEBRAS

103

pose that NI is an ideal included in N. Since NI l c NI n I and I is minimal, we have either that I c Nt, in which case NI = N, or else NI l = 0. In the latter case NI (IH) = NI N = 0 and NINI * c NI N = 0, so that NI = o. Thus N has no proper subideals and is a minimal ideal. I t follows that N2 = N, for N2 C N and N2 = NN* � o. Now let NI and N2 be distinct minimal ideals. Then NIN2 = N2 NI = 0, for NI N2 C NI n N2 and hence either NI = NI n N2 = N2 or NI N2 = o. I t follows that, if x, y E=: NI and z E=: N2 , then (xy, z) = (y, x *z) = 0, so that NI = NI 2 ..l N2 . Thus dis­ tinct 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 gen­ erating idempotent e, then eHe is isomorphic to the complex num­ ber field. Proof. This is a classical proof from the Wedderburn struc­ ture theory. If ° � x E=: He, then ° � 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 ele­ ment 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, ot course, that every element of eHe is of the form Ae.

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) { eaH} is a collection of mutually orthogonal minimal right ideals spanning N. Proof. The equivalence of (a) and (b) follows from the fact that Hea -L He{3 if and only if ea -L e{3, and 27B. Similarly for (a) and (c) . The existence of such a rnaximal collection is guar­ an teed by Zorn's lemma. 27E. Given { ea } and N as above, we choose a fixed e l and consider any other ea. Since HeaH generates N, e l HeaHe l =;6. 0 and hence e l Hea =;6. o. If e la is one of its non-zero elements, then elae la * is in e l He l and e la can be adjusted by a scalar multiple so that e lae la * = e l . Then e la * C (e I Hea) * = eaHe I , and e la *e la can be checked to be an idempotent in the field eaHea and hence equal to ea. We define ea l = e la * and ea{3 = ea l e l{3. The formulas

ea{3e-y8 = ea8 if {3 = I' = 0

( ea{3, e-y8) = 0

if {3

=;6. I'

unless a = I' and {3 = 0

(ea{3, ea(3) = (e b e l )

follow immediately from the definitions. Thus (ea{3, e-y8) = (ea l e l{3, e'Y l e I 8) = (e l{3e8 b e lae'Yl ) = 0 unless {3 = 0 and a = 1', in which case it equals (e b e l ) . The set eaHe{3 i s a one-dimensional space consisting of the scalar multiples of ea{3, for, given any x E:: H, eaxe{3a E:: eaHea :::} eaxe{3a = cea for some c :::} eaxe{3 = cea{3. We know that ye{3 is the projection of y on He{3, and eaY that of y on eaH, for any y E:: H. Therefore, if x E:: N we have x =

SOME SPECIAL BANACH ALGEBRAS

105

Li3 xei3 = La,i3 eaxe{3 = Lai3 Cai3ea{3, proving that the set of ele­ ments eai3 is an orthogonal basis for N, and that eaxei3 is the pro­ jection of x on eaHei3. The Fourier coefficient Ca{3 can be com­ puted in the usual way, Cai3 = (x, eai3)/ 1 I eai3 1 1 2 . In view of the displayed equations above, i t is clear that we have proved the following theorem :

Theorem. The algebra N is isomorphic to the algebra of all com­ plex matrices { ca{3 } such that La,i3 I Cai3 1 2 < 00, under the corre­ spondence x { ca{3 } where x = La ,{3 Ca{3ea{3 and Ca{3 = (x, eai3)/ 2 e . II l 11 The following theorem is implicit in the above analysis. �

Theorem. Two minimal left ideals of H are (operator) iso­ morphic if and only if they are included in the same minimal closed two-sided ideal N. Proof. Let II and 12 be the two ideals, with self-adjoint gen­

erating idempotents e l and e2 and suppose that I I and 12 are both included in N. If I I and 12 are not orthogonal, then the mapping x � xe l maps 12 into a non-zero subideal of Il ) and since I I and 12 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 l ) = (yx)e l . If I I and 12 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 � xe2 1 furnishes the required isomor­ phism. Conversely, if I I and 12 are included in orthogonal minimal two-sided ideals Nl and N2 , then they cannot be operator iso­ morphic since Nl (12 ) = 0, whereas Nl (II ) = I I . 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 = La ea is a self-adjoint idempotent such that, for every x E: N, x = La xea = xe and x = La eax = 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 = L ea. 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 caea for some constant Ca, and x = L caea. Then c{3efJ� = xefJ� = efJ�x = c�efJ�, so that CfJ = c� 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. I t is clear that (xY )a = Xa)' = Xa)'a, and it follows in particular that Xa is central if x is central : Xa)' = (XY)a = (yx)a = yXa. Therefore, given either Xa = 0 for every central element x or else Na has an iden ti ty ea. We thus have the following corollary. lX,

Corollary. If Ho is the subspace of H which is the direct sum of its finite dimensional minimal ideals, then every central element x is in Ho 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 i ts identity ea. The minimal ideals Na are the orthogonal com plemen ts of the maximal ideals Ma, and the homomorphism x � x(Ma) is given by x(Ma) = (x, ea) / 1 1 ea 1 1 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 HIMa and the complex number field is given by HIMa Na = { cea } , and cea c. The continuous function C{3 thus has the value 1 at MfJ and is otherwise zero, so that the space mz; of maximal ideals has the discrete topology. "-J

�

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 RjI of the reals modulo 1 and the Cartesian product Rn (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 poin t the measure 1 . In the case of an infinite Cartesian product of unit intervals (groups RjI) 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 giv­ ing rise to the measure. A general non-structural proof of the existence of an invariant measure is therefore of the utmost im­ portance. 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 ele­ mentary theorems on represen tations are discussed in § 3 1 and § 32, and the chapter closes with the theory of invariant measures on quotient spaces in § 33 . 107

108

THE HAAR INTEGRAL

§ 28.

TH E TO POLOGY O F L O CA L LY

COMPACT G R O U P S

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 , (Rjl) � men­ tioned 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 > o. 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 V is any neighborhood of e, then aV is a neigh­ borhood of a, and if U is any neighborhood of a, then a - I 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 poin t. The mapping x � xa is another homeomorphism taking e into a. The inverse mapping x � X -I is also a homeomorphism of G onto itself. 2) Every neighborhood U of e includes a symmetric neighbor­ hood W of e, that is, one such that W = W- I • In fact W = U n U- I has this property. Similarly if J is continuous and vanishes off W, then g(x) = J(x) + J(x- I ) is symmetric (g(x) =

109

THE HAAR INTEGRAL

g(x - I » and vanishes off W, while h ex) = f(x) + f(x - I ) is Her­ mitian symmetric (h(x) = h(x - I » . 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 continu­ ous functions xy - I and x - Iy, we can find V such that VV- I c U or such that V - I V c U. We notice that V- I V c U if and only if a V c U whenever e €: aVo 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) . 5) If A is any subset of G, then A = n v AV, the intersection being taken over all neighborhoods V of the identity e. For if . y � A and V is given, then y V- I is an open set containing y and so contains a point x � A. Thus y €: xV c AV, proving that A c AV for every such V, and hence that A c n v AV. Con­ versely, if y c nv AV, then y V- I intersects A for every V and y €: A. Thus n v AV = A. 6) If V is a symmetric compact neighborhood of e, then vn is compact for every n and the subgroup VOO = limn vn is thus u-compact (a countable union of compact sets) . It is clearly an open set, since y €: VOO implies y V c VOO V = VOO, and since VOO c VOO V = VOO it is also closed. The left cosets of VOO are all open-closed sets, so that G is a (perhaps uncountable) union of disjoint u-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 15D). -+

00

-

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

E

E

1 10

THE HAAR INTEGRAL

Proof. Choose a compact set C such that j E::: La (i.e., such that j = 0 off C) and a symmetric compact neighborhood U of

SF

the set W of points s such that I j(sx) the identity. By j(x) I < E for every x E::: UC is open, and it clearly contains the identity s = e. If s E::: U, then j(sx) and j(x) both vanish out­ side of UC. Therefore if s E::: V = W n U, then I j(sx) - j(x) I < E for every x, q.e.d. Gi ven a function j on G we define the functions j8 and jS by the equationsjs (x) = j(sx),jS(x) = j(xs - l ) . They can be thought of as left and right translates ofj. The choice of s in one defini­ tion and S - l in the other is necessary if we desire the associative laws js t = (j8) t , jat = (ja) t . (For other reasons js is often de­ fined by j8(X) = j(S - l X) .)

Ij j E::: L, 1 � P � 00 and I is an integral on L, then j8 and ja, as elements oj LP(I), are continuous junctions oj s. Corollary.

Proof. The case p = 00 is simply the uniform continuity of j, for we saw above that I l js - j 1 100 < E whenever s E::: V. Using

this same V and choosing C so thatj E::: La, let B be a bound for I (see 1 6C) on the compact set PC. Then I I ja - j l i p = 1(1 j8 - j I p) l /p � B i /p i l j8 - j 1 100 < B I /PE if s E::: V, proving the continuity of j8 in LP. Similarly for ja. 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 space Sill, 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 Sill. The first is that the natural mapping a of S onto Sill, in which each point of S maps into the equivalence class containing i t, be continuous. Thus a subset A of Sill should not be taken as an open set un­ less a-I (A) is open in S. The second requirement is that, if j is any continuous function on S which is constant on each equiv­ alence class of II, then the related function F on Sill defined h by F(a(x)) = j(x) s ould be continuous on Sill. In order to realize this second requirement the topology for Sill is made as strong (large) as possible subject to the first requirement : a sub­ set A of sill is defined to be open if and only if a - l eA) is an

THE HAAR INTEGRAL

111

open subset of S. Then the mapping a is continuous, by defini­ tion. Moreover, iff is continuous on S and constant on the sets of II, then a-l (P-l ( U)) = f-l ( U) is open whenever U is open. Therefore, P-l ( U) is open whenever U is open and the related function P is continuous. Clearly a subset B of Sill is closed if and only if a-I (B) is closed in S. In particular, if each equivalence class of II is a closed subset of S, then each poin t of SIII is a closed set in sill, and Sill is what is called a Tl -space. These considerations apply in particular to the quotient space of left cosets of a subgroup H in a topological group G. More­ over, such fiberings GIH 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 GIH is an open mapping, that is, a(A) is open if A is open.

Proof. Suppose that A is open. We have a-l (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 GIH, q.e.d. ')

1.

Corollary If G is locally compact, then GIH is locally compact, for any subgroup H.

Proof. If e is a compact subset of G, then a( e) is a compact

subset of GIH (since a is a continuous mapping-see 2H) . If x E:: interior (e), then the above theorem implies that a(x) E:: in­ terior (a(e) ) . Thus compact neighborhoods map into compact neighborhoods, proving the corollary.

If G is locally compact and B is a compact closed subset of GIH, then there exists a compact set A c G such that B Corollary 2.

= a(A) . Proof. Let e be a compact neighborhood of the identity in G and choose points X l ) , X n such that B C U I na(xie) = a( U l nXie) . Then a-I (B) n ( U l nXie) is a compact subset of G, and its image under a is B since it is simply the set of points in UlnXie 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 T1 -space. Proof. Let H be the smallest closed set containing the iden­ tity. We show that H is a subgroup. For if y E: H, then e E: Hy - l and so H C Hy - l (H being the smallest closed set contain­ ing e) . Thus if x, y E: H, then x E: Hy - l and xy E: H. Since H- 1 is closed, it follows similarly that H c H- l . Thus H is closed under multi.plication and taking inverses ; that is, H is a subgroup. Moreover, H is normal, for H C xHx - 1 for every x and conse­ quently x- 1 Hx c H for every x, so that H = xHx- 1 for every x. Since H is the smallest subgroup of G which is closed, G/H is the largest quotient group of G which has a Tl topology. Lemma. A topological group which is a Tl -space is a Hausdorff space. Proof. If x � y, let U be the open set which is the complement of the point xy - l and choose a symmetric neighborhood V of the identity so that V2 c U. Then V does not intersect Vxy - l and Vy and Vx are therefore disj oint open sets containing y and x respectively. Every continuous function on a topological group G is con­ stant on all the cosets of the minimal closed subgroup H. I t fol­ lows that the same is true for all Baire functions. Thus the con­ tinuous functions and Baire functions on G are, essentially, j ust those on G/H, so that from the point of view of in tegration 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 as­ sumed for all groups from now on, although many results (for example, those of the next three numbers) will not depend on it. § 29.

TH E HAAR I NTEGRAL

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

THE HAAR INTEGRAL

1 13

in the existence proof, here in the guise of choosing a point in a Cartesian product space, and then is demonstrated to be theo­ retically unnecessary when the chosen functional is found to be unique. There are proofs which avoid this difficulty by simul­ taneously 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 Ci and points Si, i = 1 , " ' , n, such that

2:1 Cig(SiX) . For example, if mf and m g are the maximum values off and g re­ spectively, then the Ci can all be taken equal to any number greater than mf/mg. The Haar covering function (f; g) is defined as the greatest lower bound of the set of all sums 2:1 Ci of coefficients of such linear combinations of translates of g. The number (f; g) is evidently a rough measure of the size of f relative to g, and the properties listed below, which depend di­ rectly upon its defin ition, show that if g is fixed it behaves some­ what like an invariant integral. f(x)

(1)

(2) (3)

(4) (5) (6)

;£

(fs ; g) = (f; g) . (f1 + f2 ; g) ;£ (f1 ; g) + (f2 ; g) . (cf; g) = c(f; g) where c > o. f1

(f1 ; g) ;£ (f2 ; g) . (f; h) ;£ (f; g) (g ; h) . (f; g) � mf/mg •

;£ f2

=>

In order to see (6) we choose x so thatf(x) = mh and then have mf ;£ 2: Cig(SiX) ;£ (2: ci) mg, so that (L Ci) � mf/mg• (5) fol­ lows from the remark that, if f(x) ;£ L Cig(SiX) and g(x) � L djh(tjx) , then f(x) ;£ Li,j cidjh (tjsix) , Thus (f; h) � glb L Cidj = (glb L ci) (glb Ldj) = (f; g) (g ; h) . The other proper­ ties are obvious. 29B. In order to make (f; g) a more accurate estimate of rela­ tive size, it is necessary to take g with smaller and smaller sup-

1 14

THE HAAR INTEGRAL

port, and then, in order that the result be an absolute estimate of the size of j, we must take a ratio. We accordingly fix jo E: L + (jo -:;e. 0) for once and for all, and define JqJ (j) as (j; cp)/(jo ; cp) . I t follows from (5) that (7)

so that JqJj is bounded above and below independently of cpo Also, it follows from (1), (2 ) and (3) that JqJ is left invariant, sub­ additive and homogeneous. We now show that, for small cp, JqJ is nearly additive.

Lemma. Given j1 and j2 in L + and E > 0, there exists a neigh­ borhood V oj the identity such that

(8)

jor all cp

E:

L +y.

Proof. Choose j/ E: L + such that j/ = 1 on the set where j1 + j2 > O. Let 0 and E / for the moment be arbitrary and set j = j1 + j2 + oj/, hi = jdj, i = 1, 2 . I t is understood that hi is defi n ed to be zero where j = 0 and it is clear that hi E: L +� Choose V such that I hi(x) - hi(y) I < E ' whenever x - 1y E: V (by 2 8B) . If cp E::: L +y and j(x) � L Cjcp(Sjx), then cp(Sjx) -:;e. 0 ==} I hi(x) - hi(sj - 1 ) I < E ', and ji(X) = j(x)hi(x) � Lj Cjcp(sjx)hi(x) � Lj CjCP(Sjx) [hi(sj - 1 ) + E /] (ji; cp) � Lj cAhi(sj - 1 ) + E /] (j1 ; cp) + (j2 ; cp) � L cAl + 2 E/]

Since L Cj approximates (j; cp), we have

JqJjl + JqJj2 � JqJj[l + 2 E /J � [JqJ (jl + j2 ) + oJqJ(j')] [l + 2 E'] . The lemma follows if 0 and E/ are initially chosen so that 2 E ' (j1 + j2 ; jo) + 0(1 + 2 E /) (j' ; jo) < E . 29C. We have seen above that JqJ becomes more nearly an in­ variant integral the smaller cp is taken, and it remains only to take some kind of generalized limit with respect to cp to get the desired invariant integral.

THE HAAR INTEGRAL

115

Theorem. There exists a non-trivial non-negative left invariant integral on L. Proof. For every non-zerof C L + let SI be the closed interval [1 /(/0 ; f), (f; fo)], and let S be the compact Hausdorff space III SI (the Cartesian product of the spaces SI) . For each non­ zero cp C L + the functional II{' is a point in S, itsf coordinate (its projection on Sf) being Il{'f. For each neighborhood V of the iden ti ty in G, let Cv be the closure in S of the set { II{': cp C Lv + } . The compact sets Cv have the finite intersection property since CV1 n . . . n CVn = C(Vl Vn) . Let I be any point in the inter­ section of all the Cv. That is, given any V and given fh " ' , fn and > 0, there exists cp C Lv + such that I I(fi) - II{'(fi) I < i = 1 , " ' , n. I t follows from this approximation and 29B that I is non-negative, left invariant, additive and homogeneous, and that l /(fo ; f) � . I(f) � (f; fo) . The extension of I from L + to L, obtained by defining I(f1 - f2 ) as J(f1 ) - I(f2 ) , is there­ (ore 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 (2BA, 6) and therefore the difficul­ ties which can plague non cr-finite measures do not arise here. For instance the proof that (L1) * = LOO as sketched in 15D is available. (See also 13E.) n . . .n

E

E,

29D. Theorem. The above integral is unique to within a multi­ plicative constant.

Proof. Let I and ] be two left invariant non-negative inte­ grals over L and let f C L +. We choose a compact set C such that f c La +, an open set U with compact closure such that C c U and a function f' C L + such that f' = 1 on U. Given we furthermore choose a symmetric neighborhood V of the identity such that I I fy - fz 1 100 < if y, z C V, and such that , (CV U VC) c U. The latter condition guarantees that f(xy) = f(xy)f'(x) and f(yx) = f(yx)f'(x) if y C V, and together with the former implies that I f(xy) - f(yx) I < Ef'(x) for all x. Let E,

E

. ,

1 16

THE HAAR INTEGRAL

h be any non-zero symmetric function in Lv +. The above con­ ditions, together with the Fubini theorem and the left invariance of I and J, imply that I(h) J(f) = IyJxh (y)f(x) = IyJxh (y)f(yx) J(h)I(f) = IyJxh (x)f(y) = IyJxh (y-Ix)f(y) = Jxlyh (x- Iy)f(y) = Jxlyh (y)f(xy) I I(h)J(f) - J(h)I(f) 1 � IyJx (h (y) 1 f(yx) -f(xy) I ) � elyJxh (y)f'(x) = eI(h) J(f')· Similarly, if g L' and h is symmetric and suitably restricted, then 1 I(h)J(g) - J(h)I(g) I � eI(h) J(g') , where g' is fixed, and has the same relation to g that f' has to j. Thus J(f) - J(g) e J(f ') + leg') I I I(f) I(g) I - I I(f) I(g) and, since e is arbitrary, the left member is zero and the ratios are equal. Thus if go is fixed and c = J(go) /I(go) , then J(f) = cl(f) for all f 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 L + and h* (x) = h (X - I ) , then J(h)I(f) = Jylxh (y)f(x) = Jylxh (y)f(xy) = Ix]yh (y)f(xy) IxJyh (x-Iy )j(y) = Jylxh* (y-Ix)f(y) = Jylx h* (x)f(y) = I(h*) J(f) , so that J(f) = cI(f) where c = ] (h) /I(h*). 29E. Theorem. G is compact if and only if J.L(G) < 00 (the constant functions are summabIe) . E:

�

E:

E:

=

Proof. If G is compact, then 1 E: L, and J.L(G) = 1(1 ) < 00. Conversely, if G is not compact and if V is 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

1 17

THE HAAR INTEGRAL

{Pn } of points in G such that Pn tL U �- l PiV. Now let U be a symmetric neighborhood such that U2 c V. Then the open sets pn U are all disjoint, for if m > n and Pn U n P m U � 0, then P m � pn U2 C Pn V, a contradiction. Since the common measure of the open sets Pn U is positive, it follows that the meas­ ure 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 p.(G) = 1

(II dp. = J(l ) = 1 ) . § 30.

TH E MODU LA R F U N CTI O N

30A. Left invariant Haar measure need not be also right in­ variant-in general I(ft) � I(f) . However, if t is fixed, I(ft) is a left invariant integral, I(f/) = I(ft 8 ) = I(ft) , and because of the uniqueness of Haar measure there exists a positive constant �(t) such that The function l1(t) is called the modular function of G; if l1(t) 1, s o that Haar measure i s both left and right invariant, G i s said to be unimodular. =

Lemma. If G is Abelian or compact, then G is unimodular.

Proof. The Abelian case is trivial. If G is compact and if f 1, then l1(s) = l1(s)I(f) = I(f8) = 1(1 ) = 1, so that the compact case is also practically trivial. We have already observed earlier that, if f � L, then I(ft) is a continuous function of t. Thus l1(t) is continuous. Also l1(st)I(f) = I(f8 t) = I((f8) t) = l1(t)I(f8) = l1(s) l1(t)I(f) . Thus l1(t) is a continuous homomorphism of G into the positive real num­ bers. 30B. As might be expected, Haar measure is not ordinarily in­ verse invariant, and the modular function again plays its adjust=

ing· role. We use the customary notation

ff(x) dx instead of

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

118

THE HAAR INTEGRAL

ff(X -I ) .::l (x - I ) dx = ff(x) dx.

Theorem.

Before starting the proof we define the functionf* by f*(x) = f(X -I) .::l (x - I ) . The complex conjugate i s taken s o that, a t the appropriate mo­ ment, the mapping f � f* will be seen to be an involution. The formulas follow directly from the defini tion. Proof. In proof of the theorem we first observe that the in­ tegral above, J(f) = I(f* -), is left invariant : J(fs) = I(fs * -) = .::l (s - I )I(f* -S) = I(f* -) = J(f) . Therefore J(f) = cI(f) . But now, given E, we can choose a symmetric neighborhood V of e on which I 1 - .::l (s) I < and then choose a symmetric f � Lv + such that I(f) = 1 , giving E

1 1 - c I = I (1 - c)I(f) I = I I(f) - J(f) I = I 1((1 - .::l - I )f) I < EI(f) = E. Since E is arbitrary, c = 1 , q.e.d.

1.

Corollary f* � LI if and only if f � Ll, and I I f* 1 1 1 = i . This follows from the theorem and the fact that L is f II ll l dense in L . Corollary 2.

G is unimodular.

Haar measure is inverse invariant 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 (:: LP (1 � P < (0 ) , then gs and gS, as elements of LP, are continuous functions of s. Proof. Given we choose f � L so that I I g - f l i p = I I gs - fs l lp < E/3 and by 28B we choose V so that I I f - fs l l p < E/3 if s (:: V. Then E,

I I g - gs l lp � I I g - f l ip + I I f - fs l lp + I I fs - gs l l p < E if s � V, showing that gs is a continuous function of s at s = e.

THE HAAR INTEGRAL

1 19

Since I I fS - gS l l p = A (s) l /p l l f - g l ip, the above inequality re­ mains valid when gs and fs are replaced by gS and fs if the choice of V is now modified so that I I f - f8 1 1 p < E/6 and A(S) l /P < 2 when s E=: V. The equations I I fsx - fx l ip = I I fs - f l ip and 1 / fSX - fX l ip = A(x) l /p l l fS - f I / p show that both functions are continuous everywhere, and, in fact, that fs is left uniformly con tin uous. 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 auto­ morphism of H (that is, G is mapped homomorphic ally into the automorphism group of H) , the result of applying u to an ele­ men t 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 (U l ' X l )(U2, X 2 ) = ( UI U2, U2 (X l ) X2 ) ' Furthermore, if u(x) is continuous in the two variables u and x simultaneously, then this group is locally compact in the ordi­ nary Cartesian product topology. Such a group is called a semi­ direct product of H by G. I t contains H as a normal subgroup and each coset of the quotient group contains exactly one ele­ ment of G. If .d and B are compact subsets of G and H respectively, then (uo, x o>CA X B) is the set of all pairs (uou, u(xo) x) such that u E=: .d and x E=: B . If we compute the Cartesian product meas­ ure of this set by the Fubini theorem, integrating first with re­ spect to x, we get J.l(A) v(B), where J.l and v 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 invari­ an t Haar measure for the semi-direct product group. To compute the modular function A we consider (A X B) (uo, xo) = (Auo) X (uo (B)xo), whose product measure is J.l(A)v(uo(B)) . If the automorphism Uo multiplies the Haar measure in H by a factor o(uo), then the above measure is o(uo)J.l(A)v(B) . The modu­ lar function A( u, x») therefore has the value o(u), and the semi­ direct product is unimodular only if 0 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 posi­ tive reals, with 0" (x) = O" · X. Thus (a, x) (b, y) = (ab, bx + y) and d«a, x») = a. This group is, therefore, not unimodular. 30E. It is clear that, if G and H are not unimodular, but have modular functions .1 1 and .1 2 respectively, then the above calcu­ lation will give the modular function .1 in terms of .1 1 and .1 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) (0"0, xo) = (AO"o) X (O"o(B)xo) is 0(0"0) .1 1 ( 0"0) .1 2 (xo) [1L(A) v (B)] so that .1( 0", x») = 0( 0") d 1 ( 0") d2 (X) . If in this situation we take H as the additive group of the real numbers (so that .1 2 1 ) and O"(x) = X[d 1 ( 0")] - I , then 0(0") = .1 1 ( 0") - 1 and .1( 0", x») 1 . Thus any locally compact group G can be enlarged slightly to a unimodular group which is a semi­ direct product of the reals by G. This was pointed out by Glea­ son [1 8]. ==

==

§ 31.

TH E G R O U P A LG E B RA

In this section we prove the existence of the convolution opera­ tion and show that with it as multiplication L l (G) becomes a Banach algebra. 31C-G then establish some of the special prop­ erties of this algebra which follow more or less immediately from its definition. 3 1A. We define the convolution of f with g, denoted f * g, by

f

f

[f * g] (x) = f(xy)g(y - l ) dy = f(y)g(y - l X) 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�f(x - y)g(y) dy, is obviously a special case of this

definition. It also agrees with and generalizes the classical no­ tion used in the theory of finite groups. A finite group G is com­ pact 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 complex­ valued functions on G, then f * g(x) =

If(xy)g(y -l ) dy

=

L yf(xy)g(y - l )

=

L u v=xf( 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 � LA and g � LB, then f * g � LAB. Proof. The integrand f(y)g(y - l x) is continuous as a function of y for every x and is zero unless y � A and y - l x � B. Thus f * g(x) is defined for every x and is zero unless x � AB. Also I f * g(X l ) - f * g(X2 ) I � I I fX l - fX2 1 100 · I g(y - 1 ) I dy and since fx is by 28B continuous in the uniform norm as a function of x it follows that f * g is continuous. Theorem. If f, g � CB +, then f(y)g(y - l x) � CB +(G X G) and " f * g l i p � I I f 1 1 1 · 1 1 g l i p, where 1 � P � 00. . Proof. Let f 0 g be the function f(y)g(y - l x) . Iff � L +, then the family of functions g � CB+ such that f o g � 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 � CB + such that f o g � CB + includes L + and is L­ monotone, and therefore equals CB +. In particular f o g � 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 that f(y)g(y - l x) is integrable in y for every x, that (f * g) (x) = f(y)g(y - l x) dy is

I

II

I

integrable, and that I I f * g 1 1 1 = f(y)g(y - l x) dx dy = I I f l l l · 1 I g i l l . If h � CB + then alsof(y)g(y - l x)h(x) � CB +(G X G) (see 13C) and, as above, (f * g, h)

=

Jj(y) [fg(Y -'X)h(X) dX ] dy � I I j 1 1 , 1 1 g 1 1. 1 1 h I I .

by the general Holder inequality 14C.

122

THE HAAR INTEGRAL

f g � f 11 g l ip, and, in The case f * g LP f * L Il ipand I I g 11 1LP. Corollary. f L I and g LP, thenf( g(y-Ix is summable in y for almostIallfx, (f * g) (x) LP and I I fy)* g l ip � )I I f 1 1 1 1 1 g l ip· Proof. The sixteen convolutions obtained by separating each of f and g into its four non-negative parts belong to LP by the above theorem, and their sum, which is equal to If(y)g(y- I x) 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, I I f * g l ip � I I I f I * I g I l i p � I I f 1 1 1 1 1 g l ip, by the above theorem. 3 IB. Theorem. Under convolution as multiplication L I (G) forms a Banach algebra having a natural continuous involution f � f* · Proof. We have seen earlier that the mapping f � f* of LI I t follows, using ISC, that I I particular, that E: if p = 00 is obvious. E:

E:

E:

E: E:

onto itself is norm preserving, and we now complete the proof that it is an involution. It is clearly additive and conjugate linear. Also

(f * g) *(x) If(x-Iy) g(y-I) Ll(x -l) dy Ig(y- I) Ll(y-l)f((y -Ix) - 1) Ll((y-Ix) -1) dy (g* *f*) (x) , so that (f * g) * g* */ *. The associativity of convolution depends on the left invariance +, we have of the Haar integral. If/, g, h ((/ * g) * h) (x) I[ (/ * g) (xy)]h (y-l) dy IIf(xYZ)g(Z- I )h(y-l) dy dz IIf(XZ)g(Z -ly)h (y-l) dy dz I/(xz)[ (g * h)(Z- I)] dz (/ * (g * h» (x) . =

=

=

=

E:

CB

=

=

=

=

=

THE HAAR INTEGRAL

123

The associative 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 L I is closed under convolution and I I f * g I I I � 1 / f i l l · 1 1 g 1 1 1 by 3 1A, it follows that L I 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 L I (G) . 3 1C. Theorem. L I (G) is commutative if and only if G is com­ mutative. Proof. If G is commutative, then G is also unimodular and

f

f * g = f(y)g(y - I x) dy

=

fg(xy)f(y - l ) dy = g * f·

Now suppose that L I (G) is commutative and that f, g e L. Then o = f * g - g * f = [f(xy)g(y - l ) - g(y)f(y - I x)] dy

f

f

= g(y) [f(xy - I ) Ll(y - l ) - f(y - I X)] dy . Since this holds for every g in L, it follows that f(xu) Ll (u) - f(ux) O. Taking x = e we have Ll(u) 1, so that f(xu) - f(ux) = 0 for every f in L. Since L separates points, it follows that xu - ux = 0 and G is commutative. 3 1D. Theorem. L I (G) has an identity if and only if G is dis­ crete. Proof. If G is discrete, the points of G are congruent open sets having equal positive Haar measures which may be taken as 1 . Then f(x) is summable i f and only i ff(x) = 0 except on a count­ able set { xn } and L I f(xn) I < 00, The function e(x) which is 1 at x = e and zero elsewhere is an identity : =

=

f * e(x)

=

ff(y)e(y - Ix) dy

=

L y f(y)e(y - I x) = 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 there exists an open neighborhood V of the identity e whose measure is less than E

E,

!v

and hence one such that I u (x) I dx < Choose a symmetric U so that U2 c V and let f be its characteristic function. Then E.

f

!vI

Lu

uI < u(y) dy � f(x) = (u * f) (x) = u(y)f(y - 1 x) dy = for almost all x in U, contradicting (x) 1 in U. Therefore, there is a number a > ° 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 � na for every n by choosing n disjoint non-void open subsets. Therefore, every point is an open set, and the topology is discrete. 3 1E. In any case L l (G) has an " approximate identity."

f

==

E

Theorem. Given f C LP (1 � p < 00 ) and > 0, there exists a neighborhood V of the identity e such that I I * u - f l ip < and I I u * f - f l ip < whenever u is any function of L l + such that u = ° outside of V and u = 1 . E

If h imply that Proof.

C

f

f

E

E

L q the Fubini theorem and Holder inequality

f, h) I = I ffu (y) (f(y - 1x) - f(x))h(x) dy dx I I h I I q fllfy-l - f I I p u (y) dy. Thus I I u * f - f l ip f l l fy-l - f I I pu (y) dy. If V is chosen (by 30C) so that I I fy-l - f l ip < whenever y V and if u (L l ) V +, then I I u * f - f l i p < Ef The proof for f * u is similar but somewhat complicated by fu (x - 1 ) dx, then modularity . If I (u * f -

I

�

�

C

E

U =

m =

E.

C

125

THE HAAR INTEGRAL

ff(f(xy - f(x)/m)u (y - l)h(x) dy dx I � I I h I I q I I I mjy-l - f I Ip (u(y - l ) /m) dy and I l f * u - f l ip � I I I mfy -1 - f I I p u (y - l ) /m dy. Now m � 1 as V decreases. (For m = I[u(x - 1 ) A(X - 1 ) ] A(X) dx, where fu (x - 1 ) A(x - 1 ) dx = Iu (x) = 1 , A(X) is continuous and A(e) I (f * u - f, h) I = I

= 1 .) Thus there exists V such that I I mp -l - f l ip < E if y - l � V and u � (L l )V +, giving I I f * u j l ip < E [u (y - l ) /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

I

Iuv = 1, then uv * f and f * uv converge

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

3 1F. Theorem. A closed subset of L 1 is a left (right) ideal if and only if it is a left (right) invariant subspace.

Let 1 be a closed left ideal and let u run through an approximate identity. If f � 1, then U x *f � 1. But U x *f = (u *f) x � fx, since u *f � f, and thereforefx � 1. Thus every closed left ideal is a left invariant subspace. Now let 1 be a closed left invariant subspace of L l . If 1.L is the set of all g � LOC! such that (f, g) = 0 for every f � 1, then we know (see Be) that 1 = (1.L) \ i.e., that, iff � L l , then f � 1 if and only if (f, g) = 0 for every g � /.L. But if h � L l , j � 1 and g � 1\ then (h *f, g) = h(Y )f(y - l x)g(x) dy dx = h (Y) Proof.

ff

I[j(y - lx)g(x) dx] dy = 0 (sincefy

.1

I

g for every y), which proves

by the above remark that h *f � 1. That is, 1 is a left ideal. The same proof, wi th only trivial modifi c ations, works for right ideals.

126

THE HAAR INTEGRAL

3IG. We conclude this section with some remarks about posi­ tive 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 ab­ stract Banach algebra having a continuous involution. The al­ gebra which we have before us now is L 1 (G) , and we start with the ideal LO consisting of the uniformly continuous functions of L 1 , and the function cp on L defined by cp(f) = f(e) . Then cp(f *f*) = f(x)f(x) dx = I I f 1 1 2 2 � O. Thus cp is positive and °

f

H", is L2 (G) . Since I I g *f 1 1 2 � I I g l l l · 1 1 f 1 1 2 , the operators Ug defined by UJ = g * f are bounded and the mapping g � Ug is a *-representation of L 1 (G) . I t is called the left regular repre­ sentation and will be discussed further in 32D. The functional cp is thus unitary, but is clearly not continuous (in the L 1 norm) . A t the other extreme are the posi ti ve functionals which are con tinuous in the L 1 norm.

Lemma. A positive functional on L 1 (G) is continuous if and only if it is extendable.

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

I P(f) 1 2 = lim I P(f * u) 1 2 � P(f *f*) lim P(u * * u) � I ! P I I P(f * f*),

P(f*) = lim P(f* * u)

=

lim P(u * * f) = P(f) ,

proving the lemma. Now every continuous functional P on L 1 is given by a func­ tion p � Loo , and, if P is positive, p is called positive definite. Notice that then p = p *, for (f, p) = P(f) = P(f*) = (f*, p) = (p, j* ) = (f, p *) for every f � L 1 . This notion of posi tive defi­ niteness is formally the same as that introduced in 26J, the fixed unitary functional cp being that considered in the first paragraph above : P(f) = Cf, p) = (f, p*) = (p,f*) = [p * f] (e) = cp(p * f) ·

127

THE HAAR INTEGRAL

The condition that a function p E: LOO be positive definite can be written f(x)f(y)p (xy - l ) dx dy � 0 for every f E: LI.

II

§ 32.

R E P R E S E NTATI O N S

32A. A representation T of G is a strongly continuous homo­ morphism of G onto a group of linear transformations on a com­ plex vector space X. That is, if Ts is the transformation asso­ ciated with the group element s, then Ts t = Ts Tt and Ts(x) is a continuous function of s for every x E: X. I f X is finite dimen­ sional, this requiremen t of continuity can be expressed as the continuity of the coefficients in the matrix for Ts when a fixed basis has been chosen for X. When X is infinite dimensional some kind of topology for X must be specifi e d. 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 I I Ts I I , s E: G. T is a unitary representation if X is a Hilbert space and the transforma­ tions Ts are all uni tary.

32B. If T is a bounded representation of G on a reflexive Banach space X and if T, is defined as f(x) Tx dx for every f E: L I , then

I

the mapping f � T, is a bounded representation of L I (G) . If T is unitary, then the integrated representation is a *-representation of L I (G) . Proof.

The function

I

I

F(f, x, y) = f(s) ( Tsx, y) ds = f(s)y ( Tsx) ds is trilinear and satisfies I F(f, x, y) I � I I f I I I B I i x l i l l y I I . I t follows exactly as i n 26F that there i s a uniquely determined

I

linear transformation T, such that ( T,x, y) = f(s) ( Tsx, y) ds, I I T, I I � B I I f I I h and the mapping f � T, is linear. More­ over,

128

THE HAAR INTEGRAL

IIf(t)g(t- I s) ( Tsx, y) dt ds = IIf(t)g(s) ( Ttsx, y) ds dt = Ig ( S ) ( Tf Ts X, y) ds dt = Ig(s) ( Tsx, ( Tj) *y) ds ( Tj TgX, y). = I ( TgX, ( Tf) *y)

( Tf * X, y) = g

=

Thus Tf * = TfTg and the mapping f � Tj is a homomorphism. I f X is a Hilbert space and T is a unitary representation, then g

f

( Tf *X, y) = f(S -' ) d (s - l ) ( T,x, y) ds = = ( Tjy, x)

=

(x, Tjy) .

(ff(s) ( T.y, x) dsr

Thus ( Tj) * = Tf* and T is a *-representation. Remark: No non-zero element of X is annihilated by every Tj. For, given 0 � x €: X and y €: X* such that (x, y) � 0 we can find a neighborhood V of the identity in G such that ( Tsx, y) differs only slightly from (x, y) if s €: V. I ff is the characteristic function of V, then ( Tjx, y) =

L ( Tsx, y) ds differs only slightly

from J.l(V) (x, y) , and hence Tfx is not zero.

32C. Theorem. Conversely, a bounded representation T of I L (G) over a Banach space X arises in the above way if the union of the ranges of the operators Tj, f €: L I , is dense in X. If X is a Hilbert space and T is a *-representation, then T is unitary. Proof. Let Xr be the union of the ranges of all the operators TiCf €: L I ) ; Xr is dense in X by hypothesis. Let u run through an approximate identity of L I . We know that U a * f = (u * f) a � fa, and therefore I I TuaTj - Tfa I I � O. Thus Tua converges strongly (that is, pointwise) on Xr to an operator Ua satisfying UaTj = Tfa, and since I I Tua I I � B I I U a 1 1 1 = B for every u, it fol­ lows that I I Ua I I � B and that Ua is uniquely defined on the whole of X (see 7F) . Since UabTf = lfab = T(fa)b = Ub Tfa = Ub UaTj (i.e., Uab = Ub Ua on Xr) , we have Uab = Ub Ua and

129

THE HAAR INTEGRAL

the mapping a � Ua is therefore an anti-homomorphism. Fi­ nally, fa is a continuous function of a as an element of LI, and therefore Ua T, = Tla is a continuous function of a. Thus Ua is strongly continuous on Xr, and hence on X. U is an anti-homo­ morphism, and we set Ta = Ua-l to get a direct homomorphism.

f

I t remains to be shown that ( T,x, y) = f(s) ( Tsx, y) ds for every x � X, Y � X* and f � L l . We may clearly restrict x to any dense subset of X and f to L. Thus, it is sufficient to

f

show that ( T, gZ, y) = f(s) ( Ts Tgz, y) ds for all f, g � L. Now if x and y are fixed, the linear functional J(f) = ( T,x, y) satisfies I J(f) I � El l x 1 1 1 1 y 1 1 1 1 f I I and is therefore a complex-valued integral (the sum of its four variations) . We therefore have from the Fubini theorem that *

I,

( T, * gX, v) = It =

[Jf(s)g(s -'t) ds 1 = Jf(s) U,g(s -'t) ) ds

ff(s) ( Ts TgX, y) ds

as desired. If X is a Hilbert space and T is a *-representation, we want to show that Ts is unitary for every s, i.e., that (Ts) * = ( Ts) - l = Ts -l. But ( Ts) * = Ts-l <==} ( Ts) *T, = Ts-IT, = TIs for allf � L l <==} ( U s-I) * *f � fs as U runs through an approximate identity <==} f* * U s- l � fs * g * Us-l � g*s *. Now by direct computa­ tion g*s * = �(S - l )gS and g * Us -l = �(S - l )gs * u. Since we know that gS * U � gS, the result follows. 32D. If we take X as the Banach space LP(G) for some fixed p . in [1, 00] and define the operator T, on X by T,g = f * g for every f � L l , g � LP, then we can check directly that the map­ ping f � T, is a bounded representation of the algebra L l (G) . For instance, T, Tgh = f * (g * h) = (f * g) * h = TI gh, giving T, g = T,Tg. Moreover the inequali ty I I f * g l i p � I I f 1 1 1 1 1 g l i p implies that I I T, I I � I I f I I so that the represen tation has the bound 1 . There is also an obvious group representation s � Ts defined by TsJ = fs-1, or [ TsJ] (t) = f(S - l t) . The transformations Ts are all isometries by the left invariance of Haar measure. <==}

*

*

I,

130

THE HAAR INTEGRAL

This pair of representations is interconnected in exactly the same way as in the above two theorems. Thus if g (:: LP and h (:: L q ( l ip + l lq = 1 ) , then

If(s) ( Tsg, h) ds IIf(S)g(S - lt)h (t) ds dt =

If p

=

(f * g, h) .

2, the transformations Ts are unitary by the left in­ variance of Haar measure : ( Tsf, Tag) = f(S - l t)g(S - lt) dt = =

If(t)g(t) dt

I

(f, g) . I t follows from our general theory that the representation of L l is a *-representation. Of course, this fact can also be checked directly-the equality (f * g, h) = (g, f* * h) can be computed directly from the Fubini theorem and the left invariance of the Haar integral. These representa­ tions are called the (left) regular representations of G and L l (G) respectively. They are both faithful representations (one-to-one mappings) , for if s � e we can choose g tOI vanish outside a small enough neighborhood of e so that ggs = b, giving in particular Tsg � g in L2 (G), and iff � 0 then we cail choose u (:: L l n L 2 similarly so that I l f * u f 1 1 2 < El l f 1 1 2 , giving in particular that T,u = f * u � 0 in L 2 (G) . =

-

§ 33 .

QU OTI ENT MEAS U R E S

In this section we consider invariant measure on quotient groups, and, more generally, quotient spaces, and prove the Fu­ bini-like theorem relating such quotient measures to the invari­ an t 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 Wei1 [48] . 33A. 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 J(f) is the integral over H of the restric­ tion off to H. Iff (:: L(G), it follows from the uniform continuity of f that the function f' defined by f'(x) = J(fx) = Jtf(xt) is continuous. Also f'(xs) = f'(x) for every s (:: H by the left in­ variance of J on H, so that f' is constant on the left' cosets of H.

THE HAAR INTEGRAL

131

Iff E:: Lc, then the related function F on G/H defined by F(a(x» = f' e x) = I(fx ) = Itf(xt) belongs to La(c) . The value F(a(x» = f' ex) can also be obtained as the integral of f over the coset con taining x wi th respect to the Haar measure of H trans plan ted to this coset. If H is a closed normal subgroup, then G/H is a group and has a left invariant Haar integral K. KxItf(xt) is understood to be K(F) where F(a(x» = Itf(xt) . This is equivalent to transferring the domain of K to those continuous functions on G which are cons tan t on the cosets of H and each of which vanishes off some set of the form CH, where C is compact. If g is such a function, then so is gy and the left invariance of K becomes simply K(g) = K(gy) for all Y E:: G. Then KxIt/(xt) is a left invariant integral on L (G) , KxIt/y (xt) = Kx/u' (x) = Kx/'(x) = KxItj(xt), and KxIt/(xt) = kI(/) . We are thus led to the following theorem. Theorem. Ij j E:: (B +(G) , then jx E:: (B +(H) for every x and I(/x) = Itj(xt) E:: (B +(K) . If I, I and K are suitably normalized, then KxIt/(xt) = I(j) for every f E:: (B +(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 { jn } of L-bounded functions of (B+ which converges monotonically to j, then 1(/) = lim I(/n) = lim KxIt/n (xt) = Kx (lim Itjn (xt) = KxIt(lim fn (xt) = KxIff(xt) . Thus the family is L-monotone and hence equal to (B +(G), q.e.d. Corollary. Ij j E:: L I (G) , then /x E:: L I (H) jar almost all x, Itj(xt) E:: L I (K) and KxItj(xt) = I(j) . 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. If F E:: L +(G/H), then there. exists f E:: L +(G) such I(/x) . that F(a(x» Proof. Let B be a compact subset of G/H such that F E:: LB +, let A be a compact subset of G such that a(A) = B and choose h E:: L +(G) such that h > 0 on A. Notice that I(h x) > 0 if the coset containing x intersects A, i.e., if x E:: AH = a- I (B), and =

132

THE HAAR INTEGRAL

that F(a(x)) = 0 in (AH) ' = a - 1 (B'), an open set. Therefore, g(x) = F(a(x)) /I(h x) if I(hx) > 0 and g(x) = 0 if I(h x) = 0 de­ fines an everywhere continuous function. Also g(x) is constant on the cosets of H. Therefore f = gh � L +(G) and Itf(xt) = g(x)lth(xt) = F(a(x)) , q.e.d. 33C. If H is not normal, it is still true that every element y � 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 homeo­ morphism induced by y : K(Fy) = D(y) K(F), where, of course, Fy (a(x)) = F(a(yx)) . The function D(y) is called the modulus, or the modularfunction, of K, and, like �, is seen to be continuous and multiplicative. Theorem. If K is relatively invariant, with modulus D, then the functional M(f) = KxItD(xt)f(xt) is (essentially) the Haar in­ tegral I(f) , and all the conclusions of 3 3Afollow. Proof. M(fy) = Kxl tD(xt)f(yxt) = D(y - l ) Kx [ltD(xt)f(xt)] y = KxItD(xt)f(xt) = M(f) . Thus M(f) is left invariant on L(G) and equals I(f) . The extensions to ill + and to L l are the same as in 33A. 33D. Any continuous homomorphism of G into the multipli­ cative group of the positive real numbers will be called a real character. Examples are the modular functions � and D. Theorem. In order that a real character D be the modularfunc­ tion for a relatively invariant measure K on the quotient space G/H, where H is a closed subgroup of G, with modular function 0, it is necessary and sufficient that D(s) = �(s)/o(s) for all s � H. Proof. If D is the modular function of K, then M(f) = I(f) as above and we have, for s � H, �(s)M(f) = M(fS) = KxItD(xt)f(xts - 1 ) = D(s) KxIt D(xts - 1 )f(xts - 1 )

D(s) o (s) KxltD(xt)f(xt) D(s) o (s) for every s � H. =

Thus � (s)

=

=

D(s) o (s)M(f) .

133

THE HAAR INTEGRAL

Now suppose, conversely, that D(x) is a real character such that Ll (s) = D(s) o(s) for every s � H, and define the functional K on L(G/H) by Kz(](lz)) = l(D-V) . I t follows from 33B tha t this functional is defined on the whole of L( G/H) once we have shown that the definition is unique, i.e., that, if j(fz) = 0, then l(D -V) = O. But if ]tf(xt) = 0 for all x and if g � L +( G) , then o = lz]tg(x)D - l (X)f(xt) = ]dz O (t- l )g(xt- 1 )D - 1 (x)f(x) =

Iz { D - l (X)f(x)]t[o(t- 1 )g(xt- 1 )] }

=

lzD- l (X)f(x)]tg(xt) .

In order to prove that l(D -V) = 0 it is therefore sufficient to choose g � L +(G) so that ](gz) = ]tg(xt) = 1 for every x for which f(x) � O. But if f � LA and if a function is chosen from L + (G/H) which has the value 1 on the compact set a (A) , then the required function g is constructed by the lemma of 33B. I t remains to check that the functional K thus defined on L + (G/H) is relatively invariant with D as its modular function :

Kz](fyz)

D(y)l((D -V) y)

=

l(D -Vy)

=

D(y)l(D -V) = D(y) Kz](fz) .

=

This completes the proof o f 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) = o(s) for all s � H. Taking H to be the nor­ mal subgroup on which Ll(t) = 1 , it follows that 0 1 and H is unimodular. The real character Ll is itself a homomorphism hav­ ing this group as kernel, and the quotient group is therefore iso­ morphic to a multiplicative subgroup of the posi tive real num­ bers. In a sense, therefore, every locally compact group is al­ most unimodular. Another sense in which this is true has been discussed in 30E. ==

==

Chapter J7I1 LOCALLY COMPACT ABELIAN GROUPS

The L l 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 char­ acter function (the kernel function of the Fourier transform), and much of our discussion will center around topological ques­ tions. 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, Planch­ erel, and Stone theorems. A more miscellaneous set of theorems is gathered together in § 37, such as the Pontriagin duality theo­ rem, 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.

TH E

CHARACTER G RO U P

We know that each regular maximal ideal M in L l is the kernel of a homomorphism f � j(M) of L l onto the complex numbers (see 23A, B) and that this homomorphism, as an element of the conjugate space (Ll) * of Lt, is uniquely represented by a func1 34

LOCALLY COMPACT ABELIAN GROUPS

135

f

tion aM � LrJJ, }(M) j(x)aM (x) dx (see lSC, D) . We shall see below that aM is (equivalent to) a continuous function, 1 and aM (xy) aM (x)aM (y) . Thus aM is a continu­ I aM (x) I ous homomorphism of G into the multiplicative group of com­ plex 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 � of regular maximal ideals of L 1 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 LrJJ = (Ll) *, is locally compact. And it follows rather easily that G is a locally compact group, multiplication being the pointwise mul­ tiplication of functions. The domain of the function } will be shifted from � to G through the above identification, and, as a function on G, } is now the classical Fourier transform of j. We proceed to fill in the details. 34A. Theorem. Given a fixed M � �, let j � L 1 be such that }(M) � O. Then the junction aM defined by aM (x) = }x(M)/}(M) is a character. It is independent oj j, uniformly continuous, and continuous in the two variables x and M if the parameter M is also varied. Ij u runs through an approximate identity, then u x CM) converges uniformly to aM (x) . =

==

=

Proof. That aM is uniformly continuous follows from the in­ equality I }x(M) - }y (M) I � I I jx - jy 1 1 1 and the fact thatjx as an element of Ll is a uniformly continuous function of x (30e) . The inequality I }x (M) - }xo (Mo) I � I I jx - jxo 1 1 1 + I }xo (M) Jxo (Mo) I shows that}x(M) is continuous in the two variables x and M together, and the same holds for aM (x) upon dividing by J(M) . The remaining properties of aM are contained in the equation

J(M)gx (M)

=

Jx (M)g(M)

(which comes from j * gx = jx * g) . Thus }x/J = gx/g, proving that a is independent ofj. Taking g = jy and dividing by (}) 2, we see that aM (Yx) = aM (y)aM (x) . Thus aM is a continuous ho­ momorphism of G into the non-zero complex numbers. If I aM (x O) I > 1 , then I aM(X O n ) I I aM (xO) I n --+ 00. But aM is =

136

LOCALLY COMPACT ABELIAN GROUPS

bounded ; therefore I I aM I I � 1 . Then I aM - l eX) I = I aM (x - l ) I 1 . Thus aM is a character. � 1 , so that I aM (x) I Finally, taking g = u, we get u x (M) = kaM (x), where k = u (M), and since u (M) � 1 as U runs through an approximate identity (I I uJ - J I I <Xl � I I U * f - f 1 1 1 � 0) it follows that u x (M) converges uniformly to aM(x) . <Xl

==

34B. Theorem. The mapping M � aM is a one-to-one map­ ping of � onto the set of all characters of G, and =

J(M)

If(x)aM (X) dx.

Proof. We first derive the formula. Given M, the homomor­ phism f � J(M) is in particular a linear functional on LI, and therefore there exists a E:: L<Xl such that J(M) = f(x)a(x) dx.

Then

If(x)aM(X) dx

I

= = = =

If(X) Ux-l(M) dx limu IIf(x) u (x- Iy)a(y) dx dy limu I (f * u) (y)a(y) dy If(y)a(y) dy J(M), limu

=

q.e.d. In particular, 11.;1 is determined by the character aM, so that the mapping M � aM is one-to-one. If a is any character, then the linear functional f(x)a(x) dx is multiplicative :

I

I(f * g) (x)a(x) dx IIf(xy)g(y - I )a(xy)a(y - I ) dx dy IIf(x)a(x)g(y) a(y) dx dy If(x)a(x) dxIg(y)a(y) dy e =

=

=

137

LOCALLY COMPACT ABELIAN GROUPS

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

If(x) a (x) dx If(x)aM (x) dx shows that a =

=

aM·

This theorem permits us to identify the set G of all the charac­ ters on G with the space mr of regular maximal ideals of L 1 (G) (which has already been identified with, and used interchangeably with, the subset of (L 1 ) * consisting of non-trivial homomor­ phisms) . 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 trans­ form formula is now written :

lea)

=

If(x) (x, a) dx

.

34C. The topology of G is that of uniform convergence on compact sets of G. That is, if C is a compact subset of G, E > 0 and aD €: G, then the set of characters U(C, E, aD) { a: (x, a) - (x, aD) I < E for all x €: C} is open, and the family of all such open sets is a basis for the topology of G. =

I

That U(C, E, aD) is open follows at once from SF. Moreover the intersection of two such neighborhoods of aD in­ cludes a third : U(C1 U C2 , min (E l ) E2 ), aD) C U(Cl ) E l ) aD) n U(C2 , E2 , aD) . I t is therefore sufficient to show that any neigh­ borhood N of aD belonging to the usual sub-basis for the weak topology includes one of the above kind. Now N is of the form N { a : J(a) J(ao) I < a } for somef €: L 1 and a > o. If C Proof.

=

I

-

I

I

is chosen so that r ' f I < 0/4, E = 0/2 1 / f 1 / 1 and (x, a) - (x, Je

aD)

I < E on C, then I J(a) - J(ao) I � II f(a - aD) I � fa +fa,

� 1 / f 1 / 1 ( 0/2 1 1 f i l l ) + 2 (0/4) q.e.d.

=

o.

That is, U(C, E, aD)

C

N,

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. I t 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- I is a char­ acter, and is the group inverse of a. These facts are evident, and there remains only the proof that multiplication is continuous. But if C is a compact subset of G, and e > 0, then I af3 - ao f3o I � I a - ao 1 · 1 f3 0 I + I a 1 · 1 f3 - f3 0 I < e

on C if I a - ao I < e/2 on C and I f3 - f3 0 I < e/2 on C. I t fol­ lows from 34C that multiplication is continuous. 34E. Remark: I t may seem to the reader that it would have

f

been more natural to define aM so that J(M) = f(x) aM (x) dx

f

instead ofJ(M) = f(x)aM(x) dx. The only justification for our

choice which can be given now is that it stays closer to the for­ malism of the scalar product and convolution ; we have defined J(M) as (f, 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 in­ verse Fourier transform. Notice that the character which we as­ sign to a regular maximal ideal M is the inverse of the character assigned by the other formula. § 35.

E XAM P L E S

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 onto T. The di­ rect verification of the correct topology for (; 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. 3SA. Theorem. The character group of a direct product G 1 X G2 of two locally compact Abelian groups is (isomorphic and homeo­ morphic to) the direct product (; 1 X (;2 of their character groups.

139

LOCALLY COMPACT ABELIAN GROUPS

The algebraic isomorphism is almost trivial. Any character a on Gl X G2 is the product of its restrictions to GI and G2 , a( (X l ) X2 ») = a( (X I , e2 ) )a( (e l ) X2 »), and these restrictions are characters on Gl and G2 respectively. Conversely the func­ tion a( (X l ) X2 ») = a l (x l ) a2 (x2 ) defined as the product of two characters on Gl and G2 respectively is evidently a character on G l X G2 • The one-to-one correspondence a (a l ) (2 ) thus es­ tablished between (Gl X G2 ) � and Gl X G2 is clearly an iso­ morphism. Let C be a compact subset of G containing e = (eI , e2 ) and of the form C = Cl X C2 • If I a - aO I < E on C, then (taking X2 = e2 ) I a l - a l o I < E on Cl and, similarly, I a2 - a2 0 I < E on C2 . Conversely if I a l - a l O I < E on Cl and I a2 - a2 0 I < E on C2 , then I a - aO I = I a l a2 - a l o a 2 o I < 2 E on C. I t follows from 34C that the topology of (Gl X G2 ) is the Cartesian prod­ uct topology of Gl X G2 • This fact also follows directly from the next theorem. 35B. Theorem. Ij H is a closed subgroup oj a locally compact Abelian group G, then the character group oj the quotient group G/H is (isomorphic and homeomorphic to) the subgroup oj G consisting oj those characters on G which are constant on H and its cosets. Proof. If (3 is a character on G/H, then a(x) = (3(Hx) is contin­ uous on G and a(x l x2 ) = (3(HX l X 2) = (3(HX lHx2 ) = (3 (HXI){3(Hx2) = a(x l )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 {3 defined on G/H by (3(Hx) = a(x) is a character. We still have to show that the topology induced in this sub­ group is the correct topology for the character group of G/H. We know that C = { Hx : X E:: F} is compact in G/H if F is a compact subset of G, and that every compact subset of G/H arises in this way (see 28C) . Then I (3 (u) - (3o(u) I < E for all u in C if and only if I (3(Hx) - (3o(Hx) I < E for all X in F, and the resul t follows from 34C. 35C. Theorem. Every character a(x) on the additive group R oj the real numbers is oj the jorm a(x) = eiy x, and f( is isomorphic and homeomorphic to R under the correspondence eiyx y. Proof.

�

�

�

140

LOCALLY COMPACT ABELIAN GROUPS

Proof. Every continuous solution of the equation a(x + y)

a(x)a(y) is of the form a(x) eax, and, since a is bounded, a must be pure imaginary. Thus a(x) = eiyx, for some real y . Conversely, every real number y defines the character eiy x, and this one-to-one correspondence between R and R is an isomor­ phism 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 iden­ tity character, and the set of such neighborhoods is a neighbor­ hood basis around the identity. But I eiyx - 1 I < E on [ - n, n] if and only if y is in the open interval ( 0, 0), where 0 = R is continuous (2/n ) arc sin (E/2), so that the mapping R both ways at the origin, and therefore everywhere. 3SD. Theorem. Each character on R/I is of the form e27ri nx, and (R/I) '" is isomorphic and homeomorphic to I under the corre­ n. spondence e27r i nx ==

-

-

�

�

Proof. This follows at once from 3SB and C; the character eiy x has the constant value 1 on the integers if and only if y = 27rm for some integer m, and the discrete subgroup { 2 7rm } of R is, of course, isomorphic (and trivially homeomorphic) to I. 3SE. Theorem. Each character on I is of the form e27r ixn , where o ;£ x < 1 , and 1 is isomorphic and homeomorphic to R/I under x. the correspondence e27r ixn Proof. If a(n) is a character and a(1) = e 27r iy , then a( n ) = 27r e iyn . I t is clear that every such y defines a character, and that e27r iyn is a homomorphism of R onto 1, with the mapping y kernel I. Thus 1 is isomorphic to RjI. The set {y : 0 ;£ y < 1 , N} is a basic neighborhood of and I e 27r iyn 1 I < n = 1 , e in I, and is easily seen to be an open interval about 0 in R/I. Conversely, any open interval about 0 in RjI is such a set (with N 1 and to be determined), and the two topologies are identical. 3SF. Each of the three groups R, R/I, and I is thus (iso­ morphic 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. �

�

-

=

E

E,

"

'

141

LOCALLY COMPACT ABELIAN GROUPS

The Fourier transform formula becomes for these groups the well-known formulas et) ley ) = f(x) e-iy x dx -et) j(n) = en = f(x) e-2.,nz dx

J

lex)

=

!.

1

Lf(n)e -21riXn dn

=

L: � C» Cn e -2r inx.

The fact that the Fourier transform is initially a mapping of L I (G) into e(G) reduces here to the fact that each of the above integrals is absolutely convergent if f E:= L I (which, in the last case, means L: � C» I Cn I < 00 ) , and } is continuous and vanishes at infinity (in the first two cases) . The corresponding formulas for multiple Fourier series, multiple Fourier transforms, and mix­ tures of these, can be written down by virtue of 3SA. The du­ ality theorem can also be directly verified for these product groups. § 36.

TH E

B O C H N E R AND

P LANCH E R E L TH E O R EMS

36A. We shall now apply to the group algebra L I (G) the gen­ eral commutative Banach algebra theory of § 25 and § 26, keep­ ing in mind the extra information that the space G of characters (maximal ideal space of L I (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 3 IG that a positive linear functional on L I (G) is extendable if and only if it is continuous. The Bochner theorem of 261 can thus be taken to refe:t to positive defi n ite functions P E:= Let). Moreover, the hypothesis in that theorem that f*" f" - is now automatically f(x) met, forf*A(OI) = f(x - 1 ) (x, ) dx = f(x) (X - I , ) dx =

)

f

=

f

01

-

01

(f

- = (j( ) ) . Fin ally, if is the measure on G asso­ (x, ) dX 0I ILp ciated with the positive definite function P E:= Let)(G) , we have 01

ff(x)p(x) dx = fj(0I) dILp (OI) ff(x) [ J(x, =

]

) dILp (OI) dx

01

142

LOCALLY COMPACT ABELIAN GROUPS

for all f E: L I , implying that p(x) =

x.

f(x, a) dp,p (a) for almost all

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

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

The formula p (x)

=

f (x, a) dp,(a)

sets up a norm preserving isomorphism between the convex set of all finite positive Baire measures p, on G and the convex set of all posi­ tive definite functions p E: LOO(G) . Every positive definite function p E: LOO(G) is (essen­ tially) uniformly continuous. Corollary.

Proof. This is because the measure p,p is mostly confined to a compact set C and the character function a) is continuous in x uniformly over all a E: C. Given e we choose a compact subset C of G such that J1.p (C') < e/4 (C' being the complement of C), and then find a neighborhood V of the identity in G such that I (X l , a) - (X2' a) < e/2p,p (C) ifx l x2 - 1 E: V and a E: C (by SF again) . Then

(x,

If[ (x h a)

I

-

(X2' a) ] dJ1.p (a)

when X IX2 - 1 E: V. Thus

I � i + i � �2 + �2 C

c'

f(x, a) dp,p (a) is uniformly continuous.

36B. Let (9 be the class of positive definite functions. Because l L (G) has an approximate identity, it follows that the vector space [L 1 n (9] generated by L l n (9 is dense in L l . For L l n Loo is dense in L 1 , and iff E: L l n LOO and u runs through an approxi­ mate identity (also chosen from L l n LOO) , then f * u is at once an approximation to f and a linear combination of four positive definite functions like (f + u) * (f + u) *. By exactly the same argument L 1 n (9] is dense in L2 (G) .

[

143

LOCALLY COMPACT ABELIAN GROUPS

The Bochner theorem above and the abstract Plancherel-like theorem of 26J now lead us to the following L I-inversion theorem. Theorem. Iff C [L I n (p], then } C I I and

f(x) =

f(x, a)j(a) da

for almost all x, where da is the Haar measure of G suitably nor­ malized. Proof. Let cp be the positive functional defined on the ideal LO of uniformly continuous functions of L I by cp (f) = fee) . We have observed earlier (3 IG) that a function p C L I n (p is posi­ ti ve defini te with respect to cp in the sense of the Plancherel-type theorem of 26J, and we can therefore conclude from that theorem that there exists a unique positive Baire measure m on G such that p C L I (m ) and (f, p) = jp dm whenever p C L I n (P and

f The formula p(x) = f (x, a)p (a) dm (a) is the same as

f e Ll . that of the Bochner theorem in 36A. We have left only to prove that m is the Haar measure of G. But if p C L I n (P, then a direct check shows that p (x) (x, ao) C L I n (P for any character ao, and we have

fp (a) dm (a) = p ee)

=

f

p (e) (e, ao) = p (aoa) dm (a),

where we have used the directly verifiable fact that Pao is the Fourier transform of pao. Since the algebra generated by LI n (P is dense in L 1, and the transforms p are therefore dense in e( G) , the above equation shows m t o b e translation invariant and therefore the Haar measure of G. Remark: If T is the Fourier transformation f � } from LI (G)

into e( G) , then the mapping p, � p defined by p(x) =

f(x, a) dp,(a)

from the space of all bounded complex-valued Baire measures on G (e(G) *) into LrrJ(G) ( = LI (G) *) is the adjoint mapping T*. I t must b e remembered, however, that the identification o f LrrJ with (L I ) * is taken to be a conjugate-linear mapping in order that the

144

LOCALLY COMPACT ABELIAN GROUPS

f

ordinary scalar product formula can be used : P(f) f(x)p(x) dx, where P and p are corresponding elements of (L 1 ) * and Loo. The same remark holds for the identification of e(G) * with the space of bounded Baire measures. It is this twist which is responsible for the factor (x, a) in the integrand of T* whereas (x, a) occurs in that of T. The inversion theorem above shows that T* = T- 1 when T is restricted to [L l n (p]. However the function ) occurring in the range of T is taken as an element of e(G), whereas the same func­ tion as an element of the domain of T * is considered as a measure (j(a) da) on G. This discrepancy vanishes when the L2 norms are used ; T then turns out to be a unitary mapping and the equation T* T- 1 is proper. This is the point of the Plancherel theorem, both in its general form (26J) and its group form (36D) . 36C. I t is evident that the inversion formula must be multi­ plied by a constant if a different Haar measure is used on G; that is, the inversion formula picks out a unique Haar measure on G in terms of the given Haar measure on G. In specific cases the correct determination of m on G in terms of J.L on G is an in­ teresting and non-trivial problem. A possible procedure is to calculate explicitly the transform and inverse transform of some particular, easily handled function. By way of illustration consider the group R and the function f(x) = e -x2/2• If in the formula =

=

j{y)

=

r e -z'/'e -iyz dx -00

we differentiate with respect to y and then integrate by parts we find that u }(y) satisfies the differential equation du Ce - y 2/2 • In order to determine C we ob- uy dy, so that }(y) =

=

=

L:

e - x2/2 dx. Then serve that}(O) = C = oo 211" OO e -r2/2r dr dO C2 = f f e - (x2 + y 2) /2 dx dy -00

-00

ii

00

=

0

0

=

27r,

and C (27r) 72 . Therefore if we take Haar measure on R to be dx/ (27r) 72 instead of dx the function e - x2/2 transforms into i tself. =

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 /(27r) 31. Thus ifj(x) E: [L l n P], then j(y) E: [L l n P], where 1 j(y) = � j(x)e-�YZ dx (27r) and 1 j(x) = 31 !(y) eWZ dy ; (27r) _ 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) 31, dy /X(27r) 31) . The proper pairing of measures for compact and discrete groups such as R/I and I will follow from 38B. �

foo . foo ... . -00

36D. The Plancherel theorem. The Fourier transformation j � Tj = j preserves scalar products when confined to [L i n (J>], and its L 2-closure is a unitary mapping oj L 2 (G) onto L2 (G) . Proof. The heart of the Plancherel theorem has already been

established in 26J and restated implicitly in the L l inversion theorem above, namely, that the restriction to [L i n cp] of the Fourier transformation j � Tj = j satisfies ( Tj, Tp) = (j, p ) = cp(j * p) = (j, p) and is therefore an isometry when the L2 norms are used. The remainder of the theorem is that T has a unique extension to a unitary mapping of the whole of L 2 (G) onto the whole of L 2 ( G) . That [L l n cp] is dense in L 2 has been remarked earlier; it depends on the fact that convolutionsj * g withj, g E: L l n L 2 belong to [L 1 n cp] and are dense in L 2 . Thus T has a unique extension to the whole of L 2 (G) , and the extended range is a complete (and hence closed) subspace of L 2 ( G) . vVe will be finished therefore if we can prove that the range of T is dense in L 2 ( G) .

146

LOCALLY COMPACT ABELIAN GROUPS

Now if F E: L I n L2 ( G), then f = T*(F) E: L2 (G) . For f is bounded and uniformly continuous, and if g E: L I n L 2 (G) , then 1 (g, j) 1 = 1 (g, F) 1 � I I F 1 1 2 1 1 g 1 1 2 = I I F 1 1 2 · 1 1 g 1 1 2 , proving that f E: L2 (G) and I I f 1 1 2 � I I F 1 1 2 . If also H E: L I n L 2 ( G) , then h E: L 2 (G) and hf = T*(H * F) E: [L I n (5)]. Thus H * F = T(hf) by the inversion theorem, and since such convolutions H * F are dense in L 2 ( G) we are finished. We have also, as a corollary of the above method of proof, the following : Corollary. L I (G) is semi-simple and regular.

Proof. Iff E: L I and g E: L 2, then f * g E: L 2 and T(f * g) = }g. If f � 0, the convolution operator U, defined on L 2 (G) by f is not zero (32C) and therefore the operator on L 2 ( G) defined by multiplication by } is not zero. That is, if f � 0 in L I (G), then } � 0 in e(G) , and L I (G) is semi-simple. The reader is reminded that L I (G) is regular if for every closed set F e e �nd every point a t[ F there exists f E: L I such that } = 0 on F and f(a) � o. We prove directly the stronger asser­ tion about local identities in L I (2SC) , namely, that if F and U are subsets of G such that F is compact, U is open and F e U, then there exists f E: L I (G) such that } 1 on F and f 0 on Uf• To see this we choose a symmetric Baire neighborhood V of the identity in G such that V3F c U and let g and h be re­ spectively the characteristic functions of V and a Baire open set between VF and V2F. 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 f = gh/m ( V) . 36E. We conclude this section b y translating the representa­ tion theorem of 26F to the group setting. If T (s � Ts) is any unitary representation of an Abelian group by unitary transfor­ mations on a Hilbert space H, the considerations of § 32 tell us that T is completely equivalent to a *-representation of the group algebra L I (G) . Then, as in 26F, T can be " transferred" to the algebra of transforms } and then extended to a *-representation of the algebra of all bounded Baire functions on G. The trilinear functional l(}, x, y) = ( T,x, y), for fixed x and y, is a bounded complex-valued integral on e(G), and satisfies I (Ft, x, y) = =

==

LOCALLY COMPACT ABELIAN GROUPS

147

I(F, Tgx, y) for F E::: e(G) and g E::: L I (G) . Thus ( Ts Tgx, y) = ( Tg8-1x, y) = I( (s I , a)g (a) , x, y) = I(], Tgx, y) . The operator T"'E corresponding to the characteristic function 'PE of the set E is a projection, designated PE, and (PEx, y) is the complex-valued measure corresponding to the above integral. (The correspond­ ence E � PE is called a projection-valued measure.) The above equation can therefore be written -

where we have set u = Tgx. 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 Ttl = (s, a) dPa •

f

The projections PE form the so-called spectral family of the commutative family of operators Ts. Each such projection de­ fines a reduction of the representation T into the direct sum of TPE operating on the range of PE and T(1 - PE) = TPE, oper­ ating 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 PE and is beyond the reach of this book. § 3 7.

MI S C E L LA N E O U S TH EO R EMS

This section contains the Tauberian theorem and its generali­ zations, the Pontriagin duality theorem, and a simple case of the Poisson summation formula. 37A. The Tauberian theorem, as proved in 25D, applies to any regular commutative Banach algebra, and hence (see 36D, corollary) to the L I -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

LOCALLY COMPACT ABELIAN GROUPS 1 48 L2 A and hence the elements of L2 whose transforms lie in L are dense in L2 • Since every function of L 1 is a product of functions in L 2, it follows via the Schwarz inequality that the functions of L 1 whose transforms lie in L are dense in L 1 . We can now restate the Tauberian theorem. Theorem. If G is a locally compact Abelian group, then every proper closed ideal of L 1 (G) is included in a regular maximal ideal. Corollary 1. If f E: L 1 is such that j never vanishes, then the translates of f generate L 1 .

Proof. By hypothesis f lies in no regular maximal ideal. The closed subspace generated by the translates of f 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 E: L 1 be such that k never vanishes and let g E: LOO be such that the continuous function k * g vanishes at infinity. Then f * g van­ ishes at infinity for every f E: L 1 . Proof. The set I of functions f E: L 1 such that f * g vanishes at infinity is clearly a linear subspace, and I is invariant since fx * g = (f * g) x vanishes at infinity if f * g does. I is also closed. For if f E: 1 and E is given, we can choose h E: I such that f - h i l l < E/(2 1 1 g 1 100) , and we can choose a compact set C such that I (h * g) (x) I < E/2 outside of C. Since (f * g) (x) - (h * g) (x) I � I I f - h 1 1 1 1 1 g 1 100 < E/2, it follows tha t I (f * g) (x) I < E ou tside of C. Therefore f * g vanishes at infinity andf E: I. Therefore (by 3 1F again) I is a closed ideal, and since it con­ tains k it is included in no regular maximal ideal. It follows from the theorem that I = L 1 • Corollary 3. (The Wiener Tauberian theorem.) If k E: L 1 (R) is such that k never vanishes and g E: Loo is such that (k * g) (x) � 0 as x � + 00, then (f * g) (x) � 0 as x � + 00 for every f E: L 1 . 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.

I

II

149

LOCALLY COMPACT ABELIAN GROUPS

37B. The Tauberian theorem in the general form of the theo­ rem of 37A asserts that a closed ideal is the kernel of its hull if i ts 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 L I (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 struc­ ture theory, to general locally compact Abelian groups by Kap­ lansky [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 (2SF) is valid in the L I algebra of any locally compact Abelian group. This immediately implies (by 2SF) 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.

This lemma has content only if � is not compact. We must show that for every x and f there exists y such that j has compact support and I I xy - x I I < f. Because A has an approximate identity there exists u such that I I xu - x I I < f/2, and then, by the Wiener condition, there exists y such that j has compact support and " y - u " < f/21 1 x I I . Then I I xy - xu " < f/2 and " xy - x " < Of, q .e.d. Proof.

Corollary. It then follows that, if I is a closed ideal with a com­ pact hull, then I contains every element x such that hull (I) c int (hull (x) ) . Proof. I t follows from the theorem that there exists y such that j has compact support and I I xy - x I I < But then xy E: I by 2SD, and since I is closed we have x E: I. E.

1 50

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 C such that m ( U) < 00, where m is the Haar measure of C. Let V be any second such neighborhood whose closure is com­ pact 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 v be their inverse Fourier transforms in L2 (G) . Then the function j = uv/ m ( V) belongs to L l (G) , and I I j \ I I � \ I u \ 1 2 \ 1 v \ l 2 /m (V) = \ I u \ 1 2 \ 1 v \ l 2 /m (V) = [m ( U)/m (V)] 72 < 2. Moreover if W is a neighborhood of e such that VW c U then j = u * v/m ( V) = (l /m (V)) u (a) v(a- l {3) da = 1 if {3 E: W.

I

Lemma. Given any compact set C c G and any e > 0 there exists a function j E: L l such that j 1 in some neighborhood of the identity e in C, \ I j \ I I < 2, and \ I j - jx \ I I < efor every x E: C. ==

We define j as above, taking for U any symmetric Baire open subset of the open set { a : 1 1 - (x, a) I < e/4 for all x E: C} (see SF) . Since j - jx = [u (v - vx) + vx (u - u x)]/m (V) and since, with the above choice of U, \ I u - U x \ 1 2 2 = I u (a) (1 - (x, a)) 1 2 da � m ( U) luba E: U 1 1 - (x, a) 1 2 < m ( U) (e/4) 2 if x E: C (and similarly for \ I v - Vx \ 1 2 ), we have \ I j - jx \ I I � 2[m ( U) m(V)] 72e/4m (V) < e for every x E: C, q.e.d. Corollary. If f E: L l (G) and J(e) = 0 then f * j � 0 as U de­ creases through the symmetric Baire neighborhoods of e. Proof.

I

Given 0 we choose C in the above lemma to be sym­ metric and satisfying r ' I f I < 0/8, and set e = 0/2 1 \ f 1 \ 1 . Then Proof.

Je

If(x)j(x - ly) dx If(x) (}(x - ly) - j(y) ) dx, and \ I f * j 1 \ 1 � I I f(x) I \ I jx-1 - j 1 \ 1 dx L + i, < I I f I l l e +

f * j(y)

=

=

=

(0/8)4 = o. 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 €: L I (G) such that v 0 in a neighborhood of e and such that f * v � ffor every f €: L I (G) such thatj(e) = o. ==

Proof. Let u run through an approximate identity and set v = u - j * u. Then I I v 1 1 1 � 3 and v = tt - tt} = 0 in the neighborhood of e where } = 1 . Also I I f - f * V 1 1 1 � I I f - f * U / / 1 + I l f * j / / 1 / / u / / 1 � 0, q . e.d. I t follows that Silov's theorem (25F) is valid for the group al­ gebras of locally compact Abelian groups. We restate the theo­ rem sligh tl y. Theorem. Let I be a closed ideal in L I (G) and f a function of L I (G) such that j(a) = 0 whenever a is a character at which every function of I vanishes; that is, hull (I) c hull (f), or f €: k(h (J) ) . Suppose furthermore that the part of the boundary of hull (f) which is included in hull (/) includes no non-zero perfect set. Then f €: I. Corollary. If I is a closed ideal in L I (G) whose hull is discrete (i.e., consists entirely of isolated points) then J = k(h(J) ) . 37D. W e have seen that the character function (x, a) , x €: G, a €: G, is continuous in x and a together. Also that (x, a l (2) = (x, a I ) (x, (2 ) . Therefore, every element x in G defines a charac­ ter on G, and the mapping of G into C thus defined is clearly an algebraic homomorphism. The Pontriagin duality theorem as­ serts 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 [L I n (J>J corresponds exactly to the space [l l n 8>J under the Fourier transform and forms an algebra A of uniformly continuous functions vanishing at infinity and separating the points of G (since it includes the subalgebra generated by [(L I n L2 ) * (L I n L 2 )J ; see 36A, B) . The weak topology de­ termined on G by A is therefore the given topology of G (5G) . But, as the Fourier transform of a function j €: [l l n 8>J, each such f has a unique extension to the whole of C, and under this extension [L I n (J>J is isomorphic to [L I n (J>J (C) . Since the to­ pology of C is the weak topology defined by these extended func-

1 52

LOCALLY COMPACT ABELIAN GROUPS

G

C

tions, it follows that is imbedded in with the relative topology it acquires from and that is dense in C. (Otherwise we could construct a non-zero convolution of two functions of which vanishes identically on contradicting the isomorphic extension.) Since the subsets of and on which I j I > > 0 have compact closures in both topologies and one is dense in the other, we can conclude that their closures are identical. There­ fore = finishing the proof of the duality theorem. 37E. The Poisson summation formula. Let f be a function on R = ( - 00, 00 ) and let be its Fourier transform. Let a be any positive number and let (3 = 2 /a. Then under suitable condi tions on f it can be shown that

C,

G

G, G

(LI n L2) (C)

C

E

G C,

F

7r

Va L � rrJf(na)

which is the Poisson formula. The formal proof proceeds as follows. The function = L � rrJf + 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 = and its character group, � rrJ' as a subgroup of R = R (see 35B), is the discrete group Then 1 . 1 + na) = - L:= rrJ = a a

g(x)

(x

eimx (21f'/a) eim{3x,

iae -tm{3xg(x) dx

G(m{3)

_

0

F(m{3) J�-a F(m{3) . But by the inversion theorem g (x) fore =

-

0

=

a Vi;

=

2:. : .f(na)

{m{3} iae -t.m(3'i(x dx

=

g(O)

=

2:. : .

L � rrJ

eim(3xG(m{3) .

There­

G(m{3) � 2:. : . F(m{3) , =

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

1 53

LOCALLY COMPACT ABELIAN GROUPS

arguing, but our purpose here is only to display the group set­ ting 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, H and G/H be adjusted so that r = r r , and let f be a function of [L l n
Ja JG/HJH

such that g(y) = y . Then

Lf(xy) dx is a continuous function (on G/H) of iHf(x) dx = JOr.-./Hj(a) da.

Proof. The character group of G/H is by 35B the set of char­

acters a

E:

G such that (xy, a) = (y, a) for every x

E:

H. Then

1GjH (y, a)g(y) dy = 1GjH (y, a) [ iHf(xy) dX ] dy 1GjHiH (xy, a)f(xy) dx dy = 1G (x, a)f(x) dx = j(a) .

t ea) =

But by the inversion theorem g(y) = r.-. (x, a)g(a) da. There­ JO/H fore f(x) dx = g(O) = r.-. t(a) da = r.-. j(a) da,

iH

JO/H

which is the desired formula. § 38.

JG/H

COMPACT A B E LIAN G R O U P S A N D G E N E RA LIZED

F O U RI E R

S E RIES

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 dis­ cussion will be given here. 38A. Theorem. G is compact if and only if G is discrete.

Proof. Reversing the roles of G and G we remark, first, that, if G is discrete, then L l (G) has an identity (3 1D) and G is com­ pact as the maximal ideal space of a commutative Banach alge­ bra with an identity ( 19B) . Second, since L l (G) is semi-simple

1 54

LOCALLY COMPACT ABELIAN GROUPS

and self-adjoint, it follows from 26B that, if G is compact, then an identity can actually be constructed for L I (G) by applying the analytic function 1 to an everywhere positive function j in the transform algebra, and (by 3 1D 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 I I a - I 1 100 < t is an open neighborhood of the identity char­ acter 1 (by 34C) which obviously contains only the identity character; thus the topology of G is discrete. 38B. Theorem. If G is compact and its Haar measure is nor­ malized so that p.(G) = 1 , then the inversion theorem requires that the Haar measure on G be normalized so that the measure of each point is 1 . Proof. I f the measure of a point in G is 1 , then the identity of L I (G) is the function u which has the value 1 at a = e and the value 0 elsewhere. Since convolutions on G correspond to ordi­ nary pointwise products on G, u is the transform of u 1 . Thus

u (a)

=

I(x, a) dx

==

=

1 if a

=

e and = 0 otherwise.

I

1 , this implies that p.(G) = I dx match as stated in the theorem. =

=

1 , so that the measures

The above identity can also be written

I(x, a2a l - 1 ) dx

Since (x, e)

I(x, a l ) (x, a2) dx

=

1 if a2 = a l and = 0 if a2 � a l . Thus : Corollary. The characters on G form an orthonormal set. =

38C. The characters form a complete orthonormal set in L2 (G) and the (Fourier series) development f(x) = Lr (f, a n) an of a function f C L 2 (G) is the inverse Fourier transform. Proof. If f(x) C L 2 , then j C I}, so that j(a) = 0 except on a countable set { an } and, if C n (f, a n) , then

= je a n ) =

If(x) (x,

a n ) dx

=

LOCALLY COMPACT ABELIAN GROUPS

1 55

which is Parseval's equation. The continuous function /n = Ll Ciai is by its definition the inverse Fourier transform of the finite-valued function ln equal to Ci at Oli if i � n and equal to 0 elsewhere. since ln clearly converges (L2) to l, it follows that /n converges (L2) to / and that the formula /ex) = L � c n (x, a n ) =

fl(Ol) (x,

) dOl is the inverse Fourier transform .

a

38D. Every continuous function on G can be uniformly approxi­ mated by finite linear combinations 0/ characters .

This follows from the Stone-Weierstrass theorem, for the char­ acters 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 prop­ erties required. A direct proof can be given. First/ is approximated uniformly by / * u where u is an element of an approximate identity in L1 (G) . Then, as above,/ is approximated L2 by /n = Ll Ci(X, Oli) and u by Un = Ll di(x, Oli), where the sequence { an} includes all the characters at which either l or u � O. Then / * u is approxi­ mated uniformly by /n * Un = L l Ci di(x, Oli) ' The last equation can be verified by writing out the convolution or by remembering that/n * U n = T*(lngn) ' The partial sums of the Fourier expansion of/ will in general not give uniform approximations.

Chapter FIll COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

If G is compact L2 (G) is an H*-algebra, and the whole theory of § 2 7 can be taken over. Thus L 2 (G) is the direct sum of i ts (mutually orthogonal) minimal two-sided ideals, and it turns out that these are all finite dimensional and consist entirely of con­ tinuous functions ; their identities (generating idempotents) are the characters of G. These facts, which are in direct generaliza­ tion 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 pre­ sen ts the theory of almost periodic functions on a general group, following a new approach. § 39.

TH E G R O U P A L G E B RA O F A COMPA CT G R O U P

39A. The Haar measure of a compact group is finite and gen­ erally normalized to be 1 . Then I I f 1 1 1 � I I f 1 1 2 � I I f I I O()O Moreover, on any unimodular group the Schwarz inequality im­ plies that I I f * g 1 1 «> � I I f 1 1 2 1 1 g 1 1 2 . Together, these two in­ equalities show that L«> 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 (f * g, h) = (g, f* * h), I I f* 1 1 2 = I l f 1 1 2 and f * f* � 0 if f � 0 are all directly verifiable. Thus f * f* is continuous and (f * f*) (e) = I I f 1 1 2 2 > 0 if f � 0, and the other two conditions follow from the inverse invariance of the Haar measure of G. The fact that L2 (G) is an H*-algebra 1 56

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

1 57

means that its algebraic structure is known in great detail from the analysis of § 2 7. Moreover, further properties can be de­ rived 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 re­ stating the general results of § 2 7.

Lemma 1. A continuous function h ex) is in the center of the algebra L2 (G) if and only if h(xy) h(yx) . Proof. h €: center (L 2) if and only if =

f[h (xY)f(y - l ) - f(y)h(y - 1 x)] dy

=

0.

Replacing y by y - l in the second term of the integrand and re­ membering that f is arbitrary, the result follows. This proof works just as well for the L l algebra of a unimodular locally compact group. Lemma 2. Every non-zero closed ideal I c L2 contains a non­ zero element of the center of L2 • Proof. (After Segal [44] .) Choose any non-zero g €: I and let f = g * g* ; f is continuous since f(x) = (gx, g) and since gx is a continuous function of x as an element of L 2 . Then f(axa - 1 ) is a continuous function of a if x is fixed, and, as an element of LOO is a continuous function of x (see 28B) . Thus hex) = f(axa -1 ) da is continuous, and h ( e) = fee) = I I g 1 1 2 2 > 0, so that h � O. Fi­ nally, if a is replaced by ay - l in the integral for h(yx), it follows from the right invariance of Haar measure that h(yx) = h(xy) , and hence that h is central. Moreover it is easy to see that h €: I, for f €: I andfaa €: I by 31F, and it then follows from the Fubini theorem as in 31F that h €: IJ.. J.. = I, q.e . d. I t now follows from 27F that every minimal two-sided ideal N of L2 (G) is finite-dimensional and contains an identity e. It then follows further that every function f €: N is continuous, for f = f * e, or f(x) = (fx, e) , and fx as an element of L 2 is known to be a continuous function of x.

f

1 58

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

We now state the global structure theorem from § 27 as modi­ fied above. Theorem. L 2 (G) is the direct sum in the Hilbert space sense of its ( mutually orthogonal) minimal two-sided ideals NOI.. Each mini­ mal two-sided ideal Na is a finite-dimensional subspace of continuous functions. It has an identity ea, and the projection on NOI. of any function f E: L2 (G) is fOl. = f * eOl. = ea *f. The regular maximal ideals of L 2 are the orthogonal complements of the minimal ideals, and every 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 NOI. are the scalar multiples of the identity ea, and every central function f E: L 2 has a Fourier series expansion f = L AaeOl. ' where AaeOl. = f * ea. If G is Abelian as well as compact, then each NOI. is one-dimensional and the identities eOl. 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.

" In are orthogonal minimal left ideals whose Theorem. If I b direct sum is a minimal two-sided ideal N, then 11 *, " ' , In * are minimal right ideals with the same property and Ii* n Ij = Ii* * Ij is one-dimensional. Therefore there exist elements eij E: Ii* n Ij, uniquely determined if i = j and uniquely determined except for a scalar multiple of absolute value 1 in any case, such that eij * ekl = ei l ifj = k and = 0 ifj � k, eij = eji* and e = L1 eii. The func­ tions { eij} form an orthogonal basis for N and the correspondence f (Cij), where f = L Cijeij, is an algebraic isomorphism between N and the full n X n matrix algebra, involution mapping into con­ jugate transposztton. Furthermore I I eij 1 1 2 2 = n and neik(xy) = Lj eij(x)ejk(y) . The left ideals Ii are isomorphic to each other ( under the correspondence Iieij = Ij) and to every other minimal left ideal in N. . .

�

Proof. The only new facts concern the equation neik (xy) = Lj eij(x)ejk(y) . The left ideal Ik has the basis elk, " ' , e n k, and is invariant under left translation. Therefore, given x and i

1 59

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

there exist constants Cij (X) such that eik(xy) = Lj Cij (x)ejk(Y) ' Then eij (x) = [e il * e ljJ (x) = e il (Xy)e lj (y - l ) dy = L m C im (X)

f

fem l (y)ejl (y) dy = Cij (X) I I ejl 1 1 2 2

Cij (X) I I e l l 1 1 2 2, which gives the desired result if I I e l l 1 1 2 2 = n . But I I e l l 1 1 2 4 = I I e l l 1 1 2 2e 1 1 (e) = Lj= l e l j(x)ej l (x - l ) = 2:1 1 e l j(x) 1 2 , and inte­ grating we get I I e l l 1 1 2 4 = .Li I I e l j 1 1 2 2 = n i l e l l 1 1 2 2 , or I I e l l 1 1 2 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. I t is not surprising that there is a functional equation for characters here as well as in the Abelian case. Theorem. If f E:: LOO (G), then there exists a scalar X such that jIX is a character if and only if there exists a scalar " such that f(s (t) . f(xsx - 1 t) dx = =

�

f

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

!

!

!ea(xsx- It) dx

=

k (s)ea (t) .

We evaluate k(s) by taking t = e, gIvIng ea(s) = k(s)n and ea(xsx- 1 t) dx = ea (s)ea (t) In, where n = ea (e) = I I ea 1 1 2 2 . I f

f

j = Xea, then clearly

!j(xsx -1t) dx = j(s)j(t) I", where "

=

nX.

1 60

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

Now suppose conversely thatf satisfies this identity. As above f(xsx- 1 t) dx is a central function in s for every fixed t, and it follows from the assumed identity that f is central. If Na is some minimal ideal on which f has a non-zero projection f * ea = Xea, then

I

IIf(xsx - 1t)ea (t- 1 ) dx dt XIea(xsx - 1 ) dx Xea(s) IIf(xsx -1t) e. (t- l ) dx dt f�) If(t)e. (t- 1 ) dt f(s) (f�e.) . =

=

=

=

Since (f, ea) = [f * ea] (e) = Xea(e) = Xn, we see that f(s) = 'Yea (s)/n, which is the desired result. Notice also that 'Y = Xn as before. 39D. A function f E::: L2 is defined to be almost invariant if its mixed translates fa b generate a finite-dimensional subspace of L 2 . This subspace is then a two-sided ideal by 3 1F, and is therefore a finite sum of minimal ideals. Thus f is almost invariant if and only if f lies in a finite sum of minimal two-sided ideals. The functions of a minimal ideal can be called minimal almost in­ varian t functions, and the expansion f = Lfa can be viewed as the unique decomposition of f 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 fg of two almost invariant func­ tionsf and g is also almost invariant. For if { fd is a set of mixed translates of f spanning the invariant subspace generated by f and similarly for { gj} , then the finite set of functions { figj} spans the set of all translates of fg. Thus the set of almost in.: variant functions is a sub algebra 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 u * f(x) - f(x) I = u (y) (f(y - 1 x) - f(x) ) dy I � max1l-1E:V f1l - f 1 100 , etc.), and

II

II

I

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 a n include all the indices for which fa � ° or Ua � 0, and choose n so that Then I I U *f " f - L ifa. 1 I 2 < an d " U - L i Uai 1 1 2 < L i Uai *fai 1 100 = I I (u - Li Ua.) * (f - Lifa.) 1 100 < E2, q.e.d. Another proof follows from the above remark that the vector space of almost invariant 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 L2 (G) . Since I I f I I I � I I f 1 / 2 the intersection with L2 of any closed ideal I in L 1 is a closed ideal in L2 . I f I � 0, then I n L2 � 0. In fact, I n L2 is dense (L 1 ) in I, for if ° � f E::: I and U E::: L 1 n L2 belongs to an approximate identity, then u *f belongs to I n L2 and approxi­ mates f in L 1 . Since the L 1 -closure of a closed ideal in L2 is ob­ viously a closed ideal in L 1 , the mapping I � I n L2 is a one­ to-one correspondence between the closed ideals of L 1 and L2 • The minimal ideals of L2 , being finite-dimensional, are identical with their L 1-closures and any L 1 -closed ideal I is the direct sum in the L 1 sense of the minimal ideals it includes. In order to see this we approximate f E::: I by g E::: I n L2 and g by a finite sum of its components. This approximation improves in passing from the L2 norm to the L 1 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. E

§ 40.

E.

R E P R E S E NTATION TH E O R Y

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

1 62

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

Theorem.

then [u, v]

=

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

f(Txu, Txv) dx is an equivalent scalar product under

which the transformations Tx are all unitary. Proof. It is clear that [u, v] has all the algebraic properties of a scalar product except possibly that u � 0 implies [u, u] � o. This follows from the fact that ( Txu, Txu) is a continuous function of x which is positive at x = e. Also [ Ty u, Tyv] = ( Tx Tyu,

f

f

f

Tx Tyv) dx = ( Tx yu, Tx yv) dx = ( Tx u, Tx v) dx = [u, v], so that the transformations Tx are all unitary in this scalar product. This completes the proof if H is finite-dimensional. If B is a bound for the norms { I I Tx I I : x E: G} and if II u II = (u, u) Y2, I I I u I I I = [u, u] Y2, we have at once that I I I u I I I � B I I u I I . Since u = Tx-lV if v = Tx u, we have I I u I I � B I I Tx u I I and there­ fore I I u I I � B i l l u I I I . Therefore the norms I I u I I and I I I u I I I are eq ui valen t and the proof of the theorem is complete. 40B. We shall assume from now on that T is always a rep­ resentation of G by unitary operators on a Hilbert space H. We know (see 32B, C) that T is equivalent to a norm-decreasing, in­ volution-preserving representation of L l by bounded operators on H. Since L2 e L l and I I f l 1 2 � I l f l l l if G is compact and p,(G) = 1 , we thus have a norm-decreasing representation of L2 (G) , which we shall denote by the same letter T. Our next arguments are valid for the *-representations of any H*-algebra.

Theorem. Every * -representation of an H*-algebra A is uniquely expressible as a direct sum of (faithful) representations of certain of its minimal two-sided ideals.

Let Nl and N2 be distinct (and hence orthogonal) minimal two-sided ideals of A and let H be the representation space. If f E: Nl } g E: N2 and u, v E: H, then ( Tju, Tgv) = (u, T,*gv) = 0 since f*g = o. If we normalize H by throwing away the intersection of the nullspaces of all the operators T, and for every index a set Ha equal to the closure of the union of the ranges of the opera tors T, such that f E: Na, and set Ta Proof.

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

1 63

equal to the restriction of T to Nex, then it is clear that the sub­ spaces Hex are orthogonal and add to H and that T is the direct sum of the representations Tex of the (simple) algebras Nex. Since each Nex is a minimal closed ideal, it follows that Tex is either 0 or faithful, and the kernel of T is exactly the sum of the ideals Nex for which Tex = o. 40C. The above representations Tex are not necessarily irre­ ducible 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 *-rep­ resentation T of N by operators on a Hilbert space H, it being understood that the union of the ranges of the operators T/ is dense in H. Moreover, we shall assume that N is finite-dimen­ sional ; our proof that the irreducible constituents are all equiva­ len t is not valid if N is infini te-dimensional. Given a non-zero V I E: H, there exists a minimal left ideal l e N, with generating idempotent e, such that TeV I � O. Then HI = { T/v I : f E: I} is a finite-dimensional subspace of H, in­ variant under the operators TJ, f E: N, and operator isomorphic to I under the correspondence f � T/V I ' 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 sub,.. ideal of I. That it is an operator isomorphism follows from the equation Tg( T/V I ) = ( Tg T/) V I = Tg/vr, which establishes the de­ sired relation Detween left multiplication by g on I and operating by Tg on HI ' Finally HI is irreducible, for if V = T/V I E: HI , then there exists g E: N such that gf = e, so that Tgv = TeV I and the subspace generated by v is the whole of HI ' If HI is a proper subspace of H, we take V2 E: HI J.. and notice that then ( TgV2 ' T/V I ) = ( V2 ' Tg *!> V I ) = 0, so that any minimal subspace derived from V2 in the above manner is automatically orthogonal to HI ' Continuing in this way we break up H into a direct sum of finite-dimensional irreducible parts Hex. If Tex is the irreducible representation derived from T by confining it to Hex, then the preceding paragraph and 2 7E imply that the Tex are all equivalent. We have proved : Er:ery faithful representation of a minimal closed ideal is a direct sum of equivalent irreducible representations.

1 64

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 (eij) be the matrix as described in 39B. We shall use the no­ tation T(f) instead of TI. T(ekk) is a projection since ekk2 = ekk; let Vk be its range. If V 1 is any non-zero vector of Vh then Vj = T(ej 1 ) v 1 is a vector in V" for T(ejj) vj T(ejjej 1 ) v 1 = T(ej 1 ) v 1 = Vj, and by a similar argument the vectors Vj have the property that Vj = T(eji) Vi. Since T(eij) v n = 0 if j � n, the set Vj is linearly independent, and if V' is the subspace it spans, the equation Vj = T(eji) vi shows that the coefficient matrix in the ex­ pansion of any f E::: N with respect to (eij) is identical with the matrix of T(f) with respect to the basis { Vj } in V'. Thus V' is invariant under T, and is operator isomorphic to any minimal left ideal in N. Also, since T is irreducible, V' = V. If we now use the hypothesis that T is a *-representation, i.e., that T(f *) = T(f) *, then I I Vj 1 1 2 = ( T(ej 1 ) vb T(ej 1 ) v 1 ) = (Vh (V b V 1 ) = I I V 1 1 1 2, and (Vi, Vj) = T(e 1 jej 1 ) v 1 ) = (Vh T(e l l ) v 1 ) ( T(ei 1 ) V b T(ej 1 ) V 1 ) = (Vh T(e 1 iej 1 ) V 1 = 0 if i � j. Therefore, the basis { Vi} is orthonormal if I I V 1 I I = 1 . We now again assume that T arises from a representation Tx of a compact group. Since Uxf = eX * f on N, and eX(y) = e(yx - 1 ) = L eii(yx - 1 ) = Li , j eij(y) eji(x - 1 ) /n = L [eij(x)/n]eij( Y) , it follows that the matrix for Tx with respect to the orthonormal basis { vd of V is eij(x)/n. Any other orthonormal basis for V arises from the one considered here by a change of basis trans­ formation, and under this transformation the functions eij 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 uni­ tary transformations on the finite-dimensional Hilbert space V. Let T be extended to L2 (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 = N.l.. If Cij(X) is the matrix of Tx with respect to any orthonormal basis for V, then the functions eij(x) = nCij(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

A LMOST P E RI O D I C F U N CTIO N S

In this section we offer the reader an approach to almost peri­ odic 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 com­ mutative C*-algebra (LAP) and therefore can be considered as the algebra e(�) of all complex-valued continuous functions on a compact space �, the space of maximal ideals of LAP. The poin ts of G define some of these ideals, a dense set in fact, and � can be thought of as coming from G by first identifying points that are not separated by any / E: LAP, then filling in new points corresponding to the remaining ideals, and finally weaken­ ing the topology to the weak topology defined by LAP. The critical lemma states that when G is considered as a subset of � the operations of G are uniformly continuous in the topology of � and hence uniquely extensible to the whole of �. We thus end up with a continuous homomorphism a of G onto a dense subgroup of a compact group �, with the property that the func­ tions on G of the form /(a(s)), / E: e(�), are exactly the almost periodic functions on G (i.e., that the adjoint mapping a * is an isometric isomorphism of e(�) onto LAP) . 41A. A left almost periodic function on a topological group G is a bounded continuous complex-valued function / whose set of left translates Sf = { /8' S E: G} is totally bounded under the uni­ form norm. The reader is reminded that a metric space S is totally bounded if and only if, for every > 0, S can be covered by a finite number of spheres of radius Also, a metric space is compact if and only if i t is totally bounded and complete. Since the closure Sf of Sf in the uniform norm is automatically complete, we see that / is left almost periodic i f and only if Sf 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 Sf X Sg is compact and the mapping (h, k) � h + k is continu­ ous. Therefore the range of this mapping, the set of all sums of a function in Sf with a function in Sg, is compact, and the sub­ set SI+g is therefore totally bounded. - We have proved that E

E.

1 66

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

LAP is closed under addition, and it is clear that the same proof works for multiplication. Now suppose that fn €: LAP and fn converges uniformly to f. Since I I fnx - fx 1 100 = I I fn - j 1 100 the sequencefn x converges uni­ formly to fx, uniformly over all x €: G. If, given e, we choose fn so that I I fn - f 1 100 < e/3 and choose the points X i so that the set { fnx.} is e/3-dense in Sr, then the usual combination of inequalities shows that { fxJ is e-dense in Sf. That is, j €: LAP, proving that LAP is complete. If f 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 : .

Under the uniform norm the set of all left almost periodic functions on a topological group G is a commutative C*­ algebra with an identity. Theorem.

41B. It follows from the above theorem (and 26A) that LAP is isomorphic and isometric to the space e(;m;) of all complex­ valued continuous functions on the compact Hausdorff space ;m; of its maximal ideals. Each point s €: 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 J(M) - k I < e} form a sub-basis for the topology of ;m;. Since the inverse image under a of such a set is the open set { s : I f(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 de­ fines a group structure in the image a(G) c ;m;. Supposing this question to be satisfactorily settled there still remains the prob­ lem 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.

1 67

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

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. If f €: LAP and e > 0, there exists a finite set of points { bi} such that given any b one of the points bi can be chosen so that I f(xby) - f(xbiy) I < e for every x and y.

Let the finite set { faJ be e/4-dense among the left­ translates off. Given b €: G and i, f is within a distance e/4 of fa i for some j, and letting i vary we obtain an integer-valued function jU) such that I I f - faj(i) 1 100 < e/4 for every i. For each such integer-valued function jU) let us choose one such b (if there is any) and let { bj} be the finite set so obtained. Then, by the very definition of { bj} , for every b there exists one of the points bk such that I I f f 1 1 00 < e/2 for all i. Since for any x €: G we can find ai so that I I fx - f I I < e/4, we have I I f f 1 17l) � I I fx b f 1 100 + I I f f 1 1 00 + I f f 1 17l) < e/4 + e/2 + e/4 = e. That is, for every b €: G there exists one of the points bk such that I f(xby) - f(xbky) I < e for every x and _V , q.e . d. Proof.

aib

aib

ai b -

aibk

Xb

ai

xbk

-

aib

aib

-

aibk

aibk

Corollary 1. The set { fij} defined by fij(X) dense in the set of all mixed translates { fa } b

-

-

xbk

= f(bixbj)

•

is 2e­

Corollary 2. If x and y are such that I f(bixbj) - f(biybj) I < e

for all i and j, then I f(uxv) - f(uyv) I < 5 efor all u and v

€:

G.

Proof. Given u and v we choose i and j so that I I f(uxv) f(bixbj) 1 100 < 2e. This inequality combines with the assumed in­ equality to yield I f(uxv) - f(uyv) I < 5 e, as desired. 4 1 C. We now show that multiplication can be uniquely ex­ tended from G to the whole of mr by showing that, given Ml ) M2 €: mr and setting x = a(x), if X l M2 , then Ml and X2 X;'X2 converges. What we already know is that, given f and e, there exist weak neighborhoods Nl and N2 of the points Ml and M2 respectively such that, if X l > .9 l €: Nl and X2 ) .9 2 €: N2 , then I f(X l X2 ) - f(Y l Y 2 ) I < e ; this follows from the second corollary of the paragraph above upon writing f(X l X2 ) - f(Y l Y 2 ) f(X I X2) - f(X l Y 2 ) + f(X lY 2) - f(X2Y 2) ' The family of sets { Q2 : X l €: Nl -7

-7

=

1 68

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS .

and X2 E: N2 } obviously has the finite intersection property as N1 and N2 close down on M1 and M2 respectively, and, mI being compact, the intersection of their closures is non-empty. If M and M' are two of its points, then I J(M) - J(M') I < for every > 0 and every f E: LAP by the second sentence above, so that M = M'. We conclude that, given M1 and M2 , there is a unique point M such that, for every weak neighborhood N(M), there exist weak neighborhoods N1 (M1 ) and N2 (M2 ) such that X l E: N1 and X2 E: N2 imply that �'X2 E: N. This shows first that the product M1 M2 is uniquely defined, and second, letting X l and X2 converge to other points in N1 and N2 , that the product M1 M2 is continuous in both factors. The reader who is familiar with the theory of uniform structures will realize that the above argu­ ment is a rather clumsy way of saying that �'X2 is a uniformly continuous function of X l and X2 in the uniform topology of mI and hence is uniquely extensible to the whole of mI. We still have to show the existence and continuity of inverses, i.e., loosely, that, if X � M, then x - 1 converges. But by Corol­ lary 2 of 41B, givenf E: LAP and > 0 there exists a weak neigh­ borhood N(M) such that, if x, y E: N, then I f(uxv) - f(uyv) I < for all u and v. Taking u = x - 1 and v = y - 1 this becomes I f(y - 1 ) - f(x - 1 ) I < and proves exactly in the manner of the above paragraph the existence of a unique, continuous homeo­ morphism M � M- 1 of M onto itself such that x - 1 = (X - 1 ) � for all x E: G. Thus MM- 1 = lim (XX - 1 ) = e and M- 1 is the inverse of M. We have our desired theorem. E

E

E

E

E

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

==

There is a further remark which we ought to make in connec­ tion with the above theorem, namely, that the group mI is unique to within isomorphism. For if mI' is any second such group, with homomorphism (3, then (3* - l a* is an algebraic isomorphism of e(mI) onto e(mI') 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), l' itself is an iso­ morphism. That is any two such groups M and M' are continu­ ously isomorphic. 41D. Because of the above theorem the set of (left) almost periodic functions on a group G acquires all the algebraic struc­ ture of the group algebra of a compact group. Thus any func­ tion of LAP can be uniformly approximated by almost invariant functions, that is, by almost periodic functions whose translates generate a finite-dimensional subspace of LOO. The more precise L2 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 i t has an apparent defect in that the Haar integral on the associated compact group seems to be necessary to define the scalar prod­ uct. However, von Neumann [39] showed that an intrinsic defi­ nition can be given for the mean M(f) of an almost periodic function by proving the following theorem.

The uniformly closed convex set offunctions generated by the left translates of an almost periodic function f contains ex­ actly one constant function. Theorem.

The value of this constant function will, of course, be desig­ nated M(f) . Von Neumann proved the existence and properties of M(f) by completely elementary methods, using only the defi­ nition of almost periodicity. I t will be shorter for us in view of the above theory to demonstrate the existence of M(f) by using the Haar integral on ffir. Since it will turn out that M(f) =

fJ dj.l, 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 SI is the set of finite sums L ci!(aix) such that Ci > 0 and L Ci = 1 . We have to show that M(f) , and no other constant, can be ap­ proximated uniformly by such sums. In view of the isometric imbedding of SI into e(ffir) and the fact that a(G) is dense in ffir, it will be sufficien t to prove the theorem on ffir. Given choose V so that I f(x) - fey) 1 < when xy l � V and choose h � Lv +. E,

E

-

1 70

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

Since mI is compact, there exists a finite set of points a l ) " ' , a n such that H(x) = L h (xai - 1 ) > 0 on mI. The functions gi = hailH thus have the property that gi = 0 outside of Vai and that

Igi' then I If(x) dx - L Ci!(aiY) I I LIgi(x) [f(xy) - f(aiY)] dx I < since I f(xy - f(aiY) I < when gi(X ) � Thus M(f) If is L gi(X)

==

1 . I f Ci

=

=

E

o.

E

=

approximated by the element L cifai of the convex set deter­ mined by Sf . Conversely, if k is any number which can be so approximated, " k - L ci!(aiY ) 1 / 00 < then I k - fey) dy I < follows upon E,

integrating, so that k

=

If

=

I

E

M(f) , q.e.d.

Corollary. Using M(f) as an integral on the class of almost periodic functions, every almost periodic function has a unique L2 expansion as a sum of minimal almost invariant functions.

41E. If G is Abelian, then its associated compact group Ge is both compact and Abelian, and can be readily identified. The minimal almost invariant functions on Ge are, of course, its char­ acters, 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 char­ acter on G if h is a character on Ge• The adjoint of a thus de­ termines an algebraic isomorphism between the character groups of G and Ge• Since Ge 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 GO, where GO is the character group G, but in the discrete topology. On an Abelian group the approximation and expansion theo­ rem states that every almost periodic function can be uniformly approximated by linear combinations of characters and has a unique L2 expansion as an infinite series of characters. In par­ ticular 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:ncneiXnx. This result will become

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

1 71

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

41F. Theorem. A continuous function f on R is almost periodic if and only if, for every E, there exists T such that in every interval of length T there is at least one point a such that I I f - fa 1 100 < 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, TJ such that I I fa - fb 1 100 < E. If f is assumed to be almost periodic and if the set { faJ is E-dense in Sf, then we have only to take [- T, TJ as the smallest in terval con taining the poin ts { ai } to meet the rephrased con­ dition. Conversely, suppose that T can always be found as in the re­ phrased condition. We first show thatf is uniformly continuous . To do this we choose 0 such that I f(x + t) - f(x) I < E if x � [ - T, TJ and I t I < o. Then, given any x, we can choose b � [ - T, TJ so that I I fx - fb 1 1 00 < E and have

I f(x + t) - f(x) I � I f(x + t) - feb + t) I

+ ! feb + t) - feb) I + I feb) - f(x) I < 3 E.

Thus I I f - ft I ! < 3 E if I t I < o. Now let { ah . . . , a n } be o-dense on [- T, TJ. Given a, we first choose b � [ - T, TJ so that I I fa - fb I I < E and then ai such that I ai - b I < 0, and therefore such that I I fb - fai 1 1 00 = I I fb a - f 1 100 < 3 E as above. Thus I I fa - fa. I I � I I fa - fb I I + I I fb - fa; I I < 4E, proving that the set { fa l ' . . . , fan ! i s 4E-dense in Sf and therefore that f is almost periodic. -

i

I

1 T x) dx. lim f( T T -+ 2 Proof. Given E, we choose the finite sets { cd and { ad such that I I M(f) - L: c if(x - ai) 1 100 < E. It follows upon integrating that I M(f) - (L: Ci T f(x - ai) dx)/2 T I < E. If we then T 41G. Theorem.

M(f)

=

I

00

-T

172

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

choose T > TE 1 2T

fT f(x -T

where I O i I < I

= ai

( I I f 1 100 · max I 1 2T

) dx = -

ai

ai

1 ) 1 E, it follows that

iT - ai f(y) dy - T-�

I I I f l i l T < E. Therefore 1

I M(f) - 2T

1 2T

= -

fT fey) dy + O i -T

fT fey) dy I < 2E -T

whenever T > TE, q .e.d. The alternate characterizations of almost periodicity and M(f) 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 indica­ tion 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 Sf of the set Sr of its left translates is a compact metric space. Let U/ be the restriction of the left translation operator Ua ( Uaf = fa-l) to Sf . U/ is an isometry on Sr and an elementary (but not trivial) argument shows that the homomorphism a � U/ of G into the group of isometries on Sf 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 isom­ etries of a compact metric space form a compact group under the uniform norm. Thus the uniform closure Gr of the group { U/ : a � G} is a compact metric group, and the Cartesian product IIf Gf is compact. Let G be the group of isometries T of LAP onto itself such that for every f the restriction Tf of T to Sr belongs to Gr. Each isometry in G determines a poin t of IIf Gr and i t is easy to see that the subset of IIr Gr thus defined is closed and hence compact. That is, G i tself is compact if it is given the following topology : a sub-basic neighborhood of an isometry To � G is specified in terms of a function f � LAP and a positive number as the _ set of isometries T � G such that I I Tf - To' I I < E. Also, under this topology the homomorphism a of G into G defined by a(s) = U8 is continuous. E

COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS

173

Iff � LAP, then I f(x) - fey) I � I I fx - fy 1 100 � I I Ux' - U/ I I · That is, the function F defined by F( Ux') f(x) is uniformly continuous on a dense subset of Gj and hence is uniquely exten­ si ble to the whole of Gj• Since the projection of C on to Gf ( T � Tf) is continuous, we may then consider F to be defined and continu­ ous on the whole of C. Moreover F(a(s)) f(s) by definition. Thus for each f � LAP there exists F � e(C) such that f(s) F(a(s)) . Conversely any continuous F on the compact group C is almost periodic as we have observed earlier, and F(a(s)) is therefore almost periodic on G. Thus C is (isomorphic to) the compact group associated with G. This method of constructing the compact group is essentially that given by WeiI [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 direc­ tions 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.

N O N-COMMUTATIVE TH EORY

42A. The Plancherel theorem. We start with the problem of the generalization of the Plancherel theorem. It may have oc­ curred to the reader that there is a great deal of similarity be­ tween the Fourier transform on an Abelian group and the ex­ pansion 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 top­ ological space G, and with each point a €: G there is associated a minimal translation-invariant vector space Va of bounded con­ tinuous functions on G. For each f €: L l there is uniquely de­ termined a component function fa €: Va such that, for each fixed x, fa(x) is a continuous function of a vanishing at infinity. There is a measure p, on G such that, for each f €: L l n rJ>(G) , fa (x) €: L l (p,) for each x and f(x) = fa(x) dp,(a) . Va is a (finite-dimen­

f

sional) Hilbert space in a natural way, and, iff €: L l n L2 (G), 1 74

1 75

SOME FURTHER DEVELOPMENTS

then

I I f 1 1 2 2 = III fa 1 1 2 dp.(a)

(Parseval's theorem) . Thus the mapping f � fa can be extended to the whole of L2 (G) preserv­ ing this equation, except that now fa is determined modulo sets of J.l-measure O. This is the Plancherel theorem. In each subspace Va there is a uniquely determined function ea characterized as being a central, positive definite, minimal al­ most invariant function. The component fa in Va of a function f E::: L l is obtained as fa = f * ea = Ulea, where U is the left regu­ lar representation. The subspaces Va are invariant under U, and, if Ua is the restriction of U to Va, then the function ea is also given by ea (x) = trace ( ua x) . The componentfa has a simi­ lar alternate characterization, fa(x) = trace ( uax ual) . The Abelian case is simplified by the fact that the subspace Va 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 fa by an integration process rather than a summation process. The compact group case is simpler in the latter respect ; f is an ordi­ nary sum of its components fa. But the subspaces Va are not one-dimensional. It is, of course, impossible without some experience to select from the above rich collection of properties common to the com­ pact 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 Ha, and a vector space of functions f on S such that f(a) E::: Ha for every a E: S and f(a) E::: L2 (S) .

II

II

1 76

SOME FURTHER DEVELOPMENTS

The ordinary L2-completion of this vector space is then a Hil­ bert space H of functions 1 of the above type, and we write

H

=

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

If we are given a separable Hilbert space H and a complete Boolean algebra ill of commuting projections on H, then the standard theory of unitary equivalence can be interpreted as

giving a direct integral representation of H, H that the projections of

ill

=

are exactly those projections Ee asso­

ciated with the measurable subsets C of S, Eel If H

=

fHa dp,(a), such =

il(a) dp,.

fHa 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 E:=

H, then T(x)

=

f( TaXa) dp,(a) .

=

fXa dp,(a)

One of von Neumann's basic

theorems is that, if T commutes with all the projections Ee asso­ ciated with the measurable subsets of the index space S, then T is a direct in tegral. 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 func­ tions lx , y (T) = ( Tx, y) and if it is closed under the involution T*. For any set a of operators, a ' is defined to be the set T of all operators commuting with every operator in a. Then it is a fundamental fact that a *-closed set a of operators is a weakly­ closed 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 lactor. Von Neumann and Murray's dimensional analysis of factors, with the resulting classification of all factors into types In, 100 , lIb 1100 and III, is one of the basic results of the theory, but will not be gone into here. -7

1 77

SOME FURTHER DEVELOPMENTS

on

Now let H

a

and let

be any weakly-closed algebra of bounded operators H .=

f

Ha

dp,(a) be the expression of H as 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 Tn be a se-

quence of operators in

a

which generates

a,

let Tn

=

fTn ,a dp,(OI.)

be the direct integral expression of Tn and let aa be the weakly­ closed algebra of bounded operators on Ha generated by the se­ quence Tn,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

fT dp,(OI.) a

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 fac­ tors aa'. Godement, Mautner and Segal ([21], [37], [38], [45] ; Goaement's major work 'I n th'I S ci"Irect"ton is unpublished at the time of writing) have considered Ll (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 £ and (R are the weak closures of these operator alge­ bras, then £' = (R and (R' = £. Thus the intersection £ n (R is the center of each algebra, and von Neumann's reduction theory implies that H = L2 (G) can be expressed as the direct integral of Hilbert spaces Ha in such a way that each operator A E::: £ is the direct integral of operators Aa acting on Ha, and the weakly­ closed algebra £a generated over Ha by the operators Aa is, for almost all a factor. This analysis of £ into factors is what corresponds to the analy­ sis of L l (G) into minimal parts in the simpler theories, and a Parseval formula is valid ; iff E::: LI n L 2 , then I I f 1 1 2 2 = trace u/ uj*a dOl.. However, these oversimplified statements conceal a 01.,

f

multitude of complications and the reader is referred to the lit­ erature 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 £ n ill is a commutative C*-Banach algebra and therefore is isometric to the space e(mr) of all continuous complex-valued functions on its compact Haus­ dorff space of maximal ideals. The points M E:: mr, subject to certain later identifications, are Godement's indices a. Kap­ lansky [28] has investigated a much narrower class of groups but obtains a much more satisfactory theory as a result. 42B. Representation theory. The above Plancherel theorem can be interpreted as the analysis of the left regular representa­ tion of G into factor representations, and suggests the correspond­ ing problem for arbitrary representations. This is not the same thing as analyzing a representation into irreducible parts, but seems to be in many ways the more natural objective. Very little is known as yet about the general representation theory of locally compact groups beyond the basic facts that sufficiently many irreducible representations always exist and are in general infini te-dimensional. However, there is considerable literature on the representation theory of special groups and of general groups with respect to special situations. The Russians have published a series of pa­ pers (of which we have listed only one, [14]) analyzing the irre­ ducible representations of certain finite-dimensional Lie groups. Mackey [36] has been able to subsume many of these special re­ sults under a generalization of the theory of induced represen­ tations from its classical setting in finite groups to separable uni­ modular locally compact groups. The representation theory of Lie groups has also been explored in its connection with the rep­ resentation theory of Lie algebras. We saw in § 32 how representations of groups are intimately associated with representations of the corresponding L 1 -algebras. In fact, group algebras were first introduced as an aid in the study of the representations of finite groups (see [32] for a short account) . Of course, the representation theory of general Banach algebras is a natural subject for study, quite apart from such motivations. Unfortunately, the results are meagre. We have cited two. One is the theorem of 26F on the representations of a

179

SOME FURTHER DEVELOPMENTS

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 TH EORY

43A. The Laplace transform. Mackey [34] has observed that of the classical Laplace transform can be given the kernel a characterization valid on a general locally compact Abelian group in much the same way as Fourier transform theory is by the general shifted to this setting by replacing the kernel character function The following description is an over­ simplification but will illustrate the kind of step to be taken. We replace the mapping � (x + iy)s, where x and y are fixed, by a continuous homomorphism v 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 v E::: V will be called simply a vector. The kernel is then replaced by the kernel v E::: V, E::: G, and the generalized Laplace transform is the mapping � F where

e(X+iY)8

eiY8

a(s). s

e(x +iy) S

eves>,

F(v)

=

feV(sy(s) ds.

/

s

A complex-valued function defined on a suitable subset of V will be said to be analytic at v if for every U E::: V the limit F(v, u) of of F(v + AU) - F(v) /A exists as the complex variable A approaches O. Leaving aside questions of existence, we see that formally the Laplace transform F of a function / on G is analytic and

F(v, u)

=

feV(S)u(s)/(s) ds.

Mackey has shown that in a suitably restricted L2 sense a func­ tion F defined on V is analytic if and only if it is a Laplace trans­ form. Many other theorems of Laplace transform theory have

1 80

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 sub algebras of Ll (G) which are themselves Ll spaces with respect to measures " larger than" Haar measure. The example consid­ ered 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 � G. Then the set LW of measurable functions] such that f I w < 00 is the space L 1 (J.L) where J.L(A) = w(s) ds, and the

II

i

inequality on w assumed above implies that LW is closed under convolution and is commutative Banach algebra. A simple con­ dition guaranteeing that LW is a subset of Ll (G) is that w be bounded below by 1 . Then every regular maximal ideal of L 1 (G) defines one for LW (see 24B) and the maximal ideal space of Ll (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 OM

�

I

LOO(G) ,j(M) = f(S)OM(S)W(S) ds,

and a simple argument as in 23D shows that the product OM (S)W(S) is a generalized character, that is, a continuous homomorphism of G into the multiplicative group of the non-zero complex num­ bers, and therefore that its logarithm, if i t can be taken, is what we have called in the above paragraph a vector. The transforms j of the elements f � 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

=

{ f:

II f I w < oo } has the same maximal ideal

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

SOME FURTHER DEVELOPMENTS

181

1 938 [4] for the special case o f the real line. Beurling's basic theo­ rem in this direction is roughly that, if I log w(x) 1 ---2 - dx < 00, 1 +x then L is regular and the Tau berian 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 : I I J 1 1 00 = lim I I jn I I I / n. His proof uses the prop­ erties of the real numbers in an essential way and does not gen­ eralize; 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 L I (G) is included in a maximal one, and then can be rephrased again, via orthogonality, as the statement that every non-void transla­ tion-invariant, weakly-closed subspace of LOO 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 Segal­ Kaplansky�Helson theorem, and suggests that we leave the gen­ eral Tauberian problem in the following form. It is required to find conditions which will imply that a closed ideal in a commu­ tative 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

foo

-00

W

1 82

SOME FURTHER DEVELOPMENTS

cases in which it is known that every closed ideal is the intersec­ tion 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 deriva­ tives 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 sup­ ported 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 con­ vergence on compact sets. The continuous homomorphisms of this algebra into the complex numbers are given by exponentials

eat (il(a) = feat dp,(t)) and the regular maximal ideal space is thus

identified with the complex plane, the transforms p, forming an algebra of entire functions. Every primary ideal of this algebra is specified by a complex number ao and an integer n, and con­ sists of the measures p, such that p, and its first n derivatives vanish at ao. And every closed ideal is an intersection of primary ideals. In both of these examples the presence and importance of pri­ mary 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 [3 1]. Levitan observed that the eigen-function expansions of Sturm­ Liouville differential equation theory can be regarded as a gen­ eralization of the Fourier transform theory of group algebras as discussed in Chapter VII. Specifically, let us consider the dif­ ferential equation y" (p(x) - X)y = 0 on the positive real axis, with a boundary condition at x = o. With this ordinary differential equation we associate the partial differential equation U xx - Uyy - (p(x) - p (y) ) u = 0 with initial conditions u (x, 0) = I(x), uy(x, 0) = 0, where p and I are even functions. The solu­ tion u (x, y) = [ Tyf] (x) is regarded as a generalization of the translation of I(x) through y . If generalized convolution is de­ fined in the obvious way an L l convolution algebra is obtained -

SOME FURTHER DEVELOPMENTS

1 83

which has very far-reaching analogies with the L l group algebras. The following topics can be developed : generalized characters (eigen-functions) , posi ti ve definite functions, Bochner's theorem, the L 1 -inversion theorem and the unique measure on the trans­ form 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 i tems expl icitly referred to in the text. For a more complete bibliography see the book of Weil ([48] below) for the years preceding 1 938, and the address of Mackey ([35] below) for subse­ quent years. 1. W. AMBROSE, Structure theorems for a special class of Banach algebras, Trans. Amer. M ath. Soc. 57, 364-3 86 ( 1 945) . 2. R. ARENS, On a theorem of Gelfand and Neumark, Proc. Nat. Acad. Sci. U.S.A. 32, 237-239 ( 1 946) . 3. S. BANACH, Theorie des operations lineaires, Warsaw, 1 932. 4. A. BEURLING, Sur les integrales de Fourier absolument convergentes et leur application a une transformation fonctionnelle, Congres des Mathematiques a Helsingfors, 1 93 8 . 5 . A. BEURLI NG, Un theoreme sur les fonctions bornles et uniJormement con­ tinues sur l'axe riel, Acta Math. 77, 1 27-1 36 ( 1 945) . 6 . H . BOHNENB LUST and A . SOB CZYK, Extensions of functionals o n complex linear spaces, Bull. Amer. Math. Soc. 44, 9 1-93 ( 1 93 8 ) . 7 . H . CARTAN, Sur la mesure de Haar, C. R. Acad. S c i . Paris 2 1 1 , 759-762 ( 1 940). 8 . H . CARTAN and R. GODEMENT, Theorie de la dualite et analyse harmonique dans les groupes abelians localement compacts, Ann. Sci. Ecole Norm, Sup. (3) 64, 79-99 ( 1 947) . 9. P . DANI E LL, A generalform of integral, Ann. o f Math. 1 9, 279-294 ( 1 9 1 7) . 1 0. V . DITKIN, O n the structure of ideals i n certain normed rings, Ucenye Zapiski Moskov. Gos. Univ. Mat. 30, 83- 1 30 ( 1 939) . 1 1 . J. DIXMIER, Sur un theoreme de Banach, Duke M ath. J. 15, 1 057-107 1 ( 1 948) . 1 2. I. GE LFAND, Normierte Ringe, Rec. Math. (Mat. S born i k) N.S. 9, 3-24 ( 1 94 1 ) . 1 3 . I . GE LFAND and M. NE UMA RK, O n the imbedding of normed rings into the ring of operators in Hilbert space, Rec. Math. N.S. 12 (54) , 1 97-2 1 3 ( 1 943) . 14. I. GE LFAND and M. NE UMA RK, Unitary representations of the Lorentz group, Izvestiya Akademii Nauk SSSR. Sera Mat. 1 1 , 4 1 1 -504 ( 1 947) . 1 5 . I. GE LFAND and D. RAIKOV, On the theory of character of commutative topological groups, C. R. (Doklady) Acad. Sci. URSS 28, 1 9 5- 1 9 8 ( 1 940) . 1 6. I. GE LFAND and D. RAIKOV, Irreducible unitary representations of locally bicompact groups, Rec. M ath. (Mat. Sbornik) N.S. 13, 30 1-3 1 6 ( 1 943) . 1 7 . I . GE LFAND, D . RAIKOV, and G . SI LOV, Commutative normed rings, Uspehi Matem aticeskih Nauk. N.S. 2, 48-1 46 ( 1 946) . 1 85

1 86

BI BLIOGRAPHY

tt1B. A. GLEASON, A note on locally compact groups, Bull. Amer. Math. Soc. 55, 744-745 ( 1 949) . 1 9. R. GODEMENT, Les fonctions de type positif et la theorie des groupes, Trans. Amer. M ath. Soc. 63, 1 -84 ( 1 948) .

20. R. GODEME NT, Sur la theorie des representations unitaires, Ann. of M ath. (2) 53, 68- 1 24 ( 1 9 5 1 ) . 2 1 . R . GODEME NT, Memoire sur la theorie des caracteres dans les groupes 10calement compacts unimodulaires, Journal de Math. Puree e Applique 30, 1 - 1 1 0 ( 1 95 1 ) . 22. A . HAAR, Der Massbegriff i n der Theorie der kontinuierlichen Gruppen, Ann. of M ath. 34, 1 47- 1 69 ( 1 933) . 23. P. HALMOS, Measure Theory, Van Nostrand, New York, 1 950. 24. H . H E LSON, Spectral synthesis of bounded functions, Ark. Mat. 1, 497502 ( 1 95 1 ) . 2 5 . N . JACO BSON, The radical and semi-simplicity for arbitrary rings, Amer. J . Math. 67, 300-320 ( 1 945) . 26. I. KAP LANSKY, Dual rings, Ann. of M ath. (2) 49, 689-701 ( 1 948) . 27. I. KAP LANS KY, Primary ideals in group algebras, Proc. Nat. Acad. Sci. U.S.A. 35, 1 33-1 36 ( 1 949) . 28. I. KAPLANS KY, The structure of certain operator algebras, Trans. Amer. Math. Soc. 70, 2 1 9-25 5 ( 1 95 1 ) . 29. M . KREIN, Sur une generalisation du theoreme de Plancherel au cas des integrales de Fourier sur les groupes topologiques commutatifs, C. R. (Doklady) Acad. Sci. URSS (N.S.) 30, 484-4 8 8 ( 1 94 1 ) . 30. M . KREIN and D . MI LMAN, O n extreme points of regular convex sets, S tudia Mathem atica 9, 1 33- 1 3 8 ( 1 940) . 3 1 . B. M. LEVITAN, The application of generalized displacement operators to linear differential equations of the second order, Uspehi M atern . Nauk (N.S.) 4, no. 1 (29) , 3-1 1 2 ( 1 949) . 32. D. E. LITTLEWOOD, The Theory of Group Characters and Matrix Representa­ tions of Groups, Oxford Univ. Press, New York, 1 940. v3"3. L. H. LOOMI S, Haar measure in uniform structures, Duke M ath. J. 16, 1 93-208 ( 1 949) . v3"4. G. MACKEY, The Laplace transform for locally compact Abelian groups, Proc. Nat. Acad. Sci. U.S.A. 34, 1 56-1 62 ( 1 948). 3 5 . G. MACKEY, Functions on locally compact groups, Bull. Amer. Math. Soc. 56, 3 8 5-4 1 2 ( 1 950) . 36. G. MAC KEY, On induced representations of groups, Amer. J. M ath. 73, 576-592 ( 1 95 1 ) . 37. F . 1 . MAUTNER, Unitary representations of locally compact groups I, Ann . of M ath. 5 1 , 1-25 ( 1 950) . 3 8 . F. I. MAUTNER, Unitary representations of locally compact groups II, Ann. of Math., to be published. 39. J . VON NEUMANN, Almost periodic functions in a group, Trans. Amer. Math. Soc. 36, 445-492 ( 1 934) . 40. J. VON NEUMA NN, On rings of operators. Reduction theory. Ann. of M ath. 50, 40 1-485 (1 949) . 4 1 . D. A. RAIKOV, Generalized duality theorem for commutative groups with an invariant measure, C. R. (Doklady) Acad. Sci. URSS (N. S.) 30, 5 89-59 1 ( 1 94 1 ) .

BIBLIOGRAPHY

1 87

42. D. A. RAlKOV, Harmonic analysis on commutative groups with the Haar measure and the theory of characters, Trav. I nst. Math. Stekloff 14 ( 1 945) . 43. L. SCHWARTZ, Theorie generale des fonctions moyenneperiodiques, Ann. of

Math. 48, 8 57-929 ( 1 947) . 44 . I . E. S EGA L, The group algebra of a locally compact group, Trans. Amer. Math. Soc. 61, 69-105 ( 1 947) . 45. I. E. SEGAL, An extension of Plancherel's formula to separable unimodular groups, Ann. of Math. (2) 52, 272-292 ( 1 950) . 46. M. H. STONE, The generalized Weierstrass approximation theorem, Math. Mag. 21, 1 67-1 84, 237-254 ( 1 948) . 47. M. H. STONE, Notes on integration: II, Proc. Nat. Acad. Sci. U.S.A. 34, 447-45 5 ( 1 948) . 48. A. WEl L, L'integration dans les groupes topologiques et ses applications, Paris, 1 93 8 . 4 9 . H . WHITNEY, On ideals of differentiable functions, Amer. J. Math. 70, 635-658 ( 1 948) .

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

Absolute continuity,

40

Adj oint, 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, Annihilator, 20 Approximate identity, 124 Axiom of choice, 2 Baire function,

78

32

Banach algebra, 16, 48 C* algebra, 28, 88, 91 H* algebra, 100 regular Banach algebra, 82 self-adj oint Banach algebra, semi-simple Banach algebra, 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, Carrier, 84 Cartesian product, 10

88 76

88, 91

Central element (function) ,

157, 175

105,

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 Conj ugate space, 18 Continuity, 4 Convolution, 120-122 Daniell integral,

29 ( § 12)

Directed set, 1, 2 Direct integral, 176 Direct sum, 16 Di tkin, condition of, Dual algebra, 161

Equivalent functions,

86 36, 37

Extendable linear functional, Factor, 176 Finite intersection property,

Fourier series, 153 Fourier transform,

188

(§ 38) 71, 135

96 5

1 89

INDEX

Fubini theorem, 44 Function algebra, 53 inverse closed, 54 regular, 57 self-adj oint, 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 Kernel,

56

Laplace transform,

74, 179

L-bounded function, 33 Linear functional, bounded,

18

Linear functional (Cont.) extendable, 96 positive, 96 unitary, 99 Linear order, 2 Linear transformation, bounded, 15 self-adj oint, 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 (§ Mean value of von Neumann,

23) 169 function) , 36

Measurable set (and 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, L-monotone family, 33 Neighborhood,

4

32

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 Normed linear space, 13 Null function, 35 Null set, 35 Null space, 16 One point compactification, Open set, 3 Orthogonal complement, 25

7

1 90

INDEX

Parallelogram law,

23

Parseval's equation, 155, 175 Partial order, 1 Plancherel theorem, 99, 145, 174 Poisson summation formula, 152 Pontriagin duality theorem, 140,

151

Positive definite element, 98 Positive definite function, 126, 142 Positive linear functional, 96 Primary ideal, 181 Proj ection, onto a coordinate space, 11 in Hilbert space, 25, 26 Pseudo-norm, 52 Quotient space, Radical,

14, 110

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 Scalar product, 24 Schwarz inequality,

24

Self-adj oint element, 90 Semi-direct product, 119 Semi-simple ring (algebra) ,

76

Silov's theorem, 151 Spectral norm, 75

62,

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 Tb 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 fun'ctional,

40

Weak topology, 10 on a conjugate space, 22 Weight function, 74, 180 Wiener Tauberian theorem,

148, 180

Zorn's lemma,

2

85,