Measure and Integration Theory on Infinite Dimensional Spaces

MEASURE AND INTEGRATION THEORY ON INFINITE-DIMENSIONAL SPACES

This is Volume 48 in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAULA. SMITHAND SAMUELEILENBERG A complete list of titles in this series appears at the end of this volume

Measure and Integration Theory on Infinite-D imensional Spaces ABSTRACT HARMONIC ANALYSIS

XIA DAO-XING FUDAN UNIVERSITY SHANGHAI

Translated by Elmer 1. Brody DEPARTMENT OF MATHEMATICS THE CHINESE UNIVERSITY OF HONG KONG

@

ACADEMIC PRESS

NewYork and London

1972

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CONTENTS

FOREWORD PREFACE

Chapter I Some Supplementary Background in Measure Theory

vii ix

1

2 24 30

$1.1. Some Measure-Theoretic Concepts $1.2. Localizable Measure Spaces $1.3. The Kolmogorov Theorem $1.4. Kakutani Distance

41

Chapter II Representation of Positive Functionals and Operator Rings

49

92.1. $2.2. $2.3. $2.4.

50 62 71 79

Topological Algebras with Involution: Fundamental Concepts Representation of Positive Functionals on Seminormed Algebras Weakly Closed Operator Algebras: Fundamental Concepts Representation of Commutative Weakly Closed Operator Rings

V

vi

CONTENTS

Chapter Ill Harmonic Analysis on Groups with Quasi-Invariant Measures

103

$3.1. Basic Properties of Quasi-Invariant Measures $3.2. Characters and Quasi-Characters $3.3. Integral Representation of Positive Definite Functions on Groups $3.4. La-Fourier Transforms

105 135 164 185

Quasi-Invariant Measures and Harmonic Analysis on Linear Topological Spaces

213

Quasi-Invariant Measures on Linear Topological Spaces Linear and Quasi-Linear Functionals on Linear Spaces Continuous Positive Definite Functions on Linear Topological Spaces

214 232 254

Chapter IV $4.1. $4.2. $4.3.

Chapter V

Gaussian Measures

$5.1. Some Properties of Gaussian Measures $5.2. Equivalence and Perpendicularity of Gaussian Measures 55.3. Gaussian Measures on Linear Spaces $5.4. Fourier-Gauss Transforms

Chapter VI

Representation of Commutation Relations in Bose-Einstein Fields

280 28 1 295 31 1 326

335

Representations of the Commutation Relations in Quantum Mechanics Quasi-Invariant Measures Applied to Representations of the Commutation Relations in Bose-Einstein Fields $6.3. The Relation of Gaussian Measures and Rotationally Invariant Measures to Conventional Free-Field Systems

369

Appendix I Background Material on Topological Groups and Linear Topological Spaces

381

$1.1. Pseudometrics, Convex Functions, and Pseudonorms $1.2. Some Properties of Semicontinuous Functions $1.3. Countably Hilbert Spaces and Rigged Hilbert Spaces

381 384 387

Appendix II Background Material on Functional Analysis in Hilbert Spaces

394

$6.1. $6.2.

336 352

$11.1.

Operators of Hilbert-Schmidt Type, Nuclear Operators, and Equivalence Operators $11.2. Tensor Products of Hilbert Spaces $11.3. Unitary Representations of Groups

394 404 409

Notes and References to the Literature

41 3

Bibliography

417

INDEX

421

FOREWORD

T h e present book is a compendium of results which are mostly of fairly recent vintage, and the theory discussed herein is very much in a state of flux. Moreover, the original book seems to have been compiled and published rather hurriedly. Thus, there were a great number of inaccuracies in the original, ranging from typographical errors to very substantial gaps in the mathematical reasoning. I have made some effort to correct these inaccuracies; in many cases, I have altered and expanded proofs without burdening the reader with a tedious explanation of how and where the revised version deviates from the origina1. I n many cases where doubts or difficulties still remain, I have called attention to these by footnotes. However, I cannot claim either completeness or consistency in this editorial work. Especially as regards Chapters I11 and IV, I feel dubious as to how much expenditure of effort would be justified in revising or developing the theory, at least until such time as more applications may be demonstrated; in this connection, the appearance of a subsequent volume, as indicated in the author’s preface, would be most enlightening.

ELMERJ. BRODY vii

This Page Intentionally Left Blank

PREFACE

T h e study of measures and integrals on infinite-dimensional spaces arose from the theory of stochastic processes, particularly the theory of Wiener processes. I n recent years, the subject has been intimately connected with research on characteristic functionals, limit theorems, sample spaces, and generalized stochastic processes. Even more noteworthy is the fact that questions of integration on infinite-dimensional spaces have, during the last ten-odd years, appeared in many scientific fields, such as quantum mechanics, quantum field theory, statisticaI physics, thermodynamics of irreversible processes, turbulence theory, atomic reactor computations, and coding problems. However, the application of integration on infinite-dimensional spaces to these fields has encountered many profound difficulties, and a lack of adequate techniques. Thus, it seems that further study of this new subject is amply justified. Heretofore, there have appeared no introductory books on this topic, either in this country or abroad. As far as the author knows, there has been only a volume of lecture notes, “Integration of Functionals,” written by K. 0. Friedrichs, H. N. Shapiro,et al., 1957, and still unpublished. Moreover, except for Wiener integrals, the mathematical theory of measures and integrals on infinite-dimensional spaces largely began to develop only after 1956. As the mathematical background involved in the literature of this theory is rather extensive, the novice is likely to find the going somewhat difficult. Therefore, the author

X

PREFACE

has been so bold as to write the present book with the hope of smoothing the way for Chinese comrades undertaking research in this direction. This volume is primarily devoted to introducing abstract harmonic analysis. It essentially consists of three parts. T h e first part is concerned with the representation of positive functionals and operator rings (Chapter 11), which constitutes the basis of abstract harmonic analysis. Although this topic cannot be regarded as lying entirely within the domain of infinite-dimensional measure and integration theory, the two are intimately related. The second part deals with abstract harmonic analysis on pseudo-invariant measure spaces (Chapters I11 and IV) ; except for just a few theorems, the results given here were, for the most part, obtained in China. This kind of harmonic analysis may provide tools for the further investigation of measure and integration on infinite-dimensional spaces. In the third part, we discuss a mathematical problem arising in quantum field theory, i.e., the representation of commutation relations in Bose-Einstein fields (Chapter VI) ; here, applications of the theory developed in the first two parts are given. In addition, one chapter (Chapter V) is devoted to another important example of measure theory on infinite-dimensional spaces, i.e., Gaussian measures. In a subsequent volume, we shall deal with the so-called continual integral problems which appear frequently in the applications of integration theory on infinite-dimensional spaces, as well as functional variational equations and various other applications. We assume that the reader is familiar with the treatise of Halmos [I], or its equivalent, and has the basic knowledge of functional analysis which may be found in ordinary textbooks on that subject. It is also expected that the reader has some acquaintance with the basic notions of topological spaces, topological groups, and linear topological spaces ; in this connection, he may consult, for example, Guan Zhao-zhi [I]. Chapter I and Appendices I, I1 of the present book also provide some supplementary background material. Owing to the author’s limitations, and the rather short time taken to write this book, its shortcomings are undoubtedly numerous, and errors inevitable. The reader’s criticisms will be welcomed. Part of the manuscript of this book was read by Professor Zheng Ceng-tong of Zhongshan University, who offered valuable comments. T h e teachers and research students of the Functional Analysis Group, Function Theory Teaching and Research Section, Fudan University Mathematics Department, also offered valuable opinions, especially Comrade Yan Shao-zong. For these contributions, I hereby express my thanks. XIA DAO-XING

CHAPTER

SOME SUPPLEMENTARY BACKGROUND

IN MEASURE THEORY

The measure-theoretic concepts and results used in this book may, for the most part, be found in Halmos’ Measure Theory, and will be directly applied in the sequel without additional explanation. However, certain supplementary measure-theoretic results, not included in Halmos’ book, will be introduced in the present chapter; these results will also be essential in the subsequent chapters. At some points in this book, we shall require the discussion of measures which are not a-finite.1 However, non-a-finite measures in general are not well behaved (e.g., the Radon-Nikodyn theorem is not generally valid for such measures). Therefore, we shall in $1.2 investigate localizable measures, which are not necessarily a-finite, but which do retain certain desirable properties of a-finite measures. The measures ordinarTranslator’s note: The term o-finite, as used by the author, means totally o-finite in the sense of Halrnos [l]. This distinction is an important one in certain parts of this book. For example, according to the author’s terminology, a Haar measure is pseudou-tinite, but not necessarily a-finite. 1

2

I.

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

ily used on groups are localizable, so that localizable measures, in fact, constitute a fairly broad class. Some rather deeper properties of localizable measures will be introduced in $2.4. In $1.3 we shall introduce the Kolmogorov theorem. This is a fundamental theorem concerning the construction of measures on infinite-dimensional spaces from given measures on finite-dimensional spaces. We shall present this theorem in a very general form, related to the notion of a projective limit of locally convex linear topological spaces; in this form, it can be used for the construction of measures on locally convex linear topological spaces, starting from given measures on Banach spaces. In $1.4, we introduce Kakutani inner measure, which plays an important role in the study of equivalence of measures on product spaces, as well as in the study of quasi-invariant measures.

.

51.l Some Measure-Theoretic Concepts

l o Extension and Restriction of Measures We shall introduce certain generalizations of the usual notion (see Halmos [I]) of “measurable set.”

Definition l,l.l.2 Let ( G , 23) be a measurable space. Let A C G, and suppose that, for every B E 8, we have A n B E 8 . We then say that A is measurable with respect to (G, 8).We denote the totality of such measurable sets by 8. Clearly, 8 C 8,and b is a a-algebra on G. If 8 is an algebra, then 8 = B. Let f be a real (complex) function on G. If, for every Bore1 set A of the real line (complex plane), we have {g If (g)E A} E @, we say that f is a measurable function on ( G , 23). Definition 1.1.2. Let (G, S,p) be a measure space. Define a set function p on (G, 8), as follows. For A E %, @(A)= S

U p~( A

n B).

BE

We call j i the extension of p. It is easily seen that, if A E 8,then @(A)= p ( A ) . Consequently, we shall, in the sequel, denote @ simply by p, without danger of confusion. Translotor’s note: It should be recalled that, in Halmos’ terminology, a o-ring need not be a o-algebra, that is, B need not contain G itself.

8

1.1. Some Measure- Theoretic Concepts

3

In what follows, any measure space (G, 8,p) will, whenever necessary, be extended to (G, %,-p). Again, extend (G, ‘23, p) to a complete measure space (G, 8*,p*). Iff is measurable with respect to (G, 8*), we say that f is a measurable function on (G, 8 , p). If B E B and p ( B ) = 0, we call B a p-null set, or simply a null set.

Definition 1.1.3. Let (G, 8, p) be a measure space, A C G, and let 8, = { E n A I E E % } . We call 8, the restriction of 8 to A. If there exists a C E 8 such that the inner measure

we may define a set function pA on 8, as follows. For E E 8, nE)

= p(E n C).

(1.1.2)

We call pA the restriction3 of p to A.

Lemma l.l.l+4 Let (G, 8,p ) be a measure space, and let A be a subset of G satisfying condition (1.1.1) for a given C E 8. Then the restriction pa of p to A is well defined, and (A, 8,, p,) is a measure space. PROOF. We need only prove that pa is well defined; the rest is obvious. Let E, F E 8,with A n E = A nF. T o justify the definition of pA , we need only show that p ( E n C ) = ,u(F n C ) .

( 1.I .3)

We may assume that E C F, for otherwise, we could replace F by E U F. Then, from A n E = A n F and E C F, it follows that A n (F - E )

= 0,

whence C - A 3 (C n F ) - (C n E). But p*(C - A) p ( ( F n C) - ( E n C)) = 0, SO that (1.1.3) holds. 3 Translator’s note: Notice that See Halmos [l].

depends upon the choice of C .

= 0,

therefore

4

I.

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

2O The Function Space f?kz(Q) We shall have occasion to use certain abstract functions taking values in a Hilbert space. We first introduce the following notions.

Definition 1.1.4. Let H be a Hilbert space, 52 = (G, B, p) a measure space, and f an abstract function on Q such that (i) for every g E G, f (9) E H, (ii) for every u E H, the numerical valued function (f (g), u), g E G, is a measurable function on 52, and (iii) the range of values {f (g) 1 g E G) is contained in a separable subspace of H. We say that such an f is measurable, and denote the totality of such functions by M ( H , Q). I t is easily seen that M ( H , Q) forms a linear space with respect to ordinary addition of functions and multiplication by constants. Lemma 1.1.2. Let {eA, h E A} be a complete orthonormal system in the Hilbert space H. Then, a necessary and sufficient condition for f to belong to M ( H , Q) is that there exist a sequence {A,} C A and a sequence of measurable functionsAn on 52 such that (1.1.4)

PROOF. Assume that f satisfies the above condition. Then, the values off are contained in the separable subspace spanned by {eAn,n = 1,2, ...}, and (f (g), u ) = C fAn(g)(ek, u ) is measurable. Conversely, suppose that f is measurable, let M be a separable closed linear subspace containing the range off, and let {Vk) be a complete orthonormal system in M. For each k, there is a sequence (hLk') C A, such that

Therefore, the range off is contained in the separable closed linear subspace spanned by { e A p , k, n = 1, 2, ...}. Since (f,e A y ) is a measurable function on Q, and since m

the condition of the lemma is satisfied. ]

Corollary 1,1.3. If V, f E M ( H , a), then ( f ( g ) , rp(g)) is a measurable function on 52. I n particular, 11 f (g)llzis a measurable function on 52.

I . 1. Some Measure- Theoretic Concepts

5

PROOF. By Lemma 1.1.2, there is a sequence (e,,) such that (1.1.4) holds, therefore, ( f ( g ) ,d g ) ) =

1(f(g),e*n)(v(g),eAn),

whence it follows at once that ( f ( g ) , y ( g ) ) is a measurable function. ]

Definition 1.1.5. Let H be a Hilbert space, and let Q = (G, 23, p) be a measure space. Let 2 2 ( H , be the totality of functions in M ( H , SZ) which satisfy the condition

a)

( 1.1.5)

and define an inner product on !i?2(H,SZ) as follows5:

(f,PI =

1,

(1.1.6)

( f ( g ) ,9 w 449.

We let L2(SZ)(or L2(SZ))denote the usual space of measurable quadratically integrable functions on a.

Theorem 1.1.4. Let {e, , h E A } be a complete orthonormal system in H , and let HA = { f ( g )e, I f EL,(SZ)}. Then (1.1.7)

PROOF. Let f~ !P(H, SZ). By Lemma 1.1.2, there is a sequence {An} C A such that (1.1.4) holds. Since IfAk(g)I Ilf(g)ll, it follows that hk€L,(Q), that is, h k ( g )eAkE HAk. Therefore, f E W @ HA,. This shows thatf belongs to the right-hand side of (1.1.7). ] Notice that, if f,,(*) eAkE HA,, k = 1, 2 ,..., and if

<

c I l f A * ( h k 112 < a, then, forming f

1

Ja

EM(H,

llf(g)//2 &(g)

Hence, f~

!i?2(H,Q).

SZ) in accordance with (1.1.4), we have

=

1

R

IfAk(g)l2d d g ) =

//fAk(*)eAk

/I2 <

From this, we easily deduce the following result.

By Corollary 1.1.3, (f(g), y(g)), g E G, is a measurable function on Q. Moreover, by condition (1.1.3, IIf(g)ll I1 ddll M g ) < a,whence

SG

so that (f,p) is well defined. It is then easily verified that (f,p) is an inner product on P ( H , Q).

6

I.

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

Corollary 1.1.5.

f?(H, Q) is a Hilbert space.’ Clearly, the concrete form of H has little bearing upon the properties of Q2(H, Q); what is important is the dimension of H, that is, the cardinality of a complete orthonormal basis for H. If H is k-dimensional, we shall write Hk for H and denote !P(H, by f?k2(Q). In particular, when k = 1, H , may be identified with the real line (or the complex plane), and !2,z(Q) is simply L2(Q). As usual, D(Q)(or LJQ)), p 2 1 , will denote the Banach space of all pth power integrable measurable real-valued (or complex-valued) functions on Q, with the usual linear operations and the norm

a)

Also, &(a) (or La(Q)) will denote the Banach space consisting of all the essentially bounded measurable functions on Q, with the usual operations and the norm

3O

Determining Sets

Definition 1.1.6. Let (G, b) be a measurable space, where b is a a-algebra. Let 2) be a family of measurable functions on (G, b), and suppose that there exists no o-algebra b, C b, b, # 23, such that 9 constitutes a family of measurable functions on (G, b,). We then say that B is a determining set (of functions) on (G, b), and that b is the o-algebra determined by 3 on G. It is easily seen that, if G is a set and B is any family of functions on G, then there exists a unique o-algebra 2 3 determined by 9. I n fact, we need only let 8 be the smallest a-algebra which contains all sets of the form f -l(C), where f E 9 and C is a Bore1 set in the complex plane. Definition 1.1.7. Let Q = (G, b,p) be a measure space and 9 a family of measurable functions on Q. For any a-finite set A of Q, let 23, denote the o-algebra determined by 9 on A. Suppose that, for every such A, every measurable set of Q which is contained in A differs from some set of 23, by a p-null set. We then say that 9 is a determining set (of functions) on $2. @

Translator’s note: Of course, Theorem 1.1.4 is meaningful only in the context of

this corollary.

1.1, Some Measure- Theoretic Concepts

7

Clearly, if ID is a determining set on (G, b),then ID is also a determining set on (G, d,p).

Lemma 1.1.6. Let ID be a family of bounded measurable realvalued functions on the measure space 52, such that ID, with respect to the usual operations, forms an algebra containing the unit element 1. Suppose also that ID is a determining set on 52. Choose any p ELI(Q),p 3 0, and let L2(52,p ) be the space consisting of all measurable real-valued functionsf on Q which satisfy the condition

Then 3 is dense in L2(Q,p ) with respect to the norm PROOF. Let 6 be the totality of sets of the form'

Ilfll.

n

n {x I m)

E (aj

b,i~

fj

E

(1.1.8)

ID,

j=1

and let 5 be the collection of all finite unions of sets in 6. Then 5 is an algebra. In fact, it is obvious that G E 5, and that the union and intersection of any finite number of sets of 5 also belongs to 5.T o show that 5 is an algebra, it only remains to prove that the complement of any set of 5 also belongs to 5.I t obviously suffices to prove that the complement of any set of G belongs to 5,but this fact follows at once from the formula

n

=

u

n

{x I fj(x) E

(-a, ajI>

u

5=1

j=1

{X

I fd.1

E (bi 9

a)).

Hence, 5 is an algebra. Let IDo be the closure of ID in L2(Q,p). Then, since 3 is linear, IDo is a closed linear subspace. We now proceed to show that, for every E E 5, the characteristic function C , of E belongs to Do. Let fi,...,fm E ID. Then, there is a positive number .$ such that, for all g E G, I fj(g)l .$, j = 1, 2 ,..., n. On the interval [-5, 51, define the functions

<

'

Here, a, < b, , and, when b, greater than a , .

=

co, then ( a j ,b,] means the totality of real numbers

8

I.

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

It is easy to show that there exists a sequence of polynomials {pmi ;j = 1,..., n; m = 1, 2 ,...}, such that

Eyt I P m i W < 2,

(1.1.9)

and, for every j , and every x E [-t, t],

If E is the set defined by (1.1.8), it is easily seen from (1.1.10) that n

n

fl

lim Pmi(fj(g)) m-m . 3=1

=

n #dfi(g))

(1.1.11)

= cE(g)*

j=1

Since D is an algebra containing the unit element 1, we have n Vm(g) = rIj=lPm,(f,(gN E 3. BY (1.1.91, I vm(g)l 2". Hence, by the Lebesgue dominated convergence theorem, we obtain

<

Thus, if E E 6, then CEE Do. If E l , E, by the formula CE,,E~= CE,

+

cE,

E

6, then El n E,

- cE,nE,

E

6. Hence,

9

and the linearity of Do,it follows that CE1,E, E Do.Consequently, CEE Dofor every E E 5. Suppose that Do #L2(SZ, p). Then there is a nonzero vector y EL,(SZ,p ) such that v IDo.Hence, for every E E 8,

J

VP dP =

J CEVP dP

= 0.

( 1.1.12)

By virtue of the countable additivity of the integral, it follows that (1 .l.12) holds for every set E in the smallest a-algebra g1 containing 3. Let A = {g I p(g) > 0); then A is a a-finite set of 52. Since D is a determining set, there exists, for every measurable set F C A, a set E E g1 such that E n A differs from F by a p-nu11 set. Hence, by (1.1.12), we obtain

J

F

VP dP =

J

EnA

VP dP

=

J

V f dP = 0.

Therefore, y(g) = 0 for almost all g E A, that is, g~ is the zero vector of L 2 ( 0 ,p). This contradiction proves that Dc= L2(Q,p). 3

1.1. Some Measure- Theoretic Concepts

9

40 Measures on Product Spaces Notice that, although Halmos [ 11 considers only products of countably many measure spaces, his method of treatment can also be used to define the product of arbitrarily many measure6 spaces. We shall not give a detailed account of this construction here. I n what follows, the product of the family of measure spaces (52, = (G, , d, pa),01 E 2l) will be denoted by Xaca 52, 9 or by (XoreaG, , Xnea d a 7 Xu=% pa). We proceed to mention a few obvious facts concerning such products. Let (G, d,pk), K = 1,2, and ( H , 8, v k ) , K = 1, 2, be measure spaces. Then, for the measure pl x v , on (G x H , 23 X 5 ) to be absolutely continuous with respect to p, x v 2 , it is necessary9 and sufficient that p, p, , v1 v, . I n this case, we havelo

<

<

(1 .l. 13)

Let (52, = (G,, B , , p,), n = 1, 2,...} be a sequence of probW ability measure spaces, and let 52 = (G, 23, p) = X,=l 52, . Let lZn) = x;=, 52, = (G(n),d ( n ) , p(,)). For everyfEL2(Qn), define a function on 52 by means of the correspondence g + f ( g ( " ) ) , where g = {g, , g, ,..., g, ,...} E G and gcn) = {g, ,...,g,} E G(,). This function clearly belongs to L2(52). I n this way, L2(Li""))is imbedded as a closed linear subspace of L2(sZ).Let P, be the operator projecting L2(52)onto L2(52(")). We then have the following lemma.

Lemma 1.1.7. {P,} converges strongly to the identity operator I . PROOF.Since we obviously have P, ,< P, < ,< P, < we need only prove that Q = U:=lL2(52(n))is dense in L2(52).Let TI denote the totality of bounded measurable real-valued functions in Q; clearly ID is a real algebra containing the unit element 1. Now, for any n, and any n-dimensional Bore1 set E, let -.a,

= {g

Ig

= ( 8 , ,g, 9 . -

>

E

G,

{g,

I..',

g7J

E

a.

Then, the characteristic function Cg E L2(52(")),hence Cg E ID. Moreover, the totality of such sets E generates 93. Consequently, 3 is a determining set on 52. Therefore, by Lemma 1.1.6, any real-valued function in L2(52)can be approximated by elements of TI. Since ID C Q, it follows easily that, in either the real or complex case, Q is dense in L2(52). ] Translator's note: If the number of measure spaces is infinite, then, clearly, one must require that all but a finite number of them be probability spaces, that is, pa(Qa) = 1. Translator's note: If, say, p1is identically zero, we can draw no conclusion regarding vl. lo Translator's note: Presumably, all the measures concerned are assumed to be o-finite.

I.

10

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

50 Direct Sums of Measures

Definition 1.1.8. Let Q, = (G, , 9, , pa), 01 E be a family of measure spaces, where {G, , 01 E a} is a family of pairwise disjoint sets. Let G = UaEX G, . Let 9 be the totality of sets of the form

u A,,,, W

A

=

A,,,E~,,,, {al, a 2,..., a,

,...}CW.

(1.1.14)

v=l

Define the set function p on (G, 9) as follows: if A is of the form (1.1.14), then m

A4

=

c

CLoJAay).

"4

Then we say that SZ = (G, 23, p ) is the direct sum of the measure spaces {Q, , a E a). If, moreover, p,(G,) < co for all 01 E 2l, we say that {G, , a E a}is a partition of SZ. Clearly, the SZ described in Definition 1.1.8 is a measure space.

Lemma 1 .I .8. Let Q = (G, 9, p ) be the direct sum of the family of measure spaces {(G, , 9, , pa), a E a}.Then B E 8 if and only if, for every

01

E

a,

B n G,

E

6,,

(1.1.15)

and, in that case,

0) = 1

n Gal.

(1.1.16)

as%

PROOF. If B satisfies condition (1.1.15), then, for any A of the form (1.1.14), we obviously have m

A nB

=

U (A," n B), v=l

whence, by (1.1.19, A," n B E versely, if B E 8,A , E 23, , then ( B n G,) n A,

C 9, and therefore B E 8. Con=B

n A, E 9,

but B n A, C G,, hence B n A, E B , , thus, (1.1.15) holds. It remains to prove (1.1.16). Let A E 9 be of the form (1.1.14). Then P(B n 4 =

whence we get

2 hi(B n 4i')< c I l a P n G,),

c

CLW = ASUP CL(B n 4 < CL4B n GJ. EB

(1.1.17)

1.1. Some Measure- Theoretic Concepts

11

If the right-hand side of (1.1.17) is a finite number, then there are at most countably many indices a such that pa(Gan B ) > 0; denoting these indices by a1 ,..., a, ,..., we have

1pa(Ga n B ) = C pai(Gai n B).

(1.1.1 8)

rn

If the right-hand side of (1.1.17) is co, one can also find indices a1 ,..., a, ,... E % such that (1.1.18) holds. However, pai(Ga, nB) = p(G,, n B), hence P(B)2

C p(Gq n B ) 2 1p a i ( G a d n B)*

(1.1.19)

"i

Combining (1.1.17), (1.1.18), and (1.1.19), we obtain (1.1.16). ]

Corollary 1.1.9. Under the conditions of Lemma 1.1.8, let B E $3; then B is a p-null set if and only if, for all a E %, pe(B n G,)= 0. Example 1.1.1. Let D = (GI 23, p) be a measure space. If there is a sequence G,, rz = 1, 2,..., of disjoint sets of 23, such that 00 G = Unel G, , then {G, , n = 1, 2,...} is a partition of SZ. In general, let L? = (G, 23, p) be a measure space, and let {G, , a E 'ill} C 23 be a family of disjoint sets such that G = UaEX G, . If % is not countable, then {G, , a E %> is not necessarily a partition of D.For example, if G is the interval [0, I], and p is the ordinary Lebesgue measure on G, then {{a}, a E [0, I]} is not a partition of [0, 11. Let D, = (G, , 23, , pa), a E '2I be a family of measure spaces, and let SZ = (G, 8,p) be the direct sum of this family. For each a E %, let fa be a given measurable function on Q, = (G, , Brn , pa).Define a function f on G as follows: if g E G, , then f ( g ) =fXg).

For any Bore1 set A and any a E %, the set { g If(g)

E

A ) n Ga = (g I f a ( g )

E

A)

is measurable, hence f is measurable. Moreover, it is easy to prove the following lemma.

Lemma 1.1.10. Let D = (G, 8,p ) be the direct sum of the family of measure spaces {D,= (G, , 8,, pa), a E a}.Extend each fa eL2(Da)to a function on G by defining its values on G - G, to be zero. Then,

12

I.

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

60 Measures on Groups

I n the sequel, we shall frequently make use of the following type of measure on a group.

Definition 1.1.9. Let G be a group, 8 a a-ring of subsets of G, SZ = (G, 8, p ) a measure space. Suppose there is a subgroup Go of G such that each of the left (right) cosets {G, , ~1 E 'Ql) of Go in G belongs to 8 and is a-finite. Let b,, pa be the restriction of 8,p to G,, and suppose that SZ is the direct sum of the {(G, , 8, , pa), cy E a}. We then say that SZ is pseudo-a-finite. Definition 1.1.10. Let G be a topological space, (G, 8, p) a measure space. If, for every xo E G, there is a neighborhood V of xo such that V E 8 and p( V) < 03, then (G, b,p) is said to be locally finite. For example, if G is a locally compact group, the Haar measure space on G is locally finite. If (G, 8,p) is a locally finite measure space, then, for any compact subset C of G, there is an open set 0 E 8 such that C C 0 and p( 0) < 03. In fact, for each x E C there is a V, E 8 such that p( V,) < 03. By the compactness of C, there exist x1 ,..., x, such that 0 = VZlu

**.

u V Z " 3c.

< 00.

Then 0 E b and p( 0 ) \< Zrc1p( V,y)

Lemma 1.1.11. Let G be a locally compact group, 8 the a-ring generated by the totality of compact subsets of G. If (G, 8, p) is a locally finite measure space, then it is also pseudo-a-finite. PROOF. Choose any neighborhood V of the identity e of G, such that the closure C of V is compact, and C = C-l. Let Cidenote the set

cc c, I

...

and form the subgroup

u C'. m

Go =

i=l

Since C is compact, and C2is the image, under the continuous mapping (x, y) +xy, of the compact subset C x C of the product space G x G, it follows that C2 is compact. Similarly, every Ci is compact, so that the measure of Ci is finite, and hence Go is a-finite. Let {G, , E %} be the left coset system of Go in G. Clearly, each G, is also the union of countably many compact sets, and hence is

1.1. Some Measure- Theoretic Concepts

13

o-finite. We shall now prove that (G, 23, p) is the direct sum of the measure spaces {(G, , 23, , p,), a E a},where 23, , pa denotes the restriction of 8,p to G, . Since 23 is generated by the totality of compact subsets of G, we need only prove that, for any compact set K C G, P(K) =

c

P,(K

f-l

G,).

Since p,(K n G,) = p ( K n G,), it suffices to prove that there are only countably many a such that p(K n G,) > 0. In fact, since K is compact, there exists a finite set x1 ,..., x, such that

6

( x v V ) 3K.

"=l

Let x,

E

G,,

. Since C = C-1, GayC= G,,

, therefore

KC(JG,". "A

Hence, if a # a1 ,..., a,, then K n G, = 0, so that p(K n G,)

= 0.

]

Corollary 1.1.12. The Haar measure on a locally compact topological group is pseudo-a-finite. We shall next introduce some results concerning integrals with respect to Haar measures on locally compact groups. In the following three propositions, we assume that G is a locally compact group, and that SZ = (G, 8,p) is the left invariant Haar measure space on G. (orL2(Q)). Let Lemma 1.1.13. Let a €L1(SZ), (EL~(SZ)

* m)= JG 4g1) &;%) 4 4 g d

(a

(1.1.20)

*

Then a 5 EL~(SZ)(L~(SZ)). Moreover, if G is commutative, then a 5 = ( a. PROOF. If 5 €L1(SZ), then, by the Fubini theorem,

*

*

J, (J

G

I 4g1) 5(g;'g)l

dP(Zl)) d d g )

J, (J I &l) t(g;lg)I 4 4 ) ) M g l ) = J, Ia(g1)l M g 1 ) J I E(g1)l dP(&), G hence a * f EL1(Q), and II a * 5 < II a II t =

G

1 1 1

1 1 1

Ill *

(1.1.21)

r . SUPPLEMENTARY

14

BACKGROUND IN MEASURE THEORY

If 4 E L2(9),then, by the Cauchy inequality,

Also,

f2 E

L1(9),hence, substituting t2for 4 in (1.1.21), we get

Integrating both sides of (1.1.22) with respect to g, and using (1.1.23), we obtain

I, (1 I

a

471) t(gg;'g)I W l ) )

G

4.49

e II a 1:

II 5 1;

-

Consequently, a * g E L2(Q),and I1 a * 4 112 < I1 a \I1I1 4 112 * If G is commutative, then p is also right invariant, and dp(gl) = dp(gll). Setting g' = ggi', (1.1.20) becomes

whence a

*f

=4

* a.

]

Lemma 1.1.14. Let 5 , EL2(Q).Then ~ thereexist{a,} C L l ( 9 )nL2(Ln) such that (1.1.24) lim 11 an * 5 - 6 11 lim )I an * q - q 112 = 0. n+w - n+m PROOF.Let U be a neighborhood of the identity in G such that co. Form the function

p( U ) <

Then Zu EL'(Q)n L2(9),and JG Z,(g) dp(g) = 1, hence

Write [(gi'g) = &Bl(g).Substituting 2, for a and replacing ((grlg) by

1.1. Some Measure- Theoretic Concepts

15

Now, given any positive number E , there existsll a continuous function t', with a compact support, such that 11 5' - 4 (I2 < 4 3 . Furthermore, there exists12 a neighborhood V , of the identity such that, if U C V , , g1 E U, then - 5'

II

Therefore, for such a U, using 11 Z ,

II 2,

112

<43.

Ill

= 1, we get

* 5 - E < II 2, * ( 5 - 5'1112 + II 5 - 5' 112

< 2 I1 E - I'

112

+ n$;

II E5,

112

- 5'

+ /I 2" * t' - 5'

112

112

< E.

Similarly, for 9 €L2(J2)and E > 0, there exists a neighborhood W, of the identity such that, if U C W, , then

I1 2,

* 7) - 7) < c* I12

Taking a, = Zvl/,nwl/,, we have 1 I/~n*E-5112

lI%*7l-7llh

1

<;.

Thus, we obtain (1.1.24). ]

Corollary 1.1.15.

Let

6 E L ~ ( Q )If, .

for every a €Lf(Q) nL2(Q),

a

* [ = 0, then 6 = 0.

a,

PROOF. Choose {a,} C L1(Q)n L2(J2)such that (1 .I .24) holds. Then * 6 = 0, whence, by (1.1.24), we obtain 6 = 0. 3

7O

Regular Measures on Topological Spaces

In the present book, we shall often restrict our considerations to regular measures on topological spaces (not necessarily locally compact). l1 la

Translator's note: See Nairnark [l], p. 137. Translator's note: See Naimark [l], p. 373.

16

I.

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

Definition 1.1.11. Let G be a topological space, (G, b,p) a measure space. If, for every E E b, p ( E ) = inf{p( U ) I U E 113,

U 3 E,

U open},

then (G, 23, p ) is said to be outer regular. If, for every E E 23, p ( E ) = sup{p(C) I C E b, C C E,

C closed compact},

then (G, 113, p) is said to be inner regular. (G, b,p) is regular if it is both inner regular and outer regular. We proceed to establish some conditions for the regularity of measure spaces.

L e m m a 1.1.16. Let G be a topological space, (G, 9, p) a finite measure space.13 Then (G, 23,p) is outer regular if and only if, for every E E b, p ( E ) = sup{p(F) I F E 23, F C E, F closed}. (1.1.25) PROOF. We know that (G, b,p ) is outer regular if and only if, for every E E 23, there is a sequence of open sets (0,) C 23 such that 0,3 E, p(0,) +p ( E ) . Letting G - E = E', this condition is equivalent to the existence of F,( = G - 0,)C E' such that p(F,) -P p(E'). But this is just condition 1.1.25. ] L e m m a 1.1.17. Let G be a separable metric space, (G, 23, p ) a finite measure space, and suppose that every ball belongs to 23. Then, for any positive number E, there exists a totally bounded set C E23, such that p(G - C) < E. PROOF. Let {x,} be a sequence of points dense in G. Let S(x, a ) denote the closed ball with center x and radius a. Then, for any natural number K,

Consequently, there is a natural number p(K) such that (1.1.26)

Form the set

c=n u m

dk)

k = l n=l

l3 Here,

b is assumed to be an algebra.

1 S ( ~ ~ , ~ ) ~ B .

1. I. Some Measure- Theoretic Concepts

17

From (1.1.26),it is easily seen that

But, by its very construction, C is totally bounded. In fact, given any S > 0, choose a natural number k such that Ilk < S; then x1 ,..., xp(k) is a 8-net for C . ]

Theorem 1.1.18. Let G be a complete separable metric space, b the o-algebra generated by the totality of closed sets of G, and p a finite measure on (G, 23). Then (G, 23,p) is a regular measure space. PROOF. By virtue of Lemma 1.1.17, for every natural number n, there exists a compact set C, such that (1.I .27)

Since C, is a compact Baire set, the restriction of p to C, is regular. Hence, for any E E B , there is a compact set F , E B such that F, C E n C , and 1

P((E n Cn) - Fn) <

Since E - F, C G - C , and (1.1.28) that

+ [ ( E n C,)

- F,],

-

(1.1.28)

it follows from (1.1.27)

+ 2;

P(E) < I*.(Fn)

+

Thus, we obtain a sequence of closed compact sets F, C E, such that p(F,) + p ( F ) . This proves that (G, b,p) is inner regular. By Lemma 1.1.16,p is also outer regular. ]

80 Some Probabilistic Terminology We now briefly present some terminology from probability theory which will be used in the sequel. Let S = (Q, 23, P ) be a probability measure space, and let X ( w ) , w E Q, be a measurable function on S. We call X ( w ) a random variable14 on S, and &p(’)t)

=

J p(w)tdpw ( ),

--co
n l4 In the present book, we shall, in general, consider only the case of real-valued functions.

I.

18

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

the characteristic function of X . If X ( E L1(S),then a)

E(X) = J X(w)dP(w) n

is called the mean value or mathematical expectation of the random variable X . If X E L2(S),then .(X)

=

E((X - E(X))2) = J ( X ( w ) - E(X))2dP(w) R

is known as the aariance of X . If X , Y are two random variables such that X(.), Y ( * €L2(S), ) we call E(XY) =

1X ( w )

Y ( w )dP(w)

the correlation number of X and Y . Let X , ,..., X , be n random variables on S . Form a Borel measure p on n-dimensional real space R, ,as follows: if E is a Borel set in R, , define

dE) = P({w I (X1(w>,-, Xn(,>> E El). This p is a probability measure, and is called the joint (probability) distribution of Xl ,..., X , . Sometimes, the function F(xl

,...,x,)

= P({w I X J w )

< x, ,

Y =

1,2,.*., n))

is called the joint distribution function of XI ,..., X , . I n particular, when n = 1, X = X , , p (or the corresponding F ) is known as the probability distribution (or distribution function) of the random variable X . It is easily seen that, if p is the joint distribution of Xl ,..., X , , then, for any bounded Baire function f,we have

90 Absolute Continuity and Singularity Relations

among Measures In this subsection, we shall prove a theorem concerning limits of Radon-Nikodym derivatives. For this purpose, we first introduce the following concept.

1.1. Some Measure- Theoretic Concepts

19

Definition 1.1.12. L e t s = (52, b,P)beaprobabilitymeasurespace, dl,...,23, a finite sequence of o-algebras such that 23, C 23, C C 23, C 23. Let x,(w), ..., xn(w) be (real) random variables on S such that xi is measurable with respect to (52, Bi).If, for any i , 1 f i < n, and any LIES,,

j

.i(W)

W W )

d

A

J

Xi+l(W)

(1.1.29)

W W ) ,

A

then {xi,23, , i = 1,..., n} is said to be a semimartingale on S . Clearly, if {xi,23, , i = 1,..., n} is a semimartingale, then, for any real number c, and any A E 23, , we have

Consequently, (max(xi(w) - c, 0), B 6 ,i = 1,..., n} is also a semimartingale. We shall also make use of certain quantities defined as follows. Let ,$, ,..., 5, be a finite sequence of real numbers, and let Y , , Y , be a pair of real numbers with rl < Y , . Let v1 be the smallest index j such that tj rl , let v2 be the smallest index j such that f j 2 r 2 ,j > v, , let v3 be the smallest index j such that rl ,j > v2 , and so on. Continuing in this manner, we get

<

sj <

tVI< y1 >

tq

3 r2

9

t,, < y1 5", 3 y2 9

T.7.

'

Let m be the greatest integer i such that vd can be defined in this manner, and let /3 denote the greatest integer not exceeding m/2. We call /3 the number of times the sequence ,..., 5, crosses the interval [ r l , r,].

el

Lemma 1.1.19. Let {xi(w), Bi, i = I , ..., n} be a semimartingale on the probability measure space S = (SZ, 23, P).Let I , , Y , be two real numbers, r, < r2 . For each o E SZ, let B(w) denote the number of times that xl(w),..., x,(w) crosses [ r , , r 2 ] . Then

( 1.1.30)

20

I.

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

PROOF. The second inequality in (1.1.30) is obvious; we proceed to prove the first. Firstly, set y1 =

xI(w) 2

0,

0,

i = 1 ,,..,n,

w E SZ.

(1.1.31)

For each w E 52, we take xl(w) ,..., xn(w) as the t1 ,..., tnin the procedure described above, thereby obtaining numbers vl(w)(= l),..., Y ~ ( ~ ) ( Wand ) /3(w). It is easily seen that p ( w ) is a measurable function. For those indices j such that m(w) < j n, we shall, for convenience, define vj(w) = n 1. The vl(w), ...,vn(w) so obtained are also measurable functions on S . We further define measurable functions u2 ,...,u, and x as follows:

<

+

z+(w) =

I

j B YI(W), V<(W) < j ,< V , + ~ ( W ) , i even, v I ( w ) < j ,< ~ ~ + ~ ( w ) i, odd,

1, 1,

0,

+c n

4 w ) = x1(w)

u,(w)(&J)

- Xj-l(W))*

5=2

Then,

The set on the right-hand side of (1.1.32) is determined by E Bj-l . Using (1.1.29), we get ..., ~ , - ~ ( w )Consequently, .

xl(w),

M w ) - xj-dw)) dP(w)

and hence

On the other hand, x(w)

therefore,

< xn(w)

- 12,B(w)i

t 0,

1.1. Some Measure- Theoretic Concepts

21

Thus, (1.1.30) holds under the assumptions (1.1.31). As for the general case, we need only consider xi'(w) = max(xj(w) - T , , 0).

As we pointed out above, {xj', Bj,j = 1,..., n} is also a semimartingale. The number of times that the sequence x,', ..., x,' crosses [0, y2 - r,] is equal to the number of times that x1 ,..., x, crosses [ r , , y 2 ] . But xl', ..., x,' and [0, r2 - r,] satisfy conditions (1.1.3 l), hence, by (1.1.33), we have

which is just (1.1.30).

3

Theorem 1.1,20. Let Q be a set, and let B1,..., 23, o-algebras in Q such that 23, c 23,

,... and

23 be

c *..c 23, c c 23, **.

uz=l

and such that 23 is generated by 23, . Let P , Q be two probability measures on (Q, 23), with Q P.16 Let P , , Q, be the probability measures on (Q, 23,) defined by

<

Pn(E) = P(E),

Qn(E) = Q(E),

EEb

n *

<

Obviously, Q, P , . Let dQ,(w)/dP,(w) and dQ(w)/dP(w)denote the appropriate Radon-Nikodym derivatives. Then, for almost all o E Q (with respect to P ) , we have

PROOF. Set x,(w) = dQ,(w)/dP,(w), x ( w ) = dQ(o)/dP(w).Then x, is measurable with respect to 23, , moreover, for any A E 8 , , we have Qn(A> = Qn+l(A) = Q ( A ) and JA x n ( w ) dP(w) =

JA

xn(w) @n(w) = Qn(A),

whence

=

l6

JA

x ( w ) dP(w) = Q(A).

That is, Q is absolutely continuous with respect to P.

( 1.1.34)

22

I.

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

It follows from (1.1.34) that, for any n, {xi , 8, , i = 1,..., n) is a semimartingale. We shall now prove that x,(w) exists almost everywhere. Let X*(W)

Now, for any c

=

lim X,(W), n+

x*(w) =

m

lim x,(w). n+m

> 0, we have

Consider the set

n (J n m

Am = {W 1 rim x,(w) = a}= n+m

m

m

{W

I x,(w) > i } .

i=l m=l n=m

<

Since, by (1.1.35), P({w 1 xn(w) > i)) l/i, it follows that P(&) = 0. Consequently, if limn+a x,(w) did not exist almost everywhere, there would exist positive numbers rl < r2 such that the set A

= (w

1 x*(w) > y2 > rl > x,(w))

has positive probability measure. Let /3,(w) be the number of times that xl(w), ..., xn(w) crosses [rl , r2]. If w E A, then /3,(w) = co. But, by Lemma 1.1.19, 1

E(t3n) G

<

(1

+ Id.

< <

<

(1.1.36)

Moreover, since pl(w) &,(w) /3,(w) it follows by the Levi lemma and (1.1.36) that @,(w) is finite almost everywhere. This is a contradiction. Therefore, there exists a finite-valued measurable function xm(w) on D such that lim x,(w) = x,(w)

n+ m

as.,

(1 .I .37)

holds almost everywhere. We shall now prove that the sequence of integrable functions (xn(w)} has uniformly absolutely continuous indefinite integrals on Q. I n fact, for any positive number E, there is a positive number 6 such that, when A € 23, P ( A ) 6,

<

Q(A) =

J

A

x(w)

dP(w) < 5 . 2

1.1. Some Measure- Theoretic Concepts

23

On the other hand, by (1.1.35), we have P((w I x,(w) Hence, if P(A) d/2, then

<

2 1fS))

< 8.

This proves that (x,(w)} has uniformly absolutely continuous indefinite integrals. Therefore, by (1.1.37) and the well-known Vitali theorem, it follows that xm(w) €L1(SZ),and, for every A E 23,

However, by (1.1.34), we have, for A E 23, xm(w)

Consequently, for any A

E

dP(w) =

uzZl8,

L

, m >, n,

x(w)

dP(w).

, (1.1.38)

But 23 is the a-algebra generated by Ufl 23, , hence (1.1.38) also holds for any A E B . This proves that x,(w) is equal to x(w) almost everywhere. ] We next establish a condition for the mutual singularity of two measures.

Lemma 1.1.21. Let Pl, Pz be two measures on the measurable space (G, 23). If, for any positive number E , there exist disjoint sets A, E 23, B, E 23, such that G = A, v B, and Pl(Af) <

€9

PZ(Bf) <

€9

then PI and P, are mutually singular. PROOF. Form the sets m

A

=

m

n u Al,Z,’

m=l n=m

m

=

m

u n Bl,P*

m = l n=m

24

I.

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

Then, A uB

=

G,

A nB

=

empty set,

moreover,

Thus, Pl and P, are mutually singular. ] 51.2. Localizable Measure Spaces

10 A Partial Ordering in the Collection of Measurable Sets

Definition 1.2.1. Let Q = (G, 8,p ) be a measure space, 6 the totality of measurable sets of Q, 8,the totality of null sets. If E, F E % and ( E - F ) u (F - E ) E b, , we say that E and F are equivalent, and write E - F . If E, F E 6 and E - F E b,, we write E < F. Clearly, the relations and < have the following properties (for N

E, F, H E 6 ) : (i) E E; (ii) if E N F , then F E; (iii) if E N F , F H , then E N H ; (iv) if E M F , then E < F ; (v) if E < F , F < E, then E - F ; (vi) if E < F, F < H , then E < H . Consequently, if equivalent elements of 6 are identified, then % N

N

N

becomes a partially ordered set with respect to the relation <, We now introduce the usual notions of upper and lower bounds, least upper bound and greatest lower bound in this partially ordered set.

Definition 1.2.2. Let 8 C 8,K E 8.If, for every E E 8, we have < K, we call K an upper bound for 5.If KOis an upper bound for 8 and, for any upper bound K for 5,we have KO< K, then we call KO a least upper bound for 5,and write KO= V E E E. S I n a similar fashion, E

we define lower bound and greatest lower bound; we denote any greatest lower bound for 8 by AEEs E. It is easily seen that, if 8 is a finite or countable collection

1.2. Localizable Measure Spaces Let 8 be a subcollection of easily seen that

6 ;if K E 6,then Q - K E 8,hence, it is

V

Q-

25

A

EN

(1.2.1)

(Q-E),

EE 5

EE5

(1.2.2)

Lemma 1.2.1. Let E be a set of finite measure, 8 a family of measurable subsets of E. Then, 8 has a least upper bound, and there exists a W sequence of sets En E 8, n = 1, 2,..., such that UnX1 En is a least upper bound for 8. PROOF.Let W

81= (

~ F , I F , E ~ ,n =

).

1 , 2,...

n=l

Then, any countable union of sets of

g1 also belongs to Z1.Let (1.2.3)

<

Then OL p ( E ) < a.Hence, there is a sequence such that 01

Let E, = UzZl An have

E

S1 ; then,

=

01

An E El,n

,...,

= 1,2

(1.2.4)

lim p(An). n+w

>, p(E,) 2 p(An), and by (1.2.4), we

dE0)

= O1.

vFEs

We shall now prove that E, = F. Obviously, if F E 8, then F E , hence E,, uF E & ' . But, by (1.2.3), we have p(Eo uF ) 01 = p(E,), whence E, uF - E, is a null set. Thus, E, is an upper bound for 8. On the other hand, if K is any upper bound for 8, then, since E, is the union of countably many sets En E 8, n = 1, 2,...,

<

m

E,-KC~(E,-K) n=l

is a p-null set. Thus, E,

< K , so that

E, is a least upper bound for

8. 1 Corollary 1.2.2. Let E be a u-finite set, 8 a collection of measurable subsets of E. Then 8 has a least upper bound, and there exists W a sequence En E 8, n = 1 , 2,..., such that UnX1 En = V F E S F .

I.

26

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

PROOF. Decompose E into the union of a sequence of mutually disjoint sets E(,) having finite measure, n = 1, 2, ... . Let

5(n)= { A n E(7")I A E 5). By Lemma 1.2.1, there exists, for every n, a sequence F,, E 5, = 1,2,..., such that

K

m

I t is easy to show that m

u

v A*

Fkn =

k,n=l

1

AE5

Lemma 1.2.3. Let SZ = (G, B, p) be a measure space, and let 5 be a directed collection of sets of 6,that is, for any A, B E 8, there exists C E 5 such that A < C, B < C. If E is any least upper bound for 5,then

<

PROOF. If A E 8, then, since A < E , we have p ( A ) p ( E ) . Hence, it suffices to prove that p ( E ) supAEE p ( A ) whenever supAE% p(A) = a < 00. Now, in this case, there are sets A, E 5, n = 1,2,..., such that p(A,) -P a. Since 8 is directed, we may assume that

<

A,

< A , < < An < **.. ***

m

Let A = Un=l A, ; then, p ( A ) = a. We shall now prove that A is an upper bound for 8. If it were not, then there would exist K E 8 such that p(K - A ) > 0. Choosing B, E 5 such that K
and letting n

---f

01

a,we get

3 I*(Bn)3 P(K -+

a contradiction.

Lemma 1.2,4,

+ An)

p(K - A )

= P(K - An)

+

01

+ pL(An)

> 01,

] Let SZ = (G, 23, p ) be a measure space, and let

1.2. Localizable Measure Spaces

27

g C 6 be such that, for any two distinct sets F , F’ of p(F n F’) = 0. If E is a least upper bound for 5,then

8, we have (1.2.5)

PROOF. If the right-hand side of (1.2.5) is 03, then one can find (F, 1 n = 1,2,,,.) C 8 such that Cn=lp(Fn) = co. But since E is a least upper bound for 5,it is an upper bound for {Fn}, hence W

(u

m

m

P(E) 3 P

Fn)=

n=l

c

PPn) =

a.

n=l

If CFEs p ( F ) < 00, then there are at most countably many sets of positive F, measure in 5;denote these by Fl ,F ,... . It is easily seen that is a least upper bound for 5,therefore, E and u,F, are equivalent, whence we obtain (1.2.5). ]

u,

20 Measurable Sets and Projection Operators

Let L2(SZ)be the Hilbert space formed by the totality of quadratically integrable (real or complex-valued) functions on SZ (see $1.1). For each E E 6 ,let C Edenote the characteristic function of E:

Let H E be the closed linear subspace formed by those functions in L2(Q) which vanish almost everywhere (a.e.) outside of E, and let PE denote the projection from L2(SZ)onto H E . Then (P,f)(g) = CE(g)f(g),

fEL2(Q).

Note that the closed linear subspaces of any Hilbert space H form a partially ordered set with respect to the inclusion relation C. Any family 5 of closed linear subspaces has a greatest lower bound (g.1.b.) with respect to this ordering, namely, the intersection of all the subspaces in 5;we denote this by A M E sM . Also, 5 has a least upper bound (l.u.b.), that is, the intersection of all those closed linear subspaces which contain all the subspaces in 5;alternatively, it may be described as the closure of the linear subspace consisting of all linear combinations of vectors in U L E L. S We denote the 1.u.b. of 5 by V w E qM . Correspondingly, the (orthogonaif projection operators in H are There is a natural one-to-one partially ordered by the relation correspondence between projection operators and closed linear sub-

<.

I.

28

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

spaces, and this correspondence is order-preserving, that is, if P, Q are projection operators in H , then PH C QH is equivalent to P Q. Consequently, any family g1 of projections in H has a 1.u.b. P,defined by the relation

<

PH=

V

QH.

OE51

Denote this 1.u.b. by P = VOE51 Q . Similarly, B1has a g.l.b., denoted by A

M

E

~

~

~

Note that, if E, F E 8, then E - F if and only if HE = HF (or PE = PF),and E < F if and only if HE C HF (or PE \< PF). I n the following lemma, we assume that, for any E E S such that p ( E ) > 0, there exists F E d,F C E, such that 0 < p ( F ) < co. This is one of the conditions for localizability (see Definition 1.2.3).

Lemma 1,2.5.

Let

5 C 8,E E 8.Then E

=

VFEgFif and only if (1.2.6)

PROOF. Suppose E PF < PE. Therefore,

=

VFE5F. Then, if F E 5,we have F < E, hence (1.2.7)

Let f~ H E , and % =I , {g I f ( g ) # O}. Then iUl, is a-finite and ‘93, E is a null set. Let gr = {A n 1Dz, I A E g}. By Corollary 1.2.2, there is a sequence A, E 5, n = 1, 2,..., such that

-v

m

u (4

?%=l

( A n rnnz,).

AE5

But it is easy to verify that

v (Anmf)-(v AE5

AS5

Consequently, as N

+ 00,

Moreover, it is easily seen that

A)nm,=Enm,-m,.

1.2. Localizable Measure Spaces

29

Since V F E sH , is a closed set, it follows from (1.2.8) that f E Therefore, we have HEc

v

VFEsH , .

HF,

F E5

<

whence PE VFE8P, . Combining this inequality with (1.2.7), we obtain (1.2.6). The proof of sufficiency is left to the reader. ] Lemma 1.2.5 shows the relationship between the 1.u.b. of a family of sets and the 1.u.b. of the corresponding family of projection operators.

30 Definition of Localizable Measure Spaces, and Some of Their Properties Definition 1.2.3. Let 52 = (G, 23, p) be a measure space satisfying the following two conditions: (i) if B E 23, p ( B ) > 0,then there exists E E 23, E C B, such that 0 < p ( E ) < co;(ii) if 8 denotes the totality of measurable sets of 52, then, for any 8 C 8, there exists a 1.u.b. VAE5A E 8.Then, the measure space 52 is said to be localizable. By Corollary 1.2.2, we have the following example. Example 1.2.1. Any o-finite (or finite) measure space is localizable. The following theorem indicates an important method of constructing localizable measure spaces. Theorem 1.2.6. The direct sum of a family of localizable measure spaces is localizable. PROOF. Let 52 = (G, 23, p) be the direct sum of the family of localizable measure spaces 52, = (G, , 23, , p,), 6 E S. Let 8 C 8.We proceed to prove that 5 has a 1.u.b. in 8. Let 8, = {F n G, 1 F E 3). By hypothesis, each 8, has a 1.u.b. E , E 6,.Form the set E = UCES E , ; this is an element of 3,and is a 1.u.b. for the family of sets { E , , 6 E S}.We shall prove that E is a 1.u.b. for 8. If F E ~ then, , for each 4 ~ 8( , F-EE)nG,=FnG,-E, is a null set, hence F - E E 23, . Thus, E is an upper bound for 5. O n the other hand, if K is any upper bound for 5,then, for each F E 5 and each 6 E E, ( F - K ) n G, = (Fn G,) - ( K n G,) is a null set, that is, K n G, is an upper bound for S,, hence

is a null set. Therefore, E - K

E

23,

. Moreover,

it is obvious that any

30

I.

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

set of infinite measure must contain a set of finite positive measure. Thus, SZ is localizable. ]

Example 1.2.2. Any quasi-a-finite measure on a group, in particular, any Haar measure on a locally compact group, is localizable. T h e following concept will also be used in the sequel. Definition 1.2.4, Let (G, 8,p) be a measure space, let E E 8, and let [El denote (F 1 F E, F E B},that is, the equivalence class of E with respect to the relation in !B. Let U be the totality of such equivalence classes. If [El, [F]E U, let [El v [F] = [ E v F ] , [El A [F] = [E A F]. We call U the measurable ring associated with the measure space (G, 8,p). Define a function p on U by

-

N

CL([El) = d E ) . p is called a measure on the ring U, and the pair (U, p) is called the measure ring associated with the measure space (G, 8,p). Let SZ = (G, 23, p) and SZ’ = (G’, B’,p’) be two measure spaces, (U, p) and (U’, p’) the associated measure rings. Suppose there exists a one-to-one mapping cp from U onto U’, satisfying the following condi-

tions:

d F I ) = [GI, d5 v 7) = 43 ” d d ,

d e m p t y set11 = [empty set19

df A

7) = d.3 A d 7 ) ,

(1.2.9)

moreover, p ( [ ) = 0 if and only if p‘(q([)) = 0. Then, we say that the measure spaces 52 and SZ’ are equivafent. I n particular, if G = G’, 23 = d’,and tp([E])= [El, then this notion of equivalence reduces to the ordinary definition of equivalence between two measures p, p‘ on a single measurable space. 51.3. The Kolmogorov Theorem

Products of measure spaces are introduced in Chapter 7 of Halmos [I]. I t is clear that the treatment presented there can easily be extended to products of families of probability measure spaces indexed by sets of arbitrary cardinality. Accordingly, in what follows, all our discussion of product spaces will refer to products of an arbitrary number of spaces.

lo Projective Limits of Families of Measure Spaces We shall now effect some generalizations of the concept of a product measure space.

1.3. Theorem 1.3. The TheKolmogorov Kolmogormi Theorem

3131

(r,

, !B,,), Definition A be be aa directed directed set, set,and andletlet (FA , 23J,A EA A, E A, Definition 1.3.1+ 1.3.1. Let A be spaces. Let Let Pi': Pi':FA, FA, -+ FA FA E A, A A', A', be aa family family of of measurable measurable spaces. + , A,, A,A' A'E A ,h < be such that, that, ifif AAEE!BA, B ~ then , (F'f)-l(A) (Pi')-l(A) be aa family family of of mappings mappings such then E ~ 2 3 ~ ' [i.e., Pi' isis aa measurable measurable mapping r A c ,, % A , ) toto(FA , !&)I, such [i.e., mapping from from( (FA, (r, , @ A ) ] , and and such satisfied:if ifh A< A', A',A' A'<< that condition isissatisfied: A", A", that the the following following consistency consistency condition then then , . PAPA. A A = p;
e'

<

<

Then, A', A,A, A'A' EEA} A )isisaaconsistenijam'l3, consistent familyof ofprojecprojecThen, we we say say that that {Pi', {Pi',h
r

P;'Pg

.

(1.3.2) (1.3.2)

= PA'

Let S,}. set P;'(A), e l ( A ) ,AAE E%A SA , iscalled calleda Borel a Borel LetBA* SA*= -{Pyl(A) {P;'(A) 1I A EE S ,}.TThhee set , is A, belonging belonging totothe theindex indexA.A.Let Let6,23,denote denote cylinder cylinder in in r, 1: with with base A, thethe totality in r,that thatis, is,6, 23, -=Uhcn UAon 23,+*, andB23 the smallest %IA*, and the smallest totality of of Borel Borel cylinders cylinders in a-ring 23, . TThe pair (I', (r,6,) 23,) isiscalled calleda aprojectine projectivelimlimit containing 23,. h e pair it a-ring in in rr containing (with of projections projections( {Pi'}) thefamily familyofofmeasurmeasur(with respect respect to to the the family of P f ) )ofofthc , BA), A}. able ((rA 6>), A E A}. ablespaces spaces {(FA Notice A', thcn then %,* SA* BE,morcovcr, moreover,forforeach eachA EA A, EA, Notice that, that, ifif hX < A', CCBT', BA* isisaa a-ring. easily that that So 23, isisa aring; ring;however, however,23,23,need neednotnot SA* a-ring. IItt follows easily be beaaa-ring. a-ring. be aa nonnegative nonnegativeset setfunction functiononon4), 23, , such Definition , such Definition 13.2, 1.3.2. Let po be that, for each A E A , the restriction of po to bA* is a measure. Then restriction of to !B,* is a measure. Then popo that, for each h E isiscalled a cylinder measure on (r, 23,). called a cylinder on So). Clearly, additive on on BSoo, ,but but need neednot notbebecountably countably Clearly, po po is is finitely additive additive on 23,. So. additive on

(r,

pA), , AAEEAA bebea afamily familyofofmeasure measure Definition ,,2B 3, A , pA), Definition 1.3.3. 1.3.3. Let (rA spaces, A } aa consistent consistent family familyofofprojections projectionsin inthethe spaces, {Pi', Ah < A'A',, A, A, A' EE A} family spaces {(FA {(rA BA), wheneverh A < and family ofof measurable measurable spaces ,,BA), AAEEAA} .} .If,If,whenever A' A'and AAEE23, 23,,,we wc have havc (1.3.3) = -'A), ( I .3.3)

{c',

<

then pA,,hA EERA [or [orthe thefamily familyofofmeasure measurespaces spaces then the the family family of of measures p,, (r,,,BA , pJ, h E A] is said to be consistent with the family of projec9,, pA),h said to be cotlsis!enl with the family of projections tions{Pi'}. (P:'}.

(r,,

(r,So) 23,) be be aaprojective projectivelimit limitofofthe thefamily family Definition of of Definition 1.3.4. 1.3.4. Let (T, measurable SA), PA:T-+ r -+ FA rA theassociated associated measurable spaces spaces {(FA ((rA ,, gA), Ah EEAA}, ) , FA: , ,A AE EAAthe

I.

32

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

mappings, and po a cylinder measure on (I',So).Let pAbe the measure on ( F ABA) , defined by PA(A) = (1.3.4) for all A E bA. We then say that (I',b, , p,) is a projective limit of the A , PA), h E A . family of measure spaces (I',, b

Lemma 1.3.1. Let (I',b,) be a projective limit of the family of measurable spaces (I',, b,),h E A, and, for each h E A, let pAbe a measure on ( F A bA). , Then, the consistency of the family of measure spaces {(FA, bA, pA),A E A} with the projections {Pi'}is a necessary and sufficient condition for the existence of a cylinder measure po on (I',do)such that (I',b, , p,) is a projective limit of the family {(FA, B,,, P A ) , h E A } . PROOF.Assume that {(rA , bA, p,), h E A ] is consistent. Define a set function po on Soas folIows: for A E bA, C 1 ( A ) E B,*,let ( I .3.5)

PO(pL1(A)) = PA(A)*

First, we show that po is well-defined, that is, if A

E

bA, A'

E

bAt , and

C1(A) = P;l(A'), then (1.3.6)

= PA(A)*

In fact, there exists A'' E A such that h < A'' and A' < A". By (1.3.2), we have (Pi;)-lA' = P A * ( K I A = ' ) P,+w(clA)= (PY)-lA, whence by (1.3.3) we get PA*(A')

=

PA*((P;'PA~)

= PAM((PA

1" -1A

)

= PA(A),

that is, (1.3.6) holds. The countable additivity of p, on (f,BA*)follows easily from the countable additivity of pA. Thus, p, is a cylinder measure on (I',%,I. Conversely, let (I',b,, p,) be a projective limit of the family (23, , pA),h E A}. Take any A, A' E A , h < A', and any A E 8 , . By (1.3.2), we have P;l(A) = P;'((F'f)-lA). Hence, by (1.3.4), we get

((m,

PA(A) = PO(pL1(A)) =

1; (PA'A

= PA'((p;')-lA).

Thus, the family of measures {p, , h E A} is consistent with the projections. ] We now give an important example.

Example 1.3.1. Let (I',, b,), 01 E 'ill be a family of measurable spaces, and let A be the totality of nonempty finite subsets of 'ill. We define the following partial ordering (called the natural ordering) in

1.3. The Kolmogorov Theorem

33

E A, and h C A', then h < A'. Obviously, A is directed by this ordering. For each ~ E A form , the product measurable space (rA , 23,) = (XuoA r, , XosA23,). In particular, when h contains but the single element a, we identify rA with . If h = {a1,..., a,}, we denote by OJ, = (wal ,..., w,,), where wayE ray. If h < A', we the points of rA may suppose that A' = {a1,..., am}, m 2 n, and define the mapping Pi': FA.4I', as follows:

A: if A, A'

r,

p%Jq

,**.>

%J= (Wa1

,a*.,

%").

It is easily seen that this family of mappings {Pi'}satisfies the consistency condition (1.3.1). We call {Pi'}the family of natural projections. Let r be the product space XaeB r, . For each w E r, let w, denote the ath coordinate of w , and for each h E A, h = {a1,..., a,}, let PAbe the mapping defined by w

---f

WA

= (WO1

)...,

Wan).

PAis called the natural projection of r onto PA; in particular, when h is a singleton {a}, we denote PAby P, . Clearly, {PA}and {Pi'}satisfy the relation (1.3.2). Let 23, be the totality of Bore1 cylinders in I'. We call (F, 23,) the canoptical projective limit of the family of measurable spaces (FA, BA),A E A. In this case, (r,23) is merely the product of the family

of measurable spaces (r,, %,), a E a. For each a E 8,let pa be a probability measure on (r,, S,), and for p,, on each A E A, h = {a1,..., a,}, let pA be the product measure (r,, BA).Then the family of measure spaces ( F A 23, , , pA)is consistent. Now there exists (see Halmos [I]) a product measure p on ( F , B), that is, 8, p) = ( X a ar, 9 Xmea 23, , X,P%pa). Restricting p to 23, , we obtain a cylinder measure p, , and 23, , p,) is the projective limit of the family of measure spaces ((FA, , pA),h E A}. In this case, po is countably additive on 23,.

(r,

(r,

Regarding projective limits of consistent families of measure spaces, we shall, in general, be primarily interested in the question of whether the cylinder measure po on (r,23,) is countably additive. If p, is countably additive, then it can be extended to a measure p on the O-ring 23. For the simple case described in the above example, p, is always countably additive. I n the following, we shall investigate the countable additivity of po when the rA are topological spaces.

Definition 1.3.5, Let A be a directed set, and ( r AZA), , h EA a family of topological spaces. Let P!': r,,-+ r, , A, A' E A, h < A' be a family of continuous mappings satisfying the consistency condition

I.

34

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

(1.3.1). Then, (P:'} is said to be a consistent family of projections in the family of topological spaces {(FA , &), A E A}. Let I' be a set and let PA: FA, A E A, be a family of mappings such that P A ( r )= FAand, whenever A, A' E A , A < A', relation (1.3.2) holds. Define a topology in r as follows. For each A E A , let

r+

ZA*= {P,'(A) 1 A E 2,).

Form 2, = UAen &*.I6 Let 2 be the weakest topology containing 2, . T h e n (r,2) is said to be al' projective limit (with respect to the family of projections {Pt'}) of the family of topological spaces { ( F AZA), , A E A}. T h e projective limit (r,2) is said to be projectively complete if the following condition is satisfied: given any sequence {A,} C A ,

< < < An < ..*)

(1.3.7)

and any sequence of nonempty sets {En)such that En is closed and compact with respect to the topology 2:, E 1 3 E 2 3 ... 3En3.*.,

then

W

(1.3.8)

En is nonempty.

Lemma 1.3.2. Let (r,2) be a projective limit of the family of HausdorfT topological spaces {(FA, 2J, X E A}. Suppose that, for each sequence {A,) C A satisfying condition (1.3.7), and each sequence of elements (5, , n = 1, 2, ...} satisfying the condition =

tnErAn,

there exists ( E

tn

for m 2 n,

(1.3.9)

r such that, for every n, PA,^

=

5,

*

(r,2)is projectively complete. PROOF. Take any sequence (An} C A satisfying condition (1.3.7), and any descending sequence of nonempty sets {E,) such that each En is Then

The elements of 2, are called open cylinders in T. Translutor's note: It should be noted that the limit of a projective system, as defined here, is not unique: given any set r' and mapping P from I" onto r, one can obtain another projective limit (Z", 2') in the obvious way, that is, 2' = P-'(2). Moreover, the condition that the projections PA(and hence also Pi') be onto means that such a limit may fail to exist, even if all the given projections Pi' are onto. Thus, this definition is far too restrictive for topological work, but it seems to suffice for the purposes of this book. Similar remarks apply to Definition 1.3.1. l6

l7

1.3. The Kolmogorov Theorem

35

closed and compact in the topology 2;. We shall prove that m En is nonempty. and, for any n Since PAmEmis compact in (rAm, m, Pi; is continuous, it follows that P,*Em = P;;PAmEmis compact (and hence closed) in (r,%, Thus, the intersection of the descending sequence of nonempty closed compact sets {PAIErn, m = I, 2,...} in ( r ASAl) l, contains a point tl. Since t1E PAIErn, m = 1,2, ..., the set P
nnEl

<

It).

P,;1SlnE,3)..3P,;1SlnEm3...,

we conclude that the sets {P,2(P<1fln Em), m mon point t 2 .We then have Pi;fz =

Repeating

15, , n

=

,

f2

E

P,,Em ,

this argument, we obtain 1, 2, ...I having the property =

5, E P,,E,,,

= 2,

m

3, ...} contain a com-

=2

,... .

a sequence of

for m 2 n,

elements ( 1.3.10)

that is, {tn}satisfies condition (1.3.9). By hypothesis, there exists such that, for each n, PAnE= 5, , whence, by (1.3.10), we have PAn(E PAnE,,

But En is closed with respect to En is nonempty. 3

nL1

n

=

tEr

1,2 ,... .

%t,hence [

E

En , n

=

1, 2,..., that is,

Example 1.3.2. Let (r,, Z,), a E 2l, be a family of nonempty topological spaces, and let A be the totality of nonempty finite subsets of 2l. As in Example 1.3.1, we order (1 by inclusion, and, for each h E A, form the topological product (FA, 2,)= (Xue,I', , XaE,2,).In particular, when X is a singleton {a}, we identify r, with r, . Let (r,2)be the , 2,),a E a}. Define the topological product of the family of spaces {(re projections {Pi'} and {PA}as in Example 1.3.1. It is easily seen that (r,2) is a projective limit of the family of topological spaces { ( F AZ ,,),X E A} with respect to the projections {Pi'}. We call it the canonical projective limit.

I.

36

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

Lemma 1.3.3. Any canonical projective limit is projectively complete.

PROOF.We need only verify that the condition of Lemma 1.3.2 is satisfied. Take any sequence {A,} C A satisfying (1.3.7). We may assume that A, = (011

1

012

,..., ak,} c a.

If (5,) is a sequence satisfying condition (1.3.9), then there are elements va, E Tarnsuch that

5,

= h u 1 *--v

.~a~,>.

Define an element ( E Xaesruas follows: the amth coordinate of 5 is = 1 , 21..., and the other coordinates of .$ are chosen arbitrarily (invoking the axiom of choice). Obviously, PAn( = 5, for all n. Therefore, by Lemma I .3.2, the canonical projective limit 2)is projectively complete. ] In 94.3 we shall encounter another example in which the condition of Lemma 1.3.2 is satisfied.

varn, m

(r,

2O

Countable Additivity of Cylinder Measures

We now present the basic theorem concerning the construction of measures on infinite-dimensional spaces, stated in its most general form.

Theorem 1.3.4, Let (r,, 2,),h ELI,be a family of topological , , p,,) be a regular probability spaces, and, for each h E A , let ( F A b, measure space. Let P,"':r,,, + r,,be a family of continuous projections, consistent with respect to the family of measures {p,,}. Let 2) be a projectively complete projective limit of the family {(F, Z,,)}, with associated projections PA:r-t FA. Suppose also that (TI8, , p,) is a pro, %, , p,,)}, with the jective limit of the family of measure spaces ((r,, same associated projections P A .Then the cylinder measure po is countably additive. PROOF. By Theorem F, $9 of Halmos [I], it suffices to prove that, for any sequence of cylinders

(r,

B, 3 B, 3

- 4 4

3 B, 3

in 8,such that lim p,,(B,) = L n-1m

the intersection

n,=, €3, m

is nonempty.

> 0,

(1.3.1 1)

1.3. The Kolmogorov Theorem Now, for each n, there is a A,

37 E

A such that B,

E

Sz . We may assume ( 1.3.12)

In fact, since A is directed, there is an index A,' such that A, < A,' and A, < A,'. Then B, E St C S$g, and we may replace A, by A,'. Proceeding in the same manner for n = 3,4, ..., we obtain a sequence of indices satisfying (1.3.12). For each n, let B, = P;,'(A,), A,E 23,; By the regularity of p,, , there exists a closed compact subset C, of FA,,with C, E b,, , such that C, C A, and L tLA,,(An - Cn) < 2"+'

*

Hence, by (1.3.4), we have (1.3.13)

it follows from (1.3.13) that pO(Bn- En)< L/2. Hence, by (1.3.11), we get L PdEn) = PdBn) - tLo(Bn - En)

>2 > 0.

Therefore, En is nonempty. Moreover, En is closed and compact with respect to 2$,and El 3 E, 3 *.. 3 En 3 ..-. (1.3.14) By virtue of the projective completeness of (r,2),it follows from m m (1.3.12) and (1.3.14) that En is nonempty. But B, 3 En , m hence En is nonempty. ] The following theorem (due to Kolmogorov) deals with the case of greatest interest.

nZEl

om=,

on=,

Corollary 1.3.5.18 Let 2f be an arbitrary indexing set, and let A be the totality of nonempty finite subsets of 2f, directed by the natural ordering. For each A = {a1,..., a,} E A , let R, be an n-dimensional euclidean space, 23, the a-algebra of all Bore1 sets in R, . Denote the points of R, by x, = {xal ,...,x.,}. For each A E A, let p, be a probability measure on (R, , S,), and suppose that the measures {p, , A E A} are consistent in the following sense: given any pair A, A' E A, with l*

Regarding the terminology used here, see Example 1.3.1.

I.

38

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

h = {a1, a2 ,..., a,} C {pl , p2 ,...,),31 PA’({~A‘

I {x,, >...)x&

= A‘,

E E ) ) = pA(E)-

r = X a E xR, , 23 = XaEI23, , and

Let

x = {x,

(r,

and any E E BA, we have

denote the elements of

(1.3.15)

, a E a}; then, there is a unique probability measure

r

by p on

d), satisfying the following condition: for any h = {a1,..., a,} E A, and any E E 23,, , p(ix I { x q ).*.) +%,> = ( 1.3.16 )

(The measures pA, or the distribution functions on the respective spaces RA determined by the pA, are usually known as the finite-dimensional probability distributions of p.) PROOF. Each p A , being a Bore1 measure, is regular. We form the {PA}as in Example 1.3.1. Then (1.3.15) families of projections {Pi’}, shows that the family of measures {pA, h E A } is consistent in the sense of Definition 1.3.3. By Lemma 1.3.1, there exists a cylinder measure p o on (r,Bo),satisfying condition (1.3.16), and, clearly, p o is uniquely determined by (1.3.16). Let ZA be the Euclidean topology on I‘, , and let 2 be the product topology Xmexr, on I‘. T h e n (r,2)is the canonical projective limit of {(FA, ZJ, X E A} (see Example 1.3.2). By Lemma 1.3.3, (r,2)is projectively complete. Therefore, it follows by Theorem 1.3.4 that p o is countably additive. It is easily seen that 23 is the smallest 0-algebra containing !€lo, hence p o can be extended to a probability measure p on (r,23). T h e uniqueness of p follows immediately from the uniqueness of p o , ] Corollary 1.3.5 can be stated in another useful form. (In the following, the characteristic functionla of a probability distribution on a finite dimensional Euclidean space, will be referred to simply by the term “characteristic function.”)

Corollary 1.3.5’. Let rU be an arbitrary indexing set, and let 8 denote the totality of finite ordered20 sequences f = (a1,..., a,) l8 According to the Bochner-Khinchin theorem, a function #(x) of n variables x = (xl,..., x,) is the characteristic function of some probability distribution if and only if the following three conditions are satisfied: (i) # ( x ) is continuous; (ii) #(O) = 1; (iii) # is positive definite, that is, for any sequence of points dl’, ..., xIm) and arbitrary complex numbers ,..., [, , m v(x(k)

- %I &)) [ k e t

> 0.

k,l=l 2o

This means that, for any nontrivial permutation al’,a;, ...,ad of the elements ,..., a n ,the sequences ( a l ,a2 ,..., a,) and (a;, ma’ ,...,a,,‘) are regarded as distinct.

a l ,a2

1.3. The Kolmogorov Theorem

39

Tu, i = 1,..., n; n = 1, 2,...). Let {F, , 5 E S}be a family of characteristic functions, having the following properties: (i) if 5' = (al',..., a,') is a permutation of 5 = (a1,..., an),then (aiE

FP'(t,,'

,..', f,;)

= F,(ta19.m.9

L");

( 1.3.17)

(ii) for m 2 n,

)*.-,lan 0, O,***,0) = F(al,...,an)(tal ,..-,t,J

F(ml....,am)(tal

9

(1.3-18)

For each a € ' % , let R, be a 1-dimensional Euclidean space, 23, the totality of Borel sets in R, , r = R, , 23 = XmSa B, , and denote the points of I' by x = {x, , CL E a}.Then, there is a unique probability measure p on (r,B) such that, for any 5 = (a1,..., a,) E 8, FE(trrl ,...,to,,) =

J r exp i(f,pel+ ... + t,,,x,,,) 44).

(1.3.19)

PROOF. Let h be any nonempty finite subset of Tu, and let the elements of h be ordered in an arbitrary sequence 5 = ( a 1 ,a2 ,..., an). Since F , is a characteristic function, there exists a Borel measure pr on R, = Rmlx Rm2x . - * x Ransuch that FE(ful,*.*)

tan) =

J

exp i(falxal RP

+ ... + t,,~,,,) 4-+(xLyl,..., x,,).

(1.3.20)

If the indices comprising 5 are permuted, it follows from condition (1.3.17) that, after the corresponding permutation of the coordinate axes, we obtain the same measure. Moreover, (1.3.15) can be deduced from (1.3.18). In fact, if 5' = {a1,..., a,} 3 {a1,..., a,}, form the measure = PE'{(xal

J.*.?

'a,,,)

I (x>"'>@ 'a,) .l E)'

Then F&,

,-.-,f,,,o,..., 0)

By (1.3.18), (1.3.20), (1.3.21), and the uniqueness of the measure determined by a characteristic function, it follows that PP' = PP

*

I.

40

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

Thus, (1.3.15) holds. Using Corollary 1.3.5, we obtain a unique measure p on 23), satisfying condition (1.3.16). Since the P A ,h = {a1,..., a,}, are measurable mappings, (1.3.19) follows immediately from (1.3.16) and (1.3.20). ]

(r,

30 Sample Measure Spaces

Definition 1.3.6. Let 6 = (9,8, P) be a probability measure space, {x,( ), a E 'u} a family of (real) random variables on 6 (one sometimes calls {x,( ), a E 'u} a stochastic process on (5). Also, suppose that {xu( ), a E 'u} constitutes a determining set of functions on (9,8, P ) . For each a E 'u, let R, be a copy of the real line, and 23, the totality of Borel sets in R, . Let F C XaGa R, , and let 23 be the restriction of XUsa 123, to r. Denote the ath coordinate of x E by x, = x,(x). Suppose that p is a probability measure on (r,23) such that, for any finite set of indices a1 ,..., a n ,and any Borel set E in n-dimensional Euclidean space Rn,we have

r

P({X I (Xml *..'> Xmn) E = P({W

El)

I (%,(W>l...l

X,,(W))

E El).

( 1.3.22)

Then, (r,23, p) is said to be a sample probability measure space associated with the stochastic process {x,(w), a E %}(w E Q) on 6,and {x,(x), a E 'u} the corresponding sample process.

Lemma 1.3.6. For any stochastic process, there exists an associated sample probability measure space. PROOF. We use the notation of Definition 1.3.6. For each nonempty finite set of indices {a1,..., a,} C 'u, define a measure on R", as follows: M a

l....,mniE)= W

W

I (Xa1(~),*.*,

XNn(W))

E

El).

(1.3.23)

I t is easily verified that the family of measures {p{,l,...,,ml ,{al,..., a,} C 'u} is consistent. Let I' = XirelIR, . By Corollary 1.3.5, there exists a probability measure p on (r,123) such that (1.3.16) holds. Combining (1.3.16) and (1.3.23), we get (1.3.22). ] We conclude this section by proving a lemma, required later, which provides a criterion for the equivalence or singularity of probability measures in terms of the equivalence or singularity of their associated sample probability measures. First, we introduce a convenient modification of Definition 1.1 .7.21 a1 Translator's note: Definition 1.3.7 was introduced by the translator in order to prove Lemma 1.3.7 (which was unprovable in its original form) and to fill a gap in the author's proof of Theorem 5.2.5.

41

1.4. Kakutani Distance

Definition 1.3.7. Let QA= (G, 23, pJ, X E A , be a family of measure spaces, and 2, a family of measurable functions on (G, 23). Let 23% denote the u-algebra determined by 3 on G. Suppose that, for every set A E 23, there exists a set B , E 23% such that PA(A

nB A ) = 0

for all X E A . We then say that 2, is a joint determining set (of functions) for the family of measure spaces QA. In particular, if 2, is a determining set of functions on (G, d),then 2, is a joint determining set for any family of measure spaces (G, 23, pA).

Lemma 1.3.7. Let (r,23,p k ) , k = 1,2, be sample probability measure spaces (associated with the same stochastic process {xm(.),01 E N}) belonging to the respective probability measure spaces Gk = (52, 8, Pk), K = 1, 2. Suppose also that {xa(.), 01 E N} constitutes a joint determining set for the measure spaces 6, , k = 1,2. Then pl p2 if and only if PI Q P, , while p1 1p2 if and only if Pl 1P, . PROOF. Consider the mapping cp from Q to r defined by 9): w --f x = {xe(w), 01 E N} E r. Clearly, cp is a measurable mapping from (Q, 8) to (r,23). From (1.3.22), it is easily proved that, for any B E 23,

<

P k ( B ) = Pk(F1(B)),

k

=

1,2.

(1.3.24)

Since { x a ( - ) , 01 E N} is a joint determining set for 6, , k = 1, 2, and {cp-l(B) I B E S}is the smallest a-algebra in Q with respect to which all the functions {xol(-),c u N] ~ are measurable, it follows that, for any E E 8, there is a B E 23 such that P,((E - v-l(B))u (cp-l(B) - E ) ) = 0,

k

=

1 , 2.

(1.3.25)

Assume that pl Q pz , P2(E) = 0; we must prove that Pl(E) = 0. Let B E B be such that (1.3.25) holds. Then, by (1.3.24) and (1.3.25), p 2 ( B ) = P,(E) = 0.

Hence, p l ( B ) = 0. Again, by (1.3.24) and (1.3.25), we get P,(E)

= p l ( B ) = 0.

Therefore, PI Q P, . The remainder of the proof is left to the reader. ] 51.4. Kakutani Distance

lo Elementary Properties of Kakutani Distance We now introduce a distance function which is useful in dealing with finite measure spaces.

42

I.

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

Definition 1.4.1. Let r b e a set, b a o-algebra consisting of certain subsets of r, and rrJl(r,b) the totality of finite measures on (r,23). For any two measures m, n E m(r, B),choose some r E m(r, 23) such that both m and n are absolutely continuous22with respect to r. Define (1.4.1)

where dmldr, dnldr denote the respective Radon-Nikodym derivatives of m and n with respect to Y. The number p(m, n) is called the Kakutani inner product of m and n. Also, define

d(m, n) is called the Kakutani distance of m and n. First, we note that p(m, n) and d(m, n) are, in fact, independent of the choice of r . Indeed, for any measurable functionf(u) on a), if the integral

(r,

(1.4.2)

exists, then its value is independent of r . For, if there is another r’ such that (1.4.3)

+

exists, let r” = r r’. Then it is easy to calculate that both (1.4.2) and (1.4.3) are equal to

Therefore, (1.4.2) is independent of r , and will, accordingly, be denoted by

Jf (w)(dm(w)dn(w))’/2. Furthermore, it is easily seen that d and p are related by the formula d(m, n)2 Such an Y always exists, e.g.,

= m(r) Y =

m

+n(r)

+ n.

-

2p(m, n).

(1.4.4)

1.4.Kakutani Distance

43

I n particular, when m and n are probability measures, (1.4.4)becomes d(m, n)2

= 2(1

- p(m,

n)).

Moreover, d(m, n) is, in fact, a metric on the set m(r, d).

Lemma 1.4.1. Two finite measures m, n on (r,d) are mutually singular if and only if their Kakutani inner product vanishes. PROOF. If p(m, n) = 0, then (1.4.5)

for almost all w (with respect to the measure r ) . Let

Then

Since dm(w)/dr(w)# 0 on r - A, it follows from (1.4.5) that dn(w)/dr(w)vanishes almost everywhere on r - A. This implies that n ( r - A ) = 0. Therefore, m and n are mutually singular. T h e proof of necessity is similar. ] 20 Another Expression for the Kakutani Inner Product We shall express the Kakutani distance in a form which does not involve Radon-Nikodym derivatives. This expression will prove useful in Chapters I11 and IV.

Lemma 1.4.2, Let p, v be two probability measures on the measurable space (G, d),and let 8 denote the totality of countable partitions23 of G. Then the Kakutani inner product of p and v is given by (1.4.6)

If G is a topological space and p, v are both regular measures, then 8 23 That is, the elements of 5 are sequences {Ek} of mutually disjoint sets in that p ( G - &E,) = v(G - z r E k ) = 0.

23, such

44

1.

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

in (1.4.6) can be replaced by ’&, where &, denotes the totality of countable partitions consisting of closed compact sets in 8. PROOF. Take any partition {Ek}E 3. By the Schwarz inequality,

( d d g ) dv(g))’/2 d

( d E k ) v(Ek))”a,

k,’

whence it follows at once that f ( k v,

<1

( d E k ) v(Ek))l’z*

(1.4.7)

k

+

Consider the probability measure r = *(p v) on (G, b). The derivatives dp(g)/dr(g),dv(g)/dr(g) are nonnegative measurable functions, hence, for any number a > 1, and any integers K, 1, the set

is rnea~urable~~. We then have

(1.4.9)

From (1.4.8)-( 1.4. lo), we get

But {Ek,l } E 3. Hence, it follows from (1.4.11) that

Letting a + 1 in the above relation, and using (1.4.7), we obtain (1.4.6). Now, suppose that G is a topological space, and p, v are regular measures. Then, we notice that, for any E E 8,there is a sequence of mutually disjoint closed compact sets {F, , 1 = 1,2, ...}, such that F,C E and B4 Translator’s note: To form a partition, one must also include the two sets {g I (dpjdr) = 0, (dvldr) > 0 ) and {g I (dpldr) > 0, (dvldr) = O}. However, the proof still remains valid.

45

1.4. Kakutani Distance

F,,

=E -

C,F , is a null set with respect to both p and v. By the Schwarz

inequality,

(1.4.12)

= ( p ( E )v(E))”/”.

Hence, if {Ek)E 8,there is an (1.4.12),

{Fkl} E

( d E k ) v(Ek>)’/2

8o such that Fkzc Ek , and, by

2 1( p ( F k 2 )

k

v(Fk2>)”2*

k2

Consequently,

But go C 8, therefore, the inequality in (1.4.13) can be replaced by an equality. ]

30 Kakutani Inner Products of Product Measures We first consider products of finitely many finite measures.

Lemma 1.4.3. Let r k , k = 1, 2 ,..., 1 be sets, and, for each k , let be a a-algebra of sets in r k , and let p k , v k be finite measures on ( r k , B k ) . Let (Fl*, B,*) denote the product measurable space I r k , X k Z l Bk), and let p l * , vz* denote the product measures I 1 v k , respectively. Then Xk=l p k , %Jk

I

h Z * ,

”Z*)

=

n

h

k

I

(1.4.14)

vk).

k=l

Yk be a finite measure on ( r k , B k ) , such that p k < Yk , . Form the product measure y Z * = X kI = l Yk on (I‘l*,B,*). Since

PROOF. Let vk pk

< <

Yk Yk

writing

for each

<

K, it follows that pI* y Z * . Similarly, vI* ,..., w z ) ,w k E r k , w E I’,we have

<

Y,*.

Then,

w,* = ( w I

(1.4.15)

and a similar formula for v I * . Using these two formulas, we easily calculate (1.4.14). ]

Theorem 1.4.4. Let ( r k , B k ) , k = 1, 2 ,..., be a sequence of measurable spaces, and, for each k, let p k , vk be probability measures on

I.

46

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY m

m

= X k = l p k " = X k = l vk * Let (r,a) = (x;=lrk > X k = l S 3 k ) , Suppose that, for each k, p k and v k are equivalent. Then p and v are either equivalent or mutually singular. A necessary and sufficient condition for the equivalence (mutual singularity) of p and v is p(pk , vk) > 0 (= 0). Moreover, in either case,

(rk

9

OD

Sk).

9

nZcl

m

&I'

v) =

n

&k

I

(1.4.16)

vk)*

k=l

PROOF. Consider the measurable spaces

(rz*, bz*)and

measures

pz*, vZ*, rl*, as defined in Lemma 1.4.3.By virtue of (1.4.14), and the 1, we know that p(pz*, vz*), 1 = 1, 2,..., is a fact that p ( p k , v k )

<

monotonic decreasing sequence. Hence, the infinite product

n W

P(h*

Yk) =

9+2P(PZ*%vz*)

(1.4.17)

k=l

either converges to a positive number or diverges to zero. We consider these two cases separately. (9 b + m P(Pl*, .z*> = 0. I n this case, for any positive number that

E,

there exists an integer I such

P(PE*,.,*I <

Consider the set

in BZ*. We have

e P(PLI*, v,*)

< c.

(1.4.18)

Similarly, since

r,*- A,* = we obtain

b*I

dPZ*(WZ*) dY,*( W ,*)

I

dr,*(w,*) ' dvl*(Wz*)

47

1.4. Kakutani Distance Now, in

r, form the cylinder A, with base Az*,that is, A,

= (w

I (w1 ,..., WE) E A,*) E B,

where w, denotes the mth coordinate of w . By the definition of product measure, we have p(AJ = pI*(A,*), v(A,) = vz*(AI*). By (1.4.18) and (1.4.19), we get v(T - A,) < E . p(A,) < E, By Lemma 1.1.21, we conclude that p and v are mutually singular, moreover, by Lemma 1.4.1, &, v) = 0, so that (1.4.16) holds. (ii) limt+wp(pI*, vl*) > 0. Since p k and v k are equivalent, we may choose r k = v k . We note that, for every I, dpz*(wz*)/dv,*(w,*) may be regarded as a function of w . Moreover, if we define

<

then t,bl € L 2 ( r ,b,v). For k 1, the functions + k , t,bl are measurable with respect to Bz*, and it is easy to calculate that

Furthermore, by (1.4.13, we have

(1.4.21)

Since we have assumed that (1.4.17) is positive,

Therefore, it follows from (1.4.20) and (1.4.21) that {t,b,}, 2 = 1, 2 ,..., is

48

I.

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

a fundamental sequence in L2(I’,8, v), and hence converges in L2(F,b, v) to a limit function +. Let A,* E Sm*,let

bJI (w1, .*.,urn) E A,*), and, for I > m, let A,* = ((wl ,..., w l ) I (wl ,..., w,) A

=

E

A,a*). Then

From this, it is easily verified that, for any A E b,p ( A ) = JA Hence, p is absolutely continuous with respect to v , and

t,b2(w)

dv(o).

Similarly, v is absolutely continuous with respect to p. Thus, p and v are equivalent, moreover, P(P,V)

=I

=

5

r

tfr(w)dv(w)

p2 P(PJ*, VZ*),

hence, (1.4.16) holds. ]

CHAPTER

REPRESENTA TION OF POSITIVE FUNCTIONA LS AND OPERATOR RINGS

As is well known, the theory of integral representations of positive functionals on normed algebras is a basic tool of harmonic analysis on locally compact groups, and in fact, the integral representation of positive functionals is in itself an important problem of abstract harmonic analysis. I n this chapter, we shall use measures on infinite-dimensional products to prove representation theorems for positive functionals on some rather broad classes of linear topological algebras. In what follows, we shall discuss groups which are not locally compact. Such a group lacks the group algebra which frequently serves as a convenient tool of harmonic analysis, hence, the usual normed algebra techniques used in the study of locally compact groups are ineffective. Consequently, one must resort to the use of commutative weakly closed operator algebras on Hilbert spaces. We shall, in $2.4, show the connection between such algebras and multiplication algebras on localizable measure spaces, thus throwing further light on the properties of local49

50

11. REPRESENTATION

OF POSITIVE FUNCTIONALS A N D OPERATOR RINGS

izable measure spaces. The results obtained will find application in subsequent chapters. 52.1, Topological Algebras with Involution: Fundamental Concepts

lo Seminormed Algebras First, we recall the meaning of the term “algebra.”

Definition 2.1.1. Let R be a set, with elements denoted by x, y , z, ... . Suppose that linear operations and pairwise multiplication have been defined in R, in such a way that the following conditions are satisfied. (i) R forms a linear space (over the real or complex numbers). (ii) For any number h and elements x , y , Z E R, we have yz, h(xy) = ( W y = x ( M , ( x y ) z = x ( y z ) , ( x r)z = x z z(x y ) = zx zy. Then R is called an algebra. In deference to established usage, we shall sometimes use the term “ring” in referring to certain types of algebras. If, for every x, y E R, we have x y = y x , then R is called a commutative algebra. If there is an e E R such that, for every x E R, xe = x , then e is called a unit of R. Let R be an algebra, and let I x 1, x E R, be a real-valued function on R, not identically zero, such that, for any number A and elements x, y E R, we have

+

+

+

+

then 1 x 1 is called a seminorml on R. Let R be an algebra, and {I x la , 01 E %} (here, % is an arbitrary indexing set) a family of seminorms on R. If, for each nonzero element x E R, there is an 01 E % such that I x la > 0, then R is said to be a seminormed algebra with respect to {I x la, ci E %} (or, in some of the literature, a locally multiplicatively convex algebra). I n particular, if 2l contains but a single index, then R is called a normed algebra. If ‘$I is denumerable, for instance, if 2l is the totality of natural numbers, we call R a countably seminormed atgebra with respect to {I x I n , n = 1, 2,...}. If R has a unit e, we shall always assume that I e la = 1 for all ci E ‘$I. Trunslutor’s note: In the terminology of Naimark [I], a “symmetric pseudonorm.”

2.1. Topological Algebras with Involution

51

We note that a seminorm is, in particular, a pseudonorm (see $I.1), hence a seminormed algebra R is a linear topological space with respect to the family of pseudonorms {I x lo: , a E a}, moreover, multiplication of elements is continuous in this topology. We shall not discuss this topology in detail; for our present purposes, we need only consider the convergence of sequences in R.

Definition 2.1.2. Let R be a seminormed algebra with respect to the family of seminorms {I x jol , a E a}.Let {x,} be a sequence of elements of R. If there is an x E R such that lim 1 xm - x

lm

m+w

= 0,

for every o l Ql, ~ we say that {x,} converges to x, and write x,+x. {xn} is a sequence in R such that lim

n,m+s

I x,

-x ,

la

If

=0

for every a E N, we say that {xn} is a fundamental sequence in R. If each fundamental sequence in R converges to some element in R, we say that R is complete.2 A complete normed algebra is called a Banach algebra. Let f be a linear functional on R. If there is a nonempty finite set of indices al ,..., an E ‘u, and a positive number M , such that, for every x

E

R,

If(4 <

ma4x

lol )..*,I x,), I

then f is said to be a continuous linear functional on R. If R is countably seminormed with respect to {I x I n , n = 1, 2, ...}, we form the sequence of seminorms (1 x )’,I as follows:

I x ’,I

<

= max(I x

II ,.-, I x I,).

Then I x I ’ I x 12’ \< moreover, this sequence of seminorms induces the same topology as {I x .},I Hence, we may always assume that the family of seminorms {I x },I satisfies the condition ..a,

Ix

II

< I x < ..’ < I x In < *... I2

Furthermore, it is easy to prove that a linear functional f on a countably seminormed algebra R is continuous if and only if, for every convergent sequence {xn} in R, x, -+ x implies f ( x n ) -+ f (x). In the theory of linear topological spaces, this property is known as sequential cornpleteness.

52

11. REPRESENTATION OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

20 Involutions and Positive Functionals Definition 2.1.3. Let R be an algebra, and let x -+ x* be a one-toone mapping of R onto R, such that, for any numbers A, p, and elements x, Y E R, (i) (Ax

+ py)* = Ax* + py- *,

(ii) (xy)*

= y*x*,

(iii) (x*)*

= x.

Then the mapping x -+ x* is said to be an involution on R, and x* is called the conjugate of x . If x = x*, we say that x is self-conjugate. For any x E R, the element x*x is self-conjugate. If R has a unit element e, then e must be self-conjugate. In fact, since e* = e*e is self-conjugate, we have e = (e*)* = e*. Furthermore, (ex)* = x*e* = x*e = x*, hence (ex)** = x**, that is, ex = x, so that e also acts as a right-sided unit. For any x E R, the elements x1 =

x

+ x*

x2=-

9

~

x -x*

2i

are self-conjugate. Therefore, any element of R can be expressed as a linear combination of self-conjugate elements: x = x1

+ ix2 .

Definition 2.1.4, Let R be an algebra with an involution. Iff is a linear functional on R such that (2.1.1)

for every x E R, then f is said to be symmetric. Iff is a linear functional on R satisfying the condition f(x*x)

0,

x E R,

(2.1.2)

then f is called a positive functional. Let R be an algebra with an involution, f a positive functional on R. For any x, y E R, and any complex number A, we have, by (2.1.2), f(.*x)

+ Mx*JJ) + Xf(r*x) + m(r*r) =f((x

>

>

+

W*(X

+ AY)) > 0.

Sincef (x*x) O , f ( y * y ) 0, it follows that Af ( x * y ) Hence, by suitably choosing the value of A, we get

(2.1.3)

+ Xf(y*x) is real.

2.1. Topological Algebras with Involution

53

Also, since the form (2.1.3) is positive semi-definite, it follows that, for any x, Y E R , If(x*r>l < (f(x*4)’/z( f ( Y * Y ) ) l / z * (2.1.5) This is known as the Schwarz inequality for positive functionals. If R has a unit e, then by setting y = e in (2.1.4), we get (2.1.1). Thus, on an algebra with an involution and a unit, every positive functional is symmetric. Let R be a seminormed algebra with respect to x* on R such E rU, x E R , we have

Definition 2.1.5.

{I x la , a E a}.Suppose there is given an involution x that, for every

(Y

----f

(2.1.6)

I x * I a = 1x1,.

Then, we call R a symmetric seminormed algebra. Notice that (2.1.6) is, essentially, the condition that the involution x -+x* be continuous. For, if (2.1.6) is satisfied, then obviously x -+ x* is continuous; conversely, if x -+ x* is continuous, we may replace each I x la by the seminorm

I x :1

=ma4 x

ID

, I x*

ID),

without altering the topology of R, and obviously I x* 3O

Id’

=

I x la’.

Multiplicative Linear Functionals

Definition 2.1.6. Let R be an algebra with unit e, and let f be a linear functional on R such that (i) for any x, Y E R , f ( X Y ) = f ( 4 f( y ) ; (ii) f ( e ) = 1. Then f is said to be a multiplicative linear functional on R. Thus, a multiplicative linear functional is just a homomorphism from an algebra R into the field of complex numbers. I n this book, we shall be interested in symmetric multiplicative linear functionals on algebras with involutions. If R is an algebra with an involution, and f is a symmetric multiplicative linear functional on R , then, for any x E R, f ( x * x ) =f(.*)f(.>

= f(.)f(.)

>, 0.

Therefore, any symmetric multiplicative linear functional is positive. Later on, we shall represent positive functionals, defined on certain

54

11. REPRESENTATION

OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

types of countably seminormed algebras, in terms of symmetric multiplicative linear functionals. In the ensuing discussion, the letter llJz will denote the totality of symmetric multiplicative linear functionals on a given algebra with an involution. We shall represent symmetric multiplicative linear functionals by points of an infinite dimensional product space. First, we begin by using such points to represent real linear functionals. Let @ be a linear space, and let {x, , /3 E B} be a system of linearly independent vectors in @ such that every vector in @ is a finite linear combination of vectors in (x, , ,8 E B}. Such a system does exist (although it is not unique). In fact, let 8 denote the family of all linearly independent sets 4 of vectors in @. Define a partial ordering in 8 as follows: if 4, 7 E 8 and 5 C 7,then 5 < v. Using Zorn’s lemma, one can prove that 8 contains a maximal element, and this maximal element satisfies our requirements. Such a system of vectors {x, ,/3 E B} is called a linear basis for @. Having selected a linear basis {x, ,3!, E B ) for a linear space @, we obtain, for each Iinear functional f on @, a system of numbers { f ( x , ) , /3 E B}. Conversely, given any system of numbers {us , ,8 E B}, there is a unique linear functional f on @, such that f ( x , ) = 248 , /3 E B; this functional is given by the formula fk3,XBI

+ ... +

ce,xo,)

+ ... +

= q31%l

CB,%,

*

(2.1.7)

Thus, we get a one-to-one correspondence between linear functionals and systems of numbers {u, , ,8 E B}. I n particular, let R be an algebra with an involution and a unit e, and let @ be the totality of self-conjugate elements of R. Then @ is a linear space over the real numbers, and e E @ . For each symmetric linear functional f on R, the restriction f I @ is a real linear functional on @. Conversely, any real linear functional f on @ can be extended to a symmetric linear functional on R by defining

Choose a linear basis {x, , /3 E B} for @; we may assume that xo = e (the unit element), Let A = B - (0). For each index

0 E B and ,8 E A , let

E, be a copy of the space of real numbers, and let Eo be the singleton containing only the real number 1. Form the Cartesian product (2.1.8)

2.1. Topological Algebras with Involution

55

and denote the points of E by u = (246, /3 E B}, where 246 E E6 is the pth coordinate of u. In accordance with (2.1.7), we have a one-to-one correspondence between points u E E and symmetric linear functionals f on R which satisfy the condition f (e) = 1. By identifying every f with the corresponding u, we may regard the totality W of symmetric multiplicative linear functionals as a subset of E. Consider any two basis elements x6 , x6‘ , /3, /3’ E B; since x8 , x@’E R, the product XBx6’ is of the form y iz, where y, z E @. Hence, we have a system of complex numbers {c,”,,,. , k E B}, of which only finitely many are nonzero, such that X6xB‘ = 2 c:6‘B’xk . (2.1.9)

+

k

Accordingly, for each pair of indices /3, /3‘ E B, we form the subset of E, consisting of all points {ug , /3 E B } which satisfy the condition

%8,Br

(2.1.10)

I n particular,

= E.

Lemma 2.1.1. W = ~ S , ~ ‘ E B W U Z , , . ~ . PROOF. Let u E n 6 , B t ~ B i D Z 6 , B ’. Let f be the linear functional corresponding to u [see (2.1.7)]. For any two elements x, y E R, we have two systems of numbers, {h, , /3 E B), {& , /3 E B) (only finitely many of them being nonzero) such that

Using (2.1.9), we get xy

CiB’h,f@’Xk

.

8,8’.k

Since we have assumed that u satisfies (2.1.10) for all /3, /3’, it follows by (2.1.7) that

Thus, f is a symmetric multiplicative linear functional such that f (e) = 1, that is, u E W. Conversely, if u E W, then the corresponding functional f is multiplicative. Applying f to both sides of (2.1.9), we get (2.1.10). Thus, u E W6,B’ for all /3, /3’ E B, hence W C n6,f W6,B‘ . 3 Now, let each E6 have the Euclidean topology (i.e., the topology induced by the Euclidean metric) and let E have the product topology.

56

11. REPRESENTATION OF

POSITIVE FUNCTIONALS AND OPERATOR RINGS

Then,

1uz is a closed subspace of E.

(2.1.1 1)

In fact, for each fl E B, the projection u -+ 248 is a continuous function on E, hence the correspondence u --f u&' - &B c ; @ a k is also a continuous function, and so its null space mJzB,fis closed. Hence, by virtue of Lemma 2.1.1, we get (2.1.1 1). The relative topology in % as a subset I ,of E, is known as the weak topology. Clearly, this topology is Hausdorff. Henceforth, unless otherwise stated, we shall always suppose that '33 is given the weak topology. The idea of imbedding a linear space in its second conjugate space is well known. In a similar fashion, we shall regard the elements of R as functions on 1uz. More precisely, for each x E R, we form the function 4(f), f~ 1uz, defined by

w-1 Let 8 be the totality of such functions. 8 is an algebra with respect to the ordinary linear operations and multiplication of functions. We also introduce an involution 4 -+ ff* on 8, as follows:

The correspondence x --t ff is then an isomorphism of R onto 8, moreover, it is consistent with the involutions in R and 8:

(a)* = 9*. For convenience, we shall henceforth write x ( f ) in place of ff(f). When 1uz is given the weak topology, every element 4 of 8 is a continuous function on W. I n fact, given any x E R, there is a finite set of indices fll ,...,Pn E B , and a set of complex numbers cg, ,..., Cg, , such n that x = Cy=Icfl,xB,, hence n

.(f)

=

c8vu8v

3

3,=f(%Y)*

"=l

But this is a continuous function of 248, , 248, ,..., 248, . Therefore, the function ~ ( f )f, EW,is continuous in the weak topology. 40 Some Examples

We next give some examples which will be used in the sequel.

2.1. Topological Algebras with Involut~on

57

Example 2.1.1. Let A, be the totality of entires functions x(z) of n complex variables z = (zl ,..., z,). With respect to the ordinary linear operations and multiplication of functions, A, is a commutative algebra with the unit element 1. For each x E A,, we define the function x* as follows: x*(zl

,..., z,)

= x(gl

,..., Z),.

It is easily verified that the correspondence x -+ x* is an involution in A,. We now proceed to find the totality of symmetric multiplicative linear functionals on A,. For an arbitrary but fixed n-tuple of real numbers zo = (z:, ..., zn0), consider the functional on A, defined by fzo : x + x(z0)). I t is easily seen that f,. is a symmetric multiplicative linear functional on A,. Conversely, let f be any symmetric multiplicative linear functional on A, . For the functions z, E A,, v = 1 , 2,..., n, we have z,* = z, , hence, if : 2

=f(zJ,

then the z,O, v = 1, 2 ,..., n, are real numbers. Now, for each function ~ ( zin) A,, form the power series expansion

which may also be written in the form

* That is to say, analytic on the entire n-dimensional complex number space.

58

11. REPRESENTATION OF

POSITIVE FUNCTIONALS AND OPERATOR RINGS

where zo = (zlo,..., zn0).Therefore, the correspondence

defines a one-to-one mapping from real n-dimensional space R, onto 5WI. If we identify R , and 5WI by identifying zO and fro, then, when f =f,o, we have x ( f ) = X(Z0)).

We define a sequence of seminorms4 {I x follows

’

Im

max

=

[eul<.,u=1,2

,...,n

I.,

a: =

1, 2,...} on A,, as

I +)I.

Then, A, is a countably seminormed algebra with respect to {I x I., a: = 1, 2,...}. Moreover, a sequence {xk} in A, converges to x if and only if the functions {xk(z)} converge uniformly to x(z) on every compact subset of n-dimensional complex number space. It is easily proved that A, is symmetric and complete. We note that A, is a universal commutative symmetric complete seminormed algebra, in the following sense. Let R be an arbitrary commutative symmetric complete seminormed algebra with a unit. Choose any y1 ,..., y, E R. Let q~ E A,, and expand q~ in a power series p(z) = Y,.

c...

.

aul...u”z:’ * . * z’,”

,u,

Since q~ is an entire function, it is easily shown that, for any set of n positive numbers c1 ,..., c, , we have

Hence, for any a E %,

ul..

c..

I aul...””I I Y1 : 1 ... I Y n I? <

.un

By virtue of the completeness of R, it follows that the series

Y1..

In this case, the seminorms x

c..

u u l . . . u , y... ~ yynn

.Yn

Jm are,

in fact, norms.

2.1. Topological Algebras with Involution

59

converges to some element of R, which we denote by d Y 1 !*.*,

m).

We then have d Y 1

,***3

m)*= P)*(Yl*,..., m*).

I n particular, when y1 ,...,yn are self-conjugate elements of R, the mapping Q, + Q,( y1 ,...,y,) is an involution-preserving homomorphism from A , into R. We shall denote the image of this homomorphism by Rv, *...,ti" * Example 2.1.2. Let H be an inner product space, and g ( H ) the totality of bounded linear operators in H. Then g ( H ) is an algebra with respect to the ordinary addition, scalar multiplication and composition of operators, and the identity operator I is the unit of g ( H ) . For any A E 9?(H), let A* denote the operator conjugate to A . Then A +A* is an involution in 9?(H).Given this involution and the ordinary operator norm, g ( H ) becomes a symmetric normed algebra. For any 8 E H , F,(4

=

( A t ,51,

A E m-q

is a continuous positive functional on @(H). We shall refer to g ( H ) and its subalgebras as operator algebras (or, sometimes, operator rings) over the space H . The norm and involution in such algebras will always be as stated in Example 2.1.2. The most important type of positive functional to be considered in this book is that described in the next example.

Example 2.1.3. Let S be a compact topological space, C ( S ) the totality of continuous complex-valued functions on S . With respect to the ordinary linear operations, the norm

IxI

=m a ses

I @)I,

x E C(S),

and the involution x + x* defined by x*(s) =x(s),C ( S )is a commutative symmetric Banach algebra with the unit element 1. Let b be the o-algebra generated by the totality of closed subsets of S , and let p be a (nonnegative) finite measure on ( S , b).Then

f ( 4= Js 4s) 444,

E

C(S)

is a continuous positive functional on C ( S ) .In actual fact, all positive functionals on C ( S ) are of this form.

60

11. REPRESENTATION

OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

Example 2.1.4. Let R be an algebra with an involution and a unit. Let !Dl be the topological space formed by the totality of symmetric multiplicative linear functionals on R, with the weak topology. Let be a compact subset of !JJl, 8 the o-algebra in ID generated by the closed subsets of ID, and p a (nonnegative) finite measure on (ID, b).Then the functional

is well defined. I n fact, for any x E R, the function x ( M ) is continuous on %Jt, hence continuous on a, and therefore bounded on ID. Consequently, x ( M ) , restricted to a, is a bounded measurable function on (a, b), so that the integral on the left of (2.1.12) exists. Obviously, f(x*x) =

I, I

x(M)I2dp(M) >, 0,

x E R.

Therefore, f is a positive functional on R. I n the following section, we shall prove that, under certain conditions, (2.1.12) is the general form for positive functionals on R.

Example 2.1.5. Let Q = (G, b,p) be a measure space, Lm(SZ)the totality of essentially bounded measurable functions on SZ. Then Lw(SZ) is an algebra with respect to the ordinary linear operations and multiplication of functions. For any x EL~(SZ), let

(1 x 11

= essential supremum I x(g)l; BEG

-

x*(g) = x(g),

g E G.

Given this norm and involution, Lm(SZ)becomes a symmetric Banach algebra with a unit. 5O

Algebras of Continuous Functions

First, we introduce the following notation. I f f and g are two real functions, we write

f " g = ma@,

g),

f A g = m w , g).

Next, we state and prove a general approximation theorem for spaces of continuous functions.

Theorem 2.1.2. (Stone-Weierstrass). Let !Dl be a compact topological space, and let R(LUI) be the Banach algebra formed by the totality of real continuous functions on!Dl with the norm 11 v 11 = maxMEmI v(M)I.

2.1. Topological Algebras with Involution

61

Suppose that Q is a closed subalgebra of R(%X),satisfying the following conditions: for any two distinct points M , N of 1)31, there is a p E Q such that p(M) = 0 and p(N) = 1. Then, Q = R(W). PROOF. (1) First, we prove that if g, EQ, then 1 p I EQ. Let E be an arbitrary positive number. There exists a polynomial P such that

<

whenever I t I 11 p 11. Let Po(t) = P ( t ) - P(0). Then Pois a polynomial without a constant term. Since Q is an algebra, Po(p)E Q. By (2.1.13), we have

Consequently, since 1 g,(M))\< )Ig, I), we get

for all M~1)31,whence

11 1 g, 1 - Po(p)ll < E. But Q is closed, hence

I'PIEQ.

+

+

+++

I t now follows that, if v, E Q, then v V = *(g, 1 - $ I) EQ, and similarly p A E Q. (2) For any two distinct points M , N of W,there exists, by hypothesis, an element p E Q such that p(M) = 0, p(N) = 1. Reversing the roles of M and N , we can also choose $ E Q so that $ ( N ) = 0, $ ( M ) = 1. Thus, for any two real numbers a, b, we have ag, b$ EQ, and (q W ) ( M )= 6 , (as) W ) ( N )= a. Now, take any x ( M ) E R(W). Then, for any M , N E 1)31, there is an x ~E Q, such ~ that x M , N ( M ) = x ( M ) , x M , N ( N ) = x(N). Hence, each of the open sets

+

+

+

+

< X ( U ) + €1, vM.N = {U I x M , N ( U ) > X(a) - €1 contains both M and N . Thus, { U M , NI M E 1)31} is an open covering of UM.N= {U I X M , N ( U )

1)31. By the compactness of 1)31, there exist M , ,..., M , ~ 1 ) 3 1 such that UM1,N UMM,,N 3 1)31. By the result proved in part (1) above, we

+ +

have XN

= xM1,N A xM,.N

A

..*

A xM,.N

EQ,

It can be proved that this condition is equivalent to the following. For any two distinct points M , N of %R, there is a p E Q such that p(M) # p(N), and, for any M E m, there is a $I E Q such that $(M) # 0.

62

11.

REPRESENTATION OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

+

and, for every u E YJl, xN(u)< x(u) E . Let V, = VMMy,, . Then V, is a neighborhood of N , and x,(u) > x(u) - E for every u E V , . Again, by compactness, there exist N l ,...,N , such that VN1u u J", 3 %TI, and, by part (1) E = xN1v x ~ , v,

.**

v xNm E Q.

Moreover, x(u) - €

whence 11 x" - x 11 that x E Q . ]

< E.

< q u > < x(u) + f,

u E 93,

Since Q is closed, and

E

is arbitrary, it follows

Corollary 2.1.3. Let !Ui be a compact topological space, and let C(!Ui)be the symmetric normed algebra formed by the totality of continuous complex-valued functions on !Ui (see Example 2.1.3). Suppose that S is a closed symmetrics subalgebra of C(23), with the property that, for any two distinct points M , N of m, there is a v E S such that y ( M ) = 0, y ( N ) = 1. Then S = C( rol). PROOF. Let Q be the totality of real functions in S . Clearly, Q is a closed subalgebra of R(!Ui). Given any two distinct points M , N E ~ , there is, by hypothesis, a function rp E S such that v ( M ) = 0, v ( N ) = 1. Since S is symmetric, $? E S , hence # = g(cp $?)EQ. Also, # ( M ) = 0, $ ( N ) = 1. Thus, Q satisfies the conditions of Theorem 2.1.2, therefore, Q = R(9Jt). For any x E C(m), the elements x1 = (x %)/2, x2 = (x - %)/2i both belong to R(m) (i.e., belong to Q), hence x = x1 ix, E S . Thus, S = C(!Ui). 3

+

+

+

52.2. Representationof Positive Functionalson Seminormed Algebras

We shall first prove that any positive functional, defined on a complete symmetric countably seminormed algebra with unit, is continuous. Here, continuity of a linear functional f means, as usual, that, for any sequence {xn} converging to the zero element, the values f ( x n ) converge to zero.

Lemma 2.2.1. Let R be a complete symmetric countably seminormed algebra with unit e. Let f be a positive functional on R. Then f is continuous. PROOF. Iff(e) = 0, it follows easily from (2.1.5) that f is identically zero. Consequently, we may assume thatf(e) = 1. Suppose that f is not That is, if x

E

S,then x *

E

S.

2.2. Positive Functionals on Seminormed Algebras

63

continuous. Then, there is a sequence {x,} C R such that x, + 0 and -+0. Taking a subsequence of {x,}, if necessary, we may suppose that

f(xn)

=

E

Setting x

,y

= x,

=e

> 0.

inf If(xn)l n

in the Schwarz inequality (2.1.5), we get

3 e2. Also, since x,

whence it follows by (2.2.1) thatf(x,*x,) have x,* + 0. Let yn = x,*x,/f(x,*x,). Yn* = Y n

Yn

7

-

= Nn(zi)

, l).' Since yn --+ 0, we can pick (Nn(zn-1)

(Nn(z2)

(Nn(z3)

+I

zn

1V' ...I2

(Nnkn-2)

< 1/2".

..

Nn(zn-,)2)2 .)2}

we

f ( m )= 1.

+ (Nn(zz)+ (Nn(zs) + (*.*+ + + + (... +

- Wn(z1)

+

-+ 0,

Then

0,

For any x E R,let N,(x) = max(1 x , 1 a subsequence {z,} of {y,} such that Qn

(2.2.1)

(2.2.2)

I n fact, suppose that z1,..., zl-lhave already been chosen, z, = yk, , and that (2.2.2) holds for n = 1, 2 ,..., I - 1. Then, we need only choose k, > kz-, so that I Y k , I is sufficiently small, take zz= yk,, and (2.2.2) will also hold for n = I (since limn+mI y, I = 0, such an index k, can certainly be found). Thus, we obtain the required subsequence {z,}. Consider the element

,

An

= z1

+ (z2 + (z3+ (... +

zn2)2 -.)Z

(2.2.3)

of R. Observe that A, is a polynomial in z1,..., z, , with positive coefficients. Let P,(tl ,..., t,) denote the polynomial obtained by repl9cing every z, in (2.2.3) by the complex variable t,. It is easily seen that A, - A,-l is also a polynomial in z1,...,z, , with positive coefficients; replacing each z, by t , , we denote this polynomial by Sn(tl ,..., t,). Then we have

I An

- An-1

In

< &(I <

z 1 In

!**.)

I 2,

In)

I zn In) = Pn(Nn(z1),***, Nn(zn-J, I X n In) &Z(Nn(~1),*-, Nn(zn-1),

- ~ n - l ( N n ( z l > , - - VN n ( z n - 1 ) )

= Qn

' Here, {I x I, , n

=

< 1/2n.

1, 2,...} denotes the given family of seminorms on R.

(2.2.4)

64

11. REPRESENTATION OF POSITIVE FUNCTIONALS

AND OPERATOR RINGS

1 < -2" . The final step above follows from the fact that N,(x) >, 1. Again, by completeness, there is a xz E R such that 'A, --f xz , and x2 = xz*. Moreover, since A, = x1 Ana, we have

+

XI = %

+

XZ2*

Continuing in this manner, we obtain a sequence of elements such that ~ n == * Xn Xn = ~n ~ 2n + vl n = 1,2,*-* *

{x,}

+

9

in R, (2.2.5)

Since xn* = xn ,f ( e ) = 1, the Schwarz inequality (2.1.5) yields

f(xnI2 G f ( x n ? *

(2.2.6)

From (2.2.5) and (2.2.6), sincef(z,) = 1, we conclude that

f ( x A 2 1 +f(xJ2 z 1 + (1 2 1

+f(x3>")"

+ (1 + (1 + + (1 '**

2

+f(Xn)2)2

.-*

*.*)2

for all n. But this is impossible. Hence, f is continuous.

>, n

3

Corollary 2.2,2. Under the hypotheses of Lemma 2.2.1, if the seminorms {I x la , OL = 1, 2, ...} of R satisfy the condition I x Il I x l a ***, then there is a natural number 01 and a positive number c such that

<

I m l
lo1

<

(2.2.7)

2.2. Positive Functionah on Seminormed Algebras

65

PROOF. If this were not so, then SUP^,^ If(x)I/I x la = co. Consequently, for every n, there would be some x, E R such that f ( x , ) = 1, I x, , 1 < l/n. But obviously, such a sequence {x,} converges to zero. This contradicts the continuity off. ] There are examples which show that this lemma cannot be extended to algebras with uncountable families of seminorms. Inequality (2.2.7) can be made more precise, that is,

If(4l < f M I x la

(2.2.8)

*

In fact, by repeated application of the Schwarz inequality (2.1.5), and using (2.2.7), we get If(x)l < f ( e ) 1 / 2 f ( x * x ) 1 / 2

++*f((x*~)2)~/~

<

f(e)t+f+"'+l/2"f((X*~)2m-1)1/2m

<

f(e)1-(1Prn)c1/2"( ( ( x * ~ ) 2 " - ~

Letting m -+00, we get If(x)I < f ( e ) m-tm Iim ( I ( X * X ) ~ ~ -lu)1/2"'. '

(2.2.9)

<

However, \ ( X * X ) ~ " ' - ' la I x* 1T-l 1 x I?-' = I x If, whence (2.2.8) follows at once from (2.2.9). I n particular, when R is a normed algebra, (2.2.8) reduces to

If(x)l G f k ) I X 1%

(2.2.10)

x E R,

where I x I denotes the norm in R.In this case, a simple proof of (2.2.10) can be given, as follows (see Naimark [l]). 1. Consider the binomial expansion Let x be self-conjugate and 1 x 1

<

The right-hand side of (2.2.11) converges for X = 1. I t follows easily that the series x2 1 . 3 *.- (2n - 3) Y = e - x- - __ - *.. n! 2" X" E R (2.2.12) 2 2! 22 ' 1 .

converges in the norm to an element y of R. Since x is self-conjugate, so are the partial sums in (2.2.12), hence, since I x I = I z* 1 for all z E R, it follows that y is self-conjugate. Moreover, it is easily verified that y2 = e - x. By the positiveness off, we havef(y2) = f ( y * y ) >, 0, hence f ( x ) Gf(e)*

66

11. REPRESENTATION

OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

It follows from this that (2.2.10) holds for all self-conjugate elements of R. Then, by virtue of the Schwarz inequality and the self-conjugacy of x*x, we have, for any x E R ,
If(x)Iz < f ( e ) f ( X * X )

I x*x I < f ( e ) z I x le,

that is, (2.2.10) holds. However, this method of proof does not generalize to countably seminormed algebras.

Lemma 2.2.3. Let A, be the algebra of entire functions of n variables (see Example 2. I. l), and let f be a positive function on A, , with f ( 1 ) = 1. Then, there is a bounded closed subset D of n-dimensional real space R , , and a probability measure p on D, such that, for any TEA,,

Moreover, the p satisfying (2.2.13) is unique. PROOF. Let {I la , a = 1,2,...} be the sequence of seminorms specified in Example 2.1.1. Since A, satisfies the conditions of Corollary 2.2.2, there is a positive number c and an integer a such that inequality (2.2.7) holds for all x E A, . For any real n-tuple t = (tl ,...,t,), and any complex n-tuple z = (xl ,..., x,), we write

+ ... + tnzn. Consider the function #(t) =f(exp(it - x)), T h e continuity off t . z = tlzl

implies that # ( t ) is continuous with respect to t . Also, #(O) = 1 . Let t ( l ) , 1 = 1, 2,..., k, be any k points of R,, let 4, ,..., tk be any k complex numbers, and let k

x(z) =

C tZexp(it(l)

*

z).

z=1

Then

c

- t ' m ) ) tZtrn = f ( x * x ) 2 0,

I&($)

E.m

+

where t ( l )- t ( m ) = (till - tim),..., tLl) - tkm)). Thus, is a positive definite function. By the Bochner-Khinchin theorem, there exists a unique probability measure p on R, , such that

(2.2.14)

2.2. Positive Functionals on Seminormed Algebras

67

We shall now prove that p is concentrated on the subset D={zI~z,I < a + l , v = 1 , 2

,..., n}

of R, . Form the sequence of functions

One easily calculates that .m

+ m(z, -

-m

which belongs to A, . Notice that the integral in (2.2.15) converges in the topology of A, . Thus, by taking t = ( t , , 0,..., 0) in (2.2.14), then multiplying both sides of (2.2.14) by g,(tl), integrating with respect to t , , using the continuity off on the left-hand side, and interchanging the order of integration on the right-hand side, we get

<

Since I h, IoL exp a2]2, it follows from (2.2.7) that the left-hand side of (2.2.16) remains bounded as m + 00. But the right-hand side of (2.2.16) cannot be bounded as m -+ 00 unless

+

1 is zero. Consequently, the p-measure of the halfspace x1 >, c1 Applying the same reasoning to the other coordinates, we conclude that the measure p is concentrated on D . Therefore, we may replace R, by D in the integral (2.2.14). We now apply the operator ak,+...+k,

atp

... at?

to both sides of (2.2.14). It is easily shown that this operator may be taken within the parentheses on the left of (2.2.14), and under the integral sign on the right of (2.2.14). Then, setting t = (0,..., 0), we see that (2.2.13) holds for any function of the form ~ ( x = ) flyz1 x : ~ ,and therefore holds for any polynomial. Since the totality of polynomials is dense in A,, it follows by the continuity off that (2.2.13) holds for all

68

11. REPRESENTATION OF

POSITIVE FUNCTIONALS AND OPERATOR RINGS

v E A,.

The uniqueness of the measure p can be deduced from the uniqueness of the measure in the Bochner-Khinchin theorem. 3 Using the description of symmetric multiplicative linear functionals given in 92.1, and the Kolmogorov theorem, we can extend Lemma 2.2.3 to more general algebras. This is done in the following theorem, which is, at present, the most general abstractly formulated result of this kind.

Theorem 2.2.4. Let R be a symmetric commutative complete seminormed algebra with a unit e, and let iUl be the topological space formed by the totality of symmetric multiplicative linear functionals on R. Let f be a positive functional on R,withf(e) = 1. Then, there exists a compact subset !Illl of iUl, and a unique measure p on %Rf, such that (2.2.17) for all x E R. PROOF. As in 52.1, choose a linear basis {xg , fi E B} for the selfconjugate elements of R. For any fixed set of indices fil ,..., fin in B, and any function q ( x l ,..., z,) in A,, form the element rp(xBl,..., xg,) in R. Consider the functional on A, defined by p- f ( d x B l

!*.*)

x6,))*

(2.2.18)

Since the mapping q -+ q(xBl,..., xE,) is an involution-preserving homomorphism from A, into the ring Rxs,,...,xBn (see 92.1 for the definition of the latter), it follows that (2.2.18) is a positive functional on A,. By Lemma 2.2.3, there exists a closed bounded subset DB1,...,6, of the n-dimensional real space EO1 x -.-X E6,, and a unique probability measure pol,...,8, , concentrated on Do1,..,,B , , such that, for any tp E A, ,

f(dxBl

)-**)

When m 2 n, we have

x6,))

2.2. Positive Functionals on Seminormed Algebras

69

hence {pol,...,B , , ,..., 3/, E B} forms a consistent family of finitedimensional probability measures. We may assume that the set aB1 in (2.2.19) (taking the case n = 1) is a closed interval of the form I uBl I qB, , and similarly for the remaining indices. By consistency, it follows easily that the measure uB,,.. is concentrated on the product qs, , v = 1,..., n. Now, by the Kolmogorov theorem (Corolset I uB,I lary 1.3.5'), there exists a unique probability measure p on E = XBEB ED, such that, for any Bore1 cylinder & with base Q, corresponding to the indices /3, ,..., 3/, we have p(&) = pol,...,B,(Q). Furthermore, if we let D = XBEBIDB, then it is easily seen that the outer measure p*(D) = 1. Also, from (2.2.19), we get

<

,

<

f(cp(xi3,9 * * * 7

S,H

=

j

E

cpP(.B,

%'..9

UD,)

444.

(2.2.20)

For any /3, /3' E B, let {c&.} be the system of numbers appearing in (2.1.9). By (2.1.9) and (2.2.20), we have

jE

(UBUB'

-

1c;B.".)'

= f ((XBXB. -

d&)

1

2

C;@'xk)

) = 0.

(2.2.21)

But the measurable set E - !JJIB,B, (see 52.1) is just the totality of points u such that the integrand in (2.2.21) is positive. Hence, p ( E - !JJlB,B,) = 0.

(2.2.22)

Regarding )IJz as a subset of E, we proceed to prove that the outer measure p(:D n !JJl) = 1. I t suffices (see Halmos [I]) to prove that, if I' = Unclr, is any countable union of open cylinders such that r3Dn!JJl,

(2.2.23)

then p ( r ) = 1. Now, by the Tychonoff theorem (see, e.g., Guan Zhaozhi [l]), 3, being a product of compact intervals, is compact, and by (2.1.1 l), 9Jl is closed in E. Hence a n !JJl is compact, and we may assume that r itself is an open cylinder. Then, A = E - r is a closed cylinder, and, by (2.2.23), we have Dn!JJlnA=O. (2.2.24) We assert that there is a finite sequence of sets illlBl,Bl',..., !JJIB,,B; such that

a n A nn 2

v=l

= 0.

(2.2.25)

70

11.

REPRESENTATION OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

For otherwise, any finite number of the sets {a n n m,,' , p , /? E B), which are closed in 3, would have a nonempty intersection. Since 3 is compact, this implies that 3 n A n mflz,,'is nonempty, which, in view of Lemma 2.1.1, contradicts (2.2.24). Hence (2.2.25) holds for some finite subcollection of {W,,f}. But, by (2.2.22), we have 1 I p(E mZR,,8,,) = 0, and since A n 3 C E - r)v=lmZuzB,,B,' , it follows that p*(A n 3 ) = 0.

n,,.

But p * ( 3 ) = 1, hence p ( A ) = p*(A n 3) = 0, and therefore p(F) = 1. This proves that p * ( 3 n mZ) = 1. Now, using the method described in $1.1, I", we concentrate p on the compact set mZ, = 3 n W. Thus, from (2.2.20), we obtain (2.2.17). T h e uniqueness of p follows directly from ] the uniqueness of the measures pB1,,..,8,. By formula (2.2.17), it follows that the positive functionalf in Theorem 2.2.4 satisfies the inequality (2.2.26)

Next, we shall use inequality (2.2.10) to derive a result concerning the space of symmetric multiplicative linear functionals on a symmetric Banach algebra.

Theorem 2.2.5. Let R be a symmetric Banach algebra with a unit, and W the totality of symmetric multiplicative linear functionals on R. ThenmZ is compact in the weak topology. PROOF. Using the procedure of 92.1, imbed mZ in the space E. As we remarked in 92.1, every element of mZ is also a positive functional on R. Hence, it follows by (2.2.10) that, for every fi E B , IU,I

G

(2.2.27)

IX,I.

I

<

Let c, denote the compact subset (uR 1 u8 1 1 x, 1) of E , . By the Tychonoff theorem, C = XBEB is a compact subset of E, and, by (2.2.27), 91 C C. By (2.1.1 l), mZ is closed in E, hence mZ, as a closed subset of a compact set, is itself compact. ] Next, we use Theorems 2.2.4 and 2.2.5 to characterize the structure of commutative symmetric operator rings on Hilbert spaces.

c,

Theorem 2.2.6. Let H be a Hilbert space, and let R be a commutative symmetric complete algebra of bounded linear operators on H , such that the unit operator I E R. Let W denote the compact Hausdorff

2.3. Weakly Closed Operator Algebras

71

space formed by the totality of symmetric multiplicative linear functionals on R, and let C(im) be the ring of all continuous complex-valued functions on W. Then, the mapping A+A(M),

(2.2.28)

MEW

is a symmetric norm-preserving isomorphism from R onto C(W). PROOF. If A E R, then the function A(&?), M Eim, is an element of C(2JI);let k denote the totaIity of such elements. For any x E H , define a positive functional on R, as follows: f z ( A )= ( A X x), ,

A

E

R.

By (2.2.26), we have fn(A*A) G $% I A(M)12fn(I),

hence

On the other hand, by (2.2.10), we have I A(M)I Hence,

< 11 A )I for all M EW.

Therefore, (2.2.28) is a norm-preserving isomorphism from R onto fi. Since R is symmetric and complete, fi is also symmetric and complete, and therefore constitutes a symmetric closed subalgebra of C(W). For any two distinct points M , and No in W, there exists an A E R such that A(M0)# A(N0). Let A(M0)I - A 9J = A(M o )- A(&)

*

Then y E R, the function y ( M ) E R, and cp(Mo)= 0, ?(No) = 1. Hence, by Corollary 2.1.3, fi = C(!JJl). ]

52.3. Weakly Closed Operator Algebras: Fundamental Concepts

l o Various Topologies on the Algebra g ( H ) Let H be a Hilbert space, and let g ( H ) be the algebra formed by the totality of bounded linear operators from H to H , with respect to the usual operations. The correspondence A 4 A*, where A* denotes the

72

11. REPRESENTATION OF

POSITIVE FUNCTIONALS AND OPERATOR RINGS

adjoint of A, is an involution on g ( H ) . We proceed to introduce various topologies on g ( H ) , as follows: I. Weak topology. For any A, E g ( H ) , we arbitrarily choose a positive number z and elements fi ,...,f, , v1 ,..., qn of H. We call the set U(Ao ;fi

i * * * i f n 9 ~1

= {A

Vn I c)

>...i

I [((A- A O ) f k , V k ) I <

€9

= l,..., n>

a weak neighborhood of A,. Taking the totality of sets of this form as a neighborhood basis, we obtain a topology on a ( H ) , which we call the weak topology. I n general, this topology does not satisfy the first axiom of countability. For any S C B ( H ) , we shall denote the closure of S, in the weak topology, by S1. It is easily seen that, given the weak topology, g ( H ) becomes a linear topological space. For any fixed B E g ( H ) , the multiplication operation A 3 BA, A E g ( H ) (and A + AB, A E g ( H ) ) is continuous; however, the operation (A, B ) + AB, from B ( H ) x g ( H ) to B ( H ) , need not be continuous with respect to the weak topology (see Example 2.3.1). On the other hand, since

I(@

- A0)fk

7

Vk)l

= I(fk

I

(A" - AO*)Pk)I,

the involution A -+ A* is continuous in the weak topology. 11. Strong topology. For any A , E @ ( H ) ,we arbitrarily choose a positive number E and finitely many vectors fl ,...,f, in H. We call the set V(A0 if1 ,...lfn ; €1 = { A 1 ll(A - A O ) f k

II < 6, l.2 = 1,2,.. ., n}

a strong neighborhood of A,. Taking the totality of sets of this form as a neighborhood basis, we obtain a topology on g ( H ) , which we call the strong topology. I n general, this topology does not satisfy the first axiom of countability. For any S C B ( H ) , we shall denote the closure of S , in the strong topology, by S2. g ( H ) is also a linear topological space with respect to the strong topology. Again, the multiplication operations A --+ A B and A --t BA are continuous for any fixed B, but the operation ( A , B) .+ AB from g ( H ) x g ( H ) to B ( H ) need not be continuous in the strong topology (see Example 2.3.1). Moreover, in the strong topology, the involution A -+ A* need not be continuous (see Example 2.3.1). 111. Uniform topology. We have already mentioned (in 52.1) that g ( H ) , given the ordinary operator norm, forms a symmetric Banach algebra. This norm determines a topology on g ( H ) , such that, for any A, E $(H), the sets { A I II A - A, II < 4

2.3. Weakly Closed Operator Algebras

73

form a neighborhood basis at A,. This is called the uniform topology. The uniform topology satisfies the first axiom of countability, furthermore, in this topology, the addition, multiplication, and involution operations are all continuous. For any S C B ( H ) , we shall denote the closure of S,in the uniform topology, by S3. Clearly, the weak topology on B ( H ) is weaker than the strong topology, and the strong topology is weaker than the uniform topology. I n general, these three topologies are all distinct (see the following example). For any S C B ( H ) , we obviously have

s3c s2c S'. Example 2,3.1. Let H be a separable Hilbert space, {e,), v = 1 , 2,..., any complete orthonormal system in H. Define an operator U as follows. For any x E H, let

c (x, m

ux

=

ev+,)e,

*

"=l

Obviously,

11 U 11

< 1. Let A ,

=

Un. Then, for any x E H,

Anx

=

1(x,

m

ev+n)ev

,

v=l

hence 11 A,x ] I 2 = C,"=,+,I(x, e,)I2. Since follows that, for every x E H, we have

co

I(x,

xk=l

ek)I2 =

I( x 11' < 00, it

that is, {A,} converges to zero in the strong topology. However, one easily calculates that the adjoint of A , is m

An*x

=

1(x,

ev)ev+n

7

"=l

hence 11 A,*x 11 = 11 x 11, and so {A,*} does not converge to zero in the strong topology. This shows that (i) the mapping A -+ A* is not continuous in the strong topology, and (ii) the strong and weak topologies on B ( H ) are distinct. Since {A,} also converges to zero in the weak topology, it follows that {A,*} converges to zero in the weak topology (since the mapping A -+ A* is always continuous in the weak topology). But A,A,* = I, the identity operator, so that {A,A,*} does not converge to zero. This shows that the multiplication operation ( A , B) -+ A B is not continuous in the weak topology.

74

11. REPRESENTATION

OF POSITIVE FUNCTIONALS A N D OPERATOR RINGS

Next, we proceed to show that the multiplication operation ( A , B ) - +AB is not continuous in the strong topology. Choose any f~ H , l l f l l = 1, and any positive number e < 1. Consider the strong neighborhood V(O;f,E) in g ( H ) . It suffices to prove that, for any two neighborhoods of zero, V(0;fl ,...,f,, el)

V(0; v1 ,.-, pa , 4,

and

one can find operators A E V(0;fl ,...,f, , el) and B E V(0;v1 ,..., vn ,E ~ ) , such that AB E V(0;f, E). Choose a positive number 8 < Ez/maxk(1 v k 11. Then )I 8A,*fpk 11 = 611 Tk 11 < e2 , k = 1,..., q, hence, &A,* E V(0;yl

,..., yo , c2),

71 =

1, 2,... .

Choose no sufficiently large, so that

1 s1

Anof,

1<

€1

=

1,2,***,P.

Then, 1

An, E V(0;fi

j***,fD t €1).

But

I

=

(f Aao)

*

z V(0;f,

€).

This proves our assertion. 20 von Neumann Algebras

Definition 2.3.1. A subalgebra R of g ( H ) which is symmetric (i.e., A E R implies A* E R ) , weakly closed, and contains the identity operator I is called a weakly closed operator ring (or weakly closed operator algebra)8 over H . If S C a(H ) , the smallest weakly closed operator ring containing S is called the weakly closed operator ring generated by S, and will be denoted by R(S). Since the intersection of any collection of weakly closed operator rings is also a weakly closed operator ring, it is clear that R ( S ) is just the intersection of all the weakly closed operator rings which contain S. Let S C g ( H ) , and let S* = {A* 1 A E S}. Let S' denote the totality of bounded linear operators which commute with all the operators in * In some of the literature, such an R is called a won Neumann algebra. Also, some authors use the term weakly closed operator ring (or W*-algebra) in reference to a weakly closed symmetric operator algebra R (without requiring that I E R).

75

2.3. Weakly Closed Operator Algebras

S U S*.I t is easy to verify that S' is a weakly closed operator ring, and that (i) if S, C S, , then S,' C S,'; (ii) S C (S')'. Writing S" for (S')',S"' for (S")',and so on, we have p i ) S' = S"' = s v = ..., 8'' = S ' V = SV' = ... In fact, setting S, = S and S , = S" in (i), we get S"' C S'. Substituting S' for S in (ii), we get S' C S"'. Combining these results, we get S' = S"'. Continuing in this fashion, we obtain (iii). We use R,(S) to denote the smallest symmetric, uniformly closed0 operator ring which contains S and the identity operator. Starting from the elements of S u S* and the identity operator, and performing all possible finite sequences of algebraic operations, we obtain an operator algebra; the closure of this algebra, in the uniform topology, is R,(S). I t is easily proved that (iv) S' = (R,(S))' = (R(S))'. By virtue of (ii), S" is a weakly closed operator ring containing S , hence (v) R(S)C S " .

Theorem 2,3.1. Let S C B ( H ) . Then, R(S) is the closure of R,(S) in both the strong and weak topologies. Moreover, (2.3.1)

R ( S ) = S".

PROOF. Choose any A, E S". We shall prove that A, E (R,(S))2.That is, we shall prove that any strong neighborhood of A,, say, V(Ao; f,,,...,fnO, c), contains an element of R,(S). Consider the direct sumlo That is, closed in the uniform topology. general, let H be a Hilbert space, and let n be an arbitrary cardinal number. Take any set A of cardinality n. Let x = { x A , X E A} be any vector function on A with values in H (i.e., x A E H) such that lo In

11 XA Ila < m. A€A

Let H,, denote the totality of such functions x . Define linear operations in H,, as follows: if a, fi are scalars, then a{xA , X E A } fi{yA, X E A } = {axA byA, h E A}. Then H , becomes a linear space. Define an inner product in H,, by the formula

+

+

It is easily verified that H,, forms a Hilbert space with respect to the linear operations and inner product defined above; it is called the direct sum of n copies of H. When n is a finite ,..., x,,}. number, we write A = (1, 2,...,n) and { x A, X E A } = {XI

76

H,

11. REPRESENTATION =H @H @

OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

@ H. For each B E R,(S), form the vector f a = (Bfi",Bf2",*.*, BfnO)

in H , . The totality of such vectors fB , B E R,(S), forms a linear subspace E of H , . Let Eo be the closure of E (with respect to the norm of H,); Eo is a closed linear subspace of H , . Let P be the projection of H , onto EO. Form an operator P,, on H, as follows. For f E H, let P,,f be the Zth component of the vector P(0, 0,...,0,f , 0

,..a,

0).

nrl

It is easily seen that Pm1E SY(H). Next, we shall show that P,, E S'. Choose any C E R,(S), and define an operator c on H , , as follows:

Then, for any C, B E R*(S),we have (?fB =fcB E E. Thus, the subspace E is invariant under (?, hence, EO is also invariant under c, and therefore

On the other hand, C* also belongs to R,(S), and c* = hence, if we substitute C* for C in the above relation, and take the adjoint of both sides, we get P(?P = PO. Consequently, whenever C E R,(S), we have CP = PC,

c*,

and hence {CPmlf, CPrnzf,**.7 Cpmnfl =

CP(0,...,0,f , 0,..., O}

= PC(0,...,0,f , 0,..., 0)

I _

m-1

= P(0

,...,0, Cf, 0,..., O} = (PmlCf,...,PmnCf}.

Thus, we conclude that P,,C = CP,, for all C E R * ( S ) , that is, E ( R J S ) ) ' = S'. Since A, E S",it follows that P,,A, = A,P,, . Now, I E R,(S), f O =fiE EO, hence, Pf O =f O, that is, for every 1,

P,,

2.3. Weakly Closed Operator Algebras

77

Applying the operator A,, , we get n

'of?

=

2

n

AoPm,fmo =

m=l

for every 1, whence PfA, a B E R , ( S ) , such that

=f A o

1

PrnAfmO

m=l

, that is, f A o E Eo. Consequently, there is

n

2

ii(AO

-

B)f? I/'

= IlfB - f A o

11' <

z=1

This proves that A,

E

( R , ( S ) ) 2C (R,(S))l C R ( S ) . Therefore, we have S" c R(S),

whence, by virtue of property (v), we get S" = R(S). Since S" C (R*(S))2C ( R , ( S ) ) l C R ( S ) , it follows that (R*(S))2= (R,(S))l = R(Sh 1 Corollary 2.3+2. Let R be a symmetric subring of g ( H ) such that I E R. Then R1 = R2. PROOF. By Theorem 2.3.1, R1 = (R,(R))2= (R3)2= R2. 1

Corollary 2.3.3. If M is a weakly closed operator algebra, then = M". PROOF. Since M is a weakly closed operator algebra, R ( M ) = M . T h e conclusion then follows from (2.3.1). ] M

Corollary 2.3.4. Let A be a bounded self-adjoint operator on a Hilbert space H and let {PA}denote the resolution of the identity associated with A. If M is a weakly closed operator algebra over H , then A E M implies that PAE M , -a < h < co. If M is an operator algebra which is closed in the uniform topology, then PAE M, -co < h < a, implies that A E M . PROOF. If M is weakly closed, A E M, B E M ' , then since B commutes with A, it follows that B commutes with every P A ,hence, by Corollary 2.3.3, PAE M" = M . If M is closed in the uniform topology, and PAE M for all A, then, since there is a sequence of partitions of the real line, { h c ) ,hin),...,,}: A: such that the sequence of operators

converges uniformly to A , it follows that A E M . ] For any weakly closed operator algebra M , we denote by M P the

78

11. REPRESENTATION OF

POSITIVE FUNCTIONALS AND OPERATOR RINGS

totality of projection operators belonging to M. As an immediate consequence of Corollary 2.3.4, we have:

Corollary 2.3.5. If M and N are weakly closed operator algebras, then M = N is equivalent to MP = N p , moreover, M = R(MP). For any weakly closed operator algebra M, we denote by M U the totality of unitary operators belonging to M. Corollary 2.3.6. If M is a weakly closed operator algebra, then R(MV)= M. If N is also a weakly closed operator algebra, then MU= N U is equivalent to M = N. PROOF. Obviously, R(MU) C M. If P E Mp, then I - 2P

But I E

M

MU,

= (I -P)

+ (-P)

EMU.

hence P E R(MU). Thus, Mp C R(Mu). Consequently,

= R ( M P )C R(MU), whence we conclude that R(MU) = M. T h e

remaining assertion then follows trivially. ] Let M be a family of linear operators on H, and let G be a linear subspace of H. If, for every ( E G and every A E M, we have A[ E G, then the subspace G is said to be invariant under M, or simply an M-invariant subspace. If M C &?(If), and G is an M-invariant subspace, then, clearly, the closure Go of G is also an M-invariant subspace.

Lemma 2.3.7. Let G be a closed linear subspace of H, and let P: H -+ G be the corresponding projection operator. Then, G is M-invariant if and only if P E M'. PROOF. If P E M', then, for any A E M, A P = PA, hence A P = PAP, which shows that G is invariant under A. Conversely, if G is M-invariant, then, for any A E M, we have A P = PAP. But since A* E M, we also have A*P = PA*P, whence, by taking complex conjugates, we get P A = PAP. Therefore, A P = PA. 'J 30 Center, Commutative Operator Algebras, and Factors Let M be a weakly closed operator algebra. Then, 3 ( M ) = M n M' is called the center of M ; it consists of those operators in M which commute with every operator in M. T h e following two cases will be of special interest to us. (1) 3 ( M ) is maximal, that is, 3 ( M ) = M. This means that M C M', in other words, M is a commutative weakly closed operator algebra. (2) 3 ( M ) is minimal, that is, 3 ( M ) contains only the operators {dI 01 E F}, where F denotes the coefficient field of H; in this case, we say that M is a factor.

2.4. Representation of Commutative Weakly Closed Operator Rings

79

Notice that, if M is a commutative weakly closed operator algebra, then M‘ is not necessarily commutative. However, if M is a factor, then M’ is also a factor, and conversely. This fact follows immediately from the relation 3 ( M ) = M n M’ = M’ n M ” = 3(M’).

Example 2.3.2. ( g ( H ) ) ’ = {a1 I ~ E F }{a1 , 1 0 1 EF}’ = g ( H ) . I n fact, {OJ I a E F}is clearly a weakly closed operator algebra, and, obviously, {al}’ = g ( H ) . Therefore, (2qH))‘ = {orl}”= {al}.

Consequently, the largest weakly closed operator ring g ( H ) and the are both factors. smallest weakly closed operator ring {d} Section $2.4 will be devoted exclusively to the discussion of commutative weakly closed operator rings.

4O Unitary Equivalence; Restriction Definition 2.3.2. Let H and G be Hilbert spaces, U a unitary operator from H onto G. If M is a weakly closed operator algebra over H, then N = UMU-’ = {UAU-l I A E M } is a weakly closed operator algebra over G. We say that M and N are unitarily equivalent. I n this case, we clearly have N = UM‘U-l, 3 ( N ) = U 3 ( M ) U-l. Consequently, if one of two unitarily equivalent operator algebras is commutative, then the other is also commutative; if one is a factor, then the other is also a factor. Let M be a weakly closed operator ring over H , and let G be a closed linear subspace of H. Let MGdenote the totality of operators in M which are reduced by G. Then MG may be regarded as a ring of operators on G, which we call the contraction of M to G. It can be proved that the contraction of M to G is a weakly closed operator ring over G. 52.4. Representation of Commutative Weakly Closed Operator Rings

Throughout the present section, H will denote a Hilbert space and B ( H ) will denote the algebra of all bounded linear operators on H. First, we prove a lemma which will be frequently used in the sequel.

Lemma 2+4.1. Let S C g ( H ) , and suppose that S* = S C S’. Then R(S) is commutative.

80

11. REPRESENTATION OF POSITIVE FUNCTIONALS

AND OPERATOR RINGS

PROOF. The totality of elements obtainable from S and I by a finite sequence of algebraic operations constitutes a commutative symmetric operator ring E. Furthermore, R(S)= R,(S)l = (E8)l = El. However, it is easily proved that the weak closure of a commutative operator ring is also commutative, hence, R(S)is commutative. ] l o Maximal Commutative Weakly Closed Operator Rings

Definition 2.4.1. Let M be a commutative weakly closed operator ring over H. If the only commutative weakly closed operator ring over H, which contains M, is M itself, then M is said to be maximal commutative weakly closed. Lemma 2.4.2. An operator ring M is maximal commutative weakly closed if and only if M

= M‘.

(2.4.1)

PROOF. Let M be a maximal commutative weakly closed operator ring. Since M is commutative, M C M’. We shall now prove that M ‘ C M. Let A be any self-adjoint operator in M‘, and consider the set S = M u {A}. Since S is symmetric and commutative, it follows by Lemma 2.4.1 that R(S)is commutative. But R(S)3 M, hence, by the maximality of M, we must have R(S)= M, so that A EM. Using Corollary 2.3.5, we deduce that M ‘ C M, and therefore M = M’. Conversely, assume that (2.4.1) holds, and let N be a commutative weakly closed operator ring containing M. Then, by property (i) of S’ in $2.3, we have N ’ C M’. (2.4.2) Since N is commutative, N C N’, hence, from (2.4.1) and (2.4.2), we obtain NCM,

Therefore, M is maximal. ] Definition 2.4.2. Let M be a commutative weakly closed operator ring. Suppose there exists a family {HAI X E A} of M-invariant closed linear subspaces of H such that that is, N

= M.

H =

c OH,,

AEA

the restriction MAof M to HAis maximal commutative weakly closed for every X E A, and all the MAare unitarily equivalent. We then say

2.4. Representation of Commutative Weakly Closed Operator Rings

81

that M has uniform multiplicity k, where k is the cardinal number of A . I t can be proved that the cardinal number K is uniquely determined, that is, if M satisfies the above definition for both of the cardinal numbers K and k', then k = k'. I n particular, a maximal commutative weakly closed operator ring is just a commutative weakly closed operator ring having uniform multiplicity 1. I. E. Segal has proved the following decomposition theorem for commutative weakly closed operator rings. Since this theorem will be used only once in the present volume, the proof, which is rather lengthy, will not be given here; we refer the reader to Segal [2].

Theorem 2.4.3. Let M be a commutative weakly closed operator ring over H. Then, there is a correspondence which assigns an M-invariant closed linear subspace H, of H to every cardinal number" n, such that H=C@Hn n

and, when H , # {0}, the restriction of M to H, has uniform multiplicity n. Moreover, the family of subspaces {H,} is uniquely determined; more precisely, if {Hn'} is another family of M-invariant closed linear subspaces satisfying the above conditions, then H , = H,' for every n. Suppose that M and N are two unitarily equivalent weakly closed operator rings. If M is maximal commutative weakly closed (i.e., M = M'), then we see from $2.3, 40,that N = N', that is, N is also maximal commutative weakly closed. Furthermore, if M is commutative and has uniform multiplicity n, then N clearly has the same property.

Example 2.4.1. An important type of commutative operator ring is the multiplication algebra, defined as follows. Definition 2.4.3. Let SZ = (G, 23, p) be a measure space, and let i!,z(SZ) be the Hilbert space formed by the totality of quadratically integrable vector-valued functions on SZ, taking values in a k-dimensional Hilbert space Hk (see $1.1, 2 O ) . Iff(g), g E G, is any bounded measurable function on SZ, then the mapping T, : ((g) +f(d

w,

g E G, I E

2k2(Q)

is a bounded linear operator in Pk2(.R); we call T, the multiplication operator corresponding to the function f. T h e set of all such multiplical1 Translator's note: Of course, H , = {O] for all sufficiently large cardinal numbers n. It is clear that logical difficulties can be avoided by a more meticulous wording of the

theorem.

82

11. REPRESENTATION OF POSITIVE FUNCTIONALS

AND OPERATOR RINGS

tion operators forms an operator ring W,(Q), which we call the multiplication algebra over i!k2(Q). It is easily seen that the correspondence f- T f is a norm-preserving12 isomorphism from Lrn(Q)l3onto W,(Q). In particular, when k = 1, Wl(Q) is just the multiplication algebra over L2(Q);we shall sometimes write simply W(Q) in place of !JJI,(Q). I n what follows, we shall prove that, if Q is a localizable measure space, then W,(O) has uniform multiplicity K, and, in fact, that all commutative weakly closed operator rings are essentially of this type.

Lemma 2.4.4. Let Q = (G, 9, p) be a measure space, (5 a determining set of bounded measurable functions on Q. Let @ = { T f I f E @}. Then R(6)3 W,(Q). PROOF. Let a be the smallest symmetric closed subalgebra of L"O(Q) containing @ and 1; obviously, a is also a determining set. It is easily seen that I?,(@) = { T , I T E a}. Choose any ~ E L ~ ( Q ) , fl ,...,f, E i!k2(Q), and form the function

in Ll(Q2).Then v E L2(Q, p ) . By Lemma 1.1.6, for any positive number 6, there exists y5 E 3 such that

Thus, T , E V(T , ;fl ,...,f, , 6). This shows that T+,E R,(6)2. Hence, by Theorem 2.3.1, we have T , E = I?(@). ] 2O

Representation of Maximal Commutative Weakly Closed Operator Rings

Let H be a Hilbert space, 2l a maximal commutative weakly closed ring of bounded linear operators on H , and r the compact Hausdofi space consisting of all symmetric multiplicative linear functionals on 2l. Let b be the o-algebra generated by the totality of closed subsets of F. Given a measure p on (r,9), we shall write Q = (TI9, p), and W(Q) will denote the multiplication algebra over Lz(Q).With this notation, we have the following theorem. l2 Trunslator's note: This correspondence is norm-preservingif and only if condition (i) of Definition 1.2.3 is satisfied. lS Regarding Lm(sZ)as an algebra (see Example 2.1.S).

2.4. Representation of Commutative Weakly Closed Operator Rings

Theorem 2.4.5. There exists a localizable14 measure p on and a unitary operator 'p from H onto L2(Q)such that (vAVJ-l)f(Y) = A(Y)f(Y),

for all A

E %,

83

(I', 23)

fGL2(Q)

and the correspondence A

-+

PAP-'

is an isomorphism from % onto %N(Q). T o prove this theorem, we require the following lemma.

Lemma 2.4.6. Let % be a symmetric operator ring over a Hilbert space H , such that I E %. Then H permits an orthogonal sum decomposition (2.4.3)

where H , is the smallest %-invariant closed linear subspace of H containing [. ( H , is called a cyclic subspace of H relative to %, and ( is said to be a cyclic element of H , relative to %.) PROOF. (1) Assuming that H # {0}, choose any element t1 E H , El # 0. Let H,, denote the closure of the set { A t , I A E %I.

(2.4.4)

Then H,, is a closed linear subspace of H , and since (2.4.4)is %-invariant, so is H E 1 Thus, . [,is a cyclic element of Htl relative to %. (2) If the subspace M is %-invariant, then so is its orthogonal complement H 0M . I n fact, let x E H 0M , A E %; then, for any y E M ,

(Ax,r>= (x,A*y) = 0, hence Ax E H 0M . (3) Let m be a family of closed linear subspaces of H , satisfying the following conditions. (i) Every M in m is cyclic relative to %. (ii) If M , N E m and M # N , then M 1N . Let 8 denote the totality of such families m. It follows from part (1) l 4 Translator's note: Scrutiny of the proof reveals that, in general, the localizability requirement cannot be satisfied (at least, not via the author's method of proof) unless p is defined on a certain o-algebra 8' which may be much larger than 8.However, if H is separable, then one can take 8' = 8,and in this case p is, in fact, o-finite. See footnote 16.

84

11. REPRESENTATION OF

POSITIVE FUNCTIONALS AND OPERATOR RINGS

that 5 is nonempty; in fact, m = {HE,}is one such family. 5 is partially ordered by inclusion, that is, m < m' provided that m C m'. Clearly, every totally ordered subset 5' of 5 has a supremum mE8.

m, = 7nE

8'

Therefore, by Zorn's lemma, 5 has a maximal element mo . (4) Let H' = CMMEmo 0M . We shall prove that H' = H . Obviously, the subspace H' is %-invariant. If H' # H , then H 0H' # (0). By part (2), H 0H' is also %-invariant. Choose any element toE H 0H', to# 0; as in part (I), we obtain a cyclic subspace H f o. Since Hto IM for every M E m, , it follows that the family m, w (HEJalso belongs to 5, arid is strictly greater than m, . But this contradicts the maximality of m,. 3 2.4.5. Let { H E, 5 E 9 ) be a family of subspaces PROOFOF THEOREM of H satisfying the conditions of Lemma 2.4.6, and let P, denote the projection operator from H onto H , . Since H Eis invariant under %, it follows by Lemmas 2.3.7 and 2.4.2 that P, E %' = %. For each f E 9, let

r, = {r I P,(Y) = 1). If t # t',then H , 1H,, , that is, PEP,. = 0, hence P,(y) - P,,(y) = 0 for every y E r, so that r, n is empty. For each 5 E S, define a positive functional FEon %, as follows:

0,

FE(4 = (&,

A E a.

Now, by Theorem 2.2.6, the correspondence A + A ( y ) , y ~ r is, a symmetric isometric isomorphism from % onto C ( r ) . Thus, F, may be regarded as a positive functional on C(I'), hence, by a theorem of Riesz (see, e.g., Halmos [I], §56), there exists a unique regular finite Bore1 measure pE on (r,b) such that F,(4

=

14 4 r

dCL,(Y)*

Since P,t = t, we have FE(P,A)= F,(A) for all A F,(4

=

1 PdY) 4) r

dCLE(Y) =

E

%, hence

1 4)44Y). re

(2.4.5)

Since C ( r , )is dense inL1(r, 23, p,), it follows easily that p L r ( r T,) = 0. Let B E= { B I B E b,B C rE}. We thus obtain a family of finite measure spaces Q, = (r,, 23, , pE),5 E 9.

2.4. Representation of Commutative Weakly Closed Operator Rings

85

Consider the dense linear subspace M , = { A t I A E '%I of H , , and the space C(F,) of continuous complex-valued functions on F,. We define a mapping ye from M , onto15 C(F,), as follows: if A E 8, then

By (2.4.5), (2.4.7)

for any A , B E 9I. If we regard C(T,) as a subspace of L2(9,),then relation (2.4.7) means that y, is isometric. Since M , is dense in H , and C ( r , ) is dense in LZ(SZ,) (see Halmos [I]), it follows that ye can be uniquely extended to a unitary operator from H , onto L2(Q,). Let A , = ycAcpcl. Then

In fact, from (2.4.6) and the relation ( A B ) ( y )= A ( y ) B(y), we see at once that (2.4.8) holds when f ( y ) = B ( y ) , B E 9I. By virtue of the density of C(f,)in L2(Q,), this implies that (2.4.8) holds for all f E L2(Q,). Let

ro= r - C€B u r, ,

6,= { B I B E 6 ,B c ro},

Form the direct sum SZ = (r,6, p ) of the measure spaces16 {SZ, , 6 E E} and SZ, . By Example 1.2.1 and Theorem 1.2.6, SZ is localizable. Moreover, by Lemma 1.1.10, (2.4.9)

Since F - r, = { y 1 P&) = 0). it follows that F, is both open and closed in F. Hence, any function in C ( r , ) can be extended to a continuous function on r by simply = IC } (r), assigning the value zero to all points of r - I', . By Theorem 2.2.6, (A(y)l A E ? therefore the image of vt does cover all of C ( r , ) . l8 Tronslutor's note: There is a difficulty here. If the indexing set S is uncountable, then the o-algebra obtained by taking the direct s u m of these measure spaces will, in general, be larger than 23. If one regards each p, as defined on 9, and simply defines p = &E p, , then one obtains the desired result, except that p may not be localizable. Of course, if H is separable, then 8 is countable, and p is, in fact, o-finite.

86

11. REPRESENTATION

OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

We now combine the family of mappings mapping y , as follows. If x E H, define

{vS, 5 E S} to form

a single

(2.4.10)

I t is clear from (2.4.3) and (2.4.9) that 9 is a unitary operator from H to L2(Q). When f E L2(QS), we have q r ' f = q~ry, hence, by (2.4.8),

But since the operators appearing on both sides of (2.4.1 1) are bounded, it follows that (2.4.1 1) holds for all f E L2(B). Let C = (p4v-l I A E W}; then C C %l(B).Since (2.4.10) is a unitary operator, it follows from the remarks in part 4" of $2.3 that C is a maximal commutative weakly closed operator ring over L2(sZ).Obviously, %l(Q) is symmetric and commutative, hence, by Lemma 2.4.1, the weakly closed ring R(%l(sZ))generated by 'iUl(sZ) is also commutative. Since R(%l(Q))3 9l(Q)3 C, the maximality of C implies that R('iUl(sZ))=

m(Q)= c. ]

While continuing to use the notation of Theorem 2.4.5, we now consider the case of uniform multiplicity k.

Corollary 2.4.7. Let W be a commutative weakly closed operator ring having uniform multiplicity k over H . Then, there exists a measure p on (T, b) such that B = (T,b,p ) is a localizable measure space, and there is a unitary operator q~ from H onto Qk2(Q) such that, for every

A E W, (vAv-l)f(y)

= A(y)f(y),

f E Qk2(Q),

moreover, the correspondence

is an isomorphism from Y l onto %Jlk(Q). T h e above corollary may be easily deduced from Definition 2.4.2, Theorem 2.4.5 and Theorem 1.1.4; the details are left to the reader.

Corollary 2.4.8. Let W be a commutative17 weakly closed operator ring over the Hilbert space H. If '3 has a cyclic element in H, then there l7

Notice here that we do not hypothesize the maximality of N.

2.4. Representation of Commutative Weakly Closed Operator Rings

87

exists a finite measure p on (I',23) and a unitary operator y from H onto L2(F,23, p ) such that ?A?-'f(Y)

=

A(Y)f(Y)

for every A E 21 and f E L 2 ( r ,23, p), moreover, the correspondence A -+ rp4v-l is an isomorphism from 'u onto 'iUl(I',23, p ) . PROOF. By assumption, H = Ht , where 5 is a cyclic element of H relative to a.Thus, in the proof of Theorem 2.4.5, P , = I , and we obtain a finite regular measure space 52 = (I',B, p ) and an isomorphism of 2l onto C(I'), regarded as a subring of the multiplication algebra 'iUl(sZ). Using the density of C(I') (qua function space) in L2(9),and the fact that C ( r ) (qua operator ring) is weakly closed in 23(L2(sZ)),it is easily verified that C(I') = 2Jl(52). 3

Corollary 2.4.9. Let 2l be a maximal commutative weakly closed operator ring over a separable Hilbert space H . Then '2I has a cyclic element in H . PROOF. Since H is separable, the indexing set E in the proof of Theorem 2.4.5 is, in this case, finite or countable. Consequently, Q is the direct sum of finitely many or countably many finite measure spaces (r,, 23, ,p,), k = 0, 1, 2,..., and hence is o-finite. Thus, (p4v-l I A E a} has a cyclic element in L2(Q),namely,

where Crk(,) denotes the characteristic function of I',. It follows immediately that '2I has a cyclic element in H . ]

Lemma 2.4.10. If 52 = (G, 8,p) is a finite measure space, then the multiplication algebra il.R(sZ) over L2(Q) is maximal commutative weakly closed. PROOF. Let u E (ml(Q))'; then ul cL2(Q).Let ul be represented by the measurable function u(g) on Q. For any bounded measurable function q on Q, we have uv

=

(uv)]

=

(v4l

=

(2.4.12)

P(d 4d9

hence JG / u(g)T(g)l2dp(g) < // u 112 I! q j 112. Given any positive number E , let q ( g ) be the characteristic function of the set {g 1 1 u(g)I2 2 11 u 112 c} = E. Applying the preceding inequality, we have pfE)(llu /I2 €1d I1 u 11' *P(E), hence p ( E ) = 0. Thus, / u(g)! < (1 u // almost everywhere. Consequently

+

+

88

11. REPRESENTATION OF

POSITIVE FUNCTIONALS AND OPERATOR RINGS

for any q~ €L2(Q),both sides of (2.4.12) belong to L2(f2). Since the totality of bounded measurable functions q~ is dense in L2(Q),it follows that uqI(g) = u(g)qI(g)for every q~ EL~(Q),that is, u E mZ(Q). ] We can now state a sufficient condition for the maximality of a commutative weakly closed operator ring.

Corollary 2.4.1 1. Let 2l be a commutative weakly closed operator is ring over a Hilbert space H . If 2l has a cyclic element in H , then QI maximal commutative weakly closed. PROOF. By Corollary 2.4.8, M is unitarily equivalent to a multiplication algebra 9X(Q), where 52 is a finite measure space. By Lemma 2.4.10, %R(Q)is maximal commutative weakly closed, therefore (11 is also maximal commutative weakly closed. ] We now consider a more general situation.

Theorem 2.4.12, Let Q = (G, 8 , p) be a localizable measure space. Then the multiplication algebra m(Q) over L2(Q)is a maximal commutative weakly closed operator ring. PROOF. Obviously,m( Q) is a symmetric operator algebra, and isclosed in the uniform topology. We need only prove that (!JJl(Q)’)p C m(Q). For then, given any self-adjoint operator A E ‘3n(sZ)’, we know by Corollary 2.3.4 that all the projections PA associated with A belong to (%R(Q)’)p, and hence to m(Q),so again, by Corollary 2.3.4, A E~X(Q). But since rol(Q) is symmetric, this implies that YX(Q)’ C %Jl(Q); obviously, since B(Q)is commutative, %(Q) C%(Q)‘. Thus, we obtain ~ ( Q= ) %R(Q)’, whence it follows by Lemma 2.4.2 that W(Q) is a maximal commutative weakly closed operator algebra. Let P E ( ~ ( Q ) ’ ) p , and consider the collection of sets

5

= {E

I E E B , p ( E ) < 03).

For each E E 5, let C E denote the characteristic function of E, and PE the multiplication operator corresponding to C, : (pEf)(g) = CE(g)f(g),

g

G,

fEL2(’)*

Denote the restrictions of 8 , p to E by B E , p E , respectively. Since p ( E ) < 03, Q, = (E, B E ,p E ) is a finite measure space. Regarding the functions in L2(sZE)as functions in L2(Q) which vanish outside of E, we have L2(QE)= PEL2(sZ),and it is clear that m(52,) is then just the restriction of %R(Q) to L2(Q,).Since P E (%R(Q)’)P, the projections P and P , commute, hence PPE is a projection operator on L2(9,), moreover, since P E ~ m ( Q E and ) PE%R(Q)‘,we have P P E ~ m ( Q E )However, ‘. by

2.4. Representation of Commutative Weakly Closed Operator Rings

89

Lemma 2.4.10, )132(QE)= )132(QE)',hence PPE E '9X(QE), that is, there exists a set A, E B E such that (ppEf

) ( g ) = CA,&)f

(g),

g

E,

f EL2(QE)*

Obviously, the above formula is also valid for any f €L2(Q),therefore PPE is the projection operator corresponding to AE E b. Now, G VAE% A , and so, by Lemma 1.2.5, I = V E E %PE , hence, P = V E E PPE S ; moreover, by the localizability of Q, there exists Q E 23 such that

-

Q=

V ~ E . EE5

Again, by Lemma 1.2.5, P is the projection operator P , corresponding to the set Q, hence P~)132(Q).]

Corollary 2.4.13. Let Q = (G, b,p) be a localizable measure space,

K a cardinal number. Then the multiplication algebra YlIm,(Q)over f!,z(Q) is a commutative weakly closed operator ring, and has uniform multiplicity k. This corollary is a direct consequence of Theorem 2.4.12; the details of the proof are left to the reader.

30 Some Properties of Localizable Measure Spaces We shall now make use of Theorem 2.4.12 to study in further detail the properties of localizable measure spaces.

Theorem 2.4.14. Let Q = (G, b,p) be a localizable measure space, and let F be any continuous linear functional onLl(Q). Then, there exists an essentially bounded measurable function F(g), g E G, such that

moreover, J j F 11 = 1) F Ijm. PROOF. Define a bilinear functional on L2(sZ),as follows: F(v, 4)= F(v$),

If

V?

4 EL2(Q)*

v, # E LZ(Q),then v$ E L1(Q), hence F ( y , #) is well I F(v, 411

< II F II I/ v

112

II

*

(2.4.14)

defined. Moreover,

112 >

hence F ( y , #) is continuous. Therefore, by a well known theorem, there

90

11. REPRESENTATION

OF POSITIVE FUNCTIONALS A N D OPERATOR RINGS

exists a bounded linear operator A on L2(Q)such that 11 A F(P, 1CI)

=

(A%9%

7 1

I( < 11 F I( and

ttr EWQ2)-

Let B E 'Dl(52), and suppose that B corresponds to the bounded measurable function b(g). Then, for any rp, I,4 E L ~ ( Q ) , (A%,

$1 = F ( b &

= (AT,

64) = (BA9J94 1 9

that is, A and B commute. Thus, AE%T(Q)'.By Theorem 2.4.12, %T(52) is maximal commutative weakly closed, hence A E %T(Q). Therefore, there exists an essentially bounded measurable function F ( g ) , g E G such that Arp(g) = F ( g ) q ( g ) , whence F(&)

=

1 F(g) G

(2.4.15)

d g ) $(g) d P k )

for any v, th, E L ~ ( SBut, ~ ) .for anyfELl(Q), the functions I,4 = If l /z and If I-lI2 belong to L2(Q).Substituting these functions into (2.4.15), we obtain formula (2.4.13). From (2.4.13), it follows at once that IIFII I(FII, , and since /IF/loo = /I A 11 /IF11, we get /I F /I = I1 F Ilm * 1 rp = f *

<

<

T h e reader may also prove for himself that, if Q is a measure space and F is a continuous linear functional on L P ( Q ) , 1 < p < m,1* then there exists a function F( E L*(Q),(1/ q ) (1/ p ) = 1, such that a)

F(v) =

J

G

+

d g ) F ( g )dP(d9

9) EL"(J-4,

and IlFll = IIFII, . T h e following theorem extends the scope of the classical RadonNikodym theorem to localizable measure spaces.

Theorem 2,4.15. Let 52, = (G, 23,p,), k = 1,2, be localizable measure spaces, with p1 pz . Then, there exists a measurable function, which we denote by dpl(g)/dpz(g),having the following properties: (i) 0 d dPl(g)ldPz(g) < 00, (ii) for any y €L1(Q1), we have ( d p l ( g ) / d p 2 ( g )rp) E L ~ ( Q and ~),

<

(2.4.16)

We call d p l ( g ) / d p z ( g )the Radon-Nikodym derivative of p1 with respect to Pz * l8

The result stated here cannot be extended to the case p =

CO.

2.4. Representation of Commutative Weakly Closed Operator Rings

91

+

PROOF. (1) Consider the measure p = pl p2 on (G, 23). I t is easily verified that the measure space 8 = (G, 23, p ) is also localizable, and since pk p , K = 1,2, we have L'(8,) 3L1(52),K = 1,2. Define two linear functionals Fl ,F, on L1(8), as follows:

<

Fk(d =

1,dd

dPkk),

T EL'(Q),

k

=

1,2.

(2.4.17)

<

Obviously, IIFk 11 1. By Theorem 2.4.14, there exist bounded measurable functions on 8, which we denote by dpk(g)/dp(g),k = 1,2, such that (2.4.18)

(2) We assert that dp2(g)/dp(g)is positive almost everywhere. For otherwise, there would exist a set E E 23, 0 < p ( E ) < co, such that dp,(g)/dp(g) 0 on E. If, in formulas (2.4.17) and (2.4.18), we let rp be the characteristic function of E, then, for k = 2, we obtain p 2 ( E ) 0, hence also p l ( E ) = 0; but this contradicts p ( E ) > 0. Thus, we may assume that dp2(g)/dp(g)> 0 for all g E G. A similar argument shows that we may assume dpl(g)/dp(g) 0 for all g E G. We now form the nonnegative finite-valued measurable function

<

<

(3) Let $ be a nonnegative finite-valued 23-measurable function on G such that the set ( g I $(g) # 0) is a-finite with respect to the measure p. Then there is a monotonically increasing sequence of functions {$,(g)} #,(g) = $ ( g ) for all g E G. Substituting in L1(8)such that rp = $, in formulas (2.4.17) and (2.4.18), then letting n -+ crc) and applying the Levi lemma, we get

Let

Using an argument similar to that given in part (2) above, one may show that pl(Go) = 0, and that, if E C G, , p ( E ) < 00, then p l ( E ) 2 p(E)/n. Consequently, any measurable subset of G - Go = U l l G, which is a-finite relative to the measure plis also a-finite relative to the measure p.

92

11. REPRESENTATION

OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

Now, consider any nonnegative function cp E L1(Ql). Since the set { g I cp(g) # 0} is a-finite relative to p l , it follows from the foregoing remark that the set { g I cp(g) # 0} n (G - Go) is a-finite relative to p. Form the function (2.4.20) Then, the set { g I +(g) # 0) is a-finite relative to p. Hence, applying = 1, we obtain

(2.4.19)with K

(2.4.21) Since dpl(g)/dp(g) = 0 on G o , it follows from (2.4.21) that (2.4.22) On the other hand, if we choose +(g) = cp(g) dp,(g)/dp2(g), then the set {B

I

f

01 = { g I d g ) f 01 n (G - Go)

is also a-finite relative to p, hence, applying (2.4.19) with

K

= 2,

we get

Combining (2.4.22) and (2.4.23), we conclude that whenever cp E L1(Q1) and cp >, 0, relation (2.4.16) holds and so cp(g)(dpl(g)/dp2(g))eL1(Q2). The proof is completed by observing that any function cp in L1(SZl) can be expressed as the difference of two nonnegative functions in L1(Ql). 3

Corollary 2.4.16, Under the hypotheses of Theorem 2.4.15, (2.4.24) for any E E 6.Moreover, if E is u-finite relative to pz , then E is also a-finite relative to p1 .19 PROOF. If E E 6,p l ( E ) < co,then, in (2.4.16),we let cp be the characteristic function of E, and so deduce that (2.4.24) holds in this case. l* Translator’s note: The statement in the original text was (perhaps unintentionally) the converse of this, that is, that o-finiteness relative to p1 implies a-finiteness relative to pa ; the reader should have no difficulty in constructing both trivial and nontrivial counterexamples.

2.4. Representation of Commutative Weakly Closed Operator Rings

93

On the other hand, if p l ( E ) = co, then there exists a sequence of measurable sets En C E, El C E, C E , C ... such that pl(En) < co and pl(E,) -+ 03. Setting E = En in both sides of (2.4.24) and letting n -+ co, we get

whence, a fortiori

Next, suppose that E is a-finite with respect to pz . Then, there exist measurable sets E, , i = 1, 2, ..., such that E = (Ji=l E, and pz(E,) < ~ 0 Let W

Then, by (2.4.24),

Since E is the union of the countable collection of sets EinF,,,

i , n = 1 , 2 ,...,

this shows that E is also o-finite with respect to pl. ] Theorem 2.4.17. Let Q, = (G, 23, pk), K = 1, 2, 3, be localizable measure spaces such that p1 pz , pz p, , and let d p l ( g ) / d p z ( g ) , dpl(g)/dp3(g), dpz(g)/dp3(g)denote the appropriate Radon-Nikodym derivatives. Then,

<

<

(2.4.25)

~ ) , by Theorem 2.4.15, PROOF. If q E L ~ ( Q then,

whence, applying Theorem 2.4.15 to the measures p, , p 3 , we obtain

(2.4.26)

.

94

11.

REPRESENTATION OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

Again, applying Theorem 2.4.15 to the measures pl , p3 , we know that, if "('l), then 9)(dpl(g)/dp3(g))L1(Q3)9 and

(2.4.27) Combining (2.4.16), (2.4.26), and (2.4.27), we see that

(2.4.28) for any

~ L l ( 9 ~Now, ) . let

E

be an arbitrary positive number, and let

Suppose that p 3 ( E ) > 0; this implies the existence of a measurable set A C E such that p 3 ( A ) > 0 and p l ( A ) < co. Let C, denote the characteristic function of A ; since C, EL'(Q~), we may substitute p = C, in (2.4.28), and thus obtain a contradiction. Therefore, p 3 ( E ) = 0; since E > 0 is arbitrary, we conclude that (2.4.29) for almost all g E G (relative to p3). An entirely similar argument shows that the reverse inequality also holds for almost all g . Hence, (2.4.25) is valid for almost all g E G (relative to the measure p3). ]

Corollary 2.4.18, Let Q, = (G, 8 , pL), k = 1,2, be equivalent localizable measure spaces. Then the Radon-Nikodym derivative dpl(g)/dpz(g)of p1 with respect to pz may be chosen so that

PROOF. Taking p3 = p1 in Theorem 2.4.17, we may choose d k ( g ) / d p 3 ( g )= hence 3

(2.4.30) g )co, it follows from (2.4.30) that for almost all g. Since d p ~ ( g ) / d p ~ (<

2.4. Representation of Commutative Weakly Closed Operator Rings

95

dpl(g)/dpz(g) > 0 almost everywhere. Thus, we need only alter the values of dp,(g)/dp,(g) on a null set and so obtain the required function. ]

Corollary 2.4.19.

Under the hypotheses of Theorem 2.4.15, the

mapping T

2

'p

-

p?(dcLlldr.L2)'/2

is an isometric operator from L2(Ql) into L2(Qz). PROOF. Let y , E L ~ ( Qthen ~);

However, y J

E

L1(Q,), hence, applying (2.4.16),we obtain

=

(TV, T#),

that is, T is isometric. 3 We shall now establish the general form of an operator in [2Jl,(f2)']u, where Q is a localizable measure space. For convenience, we shall explicitly discuss only the case k No. Let {eA, h E A } be an orthonormal basis for the k-dimensional Hilbert space H , , let HA = {yeAI y E L2(Q)},and let PAbe the projection operator from Qkz(Q) onto HA. I t is easily verified that PAE [YJl,(Q)]', moreover, by (1.1.7), we have

<

c P A = I AEA

in the strong topology. and write UAA, = PAUP,, . We assert that there Let U E "2JID1,(B)']~, exists a bounded measurable function uAA,(g) on Q such that UAAT(VeA') =z uAA'VeA

(2.4.31)

for every y eL2(B).I n fact, let $ be an arbitrary bounded measurable function on B; since UAA, E [%Nk(Q)]', we have UAA,(+VA)

~(UAA*(WA)).

Define an operator uAA,from L2(Q)to L2(Q),as follows: u A A ~ ' ~=

( U , u * ~ e ,eA), , 9

P €L2(Q).

(2.4.32)

96

11. REPRESENTATION

OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

Using (2.4.32), it is easily seen that uAA’ E [%T(Q)]’, hence, by Theorem 2.4.12, uAA,E YX(Q), that is, there exists a bounded measurable function tlAh’(g), g E G such that (uAA’P)(g) = uAA’(g)d g ) .

Thus, we obtain (2.4.31). Now, any f E Hk can be expressed in the form f = C &?A (where f A# 0 for at most countably many indices A), and if tp €L2(Q),then tpt E !i?k2(Q), hence (2.4.33)

Therefore, since U is unitary,

(2.4.34)

Consider the subspace 2, =

16 1

[=

1 &en

E

1

HI, , f Arational, [ A # 0 for only finitely many h .

Clearly, 2, is countable and dense in Hk . Using (2.4.34) and the countability of a, we see that there exists a p-null set E in G such that (2.4.35)

for all g E G - E, [ E 2,. Then, using the density of a, it follows easily that (2.4.35) holds for all f E H k , g E G - E. Now, for each g E G - E, define an operator U(g) as follows: if f = {&) E Hk , then u(g)f= Z A b ‘ 6 , where eA’

=

1uAA‘(g)6Ar

*

A’

We see from (2.4.35) that U ( g ) is an isometric linear operator from Hk into H k . Moreover, using (2.4.33), one easily verifies that, if & *) E i!k2(Q), then (US)(g)= U ( g )

for almost all g E G. Applying the foregoing argument to the unitary

2.4. Representation of Commutative Weakly Closed Operator Rings

97

operator U-l, we obtain a p-null set E’ C G and, for each g E G - E‘, an isometric linear operator U-l(g) from Hk into Hk , such that, given any f ( - )E Qk2(Q), the relation (U-lt)(g)

=

Y g ) t(d

holds for almost all g E G. Using the relation U-lU = UU-I the separability of Hk , it is easily proved that

=I

and

that is, that U ( g ) is a unitary operator from Hk onto H k , for almost all E G. Moreover, by supplementing or altering the definition of U ( g ) on a p-null set, if necessary, we may thus ensure that U ( g )is defined and unitary for all g E G. Furthermore, for every [(*) E M ( H , , we have U(.)((.)E M ( H k ,Q).

g

a),

Definition 2.4.4. Let Q = (G, 8,p ) be a measure space, and Hk a k-dimensional Hilbert space. Suppose that, to every g E G, there corresponds a unitary operator U ( g ) from Hk to H k , and that U(.)((.)E M ( H k , Q) whenever ((*) E M ( H k , Q). Then, U(.) is said to be a measurable k-dimensional unitary operator-valued function on Q. T h e foregoing result may then be stated as follows.

<

No, Lemma 2.4.20. Let Q be a localizable20 measure space, k and let U E [%N,(Q)’]. Then, there exists a measurable k-dimensional unitary operator-valued function U(.) on Q, such that

( U t ) ( g ) = U(g)t ( g )

for every ( E Q k 2 ( Q ) . Next, we consider a condition for the equivalence of two localizable measure spaces (see Definition 1.2.4).

Theorem 2.4.21. Let Q = (G, 8, p ) and Q’ = ( G , 8’, p’) be localizable measure spaces. Then Q and Q’ are equivalent if and only if there exists a unitary mapping Q from Q k 2 ( Q ) onto Qk2(Q’) such that the correspondence A

+ QAQ-’

defines an isomorphism of %Nk(Q) onto y x k ( Q ’ ) . 2o Translator’s note: I n the original statement, f2 is assumed to be o-finite. However, the above proof appears to be valid for any localizable measure space.

98

11. REPRESENTATION

OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

PROOF. I n the sequel, we shall only use the sufficiency of this condition, therefore, we omit the proof of necessity. Let ( M , p) and (M’, p’) be the measure rings corresponding to 52 and Q’, respectively. Given any E E 6 , denote the characteristic function of E by cE(g), g E G, and let P, = TcE. Since QPEQ-l E !Ulm,(f2’), there exists a function f ~ L ~ ( 5 2such ‘ ) that T, = QPEQ-’. Moreover, since Q is a unitary mapping, the relations PE* = PE, PE2= P , imply that

fC) =f(g),

f(gY =

fW

Consequently, there exists a set E‘ E 6‘ such that QPEQ-’ = PE*. Define a mapping q from M to M‘, as follows:

Using the fact that Q is unitary, one may easily verify that ‘p is a one-toone mapping of IM onto M’ and satisfies conditions (1.2.9).Furthermore, p ( [ E ] )= 0 is equivalent to PE = 0, which is equivalent to PE,= 0, which is in turn equivalent to p ( y [ E ] )= 0. Thus, 52 and 1;2’ are equivalent. 3 4O

Spectral Measure Spaces

(r,

23) be a measurable space, where 8 is a Definition 2.4.5. Let o-algebra in I‘, and let H be a Hilbert space. Suppose that, for every E E 8, there corresponds a projection operator P(E) on H , and that the following conditions are satisfied. ( i ) P(0) = 0, P(r)= 1. (ii) If E = u:=lE,, E, E 8 , and E , n E, = 0 when p # v, then, for any 5, q E H , m

(P(E)E,7)

=

c (P(E”)E,7).

”=l

T h e abstract set function E -+ P(E) [simply written as P ( . ) ] is then said to be a spectral measure on (r,8 ) over the Hilbert space H , or, alternatively, (F, 23, P ) is said to be a spectral measure space over H . Let us record some simple properties of spectral measures: (iii) If the sets E l , E, ,..., E, E 8 are pairwise disjoint, then P(El

(iv)

+ ... + En)= P ( E l ) + + P(E,).

For any E l , E,

E 8, we

have P(El)P(E2)= P(El n E2).

2.4. Representation of Commutative Weakly Closed Operator Rings

99

I n fact, from (iii), one easily deduces that, if Fl ,F2 E 23, F, n F, = 0, then P(Fl) P(F,)

Hence, for any E l , E,

E 8 , we

= 0.

have

I n particular, (iv) shows that '$3 = { P ( E ) I E E 23} is a commutative family of operators. If P ( E ) H cannot be decomposed into an orthogonal sum of uncountably many '$3-invariant nonzero closed linear subspaces, then P ( E ) is said to be countably decomposable. If, for every F E b such that P(F) # 0, there exists a set E E 23, E C F such that P ( E ) is nonzero and countably decomposable, then the spectral measure P(.) is said to be normal.

Theorem 2.4.22. Let (r,23, P) be a spectral measure space over the HiIbert space H , and let 2l be the weakly closed operator algebra generated by { P ( E )1 E E s}. If 2l is maximal commutative weakly closed, then there exists a measure p on (r,23), and a unitary mapping U from H onto L 2 ( r ,23, p), such that every projection Q ( E ) = U P ( E ) U-l, E E 8, takes the form (Q(E)V)(Y) = C E b ) V ( Y ) ,

V EL2(C 23, I.),

(2.4.36)

where C , is the characteristic function of the set E. PROOF. I n accordance with Lemma 2.4.6, we decompose H into an orthogonal sum of closed linear subspaces { H , , E E}, where 5 is a cyclic element of 2l in H , . For each ( E E, define a set function pE on 23, as follows: PAE) = ( P ( E ) t ,0,

E E 8.

Since P(.)is a spectral measure, it follows easily that Q, = (r,8,pE)is a finite measure space. Next, we construct a mapping U , from H to L2(sZ,),in the following manner. For any vector in H , of the form (2.4.37)

100 11. REPRESENTATION

OF POSITIVE FUNCTIONALS A N D OPERATOR RINGS

(where the Ek belong to b and the A, are numbers), we define I I

(2.4.38)

Obviously, U,q EL~(Q,), and it is easily verified that the correspondence q -+ U,q is linear and isometric; since the totality of vectors of the form (2.4.37)is dense in H , , and the totality of vectors of the form (2.4.38) is dense in L2(Qn,), it follows that this correspondence extends uniquely to a unitary operator U , from H , onto L2(Q,). Let Q,(E) = U,P(E) Ur', E E b.I t is easily seen that (Qo(E>F')(Y> = CE(Y)'?h')~Y E

for every q~ E L2(QE). We now proceed to prove that if 5, 5' E 8, 5 # t', then p, and pcj are mutually singular. Suppose this were not the case. Then, there would exist a set A E b such that the restrictions of p, and p,' to A are equivalent (and not identically zero). Let M CL2(Qn,) and M' CL2(Q,,)be the closed linear subspaces defined by those functions in L2(Q,)and L2(Q,l), respectively, which vanish outside of A. For every function f E M , let

Since pf and pn,#are equivalent on A, the correspondence f -+ Vf is clearly a unitary operator from M onto M'. Let P, denote the projection of H onto H , ; by Lemmas 2.3.7 and 2.4.2, we know that P,E a' = %. Write U = U;;'VQ,(A) U,P,, and let E be any set in 23; from (2.4.39), we obtain W E ) = U?lVQe(A>QdE) U P , =

UF'Qg(E) VQ,(A) UtP,

Thus, U E { P ( E ) I E E 23)' = a', hence U since p , ( A ) # 0, p g ( A ) # 0, we have

= P(E)U.

= PgU =

UP,,

(2.4.40) = 0.

But

This is a contradiction, hence, we conclude that the measures {p, , 5 E E } are mutually singular. Let 23, denote the totality of sets E €23 such that P ( E ) is countably

2.4. Representation of Commutative Weakly Closed Operator Rings

101

decomposable. Clearly, if E E 23,, then there are at most countably many indices 5 such that p t ( E ) > 0. We define a measure p on 23, , as follows:

c

P(E) =

PLp(E)*

u g w #O

(r,23, , p) is a measure space. Since % is maximal commutative weakly closed, the spectral measure P(.) is normal2I. Moreover, it is easily proved that21 p can be extended to 23 and that21 L 2 ( r ,23, p) = @L2(sZ,). Let U = U,P, ; then U is a unitary operator from H onto L 2 ( r ,23, p), and (2.4.40) follows easily from It is easily verified that

zLpEE

zLp

(2.4.36). ] Let H be a Hilbert space, let (r,23, P) be a spectral measure space over H , and let 'u be the weakly closed operator algebra generated by { P ( E )I E E 23}. Suppose that 'u has uniform multiplicity k. Then, there exists a measure p on (r,23) and a unitary operator U from H onto Qk2(F,23, p) such that every operator Q ( E ) = UP(E) Up1, E E 23, assumes the form

Corollary 2.4.23.

v E Q k V , 8, P I ,

(Q(E)VP)(Y)= C d Y ) dY),

where C , denotes the characteristic function of the set E. PROOF. Since % has uniform multiplicity k, the Hilbert space H can be decomposed into an orthogonal sum of k closed linear subspaces HA , h = 1, 2,..., k, k

H=

@HA,

A=l

such that each HA is %-invariant, and all the restrictions I\

'u I HA,

=

1, 2,...,k,

are maximal commutative weakly closed and unitarily equivalent. We arbitrarily choose an index A, k, say, A, = 1. By Theorem 2.4.22, there exists a measure p on (I', 23) and a unitary operator U, from H I onto L2(r,23, p ) such that every operator Q(E) = U,P(E) Ui', E E 23, takes the form ( Q I W d Y ) = CE(Y)dY), VJ EL2(C PI.

<

8

9

21 Translator's note: More detailed proofs of these assertions would be welcome. Of course, if H is separable, then, by Corollary 2.4.9, the set ,F reduces to a single index 5, and the proof of the theorem offers no difficulty. 22 Translator's note: If one accepts the validity of Theorem 2.4.22, then the assumption that P(.)is normal appears to be redundant.

102 11. REPRESENTATION

OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

<

For each h k, let UA1:HA H l be a unitary operator which effects an equivalence between 'ill I Hl and 'ill I HA , and let U,, = U,U,, . Then, a trivial calculation shows that every operator Q,(E) = UAP(E)U:', E E 8 takes the form --+

v E W ~8, ,PI.

(QA(E)) V ( Y ) = CE(Y)v(Y),

Let 5k be a k-dimensional Hilbert space, let {e, , h = 1, 2, ..., k} be a complete orthonormal set in jjk, and let HA= {f(.)e, l f € L 2 ( F ,23, p)). According to Theorem 1.1.4, we have k

gk2(r,8,p )

@ gA

=

*

A=l

Using the family of operators {U, , h = 1,2, ..., k}, we construct a unitary operator U from H onto e k z ( F ,8,p), as follows. If h E H , h = C h, , h, E HA, then k

( U ~ ) ( Y=)

C (UA~,)(Y)~A 3

Y

E

r.

A=1

It is easily verified that U is a unitary operator having the required property. 3

CHAPTER

HARMONIC ANALYSIS ON GROUPS WITH Q UASI-INVARIANT MEASURES

As is well known, every regular measure on n-dimensional Euclidean space which is invariant under all translations is just Lebesgue measure multiplied by some constant factor, and this translation invariance forms the basis of ordinary harmonic analysis (e.g., the theory of Fourier transforms). T h e theory of invariant measure and harmonic analysis on n-dimensional Euclidean space has long since been extended to locally compact topological groups (especially locally compact topological Abelian groups), and abundant results have been obtained. I t is natural to ask whether such a theory of translation invariant measure, and corresponding results in harmonic analysis, can be established on infinite-dimensional spaces. Actually, in dealing with the so-called continual integrals of quantum field theory, physicists have freely manipulated Fourier transforms on infinite-dimensional spaces.' UnforBrush [l] gives an extensive bibliography on the use of infinite-dimensional integrals in quantum field theory.

103

104

111.

GROUPS WITH QUASI-INVARIANT MEASURES

tunately, however, there can be no hope of extending the theory of invariant measure from finite to infinite-dimensional spaces. For example, one may easily prove that no reasonably well-behaved translation invariant measure exists on any infinite-dimensional Hilbert space; more specifically, for any translation invariant measure on an infinite-dimensional Hilbert space such that all balls are measurable sets, there must exist many balls whose measure is either zero or co. Consequently, we are compelled to relax the requirement of translation invariance. We note that if a o-finite measure on n-dimensional Euclidean space is such that any translate of a null set is also a null set (this property is called quasi-invariance), then this measure must be equivalent to Lebesgue measure (see Theorem 3.1S ) , and mutually equivalent (and, e.g., localizable) measures can be related by means of the RadonNikodym derivative. Thus, we may consider generalizing this quasiinvariance property of Lebesgue measure. But again, for the most useful infinite-dimensional spaces (e.g., infinite-dimensional Hilbert spaces), o-finite quasi-invariant (i.e., all translates of any null set are null sets) measures do not exist. Therefore, we must relax our requirements still further; we shall consider measures such that null sets are transformed into null sets, not by all translations, but only by translations corresponding to the vectors in some fixed linear subspace. T h e notion of quasi-invariance in this form was introduced by Gel’fand [l] in 1959; he used measures on rigged Hilbert spaces to obtain preliminary results concerning representations of the commutation relations in quantum field theory (see Gel’fand and Vilenkin [l]). I n this connection, Segal [5] had somewhat earlier (1958) introduced the notion of a quasi-invariant weak distribution on a Hilbert space; this notion is essentially equivalent to that of a quasi-invariant measure. However, the above-mentioned authors did not establish a theory of harmonic analysis for quasi-invariant measures. Clearly, in order to give a thorough treatment of quasi-invariant measures, one should first discuss quasi-invariant measures on groups. Accordingly, we shall, in this chapter, develop a theory of harmonic analysis for quasi-invariant measures on groups. Quasi-invariant measures constitute a very broad class, with general properties much weaker than those of invariant measures. Consequently, one cannot expect to create a general theory of harmonic analysis for quasi-invariant measures which is as orderly and complete as the theory of harmonic analysis on locally compact groups. Nevertheless, some of the problems to be discussed below are closely related with the corresponding problems in harmonic analysis on locally compact groups. However, the methods to be used here are radically different from those

3.1. Basic Properties of Quasi-Invariant Measures

105

ordinarily used in harmonic analysis on locally compact groups, that is, we lean heavily toward measure-theoretic techniques and the theory of weakly closed commutative operator rings over Hilbert spaces. Some of the theory and methods developed here are intended to form a basis for the further study of the many problems associated with the “continual integrals” of quantum field theory. I n $3.1, we consider some of the basic properties of quasi-invariant measures. I n $3.2, we introduce the notion of quasi-characters, which is essential for the study of quasi-invariant measures on groups, and proceed to discuss quasi-character groups in considerable detail. I n $3.3, the integral representation theorem for positive definite functions on locally compact groups will be generalized to groups with quasi-invariant measures. I n $3.4, we present the basic theory of Fourier transforms. 53.1. Basic Properties of Quasi-Invariant Measures

lo The Concept of a Quasi-Invariant Measure We first generalize the idea of a translation invariant measure.

Definition 3.1.1. Let 52 = (G, 8,p) be a measure space, and let h be a mapping from a measurable subset B ( h ) of G into G, where G - D(h) is a null set. Also, suppose that, for every A E 23, h-*(A) = {g I hg E A ; g E q h ) } E 23.

Then, we say that h is a measurable transformation in 52. I n this case, we write p&4) = p(h-lA), A E 8. (3.1.1) Clearly, ph is a measure on (G, a). Let h be a measurable transformation2 of 52 which has the following properties: (i) the range %(h) of h is measurable and G - %(h) is a p-null set; (ii) h is one-to-one from D(h) onto %(h); (iii) the measures ph and p are equivalent; (iv) h-l is a measurable transformation. We then say that SZ is quasi-invariant under h, or h-quasi-invariant; the totality of such transformations h will be denoted by 8(52). If h, h’ E B(SZ), we may define a mapping hh‘ as follows: the domain of hh‘ is W h ’ ) = {g I g E W’), h’g E W)), The notion of a measurable transformation, as defined here, differs somewhat from that used by Halmos [l].

111.

106 and, when g

E

GROUPS WITH QUASI-INVARIANT MEASURES

2,(hh‘), (hh’)g = h(h’g).

Lemma 3.1.1. If h, h’ E 6(sZ),then hh‘ E 6(sZ). PROOF. Obviously, 9(h‘) - b(hh’) = h’-l(G - B(h)),but G - D(h) is a p-null set, hence also a ph,-null set, and therefore 2,(h’) - 2,(hh’) is a p-null set. T h e range %(hh‘) of hh‘ is the set h(%(h’) n B(h))and h(%(h’) n 2,(h)) 3 %(h) - h(D(h) - %(A’)).

Since D(h) - %(A’) is a p-null set, and since p, p h are equivalent, h(2,(h) - %(h’))is a p-null set, therefore G - %(hh’) is a p-null set. Obviously, hh’ is one-to-one, and hh’ is a measurable transformation, moreover, since phh’ = (ph’)h , it is clear that p h h ’ and ph are equivalent. Thus, we have hh’ E @(a).] Let h, h’ E 6(sZ), and suppose there exists a measurable set 2, C b(h) n 2,(h’), such that G - 2, is a null set, and such that hg = h‘g for all g E 2,. We then say that h and h’ are equivalent, and write h = h’; this is clearly an equivalence relation in the usual sense. Moreover, it is easily seen that, if hj E 6(sZ), j = 1, 2, 3 , 4, and h, = h, , h, = h, , then h,h, = h,h, . Therefore, if we identify equivalent eIements of B(Q), then, using Lemma 3.1. I , we easily deduce the following result.

Corollary 3.1.2. 6(Q)forms a group with respect to composition. We call 6(Q) the quasi-invariant transformation group of SZ. Let h E @(a), and suppose that both h and h-l transform sets of finite measure into sets of finite measure; we then say that sZ is strongly quasiinvariant under h, or strongly h-quasi-invariant. If h and h-l transform o-finite sets into a-finite sets, we say that SZ is weakly quasi-invariant under h, or weakly h-quasi-invariant. If 6 is a subgroup of 6(sZ), then we say that SZ is quasi-invariant under 6, or 6-quasi-invariant; if sZ is strongly (weakly) quasi-invariant under every h E 6, we say that sZ is strongly (weakly) quasi-invariant under 6, or strongly (weakly) 6-quasiinvariant. In particular, if p(hB) = p ( B ) for all h E 6 and all B E 23 such that B C ’D(h),then 52 is said to be invariant under 6, or %-invariant. In what follows, we shall be primarily interested in measures defined on groups. Let G be a group, 6 a subgroup of G, and (G, 8 )a measurable space. If hB = {hg l g € B } € 2 3 for every B E 23, h E 6, then we say that the measurable space ( G , 23) is left quasi-invariant under 6, or left 6-translation quasi-invariant. If

3.1. Basic Properties of Quasi-Invariant Measures

107

p is a measure on (G, 23) such that p ( E ) = 0 and p(hE) = 0 are equivalent for all B E 23, h E 6, then we say that the measure space (G, 23, p)

is left quasi-invariant under 6, or left 6-translation quasi-invariant. Concisely stated, (G, 23, p ) is (strongly, weakly) left quasi-invariant under the subgroup 6, provided that every translation g + hg, g E G, h E 8 , transforms measurable sets and null sets (sets of finite measure, a-finite sets) into measurable sets and null sets (sets of finite measure, o-finite sets), respectively. I n particular, if p ( h E ) = p(E) for all h E 6 and E E 23, then (G, 23, p ) is said to be left invariant under 6, or left-6-translation invariant. For example, in every locally compact group G, there exists a so-called Haar measure which is left invariant under the entire group G (see Halmos [l]). Let (G, 23, p ) be a measure space which is quasi-invariant under the transformation group 6, and let v be a measure on (G, 23) which is equivalent to p ; then (G, ‘13, V ) is also quasi-invariant under 6. Let f be a measurable function on (G, 23) such that 0
then (G, 23, v) is also weakly quasi-invariant under 8 . Of course, any measure space 52 = (G, 23, p ) is, trivially, invariant under the group 6 which contains only the identity transformation of G. Let G be a group, 52 = (G, 23, p ) a measure space, and h E G; if the left translations T h : g -+ hg and ~ ~ are both 1 measurable transformations of (G, 23, p), and if the measures p h (see (3.1.1)) and p are equivalent, then we say that h is a left quasi-invariant element of 52. Clearly, the totality of left quasi-invariant elements of 52 form a subgroup 8 of G; we call 8 the maximal quasi-invariant left translation group of 52. Naturally, all the above definitions are also applicable if the word “left” is replaced by “right” throughout, and of course, in the case of commutative groups, the words “left” and “right” may be omitted. If (G, 23, p ) is a a-finite measure space which is quasi-invariant under the transformation group 6, then there exists a finite measure v on (G, 23) which is equivalent to p and is, therefore, also quasi-invariant under 6. I n fact, let B, , K = 1, 2, ..., be a sequence of pairwise disjoint sets such that m

BkE23,

O
G=(JBk* k=l

111.

108

GROUPS WITH QUASI-INVARIANT MEASURES

Construct the measure v as follows: “

v(E) =

1

221”1.0

p ( E n Bk), E E b.

It is easily seen that v(G) = 1, and that v , p are equivalent. In the remainder of this book, the emphasis will be on localizable quasi-invariant measures, and, in particular, on finite quasi-invariant measures. 20 Measures Quasi-Invariant under the Entire Group G

Throughout this subsection, we shall assume, without further explicit mention, that G is a group, that (G, b) is a measurable space, and that the correspondence (g,h) +gh-l is a measurable mapping from the product measurable space (G x G, b x b) to the measurable space (G, 8). Lemma 3.1.3. Let (G, b,p k ) , k = 1,2, be finite measure spaces which are right and left quasi-invariant, respectively, under G. If p1 is not identically zero, then p2 is absolutely continuous with respect to p1 PROOF. Form the product measure space ( G x G , b x b,pl x p2). For each E E b x 23, let xE(x,y ) denote the characteristic function of E. I t is easily seen that, for any E E 8 x b,

.

is a measurable function on (G x G, B x b).We define a finite measure v on (G x G, B x B), as follows:

We shall now prove that v is absolutely continuous with respect to p1 x p 2 . Take any E E b x b such that pl x p 2 ( E )= 0, and let Ex = (y I (x, y ) E E}. Then,

hence, there is a set d E B such that pl(d) = 0 and p 2 ( E z ) = 0 whenever

3.1. Basic Properties of Quasi-Invariant Measures

109

x C A . By the left quasi-invariance of p 2 , it follows that p2(xEz)= 0

whenever x Z A , hence, using Fubini's theorem, we obtain

Having proved that v is absolutely continuous with respect to pl x p 2 , we know that there exists a nonnegative measurable function g(x, y ) on ( G x G, 23 x 23, pl x p2), such that

On the other hand, given any A mapping, we have

E

23, since (x, y ) + xy is a measurable

E = { ( x , y ) l x y € A } € ! B x23.

Evaluating (3.1.2) for this set E , we obtain

but from (3.1.3), using Fubini's theorem, we get

(3.1.4) Consequently, if p l ( A ) = 0, then, by the right quasi-invariance of p l , we have p1(Ay-l) = 0, whence, by (3.1.4), we get v ( E ) = 0, and so p 2 ( A ) = 0. Thus, p2 is absolutely continuous with respect to p l . ]

Corollary 3.1.4. Let (G, 23, p) be a finite measure space which is right G-translation quasi-invariant, and let p".-l be the measure on (G, 23) defined by P _ ~ ( E= ) p ( E - l ) , E E 23. Then p and pel are equivalent. PROOF. It is easily seen that the measure p-l is left G-translation quasi-invariant. Therefore, by Lemma 3.1.3, p-l is absolutely continuous with respect to p, that is, if E E !B and p ( E ) = 0, then p(E-l) = 0. Hence, if E E 23 and F - ~ ( E= ) p(I3-l) = 0, then p ( E ) = F - ~ ( E - ~ =) 0, that is, p is also absolutely continuous with respect to p-l . Thus, p and p-l are equivalent. ]

110

111.

GROUPS WITH QUASI-INVARIANT MEASURES

Let (G, 23, p ) be a a-finite measure space which is left (right) quasi-invariant under G; then (G, 23, p ) is also right (left) quasi-invariant under G. If (G, 23, pk), K = 1, 2, are o-finite nontrivial3 measure spaces which are both (right or left) quasi-invariant under G, then p, and p2 are equivalent. PROOF. By virtue of the remarks in the preceding subsection, the o-finite measures under consideration may be replaced by equivalent finite measures. Assume that (G, 23, p ) is right G-translation quasiinvariant; then p-, is left G-translation quasi-invariant. But, by Lemma 3.1.4, p and p-, are equivalent, hence p is also left G-translation quasi-invariant. Conversely, assuming that (G, 23, p) is left G-translation quasi-invariant, then p-, is right G-translation quasi-invariant. Hence, by what has already been proved, p-, is also left G-translation quasiinvariant, therefore p is right G-translation quasi-invariant. Let (G, 23, pk), k = 1, 2, be finite nontrivial measure spaces which are both G-translation quasi-invariant. By Lemma 3.1.3, p, is absolutely continuous with respect to p 2 , and likewise p2 is absolutely continuous with respect to pl . J

Theorem 3.1.5.

Remark. Theorem 3.1.5 can be extended to pseudo-a-finite measures (see Definition 1.1.9). For example, if (G, 23, p ) is a pseudo-o-finite measure space which is left G-translation quasi-invariant, then (G, 23, p) is also right G-translation quasi-invariant. I n fact, let E E 23, p ( E ) = 0, h E G; then there exists a subgroup G, of G, containing E and h, such that G, belongs to 23 and is finite.^ Let 23, and p1 denote the restrictions to G, of 23 and p, respectively. It is easily seen that (G, , 23, , p l ) is a left G,-translation quasi-invariant o-finite measure space. By Theorem 3.1.5, p, is also right G,-translation quasi-invariant. Consequently p ( E ) = 0 implies that p(Eh) = 0, that is, (G, 23, p ) is right G-translation quasi-invariant . Similarly, one can prove that if (G, 23, pk), k = 1,2, are pseudoa-finite measure spaces which are both G-translation quasi-invariant, then p1 and p2 are e q ~ i v a l e n t . ~ Using the preceding remark and the results of $1.1, 6", we immediately obtain the following corollary. That is, the measures are not identically zero. note: Such a subgroup certainly exists if the subgroup Go in Defini&'Go obviously satisfies tion 1.1.9 happens to be normal in G, for in that case GI = the required conditions. If Go is not normal, then it is not clear how GI is to be constructed. Translator's note: This is easily proved if the same group Go (see Definition 1.1.9) works for both pl and p2 (fortunately, this is the case in Corollary 3.1.6). Otherwise, the proof is not obvious.

* Translator's

uz=-,

3.1. Basic Properties of Quasi-Invariant Measures

111

Corollary 3.1.6. Let G be a locally compact group, and let 2'3 be the a-ring generated by the totality of compact subsets of G. Suppose that (G, d,p) is a (left or right) G-translation quasi-invariant locally a-finite' measure space. Then p is equivalent to any Haar measure on G. 3 O A Fundamental Property of Quasi-Invariant Measures

Definition 3.1.2. Let G be a topological space, and let 6 be a family of continuous mappings of G into G. Suppose that (ti also has a topology, and that the following condition is satisfied: given any h, E 6 , any compact subset K of G, and any open set U containing h,K, there exists a neighborhood V of h, such that hK C U for every h E V . Then, we say that the topology of (ti is admissible. We now consider a particular case which occurs frequently in the sequel. Lemma 3.1.7. Let 6 be a subgroup of a topological group G, and let each h E 6 be identified with the corresponding left translation g + hg, g E G. Then, the relative topology on 6, induced by the topology of G , is admissible. PROOF. Let h, E 6 , let K be any compact subset of G, and let U be any open set containing h,K. For each g E h,K, choose neighborhoods U, and V, of the identity in G such that U,g C U and V,V, C U, ; then h,KC

u

VggCU.

gEh,,K

Since h,K is compact, there exist elements g ,

,..., g , E h,K,

such that

n

hoKC

u VU"&

*

V=l

0bviously, n

v = "=l n vgv is a neighborhood of the identity. We assert that hK C U whenever h belongs to the neighborhood Vh, of h, . I n fact, for any g E hK, h,h-lg E h,K, hence, there is an index v such that h,h-lg E VgYgv , and so g E hh,'vu"g"

c VVoyg,c UO& c U ,

Translator's note: The definition of a locally o-finite measure (which the author has neglected to state explicitly) is an obvious modification of Definition 1.1.10, and it is clear that the proof of Lemma 1.1.11 goes through without any changes if the local finiteness hypothesis is replaced by local a-finiteness.

112

111.

GROUPS WITH QUASI-INVARIANT MEASURES

which proves that hK C U . Thus, when 8 is regarded as a group of transformations of G, the relative topology on 8 is admissible. 3

Lemma 3.1.8. Let G be a topological space, let (G, b, p) be a regular measure space, and let 8 be a topologized family of continuous 23-measurable mappings of G into itself. If the topology of 8 is admissable, then, for any fixed compact set K E b, the value p(hK),h E 8,is an upper semicontin~ous~ function on 8. PROOF. Choose any h, E 8. If p(hoK)= m, then the function p ( h K ) is obviously upper semicontinuous at h , . Hence, suppose that p(h,K) < co,and let E be an arbitrary positive number. By the regularity of p, there exists an open set U 3 h,K such that P(U)

< Pr(h0K) + E .

(3.1.5)

Since the topology of 8 is admissible, there exists a neighborhood V of h, such that hK C U for all h E V. Hence, since hK E b, it follows from (3.1.5) that P ( W

< P(h0K) +

for all h E V . Consequently, the upper bound7 S(h,, p(hK)) of the function p ( h K ) at h, satisfies

Letting E -+ 0, we obtain p(h,K) = S(h, ; p(hK)). Thus, the function p(hK) is upper semicontinuous at the point h = h, . ] As an immediate consequence of Lemmas 3.1.7 and 3.1.8, we obtain the following result.

Corollary 3.1.9. Let G be a topological group, let (G, b, p) be a regular measure space, and let 8 be a subgroup of G such that hB E b for every h E 8 , B E b. Then, for any fixed compact set K E b, the function p(hK), h E 8, is upper semicontinuous with respect to the relative topology of 8 . In the following lemma, we establish a sort of continuity property of regular measures with respect to transformations; this, in turn, will be used to derive an important property of quasi-invariant measures. Corollary 3.1.10. Let (G, 23, p) be a regular measure space, let 8 be a topologized family of continuous b-measurable mappings of G

' For the definition of an upper semicontinuous function, along with the relevant notations and theorems, see Appendix I (51.2).

3. I. Basic Properties of Quasi-Invariant Measures

113

into itself, and suppose that the topology of 0 is admissable (see Definition 3.1.2). If the topological space Q is of the second category, then, for any fixed compact set K E 8,there exists a point ha E 6 such that the function p(hK), h E 8,is continuous at h a . PROOF. By Lemma 3.1.8, the function p(hK),h E 8,is upper semicontinuous on 6. Therefore, since p(hK) is nonnegative, and since Q is of the second category, it follows easily from Lemma 1.2.28 of Appendix I that Q contains a continuity point h, of the function p(hK). 3 Applying Lemmas 3.1.7 and 3.1.10, we get the following result.

Corollary 3.1.1 1. Let G be a topological group, and let 8 be a subgroup of G. Suppose that 6 itself is a topological group of the second category with respect to a topology which is stronger than the topology induced by G. If (G, 23) is a measurable space which is left (or right) quasi-invariant under 8,and p is a regular measure on (G, S),then, for any fixed compact set K E 23, there exists an ha E Q such that the function p ( h K ) [or p(Kh)],h E 8,is continuous at h, . Next, we use Lemma 3.1.10 to obtain a property of quasi-invariant measures which will be required in the ensuing discussion. Lemma 3.1.12. Let (G, 23, p) be a regular measure space which is quasi-invariant under a group Q of continuous measurable transformations. Also, suppose that Q has an admissible topology which makes 8 a topological group of the second category. Then, given any compact set K E 23, .p (. K.) > 0, there exists a neighborhood V of the identity in 8 such that (3.1.6) p(K nhK) > 0 for every h E V.9 PROOF. By Lemma 3.1.10, there exists an element h, E 6 such that the function p ( h K ) is continuous at h, . By quasi-invariance, we know that p(h,K) > 0, and we may assume thatlo p(h,K) < 00. Since p is regular, there is an open set U E 23 such that U 3 h,K and P(U) +P(h,K). ==l

(3.1.7)

* Translator’s note: Lemma 1.2.2 is valid only for finite-valued real functions. Therefore, in this lemma and the subsequent results which depend upon it, it seems necessary to assume that the measure of every compact set is finite, or some other suitable finiteness condition. Translator’s note: T h e following simple counterexample shows that this conclusion is not valid unless one imposes some suitable finiteness condition on the measure p. Let G be the additive group of reals with the usual topology, let B be the totality of Bore1 sets in G, let p ( E ) = 03 for all nonempty sets E E 8, let 6 = G, operating on G by translation, and let K be any singleton in G. lo Translator’s note: T h e justification for this remark is obscure. See footnotes 8 and 9.

111.

114

GROUPS WITH QUASI-INVARIANT MEASURES

Since the topology of 8 is admissible, there is a neighborhood Vh, of h, such that hK C U whenever h E Vh,. Moreover, Vh, may be chosen sufficiently small so that

I P(hW

- P(h0K)I

< MhOK)

(3.1.8)

for all h E vh, . Since hK C U and h,K C U , we have P(hK n hoK) 2 P @ K )

+ P(hoK) - P ( W ,

whence, using (3.1.7) and (3.1.8), we see that P(hK n hoK) 2 BP(h0K) > 0

(3.1.9)

for all h E Vh, . Again, using the quasi-invariance of p, it follows from (3.1.9) that, if g = ho'h E h;'Vh,, then p ( g K n K ) > 0.

Thus, V = h;lVh0 is a neighborhood of the identity in 8 which has the required property. ]

Corollary 3.1.12'. Let G be a topological group, let 8 be a subgroup of G, and suppose that 8 itself is a topological group of the second category with respect to a topology which is stronger than that induced by G. If (G, b, p ) is a regular measure space which is left (or right) quasi-invariant under 8, then, for any compact set K E B,p ( K ) > 0, there exists a neighborhood V of the identity in 8 such that p(K n h K ) > 0 (or p(K n Kh) > 0) for all h E V . Using the above results, one may derive a simple necessary condition for the existence of quasi-invariant measures, as stated in the following theorem. Theorem 3.1.13. Let G be a topological group, let 8 be a subgroup of G, and suppose that 8 itself is a topological group of the second category with respect to a topology which is stronger than the relative topoIogy induced by G. If there exists a nontrivialll regular measure space (G, d,p ) which is quasi-invariant under 8 , then there is a neighborhood of the identity in 8 which is contained in a compact subset of G. PROOF. Since p is regular, there exists a compact set K E b such that p ( K ) > 0. Moreover, by Corollary 3.1.12', there is a neighborhood V of the identity in 8 such that K n hK # 0 for all h E V , that is, l1

T h a t is,

p

is not identically zero.

3.1. Basic Properties of Quasi-Invariant Measures

115

V C KK-l. We assert that KK-’ is a compact subset of G. I n fact, by the Tychonoff theorem, K x K is a compact subset of G x G; moreover, since G is a topological group, the correspondence (x,

-

r>

2y-l

is a continuous mapping from G x G to G. Hence, KK-’, being the image of K x K under this mapping, is compact. 3

Corollary 3.1.14. Let G be a topological group of the second category. Then, the local compactness of G is a necessary and sufficient condition for the existence of a regular measure space (G, b,p) which is left quasi-invariant under G. PROOF. T h e necessity follows at once from Theorem 3.1.13. Conversely, if G is locally compact, let 23 be the a-ring generated by the totality of compact subsets of G, and let p be a left-invariant Haar measure on G. Then the measure space (G, 23, p ) has the required properties. J 40 Functions on Topological Groups

with Quasi-Invariant Measures Throughout the present subsection, we shall assume that (G, 23, p ) is a regular measure space which is quasi-invariant under a group of measurable continuous transformations 0,and that 8 is a topological group of the second category with respect to a certain admissible topology (see Definition 3.1.2); these assumptions will not be explicitly repeated. Let p ( g ) be a %-measurable real-valued function on G, and, for each h E 0, define p*(h) = essential infimum(p(g) gee

+ p(h-lg)).

(3.1.10)

Here, the term “essential infimum” means that one evaluates the infimum of the function on the complement of each p-null set, and then takes the supremum of all these values.

Lemma 3.1.15. Let p ( x ) be a nonnegative %-measurable function on G, and suppose that p(x) is finite on some set A E 23, 0 < p(A). Then, there exists a neighborhood V of the identity in 8 such that p*(h) is bounded on V . PROOF. Since the set {x I p ( x ) < a}contains A , there is a positive a, x E A } has positive number a < co, such that the set (x I$(.) p-measure, and since p is regular, there is a compact set K E 23 such

<

111.

116

GROUPS WITH QUASI-INVARIANT MEASURES

<

that K C {x I p ( x ) a, x E A } and p ( K ) > 0. By Corollary 3.1.12', there is a neighborhood V of the identity in 8 such that p(K n h K ) > 0 for all h E V . Now, if g E K n hK, then p ( g ) a, p(h-'g) a. But when p(K n h K ) > 0, the complement of every p-null set E contains a point g E K n hK, hence

<

<

<

Consequently, p*(h) 2a for all h E V. ] With a view to the applications of Lemma 3.1.15, it is convenient to introduce the notion of a quasiconvex pair of functions, as follows.

Definition 3.1.3. Let (G, %, p) be a measure space which is quasiinvariant under a group of transformations 8 . Let p ( g ) be a %-measurable extended12 real-valued function on G, let $(h) be a real-valued function on 8, and suppose that the following conditions are satisfied: CQ; (ii) given any h E 8,the inequality (i) 0 \
<

P ( 4 G P(E) + p(h-lg) holds for almost all g E G; (iii) j ( h ) is a convex function on 8 (see $1.1 of Appendix I). Then, we say that ( p , j) is a quasiconvex pair of functions over ((G, %, p), 8 ) . In particular, if p is a measurable convex function on G, then the conditions of Definition 3.1.3 are satisfied if one chooses $(A) = p(h).

Corollary 3.1.16. Let ( p , j) be a quasiconvex pair of functions over ((G, 9, p), 8 ) ,and suppose that p ( g ) is finite on some set A E 8, with p ( A ) > 0. Then $(h) is locally bounded (i.e., every point in 8 has a neighborhood upon which j is bounded). PROOF. By Lemma 3.1.15, there exists a neighborhood V of the identity in 8 such that p*(h) 2a, and hence j ( h ) 2a, for all h E V. Now, let h, be any element of 8 ;if h E h,V-', that is, if h = hohi', h, E V , then, by the convexity of j,we have

<

P(h) < P(h,)

+

<

< P@,)

+ 2@.

Thus, $ is bounded on the neighborhood h,V-l of h, We may now sharpen Theorem 3.1.15, as follows.

.

]

Theorem 3.1.17, Suppose that the topological group 8 satisfies the first axiom of countability, and let A E 23, with p ( A ) > 0. Then, there la That

is, infinite values are also allowed.

3.1. Basic Properties of Quasi-Invariant Measures

117

exists a neighborhood V of the identity in 6, and a positive number c, such that the inequality (3.1.1 1)

holds for every nonnegative %-measurable function p on 0 [see (3.1.10) for the definition of p*]. PROOF.If J A p ( g ) dp(g) = 0, then an argument similar to that used in the proof of Lemma 3.1.15 shows that p*(h) = 0 on some neighborhood V of the identity in 6. Hence, we may restrict our considerations to functions p such that JA p ( g ) d p ( g ) > 0. Let {p,] be any sequence of nonnegative %-measurable functions on G such that

Let 2 denote the totality of real number sequences A = {A, , A, satisfying the condition

,..., A, ,...} (3.1.13)

2’ forms a Banach space with respect to the usual linear operations and the norm [I 11 defined by (3.1.13). For each h E 9, define a function p ( g ; A) on G, as follows:

-

W

P(g; A) =

2I

An

IPnk).

n=l

Obviously, p ( g ; A) is nonnegative and %-measurable. Also, consider the function m

F(h; A )

=

2 1 An Ifn*@),

lzf5 6.

n=l

Obviously, p*(h; A) 2 F(h; A) for all h E 6. Now, since A E 2,it follows from (3.1.12) and the Levi lemma that

for almost all g in A. Thus, if we let A, denote the totality of eIements g E A which satisfy (3.1.14), we have p ( A J = p ( A ) > 0. Hence, by

111.

118

GROUPS WITH QUASI-INVARIANT MEASURES

Lemma 3.1.15, the function p*(h; A) is bounded on some neighborhood of the identity in 6, therefore F(h; A) is also bounded on this neighborhood. Let {Vm},m = 1, 2, ..., be a countable base for the neighborhoods Thus, for every of the identity in 6 ;we may assume that V , 3 V , 3 A E 9, there exists an integer m (possibly depending upon A) such that F(h; A) is bounded on V , . Notice that, for each fixed h, F(h; A) is the supremum of a sequence of continuous convex functionals on 9, namely, Ci=lp,*(h)I A, I, k = 1, 2,... . Therefore, F(h; A) is a lower and hence semicontinuous convex functional on 9, a*..

moreover, for is also a lower semicontinuous convex functional on 9, every A, there is an m such that q,(A) < co. Furthermore, since V ,3 V,3 we have em.,

It follows by Theorem 1.2.4 of Appendix I that there exists an integer m and a positive number a such that 4m(4

< a I/

I/

(3.1 .I 5 )

for all A in 9. From this, we deduce that

I n fact, one need only let A in (3.1.15) be the sequence with nth term A, = 1 and all other terms zero. Now, if the theorem were false, then, for every positive integer n, there would exist a nonnegative %measurable function p , on G such that 0

< j AP 7 k ) dtt(g) < 00

and

which contradicts (3.1.16).

]

Corollary 3.1.18. Suppose that G satisfies the first axiom of countability, let A be any set in B such that p ( A ) > 0, and let h, be any element

3.1. Basic Properties of Quasi-Invariant Measures

119

of 8 such that13 p ( A n h,A) > 0. Then, there exists a neighborhood V of h, in 8 , and a positive number c, such that (3.1.1 7)

for every quasiconvex pair of functions ( p , j)over ((G, 23, p), 8 ) satisfying the condition13 JA p ( g ) dp(g) > 0. PROOF.We proceed as in the proof of Theorem 3.1.17, but in place of F(h; A) we consider the function

I t is easily checked that the pair of functions p ( g ; A), j ( h ; A) has all the properties specified by Definition 3.1.3, except possibly that $(h; A) may not be everywhere finite; however, using (3.1.14), together with the inequality P(h; 4

< p ( g ; 4 + p(h-lg; A)

and the condition p ( A n h,A) > 0, it is easily shown that P(h,; A) < 00, hence the proof of Corollary 3.1.16 shows thatj(h; A) is bounded on some neighborhood V of h, . Then, using this result instead of Corollary 3.1.15 in the argument following formula (3.1.14), the desired result follows without difficulty. ] The results obtained in the present subsection will find application in $3.2 and $4.2. 50 k-Pseudometrics on Transformation Groups

In this subsection, we restrict our considerations to finite measure spaces.

Definition 3.1.4.

Let Q = (G, 8,p ) be a finite measure space, and 8 a group of measurable tran~formationsl~ on Q. Form the convex function = 4 P h , I”.), h E 6, (3.1.18) l 3 Transtator’s note: These two requirements were omitted in the original statement; however, it is easy to construct examples showing that the conclusion may be false in the absence of such conditions. l4 Here, we assume that the transformations h in 8 are all one-to-one mappings of the entire set G onto itself.

120

111. GROUPS

WITH QUASI-INVARIANT MEASURES

where d denotes the Kakutani distance (see 51.4) and ph is defined by (3.1.1). We call M,(h) the k-pseudonorm on 8 defined by SZ. The corresponding left-invariant pseudometric,

will be called the k-pseudometric on 6, and the topology on 8 induced by the K-pseudometric (see Definition I. 1.I) will be called the k-topology. To verify that M , is indeed a convex function, pick any two elements h, k E 6, choose a finite measure v on (G, b) such that the four measures p, ph , pk , pkh-1 are all absolutely continuous with respect to v, and let 11 [I2 denote the norm in L2(G,b,v). Then, using the Minkowski inequality and the change of variable g, = hk-lg, we have

-

=W ( h )

+ Ml(Q

(3.1.20)

It is easy to calculate that the pseudometric d, defined by (3.1.19) may also be expressed as (3.1.21) ddh, 9 h2) = &h, ,P h J , where d again denotes the Kakutani distance.

Lemma 3.1.19. Let SZ = (G, b, p) be a finite measure space, and 6 a group of measurable transformations on SZ. Then, for each fixed E E b, the function p(h-IE), h E 6, is continuous with respect to the k-topology on 6. PROOF. For any h, , h, E 6, applying the Schwarz inequality, we get

jE( d P h l ( g ) Y 2 (dPhs(g))1'2< (CL(hlW P ( h 2 w / 2 * whence, using (3.1.21), we obtain I ( p ( h ; l W 2 - (P(hilw/z

I

<

(s,

112

((~Pn,(g))l/z- ( d P dg N T )

= ddhl 9 h2)*

1

3.1. Basic Properties of Quasi-Invariant Measures

121

Lemma 3.1.20. Let SZ = (G, 8 , p) be a finite regular measure space, let 8 be a group of continuous measurable transformations on SZ, and suppose that 8 has an admissible topology 2 (see Definition 3.1.2). Then, the convex function Ml(h), h E 8 , is lower semicontinuous on

(8,q.

PROOF. Let h E 8 , and let godenote the totality of countable partitions of G (relative to p, ph-l) consisting of compact sets in 23;then, by Lemma 1.4.2, &h-I

=

T

(p(hEk) p(Ek))1’2*

Hence, using (3.1.18) and (1.4.4), we obtain Ml(h)2 = 2(P(G)- P(Ph-1 P I ) 9

In view of (3.1.22), it suffices to prove that, for any partition {Ek}, the quantity Ck(p(hE,) P ( E ~ ) )is~ an / ~ upper semicontinuous function on

(8,Z). By the Schwarz inequality,

hence,

2

(p(hEk) d E k ) ) 1 ’ 2

F

< (dG))’”(

dEk))l”

k=n

k=n

for every n. Now, since m

1

d E k ) =

P(G) < a,

k=l

it follows that m

c

p(Ek)

--f

k=n

as n + co. Hence, given any

E

> 0, there exists an integer N such that

for every n > N . On the other hand, by virtue of Lemma 3.1.8, we N know that &=l p(hEk)p ( E k ) ) 1 / is 2 an upper semicontinuous function on

111.

122

GROUPS WITH QUASI-INVARIANT MEASURES

(8, 2).Therefore, given any h, E 6, there exists a neighborhood V of ho such that

c N

E

(p(hEk)p(Ek))”2

<2 f

k=l

c N

(p(h&k) p(Ek))l’z

k=l

for all h E V. Thus, when h E V ,

I n view of Lemma 3.1.7, we immediately deduce the following result.

Corollary 3.1.21, Let G be a topological group, let (G, 8,p) be a finite regular measure space, let 8 be a subgroup of G, and let 2 denote the relative topology on 8,as a subset of G. If 8 is regarded as operating on G by left translation, and if hE E 23 for all E E 8, h E 8, then the convex function M,(h), h E 8, is lower semicontinuous on (8, 2). Lemma 3.1.22. Let G be a topological group, (G, 8,p) a finite regular measure space, and 8 a subgroup of G such that hE E b for all E E 23, h E 0.Then, given any h, E 8, there exists a neighborhood V of h, , in the k-topology, and a compact subset K of G, such that V C K. PROOF. Since p is a finite regular measure, there is a compact set E E 23 such that tL(hOlE) > MG).

By Lemma 3.1.19, there is a neighborhood V of ho , in the k-topology, such that

I/WE)

-

cL(hOlE)I

<MG)

for all h E V . Therefore, if h E V , then

hence, p(h-lE n h;*E) > 0, consequently, there exists an element g E h-lE n hClE, and so h E Eg-l C EE-lh,

.

3.1. Basic Properties of Quasi-Invariant Measures Thus, writing K

= EE-lh,

123

, we have V C K.

But since (x, y ) --t xy-1 is a continuous mapping from G x G to G, the set EE-l is compact, hence K is compact. ]

Corollary 3.1.23.16 Let G be a topological group, and let (G, 23, p) be a finite regular measure space which is left invariant under G. Then G is compact. PROOF.Since p is left invariant, d(h, , h,) = 0 for all h, , h, E G. Consequently, the k-topology on G has but a single nonempty open set, namely, G itself. Hence, by Lemma 3.1.22, G is contained in a compact subset of G, that is, G is compact. ] Note that the condition “invariant” in the hypothesis of Corollary 3.1.23 cannot be weakened to “quasi-invariant.” For example, consider the additive group of real numbers G, with the Euclidean topology; every finite Bore1 measure on G which is equivalent to Lebesgue measure is also regular and quasi-invariant, and yet G is not compact. Theorem 3.1.24. Let G be a topological group, let (G, 23, p) be a finite regular measure space, and suppose that g E E 23 for all E E 23, g E G. If G is regarded as a group of left translations, then G is complete relative to the k-pseudometric dl(hl , h,). PROOF. Let {h,} C G be a sequence which is fundamental relative to the pseudometric dl . By Lemma 3.1.22, there exists a compact subset K of G such that K contains a set of the form {h 1 Ml(h) < E } , where E is some positive number. By assumption, there is an integer N such that M,(h;’h,) < E when n 3 N . Hence, the elements h, , n = N , N 1,..., are contained in the compact set h,K, and it follows that there exists an element h, E G such that every neighborhood U of h, contains a subsequence of {h,}. Now, for any fixed m,by Corollary 3.1.21, given any positive number 6, there exists a neighborhood V of h;’h, such that

+

MI(h30)

< Mlk) + 8

whenever g E V . I f we choose g = h$h, where h is any element of the subsequence of {h,} which is contained in the neighborhood h,V of h, , we deduce that M1(hi1ho)<

+ 6.

M1(h21hn)

This theorem is, of course, well known, and may also be established by a direct proof.

111.

124

GROUPS WITH QUASI-INVARIANT MEASURES

But since {h,} is fundamental relative to dl m > N implies nlim ~ M m 1(h-lh m n) < 8.

, there is an N such that

~

Therefore, dl(h,, h,) < 26 whenever m > N . Thus, we have proved that {h,} converges to h, relative to the pseudometric 4. 3 Let (G, 9, p) be a finite measure space and 6 a group of measurable transformations of (G, 9, p). Form the pseudometric ddhl h,) 7

= M I ( h 1 f h2)

+ w;', h3)

on 6. Notice that, if 0 is commutative, then dl(hl , h2) = Ml(h,lh,)

= M,(h,h,l) = dl(h;',

and hence 9

A,)

= dl(h1

9

(3.1.23)

hzl.

Lemma 3.1.25. Let 52 = (G, 9, p ) be a finite measure space and 6 a group of measurable transformations of SZ. Let T denote the totality of transformations h E 6 such that 52 is quasi-invariant under h. Then T is a closed subgroup of 6 relative to the pseudometric do . PROOF. It is easily seen that T is a subgroup of 6. Thus, it remains to prove that T is closed. Suppose that {h,} C T , h, E 6, d,(h, , h,) -+ 0, then dl(hn , h,)

--f

0,

dl(h,l, h,l)

+ 0.

For any E E d,we know by Lemma 3.1.19 that lim p(hnE) = p(h,E) n+ m

(3.1.24)

(3.1.25)

We shall now show that 52 is quasi-invariant under h,. If p ( E ) = 0, then, since h, E T , p(h,E) = 0, hence by (3.1.24) we have p(h,E) = 0. Conversely, if p(h,E) = 0, then p(h,'(h,E))

= 0.

Hence, substituting h,E for E in (3.1.25), we get p ( E ) = 0. Thus, 52 is quasi-invariant under h, , that is, h, E T. ]

Theorem 3.1.26.

Let G be a topological group, let (G, B,p ) be a

3.1. Basic Properties of Quasi-Invariant Measures

125

finite regular measure space, and suppose that g E E 8 for all E E 8, g E G. Then, the maximal quasi-invariant left translation group T of (G, 8, p) is complete relative to the pseudometric d o . PROOF. Let {h,} C G be a fundamental sequence relative to do . Then, {h,} is fundamental relative to both of the pseudometrics d,(h, , h,), h, , h, E G, and d1(hr1,h;'), h, , h, E G. I n the proof of Theorem 3.1.24, it is shown that {h,} converges to a certain element h, , relative to the pseudometric dl(h, , A,), h,, h, E G, and a trivial modification of the proof shows that {h,} also converges to the same element h, , relative to the pseudometric dl(hyl, h;'), h, , h, E G. Hence, {h,} also converges to h, relative to do . Thus, G is complete relative to do . By Lemma 3.1.25, T is a closed subset of G relative to d o , therefore T itself is complete relative to d o . ] 60 p-Topologies on Transformation Groups

Throughout the present section, we shall assume that 52 = (G, 9, p) is a localizable measure space which is quasi-invariant under the transformation group 8. We let L2(52) [or, sometimes, L2(p)] denote the totality of 9-measurable complex-valued functions on G which are quadratically integrable with respect to p. If one identifies functions which coincide almost everywhere, then L2(sZ)forms a Hilbert space relative to the usual linear operations and the inner product

For each h E 8,we form the measure ph on (G, 8 ) : ph(E) = p(h-'E),

E E 23.

Since 52 is quasi-invariant under 8,the measures ph and p are equivalent, and it is easily verified that ph is also localizable. Hence, by Corollary 2.4.18, there exists a Radon-Nikodym derivative dphldp such that 0 < dph/dp < CO. Define an operator U ( h )as follows: if 97 EL,(Q), then

Clearly, U(h)v is a measurable function on 52, moreover,

126

111.

GROUPS WITH QUASI-INVARIANT MEASURES

thus, U(h)is an isometric linear operator from L2(Q)into L2(Q).Furthermore, by Theorem 2.4.17,

from which we deduce that, if h, , h, E 6, then U(h,h,) = U(h,) U(h,). Obviously, U ( e ) = I , hence it follows that U(h)-l = U(h-l), and therefore U(h) is a unitary operator on L2(Q).Thus, the correspondence h --t U(h),

hE6

(3.1.26)

is a unitary representation of 6 in L2(Q).T h e group of unitary operators U = { U(h) I h E S] will be called the transformation group of L2(Q) corresponding to 6. This group plays an important role in harmonic analysis on quasi-invariant measure spaces. We let N(Q,6) denote the smallest weakly closed operator algebra containing U, and call it the algebra over L2(Q)corresponding to 6.

Definition 3.1.5. Let Q = (G, 23, p ) be a localizable measure space which is quasi-invariant under a group of measurable transformations 6. Let 2 be the weakest topology on 6 such that the unitary representation (3.1.26) is strongly continuous. We call 2 the p-topology on 6. Actually, the p-topology is just the topology on 6 induced by the family of convex functions {M,(h), rp E L2(Q)},where

I n fact, let 2, denote the topology induced by the family of convex functions (M,(h), rp EL~(Q)). Then, given any y €L2(Q),h, E 6, and positive number E, there is a %,-neighborhood of ho , namely, { h I M,(h-lho)

< 4,

(3.1.28)

such that, if h belongs to this neighborhood, then

thus, the unitary representation U is strongly continuous on (6,2,). Conversely, let 2,’ be any other topology on 6 such that U is strongly

3.1. Basic Properties of Quasi-Invariant Measures continuous on (6,%,’). Then, for each 9) €L2(Q),h, there exists a %,‘-neighborhood V of h, such that MAh-lho)

=

127 E

6, and

E

> 0,

II ( U ( 4 - U(h0))V II < E

whenever h, E V. This shows that 2,‘3 2, . Thus, 2, is just the p-topology on 8. We now introduce still another family of convex functions on 6. Let 8 denote the totality of sets A E 8 such that 0 < p ( A ) < co. Write

We thus obtain a family of convex functions {%,(it), A E g} on 6. If we let C, denote the characteristic function of the set A , then C, EL’-(Q) for every A E 5, and one may easily calculate that MA(h)

= MC~(h)’

(3.1.29‘)

Lemma 3.1.27. Let Q = (G, 8,p ) be a localizable measure space which is quasi-invariant under a transformation group 6. Then the p-topology on Q is induced by the family of convex functions {%A(h), A g}* PROOF. Let 2, denote the topology on 6 induced by { N , , A E g}. By virtue of (3.1.29’), we have { N , ; A E g} C { M , , 9) EL’-(Q)}, hence 2, is weaker than the p-topology. On the other hand, given any 9) €L2(Q) and any positive number E, there exist sets A, ,..., A, E 8 and numbers A, ,..., A, such that (3.1.30)

(see Halmos [I]). For any h, h,

E

6, it follows from (3.1.30) that

n

8

< ‘12

1I

’k

I.

k=l

Then, we see from (3.1.27) and (3.1.31) that ll(U(h) - U(h,))V

II < Et

111.

128

GROUPS WITH QUASI-INVARIANT MEASURES

provided that h belongs to the %,-neighborhood {h I MA*(h-'h0) < 6, K = 1, 2,..., n}

of h, . This shows that the unitary representation h --t U(h) is strongly continuous on ( 8 ,Z,), hence, 2, is stronger than the p-topology. We conclude that the p-topology is just 2,. 3

Lemma 3.1.28. Let SZ, = (G, 2' 3, p,), k = 1, 2, be localizable measure spaces which are both quasi-invariant under the transformation group 8. If p, is absolutely continuous with respect to p z , then the p,-topology on 8 is weaker than the pz-topology. PROOF.Since p1 is absolutely continuous with respect to p 2 , there exists a Radon-Nikodym derivative dp,(g)/dpz(g),moreover, by Corollary 2.4.19, T:

tp -+ tp

(+)"', tp E L ~ ( Q , )

is an isometric operator from L2(fzn,)to L2(Qz).Let U, : k .+ Uk(h)be the unitary representation of 8 in L2(SZk),k = 1, 2. It is easily verified that Udh) Ttp = TUl(h)s,

for all y E L ~ ( Q , )h,E 6.Hence, if s, €L2(Q1),h, h, E 01, then, writing Ts,EL~(SZ~), we have

ab, =

This shows that the family of convex functions which defines the pl-topology is included in the family of convex functions which defines the pz-topology. Therefore, the p,-topology is weaker than the pztopology. 3 I n particular, if the measure spaces fz, and Q, in Corollary 3.1.28 are equivalent, then the p,-topology and the pz-topology are identical.

Lemma 3.1.29. Let G be a locally compact topological group, 23 the a-ring generated by the compact subsets of G, and p a leftinvariant Haar measure on G. If G is regarded as a group of left translations of itself, then the p-topology is weaker than the original topology of G. PROOF.Notice that, since p is invariant, 91Z,(h)= ( 2 p ( A - hA))'I2, A E 8. Now, since every Haar measure is regular (see Halmos [l]), we know that for any A E 5, there is a compact set C C A such that p ( A - C)

3.1. Basic Properties of Quasi-Invariant Measures

129

is less than the prescribed positive number ~ / 2 ,moreover, there is an open set U 3 A such that p( U - A) < ~ / 2 . Furthermore, by Lemma 3.1.7, there is a neighborhood V of the identity in G such that hC C U for all h E V , Therefore, since A - hA C U - hC, it follows that if h E V, then

This shows that aA(h) is a continuous function on G, which means that the p-topology is weaker than the original topology of G. ]

Lemma 3.1.30. Let G be a topological space, and let (G, 3,p ) be a finite regular measure space which is quasi-invariant under a topological group of continuous measurable transformations 8 . If the topology 2 of Q is admissible, then 2 is stronger than the p-topology if and only if the function

is continuous relative to 2. PROOF. Let A E ‘13. By the Cauchy inequality, we have

< Ml(h) + ( p ( h - l A ) + p ( A ) - 2p(A n h-lA))lI2.

(3.1.33)

Assume that M,(h) is continuous on (8,Z). Then, for any B > 0, there is a %-neighborhood V of the identity in 8 (we may suppose that V = V-l) such that MI@) < (3.1.34) for all h E V. From (3.1.34) and the last inequality in the proof of Lemma 3.1.19, it follows that l(P(h-1B))112- (P(B))l/zI

<

E

for all B E ‘13, h E V . Multiplying both sides of the above inequality by (p(h-lB))lIP (p(B))l12,we get

+

I tL(h-lB)

- P(B)I

< 24P(G))1’z.

(3.1.35)

111. GROUPS

130

WITH QUASl-INVARIANT MEASURES

By the regularity of p, there exists an open set U E 23 and a compact setKEbsuchthat U 3 A 3 K a n d p(U - K ) < E .

(3.1.36)

Since the topology of 6 is admissible, we may assume that V is sufficiently small so that h-lK C U whenever h E V. Now, from (3.1.35) and (3.1.36), we see that p(h-l(A - K ) )

+ p ( A - K ) < 2~ + 2 ~ ( p ( G ) ) ’ / ~

(3.1.37)

for all h E V. Furthermore, if h E V , then h-lK - ( A n P A ) C h-lK - ( K n h-lK) C U - K ,

and hence p(h-lK) - p ( A n P A ) < 6.

(3.1.38)

Similarly, if h E V , then, using (3.1.35) and (3.1.38), we obtain p ( ~-)

n h-lA)

= p ( K ) - p(h-lK)

<E

+ p(h-lK) - p ( A n F A )

+ 24p(G))lI2.

(3.1.39)

Combining (3.1.33), (3.1.34), (3.1.37), (3.1.38), and (3.1.39), we deduce that 4 ( h )<6

+ 2[41 + (P(G))”~)]”~

for all h E V. This shows that the p-topology is weaker than 2. Conversely, if the p-topology is weaker than 2, then obviously Ml(h) = &(h) is continuous on (8,2).]

70 Ergodic Measures We now introduce the concepts of quasi-invariant sets and ergodicity.

Definition 3.1.6. Let (G, 23,p) be a measure space which is quasiinvariant under a transformation group 8.If B E b, and if B - hB is a p-null set for all h E 6, then we say that B is quasi-invariant under 6, or 6-quasi-invariant, or (if the group 8 has been clearly specified) simply quasi-invaviant. We note the following simple facts. The totality of 6-quasi-invariant sets of (G, 23, p ) forms a o-subring 23, of S ;if b is an algebra, then 23, is also an algebra. Any set which differs from a quasi-invariant set by a p-null set is also quasi-invariant. Any quasi-invariant measure space

3.1. Basic Properties of Quasi-Invariant Measures

131

(G, 8,p ) has at least one class of quasi-invariant sets, that is, the p-null sets; if 23 is an algebra, then there is at least one more class of quasiinvariant sets, namely, those sets which differ from G by a p-null set. Any set which belongs to one of these two classes is said to be trivially quasi-invariant.

Definition 3.1.7. Let (G, 8,p ) be a measure space which is quasiinvariant under a transformation group 6. If there exist no nontrivially quasi-invariant sets, then (G, 23, p ) is said to be ergodic with respect to 6, or 6-ergodic. If there exist no nontrivially quasi-invariant sets of finite measure (o-finite measure), then (G, 23, p ) is said to be weakly (strongly) ergodic with respect to 8, or weakly (strongly) (fi-ergodic.16 Example 3.1.1. Let (G, 23, p,), k = 1, 2, be mutually singular measure spaces which are both quasi-invariant under a transformation group 6. Choose sets B, E 23, k = 1,2, such that p, is concentrated on B, and p,(BI) = 0 for k # 1. Then B, is 6-quasi-invariant relative to p l for any k, 1. Let p = (p, pl))/2.Obviously, the measure space (G, 23,p) is also quasi-invariant under 8, and the sets B, , k = 1, 2, are quasi-invariant relative to p. If p, 0, k = 1, 2, then it is easily seen that B , and B, are nontrivially quasi-invariant relative to p, hence (G, 8,p ) is not ergodic with respect to 8.

+

+

Lemma 3.1.31. Let G be a group, and let 23 be a o-ring consisting of certain subsets of G, such that gE-lE23 whenever g E G, E ~ 2 3 . Let p be a left-invariant u-finite measure on (G, 23)) and suppose that there exists a right-invariant o-finite measure v on (G, 23) such that v is equivalent to p. Then (G, 23, p ) is ergodic with respect to the left translation group G. PROOF.Suppose, on the contrary, that p were not ergodic. Then, there would exist disjoint quasi-invariant sets E, F, with p ( E ) > 0 and p ( F ) > 0.” Since hE - E is a null set, we have p ( h E nF )

= 0.

(3.1.40)

Moreover, since v and p are equivalent, v(E) > 0. l6 Trunslutor’s note: Note the unfortunate fact that strong ergodicity, as defined here, is a weaker property than ergodicity. l7 Translutor’s note: This inference seems to require that 2 3 be a o-algebra. Alternatively, one could modify the definition of ergodicity as follows: a measure space is ergodic provided that there do not exist two disjoint quasi-invariant sets which are both of positive measure.

132

111.

GROUPS WITH QUASI-INVARIANT MEASURES

Consider the measure pI on (G, B), defined by /Ll(A) = v(A-I),

A E 8.

As usual, we denote the characteristic function of any set ID by Ca One easily calculates that

.

By Fubini's Theorem (see Halmos [l], $36, Theorem B),

(3.1.41)

But, in view of (3.1.40), the right-hand side of (3.1.41) is zero, which contradicts v(E) p ( F ) > 0. We conclude that p is ergodic. 3 Remark. As particular cases of Lemma 3.1.31, we have (i) any leftinvariant Haar measure on a locally compact topological group, and x pLnof left-invariant Haar measures p1 ,..., p, (ii) any product p1 x on locally compact topological groups G, ,..., G, , respectively. Next, we introduce a construction which will be used in the sequel.

Theorem 3.1.32. Let {G, , a E 2l} be a family of locally compact topological groups, let 8, be the o-ring generated by the totality of compact sets in G, , and let Q, = (G, , B , , p,) be a probability18 measure space which is left quasi-invariant under G, . Let G = Xaal G, ; then G is a group with respect to the multiplication operation defined as follows: (go!, a E W

h ,

9

01

w

= {g,h,

, a E 2%

Let 0 be the subgroup of G consisting of all elements g = {g, , a E 2l} such that g, differs from the identity element of G, for at most a finite number of indices a. Let 8 = XaElb,, p = Xaal pol; then, the measure space Q = (G, 8 ,p ) is ergodic with respect to the group of left translations corresponding to 0. In particular, this implies that b, is a o-algebra.

3.1. Basic Properties of Quasi-Invariant Measures

133

PROOF. Let h = {a1 ,..., a,} be an arbitrary finite subset of CU. Form the Cartesian product G,

=

G,, x G,, x ... x Gmm,

and define multiplication in GA by the rule {g&1

)'''9

gmn>(h,l

>.'.I

=

{gcYlhml

>-"f

gcXfihmn}.

If GA is given the product topology, then it is easily shown that GA is also a locally compact topological group. Let 23, = Bml x 8,,x x Ban, PA = pa, x " * x pan; then, using Fubini's theorem, it is easily proved that SZ, = (G, , 23, ,p,) is a probability measure space which is left quasi-invariant under G, . By Corollary 3.1.6, each p, is equivalent to a left-invariant Haar measure v, on G, . Hence, p,, is equivalent to the measure vU1x - - - x vNnon 23, . By the remark following Lemma 3.1.3 1, vU1x x van is ergodic with respect to the left translations of G A , hence the same is true of pA. If CU is finite, then G, = 6 ,hence, it only a * .

remains to consider the case where 'u is infinite. Suppose that E is any set in 23 which is quasi-invariant under 6. Now, there is a sequence {a,} C CU, such that E is the direct product of m )(olg(,n) G, and some set in b a n .Consequently, we may as well suppose that CU itself is countable, for example, CU = (1, 2,..., n ,...}. Let A, = (1, 2,..., n}. If f(gA,> EL2(QAn),

then we may regardf(.) as a function in L2(Q),that is, g +f(gAn), g E G, {gAn = (g, ,g, ,..., g,)}. Thus, L2(QAn) is imbedded as a closed linear subspace of L2(Q);let P n )denote the corresponding projection operator. If S E L2(!2),X E L2(QAn), then, since P(,) is self-adjoint, we have

p'"'SkAn)

xx) 1 dpA,(ghn) =

C

p'n'S(gAn)x(g,n) 'pk)

I n particular, when the function S is nonnegative, formula (3.1.42) holds for any nonnegative measurable function X on Q A - ,provided that (Pcn)S)X ~ L l ( i 2 , ~I n) .fact, for such an X , we have X,

= min(X,

m)€L2(QAn),

134

111.

GROUPS WITH QUASI-INVARIANT MEASURES

hence, setting X = X , in (3.1.42), and letting m + 03, we obtain the desired result. Now, let S be the characteristic function of the set E; obviously, S E L2(Q). Write S(,) = P n ) ( S ) . Choose an arbitrary element h = {h, ,..., h,} E GAn,let Y be any nonnegative measurable function on QA such that Scn)(hgAn) Y(gAn)€L1(QAn), and define

we have Scn)X E L1(QAfl). Hence, using (3.1.42) and (3.1.43), we get S'n)(hgAn)'@An)

dpA,(gAn)

Let k = {ha , a E m} be the element of 8 whose coordinates are k, = h, for a = I, ...,n and k, = e, (the unit element of G,) for 01 > n. Then, by virtue of (1.1,13), the above relation may be written as

(3.1.44)

-

Since E is a quasi-invariant set, the functions S(g) and S(kg)are equal almost everywhere. Therefore, applying the change of variable g hg in (3.1.44), we obtain S(n)(hgAn)'(gAn) dpAn(gA,) =

I,

=

I,

S(g) y(gAn) dr*.(g) S'"'(gAn)

dpAflkA,),

3.2. Characters and Quasi-Characters

135

whence it follows easily that Scn)(hgAn) and S(")(g,,,)are equal almost everywhere. Since h is an arbitrary element of G A n ,and since SZAn is ergodic with respect to the left translations of G,,, , it is not difficult to deduce that S(")(gAn) is almost everywhere equal to some constant M, . But according to Lemma 1.1.7, the sequence of projections {Pn)} converges strongly to I, hence S

=

lim P(n)S = lim M,, , n+m

n+m

which is obviously also equal to some constant almost everywhere. This means that C, is either equal to zero almost everywhere or equal to 1 almost everywhere, that is, p ( E ) = 0 or p(G - E) = 0. Thus, SZ is ergodic with respect to the group of left translations 8 . ]

53.2. Characters and Quasi-Characters

lo Definition and Basic Properties of Characters The class of functions known as characters is of basic importance for harmonic analysis on groups. Their definition is as follows.

Definition 3.2.1. A complex-valued function a on a group G is said to be a character, provided (i) I a(g)l = 1 for all g E G, and (ii) a(gh) = a ( g ) a(h) for all g , h E G. If we let e denote the unit element of G, then any character a of G satisfies the condition (iii) a(e) = 1. I n fact, by (ii), a(e)2 = a(e2) = a(e), and by (i), a(e) # 0, hence a(e) = 1. In the present section, we shall use C to denote the multiplicative group of complex numbers of unit modulus; thus, a character a of a group G is just a homomorphism of G into C: a :

g+a(g),

gEG.

Let a and /3 be any two characters of the group G. We define the product a/3 by the ordinary rule for multiplication of functions:

Obviously, a/3 is also a character of G. The totality of characters of G clearly forms a commutative group with respect to multiplication. This group is known as the character group of G (or the algebraic dual of G ) ; we denote it by G'. The unit element of G' is the constant function 1.

136

111. GROUPS

WITH QUASI-INVARIANT MEASURES

Example 3.2.1. Let I be the additive group of all integers. For each c in C, define a character /Ic of I as follows: /3&)

= cn,

n E I.

The correspondence c -+ /Ic is obviously an injective homomorphism from C into I'. Moreover, given any a E 1', we have a = / I c , where c = a( I). Thus, the correspondence c -+ /Ic is an isomorphism from C onto 1'. Using this isomorphism, we may also consider C and I' as identical: I' = C.

Theorem 3.2.1. (Character Extension Theorem). Let G be a commutative group, GI a subgroup of G, and a, a character of G I , Then, there exists a character a of G such that a(g) = al(g) for all g E Gl. PROOF. If G # G,, choose any g, E G - G,, and let G, be the smallest subgroup of G containing g, and G, . We distinguish two cases. (i) For every natural number n, gln 2 G, . I n this case, every element of G, is uniquely expressible in the form ggln,

gE Gl 1 n

= 0,

&I, Z t L .

*

Choose any c1 E C, and define a character a , on G, as follows: 4gg1")

=

al(g)Cln-

(3.2.1)

I t is easily seen that a2 is an extension of a , . (ii) There is a natural number n such that glnE G I . Let n, be the smallest such number. Then, every element of G, is uniquely expressible in the form gglR,

gEG1, n = 0 , 1 , 2 ,..., n1-1.

Choose a complex number c1 such that cy1 = a,(g?), and define a character a, on G, as in formula (3.2.1). Again, it is easily verified that a, is an extension of a, . Thus, in either of the two possible cases, a, can be extended to G,. Using Zorn's lemma, we can extend a, to a character a on G. ]

Example 3.2.2. Let 9 be the additive group of all rational numbers, let t be an arbitrary real number, and consider the character of B? defined by g E B. q(g) = e i t g , The multiplicative group of all characters of this form is obviously isomorphic with the additive group of real numbers R. However, this

3.2. Characters and Quasi- Characters

137

group is not the whole of 9'. I n fact, choose any sequence of natural numbers (9,) such that qn divides qn+l, 29, < qn+l, and such that every natural number divides at least one of the q , . Clearly, such a sequence does exist. Define another sequence of natural numbers {p,} by the recursion formula

po Obviously, p , integer k,

-+

= 0;

p,

= pn-l

+ 2qn_,,

co.Now, form a character

n 01

=

f , 2,... .

on 9 as follows: for each

We must first show that 01 is well defined. If k/q, = k'/q,# n < n', then p,. - p, is an integral multiple of qn , hence

&k'

=&k

4n'

4n

=&

, and, say,

k + r,

4n

where r is an integer. Thus, 01 is well defined, and now one may easily verify that 01 is a character. Now, we assert that this 01 cannot be equal to any a t . For suppose, on the contrary, that a: = tyf for some t. Then, since a(l/qn)= a f ( l / q n ) it , follows that

is an integer. However, when n is sufficiently large, we have I t / 2 I~< i q , , and hence I p, - (t/277)1/qn< 1 , which is possible only if p, - (t/277)= 0. But this cannot hold for two different values of n. We conclude that 01 # cxf for any t, and thus 9'# R.

Example 3.2.3. Let R be the additive group of all real numbers. For each fixed real number t, we have the character at(.)

=

eitz,

x E R.

Denote the totality of such characters by R*. We assert that R" # R'. I n fact, in Example 3.2.2 we constructed a character 01 on the subgroup 9 of R,such that 01 is not of the form a ffor any t E R; using the character extension theorem, we may extend this 01 to a character of R, and of course this extension is also different from every a t . Actually, one can even extend a character of the form a t on 9 to a character on R which is not of the form a t . For example, choose any irrational number xo ,

111.

138

GROUPS WITH QUASI-INVARIANT MEASURES

and let G1be the subgroup of R consisting of all numbers of the form mxo+x,

Let /3 be the character of

mEI, X E W .

G1defined by B(m0

+ x)

=

(-1)".

Then, any extension of jl to a character of R is obviously not of the form a f . In a similar manner, one can prove that C' # I , that is, there exist characters of C which are not of the form ol,(a)

= an,

n €1.

When G is a topological group, we shall be primarily interested in continuous characters of G, that is, characters which are continuous functions on G. T h e totality of continuous characters of G is called the dual of GI and will be denoted by G*. Obviously, G* is a subgroup of G'. Of course, if G is a discrete topological group, then every character of G is continuous, so that G* = G'. However, for an arbitrary topological group GI in general, G* and G' do not coincide.

Lemma 3.2.2. Let /3 be a character of the topological group G. If jl is continuous at the identity e of GI then /3 E G*. PROOF. Let ho be any element of G, and let E be an arbitrary positive number. By hypothesis, there exists a neighborhood V of e such that I jl(g) - /3(e)l < t for all g E V . Hence, if h belongs to the neighborhood Vho of h a , then

I B(h) - P(h0)l = I P ( h W - B(e)l < E. Thus, ha is also a continuity point of /3. J Let 23 be a o-algebra consisting of certain subsets of a group GI and let G' denote the totality of 23-measurable characters of G. Obviously, G' is also a subgroup of G'. If 23 consists of all the subsets of G, then of course G' = G'. On the other hand, if 23 contains only the null set and G, then G5 consists of just the single character 1. If G is a topological group and if 23 contains all the closed subsets of G, then every continuous character of G is %-measurable, that is, G* C G'. I n the following example, we actually have G* = GB.

Example 3.2.4. Let R be the additive group of real numbers with the Euclidean topology. Then, every continuous character on the topological group R is of the form m t ( x ) = eifz, x E R. Moreover, if 23

3.2. Characters and Quasi-Characters

139

is the 0-algebra generated by the totality of closed subsets of R, then

RB = R*. In fact, let a E RB,and consider the Lebesgue integral

1

m

X

=

a(x)e-”dx.

0

Since

1

m

a(h)X =

a(h

+ x)e+

m

dx

=

eh

a(x)ecz dx,

(3.2.2)

h

0

it follows that a(h) is continuous. Thus, RB = R*. Furthermore, using (3.2.2), we obtain

hence, as h -+ 0, a(h) - 1 +I-- 1 h A’

therefore a’(0) exists. Moreover, since

a’(g) exists and is equal to a‘(0) a(g). Integrating, we obtain a(g) = e a ’ ( O ) g .

Therefore, since I a(g)l = 1, it follows that a’(0) is a pure imaginary number, which we call it, t E R. Thus, every measurable character of R is of the form a t . The preceding example may also be regarded as a special case of Lemma 11.3.2 of Appendix 11.

2* Quasi-characters In what follows, we shall require a class of functions more general than measurable characters, namely, quasi-characters. We shall not describe the concept of a quasi-character in its most complete generality; for our purposes, it suffices to restrict our considerations to groups.

Definition 3.2.2. Let (G, B, p ) be a measure space which is quasiinvariant under a group of measurable transformations 8 . Let a be a

140

111.

GROUPS WITH QUASI-INVARIANT MEASURES

measurable function on (G, ‘B), satisfying the following conditions. (i) I a(g)l = 1 for all g E G; (ii) for every fixed h E 6, the function gEG

4g)/a(g),

is equal to a certain constant &(h) almost everywhere (relative to the measure p). Then, we say that a is a puasi-chara~ter~~ of (G, 8,p) relative to 6 . First, we observe that the function B(h), h E 6, is a character of 0. In fact, we obviously have I 8(h)l = 1, moreover, for any h, x E 6,

hence, it follows from the quasi-invariance of p that 8(hx) = 8(h)~ ( X ) . ~ O We call B the character of 6 induced by the quasi-character a; the totality of such induced characters will be denoted by 6”.We also refer to 01 as a quasi-character corresponding to 8. We shall regard two quasi-characters which are equal almost everywhere (relative to p ) as identical. If two quasi-characters differ only by a constant factor (i.e., if the ratio of these two functions is equal almost everywhere to some constant), then we say that they are similar. We proceed to state some simple properties of quasi-characters. I. Similar quasi-characters induce the same character. 11. If a is a quasi-character and c is a complex number with I c I = 1, then ca is also a quasi-character. 111. If the measure p is ergodic with respect to 6, then quasicharacters which correspond to the same character are similar. In fact, suppose that the quasi-characters a1 and a2 correspond to the same character of 6, and consider the function a = ala;l. Then, for any h E 6, we have a(hg) = a(g) for almost all g E G. Consequently, l o A quasi-character can be regarded as an eigenfunction of a certain operator. In fact, consider the totality U ( G ) of %-measurable complex-valued functions 01 on G such that I a(g)l = 1 for all g ; functions in U ( G )which are equal almost everywhere ( p ) are to be regarded as identical. For each element h of 6, define an operator Thon U(G)as follows:

ThO1(g) = a(h-’g),

01

E

U ( G ) , h E 6.

Then, the correspondence h + T his a representation of 6 in U(G)[i.e., a homomorphism of 6 into a group of operators on U(G)].A quasi-character 01 is just a common eigenfunction of all the operators T h, h E 6, and a(h) is the eigenvalue of the operator Th-1, corresponding to the eigenfunction 01. Translator’s note: To draw this conclusion, one must assume that the measure p is not identically zero.

3.2. Characters and Quasi-Characters

141

if y is any arc on the unit circle of the complex plane, then { g I a(g) E y } is a quasi-invariant set. But if a(g) were not equal to some constant almost everywhere, then there would exist two disjoint circular arcs y1 ,yz such that both of the (disjoint) sets ( g I a(g) E yl>and ( g I a(g) E y2) are of positive measure. This contradicts the ergodicity of p. IV. If all the quasi characters corresponding to any given character of 6 are similar, then p is ergodic with respect to 6. I n fact, if p is not ergodic, then let A be a quasi-invariant set in G such that p ( A ) > 0 and p(G - A) > 0.21 Define a function a: as follows: a:(g) = -1

for g E A ,

.(g) = 1

for g E G - A .

I t is easily seen that a: is a quasi-character which induces the character 1 on 6. But 01 is not similar to the quasi-character 1 of G. We shall denote the totality of quasi-characters by Gu. Clearly, Gu forms a group with respect to ordinary multiplication of functions. T h e totality of quasi-characters which correspond to the character 1 of 6 will be denoted by '$I. It is easily verified that % is a subgroup of G.. Moreover, it follows from properties I-IV that '$I is the tbtality of constant functions c, c E C, if and only if p is ergodic with respect to G. Consider the factor group Go&= G'/%. For each 77 E GOP, choose a representative element a in 7.T h e mapping

s:

(3.2.3)

q+&

is clearly an isomorphism from GOuonto

6~.

Example 3.2.5. Let G be a group, let 6 be the totality of left translations of G, let (G, b,p ) be a finite measure space which is quasiinvariant under @, and suppose that the correspondence (g,h) -+gh is a measurable mapping from (G x G, 23 x 23) to (G, 23). Then, the group of quasi-characters Gu is just the totality of functions of the form c&,

C E C , & E GB.

PROOF. For any a: E GP,the functions a(hg) and a:(g), (g, h) E G x G, are both measurable with respect to 23 x 8 , hence, the same is true *l

Translator's note: Again, see footnote 17 concerning the proof of Lemma 3.1.31.

142

111.

GROUPS WITH QUASI-INVARIANT MEASURES

of their quotient ai(hg)/cx(g). Since p is finite, we may assume thatz2 p(G) = 1. By Fubini’s theorem, we know that

is a measurable function on (G, b), that is, 2 E G’. I n particular, Z(h), regarded as a function of the two variables ( g , h), is also measurable, hence

is a measurable set. Since

for every h E G, it follows from Fubini’s theorem that ( p x p ) ( E ) = 0, and therefore

for almost every fixed value of g . We arbitrarily choose an element g = go such that the above equation holds. T h e n

4ko)

=

.(go)

for almost all h in G. Writing hg, = h, , it follows by the quasi-invariance of p that

for almost every h, , that is,

a(go)/a(go). ] Example 3.2.6. Let G be a locally compact group, Q the group of all left-translations of G, and (G, b,v) a left-invariant Haar measure space. Then, given any quasi-character a of (G, 8,v) relative to 6, there exists a complex constant c E C such that a ( g ) = cZ(g) for almost every g E G. PROOF. Let Go be a subgroup of G constructed as in the proof of Lemma 1.1.1 1, and let Bo , vo denote the restrictions to Go of b, v, respectively. Then, the measure space (Go, S o ,y o ) is a-finite, hence, aa

zero.

ai

= cZ,

where

c =

Translator’s note: Again, it is implicitly assumed that the measure p is not identically

3.2. Characters and Quasi-Characters

I43

there exists a finite measure p on ( G o ,So),such that p is equivalent to y o . Now, let a be any quasi-character of (G, 8,Y ) , relative to 8. Then, the restriction of a to Go (which we also denote by a) is a quasicharacter of ( G o ,%,, y o ) , hence also a quasi-character of ( G o ,@,, p), relative to the group 6, of all left-translations of G o . According to Example 3.2.5,23 there exists a constant c, I c I = 1, such that

for almost all g E Go ; here, &(g) is the character of 8, induced by a, which is just the restriction to Go of the character of G induced by a. Consequently, if h is an arbitrary element of G , then, for almost everyg in the coset h-lG, , since hg E Go , we have .(g) = G(h-1) .(hg) = cE(h-1) d(hg) = cd(g).

I t follows that (3.2.4) holds for almost every g E G. ] Examples 3.2.5 and 3.2.6 show that the idea of a quasi-character generalizes the notion of a measurable character. We observe that, if (G, 8 ,p ) is a localizable measure space which is quasi-invariant under the transformation group 8, then any x’ E G. is continuous relative to the p-topology on 6. I n fact, given any h E 6, consider the operator U(h)defined in 93.1, 60. Then, for any appropriate choice of the vectors [, 7 €L2(G,23, p), we have2*

whence it is clear that x”(h)is continuous relative to the p-topology. We shall now make use of Lemma 3.1.12 to investigate the continuity of characters which are induced by quasi-characters. Let (G, 23, y) be a regular measure spacez5 which is quasi-invariant under a group of continuous measurable transformations 8 , and suppose that 6 is a topological group of the second category relative to an admissible topology. Then any quasi-character a of (G, 23, p ) (relative to 8 )induces a continuous character on 8.

Theorem 3.2.3.

as Translator’s note: To apply Example 3.2.5, one must make some additional assumption to ensure the measurability of the mapping (g,h) -+gh. 84 Translator’s note: Given any h, E 8, and any nonzero vector 6 E L2(G,9, p), we may choose I) = U(h,)(x[),thus ensuring that the denominator is nonzero in some sufficiently small p-neighborhood of h, Translator’s note: Again, we must require that p is not identically zero.

.

111.

144

GROUPS WITH QUASI-INVARIANT MEASURES

PROOF. Given any positive number E, we partition the unit circle of the complex plane into a finite collection of disjoint Bore1 sets A, ,..., A, , such that the diameter of each A, is less than E. Accordingly, G is the union of the disjoint sets B,

= {g

I a(g) E A k } E 8, k

=

1,2,..., n.

(3.2.5)

Therefore, at least one of the sets (3.2.5), say B k o ,has positive measure. By the regularity of p, there exists a compact set K E 8 such that K C %k0 and p ( K ) > 0; in particular, if g, ,g, E K, then

I

-4gJl

< c.

(3.2.6)

Now, a(h-lg) is almost everywhere equal to a(g) Z(h-l), where is the character of 6 induced by a. Hence, by virtue of Lemma 3.1.12,28there is a neighborhood V of the identity in 6 such that, for any h E V, there exists an element g E K n hK satisfying (3.2.7)

a(h-1g) = .(g) Z(h-1).

Thus, if 12 E V, then it follows from (3.2.6) and (3.2.7) that

I d(h) - 1 I

=

1 1 - d(h-l)l

= I .(g) - .(h-lg)I

< t.

This shows that d is continuous at the identity of 6. Hence, by Lemma 3.2.2, 2 is continuous at all points of 6. ]

Corollary 3.2.4. Let G be a topological group, let 6 be a subgroup of G, and suppose that 6 itself is a topological group of the second category with respect to a topology which is stronger than that induced by G. Let (G, 8,p ) be a regular measure space which is left (or right) quasi-invariant under 6. Then, every quasi-character 01 of (G, 8,p ) (relative to 6) induces a continuous character on 6. In particular, every b-measurable character 01 of G, when restricted to 6, defines a continuous character on 6. Remark. As a particular case of Corollary 3.2.4, if G is a locally compact group, 23 is the u-ring generated by the compact subsets of G, and p is a left-invariant Haar measure, then every measurable character of G is continuous, and every quasi-character of (G, 8,p ) (relative to the left translation group G) induces a continuous character on G. I n fact, G is then a topological group of the second category (see Guan Translator’s note: Recall, however, the additional finiteness restrictions on p which were necessary to prove Lemma 3.1.12.

3.2. Characters and Quasi-Characters

145

Zhao-zhi [I]), and (G, d,p) is invariant (hence afortiori quasi-invariant) under left translations. Applying Corollary 3.2.4, with 8 = G, the conclusion follows at once.

30 Topologies on Character Groups and Quasi-Character Groups I n the subsequent discussion, we shall require several kinds of topologies on character groups and quasi-character groups. We now proceed to describe these topologies. I. Let G be a group, and let H be a group consisting of certain characters of G. Let a. E H , let g , ,...,g, be any finite set of elements of G, and let E be any positive number. Form the set

For each ‘ Y ~ E H we, take the totality of sets of the above form as a neighborhood basis at a. . Thus, we obtain a topology on H , which we call the weak topology. This is, in fact, the weakest topology such that each g E G, when considered as a function a ( g ) , Q E H , is continuous on H . I t is easily seen that the weak topology makes H a topological group. and let H be a group consisting 11. Let G be a topological of certain characters of G. Let a. E H , let Q be any compact subset of G, and let E be any positive number. Form the set

For each a0 E H , we take the totality of sets of the above form as a neighborhood basis at a0 . Thus, we obtain a topology on H , which we call the strong topology; it may be described as the topology of uniform convergence on compact subsets of G. It is easily verified that the strong topology makes H a topological group. Since every finite subset of G is compact, it follows that every weak neighborhood2s in H is also a strong neighborhood, and consequently the strong topology is, in fact, stronger than the weak topology. On the other hand, there do exist topological groups G and corresponding groups of characters H such that the weak and strong topologies on H are actually different. 111. Again, let G be a topological group and let H be some group of Translator’s note: Actually, the definition given here is applicable even if the group operations are not continuous relative to the topology of G. 28 By a weak (strong) neighborhood, we mean, of course, a neighborhood in the weak (strong) topology.

111. GROUPS

146

WITH QUASI-INVARIANT MEASURES

characters on G. Let U be a fixed open set in G. Then, for each a. E H and positive number E, we form the set W(a0 ;

u, €1 = {a I sup I .(g) k-u

- 4g)l

< €1.

We take the totality of sets W(ao; U, E ) for all E > 0, as a neighborhood basis at ( y o . I n this manner, we obtain a topology T , on H ; we call T , the U-topology. Again, H is a topological group relative to this topology. Let Q be any compact subset of G. Then, there exist elements g, ,...,g, E G, such that (2 C x i = l g k U . Hence, if a satisfies

then

where g = gkh. Thus, we have

whence it is clear that, if the U-topology is stronger than the weak topology, then it is also stronger than the strong topology. Next, we shall consider certain topologies on groups of quasi-characters, or groups of measurable characters. IV. Let 9 = (G, 23,p ) be a measure space, and let L ( 9 ) denote the Banach space formed by the totality of integrable functions on 9. Let H be a family of bounded measurable functions on 9, functions which are equal almost everywhere being regarded as identical. (Two cases occur most frequently in the subsequent applications: either 9 is quasiinvariant under a transformation group 8 , and H is a group of quasicharacters on 9, relative to 8 , or G is a group and H is a group of %measurable characters on G.) Each element a of H defines a continuous linear functional on L ( 9 ) , as follows:

Thus, H may be imbedded29in the conjugate space of L(Q).I n this way, Translator’s note: In order that the mapping a -+ (., a) be injective, it is necessary and sufficient that Sa satisfy condition (i) of Definition 1.2.3. But in any case, we may still define the p-weak topology as the weakest topology on H such that the mapping a + (*, a) is continuous [taking the weak topology on the conjugate space of L(Q)].

3.2. Characters and Quasi-Characters

147

the weak topology on the conjugate space of L(Q) induces a topology on H ; a neighborhood basis for this topology may be constructed as follows. Let a. be an arbitrary point of H , choose any finite set of functionsf, ,...,fn inL(SZ), take any positive number E , and form the set

We take the totality of sets of the above form as a neighborhood basis at the point a o . The topology obtained in this manner will be called the p-weak topology. If H is a group, then, in general, it does not necessarily follow that the p-weak topology makes H a topological group. Further on, however, we shall prove that, under certain circumstances, H does form a topological group relative to the p-weak topology, and we shall compare the strengths of this and other topologies. V. Let (G, 23, p ) be a measure space which is quasi-invariant under a transformation group 8 , and let H be some group of quasi-character~~~ of (G, 23, p ) relative to 8 (in particular, if G is a group, then H may be just a group of %-measurable characters on G). Take any set A E 23, p ( A ) < co, and consider the function 1/2

on H . Since

for all a , /3 E H , it follows that N A ( - )is a pseudonorm on H. We use the family of pseudonorms { N A ( . )to ) define a topology on H , as follows. Given any element a. E H , we choose an arbitrary set A E 23, p ( A ) < co, and an arbitrary positive number E , and form the set Y(a0 ; A , t) = { a I

NA(aail) < c}.

We then take the totality of such sets as a subbasis for the neighborhood system of a. . T h e topology so obtained will be called the p-topology. Using the fact that N A ( . )is a pseudonorm, it is easily verified that the p-topology makes H a topological group. If p is a finite measure, then the p-topology is equivalent to the topology induced by the metric NG(a/3-I).To show this, it suffices to prove that every set of the form Y ( a o ; A ,c) is open in the topology induced by NG(a/3-l).Take any ao In particular, if 8 contains only the identity transformation of G, then H is simply a multiplicative group of 8-measurable functions 01 which satisfy the condition I a(g)l = 1.

111.

148

GROUPS WITH QUASI-INVARIANT MEASURES

Y(ao; A , E), and let E~ = E - N,(a,a;’); then, since N,(a) it is easily seen that Y ( a , ; G , €1) c W O ; A , 6).

a1 E

< Nc(a),

This shows that Y ( a o ; A, E ) is indeed an open set in the topology induced by the metric NG(aj9-l). T h e pseudonorm N , ( . ) may be used to introduce a pseudonorm on G,u. Choose any A E 23 such that 0 < p ( A ) < 03, and define a function N A ( q ) ,-q E Go@,as follows: N A ( ~=) inf NA(a). a€9

I t is easily verified that this is a pseudonorm on Go@. If F is a subgroup of Go”,and qo E F, we may take the totality of sets of the form

as a neighborhood basis at 70. In this manner, we obtain a topology on F, which we also call the p-topology. Relative to this topology, F forms a topological group. If p is a finite measure, we write NG(q)= N ( q ) , and in this case the p-topology is equivalent to the topology induced by the metric31 N(&l), 5, 7 E Gou. We now proceed to consider some properties of the topologies introduced above, and to give some examples.

Theorem 3.2.5. Let SZ = (G, b,p ) be a measure space which is quasi-invariant under a transformation group 6, and let H be some group of quasi-characters of relative to 6. Then, the p-topology on H is stronger than the p-weak topology. PROOF. Given any a. E H and any p-weak neighborhood X(a0 ;f1

,...,fn , 4,

we need only prove that there exists a set A a positive number such that Y(a0 ; A , €1)

Now, since such that

fk

E

c q u o ;f1 ,...,

fn

23, 0 < p ( A ) < 9

CO,

and

4.

E L ( G ,8,p ) , there exists a set A E b, 0 < p ( A ) < 00,

81 Since the subgroup fn is closed with respect to the metric N(a/3-’), a, /3 E G’, it follows that N([q-l), 5, q E Gou,is indeed a metric on Gou.In fact, Gou, with the metric N ( [ q - l ) , [, q E Gou$is just a quotient metric space of Gou(relative to the metric N~(ap-’), a,/3 E G”).

3.2. Characters and Quasi-Characters

149

and such that sup Ifk(g)I <

00,

k

=

1, 2 ,..., n.

Choose

This shows that Y ( a o ;Q, el) C X(a0 ;fl ,...,f,, E).

]

Theorem 3.2.6. Let G be a topological space, and let Q = (G, 23, p ) be a regular measure space which is quasi-invariant under a transformation group 8. Let H be a group of quasi-characters of Q (relative to 8 ) .Then, the p-topology on H is weaker than the strong topology. PROOF. Let a. E HI and let Y(ao ; A, E) be a p-neighborhood of a. , where E > 0 and A E 8, 0 < p ( A ) < CO. By the regularity of p, there is a compact set Q E 23 such that Q C A and €2

P(A - Q ) < 8 *

Choose

Then, if

01

JA

E

V(ao ; Q , el), we have

I .(g)

- "o(g)I2 d P k )

€2

€2

2

2

that is, NA(aa;')

< z.

<-+-=2,

Thus, V ( a o;Q , el) C Y(a0 ; A , E).

3

150

111.

GROUPS WITH QUASI-INVARIANT MEASURES

Theorem 3.2.7. Let 52 = (G, 23,p ) be a finite measure space which is quasi-invariant under a transformation group 8. Then, the group G@of all quasi-characters of Q (relative to 8)is complete relative to the metric NG(afl-l),a , fl E Gu. PROOF. Let {an>be a fundamental sequence in Gu, that is,

as m, n -+ co. Since the space L2(52)is complete, there exists a quadratically integrable function a such that

furthermore, we can select a subsequence {an,(g)} which converges to 01 almost everywhere. Therefore, since I a,,(g)( = 1, we may choose 01 such that 1 a(g)J= 1. Moreover, for any h E 6, it follows from the quasi-invariance of p that

holds for almost all g E G, that is, (Y is a quasi-character. And, by (3.2.8), limn+mNG(~na-l) = 0. Hence, Gu is complete. ] Remark. If p is not finite, but is localizable, then it can be proved that the multiply pseudometric space (for the definition, see Guan Zhao-zhi defined by the family of pseudometrics {NA(aP-l)} is sequentially complete.

Corollary 3.2.8. Under the hypotheses of Theorem 3.2.7, the group Gouis complete relative to the metric N(&-I), (, q E Gou. PROOF. Let (8,) be a fundamental sequence in Gou, that is, limm,n+mN((;l(,) = 0. Then, we can select a subsequence {(,}, k = 1, 2,..., such that N(,$;~_,f,,)< 1 / 2 k . For each k 2 2, choose an element ak E t;:-,(,, such that N(ak) < 1/2k, choose any element k 0 1 ~E ( ,, , and form the product f l k = a l. Then, obviously f l k E ( , , 1 and when 1 > k, we have flklfll = f l v = k + l a , , hence

nl=l

1

Wiglcligd <

c W E ” ) < cz z1; *

v=k+l

v=k+l

sa Translator’s note: Unfortunately, the translator does not have access to the reference cited here, and has no idea whether the definition in question has been given or discussed elsewhere in the literature. Consequently, the word “multiply” in the English rendering can only be regarded as tentative.

151

3.2. Characters and Quasi-Characters

Therefore, {Pk) is a fundamental sequence in G'. By Theorem 3.2.7, there is an element ,f3 E G' such that limk+mNG(f&'/3) = 0. Let f be the residue class containing j3:

E Then ,k?~'/?E

= (01P

I 01 E rn}.

&if, hence pz N(K;,E)= 0,

and since limm,k.+o N(&;fm) = 0, it follows easily that = 0.

lim N((-'(,)

m-m

]

Next, we turn our attention to the mapping (3.2.3).

Theorem 3.2.9, Let (G, b,p ) be a regular measure space which is quasi-invariant under a group 8 of continuous measurable transformations. Suppose that 0 has an admissible topoIogy (see Definition 3.1.2) which makes 6 a topological group of the second category satisfying the first axiom of countability. Then, for any A E 23, with p ( A ) > 0, there exists a positive number c and a neighborhood V of the identity in 8 such that sup I w4 - 1 hsV

I


II

-

I IM E )

(3.2.9)

A

holds for all 01 E Gu. Furthermore, if p is a finite measure, then there exists an open set V in 8 , such that, if 6~is given the V-topology, and G,u is given the p-topology, then the mapping S from Gouonto @Pis continuous. PROOF. Let A E 23, p ( A ) > 0. By Lemma 3.1.12,33 there exists a neighborhood V, of the identity e in 8 such that p ( A n h-lA) > 0 for every h E Vo . Furthermore, by Corollary 3.1.18, there exists a neighborhood V, of e and a positive number c, such that (3.1.17) holds for every quasi-convex pair ( p , 5) for which J A p ( g ) dp(g) > 0, and every neighborhood V of e with V C V, . We choose V = V, n V , . Now, let 01 E Gu, and let & be the character induced by 01. Consider the pair of functions (3.2.10) g E G7 P(g) = I 4 g ) - 1 I, j ( h ) = 1 G(h) - 1 I, 33

Translator's note: See, however, footnote 9.

h E 8.

(3.2.11)

111.

152

GROUPS WITH QUASI-INVARIANT MEASURES

For almost every g E G, we have

hence, ( p ,6) is a quasi-convex pair of functions over ((G, 23, p), 6). First, suppose that JA p ( g ) dp(g) = 0. Then, a(g) = 1 for almost all g E A , hence a(hg) = a(g) = 1 for almost all g E A n F A . Therefore, if h E V, so that p ( A n h-lA) > 0, it follows that a(h) = 1; consequently, both sides of (3.2.9) vanish. On the other hand, if JA p ( g ) dp(g) > 0, then, since V C V, , (3.1.17) holds, Thus, in either case, we have

where c’ = c(p(A))ll2.Let 7 = a% be the residue class containing a. Taking the supremum of the right-hand side of (3.2.12) over all OL in q, we get sup I Lqh) - 1 I < C ’ N A ( 7 ) . heV

Hence, given any point qo E Gou, and any V-neighborhood

of the point 2,

=

Sv0,there exists a p-neighborhood 2(qo ; A , E/c’) of

rl0 such that SZ(7, ; A , E/C‘)

c {a I sup I a@)- a,(h)l < hEV

which shows that S is continuous.

€},

]

Theorem 3.2.10. Let (G, 23,p) be a regular measure space which is strongly quasi-invariant and ergodic with respect to a group 6 of continuous measurable transformations. Suppose that 6 has an admissible topology which makes (5 a connected topological group of the second category. Choose any nonzero element z/ of L(G, 23,p ) such that +(g) 2 0, g E G. Form the convex function

Then, the topology on Gu induced by Nd is independent of the choice of +, and is, in fact, equivalent to the p-topology.

3.2. Characters and Quasi-Characters

153

PROOF. We shall first prove that, given any sequence {a,] C Gu such that limn+mN,(a,) = 0, there exists a subsequence {a,,} which converges to 1 almost everywhere on G. Choose a positive number a, sufficiently small so that the set A = (g I +(g) > a> has finite positive measure. Then, by virtue of the inequality

we know that {a,} converges in measure to I on the set A, hence, there exists a subsequence {an,) which converges to 1 almost everywhere on A. Consider the set E = {g I lim an,(g) = 1). n +n

Since A is contained in E (modulo a null set), we have p ( E ) > 0. We shall prove that p(G - E ) = 0. In view of the ergodicity of p with respect to 6, it suffices to prove that E is a quasi-ergodic set. Now, by the regularity of p, there is a compact set K E 8,0 < p ( K ) , such that K C E. According to Lemma 3.1.12,34 there is a neighborhood V of the identity in 6, such that p ( K n hK) > 0

for all h E V . Therefore, for each h E V , there is an element g E K n hK, such that cin,(h-lg) = En++)

holds for all n'. But g E K n hK implies that g follows that lim

n"m

&%,(A)

(3.2.13)

a,,(g) E

E, h-lg

=1

E

E, hence, it (3.2.14)

for every h E V. Since d is a character of 6, the set 6, = { h I n'+m lim En,(h) = l }

is a subgroup of 6, and, by (3.2.14), we have 6,3 V . By virtue of the connectedness of 8 , it follows that 6, = 6, that is, limn,,, d,,(h) = 1 for all h E 6. On the other hand, for each h E 6, there is a null set Ghin G such that (3.2.13) holds for all n', provided g E Gh . Consequently, s4 Translator's note: Again, recall the qualification concerning the validity of Lemma 3.1.12.

154

111.

GROUPS WITH QUASI-INVARIANT MEASURES

g E hE - G, implies that g E E ; thus, hE - E is a null set. This shows that E is indeed a quasi-invariant set, and therefore

lim a,#(g)

n’+m

=

1

(3.2.15)

holds for almost all g E G. Hence, for any set A E 23, with p ( A ) < a, it follows by the Lebesgue dominated convergence theorem that lim NA(a,,) = 0.

n‘+m

From this, we easily deduce that the topology induced by N&is stronger than the p-topology. Conversely, given any positive number E , we choose a set A E 23, 0 < p ( A ) < co,such that

and c = sup I#(g)l &!€A

Then, for any a

E {a

< 00.

I N A ( a )< r / 2 dc},we have

that is, { a I N A ( a )< ~ / dc} 2 C { a I N,(a) < E ) . Therefore, the p-topology is stronger than the topology induced by N 4 . Thus, the topology induced by N , is identical with the p-topology, and is therefore independent of the choice of 4. 3 T h e above theorem may be used as a criterion for the equivalence of two measures. I n this connection, see Theorem 4.2.14 (which is an analog of Theorem 3.2.10) and Corollary 4.2.15.

Theorem 3.2.11. Let l2 = (G, B,p) be a localizable measure space which is quasi-invariant under a transformation group 6. Suppose that 8 has been given a topology which is stronger than the p-topology. Let 6~be given the strong topology, and let Gu be given the p-weak topology. Then, the mapping cx 8, from GUto @, is continuous. PROOF. First, we shall prove that, for any fixed ~ E L ( Q )G’ , x 6, the function --f

da,h ) =

a(hg)f(g)449,

(a9

h) E GU x 6,

3.2. Characters and Quasi-Characters

of a l , then

1~

( ah), - v ( a l , hl)l

v(01,h). Now, given any

0 1 ~E

< 2.5. This

155

proves the continuity of

GF,we choose an f eL(SZ)such that

Also, we choose a sufficiently small neighborhood V of

J 4 g ) f ( g )44g) + 0 for all a E V. Thus, when

01

E

V , the character

, such that

0 1 ~

156

111. GROUPS

WITH QUASI-INVARIANT MEASURES

is a continuous function of ( a , h). I t follows that, for any positive number and any compact set Q in 6, the inverse image (under the mapping a --t a) of the strong neighborhood

E

(of d, E @) is open in Gu. From this, one easily deduces that the mapping a -+ 6 is continuous. ]

Corollary 3.2.12. Let G be a locally compact topological group, and (G, 8,p ) a left-invariant Haar measure space. Then, the p-weak topology and the strong topology on G* are identical. PROOF. By Lemma 3.1.29, the original topology of G is stronger than the p-topology. Following the argument used in the proof of Theorem 3.2.11, it is easily seen that the p-weak topology on G* is stronger than the strong topology. On the other hand, it follows from Theorems 3.2.5 and 3.2.6 that the strong topology on G* is stronger than the p-weak topology. Hence, these two topologies are identical. 3 T h e next corollary is an immediate consequence of Theorems 3.2.5 and 3.2.1 1. Corollary 3.2.13. Under the hypotheses of Theorem 3.2.11, if GY is given the p-topology, then the mapping a + d is also continuous. Note, however, that the conclusion of Corollary 3.2.13 is weaker than that of Theorem 3.2.9. Theorem 3.2.14. Let 52 = (G, 8,p ) be a localizable measure space which is quasi-invariant and ergodic with respect to a transformation group 6. Let $3 = {ca I a E Gu,0 c I}. Then $3 is compact with respect to the p-weak topology. PROOF. By Theorem 2.4.14, the conjugate space ofL(S2) is isomorphic to L"(52), the totality of essentially bounded measurable functions on 52. Moreover, since the unit ball in the conjugate space of L(52) is compact in the weak topology, the same is true of the unit ball B in L"(52). Hence, we need only prove that $3 is a weakly closed subset of B, and it will follow that $ itself is weakly compact. Let {a, , X E A } be a generalized sequence in $3 which converges weakly to an element a~ B . Since 0 ~ 6 we , need only consider the case a # 0, and so we may as well suppose that 01, # 0 for all h E A . If 01, = c,&, where 0 < C, 1 and /3, E Gu, then we write L?, = PA . We shall now prove that, for every fixed h E Q, the generalized sequence of complex numbers {a,(h), h E A } is fundamental.

< <

<

3.2. Characters and Quasi-Characters

157

Since Q is localizable and quasi-invariant under the transformation h, it follows that the measure space (G, '13, pk) [see (3.1. l)] is also localizable. Let f E L(Q). Then, by Theorem 2.4.15, we also have

Therefore, given any E > 0, since limAcA aA= a , there is an index A, such that both of the inequalities

E

A

(3.2.17)

hold whenever A,

< A.

Replacing g by hg in (3.2.17), we obtain

Since a is not almost everywhere zero, we may choose f so that

Using (3.2.16), (3.2.19), and the relation a,(hg) = a,(h) aA(g), we easily calculate that, if E is sufficiently small, and A, < A, A', then

1 d,(h) - dA*(h)l <

4E ~

f

161

Consequently, there is a complex number Z(h), of modulus I , such that lim GA(h)= &(A). A EA

Furthermore, we know from (3.2.16) and (3.2.18) that, if A,

(3.2.20)

< A,

then

Hence, using (3.2.20), we obtain (3.2.21)

158

111.

GROUPS WITH QUASI-INVARIANT MEASURES

which holds for all functions f in L ( 0 ) which satisfy (3.2.19). I t follows easily that 4hg) = 4g) (3.2.22) for almost all g E G . Since I &(h)l = 1, it is obvious from (3.2.22) that, for any two positive numbers c1 < c 2 , the set

is quasi-invariant. Consequently, there must exist a positive number c such that I a(g)l is equal to c almost everywhere, and so we may, in fact, assume that 1 cx(g)l = c for all g E G. Thus, setting /3 = a/c, we see from (3.2.22) that E Gw, that is, 01 E sj. ]

Corollary 3.2.15, Let G be a locally compact group, and let (G, 23,p ) be a left-invariant Haar measure space. Then G*, the totality of continuous characters of G, is locally compact with respect to the p-weak topology. PROOF. Let sjl = G* u (0). We shall prove that sj, is a closed subset of B (see the proof of Theorem 3.2.14). Let {aA, A E A } be any p-weakly convergent generalized sequence in B1,and write a = limAe,,0 1 ~ . If 01 # 0, then, proceeding as in the proof of Theorem 3.2.14, one can show that there is an element ,B E Gb such that 01 = $3, 0 < c < 1. Furthermore, by Example 3.2.6 and the remark following Corollary 3.2.4, there is an 2 E G* such that @ = e%. Thus, there is a complex number K = e%, with 0 < I K I < 1, such that 01

=

K&

&iG*.

T o prove that 01 E G*, we only show that K = 1. Choose a continuous function on G, vanishing outside of some compact set, and such that

Then, q EL(G, 23, p), and also

Moreover, using Fubini’s Theorem, one may easily calculate that, for every y E G*,

3.2. Characters and Quasi-Characters

159

Obviously, we may assume that aAE G*, hence, applying the preceding formula, we get

This shows that K = 0 or K = 1, but since we have assumed that 01 # 0, we must have K = 1. Therefore, 8lis closed in B, and hence compact. Moreover, it is easily seen that 5, is Hausdorff, therefore, G* is locally compact with respect to the p-weak topology. ] Using Corollaries 3.2.12 and 3.2.15, we immediately obtain the following result.

Corollary 3.2.16. Let G be a locally compact group, and G* the group of all continuous characters of G. Then, when given the strong topology, G* becomes a locally compact topological group. 4O

Characteristic and Quasi-Characteristic Functions

We now proceed to describe some commonly used a-algebras on groups.

Definition 3.2.3. set in 8 of the form

Let G be a group, and

8 a subgroup of G’. Any

{ a 1 (4g1),-v 4 g 7 8 E El,

where n is an arbitrary natural number, E is an arbitrary Bore1 set in complex n-dimensional space, and g, ,...,g, are arbitrary elements of G, is called a Borel cylinder with base E (corresponding to {g, ,...,g,}). Let 23 be the smallest a-algebra containing all the Borel cylinders in 8. The members of 23 are called weak Borel sets in 8 (corresponding to G). Similarly, any set in G of the form { g I (al(g)Y? .n(g))

E

El,

where n is an arbitrary natural number, E is an arbitrary Borel set in complex n-dimensional space, and a1 ,..., an are arbitrary elements of 8, is called a Borel cylinder with base E (corresponding to {a, ,..., a,}). Let 8 be the smallest a-algebra containing all Borel cylinders in G. T h e members of 8 are called weak Borel sets in G (corresponding to 8).

111.

160

GROUPS WITH QUASI-INVARIANT MEASURES

Definition 3,2.4. Let G be a group, let (G, 23, p ) be a finite measure space, and let & be a subgroup of GB. Then, the function P(B)

=

j

B(g) dP(d9

BE8

is called the characteristic function of (G, 8,p ) on &. I n what follows, we shall usually assume, for convenience, that y ( 1) = p( G) = 1. With reference to Definition 3.2.4, we shall frequently deal with the following type of situation. Namely, & is a group, G is a subgroup of and b is the totality of weak Bore1 sets in G. T o each h E 8,there corresponds an element CY E G', defined as follows: &I,

4g)

=

A 4 9

g E G.

Clearly, a E GB.I n this fashion, 8 is imbedded36 in GB.Identifying each h with the corresponding a ) we obtain the characteristic function of (G, 23, p) on &, defined by

Example 3,2,7. Let (R, b) be the measurable space discussed in Example 3.2.4. I n this case, RB is precisely the totality of functions of the form a I ( x )= eitz, x E R, t E R. If we identify a t with t , then the characteristic function of a probability measure space ( R , b,p ) is just the Fourier-Stieltjes transform:

Definition 3.2.5. Let (G, B,p) be a finite measure space which is quasi-invariant under a transformation group (ti. Form a complex-valued function y , on the quasi-character group G', as follows:

d4

=

J

4 g ) dCL(d9

aE

G'*

We call 9 the quasi-characteristic function corresponding to (G, b,p ) and the group 6. Again, for convenience, we shall assume that rp(1) = p ( G ) = 1. Translator's note: Of course, for the correspondence h + a to be injective, certain ) additional assumptions must be made. In any case, the characteristic function ~ ( hcan still be defined in the indicated manner.

3.2. Characters and Quasi-Characters

161

Since GBC Gu, the quasi-characteristic function is an extension of the characteristic function. If W and C are isomorphic, that is, if p is ergodic, then the absolute ] only upon the coset of W containing a. I n fact, if ‘Y, value I ~ ( c x ) depends a:’ E 7 E Gou, then there exists a number c E C, such that a(g) = cCr’(g), and hence ~ ( a= ) cs)(01’), that is, I q(01)I= 1 q(01’)I.Thus, we may regard I q(’~)las a function on Gou,and we write #(v) = I q(a)/,01 E 7. Alternatively, we may regard I q(a)l as a function on &, that is,

a E 6’,

$(E) = I Ip(a:)l,

where a: is any quasi-character corresponding to the character a. When referring to the functions #, $, we shall also use the term quasi-characteristic function, since they differ from I q~(a)lonly in having different domains of definition. Observe that the quasi-characteristic function and the norm NC(a), introduced in 30, are related by the formula N,(a:)’ = 2(1 - %Ip(a:)),

I n fact, if

01

I

E

01

E

G’.

(3.2.23)

Gu, then

1 - & ? ) I= 21

-

2%01(g)

+1

LY(g)12

=

2(1 - %‘Y(g)),

hence,

and using ~ ( 1 = ) 1, we obtain (3.2.23). I n case ill and C are isomorphic, we have

hence, from (3.2.23), we obtain

Lemma 3.2.17. Let p be a finite measure space which is quasiinvariant under a transformation group 8 . Let G,u be given the topology induced by the metric N ( & - l ) , and let 2 be any topology which makes 6% a topological group. Then, the inverse S-l of the mapping defined by (3.2.3) is continuous if and only if the quasi-characteristic function q(a) on (&, 2)is continuous at the point & = 1.

111. GROUPS

162

WITH QUASI-INVARIANT MEASURES

+

PROOF. Suppose that is continuous at the point 1. Then, given any positive number E, there is a %-neighborhood V of 1 in 6% such that 2(1 - +(a)) < 2 whenever d E V [since +(1) = 11. Hence, if d E V , it then follows by (3.2.24) that N ( 7 ) < E, where 7 = S-ld. Therefore, S-lVC Z(1; G , E ) .

(3.2.25)

Thus, S-l is continuous at the point 1. Since S-l is an algebraic isomorphism between the topological groups 6% and GOu, it follows that S-l is continuous throughout 6 ~ . Conversely, assume that the inverse mapping S-l is continuous at the point 1 . Then, given any positive number E, there is a neighborhood V of 1 such that (3.2.25)holds. Hence, if E V ,then 0 1 - +(d) < e2/2. Thus, is continuous at 1. ] We also observe that the continuity of @(a)at the point a = 1 implies continuity throughout &. I n fact, it is easily shown [see the proof of (3.3.3)] that

<

+

I d4 - Cp(~0)l < 2(1 for any a ,

a. E

%+a)

Gu. Hence, for any d , do E (@, and any c E C, we have

whence one easily deduces that

I @(a)- @(Go)l

< 2(1 - +(ddil)).

From this inequality, it is obvious that it is continuous at 1.

+ is continuous at do provided

Definition 3.2.6. Let G be a group, let (G, 23, p ) be a finite measure space, let sj be a subgroup of Gs, and let 2 be a topology on b, stronger than the weak topology, such that (9, 2 ) is a topological group. Then, p is said to be continuous with respect to the topology 2, provided that, given any two positive numbers E , 7, there exists a %-neighborhood V of the identity 1 in sj such that Pug

for all 4,

E

V.

I I 1 - P(g)l > 7)) <

6

(3.2.26)

3.2. Characters and Quasi-Characters

163

Lemma 3.2.18, Under the hypotheses of Definition 3.2.6, p. is continuous with respect to 2 if and only if the characteristic function of (G, 23,p ) on (sj, 2) (see Definition 3.2.4) is continuous. PROOF. Suppose that the characteristic function rp is continuous. Then, given any two positive numbers E and 7, there is a Z-neighborhood V of the identity 1 in fi, such that 1

-

%&3)

<

(3.2.27)

E

for all /3 E V . Also, for any /3 E 3j,

Combining (3.2.27) and (3.2.28),we see that (3.2.26) holds for all /3 E V. Conversely, suppose that p is continuous with respect to 2. Then, given any E > 0, we choose a %-neighborhood V of 1, such that Pug

II1

for all /3 E V . Hence, when /3

-

E

B ( d 2 d\/.>)
V , we have [taking p(G) = 11

+ +(d$<

E.

This shows that rp is continuous at the identity of 5, whence it follows easily that rp is continuous everywhere in sj. ]

Lemma 3.2.19. Under the hypotheses of Definition 3.2.6, if 2 satisfies the first axiom of countability, then p is continuous with respect to 2. PROOF. Let {/3,} be a sequence in 9 which converges in the topology 2 to a point /3 E sj. Since 2 is stronger than the weak topology, we have limn+m/3,(g) = /3(g) for every g E G. By the Lebesgue dominated convergence theorem, it follows that rp(/3,) + rp(P). Since 2 satisfies the first axiom of countability, this shows that rp is a continuous function on (sj,2).Therefore, by Lemma 3.2.18, p is continuous with respect to 2. ]

111.

164

GROUPS WITH QUASI-INVARIANT MEASURES

93.3. Integral Representation of Positive Definite Functions on Groups

lo Positive Definite Functions on Groups

Definition 3.3.1, Let G be a group, and let f ( g ) , g E G, be a function on G having the following property: for any natural number n, any n complex numbers x, ,..., z, , and any n elements g, ,...,g, in G, (3.3.1)

Then, f is said to be a positive deJinite function on G. I n particular, choosing n = 2, g, = e (the identity of G) and g, (an arbitrary element of G), it follows easily from (3.3.1) that

=g

(3.3.2)

for every g E G. Again, in (3.3.1), choosing n = 3, g, = e, g, = g, g, = g', ~ _ _ x1 = ( f ( g ' ) - f ( g ) ) l f ( e ) , xz = 1, x3 = - 1 , and using (3.3.2), we obtain36 (3.3.3)

From (3.3.3), we immediately deduce the following fact: if f is a positive definite function on a topological group G, then f is continuous on G if and only iff is continuous at the identity of G. I t is easy to check that any character of a group is positive definite. As we shall see below, characters may be regarded as the most basic of the positive definite functions. We note, for future reference, that if H is an inner product space, and U :g -+ U ( g ) (g E G) is a unitary representation of G in H , then, for any fixed ( E H , the function ( U ( W , 0,

hEG

is positive definite on G. This fact follows from Definition 3.3.1 by a straightforward calculation.

Definition 3.3.2. Let 0 be a topological group, 0"the dual of 0. Let r be a set such that 8*C r, and let 23 be a cr-algebra consisting of certain subsets of I'. Also, suppose that, to each h E 8, there correaa

Iff(e)

=

0,then, by (3.3.2),f is identically zero, so that (3.3.3) is trivial.

3.3, Integral Representation of Positive Definite Functions

165

sponds a %-measurable function h(y), y E r, satisfying the following conditions: (i) I h ( y ) [= 1 ; (ii) the family of functions (5 = (h(.) I h E 0} is a determining set on (r,23); (iii) 0: forms a group with respect to multiplication, and the correspondence h -+ h ( * ) is an isomorphism; (iv) when y E 0*,then h(y) is the value of the character y at the point h. If, for every positive definite continuous function f on 6, there exists a unique measure p on (r,23) such that

then we say that (r,23) is a spectral space for 8. In particular, when r is a group consisting of certain characters of 8 (Lea,F C S’), and, for each g E 6, the value of g(.) at the point y is just the value of y ( * ) at the point g [that is, g(y) = y(g)], then we call r a spectral group for 8.I n this case, setting g = e in (3.3.4), we see at once that, for any f,the corresponding measure p is finite. T h e problem of finding so-called integral representations of positive functionals is just the problem of finding appropriate spectral groups. It is not at all difficult to show the existence of spectral groups (see, e.g., Theorem 3.3.4 below); however, our ultimate objective in this direction is to find spectral groups which are as “small’’ as possible. Before pursuing this problem any further, we shall first explain the use of spectral groups in the study of unitary representation^.^^ This will demonstrate the connection between unitary representations of groups and the theory of positive definite functions.

Theorem 3.3.1. Let 0, be a topological group, let (r,23) be a for 6, and let U : h + U(h) be a weakly continuous spectral unitary representation of Q in the Hilbert space H. Then, there is a spectral measure { P ( E ) ,E E %} over H such that

for every h E 6. PROOF. Take any

6 E H , and form the function

ST Unitary representations of groups are defined and discussed briefly in Appendix 11, 511.3. Translator’s note: In the original, it was specified that B is the o-algebra of all weak Bore1 sets in T.However, the proof seems to be valid without this restriction.

111.

166

GROUPS WITH QUASI-INVARIANT MEASURES

vc is a positive definite continuous function on 8. Hence, by the definition of a spectral group, there is a unique finite measure pp on such that

(r,23) (3.3.6)

in particular, setting h

=

I , we get

Now, for each pair of vectors

5, 7 E H

and every set E

E

8, we define

Then, from ( 3 . 3 9 , (3.3.6), and (3.3.7), we calculate that (3.3.8)

Since { h ( * I) h E 8 1 is a determining set of functions on (r,b), the complex measure p ( E ; 5, 7) is uniquely determined by relation (3.3.8). Therefore, since the left-hand side of (3.3.8) is a bilinear functional on H x H , it follows that p ( E ; 5, 7) is also bilinear in 6, 7. Similarly, it follows easily from (3.3.8) that p ( E ; 5, 7) is Hermitian in 5, 7, and that 0 p ( E ; 6, E ) = p e ( E ) 11 5 [I2. Consequently, p ( E ; E, 7) is a positive definite continuous bilinear functional on H x H , and so, by a well-known theorem, there is a bounded positive self-adjoint operator P(E) on H such that

<

<

By (3.3.7), we know that p ( E ; 5, r),E E 23, is a countably additive set function on 23; obviously, P(0) = 0, and since

we have P(r)= I. Thus, to show that P is a spectral measure, it only remains to prove that each P ( E ) is a projection operator. Fix any element h, E 8, and consider the following complex measure on (r,23): (3.3.10)

3.3. Integral Representation of Positive DeJinite Functiom

167

for every h E 8, it follows by the uniqueness of the complex measure p in (3.3.7) that E E 23. (3.3.1 1) P , ( E 5, r l ) = p ( E ; U(hl)5, 77), Using (3.3.9), (3.3.10), and (3.3.11), we obtain

where C, denotes the characteristic function of the set E. Again, since {h,(.) I h, E 8}is a determining set of functions on 23), it follows from (3.3.12) that, for any bounded 23-measurable function q on r,

(r,

In particular, choosing q

=

C, , we get

Thus, P(E)* P ( E ) = P(E), and since P ( E ) is self-adjoint, we conclude that P ( E ) is a projection operator. ] From Theorem 3.3.1 and Corollary 2.4.23,39 we immediately deduce the following result.

Corollary 3.3.2. Under the hypotheses of Theorem 3.3.1, let 2f be the commutative weakly closed operator algebra over H generated by the family of operators { U(h) I h E S}.If 2f has uniform multiplicity k, 38 Translator’s note: Recall that the proof of Corollary 2.4.23 depended upon Theorem 2.4.22, and the proof of the latter, in the general case of nonseparable H , was open to question.

168

111.

GROUPS WITH QUASI-INVARIANT MEASURES

(r,

then there is a measure p on 23) and a unitary mapping Q from H onto Qkz(I', 23, p ) such that the relation (QU(h1Q-l) V ( Y ) = WY)v(Y),

YE

r

holds for every h E 8 and every rp E Pkz(I', 23, p). 20 Positive Definite Functions on Discrete Groups T h e following discussion will be restricted to commutative groups. First, we merely consider abstract groups, that is, we take all the subsets of a group G as open sets, thereby obtaining a topology on G, known as the discrete topology. All functions on G are then continuous. Let G be a commutative group, G' the totality of characters of G. We construct the group ring R(G) of G , as follows. Let rp be a complexvalued function on G, such that the set {g I rp(g) # 0} is finite or countable, and (3.3.13) I1 4pll = I rp(g)l < 03.

1

gEG

T h e totality R ( G )of such functions rp clearly forms a Banach space with respect to the ordinary linear operations and the norm (1 y (1 defined in (3.3.13). If rp, $ E R(G), the function (g'

* 3>k1=

c dg1>

+kZ>

8*82=8

is called the convolution of rp and 4, Taking convolution as a multiplication operation, R(G) becomes a commutative Banach algebra. Also, if rp E R( G), we define

-

V*(d

=

Cp(g-9.

I t is easily checked that rp* E R(G),and that the correspondence rp -+ y* is an involution in R(G). Obviously, the norm (3.3.13) is symmetric with respect to this involution. We now consider the space rXn of symmetric multiplicative linear functionals on R(G). Take any 01 E G', and define a linear functional F, on R(G), as follows: (3.3.14) F,(V) = 4 p k ) 49.

c

It is easily verified that F, ernJz. Conversely, let F E %R; for each h E G, form the function g'h(g)

=

1 10

when g = h,

when g # h.

3.3. Integral Representation of Positive DeJinite Functions Let

44

169

= F(?Jh).

Since qe (where e denotes the identity of G ) is the unit element of R(G), we have a(e) = F ( s ~ = , ) 1 Also, since yh,,* F ~ = , yhIh,, we have ~

4

1

4 = F(?Jhlh,) = F(?J*,)F(?JhJ = 4 1 )

4hd

<

<

Moreover, I a(h)l 11 F 11 11 vh11 = 1 , hence I ol(h)-l I = I a(h-l)I 1, and therefore I ol(h)l = 1. This shows that E G'. Finally, using Lemma 2.2.1, it is easily seen that F = F, . Thus, we have proved the following result.

Lemma 33.3. Let G be a commutative group, G' the character group of G, and 1111 the space of symmetric multiplicative linear functionals on the group ring R(G). Then, the correspondence T : a: --t F, [where F, is defined by (3.3.14)] is a one-to-one mapping from G' onto %)IJZ. Let 8' be the smallest u-algebra containing all Borel cylinders in G', and let 23 be the smallest a-algebra containing all Borel cylinders in %)I. Then, the correspondence T maps 8' onto 23, that is, 23 = { T ( E )I E E S'}. I n fact, let E be any Borel cylinder in G', say, E

I

= { a (&?l),..., 4 g n ) ) E

4.

Then, TE

= (F I (~(?J&.,

F(?J,"))E B).

Therefore, 23 3 { T ( E )I E E S'} = 23'. Conversely, given any v E R(G), there is a sequence { g k }C G such that v = limm+m y(gk)v g k . Hence, F(v) = p(gk)( T-lF)(gk). This shows that the function F - t F ( v ) is B'-measurable, whence it follows that 23 C 23'. Thus, 23 = d',as asserted.

xy=l

z.,"=l

Theorem 3.3.4. Let G be a group, and letf be any positive definite function on G, with f ( e ) = 1 . Then, there exists a unique probability measure P' on (G', 8') such that (3.3.1 5 )

holds for all g E G. PROOF. Using the given positive definite function f, we construct a linear functional F on R(G),as follows: Fb)=

cf(d

gsc

?J(g),

?J E R(G).

111. GROUPS

170

WITH QUASI-INVARIANT MEASURES

<

Since I f ( g ) l 1, the series C P E G f ( gy )( g ) converges for every y E R(G). Furthermore, by the positive definiteness off, we have

hence F is a positive functional on R(G). Since R(G) is a symmetric commutative Banach algebra with unit, it follows (see the proof of Theorem 2.2.4) that there exists a unique probability measure p on (!Ill, 23) such that (3.3.16) Now, by Lemma 3.3.3 and the remark following it, there is a one-to-one mapping T from G' onto !Illwhich induces a mapping from 3' onto 123. Accordingly, we may define a set function P' on 8' by letting

P'(E) = p(TE),

E E g',

and it is clear that P' is a probability measure on (GI, from (3.3.16) and (3.3.17), we obtain F(a) =

I

G

,FAT) dP'(4.

(3.3.17)

v),Moreover, (3.3.18)

In particular, choosingp, = y h in (3.3.14) and (3.3.18), we obtain (3.3.15). The uniqueness of P' may be deduced from the uniqueness of p. J Theorem 3.3.4 may also be stated in the following form.

Theorem 3.3.4'. Let G be a commutative topological group, and let G' be the algebraic dual of G. Then G' is a spectral group for G. By following a procedure analogous to the proof of Theorem 3.3.4, one can also show that the smallest spectral group for a locally compact group G is just the dual G*. However, such a proof would by itself be fairly lengthy, and since the method requires the use of either the group ring or invariant measure, it does not lend itself to further generalization. Therefore, we shall not use such an approach; instead, we shall continue our researches by measure-theoretic methods. Proceeding from Theorem 3.3.4, we see that, to find smaller spectral groups, it suffices to prove that the measure P' in (3.3.15) can be concentrated on certain subgroups of G'. For this purpose, we first require some preparatory work.

3.3. Integral Representation of Positive Definite Functions

171

3 O Positive Definite Functions on Groups with Measures We now introduce a type of topological space which will be useful in the subsequent discussion. Let G be a group, and for each g E G, let C, be a copy of the unit circle in the complex plane, with the topology induced by the usual metric

Let CG be the topological product of all the spaces C, , g E G; the elements of Cc may be regarded as complex-valued functions f on G, satisfying the condition If(g)l = 1. Take any finite set of elements g, ,..., g, E G, and any Borel set B in real n-dimensional space. T h e set

in CG is called the Borel cylinder with base B corresponding to the elements g, ,...,g, . Let 2 be the smallest o-algebra containing all Borel cylinders in CG. Observe that G' C Cc, that the topology on G' induced by CG is just the weak topology, and that the o-algebra in G" induced by 2 is just 8'. Henceforth, we shall further assume that (G, 23,p ) is a complete 1ocaIizable measure space. Let {E, , o E Z} be a partition40 of G (see Definition 1.1.8), and suppose that, for each o E Z, there exists a sequence of mappings &' : E, + E, , n = 1,2,..., satisfying the following conditions: (i) T h e values of &'(g) form an (at most) countable set {gg; , k = 1, 2 ,...}. (ii) For each k, the set {g 1 &'(g) = gg)n} belongs to 8. (iii) If rn >, n, then &'(gg)n) = gt)n , k = 1, 2,... . We let 8%denote the o-algebra generated by the totality of Borel cylinders in GB (see $3.2).

Lemma 3.3.5. Let G be a commutative group, and let f be a 8-measurable positive definite function on G, with f(e) = 1. Also, 4n Translator's note: T h e intended meaning of the term partition, as used in this subsection, is, presumably, as follows. Let E, E %, u E 2, with p ( E J < CL, let b, ,pc denote the restrictions of b, p, respectively, to E, , and let ( G , b', p') denote the direct sum of all the measure spaces ( E , , 8, , pn). Then = B and ii' = p (see Definitions 1.1.1 and 1.1.2).

172

111. GROUPS

suppose that, for every

u,

WITH QUASI-INVARIANT MEASURES

the sequence of measurableQ1functions

on E , converges to 1 almost everywhere (relative to p). Then, there exists a unique probability measure'P on (G', 3') such that (3.3.19)

€or all g E G. PROOF. We divide the proof into several steps, as follows. (1) By Theorem 3.3.4, there exists a probability measure space (G', g', P') such that (3.3.15) holds. We first extend P' to a measure on (Cc, 2).If A E 52, then, since A n G' E S', we may define Q(A) = P'(A n G ) .

(3.3.20)

Clearly, Q is a probability measure on (C", 2). From (3.3.15) and (3.3.20), we obtain

moreover, for any two elements g,g' E G, we have

(2) Every

01

E

G' is a character of G, hence, using (3.3.21), we get

= JcG I

4 3 - 4cP',"'(g))l24x4.

(3.3.22)

By hypothesis, the sequence of functions 1 - %f(yt)(g)-lg)converges 41

Translator's note: Apparently, the author has tacitly assumed that the group operation 8)to (G, 8).

(g, h) -+g-'h is a measurable mapping from (G x G, 8 x

3.3. Integral Representation of Positive Dejinite Functions

173

to zero almost everywhere (relative to the measure p), and is uniformly bounded [see (3.3.2)], therefore, using (3.3.22), we obtain

Consider the product measure space ( E , x CG,23, x 2,p x Q) (here, 23, denotes the restriction of 23 to E,). By virtue of the conditions (i) and (ii) satisfied by &)(g), the quantity a(&)(g)), regarded as a function of (g,a ) E E, x Cc, is measurable with respect to 23, x 2. Furthermore, using Fubini's theorem and (3.3.23), we obtain

Consequently, there is a subsequence { a ( ~ Z ( g ) )and } ~ ~a measurable function Vu)(g,a ) on (E,, x Cc, 8, x 2) such that lim a(pk'(g)) = U'"'(g, a ) n+ m

(3.3.24)

holds for almost all (g, a ) in E, x Cc (relative to the measure p x Q). It follows that there is a set F, E 8,, , with p(F,) = 0, such that, for each g E E, - Fa, U ( , ) ( g ,a ) is a measurable function on (Cc,2),and

I ;2 4?J,lpn'(g))=

U'"'(g,

41) = 1.

From (3.3.23) and (3.3.24), we see that

4 2 Notice that the indices {&,} actually depend upon u; however, to avoid unduly cumbersome notation, this has not been explicitly indicated.

111. GROUPS

174

WITH QUASI-INVARIANT MEASURES

Therefore, we can, if necessary, enlarge the null set F, that the equality =1 &(& I u'a'(g, 4 =

, and so ensure

holds for all g E E,, - F, . Hence, if g E E, - F,, , then &({a I

=;

.(&)(g)) = .(g)>>= 1.

(3.3.25)

(3) Next, we define four functions V,(g;a), Y = 1,2, 3 , 4 , on G x Cc, as follows. Given any (g, a ) E G x CG,let a E ,Z be the index such that g E E, , and let Vdg; 4 = lim %+k)(g)), fl+m

Vz(g; 4 = V3(K

%a(&)(gN,

4 = limm J+k)(g)), fl+

Since, for each a E 2, the restriction of V , to E, x Cc is measurable with respect to b, x 2,it follows that V , is a measurable function on (G x CG,23 x 2),v = 1, 2, 3 , 4 . Let H be the set of all a in CGsuch that the inequalities (3.3.26) hold for almost all g E G. We now proceed to prove that the outer measure Q * ( H ) is 1. 03 Let r = Urn=,I', be a union of Bore1 cylinders, with H C r;we may assume that r, is a cylinder corresponding to the elements g, ,...,g, E G, n = 1 , 2,... . For each n, there is an index a, € 2 such that g, E Earn. Let g, ,..., g, ,... denote the subsequence of {g,) consisting of all elements satisfying the condition43 g, E E,, - F., . Consider the set (1 =

n

I m+m lim

a(qy(gn,)) .(gfl,)}.

Using (3.3.25), we see that Q(A) = 1. We shall show that A C T . Given any

=

a O ~ ( ldefine ,

a function

43 The subsequence { g,,} may be empty, but in that case the intersection defining A is vacuous, that is, A = Cc, so that Q ( A ) = 1 is still valid.

3.3. Integral Representation of Positive Dejinite Functions 01

E

175

CG as follows: for every g, , let dg,)

= .o(gJ,

(3.3.27)

for every value g s m of cpe)(g),let 4g:$m)

=

%(g:$m),

(3.3.28)

and, for everyg E Eun- ( { g z m }u {g"}),let a ( g ) be any complex number of modulus 1 which satisfies the conditions

it is easily seen that such a complex number does exist. Finally, for every m EG Eun, let .(g) = 1. (3.3.31)

g

We assert that a E H . T o prove this, it suffices to show that (3.3.26) holds for this 01 and any g E G - (jz=1Fun, Ifg E G Eun, then, by (3.3.31),we have V v ( g ;a ) = 1 = %a(g), v = 1, 2, and V,(g, a ) = 0 = @ ( g ) , v = 3 , 4 , hence (3.3.26) holds. Moreover, we see from (3.3.28) and the definition of V,(g, a ) that m V,(g;a ) = V , ( g ;ao) for all g E Eu, . Hence, it follows by (3.3.29) and (3.3.30) that (3.3.26) holds if g E Eu, - ({g;tm} u {g"}). Furthermore, since the &'(g) satisfy condition (iii), we have cpg)(gtLm)= g g & whenever 12 m, hence

u:=l

+% !

(0

)

'y.(P)Bz"

(u )

(gk,;J

=

4k;;i).

Therefore, (3.3.26) also holds for g = g em . Finally, since follows from (3.3.27) and (3.3.28) that .(En,)

=

a. E A ,

it

=; .(P)y(snk)).

Hence, (3.3.26) also holds for g = g,, . Thus, we have shown that (3.3.26) holds for all g E Fun , so that a E H , as asserted. Since H C T , we must have U E rkfor some k. But r k is a Bore1 cylinder corresponding to g, ,..., g k , hence, it follows by (3.3.27) that 0 1 ~E r k . This shows that /1 C r. Since Q(/1) = 1, this implies that Q(r)= 1. We conclude that the outer measure of H is 1 .

(Jz=l

111.

176

(4) Let -O0 as follows:

=(A

GROUPS WITH QUASI-INVARIANT MEASURES

n H I A E 2), and define a set function Po on Go Qo(An H )

= Q(A),

A

E

2.

Since Q * ( H ) = I , it follows from the discussion in $1.1, lo, that Qo is a probability measure on ( H , 2,). Now, the functions V,(g, a ) on G x CG are measurable with respect to B x 2,hence their restrictions to G x H are measurable with respect to 23 x 2, . Let F = UOEzF0, If g C F , then, by (3.3.25),

Hence, if g E G - F, then

Consequently, there is a null set H , C H such that

for every a E H - Ho . But if a E H , then (3.3.26) holds for almost all g. Therefore, if a E H - Ho , then Vdg; 4 V3(g;

=

=

V,(g;

4 = J 4 g ) = V&;

4 4

(3.3.32)

holds for almost all g. Since the functions V v ( g ;a) are measurable in g for every fixed a , it follows from (3.3.32) and the completeness of the measure p that a(g) is a measurable function of g for every

H - Ho(Qo(Ho)= 0). (5) Let Hl = H - Ho ; then, H , n G‘ C GB. Since &(Ill)= 1, the outer measure Q*(Hl)= 1. By (3.3.20),we know that4s

a

E

Q*(Hl) = P’*(H, n G)l 44 Translator’s note: In, for example, the Halmos [l] version of Fubini’s theorem, it is assumed that the measures concerned are o-finite. 46 Translator’s note: The inequality Q * ( H l ) > P‘*(H, n G’) is obvious, but the reverse inequality is not.

3.3. Integral Representation of Positive DeJinite Functions

177

hence P’*(H, n G‘) = I , and therefore P’*(GB) = 1. For every A E g‘, define P B ( An GB) = P’(A). (3.3.33) Since gB = { A n GB I A E f!} and P’*(GB) = 1 , it follows again from $1.1, lo, that P B is a probability measure on (GB,gB),moreover, it is easily seen that

holds for every g E G. This, combined with (3.3.15), shows that (3.3.19) holds for all g E G. (6) It only remains to show the uniqueness of the measure PB. Suppose that PIBis another measure on (GB,gB) satisfying (3.3.19). Define a probability measure P,’ on (G’, b’),as follows: P,’(A)

=

PIB(A n GB),

A

E

3’.

(3.3.34)

It then follows from (3.3.19) and (3.3.34) that PI‘ satisfies condition (3.3.15). But, by Theorem 3.3.4, this measure is unique, that is, P,’ = P’. Hence, from (3.3.33) and (3.3.34), we conclude that P I B = PB- 1 We point out that, in Theorem 3.3.5, the condition that {%f(q:’(g)-Ig)) converges to 1 almost everywhere (relative to p ) cannot be dispensed with. This may be seen from the following example. Example 3.3.1. Let R denote the additive group of all real numbers, b the totality of Lebesgue measurable subsets of R, and p any probability measure equivalent to Lebesgue measure. It was shown in Example 3.2.4 that R B is just the totality of functions of the form at(.)

= eit”,

x E R,

hence the right-hand side of (3.3.19), in this case, takes the form (3.3.35)

where P is a probability measure on ( R , 23). It is easily seen that, if R is given the ordinary Euclidean topology, then (3.3.35) must be a continuous function on R . Consider the function f(x) =

1,

lo,

x =

x #

0, 0.

111.

178

GROUPS WITH QUASI-INVARIANT MEASURES

Obviously, f is a %-measurable positive definite function on R. But f is not continuous, and hence cannot be expressed in the form (3.3.19).

Lemma 3.3.6, Let G be a topological group satisfying the first axiom of countability, let 23 be a o-algebra in G which contains all the open subsets of G, and let (G, 9, p ) be a regular locally finite measure space. Then, there exists a partition {E, I U E C} of G, and a corresponding family of mappings {vk) I o E Z, n = 1, 2, 3, ...}, satisfying the conditions (i), (ii), and (iii) (preceding Lemma 3.3.5) such that, for every D E C, lim &)(g) n+ m

=g

(3.3.36)

holds for almost all g E E, . PROOF. Let U be a family of pairwise disjoint subsets E of G, satisfying the following conditions: E : E E 23, 0 < p ( E ) < 03, and, for any open set V , either E n V is empty or p( V n E ) > 0. Let 8 denote the totality of such families U, and let ?be j partially ordered by inclusion, that is, U, < U2 means that U, C U2 . Using Zorn’s lemma, it is easily seen that 8 has a maximal element,46which we denote by {E, 1 o E C}. We assert that any compact set K intersects at most countably many E, . I n fact, it follows easily from the local finiteness of p that there exists an open set V , having finite measure, such that V 3 K. Therefore, since

there are at most countably many E, such that p ( E , n V ) # 0, that is, at most countably many E, such that E, n V is nonempty. Hence, a fortiori, there are at most countably many E, such that E, n K is nonempty. Let R = E, . We shall prove that G - R is a null set. T o do this, it suffices to prove that, if K is an arbitrary totally bounded set, then K n (G - R ) is a null sets4’ Since K intersects at most countably many E, , we have K r\ R E 23, hence K n (G - R)E 9. Suppose that

u,

48 Translator’s note: It is not immediately clear that 5 is nonempty, unless one makes some additional assumption, for example, that every nonempty open set has positive measure. 47 Translator’s note: The justification for this assertion is obscure. The crucial point is proving that R E 8,and it is difficult to see how this follows from the argument given here. Of course, if p is o-finite, then there are at most countably many sets E, In that case, R obviously belongs to 23, and one can simply consider an arbitrary compact set K C G - R; the proof can then be completed in a manner similar to that indicated in the text.

.

3.3. Integral Representation of Positive Dejinite Functions

179

the measure of K n (G - R ) were not zero; then K n (G - R ) must contain a totally bounded set E having positive measure. Obviously, the intersection E n E, is empty for every cr E 2'. Let U , 3 U , 3 3 U , 3 be a neighborhood basis at the identity e of G; we may assume that p(U,) < cx). Since E is totally bounded, there exists, for any n, a finite set of points g p ) ,...,ggn) in E such that u 2 1 g ! n ) U ,3 E. It may happen that E n (gy'"'U,) is a null set for certain indices n, v ; let F be the union of all (at most countably many) such sets, and let E, = E - F. Then E, E B, Eo n E, is empty for every u E Z, and 0 < p ( E o ) = p ( E ) < co. We assert that, if V is any open set which intersects E, , then p ( E , n V ) > 0. I n fact, suppose g E E, n V ; since g is an interior point of V , there is an n such that g U , C V , and, for any m, since g E E, , there is a v such that g E g ~ " ) U , , that is, g-lg;"' E U;'. Hence, if m is sufficiently large, then U;' UrnC U , , and we have 1..

gj""'um CgU&lUmc v. But g E ~ ; " ) U,,g E E, , hence E n (g:"'U,) is not a null set [since E, was obtained from E by removing all those sets E n gSm'Um such that p ( E n gjm)Um)= 01. Since E, n V 3 E, n (gbm)Um),this implies that E, n V is not a null set. Thus, {E, , u E Z} u (E,} E 3.This contradicts the maximality of {EDI D E Z}. Therefore, we conclude that G - R is a null set. We may assume that G = R (for otherwise, we could incorporate G - H in one of the sets E,). Thus, {E, I u E Z} is a partition48 of G. Next, we proceed to construct the mappings {+}I.: Clearly, we may assume that U;lU, C Urn-,, n = 2, 3 ,... . Now, for each n, n = 1, 2,..., and each U E Z ,one can choose a finite or countable set of elements E E, , v = 1 , 2,..., such that p

(E,

-

u "

hFACT,) = 0.

(3.3.37)

I n fact, by regularity, E, is the union of countably many compact sets and a null set, and each of these compact sets can be covered by finitely many sets of the form hU, , where h E E, , hence, there does exist a finite or countable set (hi:;} C B such that (3.3.7) holds. Since h,!:AU, E 23, all the sets EbYL

**

=

(h?kUn

u

V-1

-

U n ) n E,

k=1

Translator's note: Again, the proof of this assertion is not obvious (unless p is o-finite).

111. GROUPS

180

WITH QUASI-INVARIANT MEASURES

also belong to 23,and if u # u', then E,!:Aand E$',L are disjoint. Moreover, if we let EkL = E, - E,!::,

u v

then, by (3.3.37), we get p(EkL) = 0.

(3.3.38)

We shall define the mappings {c#} by induction. First, define &) as follows. For each nonempty set E::: , choose any element g::! E E,!:] , and let q$)(g) = g : ] for all g E E::! . Clearly, &) satisfies conditions (i) and (ii). Now, assume that mappings &), ..., ypl have already been defined, satisfying conditions (i), (ii) and (iii) (for 1 n m I - I). Given any u, we use the letter s to denote all the indices such that let #(g) = T'&(g). If the set gEi-l E E,!:; . If g E E$' n

< < <

(u,

8

is nonempty, we arbitrarily choose a point go E F,'Pj' , and let yy)(g) = go for all g E FY(Pj . This completes the inductive definition of the mappings {&'}, and one can easily verify that they satisfy conditions (i), (ii), and (iii). Notice also that these mappings have the following property:

Consequently, if g 'C Ei:A , then03 g-l&)(g) E Ei'pA-lE:$ C U;'Un C Un-l Therefore, on the set E, - (Jn=l EEL, we have

But, by (3.3.38), 4O

.

Uz=l EgL is a null set in E, . ]

Positive Definite Continuous Functions on Commutative Topological Groups

Theorem 3.3.7. Let (G, 2) be a commutative topological group, and let B be a u-algebra of sets in G, such that b contains all the closed subsets of G. Suppose there exists a regular locally finite measure p on (G, 23). Then, given any positive definite continuous function f on G, withf(e) = I, there exists a unique probability measure PBon (G', 3B), such that (3.3.39) f(g) = JGB 4 g ) dP'B(4 for all g E G.

3.3. Integral Representation of Positive Dejinite Functions

181

PROOF. Let f be a positive definite continuous function on G. Choose any neighborhood U of the identity e in G such that p( U ) < 00. Construct a sequence of neighborhoods U, of e as follows: (3.3.40)

Then, taking {U,} as a neighborhood basis at e, we obtain a topology 2’C 2,such that (G, 2’)is a commutative topological group49 satisfying the first axiom of countability. Since 23 contains all the closed sets of (G, 2),we have 2 C 23, hence 2’C 23. Moreover, p ( U n ) < co. It is easily seen that the measure space (G, 23, p ) is also locally finite and regular50 with respect to the topology 2’. Using Lemma 3.3.6, we obtain a partition {En l a EL’} of G and a collection of mappings {&) 1 a E Z, n = 1, 2, ...}, satisfying conditions (i), (ii), and (iii), such that (3.3.6) holds (for almost all g E E,) relative to the topology 2’.By virtue of (3.3.40), the function %f is continuous at e relative to 2’.Therefore, by (3.3.36), we deduce that lim %(j(&)(g)-’g))

nim

=

1

for almost all g E E, . T h e conclusion of the theorem then follows by an application of Lemma 3.3.5. ] We now proceed to use Theorem 3.3.7, together with the properties of quasi-invariant measures, to establish a representation theorem for positive definite continuous functions on groups with quasi-invariant measures.

Theorem 3.3.8. Let G be a commutative topological group, let 6 be a subgroup of G, and let 2 be a topology on 8, stronger than the relative topology induced by G, such that (6,2)is a topological group of the second category. Let 8 be a a-algebra of sets in G, containing all closed subsets of G. Suppose that there exists a locally finite, regular and strongly 6-quasi-invariant measure p on (G, 23). Let 8* denote the dual of ( 6 , 2),and g* the totality of weak Bore1 sets in 6 * . Then, for each positive definite continuous function f on G, with f ( e ) = 1, there exists a unique probability measure P* on ( 6 * , g*) such that

48 Translator’s note: There is a serious error here. For example, iff is identically equal to 1 , then U,, = U for all n, and {U”}cannot be a neighborhood basis for a topological group in the usual sense, unless U = ( e ) . so Tronslutor’s note: The translator finds it difficult to verify these assertions.

182

111.

GROUPS WITH QUASI-INVARIANT MEASURES

moreover, P* is continuous with respect to the topology induced by G on 8. PROOF. According to Corollary 3.2.4, if a E GB, and if a' denotes the restriction of a to 8,then a' E 8*.Thus, we have a homomorphic mapping from GB into 8*:

T:

cz -+ a'.

E g*, then T-lA = {a I T a E A } belongs to 5". I n fact, let 5 ' denote the totality of sets A in 5" such that T-lA E ijB;clearly, 5' is a o-algebra. On the other hand, it is easily seen that, if A is any Borel cylinder in Q*, then T-lA E gB. Therefore, since 5" is the a-algebra generated by the totality of Borel cylinders in 8*,it follows that 5*C St. Now, let f be any positive definite continuous function on G, with f ( e ) = 1. By Theorem 3.3.7, there is a unique probability measure PB on (GB, )'5 such that f satisfies (3.3.39). We define a measure P* on &*, as follows: P*(A) = P(T-1A).

If A

It is easily seen that (Q*, 5*,P*) is a probability measure space, furthermore, if g E 6, then it follows from (3.3.39) that

Next, we prove the uniqueness of P*.Suppose that P** is another probability measure on (8*, S*), satisfying f(g) =

I,,

4 g ) dP**(a).

(3.3.42)

Let 8' be the algebraic dual of 8,and let 5'be the a-algebra generated by the totality of Borel cylinders in 8'.Then, 5*is precisely the totality of sets of the form A n 8*,where A E $j', hence we may define set functions P' and P" on g', as follows: P'(A) = P*(A n 8*), P"(A) = P**(A n 8*).

Clearly, P' and P" are probability measures on (S', 5').Also, we have the relations f(g) =

j,, 4 9 dP'(4,

g E 8,

3.3. Integral Representation of Positive DeJnite Functions

183

which follow from (3.3.41) and (3.3.42), respectively. Therefore, since the restriction f I Q is a positive definite function on the commutative group 6, it follows by the uniqueness property in Theorem 3.3.4 that

and hence P* = P**.Finally, the continuity of the measure P* may be deduced by using Lemma 3.2.18. ] Note that Theorem 3.3.8 only provides an expression for the values off on the subgroup 8. Of course, if Q happens to be dense in G, then, by continuity, the values off on 8 completely determine the values off on G.

Example 3.3.2. Let G be a locally compact commutative topological group, and let G* be the dual of G. Then, given any positive definite continuous function f on G, there is a unique finite measure P* on (G*, 5*) such that

In fact, we need only apply Theorem 3.3.8 to the Haar measure space51 1;2 = (G, 8,p), with 8 = G; the local finiteness of p follows at once from the local compactness of G. Retaining the hypotheses and notation of Example 3.3.2, we consider any function f E L1(52),and form a corresponding function [ on G*, as follows:

{ is called the L1-Fourier transform of l. We have the following result: if

6 €L1(Q),

[ ( a ) = 0,

then

5 = 0.

(3.3.45)

In fact, let 71 be any continuous function on G having a compact support. Consider the function (3.3.46) 61 TransZator's note: This must be understood as meaning the extended Haar measure space, constructed from the ordinary Haar measure space (in the sense of Halmos [I]) by the procedure described in Definitions 1.1.1 and 1.1.2.

111. GROUPS

184

WITH QUASI-INVARIANT MEASURES

It is easily shown that f is continuous on G, moreover, for any g, ...,g, E G, and any n complex numbers z1,..., z, , we have f ( g k k l ) zkHl

k,Z=l

=

11 i,

q(ggk)zk

1

2

&(g) 3 O*

Thus, f is a positive definite continuous function. Hence, there is a corresponding probability measure P* on (G*,__ g*) such that (3.3.43) holds, Now, multiply both sides of (3.3.43) by [(hg) e(h), and integrate over 0 x SZ. Since

=

I &)I2,

(3.3.47)

we get

SJm m)444 4 4 )

=

0.

Substituting (3.3.46) into the above relation, interchanging the order of integration, and performing a change of variable, we obtain

1I J I

7 k h ) 5(h) dr.L(h)I24-0)= 0.

Therefore,

1

rl(gh) 6(h)444 = 0

for almost all g E G. Since every open Bore1 set has positive measure, every neighborhood of the identity in G contains a point g satisfying the above equality. Using the fact that 9 vanishes outside a compact set, we deduce by a straightforward approximation argument that

Since the class of continuous functions with compact supports is dense in L1(Q),it follows by an obvious limiting process (using the Lebesgue dominated convergence theorem) that the last relation also holds when is the characteristic function of an arbitrary compact subset of G. Using the regularity of p, we deduce that 8 vanishes almost everywhere. ]

3.4. L2-Fourier Transforms

185

We conclude this section by proving that the points of a locally compact group are distinguished by its continuous characters.

Lemma 3.3.9. Let G be a locally compact commutative topological group. Then, for any go E G, with go # e, there exists a continuous character a. of G such that ao(go) # 1. PROOF. I t is easily seen that there exists a continuous function on G, with a compact support, such that the function

has the property f (go)# f(e) = I. Sincef is a positive definite continuous function on G, there is a corresponding probability measure P* on (G", F*) such that (3.3.43) holds. Now, if we had .(go) = 1 for all a E G, then, from (3.3.43), we would getf(g,) = 1, which is impossible. Thus, there must be an a. E G*, such that ao(g,) # 1. ] 53.4. L2-Fourier Transforms

Throughout this section, we assume that SZ = (G, 23, p ) is a localizable measure space which is quasi-invariant under a commutative group of measurable transformations 6. We shall make use of the unitary representation U of 6, defined in $3.1, 60; in the present case, since 6 is commutative, the group U = { U(h) 1 h E 6) is commutative, and hence %(Q, 6), the smallest weakly closed operator algebra over L2(Q)containing U (denoted briefly by PI), is also commutative (see Lemma 2.4.1). lo

kth Order Cyclic Measures

Definition 3.4.1. Let Q = (G, 23, p) be a nontrivial52 localizable measure space which is quasi-invariant under a commutative group of measurable transformations 6. If the operator algebra %(Q, 6) has uniform multiplicity K, then we say that the measure space Q (or the measure p) is cyclic of order k relative to 6.53 In particular, if %(a, 6) is maximal commutative weakly closed (that is, K = l), then we simply say that 52 (or p ) is cyclic relative to 6. Example 3.4.1. If 6 consists only of the identity element, and the dimension of L2(SZ)is K, then the operator algebra %(Q, 6) = { X I } (where h ranges over all complex numbers) has uniform muhiplicity k, 6* 63

+

That is, p 0. Obviously, k cannot exceed the dimension of L 2 ( 9 ) .

186

111.

GROUPS WITH QUASI-INVARIANT MEASURES

that is, Q is cyclic of order k . Thus, for any cardinal number K, there exist cyclic measures of order k . I n the ensuing discussion, we shall decompose certain measures into sums of mutually singular measures having distinct cyclic orders k . For this purpose, we introduce the following concept.

Definition 3.4.2. Let SZ = (G, 23, p) be a localizable measure space which is quasi-invariant under a commutative transformation group 6. Let B be some group of quasi-characters of f2 relative to 6. If (E constitutes a determining set of functions on 8,then we say that Q is normal with respect to E. I n particular, if Gu is a determining set of functions on Q, we say that Q is normal with respect to 8. Notice that Q can always be made into a normal measure space (with respect to E) by suitably cutting down the size of the a-ring 23, if necessary.54 Example 3.4.2. Let G be a commutative locally compact group. Then, the Haar measure space SZ = (G, 23, p ) is normal with respect to the translation group G. PROOF. We shall prove that G* constitutes a determining set of functions on SZ. Let 23, denote the smallest a-algebra which makes all the functions in G* measurable. Given any set E E 23, it follows from the a-finiteness and regularity of Haar measure that there is a sequence of compact sets Q1 CQ2 c - * * CQZnc -.*

u,"=l

in G such that E QrL is a p-null set. Let R denote the set of all a p i , (xi E G*, such that C aiai(g) is real those linear combinations for all g E G. Consider any compact set F C E ; obviously, we may assume that F C Q1 . By the regularity of p, there is a sequence of open sets {On}, O n 3F , such that p ( 0 , - F ) -+ 0. For each n, since F and Q1- 0, are disjoint closed subsets of the normal space Q1,it follows from Urysohn's lemma that there exists a continuous real-valued function f,(q), q E Ql , 0
<

Translator's note: If 23 is not a o-algebra, then it is not 8,but rather % which must be cut down to size. In any case, one replaces B by the o-algebra generated by the totality of sets of the form (modulo a null set) a-'(B), where a E (5 and B is a Bore1 subset of the complex plane.

187

3.4. L2-Fourier Transforms

prove that R is dense in the totality of real continuous functions on Q1, Consequently, for each n, there is a function cpn E R, such that

Therefore, the sequence {yn} also converges to C, almost everywhere on Q1 , whence it follows that if we define

then F and Q1 n F' differ only by a p-null set. Similarly, for each Qn , there is a set QnfE 23, such that Qn and Qn+l n Qn' differ only by a p-null set. This implies that F and Qn' n F' n E differ only by a p-null set. Again, by the regularity of p, we deduce that, for any set F C E with p ( F ) < co, there exists F" E 23, such that E n F" differs from F only by a p-null set. Since E is 0-finite, it follows easily that the same conclusion applies to any Bore1 set F C E. ] Theorem 3.4.1. Let 1;2 = (G, 8,p ) be a localizable measure space which is quasi-invariant under a transformation group 6, and let Q be a certain group of quasi-characters of 1;2 relative to 6. If 52 is normal with respect to Q, then one can assign a localizable measure space (G, 23, p k ) to each cardinal number55 k, in a unique manner, such that the following conditions are satisfied: (i) the measures pk are mutually singular; (ii) if pk # 0,then (G, 23, pk) is quasi-invariant and cyclic of order k relative to 8 , and also normal with respect to E;(iii) p = & pk . In other words, every normal quasi-invariant localizable measure space can be uniquely decomposed into the sum of a family of mutually singular normal quasi-invariant localizable measure spaces having distinct cyclic orders k. Consequently, various problems concerning general quasi-invariant measures may henceforth be reduced to problems concerning kth order cyclic measures. Before proving Theorem 3.4.1, we introduce a unitary representation I/' of Q in the Hilbert space L2(Q),as follows. If x E Q, let

nz=,

( V ( M ( g ) = 4 g )d g ) ,

g E G , 9 EL2(Q)*

Let 23 denote the totality of operators V(x),x E Q ; we call 23 the multiplication group corresponding to Q. Let C be the smallest weakly closed s6 Translator's note: Or, more appropriately, to each "sufficiently small' cardinal number (to avoid logical difficulties).

188

111.

GROUPS WITH QUASI-INVARIANT MEASURES

operator algebra containing 23; since G? is a determining set, it follows that C is maximal commutative weakly closed. Also, we note the following important “commutation relation” between the members of U and 23: for any x E E, h E 8 , we haveS6 4(h) U(h) V(X) = V(X) U(h).

(3.4.1)

I n fact, (3.4.1) is a straightforward consequence of the definitions of U, V and the properties of x. PROOFOF THEOREM 3.4.1. Since % is a commutative weakly closed operator algebra, it follows from Theorem 2.4.3 that there is a unique system of mutually orthogonal projection operators {P,} C %’, one for each cardinal number n, such that the restriction of % to P,L2(Q) has uniform multiplicity n, and P , = I . We now proceed to prove that each P, commutes with every V ( x ) , x E (5. For each x E 6,consider the correspondence T(x): A

-+

V(X)AV(x)-l,

where A ranges over all bounded linear operators on L2(Q).We assert that T ( x )maps 2l onto 2l. I n fact, if A = U(h)E U, then, by the commutation relation (3.4.1), we have T ( x ) U(h) = x”(h)U(h)E %, which shows that the commutative weakly closed operator algebra T(x)-lcU contains U. But % is generated by U, hence T(x)-l% 3 %. Similarly, T(x-l)-l % 3 %, but T(x-l)-l = T ( x ) , therefore, T ( x )% = %, and T ( x ) I 2l is an automorphism of %. Let Q, = T ( x )P, ; then Q, is a projection operator, and Qn E %’. Moreover, V ( x ) maps P,Lz(sZ) unitarily onto Q,L2(Q), and, for any A E a,the action of A on P,L2(Q) is unitarily equivalent [under V(x)] to the action of T ( x )A on QnL2(Q). It follows that the restriction of % to QnL2(Q)also has uniform multiplicity n. Furthermore, since the {P,} are mutually orthogonal, the {Q,} are also mutually orthogonal. Thus, it follows from the uniqueness of the correspondence n ---f P, that P, = Q, , that is P,V(x) = V ( x )P, . Hence, P, E (23)’ = C‘, but since C is maximal commutative weakly closed, this means that P, E C , and since C is clearly just the multiplication algebra over L2(Q), there exists a bounded %measurable function P,(g) on G, such that (f‘nF>(g) = Pn(g) d g ) ,

F ELYQ),

Since P, = Pna,we have P,(g) = Pn(g)2, which implies that P,(g) is the characteristic function of some set En E 23. If n # n‘, then P,Pnt = 0, 6*

Here, Z(h) denotes the character of 6 induced by x (see $3.2).

189

3.4. L2-Fourier Transforms that is, Pn(g)Pn,(g)= 0, hence En

n En, is a null

set. Moreover, since

xnPn = I, it follows by Lemma 1.2.5 that V n En differs from the whole of G by at most a null set. We further assert that every E, is quasi-invariant under 8. In fact, if h E 8 , then U(h)E 2l, hence U(h)Pn = PnU(h). Therefore, since dp,Jdp > 0, we obtain P n ( W d g ) = Pn(g)dd

for every rp E L 2 ( 9 ) . Consequently, the equality Pn(hg) = Pn(g) holds almost everywhere, that is, En and hE, differ only by a null set. Thus, En is quasi-invariant under 8. For each E n , we define a corresponding measure p n on (G, 8), as follows: E E 23pn(E) = P(E n E n ) ,

If n # n’, then, since p ( E n n En,) = 0, it is clear that p n and p,’ are mutually singular. Since E,&is a quasi-invariant set, it is easily seen that, if pn # 0, then pn is a quasi-invariant measure. Moreover, since G is a least upper bound for {En}, it clearly follows that E is a least upper bound for { E n En},and by Lemma 1.2.4 we obtain P.(E)= C pn(E). n

T h e normality of Qn = (G, 23, p n ) with respect to (E is obvious. Thus, a decomposition of the required kind does exist. Finally, the uniqueness of the decomposition can be deduced at once from the uniqueness of the system5’ {P,}. ] Next, we use Theorem 3.4.1 to establish a certain property of ergodic measures.

Theorem 3.4.2. Let 9 = (G, 23, p ) be a localizable measure space which is quasi-invariant under a transformation group 6. If Q is ergodic and normal with respect to 8,then there exists a cardinal number k such that 9 is cyclic of order k relative to 8. PROOF. We decompose p into the sum of a family of mutually singular quasi-invariant measures { p k } , in accordance with Theorem 3.4.1. Suppose there existed two distinct indices k, k‘ such that p k and pk’ 67 Translator’s note: In order to deduce uniqueness in this manner, it seems that one must restrict one’s considerations to systems of measures having the form pn’(E) = p ( E n E%’),where {En’}is some family of sets of the sort constructed in the foregoing proof. If the range of n is uncountable, then it is not immediately clear whether the conditions to be satisfied by {pn’} necessarily imply that it is of this form.

111.

190

GROUPS WITH QUASI-INVARIANT MEASURES

were both different from zero. Then, one easily deduces that both of the sets E, (see the proof of Theorem 3.4.1) and G - Ek have positive measure; since Ek and G - Ek are quasi-invariant sets, this contradicts the ergodicity of p. We conclude that there is an index k such that p=pk*

1

Problem. If k # 1, then, does there exist a localizable measure space which is quasi-invariant, ergodic, normal and cyclic of order k relative to some transformation group 6 ? This question may not be easy to answer. However, we can give an example of a nontrivial normal ergodic cyclic measure, as follows. Example 3.4.3. Let G be a commutative locally compact group, and SZ = (G, b,p ) a Haar measure space. Then, SZ is cyclic relative to the translation group G. PROOF. According to Example 3.4.2, Q is normal, hence, by Theorem 3.4.2, there is a cardinal number k such that Q is cyclic of order k. Suppose that k 2 2; then, there must exist two mutually orthogonal closed linear subspaces Hl and H , in L2(SZ),both invariant under %(a, 6),and a unitary mapping T from H , onto H , , such that TU(h) = U(h) T . Choose any a E H , , a # 0, and let b = Ta E H , ; then b # 0. Moreover, (U(h)a,u )

=

(U(h)u,b)

(U(h)b,b),

=

0.

+

Now, consider any E L ~ ( Qn ) L1(sZ), and let cp(h) = (+* JF U(q)+)(Is). Multiplying the above relation by ~ ( h )and , integrating over Q, we get (U(q)$* a, $ * b )

(W)$* a, $ * a) = (U(q)$ * b, $ * b),

*

=

0. ( 3 . 4 4

*

Here, both a, = $ a and b, = $ b belong to6* L1(SZ)nL2(Q). Multiplying (3.4.2) by a(q)(a E G*), integrating over SZ, and using (3.3.47),we obtain __

j cil(ci)12

=

I 6,(a)12,

ci,(ci)

&(a) =

0,

01

E

G*.

Therefore, B, E 0, 6, = 0, and hence, by (3.3.45), we see that a, = 0, b, = 0. Thus, it follows by Corollary 1.1.15 that both a and b vanish almost everywhere, which contradicts our assumption. We conclude that k = 1, that is, SZ is cyclic. ] 68 TmnsZator’s note: I t is certainly not obvious that a, and b, belong t0L’f.Q). Of course, if p ( G ) is finite, then L2(Q)CL’(Q), so that the example is at least valid for compact groups.

191

3.4. L2-Fourier Transforms 2 O Dual Measure Spaces

We have already discussed the notion of the L1-Fourier transform on a locally compact group. We now proceed to generalize this notion, as follows.

Definition 3.4.3. Let Q = (G, 8 ,p ) be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a trans$3, @) be a measure space which is formation group 6. Let 0 = quasi-invariant under a transformation group 6, and such that the following conditions are fulfilled. (i) There exists an isomorphism

(e,

(3.4.3)

ti!-+&

from

6%

onto 6 and an isomorphism h-+ h ( 6 ) ,

6€G

(3.4.4)

from 6 into @ such that & = {h(.) I h E 6} constitutes a determining set of functions on 0,moreover, for each h E 6, the character h of 6 induced by h( *) satisfies the relation h"(02)

= ti!(h),

02 E 6.

(3.4.5)

(ii) There exists a unitary mapping F from L2(Q) onto Qk2((B) (see $1.1,20), such that, for each h E 6, the unitary operator o(h)= FU(h)F-l in Qk2(0) takes the form

( w w ( J ) = h(2)cp(6>,

6 E e,

'p E e k 2 ( Q .

(3.4.6)

Then, we say that (0,G) is a dual of (Q, 6) (or that 0 is dual to a) and that F is an associated Fourier transform (from L2(Q)to Q k 2 ( Q ) ) ; we describe such a mapping F, in general, as an L2-Fourier transform. First, we prove the following existence theorem.

Theorem 3.4.3. Let Q = (G, 23,p) be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a transof (Q, 6) and an formation group 6. Then, there exists a dual (0,G) associated Fourier transform from L2(Q)to f!k2(0). PROOF. Let be the totality of symmetric multiplicative linear functionals on %(Q, 6), let !8 be the totality of Bore1 sets in %, and, for each h E 6, let h(g) = U(h)(g). Obviously, the function h(.) is ?%measurable, and I h(f)l = 1 for all E Since % has uniform multiplicity K, we know by Corollary 2.4.7 that there exists a measure

e.

192

111.

GROUPS W I T H QUASI-INVARIANT MEASURES

(e,

space 0 = @, $) and a unitary mapping F from L2(sZ)onto such that, for every A E a, FAF-W)

=

A(&

m),

g E e, f

Qk2(0)

E Qb2(@,

and, in particular, (3.4.6) holds. Let 8 be the linear hull of the family of functions & = { h ( j ) I h E Q}. If 8 be regarded as a set of multiplication operators on 2k2(Q), then F'2IF-l is just the closure of 3 in the strong topology. Now, take any A , € % , and let E be any a-finite set in 0; let E be expressed as the union of a countable family of disjoint sets {Ei) C @, with /2(Ei) < co, i = I, 2,... . If 4, is any unit vector in k-dimensional Hilbert space, then the vector-valued function f@) =

c

2-jl"(Ej)-'12

cEi(g)fO *

9

tE&'

j

[the summation being extended over those indices j for which ek(0).Hence, for every natural number n, there is a function f,(j) E 8 such that

$(Ej) > 01, belongs to

Therefore, using a diagonal process, one may select a subsequence of {f,} such that f,@) -+A,(j) for almost every j E E. Since {fn,> C 8, and since @ is, by definition, determined by the totality of functions A,@), A, E a,it follows easily that 8 is a determining set for 0,whence & is, afortiori, a determining set for 0. Let &i'(L2(Q))denote the totality of bounded linear operators on L2(Q), and, for each x E Gu, define a transformation T, : B --f V ( x )BV(x)-', B ~ a ( L ~ ( l 2 Obviously, )). T, is an automorphism of a(L2(sZ)),and TxB* = (TxB)* for every B E G?(L2(sZ)).Moreover, if h E 6, then, by (3.4.1), we have T,U(h) = 2(h) U(h). Consequently, T , induces an automorphism of a(Q, Q), and this automorphism depends only upon 2. Define (@')(A)= @(T,A), A E %(Q, 6), d E &'.

{f,.}

It is easily verified that $2 E (?,that the mapping2 --t &j is @-measurable, is an isomorphism of 6% onto that the correspondence 2-4 65 = {$I x E Gu), and that h($) = h($j)/h(j)= 2(h) for all j E e. Thus, it only remains to prove that $ is quasi-invariant under Q.

s Write r ( x ) = FV(x)F-l. Obviously, r ( x ) is a unitary operator on Sk2(fi), and, by (3.4.1), (3.4.7)

q h ) O(h) V ( x ) = V ( x ) O(h)

therefore, for any f E

5, w f ( i ((6) > = f(@)

Write that

c(g)for P(x) ((2);

w &i)-

(3.4.8)

since P(x) is unitary, it follows from (3.4.8)

Now, the two set functions PlW

=

J

II t(9)l12d&(i),

PZP) =

J

E

II f(@9)l12

d@i(i)

are finite measures on (&', 8).Since & is a determining set for fi, it is clear by Lemma 1.1.669that 5 is dense in L2((?,8,p1 p2). Hence, any bounded measurable function on &' may be simultaneously approximated in L2((?,8,p l ) and L2(&',8,p2) by a sequence of functions {fn} C 5.Therefore, it follows from (3.4.9) that

+

for any bounded measurable function f.In particular, choosingf = C, , the characteristic function of an arbitrary set E E 8,we get p 2 ( E )= p l ( E ) , that is,

J

II 5(i)l12 d & ( i ) = J II "1i)l1 d @2 i(i)

(3.4.10)

68 Translator's note: The author's proof of Lemma 1.1.6 is valid only for the real case. However, it is a simple exercise to show that the result is also valid for the complex case, provided that the algebra 3 is invariant under complex conjugation, which condition is obviously satisfied by 5 in the present situation.

111.

194

GROUPS WITH QUASI-INVARIANT MEASURES

for any E E 8.If @ ( k l E )= 0, then the right-hand side of (3.4.10) is zero, hence the left-hand side is zero for all 5 E 2k2(fi), and it follows easily that @ ( E )= 0. Since 2 is an arbitrary element of &, this shows that fi is quasi-invariant under 6. Ia0 Using the proof of Theorem 3.4.3, together with Corollary 3.3.2, one can also obtain the following result.

Corollary 3.4.4, Let Q = (G, 8,y) be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a transformation group 6. Let S denote the p-topology on 6, and let (? be a spectral groupa1 for (6,S) such that @ C (?. Let 8 be the totality of weak Bore1 sets in and take 0 = 6s. Then, there exists a &quasiinvariant measure @ on @), such that (((?, @, @),&) is a dual of (SZ, 6) with respect to an appropriate Fourier transform from L2(Q)to

c,

e k 2 G

(e,

B, 6).

Next, we proceed to find the general form of all L2-Fourier transforms associated with a fixed dual. Let U, = { U(2) 1 9 E G} be the transformation group in L2(fi) corresponding to 6. We may also regard U, as a group of transformations of ek2(0), namely, if 5 E gk2(Q),4 E 6, we define (3.4.11)

Clearly, U(2)is a unitary operator in Qk2(0). We note that the group of multiplication operators on Qk2(Q), corresponding to @, is just { o ( h ) 1 h E S}[see (3.4.6)]. A trivial calculation yields the commutation relation E(h) U(&)O(h) = O(h) U(&), h E 6, & E 0, (3.4.12)

v(a)

analogous to (3.4.1). For any (Y E GuI let W ( a )= U(&)]where aZ denotes the image under the mapping (3.4.3) of the character G induced by a ; thus, V(a) = W ( a )U(&-l), a E Gu. (3.4.13) 6 o Translator’s note: Since the mapping h -+ h(2) defined in this proof is the composition of the mappings h U(h)and U(h) + U(h)(&,and since h --* U(h)may not be injective, it is obviously impossible to prove that h h ( . ) is injective, unless further conditions are imposed. This suggests that the requirement of Definition 3.4.3 [that (3.4.4) be injective] is too strong, and that one should, instead, require the existence of an appropriate isomorphism U(h) + h(.), from U into @. Translator’s note: T h e term “spectral group” was defined with reference to a topological group. In the absence of further assumptions, (8, 2)may not be a topological group in the usual sense, since there is nothing to guarantee that the p-topology is --f

Hausdorff.

-.

3.4. L2-Fourier Transforms

195

We see from (3.4.7) and (3.4.12) that W(a)and ??(h) commute, that is, W(a)O(h) = O(h) W(a),

01

E Gu.

(3.4.14)

Since { O(h),h E 0}generates the multiplication algebra %Ilkover Qk2(L?), it follows from (3.4.14) that W(a)E { o ( h ) ,h E S]’= %Rk’.If we assume that k N o and that $ is localizable, then, by Lemma 2.4.20, we know that there exists a measurable k-dimensional unitary operator-valued function z(6; a),6 E such that

<

e,

for every

5 E 2k2(&’).Moreover, since (3.4.16)

Thus, we obtain the following result.

Theorem 3.4.5. Let Q = (G, 23, p ) be a localizables3 measure space which is quasi-invariant under and cyclic of order k ( < N o ) relative to a transformation group 8. Let (0, &) be a dual of (Q, 8 ) [with L? = (c,@, i;) localizables3], let F be an associated Fourier transform, = FV(a)F-I. Then, there exists a system and, for each a E Gu,let {z(-;a), 01 E Gu}of k-dimensional unitary operator-valued functions on 0, satisfying relations (3.4.17), and such that

v(a)

holdsad for each a E Gp. Notice that when k = 1 , that is, when Q is cyclic relative to 8, the conclusion of Theorem 3.4.5 may be further simplified, for in that case, i?kz(L?) is essentially just L2(Q),and each z ( - ;a ) is just a measurable Here, equality holds for almost all 1 E e. As in the case of Lemma 2.4.20, the original statement of this theorem hypothesized o-finiteness (in this case, for both 5d and a). The equality here holds for almost all p E &.’ se

m

111.

196

GROUPS WITH QUASI-INVARIANT MEASURES

complex-valued function whose modulus is almost everywhere equal to 1. If the measure space 52 is ergodic with respect to 6, then the totality % of quasi-characters of 52 corresponding to the character 1 of 6 is just the set of all complex constant functions c, I c I = 1. Hence, if 8, = d, , then al/az = c, and so we have

41; 4 = cz(1; .2) for almost all j E €. Using similar arguments, we may derive the general relationship between any two Fourier transforms associated with the same dual, namely,

Theorem 3.4.6. Let 52 = (G, B,p ) be a localizables6 measure space which is quasi-invariant under and cyclic of order k (k No) relative to a transformation group 6. Let (0, 65) be a dual of (52, 6) [with b = (Q, 8,F ) localizables5], and let F,F’ be any two Fourier transformss6 associated with (0, 65). Then, there exists a measurable k-dimensional unitary operator-valued function u ( j ) , j E such that

<

e,

F‘

=

(3.4.19)

u(*)F.

Conversely, given any Fourier transform F from L2(sZ)onto 2,z(fi),and any measurable k-dimensional unitary operator-valued function u(&, 6 E €, the operator F’ defined by (3.4.19) is also a Fourier transform associated with the same dual [and the same mapping (3.4.4)]. PROOF. Given the two Fourier transforms F and F’,form the unitary operator u = FF-1 (3.4.20) in 2,,(0). Since FU(h)F-l

= F’U(h)F’-l =

O((h),

h E 6,

(3.4.21)

we have uO(h) = Ir(h)U,

h E 6.

(3.4.22)

Hence, as in the derivation of (3.4.15), it follows by Lemma 2.4.20 that 66 Translator’s note: Again, in the original version, it was supposed that s;) and 8 were o-finite. Notice that the proof of the second (converse) part of the theorem does not make use of the localizability of 4. It is assumed here that the mapping (3.4.4) is the same for both F and F‘.

3.4. L2-Fourier Transforms

197

there is a measurable K-dimensional unitary operator-valued function u(&, 6 E such that, for all 5 E Qk2($),

c,

&(g)

4 2 )E(i),

=

t E Q.

Combining this relation with (3.4.20), we get (3.4.19). Conversely, given any measurable R-dimensional unitary operatorvalued function u ( . ) on 0,we form a unitary operator F’, from L 2 ( 0 ) to Qk2($), in accordance with (3.4.19). Since the operator u:

-

I ( - ) 4.1[(*I,

5 E QkYQ

satisfies (3.4.22), it follows that (3.4.21) also holds, which shows that F’ is also a Fourier transform. ] Using Theorems 3.4.5 and 3.4.6, we see that the Fourier transforms 6) and a given mapping (3.4.4) have associated with a given dual (0, the general form6’

E 6 2?2(0),

(3.4.23)

where z ( * ;a ) and u ( - ) denote the operator-valued functions appearing in (3.4.18) and (3.4.19), respectively. T h e following theorem shows that the dual measure space is, in a certain sense, unique.

Theorem 3.4.7, Let O = (G, 23,p ) be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a transformation group 8.Let $ = (c,i%, @) and $’ = (@, @’, @’) be localizable measure spaces which are quasi-invariant under the transformation groups 6 and 6‘, respectively, moreover, suppose that (0, 6) and (O‘, 6‘)are both duals of (SZ, 8). Then, the two measure spaces 0 and 0’are equivalent. PROOF. Let 8,h( .), F,O(h)and B’, h’( .), F’, l?’(h) denote the quantities in Definition 3.4.3 corresponding to the duals (0, 6) and (Q’, 6‘), respectively. Form the unitary operator Q = F‘F-1

from

Qk2($)

to

S!k2(0’).

Obviously, QO(h)Q-l

=

O’(h). Since the

O’ Translator’s note: This is something of an overstatement, since (3.4.23) does not explicitly describe the operation of F’ on the elements of Le(Q).

111. GROUPS

198

WITH QUAS1-INVARIANT MEASURES

families of functions { h ( j ) I h E S} and {h’(j‘)I h E S} are determining sets on 0 and Or,respectively, it follows that the families of operators t??(h) I h E S} and { o ’ ( h ) I h E S} generate the multiplication algebras m k and !&k’ over ! i ! k 2 ( 0 ) and !i!k2(o’), respectively. But clearly, {QAQ-I I A is a commutative weakly closed operator algebra over !i?kz($’), and contains { @(h) I h E S},hence {QAQ-’ I A €@Ik}3 @k‘* Similarly, {QAQ-l I A E a k } C !&k’, Thus, Q effects a unitary equivalence between @ k and @ILk‘. Therefore, by Theorem 2.4.21, 0 and ?A are equivalent. ] Similarly, one may prove the following result.

€ak}

Theorem 3,4,8. Let 6 be a group, and suppose that 8 acts as a group of measurable transformations on each of the two localizable measure spaces SZ = (G, 23,p ) and SZ = (G’, B‘,p’), moreover, suppose that both SZ and SZ‘ are quasi-invariant under and cyclic of order k relative to 6. If (in,6) and (Q’, 8)are equivalent,68 then they have a common dual. Remark 1. Let us consider the “uniqueness” of the mapping (3.4.4) in Definition 3.4.3. Assume that fi is ergodic with respect to 6 , and suppose that, besides (3.4.4),there is another homomorphism h -+ h’(J),

g E (2,

(3.4.24)

from 6 into &, satisfying condition (3.4.5),in other words, the quasithat is, characters h(.) and h’(6) induce the same character on (%, a+b(h),

a€(%.

Since 0 is ergodic, it follows by property 111 of $3.2, 20, that there exists a constant c(h) (independent of j ) , with I c(h)l = 1, such that h’@) = c(h) h(&,

E

(2.

(3.4.25)

Moreover, since (3.4.4) and (3.4.24) are both homomorphisms, the function c(h), h E 6, is a character of (5. Now, let F‘ be a Fourier transform associated with the homomorphism h -+ h’(.),so that every operator O’(h) = F’U(h)F’-l, h E 6, takes the form ( Q ( h ) t ) ( t )= h’M 5(&, k E G 5 E %c2(Q(3.4.26) WJ That is, there exists an equivalence between Ja and Ja‘ which is consistent with the action of 6.

3.4. L2-Fourier Transforms

199

Then, using (3.4.25), (3.4.26), and (3.4.6), one easily calculates that, if 5, rl E Q k 2 ( Q , then c(h) =

(U(h)F'-l(, F'+) ( U(h)F - q , F-17)) .

From this formula,69 we see that the character c(h), h E Q, is continuous relative to the p-topology on 0. If we assume that every character of 0 which is continuous relative to the p-topology is induced by some quasicharacter of Q (relative to a), then there is an a. E Gu such that c(h) = do(h),

h E 8.

Hence, by ( 3 . 4 3 , we get h'(k) = d,(h) h ( j ) = h(&,k).

(3.4.27)

Conversely, take any a. E Gu, and define a function h'( .) in accordance with formula (3.4.27). Then the correspondence (3.4.24) satisfies the conditions of Definition 3.4.3, and the operator F' = QF,where

serves as an associated Fourier transform. Also, consider the case where G is a commutative group, 8 is the group of all translations of G, is the group of all characters of G, @ ! is the u-algebra of all weak Bore1 sets in &, and 6 is the group of all translations of I n this case, we need not assume that fi is ergodic with respect to 6, and still (3.4.25) holds, with c E &, moreover, if there exists a Fourier transform F' corresponding to (3.4.24), then, arguing as in the proof of Theorem 3.4.7, one can show that c is a quasiinvariant point70 of a. Remark 2. Dual measure spaces are not necessarily unique. I n fact, 6) be any dual of (SZ, O ) , and let F be an associated Fourier let (0, transform. Let it' be another measure on 8)such that 0 and

e

e.

(e,

Translator's note: The author has overlooked the possibility that the denominator may vanish. However, in view of the ad hoc hypothesis in the next sentence, the question of whether or not c(h) is continuous relative to the p-topology hardly merits any special attention, since a still stronger ad hoc hypothesis could be made, if necessary. 'O Translator's note: The translator is frankly puzzled by this assertion. Presumably, it has already been assumed that sfr is quasi-invariant under the translation group G, in accordance with the requirements of Definition 3.4.3.

200

111.

GROUPS WITH QUASI-INVARIANT MEASURES

0’= (I!?,8,p’)

are equivalent and have the same u-finite sets. Then, leaving the mappings (3.4.4) (3.4.9, and (3.4.6) unchanged, we see that (&, 6)is also a dual of (Q, 8),with an associated Fourier transform (3.4.29)

Remark 3. If we take the ptopology on 8 and the $-topology on

@ = { h ( j ) I h E 0}, then the mapping (3.4.4) is a h o m e o m ~ r p h i s m ~ ~ from (li onto $. In fact, for any 4 E L Z ( Q ) ,

but the left- and right-hand sides of this equality represent arbitrary members of the families of convex functions which define the p-topology on 8 and the $-topology on $, r e s p e c t i ~ e l y Thus, . ~ ~ (3.4.4) is a homeomorphism.

Example 3.4.4. Let G be a commutative locally compact topological group, and let Q = (G, 23,p ) be a Haar measure space. Also, let G* be the dual of G (given the strong topology, G* becomes a locally compact group), and let Q* = (G*, b*,p * ) be a Haar measure space. If G and G* are regarded as operating on themselves by translation, then (Q*, G*) is a dual of (Q, G), moreover, if the measure p* is suitably normalized, then there is a Fourier transform F,,from L2(Q)to L2(Q*) such that, when f E L1(Q) n L2(sZ), (3.4.30)

where4 is defined as in (3.3.44). PROOF. As was shown in Example 3.3.2, G* is a spectral group for G. According to Example 3.4.3, Q is cyclic relative to the translation group G, and by the remark following Corollary 3.2.4, e’ = G*. Let 8 denote the totality of weak Bore1 sets in G*. Using Corollary 3.4.4, we deduce the existence of a localizable measure $ on (G*, 8) such that fi is quasi-invariant under the translation group G*, and such Translator’s note: Again, see footnote 60. For the left-hand side, this statement is obvious. As for the right-hand side, one can certainly choose 5 so that llF((i)11is the characteristic function of an arbitrary set A E $3, with P(A) < 03, hence one obtains all the convex functions defining the F-topology on (F; moreover, using the second part of the proof of Theorem 3.2.10, it is easily seen that ll(h(.) - 1 ) 11~is continuous relative to the $-topology, for any 7 E 7B

201

3.4. L2-Fourier Transforms

that (0, G*) = ((G*, @, @),G*) is a dual of (SZ, G). I n this case, the homomorphism (3.4.4) is rn~G*.

h+a(h),

(3.4.31)

Let F be any L2-Fourier transform, from L2(SZ)to L2(o),corresponding to the homomorphism (3.4.31),and let f , 7 E L2(Q).Then, for any h E G, ( U ( h ) t ,7)

=

1

G*

a 0 F t ( 4 Frl('y.) d @ ( 4

Now, take any a EL~(SZ) nL2(SZ),multiply both sides of the above formula by a(h), and integrate with respect to h; using the notation (3.3.44),we get (a

* t,7)=

1, 4.)F t ( 4

F X ) 4(4.

(3.4.32)

This shows that, if a E L ~ ( Q n L2(L?), ) then F(a * t ) ( a )= ci(a)Ft(a).

(3.4.33)

Next, take any two elements a, b cLl(J2) nL2(Q);by Lemma 1.1.14, there is a sequence { f n } CL2(SZ)such that lim 11 a * 6, - a 112 11 b n-. w - lim n+m

* En

-

b /I2

= 0.

Applying (3.4.33), it follows that

-

II & ( . ) F ( f n ) F ( ~ ) I I ~0,

II & * ) F ( f n )- F ( ~ ) I I0~ . +

(3.434)

From (3.4.34),we see that, if a, b EL~(SZ) n L2(Q),then Fa(a) &(a)= Fb(a) &(a).

(3.4.35)

Note that if f EL~(SZ),then, according to Corollary 3.2.12, f ( 0 1 ) is a continuous function on G*. Consequently, if f is not almost everywhere zero, then by (3.3.49, { ( a ) is not identically zero, and so 1 &a)( exceeds a certain positive constant E on some open subset of G*. Furthermore, note that, if f EL~(SZ) and x, 01 E G*, then

J

x(g> t ( g ) 43 4 4 g ) =

hence, for any x E G*, there is an element 5 ELI(Q)nL2(SZ),such that the infimum of I I).([ on some neighborhood of x is positive. Let $' 3'

202

111, GROUPS

denote the totality of sets E in {a,} C L1(Q) n L2(Q)satisfying

WITH QUASI-INVARIANT MEASURES

!8 such that there exists a sequence (3.4.36)

By virtue of the facts which have just been stated, every compact set in $3 belongs to b', therefore, since b' is obviously a o-ring, and since b * is the o-ring generated by the compact subsets of G*, it follows73 that 8"C b'. Now, let E E b', and choose any sequence {a,} CL1(sZ) n La@) such that E satisfies (3.4.36). We construct a measurable function uE on E, as follows. If a",(.) # 0, let (3.4.37)

In view of (3.4.35), it is clear that, apart from a set of measure zero, uE(a)is well-defined and independent of the choice of the sequence {a,}. Moreover, it is also clear from (3.4.35) and (3.4.37) that, if D is any other set in b', then %(") = U E ( 4

(3.4.38)

holds for almost all a E D n E. In what follows, we need only consider sets E which belong to b*. Define a set function p' on b * as follows:

Using (3.4.38), it is easily verified that (G", 23*,p') is a measure space. Furthermore, using (3.4.35), (3.4.37), and (3.4.39), one easily verifies nL2(Q),then that, if 4, 7 €L1(1;2)

(3.4.40)

Next, we assert that H

=

{+j I 7 €L1(Q) nL2(Q)} is dense in

Translator's note: Obviously, in order to draw this conclusion, one must establish that all compact sets are in b', and not just those which also happen to belong to 8.It is not clear how one does this.

3.4. L2-Fourier Transforms

203

L~(G*,%*, p'). I n fact, suppose that for any q €L1(Q) n L2(Q),

5 €L2(G*, 8*,p'), 6 Iff; then,

But since qh E L1(Q) n L2(Q),where q h ( g ) = q(hg), we also have

b*),it Since (a(h),h E G} is a determining set of functions on (G*, follows from Lemma 1.1.6 that ((a)q(a) is zero almost e~erywhere,'~ hence .$(a) is zero almost everywhere.74 Thus, H is indeed dense in L2(G*, B',p'). Therefore, in view of (3.4.40),the correspondence 4 -+ can be extended to a unitary mapping F,, fromL2(9) onto L2(G*,8*,p'). Moreover, when ( E L1(Q) n L2(Q), we have

(3.4.41)

=

J

((ax-1) ij(ax-1)

&'(a).

c*

From this, one easily deduces that the measure p' is invariant under translations. Furthermore, it is obvious from our earlier remarks that there exists an element 4 E L'(Q) n L2(9) such that [ [ ( a ) / > 1 on some compact neighborhood N in G*. Hence, by (3.4.40),we get

Thus, p' is locally finite, and we conclude that p' is a Haar measure. Finally, using (3.4.40),it is easily shown that F, is the required L2-Fourier transform. ]

Example 3.4.5.

Let 1 = 1 , 2,..., be a finite or countable set of

74 Translator's note: Although these assertions seem highly plausible, more detailed arguments would be welcome.

111. GROUPS

204

WITH QUASI-INVARIANT MEASURES

indices, and for each I, let Q, = (G, , 8 , ,p J be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a transformation group 6 , . For any finite sequence h = (h, ,..., hn), where h , E 8 i ,I = l,,.., n, we define a transformation by hg

= @,g,

Y...,

hng, ,gn+,

,*..I,

where g = {g, ,...,g, ,g,+, ,...} is any element of X I G , such that g,E 'D(h,), I = 1,2,..., n. Let 6 denote the totality of such transformations h, and suppose that Q, is a probability measure space for all but a finite number of indices 1. Then, the product measure space Q = x, Q, is quasi-invariant under and cyclic of order k, relative to 8.Moreover, if, for each I , (0, , 6,)is a dual of (Q, , 6,),if 0,is a probability measure space for almost all I, and if we define a corresponding transformation group 6 in the same way that 6 was defined above, then (X, 0,, 6) is a dual of (Q, 6). The proof is left to the reader.

n,

30 Duals of Ergodic Measure Spaces Lemma 3.4.9. Let Q = (G, b,p ) be a localizable measure space which is quasi-invariant under a transformation group 6. Let B E b, and let HB be the closed linear subspace of ek2(Q) formed by the totality which vanish almost everywhere outside of B. of functions in ekz(Q) Then, B is quasi-invariant under 8 if and only if H B is invariant under the group of unitary operators U = { U(h) I h E S}. PROOF. Assume that B is quasi-invariant under 6, and let y E H , . Then, for any h E 0,it is clear that y(hg) is zero for almost all g E G - B, hence

for almost all g E G - B, that is U(h)y E H E . Thus, H E is invariant under U. Conversely, let B € 8 ,and assume that H B is invariant under U. If B were not quasi-invariant, then there would exist an element h E 6 such that hB - B is not a p-null set, therefore, since Q is localizable, one could find a set E E b such that 0 < p ( E ) < 00, 0 < p(hE) < 00, E C B , hE C hB - B. Then, if C, denotes the characteristic function of E, and y o is any fixed nonzero vector in k-dimensional Hilbert space, the function y E ( . )= C,(.) y o would belong to H , , and

205

3.4. L2-Fourier Transforms

Thus, ( U ( h )cpE)(g) # 0 for allg E hE C G - B, and hence U(h)cpE E H E . But this contradicts the invariance of HB under U. We conclude that B is quasi-invariant under 8. 3

Corollary 3.4.10. Under the hypotheses of Lemma 3.4.9, let M be the smallest weakly closed operator ring containing both the multiplication algebra W(9) over L2(L?)and the group U = { U(h) I h E S}. Then, 9 is ergodic with respect to 8 if and only if M is a factor; SZ is weakly ergodic with respect to 6 if and only if the center of M contains no countably decomposable projection 0perators.~5

Theorem 3.4.11. Let 9 be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a transformation group 8.Suppose that SZ is normal with respect to 8,and let (a,6) be a dual of (L?, 6).If L? is ergodic with respect to 6, then is ergodic with respect to 6. PROOF. Assume that is not ergodic with respect to 6, and let 8 be a set in @ which is quasi-invariant under 6, with @(I?) # 0, - 8)# 0. Let H g denote the closed linear subspace of Qk2(fi) generated by the totality of vector-valued functions which vanish on - 8.Then, (3.4.42) (0) f HE # 2&9.

a

&(e

Moreover, by Lemma 3.4.9, H g is invariant under all the operators U(x), x E 6, and obviously Hg is also invariant under every operator of the form 5 4z ( * ;a) ((-). Since, by (3.4.18), P(a) = x(*;a ) U(S-l), a E G@,it follows that HEis invariant under all the operators 01 E GW. It is also obvious that HE is invariant under the operators t)(h), h E 8. Write H = F-lHg. Then, H is a closed linear subspace of L 2 ( 9 ) which is invariant under the groups U and 23. Let P denote the projection operator from L2(9)onto H ; we then have P E 23’. Since G@is a determining set of functions on 9, the group 23 generates the multiplication algebra %I over L2(sZ),and since 9 is localizable, !Ul is maximal commutative weakly closed. Hence, 23’ = !Ul’ = W, so that P EW. Let M be the weakly closed operator algebra generated by U u %I; since P E U’, it follows that P E M n M’. But by Corollary 3.4.10, M is a factor, hence P = 0 or P = I , which contradicts (3.4.42). We conclude that fi is ergodic with respect to 6. ] In general, the converse of Theorem 3.4.11 is not valid, as is shown by the following example.

r(01),

76 Translator’s note: This is clearly not a necessary condition for weak ergodicity, since it is obviously not satisfied when L2(Q)is separable, and examples of weakly ergodic o-finite measure spaces Q, such that L2(Q)is separable, are easily constructed.

111. GROUPS

206

WITH QUASI-INVARIANT MEASURES

Example 3.4.6. Let G be a group of just two elements {a, e}, where a # e, a2 = e, let b be the totality of subsets of G, and let p be the measure on 23 defined by &(a, e}) = 2, ~ ( ( u }= ) p((e)) = 1, p(empty set) = 0. Let 6 be the subgroup of G consisting of just the unit element e, and let (5 be the totality of functions from G to the complex numbers of unit modulus. Then, (G, 8 ,p ) is a finite measure space which is invariant under and cyclic of order 2 relative to the translation group 6, and is obviously normal with respect to 6. However, p is not ergodic with respect to 6, in fact, both of the singletons {e} and {u} are nontrivial quasi-invariant sets. Since the quasi-characters (3 induce just the one character 1 on 6, we see that (G, b,p ) has a dual measure space (&, B, $) defined as follows: & is a group containing only a single = 1, element 6, b consists of {6} and the empty set, $(empty set) = 0, and 6 consists of just the identity transformation. Obviously, (&, 8,$) is ergodic with respect to 6. This shows that a quasi-invariant measure space which is not ergodic may have an ergodic dual. However, when k = 1, the converse of Theorem 3.4.1 1 is, in fact, valid.

a({$})

Theorem 3.4.12. Let Q be a localizable measure space which is quasi-invariant under and cyclic relative to a transformation group 6. Suppose that SZ is normal with respect to 6, and let (0, 6) be a dual of (in, 6).If a is localizable and ergodic with respect to 6, then SZ is ergodic with respect to 6. PROOF. We follow the argument used in proving Theorem 3.4.1 1. Assume that SZ is not ergodic, and let B E B be quasi-invariant under 6, with p ( B ) # 0, p(G - B ) # 0. Let HB denote the totality of functions in L2(SZ)which vanish almost everywhere in G - B. Then, H , is a closed linear subspace of L2(Q) which is invariant under U and 2?, moreover, (0)#

HB

#L2(Q).

(3.4.43)

Let $5 = FHB ; then, $jis a closed linear subspace of Cl2((a) = L2(sfi) which is invariant under the groups 0 = { o ( h ) I h E 6) and 6= I 01 E Gu}. Let Q be the projection operator from L2(0)onto $5, and let $I denote the multiplication algebra over L2(sfi).Now, according to Definition 3.4.3, & is a determining set of functions on sfi, hence, it follows by the results of $2.4 that generates '$However, I. since 0 is localizable, $I is maximal commutative weakly closed. Thus, we have Q E a' = '3%'= $I. If 01 E Ge, and z(g; a ) is the function appearing in (3.4.18) (since k = 1, this is presently a numerical-valued function),

(r(01)

207

3.4. La-Fourier Transforms

then the operator .$ +z ( * ;a ) belongs to @. Therefore, since U(4-l) = z ( * ;a)-l P ( a ) , we see that 5 is invariant under { U ( x ) I x E 6}, and hence Q E { U(x) I x E &}'. Let i?! denote the weakly closed operator algebra generated by@ and { U ( x ) I x E 6}.By virtue of Corollary 3.4.10, the ergodicity of b implies that it? is a factor. But Q E i@ and Q E A?', hence Q is either 0 or I , which contradicts (3.4.43).We conclude that SZ is ergodic with respect to (ti. 3 ((a)

40 Strongly kth Order Cyclic Measures We shall now consider, in particular, the case where the dual is a finite measure space. Since every a-finite measure is equivalent to a finite measure space, some of the following results may easily be extended to a-finite measure spaces.

Definition 3.4.4. Let 9 = (G, 23, p ) be a localizable measure space which is quasi-invariant under a transformation group (ti. Let cp E L2(SZ), and let H, be the smallest closed linear subspace of L2(9) which contains cp and is invariant under all the operators U(h), h E (ti. We call H, the subspace generated by the cyclic element cp (and the group U). Suppose that {cpA 1 h E A} is a family of vectors in L2(52) such that

and that the function

is independent of A. Then, we say that { y hI h E A} is a 6-cyclic family of elements, that +(h) is the corresponding adjoint function, and that (G, 23, p ) is strongly kth order cyclic relative to 6 , where k is the cardinal number7s of A .

Theorem 3.4.13. Let 9 = (G, 23,p ) be a localizable measure space which is quasi-invariant under a transformation group 6 .If Q is strongly Kth order cyclic relative to (ti, then l2 is cyclic of order k relative to 6. Conversely, if L 2 ( 9 )is separable, and 9 is cyclic of order k relative to (ti, then k K O , and 9 is strongly kth order cyclic relative to (ti. PROOF. Suppose there exists a 6-cyclic family of elements { y AI h E A}, where the cardinal number of A is k. Let % denote the operator algebra corresponding to the transformation group (ti. Then each HeA is an

<

As before, when k

=

1 , we simply say strongly cyclic.

111.

208

GROUPS WITH QUASI-INVARIANT

MEASURES

%-invariant subspace, and % has the cyclic element yA in HwA, hence, by Corollary 2.4.11 , the restriction of % to HwAis maximal commutative weakly closed. Furthermore, since the adjoint function #(h) [see (3.4.44)] is independent of A, one can deduce that, for any A, A' E A , the restrictions of % to HwAand IfwA, are unitarily equivalent. I n fact, consider the mapping UAjA, from HwAto HwA,, defined as follows: if h , E 8,and z l is a complex number, 1 = 1, 2,..., n, let

Then, using (3.4.44),we obtain

1;

%U(hZ)VA

1 c 2

=

IG(h,'hd

Zl%

=

1,L'

1) 2

1r.

~ZW)TA,

(3.4.45)

Hence, UA,, is an isometric linear mapping from the dense linear subspace {Cz,U(h,)yAI h , E S} of HqA onto the dense linear subspace {Cz Z U ( h ,cph' ) I h , E S}of HvA,, and can therefore be uniquely extended to a unitary operator from HqAonto HqA,. Moreover, it is easily verified that UA,,+U(h)U;: = U(h), and it follows that the restrictions of % to HwA and IfwA,are unitarily equivalent. Thus, in accordance with Definition 2.4.2, 2I has uniform multiplicity K over L2(Q). On the other hand, suppose that L2(Q)is separable, and that Q is cyclic of order k relative to 6. T h e n there exists a family of closed linear subspaces {HA, A E A } in L2(Q) (the cardinal number of A being K ) such that each HA is %-invariant, the restrictions of 2I to the H A are maximal commutative weakly closed and unitarily equivalent, and

P(Q)=

C 0HA AEA

Since L2(Q)is separable, and each HA is nontrivial, it is obvious that k H,, , and that each HA is separable, whence it follows by Corollary 2.4.9 that % has a cyclic element in each HA. We arbitrarily choose a fixed index A, E A , and in HAo we choose any cyclic element yAorelative to %. For each A E A , A # A,, let UAbe a unitary operator from HAo onto HA such that U;;'U(h) U, = U(h), and let cpA = UAqA,. Then y A is clearly a cyclic element of H A relative to %, that is, HA = HwA, moreover,

<

(U(h)TA 9 FA) = ( U ( h )

9

uA%)

= (U(h)TA,

9

Q)Ao)?

that is, the function #(h) in (3.4.44) is indeed independent of A. Thus, {cpA I A E A } is a 6-cyclic family of elements. ]

3.4. L2-Fourier Transforms

209

I n particular, if there exists y EL~(SZ)such that L2(SZ)= H, is cyclic relative to 8 .

, then

p

Theorem 3.4.14. Let SZ = (G, 23,p) be a measure space which is quasi-invariant under and strongly Kth order cyclic relative to a transformation group 8 , and let I/ be the corresponding adjoint function. $, p), dual to Q, Then there exists a finite measure space 0 = such that I/ coincides with the quasi-characteristic function corresponding to Q, that is,

(e,

w = 1, h(k) d$(k),

(3.4.46)

h E 8.

e

Moreover, one may choose to be a spectral group for 0, (taking the p-topology on 0),and $ to be the totality of weak Borel sets in PROOF. Let { y A1 h E A } be a @-cyclic family of elements in L2(SZ), with corresponding adjoint function I/. Clearly, I/ is a positive definite function77on 6, moreover, since

#(o)

e.

- %#(h) ==

II(U(h)

(3.4.47)

- I)VA

it follows from inequality (3.3.3) that I/ is continuous with respect to the p-topology. Hence, if G. is a spectral group for 6 (the appropriate o-algebra $ being the totality of weak Borel sets in (?),78 then there $) such that (3.4.46) holds; let exists a finite measure on J-2 =

(e,

(e,8,p).

Choose a complete orthonormal system {eA I h E A } in K-dimensional Hilbert space. We construct a unitary mapping F from L2(SZ)onto i!k2(0), as follows. For any A, ,..., A, E A , hAl,..., hAnE 6, and complex numbers zA1 ,...,z,,,, define F

(2

i

(3.4.48)

zA1~(hAt)~AL) = zA1hAL(t)eAI *

1=1

1=1

F is an isometric mapping; in fact, using (3.4.46),we have

J 1 c zA,hA,(t)e~,1 d 2

=

i

~

Since { q ~1 h E A } is a 8-cyclic family of elements, the totality of elements 77

See (3.4.45). Theorem 3.3.4 ensures that such spectral groups do exist.

111.

210

GROUPS WITH QUASI-INVARIANT MEASURES

of the form zA,U(hAZ) rpA, is dense in L2(D). Also, since {h(.) I h E 6} is a determining set of functions on @), it follows by Lemma 1.1.67Q that the totality of linear combinations of these functions is dense in ,!,“(a). Hence, we see from the discussion in $ l . i , 2O, that the totality of elements of the form represented on the right-hand side of (3.4.48) is dense in ek2(Q). Thus, F uniquely extends to a unitary mapping from L ~ ( Qonto ) L?~~(Q). Let O(h) = FU(h)F - l . From (3.4.8), we see that, for any function of the form

(e,

c n

((8=

zA,hA,(6)eAt 9

1=1

the relation

(mw) 4 2 ) 5(2) =

this relation holds, and again, since these functions are dense in ek2(fi), holds for all f E Qk2((a). If B E @, then, according to the remark preceding Theorem 3.2.3, 2 is continuous with respect to the p-topology, hence, by the definition of a spectral group, we have & E &?.We take 6 = &, operating on by translation, and take 6 = 6. T h e proof may now be completed by following the reasoning used in the proof of Theorem 3.4.3. ] Next, we establish a converse to Theorem 3.4.14.

e

Theorem 3.4.15. Let Q = (G, 23,p ) be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a transformation group 8 . If D has a a-finite dual 0 = (&?,@, $), then D is strongly kth order cyclic. PROOF. Since any cr-finite measure is equivalent to a finite measure, we may assume that 0 is a finite measure space dual to Q. Let F be a Fourier transform from L2(D) onto Qkz((a), and let {e, I h E A } be an orthonormal basis for the k-dimensional Hilbert space H k . For each h E A , let cpA = F-le, ; we assert that {v, I h E A } is a 6-cyclic family of elements in L2(Q).In fact, if h # A’, h, h’ E 6, then, since U(h)rpA= F-l(h(*)e,),

h E 8,

(3.4.49)

we have

(W)% U(h’)W) = J, h(2)h’($)(e, e , \ , ) 4 w = 0. 1

I

Translator’s note: Again, Lemma 1.1.6 is valid here because the algebra under consideration is invariant under complex conjugation.

3.4. L2-Fourier Transforms

21 1

Hence, Hp, 1HpA, for h # A'. Moreover, since the totality of elements of the form zAyhV(.) e, . is dense in f?k2(O), it follows that L2(G,d,p ) = CAE,, @ Hp, Furthermore, by (3.4.49), we have

x

which is independent of A. Thus, s*, is strongly kth order cyclic relative t o 8. ] Next, we prove a lemma, concerning cyclic subspaces, which will be required in the subsequent discussion.

Lemma 3.4.16. Let l2 = (G, % p)I be , a localizable measure space which is quasi-invariant under a transformation group 6. Let y o be any fixed vector in L2(12),form the subspace H., (see Definition 3.4.4), and let {h,} be any sequence in 8. Then Me0(h,)-+ 0 if and only if the sequence of unitary operators { U(h,)) converges strongly to the identity operator on H., . PROOF. Sufficiency follows immediately from the relation Mwo(hn) = ll(u(kJ- 4%

117

'PO E

Hv,-

Conversely, suppose that MOo(hn) -+ 0. Given any cp E H., and any positive number E, there exist elements g, E 8 and numbers h k , k = I , 2,..., n, such that (3.4.50)

Choose a positive integer N such that (3.4.51)

whenever u >, N . Then, from (3.4.50) and (3.4.51), using the relation U ( k ) U g k . ) = Vg,) U(Q, we get

n

+ 1 I A,

I II(U(h")- 4Po II < E .

k=l

'Thus, U(h,) y

+y .

Corollary 3.4.17.

] Let Q

=

(G, 23,p ) be a localizable measure space

212

111.

GROUPS WITH QUASI-INVARIANT MEASURES

which is quasi-invariant under and strongly cyclic relative to a transformation group 8, with plo as 6-cyclic element. Then the p-topology on 6 is equivalent to the topology induced by the convex function M o p ) ,h E 6. PROOF. Since H., = L2(Q),it follows from Lemma 3.4.16 that, for any rp €L2(Q),M,(h) = II( U(h) - I ) pl (1 is continuous relative to M., . Hence, it is clear that the p-topology on 6 is weaker than the topology induced by M., . On the other hand, the topology induced by M., is obviously weaker than the p-topology. ]

Corollary 3.4.18. Let Qk = (G, 23,p k ) , k = 1,2, be localizable measure spaces which are both quasi-invariant under and strongly cyclic relative to a transformation group 6, and denote their respective 6-cyclic elements by plk , k = 1, 2. Write

If p1 is absolutely continuous with respect to p 2 , then the convex function M")(h) is continuous relative to M ( 2 ) ( h )In . particular, if p1 and pz are equivalent measures, then M ( l ) ( h and ) W 2 ) ( h are ) topologically equivalent. Corollary 3.4.18 is a direct consequence of Corollary 3.4.17 and Theorem 2.4.13. I t can be used as a criterion for the inequivalence of two quasi-invariant measures.

CHAPTER

IV QUASI-INVARIANT MEASURES AND HARMONIC ANALYSIS ON LINEAR TOPOLOGICAL SPACES

A linear topological space may be regarded as an Abelian group with respect to addition, hence, the theory of quasi-invariant measures on linear topological spaces may be regarded as a special case of the theory of quasi-invariant measures on groups. However, since linear topological spaces also possess the operation of scalar multiplication, certain new situations arise. I n this chapter, we shall discuss various problems relating to the concepts and results of Chapter I11 in the special context of linear spaces; in many cases, fairly complete results are obtained. I n this connection, it should be mentioned that linear topological spaces constitute the most important special case in the theory of quasiinvariant measures, or, at least, that most likely to find applications. For example, quantum field theory, stochastic processes, and the socalled generalized stochastic processes are all connected with the theory of quasi-invariant measures on linear topological spaces. 21 3

Iv.

214

LINEAR TOPOLOGICAL SPACES

Section 94.1 is primarily concerned with the study of the topology induced by the s-pseudometric on a quasi-invariant linear subspace. I n section $4.2, we discuss quasi-linear functionals, which are the analogs of the quasi-characters studied in Chapter 111. Since the theory of L2-Fourier transforms requires only a slight modification when passing from groups to linear topological spaces, this subject is touched upon only very briefly in $4.2. Section 94.3 is devoted mostly to the following three topics: (1) the countable additivity of cylinder measures on spaces of linear functionals (this topic is closely related to the concept of a commutative weak distribution, due to Segal [3, 4, 51); (2) representation theorems for positive-definite continuous linear functionals, these being among the most important results of harmonic analvsis on linear topological spaces; (3) measure-theoretic foundations of the theory of generalized stochastic processes. 94.1. Quasi-Invariant Measures on Linear Topological Spaces

lo Pseudometrics on Quasi-Invariant Linear Spaces We begin by introducing some new concepts.

Definition 4.1.1. Let 8 = (G, 23,p ) be a measure space, let 0 be a commutative group1 of measurable transformations of D, and suppose that a scalar2 multiplication is also defined on 0 in such a way that 6 becomes a linear space. Then, 6 is called a linear space of measurable transformations on Q. If, moreover, the functionall Ml(h) =

(s

((dp(g

C

+ h))1/2

-

(dp(g))1/2)2)1’z, h

€6

(4.1.1)

is well defined3 and quasi-continuous on 6, that is, for each fixed h E 0, the quantity M l ( t h ) is a continuous function of t , -GO < t < CO, then we say that the measure space 8 is quasi-continuous with respect to 6.

Lemma 4+1+1. Let 8 = (G, 23, p ) be a measure space, and let 0 be a linear space of measurable transformations on 8. Given any €L2(G,B,p), we form the convex function

g

+

Throughout the present chapter, when h E 8 and g E 8, we shall write h g (or - h) instead of dph(g). The researches in this chapter will be primarily concerned with real linear spaces. In particular, Ml(h) is always well defined if 8 is a finite measure space.

+ h) instead of hg, and d p ( g

4.1. Quasi-Invariant Measures

215

Suppose that M,(h) is quasi-continuous on 8. Define

Then (5 is a linear pseudometric space with respect to the invariant pseudometric p,(hl , h,) = R,(hl - h2), h, , h, E 8.I n particular, if D is quasi-continuous with respect to 8,then 8 is a linear pseudometric space with respect to pl(hl, h2). PROOF. It is easily verified that the three conditions in Lemma 1.1.2 of Appendix I are satisfied for M ( h ) = M,(h), therefore, 8 is a linear pseudometric space with respect to p,(., 3 Henceforth, unless otherwise stated, 8 will be regarded as a linear pseudometric space with respect to p1 (that is, rp = 1). .)a

Definition 4.1.2. Let D = (G, 8,p ) be a measure space and 8 a linear space of measurable transformations on 0.Suppose that, for each h, E 8, h, # 0, there is a real measurable function f on Q, such that, for almost all g E G, the equality

holds identically in t , -a < t < co. We then say that D is separated with respect to 8. For example, suppose G is a linear space, 8 is a linear subspace of G, and 5 is a family of linear functionals on G, such that, if h E 8,h # 0, then f ( h ) = 1 for some f E 5. Let 8 be the smallest o-algebra in G with respect to which all the functions in 5 are measurable, and let p be any measure on (G, 23). Then, (G, 8,p ) is separated with respect to the space of translations corresponding to 8 .

Theorem 4.1,2. Let D = (G, 8,p ) be a nontrivial4 finite measure space which is separated and quasi-continuous with respect to a linear space of measurable transformations 8.Then 8 is a linear metric space with respect to pl(hl , h2), h, , h, E 8. PROOF. By virtue of Lemma 4.1.1, it suffices to prove that p1 is a metric, that is, that R,(h) > 0 if h # 0. Suppose that h, E 8,h, # 0, and that R,(h,) = 0. Then, by (4.1.3) and the continuity of Ml(tho)with respect to t , it follows that M,(th,) = 0 for all t >, 0. But since Ml(-th,) = Ml(th,), this implies that, for That is, p ( C ) > 0.

216

Iv.

LINEAR TOPOLOGICAL SPACES

every real number t , Ml(tho) = 0, whence one may easily deduce that p f h , = p, that is, PL(E

+th)

(4.1.5)

= P(E)

for all - co < t < and all E E 23. Now, since D is separated with respect to 0, there is a real B-measurable function f on G such that (4.1.4) holds. Using this function, we construct a Borel measure on the real line as follows: for any Borel subset B of the real line, let

+

By (4.1.4), if B is any Borel set, then f - l ( B ) tho and f - l ( B differ by at most a p-null set. Hence, using (4.1.5), we deduce that

+ t)

for all -a < t < co,in other words, p1 is a translation invariant Borel measure on the real line. Therefore, pl is just the ordinary Lebesgue measure multiplied by some constant factor. But since p is finite (and nonzero), pl is also finite (and nonzero); this is a contradiction. We conclude that Rl(ho)= 0 is impossible if h, # 0, that is, p1 is a metric. ] I n Theorem 4.1.2, one cannot dispense with the assumption that SZ is separated with respect to 6. This is shown by the following example.

Example 4.1.1. Let G be a nontrivial linear space, let 2 ' 3 consist of just two sets, namely, G and the empty set 0, and let p(G) = 1, p ( 0 ) = 0. Then (G, d,p) is not separated with respect to the group of all translations G. I n this case, we have Rl(h) = 0 for all h E G. 20 Quasi-Invariance under Translations We proceed to specialize the situation by assuming that G is a linear space and that 0 is a linear subspace of G, operating on G by tran~lation.~ Suppose that SZ = (G, 23,p) is a measure space which is quasi-invariant (quasi-invariant and quasi-continuous) with respect to6 6, and suppose that (ti is not properly contained in any other linear subspace 6' of G such that D is quasi-invariant (quasi-invariant and quasi-continuous) with respect to 6'. Then we say that 8 is a maximal quasi-invariant (quasi-invariant and quasi-continuous) linear subspace relative to D. Henceforth, it will always be assumed, without explicit repetition, that linear subspaces are regarded as operating on their ambient spaces by translation. Or, in the phraseology of Chapter 111, quasi-invariant under 6.

217

4.1. Quasi-Invariant Measures Let N be a linear subspace of G such that P(E

+4

=P W

for every h E N and every E E 23. We then say that Q is invariant with respect to N (or invariant under N ) , or, alternatively, that N is an invariant linear subspace relative to SZ. Clearly, there exists a greatest invariant linear subspace relative to Q; we denote this subspace by 6,. I t is obvious that the s-pseudometric is identically zero on 6,, moreover, if Q is quasi-invariant and quasi-continuous with respect to the linear subspace 6, then 6 n6, = {X I p&, 0) = 0, x ~ 6 } . Using Zorn’s lemma, one may easily prove that, if G is a linear space and Q = (G, 23, p ) is a measure space, then there exists a maximal’ quasi-invariant quasi-continuous linear subspace 6 relative to Q, and obviously the greatest invariant linear subspace 6, (relative to Q) is contained in 6.

Theorem 4.1.3. Let G be a linear topological space, Q = (G, 23, p ) a regular finite measure space, and 6 a maximal quasi-invariant quasicontinuous linear subspace relative to Q. Then 6 is a complete linear pseudometric space with respect to the pseudometric pl(hl , h2), h, , h, E 6. PROOF. We need only prove the completeness of 6. Let {h,} be any fundamental sequence in 6 relative to p1 . Choose a sequence of natural numbers { n d , nl < n2 < - - * < nk < *.., such that Rdh, -A,)

when m, n and let

<

1

(4.1.6)

nk . Let p1 denote the Lebesgue measure on the real line,

Then, using (4.1.6), we get

’

Translotor’s note: The use of Zorn’s lemma here is unnecessary. In fact, it is easily verified that the linear hull of all the linear subspaces of G with respect to which R is quasi-invariant and quasi-continuous is actually the greatest linear subspace having this property.

Iv.

218

LINEAR TOPOLOGICAL SPACES

Hence, writing A = Ek.+m Ak , we have pl(A) = 0. Let Q = (0,I] - A . Then, for each t EQ, there is an integer k, such that t E (0,I] - A , , that is, 1 MOT%,+, - hnJ t )

<

for all K >, k, . Therefore, if t E Q, then

c m

Ml(t(h71b+l- hnJ)

< 00,

k=l

which implies that the sequence {thnk} is fundamental relative to the pseudometric M,(h, - h,), h, , h, E 8. T o simplify notation, we shall write k in place of n, throughout the remainder of the proof. Now, given any t , E Q , the argument used in the proof of Theorem 3.1.24 shows that there exists* an h, E G such that every neighborhood V of t,h, (in the original topology of G) contains a subsequence of {tOhk}.Hence, for any t E Q, the arbitrary neighborhood (t/t,) I/' of tho contains a subsequence of {thk}. Again, following the proof of Theorem 3.1.24, we deduce that, if t EQ, then lim Ml(t(/zk- h,)) = 0.

(4.1.7)

k+m

Let 8, denote the totality of elements (i.e., translations) of G with respect to which Q is quasi-invariant, and let Q, denote the totality of real numbers t such that tho E 6, . Obviously, 8 , is a group, and therefore Qo is an additive subgroup of the real numbers. Moreover, since th, E 8 , , it follows from (4.1.7) and Lemma 3.1.25 thate tho E 8, for every t E Q, that is, Q C Q, . Let Q1 denote the set of all real numbers t E Qo such that (4.1.7) holds. Then, we obviously have (4.1.8)

QCQ1*

We assert that Q1 is a group. I n fact, note that M I satisfies the relations Ml(h1

+ h,) < Ml(h1) + M,(h,), MI@) = M,(--h),

h, h, € 6 1 9

h €8.

>

(4.1.9)

Translator's note: At this stage, it has not yet been established that ha E 6.Therefore, it seems necessary to add the hypothesis that every translation of the space G is Bmeasurable. For otherwise, the quantities Ml(ha- hm), M ( t ( h , - ha)), and so on, would be undefined. Translator's note: Again, the hypotheses must ensure that the translation tho is 8-measurable.

4.1. Quasi-Invariant Measures

219

+

hence, t, t, E Q1 . Similarly, if t E Q1 , then -t group, and in view of (4.1.8), we have

Q - Q CQi

E Q1

. Thus, Q1 is

a

(4.1.10)

*

Since the Lebesgue measure p l is invariant, and pl(Q) > 0, the reasoning used in the proof of Theorem 3. l. 13 shows that Q - Q contains a neighborhood of zero on the real line. Hence, by (4.1.10), we see that Q1contains some neighborhood of zero, for example, the open interval (--6, 6). If t is an arbitrary real number, we choose a sufficiently large positive integer n, so that (t/n)E (--6, a), whence (t/n)E Ql ; since Q1 is a group, this implies that t = n(t/n)= (tin) (tin) € Q l . Thus, Q1 is the entire real line. Therefore, for every real number t, we have tho E 8,, and (4.1.7) holds. Let 8' be the linear hull of 8 and the set {tho I -co < t < co}. We shall now prove that Q is quasi-invariant and quasi-continuous with respect to 8'.The elements of 8' are of the form g toho , g E 8, - 00 < to < a.By (4.1.7), the sequence {tohk g, k = 1, 2 ,...} converges to toho g in the pseudometric M,(h, - h,), h, , h, E G. Since t,h, g E 8 C 8,, it follows by Lemma 3.1.25 that t,ho g E 8, . Let H , be the smallest closed linear subspace of L2(Q)containing the constant function I and invariant under all the operators {U((toh, g) t), -co < t < co, k = 1, 2, ...}. I t is easily verified that the totality of vectors cp eL2(Q),such that

+ + - - a

+

+

+

+

+

+

lim I l ( W t & k

k+m

+ g) t ) - W & o + g> 4)

QJ

I1 = 0

for all real t, forms a closed linear subspace which is invariant under all the operators (U((t,h, g) t ) , -co < t < GO, k = 1 , 2,...}, and by (4.1.7), this subspace contains 1, hence it contains Hl . This means that, for all real numbers t , the sequence of operators {U((tohk g) t ) , k = 1, 2, ...} converges strongly to U((toh, g) t ) on H , . Now, since Ml((t,hk g) t ) is a continuous function of t, we see from Lemma 3.4.16 that, for each k, the one-parameter group of unitary operators

+

+

+

{WtOhI, +g) t)l --co

+


is strongly continuous on H , . I t follows that H , is separable, and moreover, for each pair of vectors .f, 7 E H , , the function

220

Iv.

+

LINEAR TOPOLOGICAL SPACES

g) t ) 5, q), being the limit of the sequence of continuous functions (U((t,h, g) t) 4, T ) , is Lebesgue measurable. By virtue of Lemma 11.3.2, this implies that { U((t,h, g) t ) I -co < t < co} is a strongly continuous one-parameter group of unitary operators on If, . Hence, the quantity

f ( t ) = (U((t,h,

+

+

is a continuous function of t, that is, Ml is quasi-continuous on 6'. Thus, i2 is quasi-invariant and quasi-continuous with respect to the linear subspace 6' of G. Since 6' 3 6, it follows from the maximality of 0 that 8' = 8, so that h, E 6. Moreover, using (4.1.7) and the Lebesgue dominated convergence theorem, we get lim pl(hk , h,)

k+

w

=

lim R,(hk - h,) = 0.

k-rm

Therefore, 6 is complete. ]

Theorem 4.1.4. Let G be a linear topological space, (li a linear subspace of G, and (G, b,p ) a regular finite measure space which is quasi-invariant and quasi-continuous with respect to 6. If 6 is assigned the topology induced by the pseudometric pl(hl , h J , h, , h, E 6, then the functional M,(h) is continuous on 6. Moreover, if {h,} C 6, h, E 6, then pl(h, , h,) -+ 0 if and only if lim M,(t(hn - h,)) n+co

=

0,

-a

< t < co.

(4.1.11)

PROOF. We may assume that 8 is a maximal quasi-invariant quasicontinuous linear subspace relative to Q. For otherwise, we could replace 6 by a maximal quasi-invariant quasi-continuous linear subspace containing 6. Moreover, by Theorem 4.1.3 and Lemma 1.2.3, in order to prove the continuity of Ml(h), it suffices to prove that Ml(h) is lower semicontinuous. Suppose that {h,}C 6, h, E 6, and R,(h, - h,) --t 0. Choose a subsequence {h,)} of {h,} such that lim M,(h,,) = lim Ml(hn).

n'+w

n+ m

As in the proof of Theorem 4.1.3, since {ha,} is fundamental relative to p1 , there exists a subsequence {h,~}of {A,.} and an element h,' E 6, such that (4.1.12) ljm M,(t(h,e - h i ) ) = 0 n +w

22 I

4.1. Quasi-Invariant Measures

for every real number t . Since pl(hna , h,') -+ 0, we have pl(h, , A,') = 0. Hence, by the quasi-continuity of Ml , it follows that M I ( @ , - ho')) = 0

(4.1.13)

for all t . From (4. I. 12) and (4.1.13), we get 1jm Ml(t(h,- - h,))

n

--lm

= 0,

hence, Ml(h,)

< nlim (Ml(h,-) + Ml(hn+w

-

h,))

=

1i,m Ml(hn,)= lim M,(hn). +m

n

n+m

This proves that 34, is lower semicontinuous, and therefore continuous. Obviously, (4.1.1 1) is a sufficient condition for pl(h, , h,) -+ 0. On the other hand, if p l ( h , , h,) -+ 0, then, since 6 is a linear pseudometric space with respect to pl, it follows that pl(th, , tho)+.0 for every real number t . Hence, by the continuity of M , on 6, we obtain (4.1.1 1). 3 In particular, Theorem 4.1.4 shows that the topology 2, on 6 induced by the convex function M,(h) is weaker than the topology 2, induced by the pseudometric p, . The following theorem describes the relationship between 2, and 2, still more precisely.

Theorem 4.1.5. Let G be a linear topological space, 6 a linear subspace of G, and (G, 23,p ) a regular finite measure space which is quasi-invariant and quasi-continuous with respect to 6. Then the topology 2, on 6 induced by the s-pseudometric p, is the weakest among all those topologies which are stronger than the topology 2, [induced by M,(h), h E 61 and make 0 a linear topological space. PROOF.I t suffices to prove that any topology which satisfies the above two conditions is stronger than 2,.Let 2 be any such topology; then the functional M,(h), h E 6, is continuous with respect to 2. Consider any generalized sequence (h,} C 6 converging to a limit h, in the topology 2.Since 6 is a linear topological space with respect to 2, this implies that, for any real number t , the generalized sequence (th,} converges to tho in the topology 2. Hence, since M , is continuous with respect to 2, condition (4.1.1 1) holds, and therefore pl(h, , h,) -+ 0. This shows that 2 is stronger than 2, . ] Next, we examine the relationship between the topology on 6 induced by the s-pseudometric and the topology on 6 induced by the original topology of G.

Theorem 4.1.6. Let G be a linear topological space, 8 a linear subspace of G, and (G, 8,p) a regular finite measure space which is

222

Iv.

LINEAR TOPOLOGICAL SPACES

quasi-invariant and quasi-continuous with respect to 6. Then the topology 2, induced by the s-pseudometric p, on 6 is stronger than that induced by the original topology of G. Moreover, there exists a compact subset of G which contains a Z,-neighborhood of the identity 0 in 0 . PROOF. According to Lemma 3.1.22,there exists a compact subset K of G which contains a %,-neighborhood of zero in 6, say, the neighborhood { h I M,(h) < e, h E s}, > 0. But then K must contain a %,-neighborhood of zero in 6, for example, {h I pl(h, 0) < 6, h E 0 } , 6 > 0. For otherwise, one could find a sequence {h,} C 6 such that pl(h, , 0) < I /n and h, K. However, by Theorem 4.1.4, pl(h, , 0) -,0 implies that M,(h,) -,0, so that M,(h,) < E for some n, and therefore h, E K , which is a contradiction. Consequently, K does contain a %,-neighborhood of zero in 6, and it follows easily that the inclusion mapping of (5 (with the topology 2,)into G is completely continuous, hence, afortiori, continuous. Thus, the topology 2, is stronger than the topology on 6 induced by G. ]

Corollary 4.1.7. Let G be a linear topological space which satisfies some separation axiom, let 8 be a linear subspace of G, and let (G, 9, p ) be a regular finite measure space which is quasi-invariant and quasi-continuous with respect to 6. Then the corresponding s-pseudometric p1 is actualIy a metric on 6. PROOF. By Theorem 4.1.6, the topology 2, induced by p1 is stronger than the topology on 6 induced by G; since the latter satisfies a separation axiom, 2, certainly satisfies the same separation axiom. Therefore, p1 is a metric. ] 30 Topological Characterization of the s-Pseudometric

Theorem 4.1.8. Let G be a linear topological space, 6 a linear subspace of G, and (G, 9, p ) a regular finite measure space which is quasi-invariant and quasi-continuous with respect to 6. Suppose that 8 itself is a linear topological space of the second category with respect to a certain topology 2 which is stronger than the topology induced by G. Then the functionals M,(h) and R,(h) are both continuous with respect to 2 ; in other words, 2 is stronger than the topology 2, induced by the s-pseudometric p, . PROOF. Since M,(h) is lower semicontinuous with respect to the topology on 6 induced by G (see Corollary 3.1.21), M,(h) is also lower semicontinuous with respect to 2. Furthermore, by quasi-continuity, we have limn-ts, M,((l/n)h)= 0. Hence, using Lemma 1.2.3 of Appendix I, we see that M,(h) is continuous with respect to 2,that is, the

4.1. Quasi-Invariant Measures

223

topology 2, on 8 induced by M , is weaker than 2. Since 8 is a linear topological space with respect to 2, it follows immediately from Theorem 4.1.5 that 2 is stronger than 2,. ] Using Theorems 4.1.3 and 4.1.8, we can derive the following characterization of the topology induced by the s-pseudometric on 6.

Theorem 4.1.9. Let G be a linear topological space,let SZ = (G, 23, p) be a regular finite measure space, let 8 be a maximal quasi-invariant quasi-continuous linear subspace associated with 52, and let 8, be a maximallo invariant linear subspace relative to SZ, such that 8,C 8. Let p be an invariant pseudometric on 8, with the following properties. (i) 8 is a compIete linear pseudometric space with respect to p ; (ii) (5, = {x I p ( x , 0) = O } ; (iii) the topology 2 induced by p is stronger than the topology on 8 induced by G. Then, 2 is just the topology 2, induced by the s-pseudometric on 8. PROOF. Consider the quotient space 8, = 8/8,, and for any x E 6, let 2 denote the residue class x 6,. Since 8, = {x I pl(x, 0) = 0}, we may regard p and p1 as metrics defined on 8,:

+

P(2, P) = P ( X , Y ) , x E 2, Y

€9;

Pl(% 9) = Pl(% Y ) , x E 2, Y

€9.

Moreover, 8, is also a complete linear topological space with respect to both p and p,. Now, by a well-known theorem, a complete linear pseudometric space is of the second category; hence, Z satisfies the conditions of Theorem 4.1.8, and therefore 2 is stronger than 2, . From this, it follows easily that the topology 2* induced by p on 8, is stronger than the topology 2,* induced by p1 on 6,. This means that the identity mapping $ -+ 2, regarded as a linear operator from (8,, 2*)onto (8,, 2,*),is continuous, and therefore (by the Banach inverse operator theorem), its inverse is also continuous, that is, Z1* is stronger than 2*.Thus, 2* = Z1*, from which we easily deduce that

2=2,. ] Notice that the conditions (i), (ii), and (iii) in Theorem 4.1.9 are satisfied by p1 itself, so that these conditions characterize the topology 2, . Thus, the topology 2, is completely determined by 6, 8, and the relative topology of 6 as a subset of G. I n other words, to determine 2,, one need not know the explicit form of p ; it suffices to know the corresponding maximal quasi-invariant quasi-continuous linear subspace and maximal invariant linear subspace. lo Translator’s note: As indicated in footnote 7, both 8 and (5, are uniquely determined, being in fact the greatest subspaces in their respective classes, and the relation 8, C 6 always holds.

Iv.

224

LINEAR TOPOLOGICAL SPACES

Let G be a linear topological space satisfying some separation axiom, let (G, B,p ) be a regular finite measure space, and let 6 be the corresponding maximal quasi-invariant linear subspace. Let p be a metric on 6, such that, (i) 6 is a complete linear metric space with respect to p ; (ii) the topology 2 induced by p is stronger than the topology on oi induced by G. Then 2 is just the topology 2, induced by the s-pseudometric p, on 6. PROOF. By virtue of Corollary 4.1.7, we have {x I p,(x, 0) = 0} = (0). Hence, the maximal invariant linear subspace 6, reduces to {0}, and the desired conclusion follows immediately from Theorem 4.1.9. ] Next, we consider conditions for the quasi-continuity of the functional MW@). Lemma 4.1.11. Let G be a linear topological space, 6 a linear subspace of G, and SZ = (G, 23, p) a regular finite measure space which is quasi-invariant and quasi-continuous with respect to (5. Then, for each cp E L2(SZ),the functional M J h ) is quasi-continuous on 6, moreover, the topology on 6 induced by the pseudometric R,(h, - h,) is weaker than the s-topology 2,. PROOF. By Theorem 4.1.6, the topology on 6 induced by G is weaker than 2, . Hence, by Lemma 3.1.7, 2, is an admissible topology on 6. Furthermore, by Theorem 4.1.4, the functional M,(h) is continuous on (6,&), hence, it follows by Theorem 3.1.30 that 2, is stronger than the p-topology on 6. Consequently, €or each y €L2(SZ), the functional M J h ) is continuous on (6,Z,);since (6,2,)is a linear topological space, this obviously implies that M,(h) is quasi-continuous. Now, suppose that {h,} C 6, pl(h,, 0) = 0; then

Corollary 4.1.10.

lim M,(th,) n+m

= 0,

-00

< t < 00,

and therefore limR,(h,) n-m

= 0.

This shows that 2, is stronger than the topology induced by RQ(hl- h,), hl,h,E@. 1 Theorem 4.1.12. Let G be a linear topological space, let (G, 23, ,pk),

K

= 1,2, be regular finite measure spaces, with the respective maximal quasi-invariant quasi-continuous linear subspaces 6 k , K = 1, 2. Let p k denote the s-pseudometric on O k corresponding to the measure space (G, 23, , pk). If 6, C 6, , then the topology 2, on 6, induced by pz is stronger than the topology 2, on 6, induced by p1 .

225

4.1. Quasi-Invariant Measures

PROOF. By Theorem 4.1.3, 6, is a complete linear pseudometric space with respect to p , , hence (6, ,2,)is of the second category. Furthermore, (G, 8, ,p l ) is quasi-invariant and quasi-continuous with respect to 6, , and, by Theorem 4.1.6, 2, is stronger than the topology on 6, induced by G. Hence, the desired conclusion follows immediately from Theorem 4.1.8 (with p = pl). ] 40 Lebesgue-Type Measure Spaces

We proceed to investigate a type of measure space which is frequently encountered in the applications, and whose quasi-invariant linear subspaces always satisfy the quasi-continuity condition.

Definition 4.1.3. Let Q = (G, 8,p ) be a measure space such that there exists a sequence of linearly independent real measurable functions {fk} which form a determining set on a, and such that, for each n, the finite-dimensional measure corresponding to fl ,...,fn is equivalent to Lebesgue measure, that is, the measure p, on n-dimensional real space R, , defined by F n ( 4

=

CL(k I (f1(&9,-vfn(d)

E

4)

(where A denotes an arbitrary Borel set in R,) is weakly equivalent to the Lebesgue measure on R, . We then say that (G, 8,p ) is a Lebesguetype measure space. For example, let G be the Cartesian product of a sequence of real lines {R,I a E 'u}, let 8 be the a-algebra generated by the totality of Borel cylinders in G, and let p be a probability measure on (G, 23). For each a E 'u, g E G, let g, denote the ath coordinate of g, and let g, be regarded as a random variable on (G, 8,p). If, for any finite set of indices a1 ,..., a,, the joint distribution of the random variables g,% ,...,ganis equivalent to Lebesgue measure (e.g., if {g, I a E a} is a family of nondegenerate Gaussian variables), then ( G , 8 ,p ) is a Lebesguetype measure space. T h e following lemma gives a sufficient condition for a measure space to be of Lebesgue type.

Lemma 4.1.13. Let G be a linear space, 6 a linear subspace of G, and fl ,..., f, ,... a sequence of real linear functionals on G, with the following property: for each n, there exist vectors h, ,..., h, E 8 such that (4.1.14) f,(h,) = 6,, , k , 1 = 1, 2,..., n. Let 8 be the smallest a-algebra of subsets of G which makes all the

226

Iv.

LINEAR TOPOLOGICAL SPACES

functionsfk measurable, and let p be a a-finite measure on (G, 8 )which is quasi-invariant under 8. Then, (G, b,p) is a Lebesgue-type measure space. PROOF. For each n, we define a Borel measure p, on n-dimensional Euclidean space R, , as follows: if A is any Borel set in R, , then P n ( 4 =

4).

P ( b J I ( m ) ? * * * , f n ( gE) )

Now, for any y = (yl , . . . , y , )E R, , we see from (4.1.14) that there is a vector h E 8 such that fk(h) = y k , k = 1, 2,..., n. Therefore, Fin(A

+Y )

=

Pug I K(g),--Jn(gN E 4

+4 +

Since p is quasi-invariant under 6, it follows that $,(A y ) = 0 if and only if p,(A) = 0. Thus, p, is a Borel measure on R, which is quasi-invariant under translation. According to Theorem 3.1.5, p, is equivalent to Lebesgue measure.ll ]

Theorem 4.1.14. Let G be a linear space, and let Q = (G, b,p) be a finite Lebesgue type measure space which is quasi-invariant under a linear subspace 8 of G. Then, for any eL2(52),the functional M,(h)(h E 8)is quasi-continuous. In particular, 52 is quasi-continuous with respect to 8. PROOF. For convenience, we take p ( G ) = 1. By hypothesis, there is a sequence of measurable functions {fk I k = 1, 2,...} on SZ such that, for each n, the finite dimensional measure p, , corresponding to fl ,...,f, , is equivalent to Lebesgue measure. Let Fn(xl ,..., x,) denote the RadonNikodym derivative of p, with respect to the Lebesgue measure on R, ; we may suppose that F, is a Baire function such that 0 < Fn < 03. Then, for any y = (yl ,...,y,), the Radon-Nikodym derivative of the measure p,(A y ) with respect to p,(A) is

+

(4.1.15)

Let 23, denote the smallest a-algebra in G which makes the functions p, be the probability measure obtained by restricting p to b, . Then, using expression (4.1.15), we have

fl,...,f, measurable, and let

l1 Translator’s note: The hypothesis of Theorem 3.1.5 requires that both of the measures concerned be o-finite. In the present instance it is not obvious that pn is o-finite. Of course, if p is finite, then there is no difficulty.

4.1. Quasi-Invariant Measures

227

for any fixed h E 6. Let & denote the totality of Bore1 subsets of the real line R, . Then, the function

is measurable with respect to 3, x B. Now, by Theorem 1.1.20,12 for each t , there is a p-null set E , E b such that limn+mdp,(g th)/dpn(g) exists and

+

(4.1.16)

for all g E G - E , . Let E be the set of all pairs ( t , g ) such that limn+mdp,(g th)/dp,(g) does not exist. Since dp,(g th)/dp,(g) is a measurable function on ( R , x G, 5, x S),it follows that E is a measurable set. Also, for each t o , we have {g 1 ( t o ,g ) E E ) C Eta. Hence, by Fubini’s theorem, E is a null set relative to the measure p, x p (where pl denotes the Lebesgue measure on R,). Therefore, limn-rmdp,(g th)/dp,(g) exists for almost all ( t , g) (relative to the measure p, x p), moreover, this limit is a measurable function on (I?, x G, x b,pl x p), and by (4.1.16), it may be regarded as the Radon-Nikodym derivative dp(g th)/dp(g). Thus, dp(g th)/dp(g) is a measurable function on (R, x G, 5, x b,pl x p). If the function t ( g ) is .%-measurable, one can prove that t ( g th) is a measurable function13 on (R, x G, 5, x b). Using Fubini’s theorem, we deduce that, if 4, 71 eL2(SZ),

+

+

+

+

+

+

is a measurable function on (R, , 5,). Consider the one-parameter group of unitary operators {U(th) I -a

< t < co}

where U ( * )is defined as in $3.1. Since (4.1.17) is just the inner product (t,U(th)T), the unitary operator group {U(th) I -m < t < m) is Translator’s note: Theorem 1.1.20 is directly applicable only if one assumes that {jk} is a determining set for the measurable space (G, 9),and not merely for the measure space (G, 9, p ) (see Definitions 1.1.6 and 1.1.7). Is Translator’s note: This requires some additional assumption which will ensure the th, for example, the measurability of the measurability of the mapping (g,t ) + g g’ and ( g , t ) --* g t . linear operations ( g , g’) 4 g

+

+

228

Iv.

LINEAR TOPOLOGICAL SPACES

weakly measurable. But since 52 has a countable determining set, L2(52) is separable. Hence, by Lemma 11.3.2 of Appendix 11, {U(th) I -co < t < co} is strongly continuous with respect to the parameter t. In particular,

M,W

= II (U(W -

4 v II

is a continuous function of t for each rp €L2(52). ] 50 Strongly Cyclic Quasi-Invariant Measure Spaces

Lemma 4.1.15. Let 52 = (G, 23, p) be a localizable measure space which is quasi-invariant under a linear space of measurable transformations 8. Suppose that 52 is strongly cyclic relative to 8, with the %-cyclic element yo €L2(S2),and that the functional Mq0(h),h E 8, is quasi-continuous. Then, for any rp E L2(52),the functional R,(h), h E 8, is continuous with respect to the topology induced by the pseudometric R,o(hl - h,), h, 9 h2 E 8. PROOF. According to Corollary 3.4.17, if rp €L2(52),then the functional M,(h), h E 8, is continuous with respect to the topology induced by M,Jh), h E 8.Therefore, since M,o(th), - co < t < co,is continuous in t, it follows that, for any rp €L2(52),the functional M,(th), -co < t < co, is also continuous in t, so that R,(h) is well defined, moreover, by Lemma 4.1.1, 8 is a linear pseudometric space with respect to the pseudometric R,(h, - h2),h, , h, E 8. Consider any sequence {h,} C 8 such that lim RWO(h,) = 0. n+m

(4.1.18)

We shall prove that lim R,(h,)

n+ m

= 0.

(4.1.19)

If (4.1.19) were false, there would exist a subsequence {h,,} of {h,} such that lim R,(h,,) = c

n'+m

> 0,

(4.1.20)

where possibly c = +a.Now, (4.1.18) implies that, on any finite interval [0, TI, the sequence of functions {M,o(th,,)} converges to zero in Lebesgue measure. Consequently, there is a null set A C [0, co), and a subsequence {n"} C {n'}, such that lim M,,(th,-)

n"+m

=0

(4.1.21)

4.1. Quasi-Invariant Measures

229

for every t C A . Since M,(h) is continuous with respect to the topology induced by M,o(h), it follows from (4.1.21) that (4.1.22)

lim M,(th,e) = 0

n"+m

for every t C A . Using the Lebesgue dominated convergence theorem, we deduce that lim R,(hn.) = Ijm

n"+m

n

+m

(J, e-tM,(th,*)2dt) gi

112

= 0,

(4.1.23)

which contradicts (4.1.20). Thus, we conclude that (4.1.19) holds. ]

Corollary 4.1.16. Under the hypotheses of Lemma 4.1.15,14 the pseudometric R,o(h, - h J , h, , h, E 8, induces the s-topology on 8. PROOF. By Lemma 4.1.15, the s-topology is weaker than the topology 2, induced by RWo(h,- h,), and by Lemma 4.1.11, 5, is weaker than the s-topology. Hence, these two topologies coincide. ] Using the proof of Lemma 4.1.15, we may rewrite Corollary 3.4.18 in the following form. Corollary 4.1.17, Let Qk = (G, b,pk), R = 1, 2, be Iocalizable measure spaces which are quasi-invariant under and strongly cyclic relative to the linear space of measurable transformations 6, and let vk E L2(Qk),k = 1, 2, be corresponding 6-cyclic elements. Suppose that the functions

are quasi-continuous on 8, and let

If p, is absolutely continuous with respect to pz , then the pseudometric R(l)(h,- h2), h, , h, E 8 , is continuous with respect to the topology induced by the pseudometric R(2)(h,- hz). In particular, if p, and pz are equivalent, then R'l) and are topologically equivalent. Under certain circumstances, one can obtain results similar to the l4 Translator's note: Since Lemma 4.1.11 is used in the proof, it appears that the hypotheses of Lemma 4.1.11 should also be included here. In particular, the s-topology may not be defined if p is not totally finite.

Iv.

230

LINEAR TOPOLOGICAL SPACES

above, but not explicitly involving cyclic elements. For example, we have the following theorem.

Theorem 4.1.18. Let G be a linear topological space, let 8 be a linear subspace of G, and let Q, = (G, 8 , pk), K = 1,2, be regular finite measure spaces which are quasi-invariant, quasi-continuous and strongly cyclic relative to 8. Let

If p, is absolutely continuous with respect to p 2 , then R:”(h) is continuous with respect to the topology induced by Ri2)(h)on 6. If p, and pz are equivalent, then Ril)(h) and Ri2)(h)induce the same topology on 8. PROOF. I t follows from Lemma 4.1.15 that the functions MWk(h) are quasi-continuous, where F k is a cyclic element corresponding to p k , = 1, 2. Hence, Corollary 4.1.16 is applicable, so that Rik’(h)and R(k)(h) induce the same topology on 8. Thus, the conclusions of Theorem 4. I. 18 follow immediately from Corollary 4.1.17. 3 Theorem 4.1.18 may be used as a criterion for the inequivalence of two quasi-invariant measures.

k

Example 4.1.2. Let R,, 01 = 1 , 2,..., n ,... be a sequence of copies of the real line, let 8, be the totality of Bore1 sets in R, , and let 2, denote the topology on R, defined by the usual metric. Let 1I-= .:() R, , m 8,= )(,=18,. We regard I as the totality of real number sequences x = {xI ,...,x, ,...>,and define linear operations in 1 as follows. If x = {xl ,..., x, ,..,}, y = { y , ,..., y, ,...} E I, and a, b are real numbers, then ax by = (ax1 by, ,..., ax, by, ,...}.

+

+

+

Obviously, 1 is a linear space with respect to these operations. Let ( I , % ) denote the topological product of the spaces ((1, , Z,), a = 1, 2, ...}. T h e topological space (I, 2) is metrizable, in fact, 2 is just the topology induced by the metric

For each a, let pa be a probability measure on (R,, B,), equivalent to Lebesgue measure, with the probability density f a 2 , 0 < f a < 00: p m =

J

E

fU2(t)

E E 8, *

23 1

4.1. Quasi-Invariant Measures

Form the product measure p = )(a-l pa . Then (1, 23, ,p) is a regular Lebesgue-type probability measure space. For any y = {yl ,...,yn ,...} E 1, define a measure pl/ by m

+Y ) ,

P,(E) = P(E

E

6

23,

*

Using Theorem 1.4.4, one easily calculates that P(PV > P ) =

nJ m

m

k-1

--m

fk(.

+

YP)fk(.)

(4.1.24)

d..

Let 3, denote the ordinary Fourier transform of fk : fk(t) = 1.i.m. T+m

l~

fk(u)eiut du.

~

Sincef, is a real function, we havef,(t) = J(;(-t). theorem, we get

Using the Plancheral (4.1.25)

By another well-known theorem, a necessary and sufficient condition for the convergence of the infinite product (4.1.25) is the convergence of the series (4.1.26)

Again, by the Plancheral theorem,

hence, the series (4.1.26) may be rewritten as

c

I fk(t>12 --m

sin2

9

(4.1.27)

dt.

Let (ti,, denote the totality of points in 1 such that (1, 23, ,p) is quasiinvariant under the corresponding translation. Then, (ti,, is just the totality of sequences y = {yl ,..., y, ,...} such that (4.1.27) converges. Far any measure p of the type under discussion, 0, obviously includes the subspace (ti consisting of all points of the formy = (yl ,...,yn, 0, 0,...} (i.e., for some n, the coordinate y, = 0 for all k 3 n 1). Using Theorems 3.1.32, 4.1.5, and so on, one may verify that the measure

+

Iv.

232

LINEAR TOPOLOGICAL SPACES

space = (1, 8 , , p) is quasi-invariant, quasi-continuous, ergodic, and strongly cyclic1s relative to 6. I n particular, if f k is absolutely continuous on every finite interval, and ak

=

j

00

lfi(t)12

dt

< co,

K

=

1 , 2,...,

(4.1.28)

-m

then, since

we obtain

Therefore, since sin2(ty,/2) whenever

< (tYk/2)’,it follows that (4.1.27)converges m

‘kyk2 k=l

<

(4.1.29)

Thus, under condition (4.1.28), 6, contains the Hilbert space l({ak}), which is defined as the totality of all sequences y = {y”} in 1 which satisfy (4.1.29), with the inner product ( y , z ) = C where { y l )...,Y k ,***}, z = (z1 ,..., z k E l(ak). y ,..a}

g4.2. Linear and Quasi-Linear Functionals on Linear Spaces

l o Linear Functionals and Quasi-Continuous Characters on Linear Spaces Let @ be a real linear space, and let @ A denote the totality of real linear functionals on @. Define linear operations in @*as follows: if f,g E @A, and a, /3 are real numbers, then af + /3gis the linear functional

Thus, @ A forms a linear space, which we call the algebraic dual of @. I n particular, we note that @ A forms a group with respect to addition. T h e role of @” in the present chapter is analogous to that of the algebraic dual of a group in Chapter 111. Translator’s note: More details regarding the verification of strong cyclicity would be welcome.

4.2. Linear and Quasi-Linear Functionals

233

Definition 4.2.1. Let @ be a linear space and let a be a character on @. If, for every fixed g, E@, cu(tg,) is a continuous function of t , -a < t < 03, then we say that OL is quasi-continuous. Let @’ denote the multiplicative group of all quasi-continuous characters on @. For each f E @ A , define

fyV) =

(4.2.1)

VE@-

7

Since f is a real-valued linear functional, f ’ is a quasi-continuous character. Thus, we obtain a mapping from @ A to @’: (4.2.2)

A :f-ff’.

Obviously, A is a homomorphism. Moreover, A is monomorphic; in fact, iff, g E @ A and f’= g‘, then, for each q E 0,we have e i l f ( Q= ) eifg(@) for all real t , hencef(rp) = g(g,).

Lemma 4.2.1. Let @ be a linear space and let 01 be any quasicontinuous character on @. Then, there is an f E @A such that

PROOF. From our hypothesis, it is obvious that for each fixed ‘p E @, a(tg,) is a continuous character on the additive group of real numbers R (with the Euclidean topology). Hence, by Example 3.2.4, there is a unique real number f (g,) such that a(tp)) = eitf(Q),

-co < t

< co.

(4.2.3)

I t only remains to prove that the correspondence q + f ( g , ) is a linear functional on @. If s E R and v E @, then eitf(sQ)

for all t

E

=

R, hence $ ( y ) =f(sp). Similarly, if eitf(W+lL) =

a(t(V + #))

+

(4.2.4)

= eitsjw

=

.(tV) or(+)

‘p, t,b E

@, then

= eitlf(Q)+f(G))

+

for all t E R , hencef(g, +) = f ( g , ) f($). 3 Thus, the mapping A is an isomorphism from

@A

onto @’.

Definition 4.2.2. Let @ be a linear topological space, and let @+ consisting of all continuous real linear denote the linear subspace of functionals on @. We call @+ the conjugate space of @. In this case, @ is, in particular, a topological group with respect to

Iv.

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LINEAR TOPOLOGICAL SPACES

addition. The role of Qt in the present chapter is analogous to that of the dual group @* in Chapter 111.

Theorem 4.2.2. Let @ be a linear topological space. Then, the restriction to Qt of the mapping A is an isomorphism from Qt onto @*. PROOF. Obviously, A@t C @*. Moreover, if a~ @*, then, by Lemma 4.2.1, there is an f e @* such that a(p))= eif(q),y E (9. Thus, we need only prove that f is continuous. Now, by the continuity of 01, the set Dt

= {P

I 49) = 11

is closed in @ for every t E R. Consequently, the set D = r)--co
+

+

<

f(%

+

<

XCp)

f

0,

or, hf (p)) # f ( y o ) . It obviously follows that l.f(p))[ < If(yo)l whenever E W. Therefore, given any # E G and E > 0, we have

y

+

provided p) belongs to the neighborhood # (e/If(yo)l) W. This shows that f is continuous. Hence, A@+= @*. Having already proved that A is monomorphic, we conclude that A maps cPt isomorphically onto @*. ]

Definition 4.2.3. Let @ be a linear space, and let B be a a-ring consisting of certain subsets of @. Let 23, denote the a-algebra of all Bore1 subsets of the real line R , . Suppose that 8 satisfies the following conditions: (i) for each fixed Y E @ , the correspondence t-+tp) is a measurable mapping from ( R , , 23,) to (@, 23); (ii) for each fixed real number t , the correspondence y -+ tp) is a measurable mapping from (@, 23) to (@, 23). Then, we say that (@, 23) is a linear measurable space, and if p is a measure on (@,B), then we say that (@, 23, p ) is a linear measure space. We now proceed to consider the relation between measurable characters and measurable linear functionals. If (@, 23) is a linear measurable space, we let @B denote the totality of

4.2. Linear and Quasi-Linear Functionals

235

real 8-measurable linear functionals on @. Obviously, @, linear space with respect to the ordinary linear operations.

forms a

Theorem 4.2.3. Let (0, 23) be a linear measurable space. Then, the mapping A: f

-+

eif

is an isomorphism from the additive group @aonto @a(the multiplicative group of all measurable characters on @). PROOF. Clearly, A is a monomorphism from @a into GB;we need ,, then 01 is only prove that A@B = @'. First, we observe that, if 01 E @ quasi-continuous. In fact, by virtue of property (i) in Definition 4.2.3, it is obvious that, for each (PE@, the correspondence t + c r ( t p ) is a 23,-measurable character on the additive group of real numbers R, . Hence, by Example 3.2.4, a(tg))(tE R,) is a continuous function of t , that is, 01 E @'. Therefore, given any 01 E @', it follows by Lemma 4.2.1 such that that there is an f E a(?) = e i f ( a 9)

(4.2.5)

YE@,

that is, 01 = A f . Now, if A is any Bore1 subset of the complex plane, then, by the measurability of a,we know that {p' I a ( q )E A } E 8 ,hence, by property (ii) in Definition 4.2.3, for each fixed t # 0, we have

that is, a(tg?)is a measurable function on (@, 23). But, from (4.2.5),we get f ( ~= ) lim in 1 - 01 - q ~ . n+m

(

(:

1)

Thus,f(p) is the limit of a sequence of measurable functions on (@, d), and is therefore measurable, that is,fE @B . ] In analogy with Definition 3.2.4, we have:

Definition 4,2.4. Let SZ = (@, 8 , p ) be a linear measure space, with p(@) < CO, and let J be a linear subspace of @ a . T h e function

#(f) = J

0

eifCrn) MY),

f E

J

will be called the characteristic function of SZ (on J ) .

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236

LINEAR TOPOLOGICAL SPACES

20 Quasi-Linear Functionals In analogy with the quasi-characters of Chapter 111, we now introduce quasi-linear functionals.

Definition 4.2.5. Let G be a real linear space, 8 a linear subspace of G, and (G, 8,p) a measure space quasi-invariant under 8 . Let f be a measurable real-valued function on (G, 8), satisfying the following conditions: (i) f ( t g ) = t f ( g ) for every real number t and every g E G; (ii) for each h E 8 , there exists a real number f ( h ) and a set Eh E 8, p(G - Eh) = 0, such thatl6 f(h

+ td

=

tf(g) + A h )

(4.2.6)

for every g E Eh , - 03 < t < 03. Then, we call f a quasi-hear functional on (G, 23, p ) (relative to 8 ) . T h e totality of such functions will be denoted by G, . Observe that f is a linear functional on 8 . In fact, the additivity of f can be shown as in $3.2; as for homogeneity, if s is a real number, s # 0,g E Eh , then, by (4.2.6), we have

hencef(sh) = sf(h). We say that f is the linear functional induced by f , or that f is a quasilinear functional corresponding to 3. The totality of such induced functionals f will be denoted by 6, . Clearly, two quasi-linear functionals which are equal almost everywhere induce the same linear functional on 8 . In the ensuing discussion, quasi-linear functionals which are equal almost everywhere will be regarded as identical. In analogy with property I11 in $3.2, 20, we have the following result.

Lemma 4.2.4. Let G be a real linear space, 8 a linear subspace of G, and (G, 8,p ) a measure space which is quasi-invariant and ergodic with respect to 6. If fl ,f 2 E G, and f1 = f 2 , then f l - f 2 is almost everywhere equal to some constant. PROOF. Arguing as in the proof of §3.2,2",111,we see at once that, for any two real numbers c1 , c 2 , c1 < c 2 , the set { g I c1 < f 2 ( g ) -f l ( g ) < c2} l6 Translator's note: The notation f(h) seems redundant, since, by setting t = 0 in (4.2.6), one gets f ( h ) = f(h). Perhaps the author's original intention was to consider arbitrary commutative groups (or linear spaces) operating on (G, 8,p), and not merely linear subspaces of G.

4.2. Linear and Quasi-Linear Functionals

237

is quasi-invariant. By the ergodicity of p, it then follows that there exists a real number c such that the set { g I f , ( g ) - f l ( g ) = c> differs from G by a set of measure zero. ] I n particular, suppose that 8 does not contain any set E such that (i) E is quasi-invariant, (ii) p(G - E ) = 0, and (iii) for each g E G, g # 0, the line {tg I -a < t < a}intersects E in at most one point. Then, from the proof of Lemma 4.2.4, it can be seen that the abovementioned constant c must be zero, that is, fl and fi are equal almost everywhere. Let 23 denote the totality of elements of G, corresponding to the linear functional 0 in 6 ~Then, . we have the natural isomorphism between the quotient space G,, = GU/%and ,@t that is,

2:7 4, 7 E G,,,

(4.2.7)

where q denotes the residue class consisting of all the quasi-linear functionals corresponding to the same linear functional 3. Next, we derive another corollary of Theorem 3.2.3.

Corollary 4.2.5. Let G be a linear topological space, and let 8 be a linear subspace of G which is itself a linear topological space of the second category with respect to a topology 2 which is stronger than the topology on 8 induced by G. Let (G, 8,p) be a regular measure space which is quasi-invariant under 8. Then, every quasi-linear functional on (G, 23, p ) (relative to (ti) induces a continuous linear functional on (ti. I n particular, every real 8-measurable linear functional on G, when restricted to 8,is continuous with respect to the topology 2. PROOF. L e t f E G,, , and consider the function u(g) = eif(g!,

g E G.

Clearly, CY is a quasi-character on G, and by Corollary 3.2.4, 01 induces a continuous character on 8. However, by (4.2.6), if h E (ti, then eif(g+h)

= eif(A)eif(u)

holds for almost all g, hence ~ ( h= ) eif(h).

Since d is continuous, it follows by Theorem 4.2.2 thatfis continuous. ] One can also give a direct proof of Corollary 4.2.5 by the method used in the proof of Theorem 3.2.3.

238

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30 Topologies on Spaces of Linear and Quasi-Linear Functionals Paralleling $3.2, 30, we proceed to describe a number of useful topologies for spaces of linear functionals. I. Let G be a linear space, and let J be a linear space consisting of certain linear functionals on G. Given any fo E J , any finite set of elements g , ,...,g , E G, and any E > 0, form the set

Taking the totality of sets of this form as a neighborhood basis, we obtain a topology on J , which we call the weak topology. It is easily verified that, with this topology, J forms a linear topological space. If one also takes the weak topology on A ] , then A is a continuous mapping1’ from J onto A J ; this fact follows readily from the inequality I 1 - eif(f7) I I f ( g ) l . However, in general, A is not a homeomorphism. 11. Let G be a linear topological space, J a linear subspace of Gt. Given any fo E J , any bounded subset Q of G, and any E > 0, form the set

<

Taking the totality of sets of this form as a neighborhood basis, we obtain a topology on J , which we call the strong topology. I t is easily verified that, under this topology, J forms a linear topological space. T h e strong topology is obviously stronger than the weak topology; in general, these two topologies do not coincide. If G is a linear topological space satisfying the first axiom of countability, then, under the strong topology, G t forms a complete Hausdorff linear topological space satisfying the first axiom of countability.l8 111. Again, let G be a linear topological space, J a linear subspace of Gt, and U a neighborhood of zero in G such that every functional in J is bounded on U. Define

For the definition of A, see (4.2.1) and (4.2.2). T~unslator’snote: To obtain the stated conclusions, the hypothesis that G satisfies the first axiom of countability should be replaced by “there exists a bounded neighborhood of zero in G.” Note that when Gt satisfies the first axiom of countability, it follows from the theory of linear topological spaces that Gt is pseudometrizable, which, combined with the Hausdorf€ property, implies that G+ is metrizable.

4.2. Linear and Quasi-Linear Functionals

239

Clearly, J is a normed linear space with respect to the norm I ,f 1 ; the corresponding topology ZUhas the neighborhood basis

at the point fo . We call ZUthe U-topology. Observe that the U-topology on J is equivalent to the topology induced by the homogeneous convex functionls

Given any bounded subset Q of G, there exists a positive number S such that SQ C U, hence W(f0 ; U, 8 4 c

w o,

Q!

.).

Consequently, the U-topology is stronger than the strong topology; in general, these two topologies do not coincide. However, if G is a normed linear space, and if we take U = (X I I] x 11 < l } (where /I x denotes the given norm on G), then all the functionals in G+ are bounded on U. In this case, the U-topology on J is equivalent to the strong topology, and is, in fact, just the topology induced by the usual norm supzELI If(x)l = 11 f 1 . IV. Let G be a linear space, 6 a linear subspace of G, and (G, d,p ) a measure space quasi-invariant under 8. Let J be a linear space consisting of certain quasi-linear functionals on (G, 23, p ) (relative to 8 ) . For each set A E 23 such that p ( A ) < oc), define

Then, RA(.)is a homogeneous convex function on J. T h e family of functions { R A ( . )induce } a topology on J as indicated in $1.1 of Appendix I, namely, for each fo E J, we take, as a neighborhood basis at fo , the totality of sets of the form

where A E 23, p ( A ) < 00 and E > 0. We call this the p-topology. Under the p-topology, J forms a linear topological space. I n particular, if p l9

See Definition 1.1.3 of Appendix I.

240

Iv.

LINEAR TOPOLOGICAL SPACES

is totally finite, then the p-topology is just the topology induced by the metric

and the sequence {fn} converges to f in the metric R if and only if converges to f in measure. If Q C G,, , A E 23, and p ( A ) < 00, let

{fn}

T h e topology on Q induced by the family of convex functions { R A ( . )A, E 23, p ( A ) < co} will also be called the p-topology. I n particular, if p(G) < co, then the p-topology is just the topology induced by the single convex function R(y) = RG(y), y E Q , If 8 C @, A E 23, and p ( A ) < 00, let

Again, the topology on 8 induced by the family of convex functions {RA(y),A E 23,p ( A ) < co} will be called the p-topology. I n particular, if p(G) < GO, then the p-topology is induced by the single convex function R(v) = R A ( y )y, E 8. Next, the concept of quasi-characteristic function, introduced in 53.2, will be transferred to the present context.

Definition 4.2.6. Let G be a real linear space, 6 a linear subspace of G, 52 = (G, 8,p,) a finite measure space quasi-invariant under 6, and J a linear subspace of G, . T h e function

will be called the quasi-characteristic function (on J ) corresponding to 52 and 6. T h e homogeneous convex function R(f) and the quasi-characteristic function +(f)on J are related by the formula

4.2. Linear and Quasi-Linear Functionals

24 1

This may be proved by using the identity

a2

co
and Fubini’s Theorem.

Lemma 4.2.6. Under the hypotheses of Definition 4.2.6, Iet 2 be any topology on J such that (I,%) is a linear topological space. Then, the quasi-characteristic function $(f) is continuous on ( J , 2) if and only if 2 is stronger than the p-topology. I n other words, the p-topology is the weakest topology on J such that (i) ( J , 2) is a linear topological space and (ii) the quasi-characteristic function $ is continuous on ( J , 2). PROOF. The necessity of the condition follows easily from (4.2.8) and the Lebesgue dominated convergence theorem. Conversely, assume that 2 is stronger than the p-topology, and let {fa} be a generalized sequence converging to f in ( J , 2).Then {fa} also converges to f in the p-topology, that is, {fa> converges to f in measure. Since I e i f ( g ) I = 1 , it follows by the Lebesgue dominated convergence theoremz0 that $(fa)

-

$(f). 1

Lemma 4.2.6 shows that, if p is finite, then the p-topology is completely determined by the quasi-characteristic function.

Lemma 4.2.7. Let G be a linear space, 6 a linear subspace of G, and (G, 8,p) a finite measure space quasi-invariant under (ti. Then, (i) G, is complete with respect to the metric R(f - q~), f,q~ E G, , (ii) G,, is complete with respect to the metric R(f - v), f , E G,, , and (iii) 6, is complete with respect to the metric R ( 3 - fJ),3, fJ E 6,. PROOF. Let (f,> be an R-fundamental sequence, that is, lim R(fm -. fn) = 0.

m,n+m

Since the totality of measurable functions on (G, 8,p) forms a complete metric space with respect to R, it follows that there is a subsequence {f,,} which converges almost everywhere on G. We define a function f as follows: let E denote the set of all points g E G such that {fn,(g)} converges, let f ( g ) = limk,mfn,(g) for g E E, and let f ( g ) = 0 for g E G - E. We assert thatf E G, . In fact,f is a real measurable function, and since every f,, is homogeneous, we obviously have f ( t g ) = t f ( g ) for every g E G and every real number t . Furthermore, for every positive The Lebesgue dominated convergence theorem is valid for generalized sequences.

242

Iv.

LINEAR TOPOLOGICAL SPACES

integer k and every h E 6, there exists a set ELk)€ B satisfying the following three conditionsz1: (i) tEik)C ELk’, - co < t < CO; (ii) p(G - EAkl)= 0; (iii) if g E ELk’, -a < t < 00, then

+ 4 = fnnl(h)+ tfn,(d*

fn#g

n,”cl

Let ELk’ A E = Eh moreover, if g E Eh , -co f(tg

+4

-

.

(4.2.9)

Obviously, tEh c Eh and p(G - Eh) = 0;

< t < 00, then, from (4.2.9),we getzz

t f ( g ) = ;+gfn,(tg

+ 4 - tfn,(E)) = $~fnnl(h,.

Taking f ( h ) = limk+mfn’,k(h), we see that f~ G, . Since we obviously have R(fn, - f)-+ 0, this proves that G, is complete with respect to R. Finally, since 23 is a closed subspace of G,, the quotient space G,, = GJ23 is also complete with respect to R. 3

Corollary 4.2.8.

If

{fn},

f E G, , and limn-,m R(fn - f ) = 0, then

for all h E 6.

Lemma 4.2.9, Let G be a Iinear topological space, 6 a linear subspace of G, and (G, 23,p ) a regular measure space quasi-invariant under 6. Let 2 be a topology on 6 such that (6,%) is a linear topological space of the second category, satisfying the first axiom of countability,2s and suppose that 2 is stronger than the topology on 6 induced by G. Then, for every set A E B such that 0 < p ( A ) < co,there is a neighborhood V , of zero in (6,%), such that (i) every functional 3 in 6. is bounded on V , , and (ii) the topology on 6, induced by the convex function RA(q)is stronger than the V,-topology on 6,. PROOF. Let A E 23, 0 < p ( A ) < co. For each f~ G, , the two functions

a1

Translator’snote: If p is a complete measure space, we may choose

+ f,p, m < t < coj. But if is not complete, then the existence of the required Eik’ is not obvious. Translator’s note: It is not obvious why tg + h E, that is, why f(tg + h) = limk,,f,,(tg + h). Ef’

=

(g Ifnt@

+ tg)

=

tfnp

-

p

”

E

For example, ( 8 ,2)is a complete linear pseudometric space.

4.2. Linear and Quasi-Linear Functionals

243

constitute a quasi-convex pair. Therefore, by Corollary 3.1 .IS, there exists a neighborhood V , of zero in (6,Z), and a positive number cA , such that

[where cA’ = c ~ ( ~ ( A ) )holds ” ~ ] for allfE G,, . Now, given any f E G,, , we choose a sufficiently small real number t , such that cA’RA(tf)< 1. Then, from (4.2.10), we get

and it easily follows that suphEvAlf(h)l < 03. Thus, every f~ 6” is bounded on V , . Next, let the element f in (4.2.10) vary over the residue class 7 = (9’ I 9’ E G,, , $ = f}. Taking the infimum of the right-hand side of (4.2.10), we obtain (4.2.1 1)

Hence, given any positive number E < 1, if we choose a positive number 6 < E / c ~ ’1( E ~ ) ~ then, / ~ , from (4.2.1 l), we see that 3. (3I I f I v, < E } whenever 7 E (7 I R,(q) < S}. This proves that the topology induced by R , is stronger than the V,-topology. 3 Incidentally, if, in (4.2.10), we replace f by tf, 0 < t < co, divide both sides by t , then let t 40, we obtain

+

Corollary 4.2,lO. Under the hypotheses of Lemma 4.2.9, the p-topology on @ is stronger than the strong topology. PROOF. Since any V-topology is stronger than the strong topology, the conclusion follows at once from Lemma 4.2.9. ]

Theorem 4.2.11. Let G be a linear topological space, and let 6 be a linear subspace of G which is itself a complete linear metric space, such that the corresponding metric topology of 6 is stronger than the relative topology induced by G on 6; let 2 denote the strong topology on 6+.Let L2 = (G, b,p ) be a regular finite measure space which is quasi-invariant under 6. Then, @ = 6+implies that 2 coincides with

244

Iv.

LINEAR TOPOLOGICAL SPACES

the p-topology. Moreover, if $5 is a linear subspace of 6, which is dense in ((fit, 2),then the following two statements are equivalent. (i) 6% = (fit. (ii) The relative topology 2, on $ induced by the p-topology of 6, is weaker than the relative topology 2, on 8 induced by 2. PROOF. According to Lemma 4.2.7, G,, is a complete linear metric space under the p-topology, hence 6~is also a complete linear metric space under the p-topology 2,. Furthermore, (fit, under the strong topology 2,is a complete Hausdorff linear topological space satisfying the first axiom of countability, and is therefore m e t r i ~ a b l eIf . ~6, ~ = Bt, then, by Corollary 4.2.10, we have 2,3 2,hence, by the Banach inverse operator theorem, 2, = 2.Statement (ii) then follows trivially from (i). Conversely, assume that (ii) holds. Given any f E (fit, there exists a sequence { f , } C $5 which converges to f in the topology 2. Obviously, {f,} is fundamental relative to 2, , hence also fundamental relative to 2, . Since 6~is complete with respect to the p-topology (by Lemma 4.2.7), there exists an element 'p E @ such that { f , } converges to 'p in the p-topology. Hence, by Corollary 4.2.10, {f,} also converges to 'p in the topology 2. Since ((fit, 2) is Hausdorff, we conclude that f = q E 6%. This shows that 6 ~ (fit, 3 hence, by Corollary 4.2.5, we obtain (i). ] When 6, = Bt, we say that Q is a standard quasi-invariant measure space associated with 6.

Definition 4.2,7, Let Q = (G, 8,p ) be a linear measure space which is quasi-invariant under the linear subspace 6 of G. If f EL~(Q n)G, = GU1,then the integral

will be called the mean value off on Q; if

'p,

f E L2(Q)n G,

=

GU2,then

will be called the correlation number off and 'p, or, when regarded as a function of the two variables f , 'p E Gu2,the correlation functional. Clearly, the correlation functional is an inner product on GU2.

Lemma 4.2.12.

Let Q = (G, 23,p ) be a linear measure space

84 Translator's note: As the translator remarked earlier, this conclusion requires an additional hypothesis regarding 6, such as local boundedness (see footnote 18 relating to the definition of the strong topology).

4.2. Linear and Quasi-Linear Functionals

245

which is quasi-invariant under the linear subspace 8 of G. If p( G) < CO, then GU2is complete with respect to the inner product c ( q , f ) , q , f ~ Gu2. PROOF.Suppose that the sequence {f,} C GU2is fundamental relative to c ( g ) , f ) , that is, fundamental in L2(Q).Since L2(Q)is complete, there exists an element v o ~ L 2 ( Qsuch ) that {fn} converges to qo in L2(Q). Moreover, since R(f) G

(c(f,f))l’z,

is also fundamental relative to the metric R(f - g)), f, q E G, . Therefore, by Lemma 4.2.7, there exists an fo E G,, such that R ( f , - fo)-+ 0. But it is easily proved that qo and fo are equal almost everywhere, hencef, eL2(Q)n G,, . 1 Let us write 6 , = ~ I f E Gu2},and, when $ E 6,~, let

{fn}

{p

I t is easily seen that the norm (c($, $))I/, is also derivable from an inner product, which we also denote by c(q, $). Again, we call it the correlation functional on 6,~.

Theorem 4.2.13.

Under the hypotheses of Theorem 4.2.11, if

6,”= 8+, then 2 is just the topology induced by the norm (c($, $))lI2, I/ E G2w. If fi C 6,~,and fi is dense in (at, 2),then the following two statements are equivalent. (i) 6,”= 8t. $ E fi, (ii) The topology 2, induced on fi by the norm (c($, is weaker than the relative topology induced by 2 on 5. PROOF. Assume that 6 , = ~ 6+.Let 23, = L2(Q)n 2? (where, again, 23 denotes the null space of the mapping f-f). By Lemma 4.2.12, GU2is complete with respect to the norm (c(q, ~ ) ) l /hence ~ , the quotient space 6 2 u = Gu2/23, is also complete with respect to the norm(c($, $))lI2. Now, it follows easily from (4.2.12) that there exists a neighborhood V , of zero in 8 such that

Consequently, the topology induced by the norm (c($, $))l/, on 6+= G2” is stronger than 2, and it follows by the Banach inverse operator theorem that these two topologies coincide. Statement (ii) now follows trivially from (i). Conversely, following the argument used in the corresponding part of the proof of Theorem 4.2.11, one can show that (ii) implies (i). 3

Iv.

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LINEAR TOPOLOGICAL SPACES

Similar considerations apply to the subspaces G,, n LP(sZ), p 2 1; we omit the details. The following theorem is entirely analogous to Theorem 3.2.10.

Theorem 4.2.14. Let G be a linear topological space, 8 a linear subspace of G, and sZ = (G, b,p ) a regular measure space which is quasi-invariant and ergodic with respect to 6. Suppose that 6 itself is a linear topological space of the second category under a certain topology which is stronger than the topology induced by G. Choose any nonzero #(g) 2 0, and form the homogeneous convex function vector EL~(SZ),

+

Then, the topology on G,, induced by R,( f)is independent of the choice of and coincides with the p-topology. I n fact, 8,being a linear topological space, is connected. Hence, the argument used in proving Theorem 3.2.10 is also applicable here; we omit the details. We observe that the functional defined by (4.2.13) and the functional defined in Theorem 3.2.10 are related by the formula

+,

R,(f)2

=

1 “

(4.2.14)

N,(eitf(.))2 ect dt,

0

which may be proved in the same manner as (4.2.8).

Corollary 4.2.15.

Under the hypotheses of Theorem 4.2.14, let be another measure on (G, b), choose any nonzero vector +‘ €L1(G, b,p’), #’(g) 2 0, and form the homogeneous convex function

p’

R ; * ( f )=

cs,

1+ f (fg( )g2) 2 #‘(g) dp’(g))liZ,

j~ G, .

(4.2.15)

If p‘ and p are strongly e q ~ i v a l e n t then , ~ ~ R, and Rk,induce the same topology on G,, . PROOF. Let E = {g I +’(g) > O}; then, E is a-finite relative to p‘, hence also o-finite relative to p. By virtue of the (weak) equivalence of p’ and p, there exists a measurable function

When p’ and p are strongly equivalent, G,, and G,, coincide.

4.2. Linear and Quasi-Linear Functionals

247

such that g, = #'(dp'/dp) €L1(L?), y 2 0, g, does not vanish almost everywhere, and

But according to Theorem 4.2.14, R, and R, induce the same topology. Therefore, R, and Ri, induce the same topology. ] Corollary 4.2.15 may also be used to establish the inequivalence of two given measures.

Example 4,2.1. We again consider the measure space described in Example 4.1.2. Let It denote the totality of linear functionals on I having the form

for some natural number n and some set of real numbers fi ,...,f,(if we take the product topology 2 on 1, then It is just the totality of continuous linear functionals on I ) . Let 1, denote the linear subspace of I consisting of all sequences of the form {fl ,..., fn , 0, 0,...}. Then, the correspondence f -+{fl ,...,f, ,...} is a one-to-one mapping of It onto lo . Also, since every f in It is a measurable linear functional on ( I , S U ) we , have It C G, . We now consider the following special case. If 1 a 2, the function

< <

exp(-l t la),

-co

< t < co,

is the characteristic function of an infinitely divisible distribution (see, e.g., Doob [l]), furthermore, since this function is Lebesgue square integrable, there exists a function g E L2(- CQ, CQ) such that m

eitmg(x)2dx

=

exp(-l

t

la),

-co < t < co,

--m

moreover, this function g can be chosen positive almost everywhere.2s Now, in Example 4.1.2, we choose fa = g, a = 1, 2,..., thus obtaining 28

See, for example, Zhao Zhong-zhe, Shuxue Xuebao (Acta Mathernatica Sinica) 3

(1953), 177.

Iv.

248

LINEAR TOPOLOGICAL SPACES

a quasi-invariant measure pa. Let us calculate the corresponding functional R,(f), f E I+. We first calculate N,(ei"(.)).

Hence,

40 Some a-Algebras on Linear Spaces

I n this subsection, we describe certain a-algebras which will often be used in the ensuing discussion, and examine the relations between them. Definition 4.2.8, Let @ be a linear space and B a linear subspace of @". If E is a Borel set in real n-dimensional space, and y1 ,..., vn E @, the set

{f I (f(v1),*-,f(v)n>> E E, j c sj> will be called the Borel cylinder with base E corresponding to {vl ,..., y,}. If the elements y1 ,...,cpn generate the linear subspace M of 0,then we also call the above set a Borel cylinder corresponding to M, or a Borel M-cylinder. T h e totality of Borel cylinders corresponding to a fixed M forms a cr-algebra, which we denote by S ( M ) [or S(y, ,..., 4 1 , and the totality of all Borel cylinders forms an algebra S. Let 23 denote the smallest a-algebra containing S; we call the elements of 23 weak Borel sets corresponding to @, or @-weak Borel sets. I n particular, if @ is a linear topological space and sj C Ot, then we simply refer to the 23) is elements of 23 as weak Borel sets. In any case, it is clear that ($j, a linear measurable space. Let @ be a linear space and 8 a linear subspace of @*.Let 5 be the smallest a-algebra of subsets of @ which contains all sets of the form

{v IF(v) < a}, The elements of

5 will be

-co

< a < co, F E B .

called weak Borel sets corresponding to

6,or

4.2. Linear and Quasi-Linear Functionals

249

!+weak Borel sets. I n particular, if @ is a linear topological space, and $j C Ot,then the elements of 8 will simply be called weak Borel sets. If fi distinguishesz7 the points of @, then we may imbed @ in 5” by identifying each point p E @ with the element p1E $indefined by

Vl(f)

=

f(vL

f E

5.

Let = {pl I p E @}. Then the weak Borel sets in 5 corresponding to are just the weak Borel sets corresponding to @. Thus, in this case, the two concepts of “weak Borel set” described above essentially coincide. T h e following lemma shows that the weak Borel sets constitute a sufficiently wide class of sets. Q1

Lemma 4.2.16.28 If @ is a separable countably normed space, then every open (or closed) subset of @ is a weak Borel set. PROOF. Let

/I VJ Ill

< II v < ..- e /I F Iln e 112

*--

<

be the given sequence of norms on @, and let S,(r) = {p I /I p 1, r}. We shall first prove that S , = S,(1) is a weak Borel set. Since 0 is separable, there is a sequence {pm}C @ - S , , such that {pm} is dense By the Hahn-Banach theorem, in @ - S , reIative to the norm 11 g, .1, there exists a linear functional fm E Qt, such that llfrn

11-73

=

,,;t€& Ifm(v)I = 1,

fm(VJrn)

=

II vrn Iln *

(4.2-16)

Form the set m

E

=

n {F I I ~ ~ e( ~11.) I

m=l

Clearly, E is a weak Borel set. We assert that E = S, . I n fact, from (4.2.16), we obviously have E 3 S , . On the other hand, if p C S, , choose a positive number 6 < $(I[ g, ,1 - 1). Then, there exists an index m such that /I vrn- p 1, < 6, hence I fm(pm - p)I < 6, and therefore Ifm(VJ)I

3

lfm(Frn)l

- 8 = IIvmIln - 8

3 IIvIln - 28 > 1.

Thus, g, E E. This proves that S , is a weak Borel set. I n a similar fashion, one may prove that, for every p E @, S,(r) + g, is a weak Borel set.

*’

That is, for each nonzero q E @, there exists f E Sj such that f ( ~ ) # 0. Regarding the terminology used in the statement and proof of Lemmas 4.2.16 and 4.2.17, see Appendix I, 51.3. 28

250

Iv.

LINEAR TOPOLOGICAL SPACES

Now, let W be any open set in @. Then, since @ is separable, there exists a sequence {p),} C @, a sequence of positive numbers {Y,}, and a sequence of natural numbers (m,} such that m

w = u {vn +

&&l)).

?l=l

Hence, W is a weak Borel set. Since, in particular, @ itself is a weak Borel set, it follows that every closed set @ - W is also a weak Borel set. 3 Thus, under the hypotheses of Theorem 4.2.16, the totality of weak Borel sets in @ is just the a-algebra generated by the totality of open sets in @.

Lemma 4.2.17. Let @ be a separable countably normed space, with the norm sequence (11 p) \In}. Then, S-,(Y) = {f \lfI\-, r} is a weak Borel set in @+. PROOF. Choose a sequence { P ) ~which } is dense in S, , and consider the set

1

<

W

B

=

n {fiI f(v)m)~ G r).

m=l

Obviously, B is a weak Borel set in @+. We shall prove that B = S-,(r). First, since {pm}C S, , we clearly have B 3 S-,(Y). Conversely, let f E B ; if y E S, ,there exists a subsequence {p)m,> such that limk+mvm, = p) (in the topology of @). Consequently,

that is, f E S-,(Y), Thus, S J r ) = B is a weak Borel set. ] I n particular, we see that Qbnt= S-,(m) is a weak Borel set.

u:=,

50 The Existence of Quasi-Invariant Measures

Now, we shall show how inequality (4.2.12) may be used to derive necessary conditions for the existence of quasi-invariant measures. Here, we only consider Hilbert spaces and number sequence spaces, however, the methods used can, in fact, be applied in an even more general context. Moreover, all the necessary conditions given here are actually sufficient conditions, as will be shown in 95.3.

Theorem 4.2.18. Let G be a separable Hilbert space, with the inner product ( x , y ) , and let b be the totality of weak Borel sets in G. Let 8 be a linear subspace of G, and suppose that 6 itself is a complete

4.2. Linear and Quasi-Linear Functionals

25 1

countably inner product space29 with respect to the sequence of inner products (cp, +), , 7t = 1, 2,..., where (cp, y)l (9,9)2 . Also, suppose that the inclusion mapping T from 6 into G is c o n t i n u o u ~ . ~ ~ For each n, let 8 , denote the completion of 6 with respect to the inner product (cp, +), . Then, the existence of a 6-quasi-invariant finite measure p on (G, 23) implies that, for some natural number n, the mapping T can be extended to a Hilbert-Schmidt type operator from 8 , to G. PROOF. We may assume that p satisfies the condition

<

<

for otherwise, we could replace p by the equivalent measure ~L'(= E)

S,

e-(z+) dp(x),

E E S,

which certainly satisfies this condition. [Notice here that (x, x) is a measurable function on (G, S)]. Now, by (4.2.12), there exists a positive number c and a neighborhood V of zero in 8 , such that, for each continuous linear functional f on G,

Of course, V must contain some neighborhood of the form {h I (h, h), 82). Choose any complete orthonormal set {em} in G. Then, by (4.2.12'), we obtain

<

Clearly, we may assume that n is sufficiently large, so that T is continuous with respect to the inner product ( y , $), , and may therefore be extended to a bounded linear operator T, from 8 , to G. We then have31

Regarding the concept of a countably inner product space, consult Appendix I. In other words, the topology of 6 is stronger than the relative topology induced by G on 6. s1 Translator's note: The author has incorrectly squared the absolute values appearing in this inequality. It is not clear how this gap can be fixed up. ae

so

252

Iv.

LINEAR TOPOLOGICAL SPACES

I t follows from (4.2.12'), (4.2.17), and Lemma 11.1.2 of Appendix I1 that T,* is a Hilbert-Schmidt-type operator. Hence, by Lemma 11.1.3 of Appendix 11, T, is Hilbert-Schmidt. ] T h e following two cases are of greatest interest to us.

Corollary 4,2.19. Let Q C H C Qt be a separable rigged Hilbert space, and let 8' denote the totality of weak Borel sets in Q'. If there exists a @-quasi-invariant regular finite measure p on (a+,gt), then there must be a pair of natural numbers n, m(n 2 m) such that the imbedding of Q, (i.e,, the completion of @ relative to the nth inner product) in amt is a Hilbert-Schmidt-type operator. PROOF. Since Qmt E St (see Lemma 4.2.17), the sequence of sets m {Qm+} is monotonic increasing, and Qt = Qmt, it follows that p(Qm+)3 p(@'), hence, there must be an m such that p(Qm') > 0. Restricting 8' and p to Qmt, we obtain a @-quasi-invariant measure space (Qmt, Bmt,pm). Now, in Theorem 4.2.18, we take Q for 8 and Qmt for G, and the desired conclusion follows at once.32 ] Corollary 4.2.20. Let G and 6 be separable Hilbert spaces, where 6 is a linear subspace of G such that the inclusion map T of 6 into G is continuous. Let b be the totality of weak Borel sets in G. If there exists a 8-quasi-invariant finite measure p, on (G, b), then T is a Hilbert-Schmidt operator. PROOF. This corollary is just a special case of Theorem 4.2.18. ] Example 4.2.2, Let I < p < 00, let {a,} be a sequence of positive numbers, and let P({u,}) denote the totality of real number sequences g = (En} which satisfy the condition (4.2.18)

P({U%}) forms a Banach space with respect to the usual coordinatewise linear operations and the norm (4.2.18). In particular, we write ZP in place of P((1)). Let 23 be the a-algebra in P({u,}) generated by the totality of Borel cylinders

(where B represents an arbitrary finite-dimensional Borel set). If s2 Translator's note: The definition of rigged Hilbert space given in Appendix I does not include the completeness of @, which is required here.

4.2. Linear and Quasi-Linear Functionals

253

<

1 q < CO, then the existence of an l@-quasi-invariant33finite measure space (P({an}),8,p) implies that xun

< 00.

I n fact, since 11 4 (1 = limn+m(Xr-, a, I we may suppose that

(4.2.19)

5, I P ) l / P is a measurable function,

/I 5 IIP 4 4 6 ) < 00,

M=

(4.2.20)

for otherwise, we could replace p by the equivalent measure

tL’(Jq =

j exp(--ll 6 11”) E

4@),

E 8.

For each n, consider the %-measurable linear functional

By (4.2.12), there exists a neighborhood V of zero in la, and a positive number c, such that

I 5 I < a}, 6 > 0, I 5 I = (C,“=II 4,

We may suppose that V = {5 1 so that (4.2.12“) becomes

I

-. d8 J,,,,,,,

Iq)”*,

I En I ME)*

Using the Holder inequality, we deduce that

Multiplying both sides of the above inequality by a , , and summing with respect to n, we get

x <..(3” u

- p(Z”({un}))”-’M

< 00.

83 Translator’s note: More properly stated, quasi-invariant under the intersection of la and P({u,,}).However, if (4.2.19) holds, then l a C P({an}).

2 54

Iv.

LINEAR TOPOLOGICAL SPACES

60 The L2-Fourier Transform

Naturally, the theory of L2-Fourier transforms on groups, already discussed in 93.4, can also be transferred to the context of linear spaces; the relevant definitions and results require only slight modification. We do not propose to restate the entire theory in detail. Instead, we merely give the definition of the L2-Fourier transform for linear measure spaces, leaving the rest to the reader. T h e results will subsequently be applied without further preliminary explanations. Definition 4.2.9. Let 52 = (G, 23, p ) be a linear measure space which is localizable and quasi-invariant under a linear subspace 6 of G. Let fz = 8, @)be a linear measure space which is quasi-invariant under a linear subspace 6 of &', and suppose that the following two conditions are satisfied. (i) There exists an isomorphic mapping

(e,

from & onto

6, and an isomorphic mapping h+h(t),

tee,

from 8 into Go, such that 2 = (h(*)1 h E 0}is a determining set of functions on 0,moreover, for each h E 6, the linear functional induced by h(.) on 6 is

R ( f ) =f ( h ) ,

f,6.

(ii) There exists a unitary operator F from L2(Q)onto !i?,2(o) such that, for each h E 6, the corresponding operator f f ( h )= F( U(h))F-' takes the form

(li(4P>(d ) = exp(ih( t))d k ) ,

d E e,

'p E

e7cz(fZ)-

Then, we say that (0, 6) is a dual of (Q, 0) (or that 0 is a dual measure space of Q), and that F is an associated L2-Fourier transform [from L ~ ( Qonto ) !i?k2(0)]. s4.3. Continuous Positive Definite Functions on Linear Topological Spaces

I n this section, we shall, as before, consider only real linear spaces, and shall regard these spaces, in particular, as additive groups. Since any finite-dimensional To linear topological space is topologically isomorphic to a Euclidean space of the same dimension, the Bochner

4.3. Continuous Positive DeJnite Functions

255

theorem provides the general form for continuous positive definite functions on finite-dimensional linear topological spaces. However, for infinite-dimensional linear topological spaces, there still exists no comprehensive theorem concerning the representation of continuous positive definite functions. Nevertheless, there are a number of important results along this line. First, we shall consider the situation for linear spaces in general.

lo Continuous Positive Definite Functions on Linear Spaces. Cylinder Measures T h e usual notion of measure is somewhat too strong for the formulation of certain preliminary results required in the following discussion. Accordingly, we begin by introducing a weaker concept, that is, cylinder measure.

Definition 4.3.1. Let Q be a linear space, let 9 be a linear space consisting of certain linear functionals34 on 8, and let S be the algebra of all Borel cylinders in 6. Suppose that P is a set function on S having the following property: if @ is any finite-dimensional linear subspace of 8, and S ( 0 ) is the o-algebra of Borel cylinders corresponding to 0, then the restriction of P to S ( @ )is a probability measure. Then, we call P a cylinder measure35on 5. Clearly, any cylinder measure P also has the following properties: (i) P(z) 0 for all z E S; (ii) P(sj) = 1; (iii) P is finitely additive. I n fact, (i) and (ii) are obvious from the definition, while (iii) may be seen as follows. If z1,..., z, are disjoint sets in S, then it is clear from the definition of a Borel cylinder that there exists a finite-dimensional subspace @ of 8 such that z1,..., x, E s(@), hence, by the additivity of P on S(@),we have

4%+

**.

+ z,)

= P(X1)

+ ... +

P(Zn),

that is, property (iii). However, P is not, in general, countably additive. But if it happens that P is countably additive, then, using a well-known technique (see, e.g., Halmos [l]), we can extend P to a probability measure (which we also denote by P)on the c~-algebra8 generated by S. Now, let P be any cylinder measure on 5. Let U be a complex-valued function on sj, and suppose there exists a finite-dimensional linear subspace 0 of Q such that U is integrable on the probability measure 34

36

With the usual linear operations. Notice that this is, in fact, a special case of the cylinder measure concept introduced

in $1.3.

256

Iv.

LINEAR TOPOLOGICAL SPACES

space (fi, S ( @ ) ,P).36 We then say that U is integrable with respect to the S(@),P) is cylinder measure P. I n this case, the integral of U over (8, clearly independent of the choice of @, and will be called the integral of U with respect to the cylinder measure P ; we denote it by

In particular, when P is countably additive, this integral coincides with the ordinary integral of U over the measure space (fi, 8, P).

Definition 4.3.2. Let 6 be a linear space and let f be a complexvalued function on 6 such that, for every finite set ‘pl ,..., y n E 6, the valuef(t,v, t,v,) is a continuous function of the n real variables t , ,..., t , , that is, f is continuous on every finite-dimensional linear subspace3’ of 6. We then say that f is pseudocontinuous on 6.

+ +

Lemma 4.3.1. Let 6 be a linear space, let sj be a linear space consisting of certain real linear functionals on 6, and let P be a cylinder measure on fi. Then, the integral h (4.3.1)

is a pseudocontinuous positive definite function on 6, andf(0) = 1. PROOF. P ( 8 ) = 1, hencef(0) = 1. Furthermore, since

f is positive definite. Finally, given any

v1 ,...,vn E 6, we observe that (4.3.2)

hence, applying the Lebesgue dominated convergence theorem to the probability measure space (9, S(vl ’..., v,), P), we see that (4.3.2) is a continuous function of the real variables tl ,..., t , . 3 If fi contains “sufficiently many” functionals, then the converse of Lemma 4.3.1 also holds, that is, we have the following result. Here, P means the restriction of P to S ( 0 ) . Normally, it is always assumed that finite-dimensional linear spaces are given the Euclidean topology. s7

4.3. Continuous Positive DeJinite Functions

257

Lemma 4.3.2. Let 6 be a linear space and let $3 be a linear space consisting of certain real linear functionals on 8, moreover, suppose that $3 distinguishes the points of 6. Then, given any pseudocontinuous positive definite function f on 8 [with f ( 0 ) = 11, there exists a unique cylinder measure P on 8 such that (4.3.3)

PROOF. Take any linearly independent finite set of vectors cp, ,..., vn in 6, and consider the valuef(t,y, --. t,cp,), regarded as a function of the real variables t, ,..., t, . Since f is positive definite and pseudocontinuous, and f ( 0 ) = 1, it is easily verified that f (t,cpl .-- tncpn) satisfies the conditions of the Bochner-Khinchin theorem, hence, there exists a unique probability measure on real n-dimensional space R, , such that

+

+

+ +

f(tlvl

+ + t,v,) *.*

=

1 1 . * a

exp i(tlE1

+ +

t,5,)

dQ{o,)(tl

t,).

Consider the correspondence T,,, : t -+ (5(y1),..., e(yn)),5 E $3. Since $3 distinguishes the points of @, the image of Ttqj1 is the entire space R, . Define a set function P{,jl on S({yj}) as follows: if A is any Bore1 set in R, , let %J(%:)4 = Q(m,)(A). (4.3.4) Clearly, ($3, S({cpj}),P(mj)) is a probability measure space. Moreover, a straightforward calculation shows that f ' ( q j ) is independent of the particular choice of the basis y 1 ,..., cpn for the finite-dimensional linear subspace under consideration. We now proceed to prove that the family of probability measure spaces constructed in this manner is consistent, that is, if B E S({#,})n S({wj}), #, ,..., t,hrn , w1 ,..., w k E 6, then P(,,)(B)= %JJP)*

Choose a linearly independent set y1 ,..., yn

E

(4.3.5)

8, such that

"

S({A,>) S ( { 4 ) c WvJ).

We need only prove that P(Wj)(B)=

%,m

(4.3.6)

for we may then deduce by the same reasoning that P(qj)(B)= P(u,l(B),

Iv.

258

LINEAR TOPOLOGICAL SPACES

thereby obtaining (4.3.5). Obviously, we may assume that y j = &, j = 1,2,..., m,n >, m. Since

J J exp &tl + ... + L E , ) .**

dQ($,}(tl ,***,

tm)

+ .-.+ L2fu f(tlV1 + ... + ~,V, + Opm+l+ + OFn) J ... J exp i(tltl + ... + t,t,) ~ Q I ~ ,..., ~ It ,( ~~ ,~

= f(tllCI1 = =

***

it follows that Q(wpl)({(E1

9**'9

En)

for every Bore1 set D in R ,

%J{tI (t(V1)9...,

I (tl

I...)

tm)

ED))

= 8(@,dD>

, that is,

t(V?n>> E Dl) = P($J{t I (E(lCI1)YV

E(lCIm)) E

D)). (4.3.7)

But every set B in S(@,) is of the form {,$ I (,$(&),..., ~),I!( ED}, hence (4.3.7) is equivalent to (4.3.6).Thus, we have shown that the family of measures {P(@,)} is consistent. Define a set function P on S, as follows: if A E S({vj}),let 4 4 )= P ( O j } ( 4 Using the consistency of the family of measure spaces (sj, S ( { y j } ) ,P,,,)), {yj} C 6, and following the reasoning of $1.3, we see that P if unam-

biguously defined. It is then obvious that P is a cylinder measure. Next, we verify that P satisfies (4.3.3).I f g = 0, then (4.3.3)is obvious. If g # 0, then

Finally, the uniqueness of the cylinder measure in (4.3.3) may be seen as follows. T h e values of P on each S({yj}), v1 ,..., yn E 6, are determined by the measure Q(c,) in (4.3.4),and by the Bochner-Khinchin theorem, Q(w,) is uniquely determined by f. ] 2O

Continuity of Cylinder Measures

I n analogy with Definition 3.2.6, we shall now introduce a notion of continuity for cylinder measures.3* 38 In particular, applicable to probability measures on u-algebras of the type described in Definition 4.2.8.

4.3. Continuous Positive Dejnite Functions

259

Definition 4.3.3. Let (ti be a linear topological space with topology 2.Let $3 be a linear space consisting of certain linear functionals on 8 , and let P be a cylinder measure on 9. Suppose that, given any positive number E, there exists a neighborhood V of zero in 8 such that

I

> 1,tE93)) <

P((t I

E

whenever X E V. Then, we say that P is continuous with respect to the topology 2 (or simply continuous). Next, we establish the connection between continuity of positive definite functions and continuity of cylinder measures (cf. 93.3).

Lemma 4.3.3. Let (ti be a linear topological space, $3 a linear space consisting of certain real linear functionals on 6, and P a cylinder measure on 9. Then, the function

is continuous if and only if P is continuous. PROOF. Assume that f is continuous. Then, given any positive number e, there exists a neighborhood U of zero in 6, such that 1 - W ( g )<

(4.3.9)

for all g E U [since f ( 0 ) = 11. Choose a positive number a such that e-a < ~ / 8Since . 8 is a linear topological space, there is a neighborhood V of zero such that tV C U whenever I t 1 a. Thus, if g E V , then, from (4.3.9), we get

<

I

m

(1 - %.(gt))e-t dt

0


m

2e-t dt

a

< . 2

(4.3.10)

However, using Fubini’s theorem, we have

Jr

(1

-

sj(gt1)e-t dt

=

J5

JI

(1 - cos tt(g)) e+ dt d p ( 5 )

(4.3.11)

Combining (4.3.10) and (4.3.1 l), we deduce that

whenever g

E

V . This shows that P is continuous.

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260

LINEAR TOPOLOGICAL SPACES

Conversely, assume that P is continuous. Then, given any positive number E , there exists a neighborhood V of zero in 6 such that

whenever X E V . Choose a positive number 6 < 7712 such that 1 - cos 6 < €12, and let U = 6V. Obviously, U is a neighborhood of zero, and if g E U, then x = (1/6) g E V , hence P((5 I I 5(dl

> 81) = P ( ( t 1 I &)I

> 11) <

; *

Therefore, if g E U , then 1 - Wg) =

I,

(1 - cos 5(gN W

5)

< (1 - cos 8 ) + 2P({5 I 1

> 8)) < 6.

This proves that %,(g) is continuous at g = 0. But by Lemma 4.3.1, f is positive definite, hence, it follows by inequality (3.3.3) that f is continuous. 3 T h e following theorem is entirely analogous to Lemma 3.2.19.

Theorem 4.3.4. Let 6 be a linear topological space satisfying the first axiom of countability, $j a linear space consisting of certain continuous linear functionals on 6, and 5 the smallest o-algebra containing all the Borel cylinders in 5. Then, every probability measure P on (8,5 ) is continuous. T h e proof is left to the reader. The conclusion of Theorem 4.3.4 may not be valid if 6 does not satisfy the first axiom of countability (see Xia Dao-Xing [3]).

30 Representation of Pseudocontinuous Positive Definite Functions as Fourier Transforms of Measures Let 6 be a linear space, $j a linear space consisting of certain linear functionals on 6, and 5 the smallest o-algebra containing all the Borel cylinders in $5. In Lemma 4.3.2, we have already obtained a result concerning the representation of pseudocontinuous positive definite functions as Fourier transforms of cylinder measures. Now, we proceed to investigate, under what circumstances a positive definite function f can be expressed as an integral (4.3.1)

4.3. Continuous Positive Definite Functions

26 1

with respect to a measure P on (4,,5).By virtue of Lemma 4.3.1, we know that the pseudocontinuity o f f is a necessary condition for the existence of a representation of the form (4.3.1). It can also be proved that39 if (4.3.1) holds, then 4, distinguishes the points of 6. However, these two necessary conditions, taken together, still do not constitute a sufficient condition (see 95.3). But we do have the following result.

Theorem 4.3.5. Let 6 be a linear space, 6” the totality of real linear functionals on 6, and 5” the smallest a-algebra containing all the Borel cylinders in 6”. Then, for any pseudocontinuous positive definite function f on 6 [withf(O) = 11, there exists a unique probability measure P A on (6”,5”) such that f(g)

=

eif(g)

(4.3.12)

d ~ ~ ( 5 )

6”

for all g E 6. PROOF. Choose a linear basis {w, I a E CU} for 6. For each finite set of indices a1 ,..., E W, we form a function of n real variables tal ,..., t a n ,as follows:

+ +

f(,jl(?T1 ,.*** tan) = f(t,pal ..*

ta,%,).

(4.3.13)

Since f is pseudocontinuous, positive definite, and satisfies f ( 0 ) = 1, it follows at once that fq is continuous, positive definite, and satisfies j(,,}(O,..., 0) = 1. Moreover, it is easily verified that the family {f(,,}} of all functions constructed in this manner is consistent. For each cy E rU, let R, be a copy of the real line, and let denote the totality of Borel sets in R, . By the Kolmogorov theorem (Corollary 1.3.5’), there exists a unique probability measure P A on (XaEa R, , XaEa5,) such that, for any finite set of indices a1 ,..., an E CU, f(tp,,

+ ... + k”%“) = f(ajl(t,l

)**’)

exp

= x R,.

tan)

(i

j=l

ta,x,,)dPA(x),

(4.3.14)

where x, denotes the ath coordinate of the point x in XaEaR, . Now, for each x E )(,Ex R, , define a functional 3i. on 6, as follows:

W,pm1+ .*.+ tan%,)

= t,lX,l

+ + *.*

ten%”*

s8 Trumlator’s note: This remark as it stands is obviously false. Perhaps the author actually meant that if representation (4.3.1) is possible for all members of a certain class of positive definite functions on 6 ,then Sj must distinguish the points of 8.

Iv.

262

LINEAR TOPOLOGICAL SPACES

Clearly, 4 is a linear functional on 8, and the correspondence x + $ is a one-to-one mapping of R, onto 0”.Thus, if we identify x and 4, (4.3.14) becomes (4.3.12). Moreover, the uniqueness of PA in (4.3.12) follows immediately from the uniqueness of PA in (4.3.14). ] If sj is a linear subspace of which distinguishes the points of 8, then the conclusion of Lemma 4.3.2 can be deduced from Theorem 4.3.5. I n fact, if B is a Borel cylinder in W,then B n sj is a Borel cylinder in sj, and one may verify that the set function P(B n sj)

=

PA@)

is a cylinder measure which fulfills the requirements of Lemma 4.3.2. T h e following theorem may be regarded as an alternative form of the Kolmogorov theorem. the totality of Theorem 4.3.5‘. Let 8 be a linear space and real linear functionals on 6. Then, every cylinder measure on Q A is countably additive.

PROOF. Let P be a cylinder measure on 8”.Taking sj = 8“ in formula (4.3.8), we define a corresponding pseudocontinuous positive definite function f. By Theorem 4.3.5, there exists a probability measure PA on (8”, 8”)such that (4.3.12) holds for this f. Of course, PA is also a cylinder measure on 8”.Applying Lemma 4.3.2 (with sj = 8”), it follows from the uniqueness property that P ( B ) = P”(B) for every Borel cylinder B . Therefore, since PA is countably additive, the cylinder measure P is also countably additive. 3 Incidentally, notice that the measure PA in the above proof is the unique extension of P to 8”. If 8 # 8”, then cylinder measures on are not, in general, countably additive. However, we are primarily interested in countably additive cylinder measures, since they can be extended to a larger family of sets (see $4.2, 40), and provide a more effective analytical tool. We shall conclude this subsection by deriving some conditions for the countable additivity of continuous cylinder measures. The countable additivity of continuous cylinder measures is intimately connected with the notion of a spectral linear space, which we now proceed to define. Definition 4.3.4. Let (8,Z) be a linear topological space, a and 8 the totality of weak Borel sets in 8. Then, linear subspace of 8”, 8 is called a spectral (linear) space for (8, Z), provided that, for each

4.3. Continuous Positive Dejnite Functions

263

continuous positive definite function f on ( 8 ,2) [withf(O) = I], there exists a unique probability measure P on ($3, g), such that

for all h E 8.

Lemma 4.3.6. Let (8,2) be a linear topological space, and let $3 be a linear subspace of 8” which distinguishes the points of 8 . Then $3 is a spectral space for (8,2) if and only if every continuous cylinder measure on $3 is countably additive. PROOF. Assume that every continuous cylinder measure on sj is countably additive. Now, according to Lemmas 4.3.2 and 4.3.3, for each continuous positive definite function f on (8,2) [withf(O) = I], there exists a cylinder measure P on $3 which is continuous with respect to 2 and such that (4.3.3) holds. By our assumption, P is countably additive and therefore defines a probability measure which also satisfies relation (4.3.3) for the given functionf. Thus, $3 is a spectral space for (8,Z). Conversely, assume that Ij is a spectral space for ( 8 ,2).Given any continuous cylinder measure P on $3, we form the corresponding pseudocontinuous positive definite function f in accordance with formula (4.3.1). By Lemma 4.3.3,fis continuous, hence, by our assumption, there exists a probability measure P’ on ($3, 8 ) such that j(g) =

I

5

ei“g)

d~’(t),

g €8.

By the uniqueness property of the cylinder measure in Lemma 4.3.2, it follows that P’(B) = P ( B ) for every Borel cylinder B in $3. Since P’ is countably additive, P is also countably additive. ] Lemma 4.3.6 clarifies our earlier remark concerning the intimate connection between spectral spaces and countable additivity of continuous cylinder measures. If 8 is a linear space and $3 C 8”,then, by a weakly open cylinder, we mean a set of the form

{f I (f(%)>-*,f(vn)) E V , f € $31, where v1 ,..., qn E 8 and V is an open subset of R, . Using this definition, we introduce a sort of regularity property for cylinder measures.

Lemma 4.3.7. Let 8 be a linear space, sj a linear subspace of 8”, and P a cylinder measure on $3. Given any Borel cylinder A in $3, and

264

Iv.

any positive number that U 3 A and

E,

LINEAR TOPOLOGICAL SPACES

there exists a weakly open cylinder U such P(A) > P(U)- E.

PROOF. Let A E S(@),where @ is an m-dimensional linear subspace of (ti. Choose a basis y1 ,..., ym for @, and let 5 denote the totality of Borel subsets of R, . For any B E 5,let

B

fE8L

= {f I(f(P),l>,...,f(P)m))EB,

and define a set function Q on

5 as follows: B E 5.

Q(B)= P(@,

(4.3.15)

Since (6, S(@),P) is a probability measure space, it is easily verified that (R, , 5,Q) is also a probability measure space. Therefore, if D E 5 and D = A, then there exists an open set V in R, such that Y 3 D and (4.3.16) Q(0) > Q(v> - 6. Thus,

r is a weakly open cylinder,

r3

D

= A, and, by

(4.3.15) and

(4.3.16), we have P ( A ) > P(B) - E .

1

Lemma 4.3.8, Let 6 be a linear space, !ij a linear subspace of (ti", and P a cylinder measure on &. Suppose that, given any positive number E , there exists a weakly compact set C , in $3, such that P(2) < 6

(4.3.17)

for every Borel cylinder !i? which does not intersect C,. Then, P is countably additive. PROOF. We shall first prove that, if 2, ,..., z k ,... is any sequence of Borel cylinders in !ijsuch that 2, = 9, then

u;=l

m

1 P(Z,) 2 1 .

(4.3.18)

k=l

Given any positive number E , we know by Lemma 4.3.7 that, for each K, and there is a weakly open cylinder u k such that u k 3

P(2,) > P(U,) Since

u,"~

u k3

-5 . 2k

u,"~z k = !ij2 C,, it follows from the weak compact-

4.3. Continuous Positive Dejinite Functions

265

C, that uF=lU k 3 C, for some m. Thus, the Borel cylinder u s 1U, does not intersect C,. Therefore, by (4.3.17), we have

ness of $3 -

and so, by the finite additivity of P,

Hence,

Letting E -+ 0, we obtain (4.3.18). Now, let {En} be any disjoint sequence of Borel cylinders in $3, such m that Un=lEn = E is also a Borel cylinder. Let E, = $3 - E; then E, is also a Borel cylinder, and En = 9. By (4.3.18), we have

uL0

Hence, by the finite additivity of P,we get

Again, by the finite additivity of P,

hence,

Letting N -+00, we obtain

Iv.

266

LINEAR TOPOLOGICAL SPACES

Therefore,

c P(En), W

P(E) =

n-1

that is, P is countably additive. 3 T h e following special case of Lemma 4.3.8 will be used in the ensuing discussion.

Corollary 4.3.9. Let 8 be a separable countably normed space,

conjugate space of 8,and P a cylinder measure on Bt. Let (1 ( denote the minus nth norm in 8+,and let SJR) = (6 [ (1 8 (1R) (see Appendix I). Suppose that, given any positive number E , there exist n and R such that (litthe

<

P(2) < E for every Borel cylinder i? which does not intersect S,(R). Then P is countably additive. PROOF. According to a proposition in the theory of linear topological spaces, S,(R)(R < co) is weakly compact. Hence, the conclusion follows immediately from Lemma 4.3.8. ] 40 Representation of Continuous Positive Definite Functions

on Linear Topological Spaces Throughout the present subsection, 8 will denote a linear topological space, (St the conjugate space of 8, and St the smallest o-algebra containing all Borel cylinders in Bt. First, we remark that an arbitrary continuous positive definite functionfon 8 [withf(O) = 11 cannot, in general, be expressed as a Fourier transform f(g) =

J,t

eif(g)~ P Y S ) ,

g €6,

(4.3.19)

where Pt is a probability measure on (8+, St) (see 95.2). Therefore, as in 93.3, we shall impose additional conditions on the positive definite function f.We begin by bringing Theorem 3.3.7 into the present context.

Theorem 4.3.10. Let G be a linear topological space, and let B be a o-algebra consisting of certain subsets of G, including all the closed subsets of G. Suppose there exists a locally finite regular measure p on (G, S).Then, given any continuous positive definite function f on G

4.3. Continuous Positive DeJinite Functions

267

[with f (0) = I], there is a probability measure P , on (G, that

, &),

such

for all g E G. T h e proof of Theorem 4.3.10 is similar to that of Theorem 4.3.11, and is therefore omitted. T h e analog of Theorem 3.3.8 is as follows.

Theorem 4.3.11. Let G be a linear topological space, and let 0 be a linear subspace of G which is itself a linear topological space of the second category with respect to a topology 5 which is stronger than the topology induced by G on 0. Let % be the o-algebra generated by the totality of closed subsets of G, and suppose there exists a strongly 0-quasi-invariant locally finite regular measure p on (G, 23). Then, for each continuous positive definite function f on G [with f ( 0 ) = 11, there is a unique probability measure P t on (St,gt), such that (4.3.19')

Moreover, Pt is continuous with respect to the topology induced by G on 8. PROOF. According to Theorem 4.2.2, the mapping ~:.+(=eif

is an isomorphism of (St onto O*. Moreover, A clearly maps each set in 5' into a set in 5" (the totality of weak Bore1 sets in 8*),and A-1 maps each set in 5" into a set in 8'. By Theorem 3.3.8, there is a probability measure P* on ( 0 * , B*) such that the given function f is expressed in the form (3.3.41). Define Pt(E) = P*(AE),

E E 8'.

Obviously, Pt is a probability measure on (Of, Bt),and (3.3.41) is equivalent to (4.3.19') via the measurable transformation A. T h e uniqueness of Pt is easily deduced from the uniqueness of P*.Finally, since f is continuous on G, the required continuity property of Pt follows by Lemma 4.3.3. ] We shall be especially concerned with the case where 0 is a complete linear metric space, and (G, 23, p ) is a 6-quasi-invariant regular finite measure space.

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Let 6 be a (real) countably Hilbert space, with the sequence of inner products (9,+), , n = 1,2,..., and let 8 , denote the completion of 6 with respect to the norm 11 cp = ((T, ~ ) , ) l / ~We . introduce a new topology on 8, as follows. Let n be any natural number, let T be any positive nuclear operator on 6,, and let U(T,4 = {p' I (TP, p')n

< 1, v E 8 1 .

We take the totality of sets of the form U(T , n) as a neighborhood basis at zero, and so obtain a topology G on 8. The topology 6 is clearly weaker than the original topology of 6, moreover, (6, G ) is a linear topological space.40

Theorem 4+3.12. Let 6 be a real separable countably Hilbert space, and let f be a positive definite function on 6, with f ( 0 ) = 1. I f f is continuous with respect to the topology 6, then there exists a unique probability measure Pt on (Ot, St), such that f is expressible in the form (4.3.19), moreover, this measure Pt is continuous with respect to 6. The proof of this theorem depends upon Lemma 4.3.13 below. We shall employ the following notation. Let R, be an m-dimensional real., inner product space, let el ,..., em be an orthonormal basis in R , , and if x E R, , let x l = (x, el). Let b be the totality of Bore1 sets in R, , let ( R , , b,Q) be a probability measure space, and denote the characteristic function of Q by x(y) =

1

ei(u,r)d ~ ( x ) .

Lemma 4.3.13. Let Cz,=l u,,y,y, form, and let A2 = CYGlukk . If q

>,

sup

Cq,vkv,
be a positive definite quadratic

(1 - %X(Y)),

then

PROOF. It suffices to prove (4.3.20) for R = 1. First, we note the following property of Gaussian measures (see $5.1):

40

Translator's note: A detailed proof of this assertion would not be amiss.

4.3. Continuous Positive Dejinite Functions

269

Using this identity, we obtain

(4.3.21)

c

< < <

<

c

Now, if a,,yky, 1, then 1 - %x(y) 9, and if akZyky, 3 l, then 1 - %x(y) 2 2 C a k r y k y lHence, . since 1 a,,y,y, is positive definite, the inequality 1 - %X(Y)

< 7 + 2 1%YkYZ

(4.3.22)

holds for all y E R, . Using (4.3.22) in (4.3.21), and evaluating a simple Gaussian integral, we obtain (4.3.20) (for R = 1). T h e calculation for arbitrary R is entirely similar. ] PROOFOF THEOREM 4.3.12. Obviously, Q' distinguishes the points of 6. Hence, applying Lemma 4.3.2 to the function f, we obtain a cylinder measure P on 6+such that f and P satisfy (4.3.3). We need only prove that P is countably additive, for then P can be extended to a probability measure P+on (O+,%+),and (4.3.3) becomes (4.3.19). Let

By virtue of Corollary 4.3.9, it suffices to prove that, for any positive number E , there exists a ball S m

=

I

{F II F Il-n

< R1

such that the cylinder measure of any Bore1 cylinder lying outside of S J R ) is less than E . First, choose a number 9 satisfying (4.3.23)

270

Iv.

LINEAR TOPOLOGICAL SPACES

Since f is continuous with respect to the topology 6,there is a natural number n and a positive nuclear oherator T on an such that, if x E U( T, n), that is, if (Tx, x ) ~ < 1, then 1 - !)If(%)

< 7.

(4.3.24)

Let )I 2'11, denote the trace norm of the nuclear operator and choose a positive number R such that

T (see §II.l), (4.3.25)

We shall prove that the ball S J R ) fulfills the above-mentioned requirement. Let L be any Borel cylinder in 6+,say, L

= {5 I (5(el)Y., 5(em)) E B ) ,

where el ,..., em E 6 and B is a Borel subset of R, . We may assume that the vectors e, ,..., e, form an orthonormal set in 6, . If L and Sn(R) are disjoint, then the base B of L and the ball x12 xm2 R2 are disjoint subsets of R, . I n fact, if there were a point (xl ,..., x,) E B such that C xi2 R2, then the functional

+ +

<

<

on

would satisfy

hence

toE Sn(R).But, on the other hand, we get

that is, toE L , which is impossible. Define a probability measure Q on R, as follows: for any Borel set D in R, , let Q(0) = P({5 I (5(e1)9...9 I ( e , d

E 0)).

We shall apply Lemma 4.3.13 to the measure Q, the function x ( y ) = f ( X y , e , ) , and the quadratic form with coefficients ULZ =

(Te* 9 edn

-

4.3. Continuous Positive DeJiniteFunctions

27 1

By the properties of the trace norm, we have A2 = Cr=lukk Moreover, (T ( x Y k e k ) ,

xYkek),

=

< II T . 1 1 1

akZYlcYZ >

which shows that the quadratic form C akzyky, is positive definite, and 1, that C ykek 6 U ( T ,n) whenever C ak,yky, 1 . Hence, if C aklykyr then, by (4.3.24),

<

<

- sf( l Y k e k ) < 7. Now, applying Lemma 4.3.13, and using (4.3.23), (4.3.25), we get

Thus, S,(R) satisfies the required condition, and we conclude that P is countably additive. The uniqueness and continuity of Pt may be deduced by applying Lemmas 4.3.2 and 4.3.3, respectively. ]

Corollary 4.3.14. Let (ti be a real nuclear space, and let f be a continuous positive definite function on (ti, withf(0) = 1. Then, there exists a unique probability measure Pt on ((tit, S'), such that f and Pt are related by (4.3.19). PROOF. Let 2 denote the original topology of (ti. I t suffices to prove that 2 C 6, for this implies that f is also continuous with respect to G, and the desired conclusion then follows from Theorem 4.3.12. Consider any basic neighborhood of zero in the topology 2,say,

v(k,

< €1. Since (6,2) is a nuclear space, there is an n > tz such that the imbedding map Tknis a nuclear operator. Now, €)

(9's

(P)R

(9' I (9'3

= ((Tkn)*

9')k

TknviP)n

9

and, by Lemma 11.1.1 of Appendix 11, T = (Tk")*Tkn is a nuclear operator on (ti, . Thus, V(k,c) = U(T/c,n), which shows that T C G. ]

Corollary 4.3.15.

Let G be a real Hilbert space, and let

(ti

be a

272

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LINEAR TOPOLOGICAL SPACES

linear subspace of G which is itself a real separable Hilbert space with respect to an inner product (h, x), h e 0, X E 0. Suppose that the imbedding A of 0 into G is a Hilbert-Schmidt type operator. Then, given any continuous positive definite function f on G, there exists a unique finite measure Pt on (St, St),such that

moreover, Pt is continuous with respect to the topology 5 induced by G on 8. PROOF. Let [h,g] denote the inner product on G. Then, for any h,gE(fj, W,gl

=

(A*Ah,g),

and by Lemma 11.1.1, A*A is a nuclear operator. This shows that 2 is weaker than the topology G on 6 (here, we regard a Hilbert space as a special case of a countably Hilbert space). 3 Now, we give the converse of Theorem 4.3.12.

Theorem 4.3.16. Let 8 be a real separable countably Hilbert space. If Pt is a probability measure on (Ot, gt),then the function

is continuous with respect to the topology 6. PROOF.Consider the sequence of sets S, = (F 11 F 11By Lemma 4.2.17, each S, is measurable, and obviously

I

< n) in Bt.

m

0 + =(J s,. n=l

Therefore, given any positive number Pt((fit- S,) < ~ / 4Since .

E,

there is an n such that

4.3. Continuous Positive Definite Functions

273

Define a positive definite Hermitian bilinear functional T on Q,, as follows: 1 T ( g ,4 = ; 5(g) '34 dPt(6).

1

S"

Since I c(g)[

< n [[g[l,whenever I T ( g ,4

(E

S,

we easily obtain the estimate

< T1 n2 II g lln II h

Iln

*

Thus, T ( g , h) is continuous with respect to the topology of 6, . Choosing any complete orthonormal system { g k } in 6 , , we get

c

1

T(gk

9

gk)

=

T

1 11 6

1 6(gd2dPt(6) = ;

dPt(6) <

n2 *

Sn

Hence, using Theorem 11.1.6, we see that there exists a positive nuclear operator T on 6, such that T ( g ,h) = (Tg,h), . If g E U(T , n), that is, ( T g , g ) , < 1, then, from (4.3.27),we get 1 - %f(g)

< E.

This shows that %f is continuous with respect to the topology 6. Since f is positive definite, it follows that f itself is continuous with respect to 6. 3 50 Representation of Continuous Positive Definite Functions on Exact Unions of Nuclear Spaces We now proceed to establish a further generalization of Corollary 4.3.14.

Definition 4.3.5.

Let @ be a linear topological space, and let a sequence of closed linear subspaces of @; for convenience, we assume that @ ( l ) ,@(2),...,

@(m),...be

Suppose that the following conditions are satisfied. m (i) @ = Wm). (ii) Every continuous linear functional on Pm)can be extended to a continuous linear functional on 0. (iii) If, for each m, there is given a continuous linear functional and if F ( m ) ( = F(m)on ~ )FCn)(y)whenever m >, n and rp E @(,I,

274

Iv.

LINEAR TOPOLOGICAL SPACES

then there exists a continuous linear functional F on @ such that

F(v) = Fn)(p)when cp E @cn). We then say that @ is the exact union41 of the sequence of subspaces {Pm), m = 1, 2,...1. Let Qtand @(m)t denote the conjugate spaces of @ and @(m),respectively. Take the weak topology on Qt and @(m)t, m = 1,2,... . If n 2 m, then every element f ( n ) of @(n)t,when restricted to @(m),defines a continuous linear functional ( ( m ) on @fm). Thus, we have a mapping pmnpn) = f ( m )

from @ ( n ) t to @(mlt. Condition (ii) in Definition 4.3.4 implies that Pmn@(n)t = @(m)t, moreover, it is easily seen that Pma is continuous. Let A denote the totality of positive integers directed by the natural ordering; then {Pmn, n >, m,n, m E A} is a consistent family of projection operators in the sense of Definition 1.3.5. If f E Qt, the restriction of f to @(m) is a continuous linear functional f ( m ) on @ ( m ) . We thus obtain a projection operator P,: f

+ f(")

from Gt onto @(m)t. I t is easily verified that the projections {Pnb, m E A} and {P,", n 2 m,n, m E A} satisfy the mutual consistency condition (1.3.2), and that the linear topological space Gt is thereby represented as a projective limit (with respect to the family of projections {Pmn,n 2 m,n, m E A}) of the sequence of linear topological spaces {@(m)t, m = 1 , 2,... }, in accordance with Definition 1.3.5. By virtue of condition (iii), the projective limit @+ satisfies the conditions of Lemma 1.3.2, and is therefore projectively complete. Let B(m)tand Bt denote the a-algebras consisting of all weak Bore1 sets in @ ( m ) t and W , respectively. I t is easily proved that {P,", m 2 n) is a consistent family of projections in the family of measurable spaces {(@(m)t, g(m)t), m = 1, 2, ...}. Let 8 be the algebra of all sets of the form P;'(A), A E B(m)t,m = 1 , 2,... . Again, using the projections {P, , m E A}, one may represent (Dt, 8) as the projective limit (with respect to the family of projections (Prim, m >, n}) of the sequence of measurable spaces {(@(m)t, m = 1, 2,.,,}, in accordance with Definition 1.3.1. Moreover, 8' is clearly the smallest a-algebra containing 5.

s(m)t),

Theorem 4.3.17. If the linear topological space @ is the exact union of a sequence of nuclear spaces, then, for each continuous positive Trunslutor's note: The corresponding term used in Gel'fand-Vilenkin [l] has been translated as union space.

4.3. Continuous Positive DeJnite Functions definite functionfon @ [withf(O) measure P t on (@*, 5’)such that

=

f ( v )= J

275

I], there exists a unique probability

eip(0) d ~ t ( 6 )

(4.3.28)

at

for all F E @. PROOF. Let @ be the exact union of the sequence of nuclear spaces @ ( l )c @ ( 2 ) c ... c @(n) c ... . By Corollary 4.3.14, for each m, there is a unique probability measure P ( m l t on (CD(~)+, 5tmlt)such that

(4.3.29) fJ(m)t, P m j t ) is a regular for all g, E @(m). By Lemma 1.1.18, (@tm), measure space. Moreover, since the family of measure spaces {(CP(m)t, g ( m ) * , P(m)t),m = 1, 2,...I is consistent with the projections {Prim, m n}, it follows by Lemma 1.3.1 that there exists a cylinder measure p on (@?, 5) such that ( W , 5,p ) is a projective limit of the family {(CP(m)*,SCm)+,P ( m ) t )m, = 1, 2,...}. Furthermore, by Theorem 1.3.4, p is countably additive, and may therefore be extended to a probability measure P t on (@+, Bt). If g, E @(m), then, from (4.3.29), we get

Therefore, since CP = Uz=l W m ) , relation (4.3.28) holds for all F E @. Finally, the uniqueness of P t follows from the uniqueness of the p(m)t.

1

Corollary 4.3.18. Let K be the space of test functions defined in Example 1.3.3 of Appendix I. Then, the space of generalized functions Kt is a spectral space for K . PROOF. Since K is the exact union of the sequence of subspaces {K([-n, n ] ) , n = 1, 2,...}, and since each K ( [ - n , n ] ) is a nuclear space, our conclusion follows immediately from Theorem 4.3.17. ] 6O

Generalized Stochastic Processes

As we know, an ordinary stochastic process is just a family of random variables {xi(.), t E T } defined on a probability measure space 52 = (G, 8,P ) ; here, t represents the “time parameter,” and the indexing set T is a set of real numbers representing the time range under consideration, For example, the coordinate of a particle performing

Iv.

276

LINEAR TOPOLOGICAL SPACES

one-dimensional Brownian motion may be described by a stochastic process of just this kind, namely, the Wiener process. However, phenomena governed by statistical laws cannot all be described by this sort of model, for example, the velocity of a particle performing onedimensional Brownian motion cannot be described by an ordinary stochastic process. It frequently happens that one cannot directly observe the values of xI(w), rather, one observes a kind of average value X(w;d =

J,

X t b )

944 4

where the function y is determined by the instrument used for the observation. This clearly necessitates the use of a function space. In this case, the random process is not manifested in the form {xt(o), t E T, w E G), instead, we have a family of random variables {X(-; cp), y E @}, and one expects X(.; y ) to satisfy certain linearity conditions. Formalizing these ideas, we are led to the concept of a generalized stochastic process (see Definition 4.3.6). Let SZ = (G, 23, P) be a probability measure space, and let M ( Q ) denote the linear space consisting of all random variables on SZ, with the ordinary linear operations; here, any two random variables which are equal almost everywhere are regarded as identical. We introduce the metric

which makes M(Q) a complete linear topological space.

Definition 4,3.6+ Let @ be a linear (or linear topological) space, let SZ be a probability measure space, and let {X(-; y ) , v E @) be a family of random variables on SZ, such that the correspondence

u: p + X ( * ;q)),

ql E @

is a linear (or continuous linear) mapping from @ into M(SZ). Then, we say that {X(.; v),y E @) is a linear (or generalized) stochastic process on (Q, @). I n the applications, Q, is usually some space of the type K,(d), K ( d ) , K, and so on (see Examples 1.3.1 and 1.3.3 in Appendix I).

Definition 4.3.7. Let {X(-; y ) , v E @} be a linear stochastic process on (SZ, @), let Y be a linear subspace of @ A which distinguishes the points of @, let 5 be the totality of @-weak Bore1 sets in Y, and let p be

4.3. Continuous Positive DeJinite Functions a probability measure on (Y, 8). For each variable on Q’ = (Y, 8, p), as follows:

277 p) E

@, we define a random

(a‘,

Clearly, { X ’ ( . ;p)), p) E @}is a linear stochastic process on @).Suppose that, for every finite set y1 ,..., cpn E @, the two sets of random variables (4.3.31)

have the same probability distribution, that is, the equality

holds for every Bore1 set A in n-dimensional space. Then, we say that Y, and { X I ( - ;p)), p) E@} are, respectively, a sample space, a sample probability space, and a sample process for the linear stochastic process 91,9 E @I. T o facilitate our study of sample spaces of linear stochastic processes, we introduce the notion of a characteristic functional, as follows.

a’

{a*;

Definition 4.3.8, If on (Q, @), the integral

{X(*; p)), p) E @} is a linear stochastic process

will be caIIed the characteristic functional of the process. Using the linearity of X(.; p)) with respect to p), one easily verifies that the characteristic functional (4.3.32) is a pseudocontinuous positive definite function on @. Any linear stochastic process {X(-; cp), p) E @} has as a sample space. PROOF. Since the characteristic functional L(p)) of { X ( - ;p)), p) E @} is a pseudocontinuous positive definite function on 0,it follows by Theorem 4.3.5 that there exists a probability measure PA on (QA, 8”) such that

Theorem 4.3.19.

@A

Iv.

278 Define a linear stochastic process as follows. For each 'p E 0,let

LINEAR TOPOLOGICAL SPACES

{A?(*;'p), cp E @} on (aA, iJ*, PA,@),

Then L('p) is also the characteristic functional of { X ' ( - ;y ) , cp E @}. Consequently, by substituting 'p = tlcpl *.. t,y, (- 00 < t , < m) into (4.3.32), (4.3.33), and (4.3.34), one may deduce that the two sets of random variables (4.3.3 1) have the same probability distribution. ; cp E @}. 3 Thus, @ a is a sample space for { X ( *cp), Using an argument similar to the proof of Lemma 4.3.3, we obtain the following result.

+ +

Lemma 4.3.20. If @ is a linear topological space and L? is a probability measure space, then a linear stochastic process on (Q, @) is a generalized stochastic process if and only if its characteristic functional is continuous. Using Lemma 4.3.20, Definition 4.3.4, and the proof of Theorem 4.3.19, we obtain the following result.

Theorem 4.3+21, Let @ be a linear topological space, and let P ! be a spectral space for @. Then, every generalized stochastic process { X ( - 'p), ; cp E @} has !P as a sample space. Using Corollaries 4.3.14 and 4.3.18, we immediately obtain the following two corollaries. Corollary 4.3.22, Every generalized stochastic process

has K(0)' as a sample space (see Example 11.1.1 of Appendix 11).

Corollary 4.3.23.

Every generalized stochastic process

N.;4, v E K ) has K t as a sample space (see Example 1.3.3 of Appendix I). We end this chapter by pointing out the relation between linear stochastic processes and cylinder measures. The following theorem is a consequence of Lemmas 4.3.2 and 4.3.3.

Theorem 4.3.24. Let {X(.; 'p), 'p E @} be a linear stochastic process on (52, @), and let Y be a linear subspace of @ A which distinguishes the points of @. Then, there exists a cylinder measure p on !P, such that,

4.3. Continuous Positive Dejinite Functions for any finite set cpl Euclidean space,

,..., vn E @,

and any Bore1 set A in n-dimensional

P({W I (-wG P1),...,X b ; Pn)) =

279

d f I (f(n),*-,f(vn))

E

4) A f G

v.

Moreover, if @ is a linear topological space, then (X(*;p)), p) E @} is a generalized stochastic process if and only if p is continuous with respect to the topology of @. We say that the cylinder measure p in Theorem 4.3.24 is generated by the linear stochastic process ( X (-;cp), tp E @I. Conversely; cylinder measures give rise to linear stochastic processes, as shown by the following theorem.

Theorem 4.3.25. Let @ be a linear (or linear topological) space, let Y be a linear subspace of which distinguishes the points of @, and let p be a cylinder measure (or continuous cylinder measure) on Y. Then, there exists a probability measure space 52 = (G, '23, P) and a linear (or generalized) stochastic process (X(.; p)), p) E @} on (52, @), such that p is generated by { X ( . ;rp), 9 E @}. PROOF. Consider the positive definite function

and apply the reasoning used in proving Theorems 4.3.19 and 4.3.21. ]

CHAPTER

V GAUSSIAN MEASURES

Gaussian measures are the most classical category of quasi-invariant measures. In 95.1, we introduce the basic properties of Gaussian measures, and give an example of an ergodic quasi-invariant measure which is not equivalent to any Gaussian measure; we also present an introductory discussion of the most commonly used type of Gaussian measure, that is, Wiener measure. In $5.2 we derive a criterion for the equivalence or mutual singularity of Gaussian measures. A very significant problem is that of determining those types of measures on a given linear measurable space which are quasi-invariant relative to a prescribed linear subspace; the investigation of equivalence and singularity of Gaussian measures constitutes a preliminary step in this direction. In $5.3, we describe a class of Gaussian measure spaces which are useful in connection with harmonic analysis on linear topological spaces; we also explain some important problems in the theory of quasi-invariant measures, illustrating the situation by reference to Gaussian measures. I n 95.4, we discuss in detail the L2-Fourier transform for Gaussian measure spaces, and, in particular, the Fourier transform on Wiener measure spaces. 280

5.1. Some Properties of Gaussian Measures

28 1

55.1. Some Properties of Gaussian Measures

lo The Joint Distribution of Finitely Many Gaussian Variables First, we describe the nondegenerate case. Let R, be the ordinary real n-dimensional EucIidean space, denote the points of R, by x = (xl ,..., xn), and write (x, y ) = C x z y z. Given any positive-definite real quadratic form CZ,, bz,xlx, (blm = b,J, we define another inner product in R, by n

( x , Y )= ~

1

x = (x1,--.*xn),

b l r n ~ l ~ 9r n

Y

= ( ~ ,..-,~n)1

l,m=l

Let (cZm) denote the inverse of the matrix (btm). For any Bore1 set E in R, , define N(E)=

1

j

(2n-)n/2(det(b,,))1/2

e-(x-a,x-a),/2

dx

(5.1 . I )

where (x, x ) = ~ C czrnxlx,, a = (al ,..., a,) E R, , and d x denotes the Lebesgue measure d x , . * - dx, induced by the original inner product ( x , y ) . Clearly, (5.1.1) defines a probability' measure on R , . Any measure N defined in this way is said to be Gaussian. Likewise, a set of random variables x1 ,..., x, , whose joint distribution is of the form (5.1,1), is said to be Gaussian. Note that the mathematical expectations, correlations and characteristic function of a set of Gaussian variables are given, respectively, by

(5.1.3)

E(exp i(tlxl

+ -..+ tnxn)) = JR

eixtvxvdN(x) = e-(t.t)b+i(t.a)/2,

t

= (tl

,..., tn). (5.1.4)

I n the degenerate case, the quadratic form C blmxZx, is only nonnegative definite. We may then use (5.1.4) to define the corresponding Gaussian measure, since any characteristic function uniquely determines a measure. T h e explicit expression for the measure is, in this case, T o verify that N(R,,) = 1, apply an orthogonal transformation which diagonalizes the matrix ( c 3 , thus separating the variables in the integrand. T h e integral may then be evaluated by using formula (5.1.5).

v. GAUSSIAN MEASURES

282

slightly more complicated than (5.1.1). If b,, = 0, I , m = 1, 2,..., n, then the measure is concentrated at the point a, that is, N ( E ) = 1 if a E E and N(E) = 0 if a Z E. If the rank of (bl,) is I, 0 < r n, then there is an orthogonal transformation u of R, [relative to the inner product (x, y ) ] such that (x, x ) = ~ x ’ ) ~ !, where x f = ux = (xl‘,..., x),‘ and

<

(XI,

c 7

(x’,yf)0*=

A,x,~,’,

A,

> 0.

v=l

+

Let M,, = R, a’, where R, denotes the r-dimensional linear subspace of R , consisting of all vectors of the form (xl , x, ,..., x,, , 0,O,..., 0), and a’ = ( a l f..., , urn’)= ua. Then a calculation shows that

and that (5.1.2) and (5.1.3) are still valid. Relations (5.1.2)-( 5.1.4) show that a Gaussian measure is completely determined by its means and correlations (or covariances). T h e following formula is often useful when calculating integrals with respect to Gaussian measures. If %p > 0 and m is a nonnegative integer, m

Am exp(-pA2 -m

+ qA) dA = (.r/p)1/2(d/dq)m exp(q2/4p).

(5.1.5)

Notice that two Gaussian variables X , Y are stochastically independent if and only if E((X - E ( X ) ) ( Y- E ( Y ) ) ) = 0, and that X is a degenerate Gaussian variable if and only if E ( ( X - E(X))2)= 0.

Furthermore, using (5.1.5), one can calculate that a nondegenerate Gaussian variable X satisfies2 E((X2 - E(X2))2)

= 2(E(X2)2 - E(X)4). (5.1.6) Next, we consider limits of sequences of Gaussian random variables.

Here, u ( X ) = E((X - E(X))*)is the variance of X .

283

5.1. Some Properties of Gaussian Measures

Lemma 5.1.1. Let S = (SZ, 8,P ) be a probability measure space, and let {X,( *)} be a sequence of Gaussian variables on 9 which converges in L2(S)to a random variable X . Then X is also a Gaussian variable. PROOF. Since X,( E L2(S ) , naturally, X ( .) E L2(S ) , therefore a)

lim 1 E(X,) - E ( X ) / < Iim E((X, - X)2)1/2 = 0, n+m

T

n+m

Eiii I E(X,2)

n+m

- E(X2)I =

E I E ( ( X , - X ) ( X , + X))l n* m

Eiii I E(exp(iX,t)) - E(exp(iXt))I < lim E(( X , n+ m n+ m

-X

I2)lI2

*

It I

= 0.

However, by (5.1,4), the characteristic function of X , is E(exp(iX,t)) = exp(-$u(X,) t 2

+ iE(X,) t ) .

Letting n + co on both sides, we get E(exp(iXt)) = exp( -+(X)

t2

+ i E ( X )t ) .

Hence, X is a Gaussian variable. ] 20 Gaussian Stochastic Processes

Definition 5.1.1. Let S = (SZ, 8, P ) be a probability space, {Xu, a E a} a family of random variables on S. If the joint distribution of any finite subfamily X., ,..., Xu, is Gaussian, then the family {Xu, a E N} is said to be Gaussian, and is also called a Gaussian process on 2L As we have seen, the means and second moments, that is, au = E ( X u ) ,

E(XuXo) = baB

+ au%

(5.1.7)

completely determine the finite-dimensional distributions of such a process. Notice that the family of real numbers {buo, a, /3 E a} is nonnegative definite, that is, for any finite set of complex numbers f,, ,..., tun,

Conversely, given any family of real numbers { a , , a E %} and any nonnegative definite family of real numbers {buB, a , /3 E a},there exists

284

v. GAUSSIAN

MEASURES

a Gaussian process { X a, 01 E %} satisfying (5.1.7). I n fact, for any finite set of indices a1 ,..., 01% E %, form the function

By (5.1.4), this is the characteristic function of a Gaussian measure. Obviously, the family of all such functions {~al,...,an) is consistent. For every a E %, let R, be a copy of the real line, and form the Cartesian product r = XaEcU R, . Let b be the smallest a-algebra in which contains all Bore1 cylinders. By the Kolmogorov theorem (Corollary 1.3.5’), there exists a probability measure P on (r,b) such that the characteristic functions of the finite-dimensional distributions of the stochastic process {Xa(w),01 E %} [where X,(w) denotes the ath coordinate of the point w E r] are given by (5.1.8). This process is clearly Gaussian and satisfies (5.1.7). We call a measure space (I‘, 9, P) constructed in this manner a canonical Gaussian measure space. Using the Kakutani inner product, we first proceed to consider the equivalence or perpendicularity of a special type of canonical Gaussian measure space.

r

Lemma 5.1.2. Let XasSR, = r be the Cartesian product of a family of real lines, b the smallest a-algebra in r containing all the Bore1 cylinders. Let Nk , K = 1, 2, be probability measures on (r,8) such that the coordinate functions {xa(w), 01 E a} [here, xa(o) represents the ath coordinate of the point w E r]are independent Gaussian random variables on (T, 9, Nk), with mathematical expectations m:k) and variances aLk)> 0 relative to Nk , K = 1, 2. Then Nl and N2 are either equivalent or mutually singular, A necessary and sufficient condition for the equivalence of Nl and N 2 is

Moreover, when Nl and N2 are e q ~ i v a l e n t , ~

By virtue of condition (5.1.9), there are but countably many indices (Y such that up),mc) # mg),hence this product is essentially just an ordinary iniinite product of countably many factors.

02) #

285

5.1. Some Properties of Gaussian Measures

PROOF.Suppose that (5.1.9) holds. Then, the set 2l' of indices a: such that m:') # mL2)or a:') # uL2)is at most countable; for convenience, we shall denote these indices a by the numerals 1, 2, ... . Let 2l" = 1 ' 1 - 2l'. First, we point out that one may assume 2l' = CU. I n R, is the direct product of Xae%,R, and Xaea*R , . Since fact, {xa , a E 1 ' 1} are independent Gaussian variables, each measure N k is the product of the Gaussian measures Nk' and Nk" on R, and x a O w x R, , respectively. Since m f ) = mh2)and 0:') = uL2)for all a: E a", the variables {x,, , a: E a''}have the same finite-dimensional distributions relative to N;' and N i , hence N ; = N l . Consequently, in view of the facts mentioned in $1.1, 40, it suffices to prove that N,' and N,' are equivalent. If 2l' should contain only a finite number of indices, then, since u, # 0, N,' and N2' would be nondegenerate Gaussian measures on a finite-dimensional space, and so they would obviously be equivalent. Hence, we shall assume that 2l = a' is just the totality of natural numbers. Let N i k )be the Gaussian measure on R, with mean mLk) and > 0. Then NL') and Nh2)are equivalent, and their Kakutani variance inner product is

Now, Nk' is the product of the measures {N,'k),a Theorem 1.4.4,

=

1, 2,...}. Hence, by

(5.1.1 1)

Since

x (a,

-

(l/a,))2

< co is equivalent to

x (I:.z

-

(l/a:''))'

< a,

v. GAUSSIAN MEASURES

286

n.“=,

it follows from (5.1.11) and condition (5.1.9) that p(N:’), NA2’)> 0. Therefore, by Theorem 1.4.4, N,’ and N2’are equivalent. Moreover,

hence, using (1.4.15) and Theorem 1.1.20, we obtain (5.1 .lo). Conversely, suppose that condition (5.1.9) does not hold, for example, suppose that

then there exists a sequence of indices a = 1, 2, ..., such that

a€%,

which we denote by (5.1.12)

Now, in view of (5.1.11), we have

However, by (5.1.12),

nf,

p(Nil’, NL2’)= 0, that is, Hence, by (5.1.13), we conclude that that N,’ and N2’are mutually singular. The case

may be handled in a similar fashion. J Example 5.1.1. Let (I, b, ,p ) be the linear measure space discussed in Example 4.1.2, with probability densities as follows: 1 fu(t> =

(2Tuu)’f4

exp

(-

1 t2

4

uu > 0,

....

a: = 1) 2)

Then, the totality of quasi-invariant points of (I, 8 , , p ) is just 12({l/ua}).

5.1. Some Properties of Gaussian Measures I n fact, let y

E

I, y

= (yl

,...,y, ,...},

287

and form the measure

Then {x,(w), w E Z}(w = {xl(w), ..., x,(w), ...}) are mutually independent Gaussian variables on (I, 23, ,p,,), with mean values yu and variances o, . Using (5.1.9), we see that a necessary and sufficient condition for the equivalence of py and p is

that is, y

E Z2({l/oa}).Moreover,

if y E Z2({l/ou}),then, by (5.1.10), (5. I. 14)

Now, let (I, B , , P) be a canonical Gaussian measure space, with Gaussian variables x,(w), w E I [xa(w) denotes the ath coordinate of a]. The characteristic function q(f) of (I, BU,P) may be constructed as follows. For each f~ It If(.) = fuxa ,x = {xu) E I], applying (5.1.4), we obtain

) E(g(w))));this is a positive Write c ( f , g ) = E ( ( f ( w ) - E ( f ( o ) ) ) ( g ( w definite bilinear functional on It. Also, write

which defines a linear functional on It. Then, the characteristic function may be rewritten as ?(f) = e-c(f.f)+inff)/2,

Lemma 5.1.3.

f 6 p.

(5.1.15)

Let I,, be the totality of real number sequences

x = (xn) such that only a finite number of the x, are not zero; I, forms

a linear space with respect to the usual linear operations (see Example 4.2.1). Let 1 a < 2, and define a norm on I, by

<

v.

288

GAUSSIAN MEASURES

Let [x, y ] be any positive definite bilinear functional on l o . Then, the norms I x I = [x, xI1l2 and I x ,1 cannot induce the same topology on I , . PROOF. Suppose that 1 x 1 and 1 x ,1 did induce the same topology. Let I, denote the completion of I, relative to the norm I x ,1 ; clearly, I, may be identified with the Banach space consisting of all real number sequences (x,} such that 1 x ,1 = (C,"cl 1 x, /')l/a < co. Since [x, x] and I x ,1 induce the same topology on lo ,it follows that [x, y ] can be extended to a continuous inner product on I, ; we denote this extension also by [x, y ] . Moreover, 1 x 1 = [x, x]1/2 and 1 x ,1 induce the same topology on I , , hence I, is a Hilbert space relative to [ x , y ] . Therefore, there exists a positive number c such that, for all x E I, , (5.1.1 6 )

Now, for any n vectors g1 ,..., f , E I, we have

, since

[x, y ] is an inner product,

whence, using (5.1.16), we get

In particular, taking & to be the vector (0,0,..., 0, xi, 0,..., 0,...}, and substituting into (5.1.17), we find that

for any n, x1 ,..., x,. But since a # 2, this is obviously impossible. Hence, we conclude that 1 x I and I x ,1 induce different topologies on I,, . ]

<

Theorem 5.1.4. If 1 01 < 2, then the ergodic measure pa in Example 4.2.1 is not equivalent to any Gaussian measure.4 PROOF. Suppose there existed a measure N on ( I , BU),equivalent to p, and such that the variables {xy(w), Y = 1, 2, ...} are Gaussian. Form Translator's note: Meaning, in this case, a measure relative to which the coordinates are Gaussian variables.

{x,(w)}

5.1. Some Properties of Gaussian Measures

289

the characteristic function y(f) of N as in (5.1.15); then, according to (4.2.14), we have

Since we have assumed that N and p. are equivalent, it follows from Corollary 4.2.1 55 and Example 4.2.1 that the functionals

and R , ( f ) [see (5.1.19)] induce the same topology on I t . Write [f,g1 = 4f,d

+ E(f) W),

f,g

E It.

We shall now prove that R l ( f ) ,f E P , and [ f , f ] 1 / 2f, E P, induce the same topology. T o this end, consider the function m

u(a, b ) = %

(1 - exp(--at2 n

+ ibt)) ect dt,

0


03,

-03

< b < 03.

Obviously, lima+lb,+n u(a, b) = 0. Conversely, if u(a, , b,) -+ 0, then a, I b, I -+ 0. In fact, suppose that, for some subsequence, we had a, --t a, where 0 < a 00; then

+

<

On the other hand, if a,

-+

0, but b,

-

b, where b # 0, b # &a,then

1

m

lim u ( a , , n+ m

b,)

1-

cos b w t dt

> 0.

n

The case a, 4 0 , b, -+ * C O gives a similar result. T o summarize, whenever lim u(a, b ) = 0, we have lim(a + 1 b I) = 0. Consequently, in view of (5.1.19), R,( f,) 0 is equivalent to c ( f , ,f,) 1 E( fn)l +0,

-

+

Translator’s note: There seems to be a snag here, that is, the transformation group I , is not of the second category in the topology induced by I ; it is not obvious that there exists any stronger topology on Zo relative to which Zo is a linear topological space

8

=

of the second category.

v. GAUSSIAN MEASURES

290

that is, equivalent to [f, ,f,] 40. In a similar fashion, we can show that P ( f ) and If la induce the same topology on It. Thus, it follows that If la and [f,f]1/2 induce the same topology on It; but this contradicts Lemma 5.1.3. Therefore, we conclude that pa is not equivalent to any Gaussian measure. ] Thus, we have given an example of an ergodic measure which is not equivalent to any Gaussian measure.6

30 A Special Class of Gaussian Processes: Wiener Processes There is an important class of stochastic processes, called Wiener processes, which are Gaussian, Markov, and have stationary increments. For convenience, we shall only consider Brownian motion on the real line. Let Co[O, TI be the totality of continuous functions on the interval [0, TI satisfying the condition x(0) = 0. Clearly, Co[O, T ] forms a Banach space with respect to the usual linear operations and the norm

Example 5.1.2.

x(t)

Ix1

= max O
I x(t)l.

Let 23 denote the totality of weak Borel sets in Co[O, TI. Actually, !I3 can be constructed in the following manner: take any partition 0 < t,
<

= .{

I (x(&..,

@n))

E El

(5.1.20)

the Borel cylinder with base E associated with the points t , ,..., t , . Then 23 is just the a-algebra generated by the totality of Borel cylinders. Let 0 < c < 00, and let W, be a probability measure on (Co[O, TI, 8) having the following property: for any set I? of the form (5.1.20),

where to = xo = 0. Then, W, = (C,[O, TI,8,W,) is called a Wiener measure space. For each t E [0, T ] , the function x ( t ) , x E Co[O, TI, is a measurable Translator's note: This assertion must be interpreted in the context of the translator's note appended to the statement of Theorem 5.1.4.

5.1. Some Properties of Gaussian Measures

291

function on the measurable space (Co[O, TI, B), and is therefore a random variable on Wc. Moreover, we see from (5.1.21) that the variables x ( t ) are Gaussian. T h e Gaussian stochastic process { x ( t ) , t E [0, TI} on W, is called a Wiener process. Using (5.1.21), one easily calculates that the mean and correlation functions of the Wiener process are E ( x ( t ) ) = 0,

E(x(t) x(s))

C

= - min(t, s).

2

(5.1.22) (5.1.23)

< <

Now, let 0 01 1 and let Ct'EO, TI denote the totality of functions in C,[O, T ] which satisfy the Holder continuity condition

Theorem 5.1 +5. The Wiener measure space W, = (Co[O,TI, 8,W,) does exist. Moreover, if 0 01 < &, then the Wiener measure W, is concentrated on Ck)[O, TI. PROOF. Since Clp'[O, TI = Co[O,TI, it suffices to prove the theorem for 01 > 0. Clearly, we may assume that T = 1, c = 2. Let I denote the totality of numbers of the form

<

m = 1 , 2 ,..., 2"; n = 1 , 2 ,....

m 2" '

-

Obviously, I is a denumerable set. For each t E I , let R , be a copy of the real line, and form the Cartesian product R ( I ) = XlClR , , that is, the set of all real-valued functions on 1. Let 8,be the o-algebra generated by the totality of Bore1 cylinders in R(I). By virtue of the remarks at the beginning of 20, there exists a canonical Gaussian measure space ( R ( I ) ,8, , P ) such that the coordinates have mathematical expectation (5.1.22) and correlations (5.1.23), where t, s E I . Observe that, if 0 < t, < t, < < t, 1, where ti E Ifor j = 1 ,..., n, and 8 is of the form (5.1.20), then the value of P ( E ) is indeed given by (5.1.21). Write I, = I {0}, and extend the functions in R ( I ) to I , by defining x(0) = 0. For 0 < 01 1, K 2 0, let CaK(Il)denote the totality of functions x E R ( I ) which satisfy the condition

<

+

<

v. GAUSSIAN

292 Lemma 5.1.6.

MEASURES

If 0 < a: < i,then P(R(I) - C&))

(5.1.24)

= 0.

PROOF. For any positive number K, positive integer n, and nonnegative integer m, 0 m < 2n, let

<

S ( K ,fl, m>

Clearly, S(K, m,n) E !B1. We shall now prove that

R ( 4 - CaK(Z1) c

u u w, m

2"-1

m,4.

(5.1.25)

n=l m=O

Take any x E R ( I ) - CmK(ll);then, for some t ,

j

-ti

= min

< t , , t, , t , E I , , we have

1 >2n+11*

n t, - t ,

(5.1.26)

Since

we get (5.1.27)

We shall show that x E (J~==,+, U g ~Si( K , n, m).Suppose this were not true; then, x E S(K, m,n) for all n j 1, 0 m < 2", that is,

> + 0

<

< m < 2",

1z

>j

+ 1.

(5.1.28)

Let us refer to an interval of the form

as an interval of order m. By (5.1.26), we have tz - t, ,< (1/29. Therefore, ( t , ,t2] contains at most two intervals of order j 1, which we

+

5.1. Some Properties of Gaussian Measures

293

denote by I;:\, I;!\, with the understanding that if ( t l , t,] should contain only one interval of order j 1, or none at all, then one or both Ij!\ shall denote the empty set. Now, since the set of I;:\, (tl , t,] - (I!:; uI;!\) intersects at most two intervals of order j 1, it contains at most two intervals of order j 2. Continuing in this fashion, we obtain mutuaIIy disjoint sets (v = 1, 2; I = 1, 2,...), where I;:! is either an interval of order j 1 or the empty set, such that

+

+

+

+

m

(tl

, t2l

=

u (Z22 + 4Yz).

(5.1.29)

1=1

+

However, when n > j I , we have the estimate (5.1.28) for the difference of the values of x ( t ) at the two endpoints of 1:) [if 1:) is nonempty]. Therefore, using (5.1.29), we get

(5.1.30)

which contradicts (5.1.27). We conclude that (5.1.25) is true. Write b = *(l - 2-"). Using (5.1.21), we calculate that, for m

> 0,

P ( S ( K m,n))

(5.1.31)

- /3(bK2"(*-=)),

ST

where p ( t ) denotes the function (2/7~)'/~ exp(-xX2/2) dx. Similarly, we get P(S(K,0, n)) = /3(bK2"(+-@)). (5.1.31 ') However, when t 2, it is easily seen that p ( t ) 6 ( 2 / 7 ~ )e-t. ~ / ~Hence, by (5.1.25), (5.1.31), and (5.1.31'), it follows that, if K > 2*+"/b, then

n = l m=O

n=l m=O 03

< (2/n)lP 1 2" exp(--bK2"(+-")).(5.1.32) n=l

294

v.

GAUSSIAN MEASURES

Clearly, the right-hand side of (5.1.32) tends to zero as K + 00. But R(I) - Ca(Il)C R ( I ) - CaK(Il),hence, letting K -+ 00 in (5.1.32), we obtain (5.1.24). If x E Ca(Il),LY > 0, then x is uniformly continuous on Il , hence x can be uniquely extended to a continuous function on [0, 11, which we also denote by x, moreover, x E Ct’[O, I]. Identifying Ca(Il) with Cou[O,11 in this manner, it is easily seen that the restriction g(a)of dl to Ca(Il)is just the restriction of b to Cp)[O, I]. Let PCm)denote the restriction of P to Ca(Il)(i.e., to Ct)[O, 11). We thus obtain a Gaussian measure space Sa) = (Ck)[O, I], 3(“), P o ) ) , 0 < a < Q. Moreover, { x ( t ) , t E [0, l]} are clearly random variables on S a ) when ; t E Il , they are Gaussian variables with mathematical expectation and covariance function given by (5.1.22) and (5.1.23), where t , s E Il . We shall now prove that, for any t E [0, 11, x ( t ) is also a Gaussian variable on and that (5.1.22) and (5.1.23) are valid for any t, s E [0, I]. Let to E [0, 11 - Il , and choose a sequence {tn}C Il such that t , -+ to . By (5.1.23), we have

hence {x(tn)}, n = 1, 2,..., is a fundamental sequence in J ~ ~ ( S ( ~ ) ) . Moreover, since the functions x in C$)[O, 13 are continuous, limn+o3x(tn) = x(to), consequently, {x(tn)} converges to x(to) in L 2 ( S a ) ) . Hence, it follows by Lemma 5.1.1 that %(to)is a Gaussian variable, furthermore, E((x(t,) - ~ ( t ~ + ) )0,~ )whence it easily follows that (5.1.22) holds for t = to . Now, let so E [0, 11, and choose {sn} C Il such that s, +so ; then we also have E((x(s,) - ~ ( s ~ ) )---t ~ )0, therefore, since

it follows that (5.1.23) holds for s = so , t = to . If &’ E b, then obviously n Clp’[O, 11 E g(a).Define

Clearly, W, is a probability measure on (Co[O,13, 23) which is concentrated on Ct’[O, 11, and is such that (5.1.22) and (5.1.23) hold. Hence, W, satisfies (5.1.21), and is therefore the required Wiener measure. ]

5.2. Equivalence and Perpendicularity

295

55.2. Equivalence and Perpendicularity of Gaussian Measures

l o The Basic Theorem I n 95.1, we considered the question of equivalence for a rather special class of Gaussian measures. I n the present section, we shall consider a more general problem. Let (Q, b)be a measurable space, and (5, , a E a} a family of measurable functions on (Q, b). Let Pl ,P, be two probability measures on (Q, b) such that {& , cy. E a} is a joint determining set for the two measure spaces (Q, b,P,) = S, , v = 1,2. Moreover, suppose that {&(-), 01 E a} is a Gaussian stochastic process with respect to both of the measure spaces S , and S, . We shall investigate the equivalence and perpendicularity of Pl and P , . Henceforth, we shall write EL(%) =

I

dPk(W),

.(W)

s2

Dk(X) = Ek((X

- Ek(X))'),

k = 1, 2.

Let 9 denote the totality of linear combinations of functions { f a , a E a} and real constant functions on Q; obviously, 9 forms a linear space. Define two nonnegative definite bilinear functionals on 9, as follows: ==

(x, r ) k = E k ( q ) ,

1, 2-

Lemma 5.2.1. If the Gaussian measures Pl and Pz are not mutually singular, then there exist positive numbers c1 and c, such that, for all X E 9 ,

(x,

4 2

and (x, 4

1

< <

CI(X, 4 1

CZ(% 4 2

(5.2.1)

Y

-

(5.2.2)

PROOF. Suppose there were no c1 > 0 such that (5.2.1) holds; then there is a sequence {x,} C 9 such that (xn , x,),

=

1,

(Xn

7

Xn)1-+

O(n

+

I

a).

<

Write E,(x,) = a , . Consider the sets A, = {w 1 x, I (x, , x,):'~} and Q - A, = {w 1 x, I > (x, , x~);'~}.If (x, , x,)~ > 0, then, by the Chebyshev inequality'

I

' Let P(w) be a probability measure, X ( w ) a random variable and E ( X ) = J X(w)dP(w); then

v. GAUSSIAN

296

MEASURES

If (xn , x,)~ = 0, then x, = 0 almost everywhere (P,), hence P,(Q - A,) = 0. Thus, we have lim P,(Q - A,) n+ m

=

0.

(5.2.3)

Now, let {n,} be any increasing sequence of positive integers. If lim,+a a& = 1, then, writing n in place of n, ,we have Ez((x, - a,J2)---t 0, whence it follows that x, - a, converges to zero in measure (P,). Let N be an integer such that (x, , x,):I4 < and I I - an2 I < for all n 2 N . Then, for w E A,, that is, I x , 1 (x,, x,):/~, we have [ x, - a, I 2 8 for all n 2 N . But P,({w I x, - a, I i}) + O as n + 00, hence P,(A,) + 0. On the other hand, if akk does not converge to 1 as k + 00, then there is a subsequence {n,'} of {n,} such that lim,+a aik. < 1. Since any linear combination of Gaussian variables is also Gaussian, it follows by (5.1.1) that (for brevity, we write n in place of nk')

a

I

<

4

>

Thus, {n,} always contains a subsequence {n,'} such that P2(Ank,) ---t 0. Therefore, we must have lim P2(A,) = 0. n+m

(5.2.4)

But then it follows from (5.2.3), (5.2.4), and Lemma 1.1.21 that PI and P, are mutually singular, contrary to hypothesis. We conclude that (5.2.1) holds for some c1 > 0. Similarly, (5.2.2) holds for some c, > 0. ] We continue to assume that P, and P, are not mutually singular. By virtue of (5.2.1) and (5.2.2), equality almost everywhere (P,) and equality almost everywhere (P2)are equivalent for functions in 9. If we identify functions in S? which coincide almost everywhere (PI),or, what is the same, coincide almost everywhere ( P J , then 9 becomes an inner product space relative to each of the inner products (x, Y ) k , k = 1, 2. Thus, 9 may be regarded as a linear subspace of each of the Hilbert spaces L2(S,) and L2(S2).Let H , and H , denote the closures of S? in L2(S,) and L2(S,), respectively; by Lemma 5.1.1, the elements of H, are also Gaussian variables on S k , k = I , 2. Suppose that [ E Hz,

5.2. Equivalence and Perpendicularity

297

and let {xn>C 9 be a sequence of functions converging to f in the norm . Since { x n ) is a Cauchy sequence relative to (x, x), it follows by (5.2.2) that it is also a Cauchy sequence relative to (x, x ) ,~and hence converges in the norm (x, x ) ~to an element of Hl , which we call T f . It is easily seen that the correspondence ( + T f defines a one-to-one linear mapping T from H2 onto Hl such that the restriction T 1 9 is Using the mapping T , we shall identify the identity mapping on 9. the point-sets of H I and H,, i.e., we shall regard Hl and H , as the Hilbert spaces associated with the respective inner products (x, Y ) and ~ (x, y ) , defined on the same underlying linear space8, which we call H. With this understanding, inequalities (5.2.1) and (5.2.2) are valid for all x E H . (x, x),

)

Lemma 5.2.2. If Pl and P , are not mutually singular, then there exists a positive number K such that, for any finite set of vectors rll )...,rln in H satisfying the conditions

(where

is the Kronecker delta) the following inequality holds, (5.2.5)

PROOF. For convenience, we write El(rjk) = mk , &(Tjk) = “ k . Now, given any set {vI ,..., vn) satisfying the conditions of the lemma, we form the measurable functiong

Actually, for each 5 E H z , one can choose a measurable function x on (a,8)such that x represents f in H , and represents T f in H , . In fact, let {x,J be a sequence of functions in 0 such that x, converges in the norm (., to a function x’ representing 6, and also converges in the norm (., .), to a function x” representing T f . Choosing subsequences, if necessary, we may assume that x, converges almost everywhere (P2) to x’ and also converges almost everywhere ( P , ) to x”. Let x ( w ) = lim x,,(w) if this limit exists, and x,(w) = 0 otherwise. Then the function x has the desired properties. Here, we choose functions representing {vl ,..., 7), in the manner indicated in footnote 8.

v.

29 8

GAUSSIAN MEASURES

on (Q, S).It is easy to calculate thatlo

(5.2.6)

D,(f)

=

C [2 ('k---+4 k

ukz

Write

(5.2.7)

lo For example, D l ( f ) may be calculated as follows: since qk and qk' are mutually independent for k # k', we have

5.2. Equivalence and Perpendicularity

299

On the other hand, by (5.2.1) and (5.2.2), we have

If we write

then it follows from (5.2.6), (5.2.7), and (5.2.10) that there exist positive numbers b, , b, and b (independent of n and q1 ,..., qn) such that

Consequently, by (5.2.8) and (5.2.9), b 1 P”(A,) < r-. b2 m

(5.2.11)

Now, as n and {q, ,..., qn} vary, if the supremum of m were +a,then, for any given positive number E , there would exist some (7, ,..., qn} such that the corresponding number m satisfies m >, (l/b%) max(b, , b,), and so, by (5.2.1 l ) , the corresponding set A, would satisfy P,(A,)

< E,

Pz(Q - A,) < E .

I t would then follow from Lemma 1.1.21 that P , and Pz are mutually singular, contrary to hypothesis. Therefore, we conclude that m has a finite upper bound K . ] Now, let Ho denote the totality of vectors of H which are orthogonal to the vector 1 relative to the inner product (., .), , that is, H , comprises the totality of functions in H having mean value zero relative to the measure Pz . Then Ho is a closed linear subspace of H , , and hence is itself a Hilbert space with respect to the norm (., *), . We form a nonnegative definite bilinear functional on Ho , as follows:

v. GAUSSIAN

300

MEASURES

which shows that B ( ( , v ) is bounded. By a well-known theorem, there exists a bounded self-adjoint operator B on H , such that

4t,7) = (a, 17h for any (, 7 E Ho .

Lemma 5.2.3, If PIand P , are not mutually singular, then B - I (where I is the identity operator in H,) is a Hilbert-Schmidt operator on H,. PROOF. According to Lemma 5.2.2, there exists a positive number K such that -'k)2

= c ( l- B ( r i k , r i r c ) f 2 = c ( ( B - ' I ) 7 k r T / k ) 2 2 k

<

k

for any finite set of vectors q1 ,..., qn E H , which is orthonormal relative to the inner product (., *)2 , and satisfies B(7k

9

7,') =

whenever k # k', that is, ( ( B - I ) q k , nk!), = 0. But then it follows immediately by Lemma 11.1.5 of Appendix I1 that B - I is a HilbertSchmidt type operator from Ho to H, , ] By virtue of the Hilbert-Schmidt property of B - I , there exists (see $11.1) a complete orthonormal system {vu , a E a} in H, [relative to the inner product consisting of eigenvectors of B - I , or, what is the same, eigenvectors of B, moreover, if we denote the corresponding eigenvalues by a, , that is, Bq, = uuqa,then (a,

a),]

(5.2.12) I n particular, there are at most countably many indices a , say a such that urn# 1. Furthermore, by Lemma 5.2.2,

=

1, 2, ..., (5.2.13)

Lemma 5.2.4. If P , and P2 are not mutually singular, then they are equivalent. PROOF. Since 2 is a joint determining set for S, and S, , and since each function in 2 can be expressed as a sum of a function in H , and a constant function, it follows that the totality of functions in H, also forms a joint determining set for S , and S, . Moreover, since {qm , a E a} is a complete orthonormal set in H , , the totality of linear combinations

5.2. Equivalence and Perpendicularity

30 1

of the q, is dense in H , relative to the inner product (-, - ) 2 , and hence [by (5.2.2)] also dense in Ho relative to the inner product (., .),. Using this fact, it is not difficult to deduce that {q, , a E is a joint determining set for S, and S,. Let SZ, = R,, where R, is the real line for every a E rU, and let 23, be the 0-algebra generated by the totality of Bore1 cylinders in SZ,. Now, from the discussion at the beginning of $5.1, 20, we see that there exists a probability measure Nk on (SZ, , 23,) such that (a,, 8,Nk) is a sample measure space for the Gaussian process {q,(*), a E rU} on (a, , 23, , N J , k = 1, 2. Let p , denote the ath coordinate of the point p of SZ, ; then the stochastic process {p,(p), a E N} on (Q, , 23,, Nk) has the same finitedimensional distributions as the Gaussian process {qa(w),01 E rU} on s,. Consequently, the variables { p , , a t 21) are also mutually independent and Gaussian, moreover, their respective mathematical expectations and variances with respect to N l , N2 are El(v,), 0; a,, 1. Using conditions (5.2.12) and (5.2.13), it follows by Lemma 5.1.2 that N , and N , are equivalent. Hence, by Lemma 1.3.7, we deduce that P I and P2 are equivalent. ] We are now ready to state and prove the basic theorem on the equivalence of Gaussian measures.

Theorem 5.2.5. Let S , = (SZ, 8,P,), Y = 1, 2, be probability measure spaces, and let (5, , a E rU} be a joint determining set for S , and S, , which is also a Gaussian process with respect to both S, and S, . Then PI and P2are either equivalent or mutually singular. Furthermore, a necessary and sufficient condition for the equivalence of P, and P2is as follows. Let 9 be the linear space generated by the functions {t, ,01 E 2I) and the real-valued constant functions. T h e n two functions in 9 are equal almost everywhere ( P I )if and only if they are equal almost everywhere (P& and the identity correspondence between the two types of equivalence classes in 9 is induced by an equivalence operator12 T between the closure H , of the P,-equivalence classes in L2(S,)and the closure H 2 of the P,-equivalence classes in LZ(Sz).l3 PROOF. We have already proved that P, and P, are either equivalent or mutually singular (Lemma 5.2.4). Thus, it remains to prove that the above condition is necessary and sufficient for equivalence. If PI and P, are equivalent, then it follows from the remarks preceding l1 Or, more precisely, a set of functions representing the elements v,, chosen as indicated in footnote 8. la See Appendix I1 for the definition and properties of equivalence operators. Is Translator's note: In the original, the author states that the spaces H I and H z coincide, and calls T the identity mapping from Hl to H 2 ; this is not accurate.

v. GAUSSIAN

302

MEASURES

Lemma 5.2.2 that T is a bounded linear operator from Hl onto H , such that T-l exists and is bounded. Let Q denote the projection operator from H , onto H, . Then I - Q is the projection operator from H, onto the one-dimensional subspace {A I -a < A < co}. Also, let Q1 denote the projection operator from Hl onto the one-dimensional subspace {A I --oo < A < a}; obviously, Q1(C)is just the constant function E1(t). Now, we assert that QT*TQ = QBQ -tQT*QiTQ.

I n fact, take any

(5.2.14)

5, q E H, , and write t1 = Qt, q1 = Qv E H,, ; then

+

((QBQ QT*QlTQ)5, 7 ) 2

=

((B

=

+ T*QlT)51 V& 71) + (QlhQ ~ d l 9

9

= (51 9 7111 =

(T*T51

9

71112

(QT*TQ5,712 3

hence (5.2.14) holds. Using (5.2.14), it is easy to calculate that T*T - Z

=

( B -1)Q + A ,

(5.2.15)

where A

=

(T*T -I)(Z-Q)

+ ( I - Q ) ( T * T -Z)Q +QT*QlTQ.

Since I - Q and Q1 are operators of rank one, A is also of rank one, moreover, by Lemma 5.2.3, (B - I)Q is Hilbert-Schmidt. Hence, the left-hand side of (5.2.15) is a Hilbert-Schmidt operator, that is, T is an equivalence operator. Conversely, if the condition stated in the Theorem is satisfied, we form the space H,, and the operator B just as before. By (5.2.14), we have (B - I)Q = Q( T*T - I ) Q - QT*QITQ, whence it follows that B - I is a Hilbert-Schmidt type operator on H , . Again, we choose the random variables {.ra , a E rU} in the manner indicated following Lemma 5.2.3, and (5.2.12) holds as before. Since T is bounded and Q1 is of rank one, it is clear that

C El(%)' = oi

II QiT~oi1 ,"


< ~f3,

oi

that is, (5.2.13) still holds. Now, proceeding as in the proof of Lemma 5.2.4, we deduce that Pland P, are equivalent. ] Remark. Examining the proof of Theorem 5.2.5, it is easily seen that, if Ek(trX) = 0, k = 1,2, a E 'L[, then one may define 9 as the totality of linear combinations of the functions a E a>,without

5.2. Equivalence and Perpendicularity

303

introducing the constant functions, and with respect to this 2, the condition of Theorem 5.2.5 is still necessary and sufficient for the equivalence of PI and P 2 . 20 Some Concrete Examples We shall now give an example illustrating the application of Theorem 5.2.5 to certain Gaussian processes having the same mathematical expectations but different correlation functions.

Example 5.2.1. Consider the class of Gaussian processes whose mathematical expectation is identically zero, and whose correlation b, has the properties (i)-(vi) listed below. function r(s, t), a ,< s, t r(s, t ) is a continuous function of the variables s, t E [a, b ] . (i) For every fixed t E [a, b], Y ( S , t ) is an absolutely continuous (ii) s b, moreover, function of s, a

<

< <

(iii)

For almost every (in Lebesgue measure) fixed s E [a, b], the function ar(s, t)/as is absolutely continuous on the intervals [a, s) and (s, b] [whence it follows that a2r(s,t ) / a s at exists almost everywhere and is measurable on the square a ,< s, t b].

<

By virtue of condition (iii), the limits

exist and are finite for almost all s E [a, b], moreover, it is easily seen that q-(s) and q+(s) are measurable functions. We further assume the following property. (v) There exist positive numbers c l , c2 such that the function D,(s) = q+(s) - q-(s) satisfies the relation c1


c2

(5.2.17)

almost everywhere. We call D7(s) the discriminato~~~ corresponding to the correlation l4 Translator’s note: The Chinese term used by the author is the same as that expressing “characteristic function,” which seems rather inappropriate here.

v. GAUSSIAN

304

MEASURES

function r . And finally, we assume that a2r(s, t ) / a s at is a positive definite integral kernel, that is, the following property holds. (vi) For everyfEL2[a, b],

We say that a Gaussian process satisfying the conditions listed above is of type S[a, b ] , and its correlation function r(s, t ) is said to be of type 4% bl. Suppose that u(s), o(s) are real-valued absolutely continuous functions on [a,b] such that v(s) > 0 (s E [a, b ] ) ,

i:

~ ’ ( 3 )ds ~

1:

< 00,

Z(’(S)~

ds

< 00,

and that there exist positive numbers cl, c2 such that c1

< v(s) u’(s) - u(s) w‘(s) <

c2

for almost all s. Then, it is easily verified that the function

is of type16 s[a,b], and that D,(s)

= o(s) u’(s) - u(s) o’(s).

In particular, for u(s) = cs/2, o(s) r(s, t )

= 1, we have C

= - min(s,

2

t),

which is just the correlation function for the Wiener process (see $5.1).

Theorem 5.2.6. Let { ( ( t , w ) , t E [a, b ] } be a joint determining set of functions on the probability measure spaces Sk= (Q, b,Pk), k = 1,2, forming a Gaussian process of type S[a, b] relative to both S, and S, , Translator’s note: The assumptions made here are, in fact, not sufficient to ensure that condition (vi) above is satisfied. It would suffice, however, to replace the requirement that w(s) > 0 by, for example, the following condition:

5.2. Equivalence and Perpendicularity

305

with the respective correlation functions rk(s, t), and P , are equivalent if and only if

K

=

1, 2. Then PI

(i) the discriminators D J t ) , D,,(t) are equal almost everywhere on [a, bl, and (ii) rl(a, a ) and r2(a,a) are either both zero or both different from zero. PROOF. In order to use Theorem 5.2.5, we shall explicitly construct the closure Hkof 9 in L2(Sk)(K = 1, 2). 1. Let V [ a ,b] denote the totality of real-valued functions f of bounded variation on [a, b] which are also continuous on the left at every point of (a, b) and satisfyf(b) = 0. For every f E V [ a ,b] and every sequence of partitions a = t ( n ) < t(n1 0

1

< ... <’ : 2

=

6

of the interval [a, b], we form a sequence of functions {fn} in 9, as follows: n

t n ( 4 = 1 I($%w ) ( f ( t P ) )-f(t?J)* k=l

Then, one easily calculates that

as n, m -+00. Consequently, {tn}is a fundamental sequence relative to the inner product (., and hence converges to a vector in H , , which we denote by (y’. Using (5.2.18), we easily calculate that, iff, g E V[a,b ] , then - ) y ,

(5.2.19) Let K , = {(I”) I f~ V[a,b ] } ; then K , is dense in Hu . I n fact, for any to E (a, b), we form the function

v.

306

GAUSSIAN MEASURES

by the continuity of r(s, t), it is obvious that ((u, -) = limfo+a((to ; and therefore f ( u ; .) belongs to the closure K,O of K,, in H , . Similarly, ((b; .) E K,O. Since K,, is a linear space, it follows that 9 C KyO, and so a),

K,O = H, .

(5.2.20)

By condition (ii), rv(s,t ) is absolutely continuous with respect to s, therefore, using integration by parts, we find that (5.2.21 )

According to property (iii). in Example 5.2.1, for almost all fixed ,

s E [a, 61, the partial derivative ar,(s, t)/& is an absolutely continuous

function of t on [a, s), (s, b]. Consequently, when t < s, (5.2.22)

Let qy), q?” denote the functions (5.2.16) corresponding to r, . Then, letting t + s - 0 in (5.2.22), we get (5.2.23)

Similarly, when t

> s, using (5.2.23), we obtain

Substituting (5.2.22) and (5.2.24) into (5.2.21), and applying Fubini’s theorem to interchange the order of integration, we have

s” a

Y,,(s,

t ) d f ( s ) = +(a) -

Y”(u,t ) -

1; w

St

a

(r

f ( s ) ds) dv

a

f ( s ) ds -

1‘

DrY(s)f(s)ds

a

5.2. Equivalence and Perpendicularity

307

Finally, using integration by parts, we arrive at the formula

(5.2.25)

2. We now construct function spaces Lk’[a, b], v = 1, 2, in the following manner. The functions in Lg’[a, b] are the same as the functions comprising L,[a, b], however, two such functions f,g are to be identified if and only iff(t) = g ( t ) almost everywhere on ( a , b ] , and also

The linear operations in Lk)[a,b] are defined in the usual way, while the inner product is defined by

Using condition (v) of Example 5.2.1, one may verify that Lk’[a,b] is indeed a Hilbert space relative to the inner product [f,g]” . Next, we form a linear operator A , from Lp’[a,b] into itself, as follows. GivenfEL:’[a, b], let A,f be the function defined by

for t

E (a, 61,

and, in the case rv(a,a ) # 0, let

(A,f)(a)=

s”

y f c s ) ds + f ( a ) ( r , ( a , a ) - 1).

U

We proceed to prove that A, is a Hilbert-Schmidt

operator. ( I ) If

v. GAUSSIAN

308

r,(u, a) = 0, then E(a, w ) = 0, hence r(a, t ) = 0, a case, Lt’[a,b] is just L,[a, b], and

MEASURES

< t < b.

I n this

(5.2.26)

By virtue of conditions (iv) and (v) of Example 5.2.1, the correspondence J: [a2rY(s,t ) / ha t ] f ( s )ds is a Hilbert-Schmidt operator, and f -+ l/Dr, is a bounded linear operator, hence A , , being the product of these two operators, is Hilbert-Schmidt. (2) If rv(a,a) # 0, then A, is the sum of a Hilbert-Schmidt operator of the form (5.2.26) and a bounded linear operator of finite rank, hence A, is also Hilbert-Schmidt. We observe that, iff, g €L‘,”[a,b], the right-hand side of (5.2.25) is just [(I A,)f, g ] , . In particular, when f,g E V[a,b], it follows by (5.2.19) that (5.2.27) csy, 5;’)” = “1 AJf,El, .

f3

+

+

Now, for any f ~ L $ ’ [ ab], , since V[a,b] is dense in L t)[a,b ] , one can choose a sequence {f,} C V[a,b] such that {f,} converges to f in L‘,“[a,b ] . Using (5.2.27), we see at once that {ti:}is a fundamental sequence in K , . Therefore, converges in H , to a vector which we call $’, moreover, (5.2.27) then holds for all f,g ~ L g ’ [ ab,] . Since 8:’) 0, it follows from (5.2.27) that B, = I A, is a nonnegative self-adjoint operator on Lt’[a,b ] . Let us examine the null space of B, . Again, we distinguish two cases. (I) Suppose that r,(u, a) = 0. If B,f = 0, then, by (5.2.26),we have

(fr),

+

>

+

0 = (BYfTf) = (fJ) ( A Y f ? f ) f ( s ) f ( t )ds dt.

(5.2.28)

However, by condition (vi), the double integral in (5.2.28) is nonnegative, whence it follows by condition (v) that f = 0. This shows that B, has an inverse operator. Thus, -1 is not an eigenvalue of A , , therefore, since A, is completely continuous, - 1 is a regular value of A, . Hence, B;l is defined as a bounded operator on the entire space LP’[a,b ] . In other words, B, is an equivalence operator from Lt’[a,b] onto itself. (2) Suppose that r,(u, a) > 0. In this case, we assert that, if q, belong to the null space of B, and ~ ( a=) +(a), then q = +. In fact, under these assumptions, the vector f = 9 - also belongs to the null space of B, andf(a) = 0. Consequently, it can be seen from the expression for A , thatf also satisfies (5.2.28), and hencef = 0, that is, cp =

+

+

+.

5.2. Equivalence and Perpendicularity

309

It follows that the null space E, of A , is one-dimensional,le Using the fact that A, is a completely continuous self-adjoint operator, it is easy to prove that the restriction of B, to the orthogonal complement F, of E, is a homeomorphism of F , onto itself. Hence, B, I F, is an equivalence operator from F, onto F, . I n order to combine cases (1) and (2) throughout the remainder of this proof, we agree that F, is to denote Lk’[u, b] if r,(u, u ) = 0. Now, by virtue of (5.2.27),we know that the correspondence

maps F, homeomorphically onto U,F, . However, iff E E,, , then B,f = 0, that is, = 0; consequently, U,F, = U J g ) [ u ,b] 3 K , . Therefore, since F, is complete, it follows from (5.2.20)that U,F, = H , . Moreover, by (5.2.27),we have U,*U,

=

B, ,

hence U,, is an equivalence operator from F, onto H , . (3) We can now demonstrate the necessity of conditions (i) and (ii). Assume that Pl and P2 are equivalent. Then T ‘ = Ui’TU2 is the identity operator1’: T‘f =f , f E F2 . Consequently, either (1) F, = L t ’ [ u ,b] for both v = 1, 2, whencela rl(u, u ) = r2(u,u ) = 0, or (2) F, # L ~ ’ [ ub], for both v = 1, 2, whence rl(u, u) f 0, r2(u,u ) # 0. This proves the necessity of condition (ii). Furthermore, since U , , T and U , are all equivalence operators, it follows by Theorem 11.1.9 of Appendix I1 that T‘ is also an equivalence operator. I n case (1) [i.e., rl(u, a ) = r2(u,u ) = 01 we have

hence, ((T’)*T‘ - I ) f =

D,, - DT,

f.

DT2

l 6 Translator’s note: In actual fact, it has only been proved that E, is at most one dimensional. If E, = (0), then it follows just as in case ( I ) that B , is an equivalence operator. However, in this case, F , = Lg)[a,b], and this seems to invalidate the already questionable reasoning used in the proof of sufficiency (see below). I’ Translator’s note: The reasoning behind this assertion is, perhaps, based upon the inaccurate statement of Theorem 5.2.5 in the original text (see footnote 13). l a Translator’s note: See, however, footnote 16.

v. GAUSSIAN

310

MEASURES

However, it is easily shown that the above operator cannot be completely continuous unless Drl = DrBalmost everywhere. Thus, condition (i) is also necessary. A similar argument applies to case (2). (4) Conversely, assume that conditions (i) and (ii) are satisfied. Then the spaces Lil’[a,b] and Li2’[a,b ] , as well as their respective inner products, are completely identical. Hence, Fl and F, ~0incide.l~ Consider the operator

from H , to Hl . By Theorem 11.1.9, T is an equivalence operator; then T [ = [, Thus, T satisfies the conditions of moreover, if 4 E 9, Theorem 5.2.5. Hence, Pl and P, are equivalent. ]

< <

Corollary 5.2.7. Let { [ ( t ,w ) , a t b} be a Gaussian process of type S[a, b] on the probability space (52, 5, P);let ~(s,t ) be the correlation function of this process, and D,(t) ( a t b) its discriminator. Assume also that { [ ( t , o),a t b} is a determining set of functions on (52, 5, P). Then, there exists a probability measure P‘ on (52, S), equivalent to P, such that {t(t,w ) , t E [a, b]} is a Gaussian process of type S [ a , b] on (52, 8,P’),with correlation function

< <

< <

= 0 if r(a, a ) = 0, and c, = I if r(a, a) > 0. PROOF. Let (T, 23, p) be the sample probability space associated with the given Gaussian process (see $1.3, 30) and denote the corresponding mapping of 52 into r by v : w -+ = { K t , w ) , t E [a, b]}. I t is

where c,

easily verified that p is a positive definite function, that p E $[a, b ] , and that D,(q) = D,(q). In view of the remarks following Definition 5.1.1, there exists a probability measure v on (r,23) relative to which { t t ,t E [a, b ] } is a Gaussian process with identically zero mean and correlation function p(s, t ) . Since the process {ti,t E [a, b ] } has zero means and correlation function r(s, t ) relative to the measure p, and since D,,(q) = DT(9),p(a, a) = c, , it follows by Theorem 5.2.6 that p and v are equivalent. Now, for any set F E 5, there exists a set B in 8 such that P ( F A v - l ( B ) ) = 0. Define P’(F) = v(B). Tvanslatov’snote: The justification for this statement is not clear. Certainly L,f)[u,b] and Lia)[a,b] are identical, but B1and Ba need not be identical.

5.3. Gaussian Measures on Linear Spaces

311

I t is easily verified that P’(F) is independent of the choice of B , and that P’is a probability measure having the required properties. ] Corollary 5.2.7 provides a canonical formz0 (5.2.29) for Gaussian processes of type S[a, b]. As a special case, if (0, 23) is the measurable space (C,[O, I], 23)of Example 5.1.2, then any of these canonical Gaussian processes is obtainable from the Wiener process {x(s, w ) , s E [0, 11) (taking, say, c = 2) by a transformation, that is,

and

5 is a Gaussian variable, independent of all the x(s, w ) , such that

E ( t ) = 0, E(S2) = c,

.

55.3. Gaussian Measures on Linear Spaces

l o Standard Gaussian Measure Spaces When studying problems of harmonic analysis related to Gaussian measures, frequent use is made of the following type of measure space.

Definition 5.3.1. Let D be a real linear space, H a linear subspace of Q, and (., -) an inner product defined on H. Let 23 be a a-algebra formed by certain subsets of Q, and let N be a probability measure on (Q, S ) such that (Q, 23, N ) is quasi-invariant with respect to H . Moreover, suppose that, for every x E H, there corresponds an H-quasilinear functional x(w) on (Q, 23,N ) such that the linear functional on H induced by x(w) is just the inner product (x, y ) , y E H . I n other words, the identity x(tw) = t x ( w ) holds for all w E Q, -a < t < 00, and, for any given y E H, the relation x(tw y) = tx(w) ( y , x) holds for all t, -co < t < co, provided that w lies in a certain set E, E 23, N(E,) = 1. We also suppose that the correspondence between x and x(w) is linear, that is, for any x,y E H , 01, E R and w E Q, we have (ax PY)(W) = 4 w ) PY(W). Furthermore, suppose that the family of functions { x ( w ) ,x E H )

+

+

+

+

Translator’s note: That is to say, canonical only with reference to a fixed indexed set of variables { [ ( t , w), t E [a, b ] } .

v. GAUSSIAN

312

MEASURES

constitutes both a determining set and a Gaussian process on (52, 23, N ) , the mathematical expectation of this process being identically zero and its correlation function being

We then say that (52, 23, N ) is a standard Gaussian measure space (with parameter c ) relative to H , and that {x(-), x E H } is a standard Gaussian process on (SZ, 23, N ) . T o indicate the parameter, we shall sometimes write N , instead of N .

Theorem 5.3.1. Let H , J2 be real separable Hilbert spaces such that H forms a dense linear subspace of 52 and the imbedding of H in 52 is an operator of Hilbert-Schmidt type. Let 23 be the o-algebra of weak Bore1 sets in 52. Then, there exists a standard Gaussian measure space (52, 23, N ) relative to H . PROOF. Denote the inner products in H and 52 by (., -) and *)1, respectively. Since the imbedding operator T : H +52 is of HilbertSchmidt type, the operator B = T*T is nuclear. Obviously, B has the following property: for any x, y E H , (a,

(x, Y ) l = (Bx,Y ) *

Let { e n , n = 1, 2, ...} be a complete orthonormal system in H , consisting of eigenvectors of B, and let A, denote the eigenvalue corresponding m to en . Since B is a nuclear operator, A, < 00; also,

xmCl

(em

)

enh = Amam,

t

and ( e m , > 0, hence A, > 0, m = 1,2 ,... . Let 1 = Xn=l R , , where R, is the real line for every n, and let l2 denote the totality of elements x = {x,} in 1 which satisfy the condition C xn2 < co. Then, Zz is a Hilbert space with respect to the inner product m

V=I

(wherey = {yv}).Also, let 12({A,}) denote the totality of elements x = {x,} in 1 which satisfy the condition C A,xn2 < co (see Example 4.2.2). Then, l2({L}) is a Hilbert space with respect to the inner product m

(x, Y L =

C Amxmym m=l

*

5.3. Gaussian Measures on Linear Spaces

313

Define a mapping U from 52 to Z2({Am}), as follows:

Since the points of Q are characterized by the condition

it follows that U is a unitary operator from SZ to P({Am}), moreover, U maps H unitarily onto P. Consequently, we may as well assume that 52 = E2({L}) and H = Z2. Let 23, be the smallest o-algebra containing all Bore1 cylinders in 1. According to $5.1, 20, there exists a Gaussian measure N on (I, 23,) such that the coordinates {x,} (regarded as functions of x) are mutually independent Gaussian variables with mathematical expectation zero and variance (c > 0). Given any y E I, consider the measure N , on ( I , 23,) defined by N,(E)

=

N(E - y),

E E 23,

.

(5.3.1)

Then, the functions {x, , v = 1,2,...} are also mutually independent Gaussian variables on ( I , 23, , N,), with variance &. However, the mathematical expectation of x, with respect to the measure N , is given by

J x, dNY(JE)= J x, dN(x - Y)= J + rJ d w 4 (X”

=Y

Y *

By Lemma 5.1.2, the measures N , and N are equivalent if and only if E Z2. Thus, the translations under which ( I , 23, , N ) is quasi-invariant are precisely those corresponding to the elements of Z2. Next, we assert that the measure N is concentrated on 12({Am}). I n fact, using the Levi Lemma, we have

Cyy2 < co,that is, y

hence N ( { x I (x, x ) ~= co}) = 0, which means that the measure N is will concentrated on 12({Am}). The restrictions of Bw and N to Z2({L}) also be denoted by 23, and N , respectively. Thus, we obtain a Gaussian measure space (12({A7n}), 23, , N ) which is quasi-invariant relative to 12.

v. GAUSSIAN

314

MEASURES

For each y E 12, we form a sequence of measurable functions {y(")(x)) on (P({&J), b, ,N), as follows: n

Y'"'(x) =

Then, when n 2 m, n, m

--f

CYJ~ u=l

1

x E 12({ArnI)*

co,

c

c n [ y ' y x ) - y(n"'(x)]2 dN(x) = yy2 + 0.

2 v=rn+l

Hence, {y(")(x)}converges in L2(N),and therefore there exists a subsequence { y("fi)(x)) which converges almost everywhere.21 Let A, denote the set of all points x E P({h,)) such that limk+wycnr)(x) exists; then Z2({h,}) - A, is an N-null set. Now, define a measurable function y(x) on (Z2(&J), 23, , N) by letting y(x) = lim ~ ( ~ x ) ( x ) k-tm

when x E A,, and y(x) = 0 when x C A, , Then, for any x E 12, we have z ) = y("k)(x) ~ ~ ~ l y vconsequently, z,, if x E A, , then x+x~A,,and

y("k)(x

+

+

Y(X

+4

=Y(4

+ (Yl 4.

Moreover, if x E A, and t is any real number, then t x E A, , and so it follows easily that y( .) is a quasi-linear functional. By Lemma 5.1.1, the function y(x), x E P({hm}),being the limit of the sequence of Gaussian variables {y("k)(x)}, is also a Gaussian variable, moreover, if y, x E P, then,

Also,

Translator's note: According to Kakutani [l], Lemma 6, the right-hand side of (5.1.14) converges p-almost everywhere to a nonzero limit. In the present situation, this implies that ( ~ ' " ' ( x ) } actually converges N-almost everywhere, so that the selection of a subsequence is not strictly necessary.

5.3. Gaussian Measures on Linear Spaces

315

E Z2 defined by yy = a,, for n = 1, 2, ..., we see that the family of functions {y(.),y E Z2} includes all the coordinate functions (x,}, and is therefore a determining set on (Z2({&}), 23, , N). Thus, we have shown that { y ( * ) ,y E Z2] is a standard Gaussian process on (P({Am}), 23, , N). Finally, we transfer the above construction back to D and H via the unitary operator U-l, and so obtain the desired result. ] Notice that, in the case just considered, (5.1.14) yields

If we consider the elements y

D,we get

whence, returning to

*I=

exp(;y(w)-;(y,y)), 2 1

~ E H ~,

E Q .

(5.3.2)

dN(w)

Furthermore, from the proof of Theorem 5.3.1, we easily obtain the following result.

Corollary 5.3.2. In Theorem 5.3.1, the totality of points x such that the Gaussian measure N is quasi-invariant with respect to the translation w --f w x is precisely H.22

+

Corollary 5.3.3. Let H , D be real separable Hilbert spaces such that H is a linear subspace of Q and the imbedding H + Q is an operator of Hilbert-Schmidt type. Let 23 be the cr-algebra of weak Bore1 sets in D. Then, there exists a ( Gaussian23) probability measure N on (D,23) such that the translations under which (a, 23, N) is quasi-invariant are precisely those corresponding to the elements of H. PROOF. Let D, be the closure of H in D;then H is dense in 9, . Let 23, be the restriction of 23 to D, . Applying Theorem 5.3.1 and Corollary 5.3.2 to H and (12, , B,), we obtain a probability measure Nl on (0, , 23,) such that Nl is quasi-invariant under translation by elements of H (and no others). Also, let N2 be the probability measure on D @ Q, which is concentrated at the zero vector. Then, the product measure N , x N 2 has the desired properties. ] aeT~unslator’s note: This is certainly true with regard to the specific measure N constructed in the proof of Theorem 5.3.1. However, it is not immediately clear that it need be true for any measure N which satisfies the requirements of the theorem. aa Translator’s note: Presumably, the term “Gaussian measure” as used here means that there exists some Gaussian process which forms a determining set for the measure space.

v. GAUSSIAN

316

MEASURES

T h e following corollary is an immediate consequence of Corollary 5.3.3.

Corollary 5.3.4. Let G be a real separable Hilbert space, and 23 the totality of weak Borel sets in G. Let 0 be a countably Hilbert space which is imbedded as a linear subspace of G, and let 8 , denote the completion of 0 relative to the nth inner product, n = 1, 2,... . Suppose that, for some n, the imbedding 0 -+G extends to a Hilbert-Schmidt operator24from @into G. Then, there exists a finite measure p on (G, 23) which is quasi-invariant with respect to 6. Corollary 5.3.4 provides a converse to Theorem 4.2.18. Similarly, one can state a converse of Corollary 4.2.19; we omit the details. Under the hypotheses of Corollary 5.3.4, does there necessarily exist a finite measure space (G, b,p ) such that the totality of quasi-invariant translations is precisely 0 ? This question still remains unanswered. On the Equivalence of Three Conditions Theorem 5.3.5, Let 0 and G be real separable Hilbert spaces such that 6 is a linear subspace of G and the imbedding operator T : 0 G is continuous. Let 23 denote the totality of weak Borel sets in G, and 8' 2O

--f

the totality of weak Borel sets in the conjugate space 8+of 8. Then, the following three statements are equivalent. (i) T is of Hilbert-Schmidt type. (ii) There exists a finite measure on (G, 23) which is quasiinvariant with respect to 6. (iii) For any positive definite continuous function f on G, there exists a unique measure Pt on (0+, 5') such that, for any h E 8, f(h) = e i f ( h )~ P + ( E ) . (5.3.3)

1,'

PROOF. We present several ways of deducing these conditions from one another, as follows.26 (1) By Corollary 5.3.3, (i) implies (ii). (2) Statement (iii) may be deduced from (i) by using Theoand (., .) denote the inner products in G rem 4.3.12. In fact, let (-, and 6 ,respectively; then, when y , ) I E 6,

(% 411 = (T*TP 9

4)-

(5.3.4)

Translator's note: Apparently, one must also assume that this extension is injective. Translator's note: Obviously, parts (2) and (3) below could be omitted from the proof; presumably, their inclusion was merely intended to shed additional light on the situation. a4 *6

5.3. Gaussian Measures on Linear Spaces

317

By (i) and Lemma 11.1.I, the positive operator T*T is nuclear. Now 6, being a Hilbert space, is, in particular, a countably Hilbert space. Suppose that the positive definite function f is continuous relative to the topology of G, that is, continuous relative to the inner product (5.3.4). Then, given any E > 0, there exists a S > 0 such that If(y) - f ( O ) I < E whenever q E {v I (( T*T/6)v, y ) < I}. Since T*T/S is also a positive nuclear operator, this shows that f l 6 is continuous relative to the 6-topology (see 94.3). Hence, by Theorem 4.3.12, there exists a unique finite measure Pt on (6+, 5,)such that (5.3.3) holds. (3) By Corollary 4.2.20, (ii) implies (i). (4) By Theorem 4.3.11, (ii) implies (iii), ( 5 ) We now proceed to deduce (i) from (iii). Assume that (iii) holds, and consider the continuous function f ( h ) = exp(- &(A, h)l),

h E G.

It is easily verified that this function is positive definite (in fact, it is the characteristic function of a certain Gaussian measure space). Hence, by condition (iii), there exists a unique measure Pt on (6+, 8') such that f satisfies (5.3.3). Next, replace h in (5.3.3) by C:=l t,h, , h, ,..., h, E 6,and define an n-dimensional Bore1 measure P by the equation P(E) = W

t I (t(hl),...> 6(h,)) E El).

We then have

which shows that P is the probability distribution of a Gaussian variable. By (5.1.3),

j,, mJ2dP+(t)= j

Xk2

dP(x) = (h, ,h,)l ,

=

1, 2,...,n,

and if the vectors {hk} are orthogonal relative to the inner product then, by (5.1.6), we get

J,,

(6(h!J2- (A,

!

hR)1)(t(hd2- (A, > Wl) W k,Z= 1,2,...)71,

5 ) = 2(h, > h,)12 &J

where Sk,lis the Kronecker delta. If the operator T is not of Hilbert-Schmidt

(e,

-),

,

9

(5.3.5) type, then, by

v.

318

GAUSSIAN MEASURES

Lemma 11.1.1, T*T is not nuclear. Hence, by Lemma 11.1.6, given any positive integer K, there exist vectors h, ,..., h, E 8, satisfying and (h, , h),= 6,,,(hk h,)l such that (hk 9 h,) = a,,

c

Y

9

hk)l

> K.

n

=

(hk

9

(5.3.6)

k=l

Let Q(t) =

xtzl~ ( I z , ) ~Then, . using (5.3.5), we calculate that

moreover, (h, , hk), < (1 T (I2, hence, by the Chebyshev inequality,

(5.3.7) However, since the (hk} are orthonormal relative to inequality implies

(a,

(5, E ) 2 Q(0.

-), the Bessel (5.3.8)

From (5.3.7) and (5.3.8), we get pt({5NE, 5) 2 - 2 II TI1 d/h))

2 f'+({5 I Q(4) 2

- 2 )I T I1

3 pt({5I I Q(5) - I

4x1)

< 2 II TII a}3)4.

(5.3.9)

On the other hand, m

n {5 I (5, 5 ) 3 K - 2 II T II dK}

= empty set,

K=1

hence lim Pt({5 I

K-tm

(6, 6) 2 K

- 2 11

T 11 dE}) = 0.

(5.3.10)

But by (5.3.6), X > K , therefore (5.3.9) and (5.3.10) are contradictory. Hence (i) holds. ] We shall now establish the converse of Corollary 4.3.14. Corollary 5.3.6.

Let 8 be a real separable countably Hilbert space,

5.3. Gaussian Measures on Linear Spaces

319

5'

the totality of weak Borel sets in (tit. Suppose that, for any positive definite continuous function f on 6 [with f ( 0 ) = I] there exists a probability measure Pt on ((tit, 5') such that (5.3.3) holds for all h E 6. Then (5 is a nuclear space. PROOF.Let (11 * \Im} be the given sequence of norms on 6, and let (tim be the completion of (5 relative to 11 * Ilm. Let m be an arbitrary natural number, and consider the function

Now, suppose that, for every n > m, the imbedding Tmn: (ti, -+ (5, is not of Hilbert-Schmidt type. Then, using part (5) of the proof of Theorem 5.3.5 (replacing (ti by 6, and G by we obtain

I

p+({t II t 11-n =

that is, for all n

00))

=

1~pt({tI II t IIL>, K - 2 II Tmn II dKi>2

> m, Pt(6+-(ti,+) 2

*.

n= :+ ,,

But this contradicts the fact that (St - Ont) is the empty set. We conclude that, for some n > m, the operator Tmnis of HilbertSchmidt type. Therefore, 6 is a nuclear space. ] Next, for number sequence spaces, we establish a partial result analogous to Theorem 5.3.5.

Example 5.3.1. Under the hypotheses of Example 4.2.2, if we 2, then the following two statements are also assume that 1 q equivalent. (i) C a n < 00. (ii) There exists a finite measure on (P((an}),123) which is quasiinvariant with respect to P. Furthermore, let Zqt denote the conjugate space of P, and 5' the totality of weak Borel sets in Zqt. Then, if either (i) or (ii) holds, we have the following conclusion. (iii) For each positive definite continuous function f on P((a,)), there is a unique finite measure Pt on (Int, St) such that, when h E P,

< <

Here, we bear in mind the density of 6 in 6,,,and the remark following Lemma 11.1.6.

v. GAUSSIAN

320

MEASURES

PROOF. (1) It was shown in Example 4.2.2 that (ii) implies (i). (2) Assume that (i) holds. In the construction of Example 4.1.2,

choose

and form the corresponding Gaussian measure p. Clearly, Q = (I, b, ,p) is quasi-invariant with respect to 12. However, since q 2, we have lq C P, hence Q is quasi-invariant with respect to lq. Write 11 f [I = (C a , I f m I P ) l / P , t = (fl ,..., f , ,...}. Using the Levi Lemma, one easily verifies that

<

Consequently, p({f

I 11 f

I( = 00))

= 0,

PL(W4)) = P u t

that is,

I I1 5 II < 4)= 1.

Thus, restricting p to (P({a,}),b),we obtain a finite measure which is quasi-invariant with respect to lor, that is, (ii) holds. (3) Conclusion (iii) may be deduced from (ii) by applying Theorem 4.3.1I. ] It is still not known under what circumstances conditions (ii) and (iii) of Example 5.3.1 are equivalent.

Problem. Can one establish a theorem analogous to Theorem 5.3.5 when 6 and G belong to some suitable class of separable Banach spaces ? 30 More on Standard Gaussian Measure Spaces We shall now give a procedure for constructing standard Gaussian processes from Gaussian measures on linear topological spaces.

Theorem 5.3.7. Let SZ be a real linear topological space such that the conjugate space Qt distinguishes points of Q. Let 8 be the a-algebra of all weak Bore1 sets in Q, and let S = (Q, 9, N ) be a regular probability f E Qt} is a Gaussian process on S with measure space such that {f(.), mathematical expectation zero. Write jf(wfg(4

W w )=

;

(ft

g),

5.3. Gaussian Measures on Linear Spaces

321

and assume that (f,f) > 0 whenever f # 0. Let H denote the set of all vectors x in 52 such that

Then H is a linear subspace of Q, and constitutes a Hilbert space relative to the above norm2' (1 . (I. Moreover, S is quasi-invariant and quasicontinuous with respect to the translations H. Furthermore, if H is dense in 52, then there exists a system of vectors (f,}C which is orthonormal relative to the inner product (., .) and has the following property: when h E H , there is a function X(w) such that (5.3.1 1 ) in L 2 ( S ) ,and S forms a standard Gaussian measure space relative to the process {A(.), h E H}. PROOF. Without loss of generality, we may suppose that c = 2. Let h E 52, and define a measure NA on (Q, '$3) as follows: N,(A) = N ( A - A),

A E B.

We shall apply Theorem 5.2.5 to the Gaussian measures PI= N , P2 = N,, . Accordingly, we consider the totality 9of linear combinations of elements of Qt and constant functions. For any constants a, b and any f,g E Qt,

(a (a

+f,b .tgh

=

ab

+ (f,g),

+ + g)Z / +f ( w ) ) ( b+ dw)) fi

=

(a

=

[(a +f(4)+f(w)"

=

( a +f(A))(b

dNA(w)

+ g(4) + g(w)l

W

W

)

+ g(4) + (fig).

If N Aand N are equivalent, then the two inner products (-, and (., .)z are at least topologically equivalent. Hence, there is a constant K such that f ( V G ( f , f ) z G K(f,f)l = K ( f , f ) (5.3.1 2) for all f E Qt, that is, I( h 11 27

< K. Thus, X E H.

It is not difficult to see that the norm

/I . 11 is induced by an inner product.

v. GAUSSIAN

322

MEASURES

Choose any complete orthonormal system2* {f,} in Q+. Then the functions {f,(w)} are mutually independent Gaussian variables on S , with variance 1 and mathematical expectation zero, hence the {f, (w)} are also mutually independent Gaussian variables on (a,b,N,), with variance 1 and respective mathematical expectations {f,(h)}. It follows by (5.1.11) that the Kakutani inner product of N and NA is (5.3.13) However, when h E H , it is easily seen from the Parseval equality that (5.3.14) and so p ( N , N,) > 0, that is, N Aand N are equivalent. This, combined with the result of the preceding paragraph, shows that the totality of quasi-invariant points of S is precisely H . It is obvious that H is a linear subspace of Q. Furthermore, by (1.4.4), (4.1.1), and (5.3.13), we have

IM,(V

=

2(1 - exp(-& II ii”)

for any h E H , which shows that S is quasi-continuous with respect to the translations H . By Theorem 4.1.3, H is a complete linear pseudometric space relative to the pseudometric defined by the convex functional

But it is easily seen that I?,(.) and 11 * 11 induce the same topology on H , therefore, H is complete relative to 11 . 11. According to Theorem 4.1.6, the topology on H induced by 11 11 [i.e., induced by I?,(-)] is stronger than the relative topology induced by the topology of SZ. Hence, for every f E Q+, the restriction f‘ o f f to H is a continuous linear functional on H , that is, the correspondence

f -+f‘ is a mapping of Qt into Ht. If H is dense in SZ, this mapping is injective. 28 Translator’s note: Let denote the completion of Qt with respect to the inner product (., .). If SZt is separable, then one can use the Gram-Schmidt process to construct an orthonormal system in Qt which is complete relative to 8t. However, if Qt is not separable, it is not clear whether such a system can be found.

5.3. Gaussian Measures on Linear Spaces

323

Thus, identifying f with f ’, we obtain an imbedding2B: Qt C Ht.

Now, Qt C ON2,and clearly30 Q N 2 is complete relative to the inner product (., .). Hence, the closure of Q+ in Q N 2 is also complete relative to (*, .). But {fn} is a complete orthonormal system in Q+relative to the inner product (., *), hence, for any A E H , since C Ifn(A)I2 < 00 by (5.3.14), it follows by the Riesz-Fischer theorem that there is an element of Q N 2 , which we call A(.), such that (5.3.11) holds in L2(S). Since A(.) E Q N , therefore, for any y E H , the relation31 A(u

+ Y ) = A(), + C f n ( A ) f n ( Y )

=

4),

+ ( Y ,4

(5.3.15)

holds for almost all w in Q. Also, { A ( * ) 1 h E H } is a determining set on S, since it contains32 Qt. I t is now clear that {A(.) 1 A E H } is a standard Gaussian process on S. ]

Example 5.3.2. Consider the case of the Wiener process. We use the notation of Example 5.1.2. With respect to the ordinary linear operations and the norm

the set of functions Co[O, 11 forms a normed linear space, which we denote by Q. Let Vo[O,I] be the totality of real-valued functions f on [0, 11 which are of bounded variation, continuous on the right, and such that f(1) = 0. The set Vo[O,11 forms a normed linear space relative to the ordinary linear operations and the norm 1

llfll =

v (f) 0

a e Translator’s note: Under the given assumptions, this imbedding is continuous relative to the norms [(., .)Illa and 11 . )I on Jzt and H t ( = H), respectively; however, I see no proof that it is isometric. It is not clear how this imbedding is to be utilized in the proof, and the density of H in is used only in obtaining this imbedding. Likewise, it is not clear how the assumption that Jzt distinguishes points of Jz enters into the proof. Translator’snote: Presumably, the author has in mind an application of Lemma 4.2.12. Translator’s note: To establish (5.3.15), it seems that one must produce an argument, perhaps similar to that used in the proof of Lemma 4.2.7, showing that (5.3.11) can be converges to (A, w ) for assumed to hold for all w E H [using the fact that ,Y,fn(A)fn(w) allUEII]. 33 Translator’s note: This last assertion can be proved if one makes the assumption that every linear functional 8 on Ot,which is continuous relative to the norm [(., .)I1’*, is of the form [(f) = f ( x ) for some Y E H (this condition is satisfied in Example 5.3.2). It is certainly not obvious that this condition can be deduced from the original hypotheses.

v. GAUSSIAN

324 (i.e., the total variation off on [0, 11). For each a linear functional F, on Co[O,I] as follows:

s

01

E

MEASURES

Vo[O,I], we define

1

F,(x)

=

(5.3.16)

x ( t ) da(t).

0

I t is easily proved that the correspondence 01 +Fa is an isomorphism of V,[O, 13 onto Q+.Let 01, /3 E Vo[O,I]; then, applying Fubini’s theorem to interchange the order of integration, and using (5.1.23), we have

=

f

s: 1:

min(t, s) da(t) dS(s)

If we write

then the space H of Theorem 5.3.7 is, in the present case, the totality of functions x in Co[O,11 which satisfy the condition (5.3.1 7)

I t is easily verified that condition (5.3.17) holds if and only if x is absolutely continuous and satisfies

(So 1

11 x 11

=

X’(t)2

1/24

dt)

< 00.

Consequently, the inner product in H is (x, Y ) =

Now, if we get

01

E

j

1

X‘(t)Y’(t) 0

3,y E

d4

H.

Vo[O,11 and x E H, then, integrating by parts in (5.3.16),

1a(t)~ ’ ( dtt ) 1

F,(x) = -

0

=

-

5.3. Gaussian Measures on Linear Spaces hence, when A E H and A’ h(x)

=

Vo[O,I],

E

-

325

p

x ( t ) dh’(t) =

J1

0

h‘(t) dx(t).

(5.3.18)

0

Choose any complete orthonormal system of vectors {A,} in H such that A,’ E Vo[O,11 for every n. Then, by (5.3.1 l), h(x) =

c s1

h,’(t) d x ( t )

0

j1h,’(t) dh(t).

(5.3.19)

0

Henceforth, we shall formally use the notation33 (5.3.18)for any A Summarizing, we have the following result.

E H.

Corollary 5.3.8. The Wiener measure space “w;: is quasi-invariant under the translations corresponding to the linear subspace A of Co[O,13 consisting of all absolutely continuous functions A E Co[O, 11 such that A’(t) E L2[0,11. Moreover, if A(x) is defined by (5.3.19), then {A(.), A E A} is a standard Gaussian process. When A E A and A’ E Vo[O,11, then A(x) may be expressed in the form (5.3.18). We conclude this section by establishing the ergodicity of standard Gaussian measure spaces. Theorem 5.3.9. Let S = (SZ, 8,N ) be a standard Gaussian measure space relative to a linear subspace H of Q. Then S is ergodic with respect to H . PROOF. For convenience, we assume that H is separable. Choose a complete orthonormal system {x,} in H , and define a mapping U from SZ into the totality I of real number sequences, as follows:

u :w

--f

b = {xl(w)

,..., x,(w) ,...}.

m

It is easily seen that U H = 12. Construct a measure on (1, BU)such that the coordinates A, , n = 1,2, ... ( A = (A,) E t ) are mutually independent Gaussian variables on ( I , 23, , with mathematical expectation zero and variance 1212. Since {x(w), x E H } is a determining set on S, it is easily proved that, for every E E 8, there exists an E 8, such that E and U-lE differ only by an N-null set, and that

w),

e

N ( E ) = rn(E).

(5.3.20)

ss Translator’s note: Perhaps the author has in mind here his projected second volume, since no further use is made of this notation in the remainder of the present volume.

v.

326

GAUSSIAN MEASURES

If E E 113 is quasi-invariant under H , that is, for every X E H , N(E - ( E x)) = 0, then it follows easily from (5.3.20) that R(E - ( E U,)) = 0. Hence, I? is quasi-invariant under 12. However, by Theorem 3.1.32, (1, 23, , is ergodic with respect to 6, the totality of elements of 1 having only finitely many nonzero coordinates.

+ +

m)

Obviously, 8 C E2, hence we conclude that N(E) is either zero or one, and therefore N ( E ) is either zero or one. Thus, S is ergodic with respect to H . J 95.4. Fourier-Gauss Transforms

In this section, we shall consider the theory of L2-Fourier transforms based upon Gaussian measures. For this purpose, we need only specialize the theory developed in 93.4 and subsection 50 of 54.3. In all of the following lemmas and theorems, it is assumed that H is a separable real Hilbert space, that S, = (Q, b, N,) is a standard Gaussian measure space (with parameter c) relative to H , and that {x(w), x E H } is the corresponding standard Gaussian process; we shall not bother to restate these assumptions.

Lemma 5.4.1. Let Nch be the measure on (Q, b) defined by N,,(E) = N,(E - h), E E b, h E H . Then (5.4.1)

PROOF. Apply the mapping U defined in the proof of Theorem 5.3.9, and use formula (5.1.14). ]

Lemma 5.4.2. The standard Gaussian measure space S, is strongly cyclic (of degree one) with respect to H , with 1 as cyclic element. T h e corresponding adjoint function is

PROOF. Let H , denote the smallest closed linear subspace of L2(S,) containing 1 and invariant under all the operators {U(h) I h E H } (see 93.1). By virtue of Lemma 5.4.1, H I contains all the functions (5.4.3) We must prove that H I = L2(S,).Clearly, we may assume that c = 1.

327

5.4. Fourier-Gauss Transforms

Choose any h, ,..., h, E H such that (h, , h k ) = 6,k , I, K = 1, 2 ,..., n. We shall first show that, for any bounded continuous function ~ ( t,..., , 2,) on Rn 9 dh1(w),..., h,(w)) E 4

(5.4.4)

Now, h,( h,( -) are Gaussian variables, with mathematical expectation zero and covariance 6&/2. Hence, for any Baire function $(tl ,..., tn), we have $(h,(w),..., h,(w)) €L2(S1)if and only if a),...,

Let A denote the totality of such functions $; A forms a Hilbert space with respect to the obvious inner product corresponding to the norm / / * I/ defined by (5.4.5). Let B = {$ I $ E A , $(h,(w),..., h,(w)) E H,}; obviously, B is a linear subspace of A. Moreover, since p'> =

($9

(+(M.),***1

(5.4.5')

h,(*)),dhl(.)?**V h7&(*)))!

it follows that B is closed in A. We assert that B = A. For otherwise, there would exist $o E A, +o # 0, $, 1B. However, the function A,h,) 1 E H , , -co < A, ,..., A, < co, hence the funcU(A,hl tion

+ +

+(tl

,..., t,)

=

+ + Antn)] E B,

exp[-(A,t,

and therefore

=

($9

$0)

=

(5.4.6)

0.

For any positive numbers 7,

,...,q,,

exP(?ll I t l

hence, if A, W

l

,.**,

=

,...,An

I + .**

the function

+

77%

I

t,

I) E 4

are complex variables, then the function

A,)

j

m -m

exp[-(t12

+ + t,2)

- (h1tl

+ .*. + Ant,)]

,..., t,) dt,

I/Jo(t1

dt,

v. GAUSSIAN

328

MEASURES

is well-defined, and is, in fact, an entire function of A, ,..., A, . But then it follows from (5.4.6) that F is identically zero. I n particular, choosing A [ = ix, , -co < x, < co, we see that the Fourier transform of the function exp[-(t12

+ + s.0

tn2)1

4dtl ,..., tn)

is identically zero, and hence &, = 0, contrary to our assumption. We conclude that A = B, and therefore (5.4.4) holds. Now, choose a complete orthonormal set { h , , h, ,..., h, ,...} in H. Let B be the totality of functions of the form q(hl(w),...,h,(w)), where n varies over all positive integers and cp is a bounded continuous function on R, , Obviously, 9 forms an algebra, moreover, since (h(w) 1 h E H } is a determining set on S, , it is not difficult to see that B is also a determining set.34 Hence, by virtue of Lemma 1.1.6, B is dense in L2(S,).But by (5.4.4), B C Ell , therefore H I is dense inL2(S1). Since Hl is closed, we have Hl = L2(S,). Thus, we have proved that S, is strongly cyclic with respect to H , with 1 as cyclic element. Furthermore, from (5.4.3)we get #c(h)

=

(w41, 1) =

exp

;[1 Nu) - 51 (4h,] dNc(w)

We note that the quasi-characteristic function of S , on the space of quasi-linear functionals A = {h(w) I h E H } is

Translator’s note: T h e following sequence of steps may assist the reader in convincing himself of this fact. First, any trigonometric polynomial is bounded and continuous, and the totality of real trigonometric polynomials in the hi(w) determines the same measurable sets as { h i ( w ) , i = 1, 2,...}. Second, {hi(w), i = 1, 2,...} determines the same measurable sets as the totality 2’of finite linear combinations of the h,(w). Finally, since 9 is dense in La(&), the characteristic function X E of any measurable set E may be approximated in La(&) by a sequence of elements in 9,and one may choose a subsequence which converges to X E almost everywhere; from this, it can be seen E belongs to the a-ring determined by 9,modulo a null set.

5.4. Fourier-Gauss Transforms

329

Comparing (5.4.2)and (5.4.7), we see that

where T denotes the isomorphism

from H onto A. I n other words, the adjoint function of S, coincides with the quasi-characteristic function of & I , . Following the proof of Theorem 3.4.14,we obtain the following result.

Theorem 5.4.3. of S, .

The standard Gaussian measure space SI,icis the

Definition 5.4.1. The Fourier transform F, of L2(S,)onto L2(Sl,,), corresponding to the cyclic element 1, is called a Fourier-Gauss transform. We now proceed to investigate the effect of the Fourier-Gauss transform on the elements of L2(S,).By Lemma 5.4.2,we know that the set of all linear combinations of functions of the form

is dense in L2(S,).Consequently, in order to describe the operator F, , it suffices to determine the functions F,(exp(( 1 /c) h( a))).

Lemma 5.4.4,

If h E H , then

=

1

exp

[,1 (d?h(w,) + ich(w))] diVo(wl).

(5.4.8)

PROOF. By the definition of F, [see (5.4.3)and (3.4.48)],we have

G’’

36 Translator’s note: In order to satisfy the requirements of Definition 4.2.9, must be identified with H . This, in turn, requires the application of, Corollary 4.2.5. Hence, it seems necessary to add certain hypotheses, for example, that (i) s;) is a linear topological space, (ii) the imbedding H -+ f2 is continuous, and (iii) the measure space S, is regular.

v. GAUSSIAN

330

MEASURES

Furthermore, h(w) is a Gaussian variable, with mathematical expectation zero and variance &h, h), therefore

whence we obtain (5.4.8). ] Next, we establish a useful formula for the Fourier transform of a polynomial functional.

Theorem 5.45. For any polynomialP(t, ,...,t,)and anyg, ,...,g, E H, the polynomial functional P(gl( g,(.)) belongs to L2(Sc),and ,...,)s

FC(P(gd.),..., gn( *)>)(w) =

J P(4%!1(4 + icg&J),..., .\/Zg,(w1) + icgn(4) dNc(w1).

(5.4.9)

Moreover, the totality of such polynomial functionals is dense in L2( SJ. PROOF. Consider an arbitrary element g E H; for convenience, we may assume that (g, g) = I , For any nonnegative integer m, we have t m E A (see the proof of Lemma 5.4.2), hence gm(*)€L2(Sc).Furthermore, for any real number A, the sequence fn(W) =

h ugy(w) c (T) y J - 9

n

=

1, 2,...

u=o

converges in the norm of L2(S,) to the function exp((Afc)g(w)); in fact, this follows immediately from

and (5.4.5). But F, is a bounded linear operator, therefore, using (5.4.8), we obtain

331

5.4. Fourier-Gauss Transforms

Comparing coefficients of like powers of h, we see that (5.4.9) holds for all functionals of the form g(.)’. Now, for any g, ,...,g,E H , let h,g, ; then g = h,g,

+ +

dw)’

c

=

Cvl...v>;I

*-. h’,g;I(w)

***

gpd,

(5.4.10)

vl+. ..+v,=w

where cvl..

.YII

denotes the appropriate multinomial coefficient. Setting

P(g(w))= g(w)’ in (5.4.9), and comparing coefficients of like terms

x;l -..h’;,

we conclude that (5.4.9) holds whenever P is a monomial. Hence, by linearity, it follows that (5.4.9) is valid for all polynomials P. As we have shown above, every functional of the form @ ( w ) I c , g E H can be approximated in L2(S,) by polynomial functionals, and, by Lemma 5.4.2, the set of all linear combinations of such functionals @ ( w ) I c is dense in L2(S,). Consequently, the totality of polynomial functionals is dense in L2(Sc). ] Remark. By examining the proof of Lemma 5.4.2, it can be seen that the conclusion of Theorem 5.4.5 remains valid even if the choice of the elements g1 ,...,g, is restricted to some arbitrary but fixed orthonormal basis {g, ,...,g, ,...} in H. We shall establish another expression for the operator F , . T o this end, we consider the Hermite polynomials h,(t)

=

dn dtn

=

1 , 2,...;

2%! &-8m,n ,

nz, n

(-l)n exp(t2) -[exp(-t2)],

n

h,(t) = 1.

(5.4.11)

As is well known, m

1

h,(t) h,(t) exp(-t2) dt

=

=

0, 1 , 2,..., (5.4.12)

-m

moreover, the functions {exp(-t2/2) h,(t)}, n = 0, 1, 2, ..., constitute a complete orthonormal system in L2(- 00, 00). Furthermore, for any complex number h and any real number t,

(-

t2

( t - h)2

+ 21 P).

(5.4.13)

In fact, (5.4.13) follows immediately from the Taylor expansion and (5.4.1 1). On the other hand, by (5.4.12), m

hk(t)

hk

1’

exp( - P ) dt =

1( 2 h 2 ) k fi k=n

+

0

(5.4.14)

v. GAUSSIAN MEASURES

332

as m, n -+ 00. Consequently, the left-hand side of (5.4.13) converges in L2(- m, co) to the right-hand side. For any positive integer n, and any two vectors u = (ul ,...,u,), v = (v, ,..., a,) in real n-dimensional space R, , let (u, v) denote the inner product C;==, u p k . Let c be a positive parameter, let k = (k,,..., k,) be a sequence of nonnegative integers, and write k! = k,! 1 k 1 = k, *.* k, . Define

+ +

Writing du = du,

- a .

... A,!,

d u n , we have, by (5.4.12),

1 hk(u; C) hk'(u; C) eXp (CTr)n/z

[ c1 -

(24, u ) ]

du

=

0, when k # k', 1, when k = k'.

(5.4.16)

Choose any orthonormal system {gv , v = 1,2,...} in H. Then, by virtue of (5.4.5') and (5.4.16), the set of functions *({ gv}; c,

= {hk(

gl(u),*-*, gn(u); c )

I

(k

$*..)

kn)

a sequence of nonnegative integers, n = 1, 2, ...I

(5.4.17)

forms an orthonormal system in L2(Sc).

Definition 5.4.2. The set of functions 2 ( { g Y } ;c ) is called the system of Hermite polynomial functionals corresponding to {gv}. If {gv} is a complete orthonormal system in H, and f € L 2 ( S C then ) , the number ukl***k,(f;

c> = R

f ( w ) h k ( g l ( w ) , * * * , gn(u); c> dNc(u)

is known as the kth Fourier-Hermite coejicient off relative to {gv},

Lemma 5.4.6. If {gv}is a complete orthonormal system in H, then the corresponding system of Hermite polynomial functionals (5.4.17) constitutes a complete orthonormal system in L2(Sc). Moreover, for any k = (k, ,..., A,), F c ( h k ( g l ( - ) , * * * ,gn(*); c ) ) ( w ) = ( i ) ' k '

(gl(u),***, :)gn(w);

(5.4*18)

PROOF. Obviously, every polynomial is a linear combination of Hermite polynomials, hence, by virtue of the remark following Theorem 5.4.5, we see that the totality of linear combinations of elements of 8 ( { g v } ;c) is dense in L2(Sc),that is, X ( { g v } c) ; is complete.

333

5.4. Fourier-Gauss Transforms

,..., A, be any set of real numbers, and .--A k n by Ak. By (5.4.57, (5.4.8), and (5.4.13),

Let A, A?

= Fc

(exP

2 c" 1 3

-

AYS"(*)

A.21)

( w ) = exp

v=l

denote the monomial

(2
2.2 fiag.(w,))

v=l

hk (gl(W),...,g,(w); - dk12rkJ'2hk C l) (k!)lI2

=c k

Comparing coefficients of like terms hk, we obtain (5.4.18). ] It follows from Lemma 5.4.6 that, for every f € L 2 ( S C )we , have an orthogonal expansion

f(u)= 1a k ( f ;

c,

hk(gl(w),--*, gn(w);c)*

(5.4.19)

k

Since F, is a unitary operator from L2(S,)to L2(S,,,), relations (5.4.18) and (5.4.19) yield the following result.

Theorem 5.4.7.

IffEL2(Sc),then

where {ak}are the Fourier-Hermite coefficients off.

Corollary 5.4.8. F, is given by

The inverse F$ of the Fourier-Gauss transform

m!Z)17,/c(!7>, -

=

?!

E~2(&Ic)*

(5.4.21)

PROOF. LetfEL2(Sc); by (5.4.20), we have 1 (F&f); 2)

= a k ( f ; c)(i)rk'a

(5.4.22)

Setting F , ( f ) = q in (5.4.22), we get = ak(K'(4); C)(i)lk'.

(5.4.23)

v. GAUSSIAN MEASURES

334

On the other hand, replacing c by l/c and f by 4 in (5.4.22), we get 1

4 = 4 ( P ; ); (i)’”.

%(&c(q);

(5.4.24)

Taking complex conjugates of both sides of (5.4.22), using (5.4.23) and noting that ll4

= @k(%

lk),

we get a k ( m 4 ) ; 4 = %(h(!@; 4,

whence (5.4.21) follows by applying (5.4.19). 3 Finally, let us consider a special case, that is, Wiener measure.

Example 5.4.1. We use the notation of Examples 5.1.2 and 5.3.2; for convenience, we take c = 1. In this case, Wl is its own dual. Choose any system of functions {a,} which is complete and orthonormal relative to the inner product (%

5

1 1

tt) = 5

0

c p w ttw

in L2[0,I], and such that every an is of bounded variation and satisfies a,(l)-= 0. Form the functionals

1 1

gnM =

0

an(4

4 9

on C,,[O, 11. These are the quasi-linear functionals associated with a complete orthonormal system in H. The corresponding system of Hermite polynomial functionals is

Naturally, one may also consider Wiener measure and the corresponding Fourier-Gauss transform in the multidimensional case. However, the results are entirely similar; the details are left to the reader.

CHAPTER

REPRESENTATION OF COMMUTATION RELATIONS IN BOSE-EINSTEIN FIELDS

As we mentioned in Chapter 111, some of the problems which led to the study of quasi-invariant measures arose in the quantum theory of fields, in particular, boson fields or Bose-Einstein fields. In the present chapter, we shall point out the connection between Bose-Einstein fields and the theory of quasi-invariant measures. In order to assist the reader in understanding the case of infinitely many degrees of freedom, we shall first (in $6.1) discuss in detail representations of the commutation relations in quantum mechanics (i.e., the case of finitely many degrees of freedom). Comparing the situation discussed in $6.1 with the corresponding situation in $6.2 and $6.3, the reader will appreciate the difficulties of the infinite-dimensional case; these are, in fact, the crucial mathematical difficulties in quantum-field theory. T h e discussion in $6.2 is rather general, whereas in $6.3 we study more specific representations and point out their connection with Gaussian measures, and so on. Finally, although the present chapter is concerned only with representations of the commutation relations, we again emphasize that the appli335

336

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

cations of harmonic analysis on quasi-invariant measure spaces are by no means confined to representations of the commutation relations: Applications to interacting field equations will be discussed in Volume 11. 56.1. Representations of the Commutation Relations in Quantum Mechanics

l o Basic Properties When dealing with quantum-mechanical systems having one degree of freedom, one must consider operators p , q satisfying the relation

P!7 - 4P

= 1,

(6.1.1)

where I is the identity operator. First, we point out that such operators cannot be bounded. I n fact, we have the following result.

Theorem 6.1.1. Let E be a Banach space. Then, (6.1.1) is not satisfied by any two bounded linear operators p , q on E . PROOF. Suppose that p , q are bounded linear operators on E, satisfying condition (6.1.1). For any bounded linear operator B on E, let B' = Bq - qB, B" = (B')',..., H n )= (B(n-l))',.... Then p' = I and p" = 0. We shall now show that

(P")(")= %!I.

(6.1.2)

+

Since (AB)' = A'B AB', we have the following relation, in analogy with the Newton-Leibnitz formula for differentiation: (6.1.3)

Formula (6.1.2) obviously holds for 1z = 1. Assume that (6.1.2) holds for n - 1. Then, setting A = pn-l, B = p in (6.1.3), and using the fact that p" = 0, we obtain (6.1.2). Obviously, 11 B' 11 2 (1 q 11 11 B 1 . Hence, by (6.1.2), we deduce that

<

n!

< (2 II 4 It)" II P" I1 < (2 /IP II /I 4 11)"

(6.1.4)

for all a, which contradicts the fact that 1imn+a((2 11 p 11 11 q Il)"/n!)= 0. ] In what follows, we shall confine ourselves to the case where p , q are unbounded operators on a Hilbert space; this is, in fact, the situation

6.1. Commutation Relations in Quantum Mechanics actually encountered in quantum mechanics, where operators satisfying the relation

qp - pq

337

p , q are self-adjoint (6.1.5)

= il.

This relation is not essentially different from (6.1.1), indeed, it may be obtained from (6.1.1) simply by replacing q by -iq. By Theorem 6.1.1, the operators p , q in (6.1.5) cannot be bounded, hence (6.1.5) should more properly be written as

qp - pq c il.

(6.1.6)

Here, the notation A C B means that the domain nAof the operator A is contained in the domain 3, of the operator B , that 9, is dense in a,, and that A x = Bx for all x E 3, . Now, let p be a self-adjoint operator on a Hilbert space H , and let ( E A, -a < X < CO} be the resolution of the identity corresponding to p , that is,

j

a,

p =

AdE,.

-a,

Form a one-parameter group of unitary operators (U(t);- co as follows: eitAdE,

U(t)=

< t < CO},

.

-02

This construction will be indicated by the concise notation U ( t ) = eipl.

Theorem 6-1.2, Let p and q be self-adjoint operators on a Hilbert space. Then, relation (6. I .6) is equivalent to e5Pteiqs

= eitseiqseivt

-a < s, t < cr;f.

The proof of this theorem will not be given here, since it is rather lengthy and employs methods not closeiy related with the subject of the present book. Henceforth, when dealing with the commutation relations, we shall always express them in the form given by Theorem 6.1.2, thus avoiding the complications connected with the manipulation of unbounded operators. 20 Quantum-Mechanical Systems Having Finitely Many Degrees of Freedom Let H be a Hilbert space, and let p , ,...,p , , q1 ,..., q, be self-adjoint operators on H , satisfying the following commutation relations:

VI. COMMUTATION

338

RELATIONS IN BOSE-EINSTEIN FIELDS

(i) p , ,...,p, all commute with each other; (ii) q1 ,...,qn all commute with each other; (iii) p , and q,, commute when p # v, and

We then say that {p,}, {q,} satisfy the Heisenberg commutation relations. I n view of Theorem 6.1.2, {p,), (4.) satisfy the Heisenberg commutation relations if and only if the unitary operators exp(ipyt,), exp(iqus,,)( - co < t,, s, < co) satisfy the so-called Weyl commutation relations, namely, (i) all the {exp(ip,t,)} commute with each other, (ii) all the {exp(iqyS,)} commute with each other, and (iii) exp($,t,) commutes with exp(iqySy) when p # v, and eXP(ipYtY) exp(iq,s,) = exp(it,s,) exp(iqYFy) eXP(ip,&)-

We shall presently express these commutation relations in a more concise form. Let R, denote the ordinary n-dimensional real vector space of real n-tuples t = (tl ,..., t,), with the inner product (t, s) defined by n

(2,

$1 = 1 t$,

7

”=l

where s = (sl ,..., s,). Form two families of unitary operators, U(t) and V(s),parameterized by the vectors of R, , as follows:

,..., t,J, = (sl ,...,sn),

for t = ( t l

let

U ( t ) = exp(ipltl)

exp(ip,t,);

(6.1.7)

for s

let

V(s) = exp(iq,sl)

exp(iq,s,).

(6.1.8)

It is easily proved that, for {p,,}, {qv} to satisfy the Weyl commutation relations, it is necessary and sufficient that both {U(t), t E R,} and { V(s),s E R,} are weakly continuous unitary representations of the additive topological group R, in H, and that U ( t ) V(s) = ei(t*s)V(s) U(t).

(6.1.9)

We shall refer to such a system {U(s), V(t); s, t E R,} as a weakly continuous unitary representation of the commutation relations. Actually, for any n, the Weyl commutation relations can be expressed in terms of unitary representations of a single group r, . I n fact, let r, be the totality of triples (x, y , a ) , x, y E R, , a E C, where C is the set of all complex numbers of unit modulus, and define multiplication in I‘, by

6.1. Commutation Relations in Quantum Mechanics

339

r,

Clearly, forms a group with respect to this operation. Take the Euclidean topology on R, and C,and let r, = R, x R, x C have the corresponding product topology. Then T, becomes a topological group; indeed, r, is a (2n 1)-dimensional Lie group. Consider a weakly continuous unitary representation T of the topological group F, in a Hilbert space H. If T satisfies the condition

+

T(0,0, a ) = aI,

where I is the identity operator, let

Straightforward calculation then shows that { U ( x ) ,x E R,}, { V(y ) ,y E R,} are both weakly continuous unitary representations of R , in H, and satisfy relation (6.1.9). Conversely, let { U(x),x E R,}, { V(y ) ,y E R,} be any two weakly continuous unitary representations of R, in H, satisfying relation (6.1.9). Then, T(x,y , 4

=

.U(x)

W)

is a weakly continuous unitary representation of T, in H, and T(0,0, a ) = ai.

We shall now give an important example of a pair of unitary operator groups { U(t)} ,{ V(s)}satisfying the Weyl commutation relations. Let L2(R,) be the Hilbert space formed by the totality of quadratically integrable Lebesgue measurable complex-valued functions on R , , with the inner product

For any t , s E R, follows:

, form unitary operators Uo(t)and V,(s) on L2(R,), as (U,,(t)f)(x)= e*"t-"'f(x),

(~Il(W(4 = f ( x - 4. I t is easily verified that ( U o ( t ) t, E R,} and {Vo(s),s E R,} are weakly continuous unitary representations of R, in L2(Rn),and that they satisfy (6.1.9). The system { U o ( t) ,Vo(s),t , s E R } is called the Schrodinger representation of the commutation relations, or a system of Schrodinger operators.

340

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

We may also express the Weyl commutation relations in another convenient form, as follows. Let C, be the ordinary n-dimensional complex vector space, and define an inner product in C, by (z, z') = zpl'

+ + z,Z,' .'*

where z = (zl,..., z,), z' = (xl',...,x,'). Now, let { U(t), V(s),t , s E R,} be a system of unitary operators, and let ~

(

+tis) = ~ ( t~ )( s exp ) [- i ( t ,

513.

(6.1.lo)

I t is easily proved that the system { U(t), V(s),t , s E R,} satisfies the Weyl commutation relations if and only if the operators { W(z),z E C,} depend continuously (in the weak topology) on the parameter z, and satisfy

Conversely, given a family of unitary operators (W(z),x E C,}, satisfying (6.1.11), we may define U ( t ) = W(t),V(s)= W(is).Then, { U ( t ) ,t E R,} and (V(s),s E R,} are obviously one-parameter groups of unitary operators, satisfying (6.1.9), and are related to W ( z ) by the equation (6.1.10). Moreover, if W ( z )is weakly continuous in z, then {U(t),t E R,} and { V, , s E R,} are weakly continuous. Formula (6.1.1 1 ) is known as the von Neumann form of the commutation relations.

Theorem 6.1.3. Let H be a Hilbert space, W = {W(z),z E C,} a family of unitary operators in H , satisfying (6.1.11), and such that W ( z )is weakly continuous with respect to z. Then, there exists a family {Ha , a: E A } of mutually orthogonal subspaces of H , invariant with respect to W, such that H = @ H , and the restriction of W to any H , is unitarily equivalent to the Schrodinger representation. PROOF. Define a bilinear functional I,(.$,7) on H , as follows:

xa

where

, is a bounded continuous function of x E C, ,the integral Since ( W ( x ) f 7) (6.1.12) exists, moreover

I L(6,?)I G C II E II II 7 II,

6.1. Commutation Relations in Quantum Mechanics

34 1

where 1 (2r)"

c = __

J exp (-

$11

z

112)

dz = 2".

Furthermore, by (6.1.11), we have W(z)*= W(-z), whence it follows easily that L((, .I) = L(q, 5). Therefore, there exists a bounded selfadjoint operator P such that

q 5 , r l ) = (P5,rl).

(6.1.13)

We assert that, for any z E C, , PW(z)P = exp(-

In fact, for any 8, q

E

4 1) z jJz)P.

(6.1 .14)

H,

( P W 4 Pt9 7) = - q W ( z )P t , 17)

1 (24"

J exp (-

=-

1

I

=-

(2T)n 1 -(242"

exp

(-

1 4 1

~ t ,

- 11 z'

112)

( ~ ( z ' ~) ( z ) 7) dz'

11 z'

112)

~ ( 5 ~, ( z >~*( z ' ) * y )dz'

1exp (-

(11

z'

112

+ 11 z" 1 2)

(W(z')W(z)W(z")t,7) dz'dz". (6.1.15)

Again, using (6.1.1 I), we have ~ ( z '~) ( z~ )( z " = ) exp

+

(-

~ [ ( z 'z,)

+ (z', z") + (z, z")]) ~ ( +zz' + z").

+ + Z" as the variables of integration

Taking z1= z z", z2 = x x' in (6.1.19, and using the equality

342

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

+

+

+

where (ul iv, ,..., u, iv,) = z1, ( jil ,fi2 ,...,fin)= z2 z, we calculate that (6.1.15) is equal to exp(-& 11 z 1I2)(Pt,7). Hence, (6.1.14) is established. I n particular, setting z = 0 in (6.1.14), we see that P is a projection operator. Let 9JI = PH, and choose a complete orthonormal system { v a a, E ‘ill}in 93. Let Ha be the smallest closed linear subspace of H which contains g,, and is invariant under all the operators W(z),z E Cn We shall first prove that Hu 1Ha, , 01 # a‘. In fact, since Pcpa = cpa , Pg,,.= vat , it follows by (6.1.14) that

.

( w ( z )p a w(z’)?a,) 1

=

(w(z)Pva

=

(Pw(-z’) w(z)PvLy 9 v a , )

[

i

[-

51 II z - z‘ 112 - -22 3(z, 2’11 (vu, v,#)

= exp -

= exp

w(z’)%a,)

3(z, z’)] ( P W ( ~ z’)PV,

,pa,)

(6.1.1 6)

for any z, 2’ E Cn . Thus, W ( z )g,, 1 W ( d )rpa’ when 01 # a’. But Ha is the closure of the set of all linear combinations of the vectors W ( z )ya , z E C, , consequently Ha IHat ( a # a’). Next, we proceed to prove that the closed linear hull 6 of H , is the entire space H . Clearly, 6 is invariant under all the operators W(z),x E C, , and !Dl C G. Therefore, H @ 6 = G1 is also invariant under all the W(z), z E Cn , moreover, Pg,= 0 for all E 61. Suppose that GL # (0), and choose any g, E G1,g, # 0. Then W(x)g,also belongs to 61, and so PW(z)g,= 0. Hence, by virtue of (6. I. 12), (6.1.13), and (6.1. I I), we obtain

u,

for all x

.

E Cn

Since 3(z’,z) =

C (yv’x, - y,x,I),

where z = the above relation shows that the Fourier transform of exp(-t 11 z‘ [Iz)(W(z‘)g,, g,) (regarded as a function of the 2n real variables x,’, y,‘, v = I, 2,..., n) is identically zero. Consequently, ( W(x’)g,,g,) = 0, z‘ E C, . I n particular, taking z‘ = 0, we get g, = 0, which contradicts our assumption. Hence, @ H, . we conclude that H = (xl

+ ;rl ,..., x, + +,),

x‘

xa

= (xl‘

+ iyl’,..., x,‘ + iy,’),

6.1. Commutation Relations in Quantum Mechanics

343

We now consider the manner in which W(x)operates on H, . Writing W ( z )v, = T,,~ , we have, by (6.1.1 1) and (6.1.16),

w4

Pu.2

= exp

[

1

-

2 W', 41 Pm.z+z'

(6.1.17)

and (Pa., 9 P,,Z*)

for all x, x' E C, . Define a function cPs(U)

=

1 i exp - 4 II z - z' 112 - - 3(2,4 2

]

[

vz €L2(R,)(with z E C,

(6.1.18)

as parameter), as follows:

1 1 1 i =rr"/4exp [ - ~ l l ~ l 1 2 + ~ ~ ~ + r ~ ~ ~ - ~ I l (6.1.19) Y l 1 2 - ~ ~ ~ ~

+

where x, y E R, , x = x iy. It is easy to check that the system of operators Wo(x iy) = Uo(x)V o ( y )exp[-(i/2)(x, y)], corresponding to the Schrodinger operators U,,(x) and Vo(y), satisfies

+

[

i

4

W o ( 4 PZ = exp - 2 3(2',

%+2'

(6.1.20)

and

(vz, Tz,)= exp [- 1 11 z - z' 112 - -2i 3 ( z , 41.

(6.1.21)

Construct a linear operator U, from H , to L2(R,) as follows: let U,y,,, = y z , and extend U, by linearity to all linear combinations of the T,,~, z E C, . By virtue of (6.1.18) and (6.1.21), U, is isometric, and since the totality of linear combinations of the T,.~, z E C, , is dense in H , , the mapping U, can be uniquely extended to an isometry of H , into L2(R,). Moreover, it is clear that totality of linear combinations of the functions {vz , x E C,} is dense in L2(R,).Hence, U , is a unitary operator, furthermore, it follows easily from (6.1.17) and (6.1.20) that U,W(Z) UL1 = WO(Z),

that is, the restriction W [ H, is unitarily equivalent to the Schrodinger representation. ] Let H be a Hilbert space, and let {W(z),z E Rn}be a family of unitary operators on H , satisfying the commutation relations (6.1.1 1). If no closed linear subspace of H , other than (0) and H itself, is invariant under all the operators W ( z ) ,then we say that { W(x),z E R,} is irreducible in H . If U ( t ) = W(t), V(s)= W(is),then this irreducibility condition simply

344

v1. COMMUTATION

RELATIONS I N BOSE-EINSTEIN

FIELDS

means that the unitary representation1 of the group r, [i.e., T(x,y, a ) = aU(x) V ( y ) ]is irreducible in the usual sense.

Theorem 6,1.4.

The Schrodinger representation in L2(R,) is

irreducible.

PROOF. Let 9Jl # (0) be a closed linear subspace of L2(R,) which is invariant under all the operators Wo(z),z E C, . By virtue of Theorem 6.1.3, we may assume that the restriction of {Wo(z),z E C,} to %I is unitarily equivalent to the Schrodinger representation, for otherwise, we could select a nontrivial subspace of 'JJZ for which this is the case. Now, for the Schrodinger representation, there exists a vector cp [e.g., p(u) = exp(-4 1) u /I2)] such that the totality of linear combinations of the set { Uo(x)cpI x E R,} is dense in the entire space. Obviously, any family of operators which is unitarily equivalent to the Schrodinger representation must also have this property. Thus, there exists a vector $ in %TI such that the totality of linear combinations of the set {Uo(x)t,41 x E R,) is dense in 9Jl. Let E = {u I $(u) # 0}, and let L2(E)denote the totality of functions in L2(R,) which vanish almost everywhere outside of E. We assert that W = L2(E).In fact, it clearly follows from the foregoing remarks that W C L2(E).Suppose that W # L2(E),that is, that there is an f E L2(E)0'JJZ, f # 0. Then, ( U o ( x ) $ , f )=

ei@.")$(u)f(u>du = 0

, in other words, the Fourier transform of the function $(u) fo €L1(R,) is identically zero. Hence, +(u) f<.> vanishes almost for all x E R,

everywhere, and so f(u) vanishes almost everywhere in E. But f also vanishes almost everywhere outside of E, thereforef = 0 almost everywhere, which contradicts the choice of j.Thus, we must havem = L2(E). On the other hand, 9Jl = L2(E)is also invariant under all the operators Vo(y ) , y E R, . Consequently, for every y E R, , $(u - y ) vanishes almost everywhere outside of E. This means that E - ( E y ) is a null set for every y E R, , that is, E is quasi-invariant with respect to translations. But Lebesgue measure is ergodic (see Lemma 3.1.31), hence, either E or R, - E is a null set. Since'JJZ # (0),E cannot be a null set. Therefore, R, - E is a null set, and 'JJZ = L2(R,). ] From Theorems 6.1.3 and 6.1.4, we immediately obtain the following result.

+

Corollary 6.1.5. Let {W(z),x E C,} be a family of unitary operators on H , satisfying the commutation relations (6.1.11). Then H can be

6.1. Commutation Relations in Quantum Mechanics

345

decomposed into an orthogonal sum of closed linear subspaces H, , E 2l, which are invariant under { W ( z ) z , E C,}, and such that the restriction of {W(z),z E C,} to any H, is irreducible. a

Corollary 6.1.6, If a family { W ( z ) ,z E C,} of unitary operators on H satisfies the commutation relations (6.1.1 1) and is irreducible, then it is unitarily equivalent to the Schrodinger representation. Thus, an irreducible representation of the commutation relations is uniquely determined up to unitary equivalence. Let 'illbe the weakly closed operator algebra in B ( H ) generated by the operators {U(x), V(x') I x, x' E Rn}. We call 2I a concrete Weyl algebra (with n degrees of freedom) on H. In general, the structure of this algebra depends upon the representation { U ( x ) , V ( d ) I x, xf E Rn}.However, we have the following result. Lemma 6.1.7. If the representation { U ( x ) , V(x') I x, xf E R,} is irreducible in H, then the corresponding concrete Weyl algebra 2l is just b ( H ) (see $2.3). PROOF. Let P E (2l')P. Then the subspace PH is invariant under the operators { U(x), V(x') 1 x, xf E Rn),whence it follows by irreducibility that either P = 0 or P = I. Therefore, by Corollary 2.3.5, 21f = {AI I h E F } and '2I = 8 ( H ) (see Example 2.3.2). ] Lemma 6.1.8, Let H be a Hilbert space and 'p a symmetric automorphism of b(H). Then, there exists a unitary operator U on H such that (6.1.22)

?(A) = UAU-'

for all A E b ( H ) . PROOF. For any unit vector ( in H , let P, be the projection operator defined by P,x = (x, ()(, X E H . Note that, if P is any projection operator in H , then 'p(P)z= 'p(P2)= 'p(P), 'p(P)*= 'p(P*) = 'p(P), hence 'p(P)is also a projection operator; conversely, if q ( P )is a projection operator, then so is P . Since P , cannot be expressed as the sum of two nonzero projection operators, neither can q(P,). Consequently, 'p(P,)H is also a one-dimensional space, and so there exists a unit vector T E H such that q(P,) = P , . Now, arbitrarily choose a fixed unit vector (,, , and let yo be a fixed unit vector such that 'p(P,,) = Pn, . For any vector ( E H, form an operator P,,, as follows:

P,,l(Go) pE,(c) = 0

= At,

when

c 1t o .

346

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COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

We define

ut = d P t t , ) 7 0

*

I t is easily verified that U is a unitary operator, and that relation (6.1.22) is satisfied for any operator of the form P, . I t follows immediately that (6.1.22) is satisfied for any projection operator A which is of finite rank. Furthermore, since any projection operator is the supremum of a family of projection operators of finite rank, and since A B implies that v ( B ) - v(A) = v ( ( B - A)1/2)22 0, relation (6.1.22) holds for any projection operator A. Next, observe that, for any A E b(H), the spectra of A and y ( A ) coincide, hence

<

II A I1 = (I1 A*A

= (I1 dA*A)11)”2 = II rp(4II.

(6.1.23)

Applying (6.1.22) to linear combinations of projection operators, and then taking limits, it follows by (6.1.23) that (6.1.22) holds for any selfadjoint operator A , hence also for any A E d(H). ]

Lemma 6.1.9. Let {Uo(x),Vo(x‘)I x, x’ E R,} be the Schrodinger representation in L2(R,), arid let g, be a symmetric automorphism of B(L2(R,))such that ?( uo(X))

=

uo(x),

y( F‘~(x’)) = P‘o(x’),

X, X‘ E

R,

.

(6.1.24)

Then y is just the trivial automorphism A -+ A. PROOF. By Lemma 6.1.8, there exists a unitary operator U on L2(R,) such that (6.1.22) holds for all A E d(L2(R,)), and by (6.1.24), we know that U E { Uo(x),Vo(x’)I x, x’ E R,}‘. Using Theorem 6.1.4 and Lemma 6.1.7, it follows by $2.3,2O, (iv) and Example 2.3.2 that U E B(L2(R,))’= { X I I X E F } . Thus, U = X I , therefore, by (6.1.22), y ( A ) = A. ]

Theorem 6.1.10.

Let Htk), K

=

1, 2, be Hilbert spaces,

{ W ) ( x ) , V ‘ k ) ( x ’ ) I x, x‘ E R,}

unitary representations of the commutation relations in the respective spaces H ( k ) ,and ‘ill(k) the corresponding concrete Weyl algebras. Then, there exists a unique symmetric isomorphism J/,I from onto ‘ill(2) such that, for any bounded Baire functionlf, +(f(u(l)(x))) = f(U(2’(X>),

#(f(V(’)(X‘))) = f (VL2)(x’)),

x E R, X’ E

R,

(6.1.25)

I

.

(6.1.26)

The notation used here refers to the usual operational calculus (see, e.g., Riesz and Sz.-Nagy [I]).

6.1. Commutation Relations in Quantum Mechanics

347

PROOF. By Theorem 6.1.3, there exists a family of closed linear subspaces {HLk),a E A k } in Hck) such that H ( k )= C, @ Hik),each Hi k ) is invariant under { Vk), Vk)}, and the restriction of { Uk), Vk)} to Hik) is unitarily equivalent to the Schrodinger representation (K = 1,2). Let Qik)be the appropriate unitary operator from L2(R,) to Hik’, and let PLk) denote the projection operator from H C konto ) Then U‘”(X) =

c QP)U0(X)Qy-P?),

(6.1.27)

(I

(6.1.28) For any A E 23(L2(R,)), form the operator2 Ip””‘(A)=

1

Q;k)AQp-qJ;k).

a

Using (6.1.27) and (6.1.28), it is easily verified that y ( k )is a symmetric isomorphism of 23(L2(R,)) onto such that, for every bounded Baire function f, P)(f( uo(~))) = f(U ( k ) ( ~ ) ) , x E Rn (6.1.29) 9

cp‘”‘(f( V0(x‘)))= f(

X‘ E

Rn .

(6.1.30)

+

it follows at once that fulfils the requirements Taking $ = ~$~)(y(l))-l, of the theorem. Now, suppose that $’ is another isomorphism satisfying the conditions of the theorem. Consider the mapping = (+2))-1

*’Ip(1).

(6.1.31)

Obviously, y is a symmetric automorphism of 23(L2(R,)), moreover, by virtue of (6.1.25), (6.1.26), (6.1.29), and (6.1.30), y satisfies (6.1.24). Therefore, is the trivial automorphism of 23(L2(R,)), whence, by (6.1.31), $’ = $. ]

30 Another Type of Representation We shall now consider another type of representation which will be required later on. This type of representation differs from the Schrodinger representation in that it is based upon a Gaussian measure (which is only The sums appearing here and in (6.1.27), (6.1.28) are to be interpreted as strong limits.

348

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FIELDS

quasi-invariant under translations) in R, , rather than Lebesgue measure (which is invariant under translations). Let L2(R,, G) denote the totality of Lebesgue measurable complexvalued functions f on R, such that

L2(R,, G) forms a Hilbert space with respect to the ordinary linear operations and the inner product 1

(f,g) = ,n12 /f(=)g(ld) =P(- II Il’)

du-

For x, x‘ E R, , define ( U ( x ) f ) ( u )= e i ( U . 2 ) f (

4 1

T h e system { U(x), V(x’)I x, x’ E R,} clearly satisfies the Weyl commutation relations. Actually, this system is unitarily equivalent to the Schrodinger representation. I n fact, such an equivalence is realized by the unitary operator

from L2(R,, G) onto L2(R,). I t follows that the representation { U(x), V(x’)I x, x’ E R,} is also irreducible. In the present case, the corresponding self-adjoint operators p , and q, [see (6.1.7) and (6.1 A)] are defined as follows: the domain of p , is

D ( P ~= ) {V I F EL2(% G),uvdu) EL2(& G)), 9

9

where u

= (ul

,..., u,),

and, when

E

D(p,), (6.1.32)

6.1. Commutation Relations in Quantum Mechanics

349

and, when g, E D(qy),

(!?vd(4

=

- ;Uv?(4

a + . au, dU>-

(6.1.33)

I n fact, it is easily shown that (6.1.32) defines a self-adjoint operator, and, when 9) E D(p,),

Consequently, U ( x ) and ( p , ,...,pn} are related as in (6.1.7). A similar argument applies to (ql ,..., q,J. Let c, be the closure of the operator ( p , iqY)/l/Z.I t is easily verified that the domain of c, is

+

and, when

9) E

D(c,), (6.1.34)

T h e adjoint c,* of c, is the closure of the operator ( p y - iq,,)/dZ; its domain is D(cv*)=

1.

a v EL’(&, G ) ,-a% du)

€ L 2 ( R , ,G ) / ,

and, when g, E D(c,*), (6.1.35)

A simple calculation yields the relation CR*CI

- C&*

c 6,J.

Consider the complete orthonormal system {hk(U,

1);

R

=

( k , , k,

)...,kn),

R,

= 0,

1 , 2,...}

VI.

3 50

COMMUTATION RELATIONS IN BOSE-EINSTEIN FIELDS

in L2(R,, G) [see (5.4.15)]; hereafter, we shall simply write hk(u)in place of h,(u, 1). For each nonnegative integer m, let H ( m , n) be the finitedimensional linear subspace of L 2 ( R , , G) spanned by the vectors {h,(u) I I k I = m}. Then, m

1 0H(m, n).

L2(R,, G ) =

m=O

Lemma 6.1.11.

T h e operator c, maps H(m, n) into H(m

+ 1, n),

and c , ~ , ( u )= (kv‘)1’2hk,(u>,

where k’

=

(kl’,..., k,’) and k k’=

kl,

’

Ik,

=

(k, ,..., k,) are related by

+ 1,

when 1 # V, when 1 = v.

(6.1.36)

T h e operator c,* maps H(m, n) into H(m - 1, n) [letting H ( - 1, n) = (O)], and c,*hk(u) = (k,)1’2h k * ( ~ ) , where k’ = (k,’,..., k,’) and k k 1‘

=

=

(k, ,..., k,) are related by

lkz’

max(k, - 1, 0),

when 1 # v, when I = V.

(6.1.37)

PROOF. Note that the Hermite polynomials [see (5.4.I l)] satisfy the relations h,’(x)

= 2mhm-,(x),

h,+,(x) - 2xh,(x)

+ 2mhm-,(x) = 0.

Using these relations, (6.1.34) and (6.1.35), the assertions of the lemma follow immediately. ] We now form system { W(z),z E C,}, in accordance with (6.1.10).

Lemma 6.1.12. Let k = (k, ,..., k,), k’ = (k,’,..., k,’) be any two n-tuples of nonnegative integers, n 2 1, and 01 any real number. Then (W(eimx) hk(.),hk,(.))= (W(x)hk(.),hk,(.))e i u ( ~ k ~ - ~ k ’ ~ ) .(6.1.38)

PROOF. First, notice that, for any A, A’ (W(x)exp(2(h, t ) ) ,exp(Z(2,t))) =

&

E

C,

,

exp ( - d t ; x, A, V )dt, (6.1.39)

6.1. Commutation Relations in Quantum Mechanics

351

where ~ ( tz,; A,

A') = II t

112

- (ix

+ y + 2h + 2h', t )

Calculating (6.1.39) by means of the Gaussian integral formula (5.1.5), we obtain

+ 2(X, t)I, exp[-(k,

( W ( z )exp[-(k A) =

exp

[-

4

A')

+ 2(2,t)I)

+ 2(h, A') + i(h, + i(h', z ) ] . Z)

(6.1.40)

Now, the replacement of z by eiaz in the right-hand member of (6.1.40) yields the same result as the replacement of h by heia and A' by h'eciD. Consequently, (W(efaz)exp[-(h, A)

+ 2(h, t ) ] ,exp[-(J',

= ( ~ ( zexp[-e2"(AX, )

A)

+ 2(2,t)I)

+ 2ei=(h,t ) ] ,exp[-

eZia(A', A')

+ 2eia(2, t ) ] ) . (6.1.41)

Using (5.4.13), we expand both sides of (6.1.41) as power series in A, A'; comparing coefficients of like monomials XkXIk', we obtain (6.1.38). 3 40 Gradient Transformations

Let {W(z),z E C,} be an irreducible family of unitary operators on a Hilbert space H , weakly continuous in the parameter z, and satisfying the commutation relations (6.1 .I 1). Let U be a unitary operator on C, . Form the family of unitary operators W ( z )= W(Uz),

zE

c,

in H . Clearly, { W'(z),z E C,} is also weakly continuous in z , and satisfies the commutation relations

[

E

W ( z )W ( z ' ) = exp - - 3 ( z , z 7 ] W ( z + z'), 2

moreover, { W'(z),z E C,} is also irreducible in H. Therefore, by virtue of Corollary 6.1.6, { W ( z ) z, E C,} and {W'(z),z E C,} are unitarily

3 52

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COMMUTATION RELATIONS I N BOSE-EINSTEIN

FIELDS

equivalent, that is, there exists a uniqueg unitary operator I ‘( U ) on H , such that z E c, . W(Uz) = r(u)W(Z)q u y , I n particular, let us consider the case in which U is of the form eiaI : z +-eiax, x E C, , where a is a real number. I n this case, F( U ) is called a gradient transformation.

Theorem 6.1.13. If H = L2(R,, G ) and {W(x)} are the unitary operators introduced in 3O, then, for any real number a, we have (6.1.42)

where P,,denotes the projection of H onto H(m, n). PROOF. Since the operator defined by (6.1.42) has the property r(k.1)h k ( ’ )

= e-ialWzk(.),

it follows that (6.1.38) may be rewritten as (r(ei.1) W ( z )T(e”I)-l h k , hk,) = (W(e%) h k , h k ’ ) .

But {hk} is a complete orthonormal system in L2(R, , G), hence W(e%)

= r(ei.1)

W ( z )T(&I)-l.

This proves that (6.1.42)is, in fact, the required operator.

]

56.2. Quasi-Invariant Measures Applied t o Representationsof the Commutation Relations in Bose-Einstein Fields

lo Representations of the Commutation Relations: Various Equivalent Formulations We now turn to the consideration of the general case (including that of infinitely many degrees of freedom).

Definition 6.2.1. Let $3 and $3‘ be real linear spaces, and let B(x, x’)(x E 43, x’ E $3’) be a nondegenerate real-valued bilinear functional on ($3, 5’), that is, the following conditions are satisfied. (i) For any fixed x E $3, B(x, x’) is linear in x’; for any fixed x’ E &’, B(x, x’) is linear in x. Translator’s note: That is, up to a numerical factor of unit modulus.

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

353

(ii) For any x E $3, x # 0, there exists an x‘ E $3’ such that B(x, x’) # 0; conversely, for any x’ E $3’, x’ # 0, there exists an x E Sj such that B(x, x ’ ) # 0. Then, Z = ($3, sj‘, B ) is called a single-particle (state vector) system. We shall have occasion to imbed $3’ in $3” as follows: for every x‘ E Sj‘, define a functional f,. E $3” by f&(X)

=

B(x, x’),

x E $3.

Clearly, the correspondence x‘ --+ fx# is an isomorphism of $3‘ onto a certain linear subspace of $3”. Thus, by identifying every x’ with the corresponding f,, ,we may regard $3 as a linear subspace of $3”.

Example 6.2.1. Let sj be a real linear space, and $3’ a sufficient subspace of $3”. For any ~ € 5x ,‘ E $ ~ ’ ,let b(x, x’) = x’(x). Then ($3, sj’, B ) forms a single-particle system. Example 6.2.2. Let $5 be a real inner product space, let sj’ = 8, and let B(x, x ’ ) be the inner product on $3. Then (a, Sj, B ) forms a single-particle system. Definition 6.2.2. Let Z = ($3, sj’, 23) be a single-particle system, and C the totality of complex numbers of unit modulus. Let r(Z) = $3 x sj‘ x C, and denote the elements of r(Z)by ( x , x’, a),x E $3, x’ E $3’, a E C. Define a multiplication operation in r(Z)as follows: (x, x’, a ) ( y ,y‘, 8) = (x

+ y , x’ + y’, aPe-iBcu-s’)).

Then r(Z)forms a group, which we call the group associated with the system Z. Let S : y +S(y) be a unitary representation of the group r(Z) in a complex Hilbert space H , such that S(O,O, a) = aI,

where I denotes the identity operator. Suppose that, for any pair of finite-dimensional linear subspaces W C $3 and W’C $3‘, the mapping y --+ S(y) is weakly continuous on W x W’x C with respect to the product topology induced by the ordinary Euclidean topologies on W, W’and C. Then S is called a canonical unitary representation of r(Z)in H. The above definition is equivalent to the following one.

Definition 6.2.3. Let .Z = ($3, $3‘, B ) be a single-particle system, and H a complex Hilbert space. Let U : x + U ( x ) and V : x’ -+ V(x’)

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COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

be representations of sj and sj‘, respectively, in the group of unitary operators on H , such that the commutation relations U ( x ) V(x’) = eiB(zs*‘)V(x‘)U(x),

x E sj,

x‘

E Jj‘

(6.2.1)

are satisfied, Moreover, suppose that each of these representations is quasi-continuous, that is, if x(x’) is restricted to any finite-dimensional linear subspace of $($’), then the mapping x -+ U(x)(x’ -+ V(x’)) is weakly continuous (with respect to the Euclidean topology on the subspace). Then { U , V }is called a (Weyl) canonical system (in H ) over Z. In fact, given any canonical system { U,V } over Z,one may construct a canonical unitary representation of r(Z)in H , as follows: for any y = (x, x’, a ) , let S(x, x’,

E)

=

.U(x) V(x’).

We call S the canonical unitary representation corresponding to {U, V}. Conversely, given any canonical unitary representation S of r ( Z ) in H , let U ( x ) = S(x, 0, l),

V(x‘) = S(0, x’, 1).

Then { U(x), V(x’)I x E Jj, x’ E 9’) is a canonical system over Z, to which S corresponds in the above sense. Let { U , V } be a canonical system. Choosing any x E $, x’ E Jj‘, we form the infinitesimal generators I d

(6.2.2) t=0

of the one-parameter unitary groups {U(tx) [ -00 < t < 00) and { V ( t x ’ ) I -00 < t < a}, respectively (see 511.3, 2O). Consideration of the properties of these infinitesimal generators leads to the following definition.

Definition 6.2.3’. Let Z = ($, $‘, B ) be a single-particle system and H a complex Hilbert space. Let p : x -+ p ( x ) , q : x’ --t q(x’) be linear mappings of sj and Jj’, respectively, into the set of self-adjoint operators in H . Suppose that, for all x, y E 8,x’,y‘ E sj‘, (i) p ( x ) commutes4 with Here, as in 56.1, the cornmutability of two (unbounded) self-adjoint operators p ( x ) and p ( d ) means that their spectral functions (i.e., resolutions of the identity) commute, and the “sum” of p ( x ) and p(x’) means the closure of their linear sum (of course, we assume that this closure exists).

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

355

p ( y ) , (ii) q(x’) commutes with q(y’), and (iii) p(x), q(x’) satisfy the commutation relation

q ( x ‘ ) p ( x ) - p ( x ) q(x’)

c iB(x, x’)L

(6.2.3)

Then, we say that { p ( x ) ,q(x‘) I x E $j, x‘ E $’} is an infinitesimal canonical system (in H ) over Z. If { U , V } is a Weyl canonical system, then, using Theorem 6.1.2 and the facts stated in $11.3, 2O, it is easily shown that the { p , q} defined by (6.2.2) form an infinitesimal canonical system. Conversely, given any infinitesimal canonical system { p , q}, the unitary operators defined by

u(x) = p i P ( r ) ,

qx‘) = ei4(S’)

(6.2.4)

form a Weyl canonical system {U, V}.5 In what follows, we shall most often be concerned with the case mentioned in Example 6.2.2. Let A be a complex inner product space, and denote its inner product by ( z , z’), x, x’ E A. Let J : x -+ x* be a one-to-one mapping from A onto A, satisfying the conditions (x*)* = x,

(ax

+ @)* = Ex* + by*,

a, /3

complex numbers,

and

(X*,Y*) = ( Y , 4. Then Jis called an involution in A. Let $j= {x I x* = x,x E A}. Then 5 forms a real inner product space with respect to the given inner product (x,y ) , x, y E 5. Moreover, every z E A can be uniquely expressed in the form z=x+iy,

x,yE$j.

A is said to be the complexiJication of 5. We note that involutions do exist in every complex inner product space A. I n fact, one need only choose an arbitrary complete orthonormal system in A, and let 5 denote the cIosure of the set of all real linear combinations of this system. Obviously, each z E A has a unique decomposition z = x iy, x,y E $5, and we set z* = x - iy.

+

Translator’s note: The verification of equivalence between the notions of Weyl canonical system and infinitesimal canonical system (as defined by the author) involves certain technical difficulties. Since the definition is, in any case, an ad hoc one, perhaps it might be better to simply say that { p ( x ) , q(x’)} is an infinitesimal canonical system provided that {eip(z’, ei9(*‘)} is a Weyl canonical system.

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COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

Then z ---t z* is an involution. Furthermore, given any real inner product space sj, there clearly exists a complex inner product space A which is a complexification of fi, and A is unique up to an isomorphism which leaves 4, invariant. Now, if 2 = (a, $3, B), where sj is a real inner product space with the inner product B , we form the complexification A of 8,and call it the state vector space of 2.

Definition 6.2.4. Let R be a complex inner product space, with inner product (z, z‘), z, z’ E A. Let { W ( z )1 a E A) be a family of unitary operators on a complex Hilbert space H , satisfying the von Neumann commutation relations

Suppose that the mapping z -+ W ( z ) ,when restricted to any finitedimensional linear subspace %Jl, is weakly continuous (with respect to the Euclidean topology on IIJZ). Then we say that {W(z)I z E A} is a (von Neumann) canonical system over the state vector space A. If sj is a real inner product space, with inner product B(x, x’), and A is the complexification of sj, and if { U , V > is a Weyl canonical system over 2 = (@, $3, B), form the system

Then {W(z)I z E A} is a von Neumann canonical system. Conversely, if (W(z)1 z E A} satisfies the conditions of Definition 6.2.3, choose any real linear subspace sj of A such that (x, y ) is real on $3, and such that R is the complexification of sj. Let V(x) = W(x),

V(x’) = W(ix’), x, x’ E fi.

Then, it is easily verified that { U, V }constitutes a Weyl canonical system over Z = (6, fi, B), B(x, x‘) = (x, x‘).

20 The General Form of a Canonical System Definition 6.2.5. Let { U k , Vk},k = 1, 2, be canonical systems, in the respective Hilbert spaces H , (k = 1,2), over the same single-particle

6.2. Quasi-Invariant Measures and Bose-Einstein Fields system

357

Z.If there exists a unitary operator Q from H , onto H , such that QU1(x)Q-’

=

UZ(x),

QV1(x’)Q-’

=

V2(x’)

(6.2.5)

for all x E 5, x‘ E $j‘, then we say that the systems { U , , Vk},k = 1, 2, are unitarily equivalent.s Insofar as their essential properties are concerned, any two unitarily equivalent canonical systems may be regarded as identical. Accordingly, we shall proceed to describe a certain type of concrete canonical system such that every canonical system is unitarily equivalent with one of this type, and thereby establish the general form of a canonical system. Let Z = (5,b’,B ) be a single-particle system, and let { U , V } be a canonical system over Zin the complex Hilbert space H . As in $3.4, we let 2l and 6 be the weakly closed operator algebras generated by { U ( x ) I x E 5) and {V(x’)I x‘ E B}, respectively. Clearly, both % and 6 are commutative.

Lemma 6.2.1. For every cardinal number n, there exists a projection operator P, E 2l‘ n 6’such that the restriction of 2l to P,H has uniform multiplicity n. Moreover, En P, = I . PROOF. Applying Theorem 2.4.3 to the commutative weakly closed operator ring ‘2l, we see that there exists a unique system of projections (P,}satisfying all the requirements of the Lemma, except possibly the condition P, €6‘.As in the proof of Theorem 3.4.1, we consider the mapping T(x’): A V ( x ’ )AV(x’)-1, A E Iu, --f

where x’ is an arbitrary element of 5’.Using relation (6.2.1), it is easily proved that T(x’)% = ‘3. By the uniqueness of the projections {P,}, it follows that T ( x ’ )P, = P, , whence P, E 6’. ] I n view of Lemma 6.2.1, we may, in the ensuing discussion, assume that % has uniform multiplicity n in H . When this is the case, we shall say that (U(x) I x E &} has uniform multiplicity n. Also, we shall restrict our considerations to the case where H is separable, this being sufficient for applications to quantum field theory. Theorem 6.2.2. Let Z = {$, b‘,B} be a single-particle system, and { U , V>a canonical system over Z in the separable Hilbert space H. Suppose that { U ( x ) I x E $j}has uniform multiplicity n. Then, there 23, p), where SZ is a linear subspace exists a linear measure space S = (0, of $”, L? 3 b‘,23 is the totality of weak Bore1 sets in SZ, p(Q) < OC), and S is quasi-invariant under b‘,such that { U , V }is unitarily equivalent This simply amounts to the unitary equivalence, in the usual sense, of the corresponding canonical unitary representationsof the group r(Z).

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COMMUTATION RELATIONS IN BOSE-EINSTEIN

to the canonical system For any E E Qn2(S),

FIELDS

{o,P} in en2(S)which is defined as follows. x E $5,

Q ( x ) 5 ( f ) = eif(*’5(f), x’) ,”I)*( dcL

(6.2.6) x’ E 8‘.

(f)

(6.2.7)

Here, for each x’ E fi’, x( f;x’) is a unitary operator-valued measurable function’ on S, such that, for any given pair of elements x1 , x2 E fi’, the relations

z ( f ;xi

+ xJ

= z ( f ;xi)

z ( f - x i ; xz),

~ ( f0); == 1

(6.2.8)

hold for almost allf. PROOF, Since H is separable, it is obvious that n K O . Now, 2l has uniform multiplicity n, that is, there exist closed linear subspaces H, of H , a = 1, 2,..., n, such that H = @ H , , each H, is invariant under ‘u, each restriction ‘u, = % I H , is maximal commutative, and all the ‘u, are unitarily equivalent. Since H , is separable, it follows from Corollary 2.4.9 that ‘u, has a cyclic element in H, . Choose a system of cyclic elements {e, E H , , a = 1, 2,..., n} which correspond with one another under the unitary equivalences between the a,. Consider the function

<

x,

4(.)

=

( We,

9

em).

It is easily verified that +(x) is a positive definite quasi-continuous function on 6.By Theorem 4.3.5, there exists a linear measure space S = (9,23, p), where 9 is a linear subspace of $j”,9 3 &’ and y ( 9 ) < 00, such that #(x) =

1

eif‘”)

dp(f).

a

Proceeding as in the proof of Theorem 3.4.14, one can show that S is quasi-invariant under fi‘, and that there exists a unitary operator Q from H to gn2(S)such that = QU(x)Q-I. T h e remaining assertions can then be established by following the proof of Theorem 3.4.5. ] Remark. Suppose we are given a linear measure space* S = (52, 23, p), where fi’ C 52 C $”, 123 is the totality of weak Borel sets in 52, p ( 9 ) < co, and S is quasi-invariant under $5’. Also, Iet there be given a system of

o(x)

’

That is, taking values in the group of unitary operators on n-dimensional complex Hilbert space (see Definition 2.4.4). Translator’s note: Assuming only that 9 is a linear subspace of $?A and that 23 is the totality of weak Borel sets in Sa, one can then deduce that the conditions of Definition 4.2.3 are necessarily satisfied.

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

359

unitary operator-valued measurable functions z( f ; x’), f E SZ, x‘ E $3’, satisf ing conditions (6.2.8). Then, it is easily verified that the operators d efined by (6.2.6) and (6.2.7) constitute a canonical system9 over Z in Qfl2(S>. Thus, Theorem 6.2.2 gives the general form of a canonical system in the case of uniform multiplicity.

{a6

Corollary 6.2.3. Under the hypotheses of Theorem 6.2.2, (i) if $3 = $3’ is a nuclear space, B(x, x‘) is a continuous inner product on $3, and !lJ is the completion of $3 relative to B(x, x’), form the rigged Hilbert space $3 c(nl c $3+. Suppose also that x + U ( x )is weakly continuous relative to the topology of $3. Then, we may choose SZ = $3+. (ii) If $3 = $3’ is an inner product space, with the inner product B(x, x’), and if x --t U ( x ) is weakly continuous relative to the topology of $3, then we may choose as SZ any Hilbert space containing Sj such that the inclusion map of $3 into SZ is a Hilbert-Schmidt type operator. PROOF, I n the proof of Theorem 6.2.2, instead of applying Theorem 4.3.5 to the positive definite continuous function #, we apply Corollaries 4.3.14 and 4.3.15, thereby deriving conclusions (i) and (ii),l0 respectively. ] T h e case of greatest interest to us is n = 1 and z ( f , x’) = 1 (this is the case which suits the requirements of quantum field theory).

Theorem 6,2.4. Let Z = ($3, a’, B ) be a single-particle system. Let SZ be a linear subspace of containing $3’, let 23 be the totality of weak Bore1 sets in L2, and let (a, 8,p k ) , K = 1,2, be finite measure spaces which are quasi-invariant under $3’. Suppose that { u k , vk}, k = 1,2, are canonical systems over 2 inL2(SZ, 23, pk),defined by uk(X)

[ ( f ) = eif(r)Hf),

(6.2.9)

x E $3,

~ ~ ( [(f) ~ ’ =1 [(f - x’) ( d p k r ‘ ( f ))1’2,

X’

E

$3’.

(6.2.10)

dpk(f)

Translator’s note: I t appears that some additional condition on S is necessary to ensure the weak continuity of V ( x ’ ) (on finite-dimensional subspaces of $’). lo Translator’s note: Whether or not part (ii) of Corollary 6.2.3 is true, it is hard to see how it can be deduced from Corollary 4.3.15. If 5’= $ is a real separable Hilbert space which is a dense linear subspace of another separable real Hilbert space Q, and if the imbedding of $ into Q is of Hilbert-Schmidt type, then by Corollary 5.3.3 it follows that there exists a finite measure space (Q, b,p ) which is quasi-invariant under 8.Thus, one can obtain a canonical system (6.2.6), (6.2.7), but it is not clear why it should necessarily be equivalent to the a priori given canonical system ( U , V}.

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Then the measures p1 and pa are equivalent if and only if the canonical systems {Ul , Vl} and { U 2 , Vz}are unitarily equivalent. PROOF. Assume that pl and pa are equivalent. Form a unitary operator Q, from L2(Q,23, p,) to L2(Q,23, pa), as follows: (6.2.1 1)

Then, using (6.2.9), (6.2.10), and (6.2.11), it is easy to check that (6.2.5) holds. Conversely, assume that Q is a unitary operator from L2(Q,8,pl) toL2(Q, 23,pz) such that (6.2.5) holds. Write &(f) = Ql. Then, for any finite set of elements xl, x2 ,..., x, E sj and any n complex numbers A, ,..., A, , we have

Introducing the finite measure

the foregoing relation may be written in the form

Now, since 23 is the smallest a-algebra with respect to which all the functions { f ( x ) I x E sj} are measurable, the algebra

constitutes a determining set on (Q, 23). Hence, by Corollary 1.1.6, ID is dense inL2(SZ, 23,pl p3). ConsequentIy, for any 5 eL2(Q,23, p , pa), we see that

+

+

6.2. Quasi-Invariant Measures and Bose-Einstein Fields I n particular, letting E E b,we obtain

5 be the

361

characteristic function of an arbitrary set

Thus, pl Q p 2 . Similarly, p2 equivalent. ]

< pl.

Therefore, pl and pa are

Definition 6.2.6. Let ,Z be a single-particle system, and {U, V } a canonical system over ,Z in the complex Hilbert space H. If no nontrivial closed linear subspace of H is invariant under all the operators U(x), V(x’),ll then { U , V } is said to be irreducible. [This is equivalent to the irreducibility of the corresponding canonical unitary representation of the group r(Z)]. Let A be a complex Hilbert space, and let U denote the group of all unitary operators on R. In quantum field theory, one must consider not only a canonical system { W ( z )I z E R} in some Hilbert space H , but also a unitary representation r : U -+r(U ) of U in H. This representation has the property

r(U ) W(z)r(U)-1 = W(Uz).

(6.2.12)

We note that, if { W ( z )1 z E R) is irreducible, then (6.2.12) uniquely determines r ( U ) up to a scalar factor a ( U ) , where U + a ( U ) is a representation of U in the group of complex numbers of unit modulus. I n fact, suppose that U 3 r’(U ) is another unitary representation of U in H such that

r‘(U ) W(z)a(U)-1 = W(Uz) for all z E R, and let To(U ) = r’(U)-lr( U). Then

for all z E R. Hence, To( U )E { W ( z )I z E R}’. Since {W(z)I z E R} is irreducible, To(U ) must be of the form a( U ) I , where a( U ) is a complex number of unit modulus.

Theorem 6.2.5. Let ,Z = (9,9’,B) be a single-particle system. Let Q be a linear subspace of $ A such that 9‘ C Q, let b be the totality of weak Bore1 sets in Q, and let S = (Q, 8,p ) be a finite measure space l1

Or, more concisely, { U ( x ) ,V(x’)I x E $, x’

E

$’}’ = {UI X a complex number}.

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COMMUTATION RELATIONS IN BOSE-EINSTEIN FIELDS

which is quasi-invariant under 8‘. Then the canonical system in L2(S) defined by U(x) S ( f ) = e i f ( q f ) , x E sj, (6.2.13)

is irreducible if and only if S is ergodic with respect to the group of translations sj’. PROOF. Assume that { U , V } is irreducible. If S were not ergodic, there would exist a quasi-invariant set E E 23 such that p ( E ) > 0, p(Q - E ) > 0. Let M = {[ I (( f ) = 0 for almost everyfE E}. Then M is a closed linear subspace of L2(S)which is invariant under all the operators { U ( x ) ,V(x’)},and L2(S) # M # (0).This contradicts the irreducibility of { U , V } . Conversely, assume that { U , V }is not irreducible, so that H contains a nontrivial closed linear subspace M which is invariant under all the operators U(x), V(x’). Let P denote the projection operator from H onto M ; then P E rU‘ n E’. Now, since {eif(r)I x E sj} is a determining set on S , it follows by Lemma 2.4.4 that !!Icontains the multiplication algebra mZ(S); but p is finite, hence, by Lemma 2.4.10, mZ(S) is maximal commutative, and therefore !!I = W(S). Hence 9l is maxima1 commutative, and so we have P E rU’ = !!I = %R(S).Consequently, there exists a bounded measurable function v(.) on S such that P is the multiplication operator corresponding to -q(.). Since P is a projection operator, it is easily seen that q(-) is equal almost everywhere to the characteristic function of some set E E 23. Moreover, using the fact that P EE‘, it is easy to deduce that E is quasi-invariant. Furthermore, since I # P # 0, # p ( E ) # 0. Thus, S is not ergodic. ] Recall that, in the case of finitely many degrees of freedom (i.e., when sj is finite-dimensional), all irreducible canonical systems are (by Corollary 6.1.6) unitarily equivalent. However, in the case of infinitely many degrees of freedom, the situation becomes very complicated. For then, the number of mutually inequivalent ergodic measures is very great indeed (for example, there is at least a continuum of mutually inequivalent Gaussian measures on a countably dimensional space S!). Consequently, even if we confine ourselves to those having the relatively simple form (6.2.13)-(6.2.14), there are still a vast number of unitarily inequivalent irreducible canonical systems. I n this context, the import of the following problem is clear.

,(a)

Problem. Let Q, sj be Hilbert spaces such that !jj is a linear subspace of Q and the operator imbedding $5 into Q is of Hilbert-Schmidt

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

363

type. Determine the general form of all measures on (Q, 23) which are quasi-invariant and ergodic with respect to fj.12 We shall now express the von Neumann commutation relations in terms of quasi-invariant measures. Let 52 be a complex Hilbert space with inner product ( x , x’), x , z’ E R, and write I x I = (2,~ ) l /Now, ~ . R may also be regarded as a real linear space, and is, in fact, a real Hilbert space with respect to the inner product [z, 2’3 = %(z, z’),

z, z’ E R.

We denote this real Hilbert space by sj to distinguish it from the underlying complex Hilbert space R. Let S = (SZ,%, p ) be a standard quasi-invariant measure space associated with fj (see 54.2). For each z E 52, let W ( z )be the unitary operator on L2(S)defined by

Theorem 6.2.6.

(6.2.15)

where (iz)(w), w E SZ denotes the quasi-linear functional13 corresponding to the element i z of 9. Then { W(x)I z E R} satisfies the von Neumann commutation relations. PROOF.For any 2, z’ E R, we have

(w4 u.’(z‘lf)(w) = exp

(-

+

[(iz)(w) (iz‘)(w

+ z)l)f(w + z + z’> (

dp(w)

(6.2.16)

Since (id)( .) is the quasi-linear functional corresponding to id, the relation (iz’)(w + z ) = (iz‘)(w) + [iz’, z] (6.2.17) la Translator’s note: Although it is not explicitly stated, the author presumably intends that 0, 5 are real and separable, that 8 is dense in 0, that b is the totality of weak Bore1 sets in 52, and that only finite measures are to be considered. l3 Translator’s note: Unless some additional hypothesis is imposed (see Lemma 4.2.4 and the remark following it), these functionals need not be unique. However, a system (6.2.15) satisfying the von Neumann commutation relations can- be constructed by choosing any subspace W of 0, which maps isomorphically onto 9’’ under the natural homomorphism (4.2.7), and, for each z E 9, letting (iz)(.)be unique antecedent of iz in W.

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holds almost everywhere. But [id, x] = %(iz’,z ) = 3(z, z‘), hence the required conclusion follows directly from (6.2.16) and (6.2.17). ]

30 Algebras of Observables Let Z = ($3, sj‘, B) be a single-particle system and { U, V} a canonical system over Z in the complex Hilbert space H. Let M be the weakly closed operator algebra generated by % u K. If (U, V } is irreducible, then M’ = {M I h a complex number), hence M = B(H). Suppose that { U , , V,} is another irreducible canonical system over Z, in another complex Hilbert space H,; in general, there will not exist a symmetric isomorphism q~ from B(H) onto B(Hl) such that v(U(x))= Ul(x), y ( V ( x ‘ ) )= Vl(xf). For, by Lemma 6.1.8, such an isomorphism must be of the form v(A) = UAU-l, where U is a unitary operator, that is, {U, V } and {Ul , V,} would have to be unitarily equivalent. Thus, it is apparent that the above definition of the algebra M is of little use. Accordingly, we proceed to reduce the size of M, in the following manner. Let %It, %Ituz’be finite-dimensional linear subspaces of $3, sj‘, respectively, and suppose that B(x, x’) is nondegenerate on (%It, %It’) (in particular, this implies that YJl and %It’ have the same dimension). We then say that the pair (%It, %Itf) is nondegene~ate.’~ For each nondegenerate pair of finitedimensional subspaces (%It, %It’), let Wmm*denote the weakly closed ring generated by the family of unitary operators { U(x),V(x‘)I x E %It, x’ EYJ~’} on H, and let 2B be the smallest uniformly closed operator algebra which contains the union of all such rings 2Bm,mf .

Definition 6.2.7. Let ,Z be a single-particle system, { U,V} a Weyl canonical system over Z in the Hilbert space H , and {p, q} the corresponding infinitesimal canonical system. Then, the operator ring 2B described above is called the concrete WeyZ algebra over Z (on H) associated with {U, V } (or with {p, q}). The operators in 2B are called the observables of Z.

Theorem 6.2.7, Let Z = (8, sj’, B) be a single-particle system, let { p , , q,}, { p , , q,} be infinitesimal canonical systems over Z in the Hilbert spaces Hl , H , , respectively, and let a, , 2& be the associated concrete Weyl algebras. Then, there exists a unique symmetric isomorphism from a,onto 912 such that, for any bounded Baire functionf, d f ( P 1 W ) =

f(PZ(4,

d f ( q , ( x ‘ ) ) ) =f ( q 2 ( 4 ) ,

x E $3,

x’

E

5’-

l4 Note that, since B(x, x’) is nondegenerate on (5, $’), it follows that, for each finitedimensional linear subspace 1112 C $, there exists a finite-dimensional linear subspace 901’ C such that the pair (1112, !W) is nondegenerate.

a’

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

365

PROOF. Let (1131, 1131’)(1131 C fi, 1131’ C 8’) be any nondegenerate pair of finite-dimensional linear subspaces. Choose bases {el ,..., e,} and {el’,..., e i } for 1131 andm’, respectively, such that, when x = CL1 x,e, €1131 and x f = CL1 x,’e,’ E‘W, B(x, x’)

For any t

=

(tl ,..., t,), s

= xlxl’

= (sl

+ + x,x,’.

,...,s,)

u7&) = exP(iP,(e,)

tl)

Vk(4 = exp(Me1’) 51)

***

E

..* **.

R, , form the operators exP(iP,(en)

tn),

exp(iq,(en‘) sn),

= 1, 2. Then { Uk(t) I 2 E R,} and { v k ( s ) I s e R,} are weakly continuous unitary representations of R, in H k , and satisfy the commutation relations (6.1.9), k = 1, 2. Let %B$,! denote the weakly closed operator ring on Hk generated by the family of operators { Uk(t), v k ( s ) 1 t , s E R,}. By virtue of Theorem 6.1.10, there exists a unique symmetric isomorphism prm,, from 2BG:,, onto 2B!$!m, such that, for every bounded Baire function f, ml,w,(f(P,(x))) = f(P&)), (6.2.18)

k

m,wlc(f(ql(x’)))= f(q&’)).

(6.2.19)

Let A denote the totality of nondegenerate pairs of finite-dimensional subspaces (YR,1131‘). If (%, %’), (mZ, 9X‘) E A and 9X C %, ‘33’ C W’, then we write (1131,1131’) < (%, %’); the set A is directed by this ordering relation. Now, ! l B ( k ) is the uniform closure of the ring

mp =

(J

m#,. .

(YJ31.W)€A

Clearly, if (a,3 ’ ) < (1131, W‘), then ?lBRhjC 2 B ~ ~ m . We m . assert that the restriction of F ~ , , to~ !Elk,&,is just rpw,w, . In fact, since (6.2.18) and (6.2.19) are, in particular, valid for all x E %, x’ E a’, it can be seen from the proof of Theorem 6.1.10 that the restriction of pm,mJ to !lB!&, is also a symmetric isomorphism from m#,&,to $2llg,&.such that (6.1.25) and (6.1.26) hold for all x E %, x‘ E %’. But Theorem 6.1.10 states that there exists just one such isomorphism, namely, q ~ ~ . ~ , ~ ~ , We can now define a mapping qo from ‘2Bf)to mi2),as follows. Given any A E 213h1’, there exists (ml,9X‘) E A such that A E 2BDl,m,. Define TOW) = m . m n ’ ( 4

Then rpo is a symmetric isomorphism from ‘ 9 3 ~ onto ) mi2’.Since ‘lxil),

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'r23h2' are dense in 'r23D(2),respectively, v0 can be extended to a symmetric isomorphism g, from 'r23(') onto by (6.2.18) and (6.2.19), g, satisfies the requirements of the theorem. Finally, the uniqueness of g, follows at once1&from the uniqueness of the g,w,ml. ] 40 The Characteristic Functional of a Canonical System $3, B) be a single-particle system, Definition 6.2.8, Let Z = (8, where Sj is a real inner product space and B( *, -)is the inner product on Sj. Let 3 be the complexification of $3, and let { W ( z )I z E R} be a von Neumann canonical system over 2 in the Hilbert space H . Suppose there exists an rlo E H such that { W(z)rlo I z E R) generates H ; then, the canonical system { W ( z )I z E S} is said to be cyclic, rlo is called a cyclic element for the system, and

d'(.)

= ( w ( zTO ) 9 TO),

9

is called the characteristic functional of the canonical system (associated with v0). Following the proof of Lemma 2.4.6, we easily obtain the following result.

Lemma 6.2.8. Let Z be a single-particle system, and let { W ( z )I z E R} be a von Neumann canonical system over Z in a separable Hilbert space H . Then, H can be decomposed into an orthogonal sum

where each H E is invariant under the system { W ( z )I z E R}, and the restriction of ( W ( z )I z E R} to H , is cyclic. By virtue of the above lemma, we may henceforth restrict our considerations to cyclic canonical systems. If two canonical systems are unitarily equivalent, and one of them is cyclic, then obviously the other is also cyclic, moreover, the characteristic functionals associated with corresponding cyclic elements are equal. Conversely, we have the following result.

Lemma 6.2.9. Let Z be a single-particle system, and let {W,(z) I x E R), K = 1, 2, be cyclic canonical systems in the respective Hilbert spaces Hk , k = 1, 2. Suppose that there exist cyclic elements l6 Translator's note: If, in the statement of Theorem 6.2.7, one adds the condition that 'p maps each q),m, onto B3#,m,,then the uniqueness follows easily, as asserted. Otherwise, it seems difficult to prove.

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

367

v k in Hk, k = 1 , 2, such that the associated characteristic functionah are equal. Then the two canonical systems are unitarily equivalent. PROOF. Let $(z) denote the common characteristic functional associated with ql and q2 . Since

for any z, x’

E

A, it follows that

(6.2.21)

Let Mk be the linear hull of the set of vectors { wk(z)q k , z E A}, k! = 1, 2, and let Q be the linear mapping from Ml onto M , defined by Q Wl(z)ql = W,(z) q 2 . By (6.2.21), Q is an isometry, and since Mk is dense in Hk (k = 1, 2), Q can be uniquely extended to a unitary operator from Hl onto H , . Moreover,

Consequently, for any

4 E M , , we have

This proves that QW,(x) Q-’ = W2(z),z E A. 3 Next, we establish a necessary and sufficient condition for a function to be the characteristic functional of some canonical system.

Lemma 6.2.10. Let R be a complex inner product space and $(z) a complex-valued function on A. Then, #(z) is the characteristic functional of a von Neumann canonical system in some Hilbert space if and only if: (i) # is quasi-continuous, and (ii) the function

368

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RELATIONS I N BOSE-EINSTEIN FIELDS

is a positive definite kernel, that is, for any finite set of complex numbers A, ,..., A, , and any n vectors z1,..., z, E A,

PROOF. Suppose that I/J is the characteristic functional of a canonical system { W ( z )I z E A}. Then, using (6.2.20), we see that the left-hand side of (6.2.22)is equal to

Thus, condition (ii) is satisfied. Furthermore, since W ( z ) is weakly continuous on any finite-dimensional linear subspace of A, it follows immediately that t,h is quasi-continuous. Conversely, assume that zL, satisfies conditions (i) and (ii). Let Ho denote the totality of complex-valued functions on A which vanish everywhere except possibly on some finite set of points x1 ,...,z, E A. Obviously, H, is a linear space relative to the ordinary linear operations. For any f,g E H, , define

(f,g)

=

C

[ f 3(z, z')] f(z) g(z').

#(z - 2') exp -

Z.Z'ER

(6.2.23)

If we identify elements of Ho whose difference f is such that ( f , f ) = 0, then (6.2.23) induces an inner product ( f , g ) on H, . Let H be the completion of H, relative to this inner product. For each zo E A, define an operator W,(z,) on H, as follows: (WO(ZOlf)(4

= exp

[; w, a,)]f(z - zo).

(6.2.24)

I t is easily verified that (WO(ZO)f?

Wo(z0)g)= (f!g).

I n particular, if ( f , f ) = 0, then ( Wo(z,)f, W,(?,)f) = 0. Hence, Wo(zo) is an isometric operator from H, into H, . But since W,(z,) Wo(-z,) = I , it follows that Wo(zo) maps H, onto H , , and can therefore be uniquely extended to a unitary operator W(zo)from H to H. Moreover, it is easy to calculate from (6.2.24) that the operators Wo(z),and hence also W(z),satisfy the von Neumann commutation relations. Next, we must show that Wo(z)is weakly continuous when z is

6.3. Gaussian Measures and Convential Free-Field Systems

369

restricted to any finite-dimensional linear subspace of A. Since Ho is dense in H , it suffices to show that, iff, g E Ho , then the function ( Wo(z)f, g) =

$(u

- u’) exp [-

f 3(u, u‘ - z)]f ( u - z)g(u’>

(6.2.25)

is quasi-continuous. Let

Then, (6.2.25) becomes

Since +h is quasi-continuous, it is clear that (6.2.26) is also quasicontinuous. Finally, let 7 E H, be defined by ~ ( 0 = ) 1, ~ ( x = ) 0 for x # 0. It is easily verified that is a cyclic element for the system {W(z)1 z E A>, and that (W(x)q,3) = +h(x). ]

56.3. The Relation of Gaussian Measures and Rotationally Invariant Measures to Conventional Free-Field Systems

l o The Fock-Cook Free-Field System We now proceed to formulate the Fock-Cook free-field system commonly used in quantum field theory. Let A be an infinite dimensional (complex) Hilbert space. Let R‘O) denote the one-dimensional Hilbert space formed by the totality of complex numbers with the inner product (A, p) = Aji, and let denote the tensor product 0R of n copies of R (see Appendix I1 for the terminology of tensor product spaces and operators in such spaces); in particular, A(1)= R. Let M , be the space of symmetric tensors of order n over R (n 3 l), Mo = R(O), and let

n:-,

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This is a Hilbert space. We consider the following closed linear subspace of 2:

Let X denote the linear subspace of 2 spanned by all the R(,), n = 0, 1,..., and let M = X n H; M is called the symmetric tensor algebra over R. For each x E R, form the operator C,(x) on M, as follows: for z E M, z, , z k E 5Vk),define z = zo z1

+ + +

where x 0 z k denotes the tensor product of the vector x and the tensor z k of rank K, so that x @ x k E W+l).Clearly, C,(x) is a linear operator from M to M, and acts as a bounded linear operator from M , to M,+, . We also form an operator C,(x) on M, defined as follows: for z E M ,

x

= 20

+ + ". f z,, z1

zk

E fi(k),

(6.3.2)

where zk-l

E

Mk-l is such that, for all y (4-1

I

Y) = ( Z k

I t is easily seen that such a z;-l Z = s k ( x 1 @ * * * @ X k ) , then

9

E

Sk(X

Mk-l ,

0Y ) ) .

(6.3.3)

exists and is unique. For example, if

C,(x) is also a linear operator from M to M, and acts as a bounded linear operator from M , to M,.-l (n 3 I), with C,R(O) = 0. T h e physical significance of these spaces and operators is as follows. M , (n 1) represents the vector space of all n-particle states, and Mo represents the vacuum vector space; C,(x)(C,(x))represents the operator corresponding to the creation (annihilation) of a particle with wavefunction x .

>

Lemma 6.3.1.

C,(x) and C,(X) are related by

6.3. Gaussian Measures and Conventional Free-Field Systems

37 1

PROOF. I t suffices to prove that (6.3.4) holds for X E M , , Y E M,. . Since C,(x) M , C M,,, , C2(x) M,, C M,,-, , it suffices, in fact, to prove (6.3.4) for x E M k , y E Mk+, . But in this case, C,(x)z = (k 1)1/2Sk+lx0z , hence, by (6.3.3),

+

(cl(x)z,y) = ( ( k

+ l)l’’

c sk+lx

0z,y> = (z, cZ(x)y)- 3

(6.3.5)

By virtue of (6.3.5) and the mapping properties of C,(x) and C,(x), it is easily proved that the closure C(x) of C,(x) exists, and that C(x)* [the adjoint of C(x)] is the closure of C2(x). We call C(x), C(x)* the creation and annihilation operators, respectively, corresponding to x. Set

Choose any real inner product subspace $3 of 53 such that si is the complexification of Sj. We shall prove below that { P ( x ) ,Q(x’) I x, xf E Sj} is an infinitesimal canonical system on ($3,$3, B ) ;we call it theFock-Cook system. We proceed to construct a canonical system of the form (6.2.13)(6.2.14) which is unitarily equivalent with the Fock-Cook system. Let S = (Q, 23,N ) be a standard Gaussian measure space, with parameter 1, associated with 6,and let { ~ ( ux) E , $3) be the corresponding standard Gaussian process. We now define a linear mapping D of H into L2(S), as follows: arbitrarily choose a complete orthonormal system {g,} in $3; then, by Lemma 11.2.4 of Appendix 11, the vectors of H may be expressed in the form x=

1

akSlklcg:log,k” 0

k=(k,...-.k,)

I uk 1’

= 11

--*

Og?)(l k I !/k!)l’z,

1 ’ < 03*

We define16 Dx(w) =

1...

k=( k l ,

u k h k ( g l ( w ) ~ * * *g!n ( w ) ) .

,k,)

By Lemma 5.4.6, we have

11 DX 1 ’

=

1I

uk

1’

11

11’9

and, since {hk} is complete, D ( H ) = L2(S),thus, D is a unitary operator l6

Here, as in $6.1. we denote hL(u1,..., u, ; 1) by hk(u1,..., u-).

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FIELDS

from H to L2(S).It can be shown that this unitary operator D is independent of the choice of the complete orthonormal system {g,}. We denote by ‘p the totality of polynomial functionals of the form P ( g l ( w ),..., g,(w)) [where P(tl ,..., tn) is a polynomial in t , ,..., t,]; then ’$ is a dense subspace of L2(S).Also, denote by H(m) the closed linear subspace spanned by all functionals of the form hk(gl(cu),...,g,(w)), I k I = m; we call H(m) the m-particle space. In fact,

H(m) = DM,

.

We now set up a representation of the commutation relations analogous to (6.2.13) and (6.2.14). Namely, iffEL2(S), let ( U ( x ) f ) ( w )= eaZcm)f ( w ) , (V(x’)f)(w)= exp [x’(w) - &(x’, x‘)] f ( w - x’).

(6.3.6) (6.3.7)

Note that this V(x‘)can also be written as (6.3.8)

( V(X’)f) ( w >

From (6.3.6) and (6.3.81, it is easily seen that (U(x), V(x’) 1 x, x’ E $1 satisfies the-Weyl ’commutation relations (6.2.1), and that U , V are weakly continuous on any finite-dimensional linear subspace of $. We note that, in analogy with (6.1.32), the operator a dt

may be expressed as follows:

W P ( 4 = {p’ I P EL2(% x(*) d.) E w P ( X ) 9J(w> = x ( w ) PP(W)I

9JE

m I

Ww).

(6.3.9)

However, the expression corresponding to (6.1.33) is more complicated. Using again the complete orthonormal system {g,}, and the notation of (6.2.2), we let p , = p(g,), qm = q(gm).Then, by (6.3.9), we have

l W”)

= {p’ I 9J E w 3 9 g u ( * ) d.1EL2(S)l, PuPbJ) = g d w ) ?+)I PE W P J ,

and, in analogy with (6.1.33), qy is the closure of the operator qy’, where B(qy’)= ‘p and %’p’(W)

=

-k&)

dw)

a +idw), agu

P E cp.

6.3. Gaussian Measures and Conventional Free-Field Systems

373

Set 1

C”‘

= -( p ,

42

+ iq”),

1

c,*‘ = d z ( P , - iq,);

and let c, , c,* be the closures of c,‘, c,*‘, respectively. Then c,* is the adjoint of c, ,and, corresponding to Lemma 6.1.1 1, we have the following result.

Lemma 6.3.2. T h e operator c, maps H(m) into H ( m bounded on H(m),and satisfies

where k and k’ are related as in (6.1.36). Moreover, H ( m - 1),17 is bounded on H ( m ) , and satisfies

c,*

+ l),

is

maps H(m)into

where k and K‘ are related by (6.1.37). Using (6.3.1), (6.3.2), Lemma 6.3.2, the boundedness of c,on H(m) and the boundedness of C(g,) on M , , one can calculate that c, = DC(g,) D-1.

Similarly, c,*

I n general, writing c(x)

=

=

DC(gy)*D-1.

(1/42)(p(x)

c(x) =

+ iq(x)), we have

DC(x)D-1,

c(x)* = DC(x)* D-I.

Consequently,

p ( ~= ) D P ( x )D-l,

q(x) = DQ(x)D-’.

Thus, since { p , q} is an infinitesimal canonical system, it follows that (P,Q} is an infinitesimal canonical system, moreover, the Weyl form of the Fock-Cook system is unitarily equivalent to (6.3.6)-(6.3.7), via the unitary transformation D.Since the standard Gaussian measure space is ergodic, it follows as in the proof of Theorem 6.2.5 that the Fock-Cook l’

Here, H ( - 1) is to be understood as meaning (0).

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FIELDS

system is irreducible. Corresponding to Theorem 6.1.13, one can also prove the following result.

Theorem 6.3.3. Let P , denote the projection of H onto M,; then, for any real number a, W

C

r(&I) =

m

(6.3.10)

m=O

is a unitary operator on H , moreover, if { W(x)I x E R} is the von Neumann form of the Fock-Cook system, then r(eiaI)W(x)r(eial)-l = W(e%). In other words, (6.3.10) defines a gradient transformation on the FockCook system. It is easily seen that the Fock-Cook system is cyclic, with the vacuum state 1 as cyclic element. We now write out the characteristic functional of this system corresponding to the cyclic element 1. To do this, it suffices to consider the unitarily equivalent system defined by (6.3.6)(6.3.7). Then

and we may suppose that x # 0. Now, there exist mutually independent Gaussian variables U(w), V ( w ) , each having mathematical expectation zero and variance 1/2, and such that XbJ)

= I1 x I/ U ( w ) ,

Substituting (6.3.12) into (6.3.1I), we obtain (W(Z)L1)

- - ( y ,y ) -

‘i( x ,Y )

exp[-t(ll x 112

+ IIY 11”l.

1 2

=

+ VP,]

- ( u2

6.3. Gaussian Measures and Conventional Free-Field Systems Therefore, the characteristic functional of the Fock-Cook corresponding to the vacuum state 1, is

375 system,

We next proceed to formulate a class of canonical systems, similar to the Fock-Cook system, and also describable in terms of Gaussian measures. Let R be a complex Hilbert space, let $3 be the real Hilbert space formed by A with respect to the inner product [z, z’] = A(z, z’), and let S, = (Q, 23, N,) be a standard Gaussian measure space associated with $3. Using (6.2.15), for each z E R, we form the unitary operator

(6.3.14)

in the Hilbert space L2(S,).Following the proof of Theorem 6.2.6, we see that { W,.(z) [ z E R) satisfies the von Neumann commutation relations; moreover, one may verify that the correspondence z -+ W,(z) is weakly continuous on any finite-dimensional subspace of R. Therefore, {W,(z) I x E R} constitutes a canonical system. Furthermore, it is easily shown that it is irreduciblelO and cyclic, with 1 as cyclic element, and that the corresponding characteristic functional is (WC(z)1, 1)

=

1

R

exp

[-

i

1 (iz)(w)- z(w) C 2c

dNc(w). (6.3.15)

Since [z, z] = [iz,iz] = [ j z

the two Gaussian variables (6.3.15) becomes

112,

z(w),

[z, iz] = [iz,z] = %(z, iz) = 0,

(iz)(w)are mutually independent, and

(6.3.16)

I n particular, when c = 2, formula (6.3.16) reduces to (6.3.13). Thus, lo Translutor’s note: Details would be welcome, since the proof of Theorem 6.2.5 is not applicable in this case.

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by virtue of Lemma 6.2.9, the Fock-Cook system is a special case of the type under consideration. We also note that, if one sets

and writes W,(x) in place of W,(x) in (6.3.16), then (6.3.16) can be written in the form (W&)l, 1)

20

A

= exp(-

II z 112 A),

a
00.

(6.3.17)

More General Type of System

Let R be a complex Hilbert space, let U be the Definition 6.3.1. group of all unitary operators on R. Let {W(z)I x E R} be a canonical system over R in the Hilbert space H , and let I' : U -P I'(U) be a unitary representation of U in H , satisfying the condition

r(U ) W(z)r(U)-1 = W(VZ).

(6.3.18)

Suppose, moreover, that ( W ( z )I z E R} is cyclic and that there is a cyclic element 7 such that 1'(U)q = 7 for all U E U. Then we call { W(x)I z E R} a conventional free-jield system, and 7 a vacuum state vector. We observe that the system {W,(x) I x E R} defined by (6.3.14) is a conventional free-field system. I n fact, letting B denote the linear space spanned by the totality of vectors of the form Wc(z)l,Z E ~ it, can be shown by methods similar to those used in 55.4 that 2)is dense inL2(Sc). For each U E U, we define a linear operator P( U) on B as follows:

T(U)WC(X)l = Wc(Uz)l.

(6.3.19)

By (6.2.20)and (6.3.16) we have, for any x, x' E A,

(r(u)Wo(413 T(U)WC(41)

= (Wc(z')L

u.',(Z)l).

Hence, P ( U ) is an isometric mapping. Since I'(U)-l = I'(U-l), r ( U ) can be uniquely extended to a unitary operator on L2(Sc);we also denote this extension by U). From (6.3.19), it is easily seen that U r(U) is a unitary representation of the group U,and that (6.3.18)holds. I n this case, 1 is a vacuum state vector.

r(

--f

6.3. Gaussian Measures and Conventional Free-Field Systems

377

We shall now investigate the general form of the characteristic functional of a conventional free-field system corresponding to an arbitrary vacuum state vector. For this purpose, we first prove the following lemma, which provides a characterization of rotationally invariant measures.

Lemma 6.3.4. Let & be an infinite dimensional real inner product space, and let +(.$),.$ E 5,be a positive definite quasi-continuous function on & such that the value of $ ( f ) depends only upon 11 g 11 = (6, E)l/2 [here (., .) denotes the inner product in 931. Then there exists a finite measure rn on [0, 00) such that

PROOF. Since +(() depends only upon 11 5 11, it may be written in the form ~ ( 1 1 f 11). Arbitrarily choose a unit vector e in &; then q(t) = +(te), t 2 0, is a continuous function oft, hence +(t) = (~(11 6 11) is a continuous functional on &. If we choose any orthonormal sequence of vectors {en, n = 1, 2, ...} in &, then $(fiel fne,) is a positive definite continuous function of the real variables t1,..., . Let I be the totality of real number sequences, and let 23 be the a-algebra generated by the family of all Bore1 cylinders in I; by the Kolmogorov theorem (see Corollary 1.3.5’), there exists a finite measure P on (I, 23) such that

+ +

+(tiel

+ + a * *

tnen) =

tpv)dp(x),

S,exp(i

x = (xi

,.-, x n ,.--)-

”4

(6.3.21)

Let s2, be the unit sphere in n-dimensional Euclidean space, and write

Denote the coordinates of the points of Qn by forming to spherical coordinates, we have

The element of area on f i n is dQ,(w) = sinn-2vl

w = (wl

*..

,..., w,);

trans-

sin ~ p ~ - ~ d a 7dvn-l ~

,

378

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN

FIELDS

and the total area of SZ, is 2 ~ ~ l ~ / l r ( nwhich / 2 ) , we denote by I SZ, set 5, = pwy , v = 1, 2,..., n, then the value of the integral

1. If we (6.3.22)

depends only upon p and T , hence, in the evaluation, we may choose ,...,xn) = ( T , O,...,0). Therefore, (6.3.22) becomes

(xl

--.dQ,(w)

Denote the above quantity by #,(p~). Applying the integral J., to both sides of (6.3.21), we obtain

where u,(T) = P({x I CL1x,2 < 7”). Make the substitution 7 = l / nu on the right-hand side of (6.3.23); since u,(l/iu), n = 1, 2,..., is a uniformly bounded sequence of monotonically increasing nonnegative functions on [0, co], it follows by Helly’s theorem that there exists a subsequence (a,,(d/nku)} which converges at all points of [0, a]to a bounded monotonically increasing nonnegative function a(u). On the other hand, sincez0

*O

Here, we use Stirling’s formula

1-;

(-

4)

cos PTV dw

-+

r(s + 1)

( 2 7 r ~ ) ~ ~ * s ~ e s- ’ ,-+

N

exp

cos p r dv ~

W.

6.3. Gaussian Measures and Conventional Free-Field Systems

379

therefore,

Similarly, one can show that21 there exist positive numbers c, cf such that I 4 J d i i p ~ ) I c exp(--Cfp2T2). Hence, taking n = nk in (6.3.23), and letting k + 00, we obtain

<

where m(t) = ~ 4 ( 2 t ) ' / ~ ]) > . If the characteristic function of a measure is of the type described by Lemma 6.3.4, then this measure is said to be rotationally invariant. I t can be seen from (6.3.20) that any rotationally invariant measure is a superposition of Gaussian measures.22

Theorem 6.3.5. Let R be a complex infinite-dimensional Hilbert space, { W ( z )1 z E A} a conventional free-field system, and 7 any vacuum state vector for this system. Then there exists a finite measure m on co) such that the characteristic functional of {W(z)I z E R) corresponding to 7 is

[t,

(6.3.24)

PROOF. Since r(U)7 = 7, the characteristic functional +(z) has the following property: for U E U, z E R,

Now, if 11 z 11 = 11 zf 1 , there must be some U E U such that z' = Uz, hence, by (6.3.25), I,!J(z) depends only upon 11 z 11. We choose any real linear subspace $j of R such that $jis a real inner product space and R is the complexification of 9. If z is restricted to 5, the function #(z) stiI1 depends only upon 11 z 11. Since 9 is also infinite-dimensional, we know that there is a finite measure m on [0, CQ) such that (6.3.20) holds for all 5 E 9. But since +((), on the entire space R, depends only upon 11 5 11, *l Translator's note: Actually, it suffices to show that the convergence of + , , ( z / ; T p ~ ) is uniform with respect to 7 . See Umemura [ 13 for further details.

380

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

it follows that (6.3.20) holds for all f E R. Using the condition in Lemma 6.2.10 and carrying out a rather intricate calculation, it can be proved that23

M O , t))= 0, whence we obtain (6.3.24). ]

APPENDIX

BACKGROUND MATERIAL ON TOPOLOGICAL

GROUPS AND LINEAR TOPOLOGICAL SPACES

51.1. Pseudometrics, Convex Functions, and Pseudonorms

Definition 1.1.1. Let R be a set, and let p ( x , y ) , x, y E R , be a realvalued function on R x R, satisfying the following conditions: > p(x, y ) 2 0, p(x,). = 0; ( 4 p(x, r) p(x, 4 p (y, 4. Then (i) p is called a pseudometric on R . If R is also a group, and p(y, x ) = p(x-ly, e ) for all x, y E R, then p is said to be a left-invariant pseudometric on R . Right-invariance is similarly defined. Let lpo,,(Y E 'Ir} be a family1 of pseudometrics on R. T h e topology generated by the totality of sets of the form

+

<

{X

I per(X,Y) < E, k

=

1, 2,..., n},

E

a,

E

>0

Naturally, this subsumes the case in which 'u contains only a single index. This case will not be singled out for special attention in the ensuing discussion.

381

382

APPENDIX

I.

TOPOLOGICAL GROUPS A N D LINEAR SPACES

is called the topology induced by (p, , a E a}.The set R, provided with this topology, is said to be a pseudometric space. If R is a group, and p is a (left and right) invariant pseudometric on R, then R, given the topology induced by p , becomes a topological group.

Definition 1.1.2. Let G be a group, and let N ( g ) , g E G, be a real-valued function on G, satisfying the following conditions: (i) 03 > N ( g ) 2 0, N ( e ) = 0 ; (ii) N ( y x - l ) N ( x ) N ( y ) . Then N is called a convex function (or, in some literature, a pseudonorm) on G. It is easily verified that any convex function also satisfies the condition: (iii) N ( x ) = N ( x - l ) . Given a convex function N on a group G, we introduce the function

<

+

P(g, 4 = W-lg)

(1.1.1)

on G x G. I t is easily verified that p is a left-invariant pseudometric on G; we call it the pseudometric corresponding to the convex function N . Conversely, given a left-invariant pseudometric p on G, the formula N ( g ) = P(g* 4 defines a convex function on G, and p is just the pseudometric corresponding to N . A similar discussion applies to right-invariant pseudometrics. Suppose that { N , , 01 E a}is a family of convex functions on a group G, and that (pa, 01 E Qf) is the corresponding family of (left-invariant) pseudometrics. In such a case, the topology induced by (pa, a E a}may also be called the topology induced by {Na, a E a}.

Definition 1.1.3. Let R be a linear space (which is, of course, a group with respect to addition) and let N be a real-valued function on R. (i) If N is a convex function and, for any x E R and number t , W x ) =It

I N(4,

then N is said to be apseudonorm on R. (ii) If N is a convex function and satisfies the condition N(tx) < N(4,

It I

< 1,

then N is called a homogeneous convex function, and the corresponding pseudometric is said to be a homogeneous pseudometric. (iii) If N satisfies N(h) 03, N ( g h) N ( g ) N(h), N(th) = the conditions: 0 I t I N ( h ) , then N is called a conwex functional. Proceeding directly from the definition of a linear topological space, one can easily prove the following lemma.

<

<

+ <

+

I. 1. Pseudometrics, Convex Functions, and Pseudonorms

383

Lemma 1.1.1. Let R be a linear space, p an invariant pseudometric on R . Suppose that p also satisfies the following conditions: (i) if {A,} is a sequence of numbers converging to zero, and x E R, then lim p(A,x, 0)= 0;

n+ m

(ii) if {A,} is a bounded sequence of numbers, {h,} C A, and lim p(h, , 0) = 0, n-w

then limn+mp(X,h, ,0) = 0. Then, R is a linear topological space relative to the topology induced by p. We call this kind of linear topological space a linear pseudometric space. I n particular, if the pseudometric p is a metric, we call R a linear metric space. Notice that, if p is homogeneous, then condition (ii) of Lemma I. I. 1 is automatically satisfied. T h e next lemma provides a method by which, starting from certain invariant pseudometrics, one may construct pseudometrics satisfying the conditions of Lemma I. 1. I.

Lemma 1.1.2. Let 6 be a real linear space, and let M(h), h E 6 be a nonnegative bounded functional on 6, satisfying the following conditions:(i) for any h, , h, E 6, M(h, h2) M(hl) M(h,); (ii) M(0) = 0, and, for any h E 6, M ( t h ) is a continuous function of t , -co < t < co; (iii) for any h E 6, M(-h) = M(h). Let

+

+

<

Then, 6 is a linear pseudometric space with respect to the pseudometric R(h1 - h,), hl , h, E 6. PROOF. By virtue of condition (i), we clearly have

R(h1

+ h,) d R@l) + R(h,),h,

I

42 E Q.

By condition (iii), we have R(h) = R(-h). Now, take any h E 6, and let {t,} be any sequence of real numbers converging to zero. If M is an upper bound for M(h), we have ~ ( t , h ) z= R ( It , I h)2

< <

sA

e-tM(t I t ,

I h)2 dt

0

max

O=Gt=GAltnl

M(th)2

+ M2e-A

+ M2e-A.

Using condition (ii), we deduce that limn+mR(t,h)

=

0.

384

APPENDIX

I.

TOPOLOGICAL GROUPS AND LINEAR SPACES

Next, suppose that {h,} C 8 , R(h,) -,0, and let {t,} be any bounded sequence of real numbers. Then, (I. 1.2) + 1)hn) + R(hn)* Let c be a positive constant such that (I t, I + 1) < c for all n, and let h R(tnhn) Q R((I

tn

I

be an arbitrary positive number. Then,

< eAj

A

e-$M(th,J2dt

+ cM2exp

0

+

Q eA ~ ( h , ) 2

Since R(h,)

---t

exp

C M ~

(-

1

-A). c

(1.1.3)

0, it follows from (1.1.2) and (1.1.3) that

E R(tnh,J2G cM2 exp (-;A). n-, m

Letting h -+ co, we conclude that R(t,h,) = 0. Thus, by Lemma 1.1.1, 8 is a linear pseudometric space relative to the pseudometric R(h, - h,), h, , h, E 6. ] 81.2. Some Properties of Semicontinuous Functions

l o The Concept of a Semicontinuous Function First, we introduce the following notations. If f ( x ) is a real-valued function on a set M, and B a subset of M, we write

w,4

=

supf(x), X€B

I(!, B ) = inff(x). XEB

Now, suppose that M is also a topological space, and x,, E M. Let Vz, denote the neighborhood system at the point xo . The numbers

.

are called, respectively, the upper and lower bounds off at x, If S(x, ,f )= f ( x o ) , we say that f is upper semicontinuous at x,; if I(xo ,f)= f ( x o ) , we

1.2. Some Properties of Semicontinuous Functions

385

say that f is lower semicontinuous at x, . Iff is upper (lower) semicontinuous at every point xo E M , we say that f is upper (lower) semicontinuous on M . For any point x E M , we write w ( x ,f) = S(x, f ) - I(x,f ).

Lemma 12.1.

For each positive number c, the set (x 1 w ( x , f ) < c)

is open.

+

PROOF. Suppose w(xo,f) < c. Then, S(x,, f)< 03, I(xo,f ) > - 03. Hence, given any positive number c < Q(c - w ( x o ,f)), there exists a neighborhood V of x, such that

+

S(f, V ) < S(x0 ,f)

€3

Z(f3

V )>

,f)- E .

Therefore, if x E V , then

Thus, V C (x I w ( x ,f ) < c}. ]

Lemma 1.2.2. Let M be a topological space of the second category and f an upper semicontinuous function on M such that, for every point x, E M, I ( x , ,f ) > -a. Then M contains at least one continuity point off. PROOF. Let c be any positive number. We assert that the open set 0, = {x I w ( x , f ) < c} is dense in M. In fact, given any x, E M , we have I ( x , ,f ) > - 0 3 , hence there must exist a V E Vzosuch that I ( f , V ) > -a.Therefore, for any neighborhood U of x, such that U C V , we have I ( f , U ) > -a.Obviously, we may assume that I ( f , U ) < 03, and hence that f ( x > < Z(f9 U )

+c

(1.2.1)

for some point x E U. On the other hand, Z(f, U ) < Z(x,f)
=

S(x,f).

(1.2.2)

From (1.2.1) and (1.2.2), we get

that is, U intersects 0, . Thus, 0, is dense in M . The set {x I w ( x ,f ) 2 c}, being the complement of the dense open set 0, , is nowhere dense. Since M is of the second category, it follows that M # (J:=l {x I w ( x , f ) 2 lln). m Hence, the set O1ln = {x I w(x, f) = 0} is nonempty. But the points of this set are just the continuity points off. 3

on=,

386

APPENDIX

I.

TOPOLOGICAL GROUPS AND LINEAR SPACES

Lemma 1.2.3. Let 8 be a linear topological space of the second category, and let M(h) be a nonnegative, lower semicontinuous functional on 8 , satisfying the following conditions.

+

<

+

For any hl , h2 E 8, M(hl hz) Wh1) M(hJ. For any h E 8, M(h) = M(-h). For any h E 8, limn+mM(h/n) = 0. M(h) is continuous on 8. PROOF.Given any positive number E , form the set E, = {h I M(h) E}. Since M(h) is lower semicontinuous, E , is closed. Now, by condition (iii), for any h E 8, there exists an integer n such that M(h/n) E , that is, h E nEf . Hence, (i) (ii) (iii) Then,

<

<

u m

8

=

nE,.

n-1

Therefore, since 8 is of the second category, there exists an n such that

nEf is not a nowhere dense set. Moreover, 8 being a linear topological space, the mapping h -P h/n is a homeomorphism of 8 onto itself, consequently, E, itself cannot be nowhere dense. As E, is closed, it follows that E, contains a nonempty open set V. Let U = {x - y I x, y E v>; U is a neighborhood of 0 in G. By condition (i), when h =x--yEU(X,yEV),

we have M(h) d M ( x )

+ M ( y ) < 2E.

+

Now, for any xo E 8,xo U is a neighborhood of xo , and if x E x, then, by condition (i) and (1,2.3),

I M ( x ) - M(x,)l

(1.2.3)

+ U,

< M(" - xo) < 2E.

This proves that M(h) is continuous. ] 2O

Sequences of Lower Semicontinuous Convex Functionals on Banach Spaces

We limit the statement and proof of the following result to the case which finds application in the present book. The reader is referred to Gel'fand and Vilenkin [I] for a more general formulation.

n = 1, 2,...1 Theorem 1.2.4. Let @ be a Banach space, let {pn(v), be a sequence of lower semicontinuous (but not necessarily everywhere finite) convex functionals on @, and suppose that, for every V E @, there is an n such that p,(q) < co. Then, there exists an no such that

1.3. Countably Hilbert Spaces and R+ged Hilbert Spaces

387

pn0(v)is finite and

continuous on @. I n other words, there exists an no and a positive number c < m, such that Pn,(P)

where 11

< c I1 P 11,

P E 0,

- 11 denotes the norm in @.

<

PROOF. Consider the closed sets A, = (9, I p,(v) I} in @. For each v E @, there is an n such that p,(p') < 00, and hence, for some positive integer k, pn(v) k, that is, E kA, . Thus,

<

m

0 = (J (kAn). k.n=l

Now 0,being complete, is of the second category, hence, for some k and n, kA, is not a nowhere dense set. Therefore, since kA, is closed, it contains an open ball {p' I 11 p' - v0 11 < e}. Hence, if $ E @, 11 $11 < e/k, then q0 k$ E (p' I 11 p' - vo 11 < e} C kA, , whence it follows that $ + (1/k)qoEAn,andso

+

Therefore, for any $ E @, we have

91.3 Countably Hilbert Spaces and Rigged Hilbert Spaces

lo Countably Normed Spaces Definition 1.3.1. Let @ be a linear space and (11 q~ lln, n = 1, 2, ...} a sequence of pseudonorms on @. Introducing the topology determined by the pseudometric (1.3.1)

@ becomes a linear pseudometric space. We say that @ is a countably pseudonormed space, with the topology determined by the sequence of pseudonorms {[I 9 1,( , n = 1, 2 ,...}. We may, without loss of generality, assume that

388

APPENDIX

For, we may replace

I.

{I1 v 1, , n

TOPOLOGICAL GROUPS AND LINEAR SPACES =

1, 2,...} by

the pseudonorms {[ y },I satisfy condition (1.3.2), moreover, it is easy to verify that the pseudometrics derived from the respective sequences {I v },I and (11 v ,}[I in accordance with (1.3.1), define the same topology. Also, notice that, in the topology of Definition 1.3.1, a sequence {y,} C @ converges to y E @ if, and only if, for every natural number k,

Similarly c{p),

is a fundamental sequence in @ if, and only if, for every k,

Definition 1.3.2. Let @ be a linear space, and let 11 Ill and 11 112 be two norms on @. Suppose that, for any sequence {y,} C @ which is fundamental with respect to both 11 * [ I 1 and 11 /I2 , we have 11 y , Ill -+ 0 if and only if (1 y , 112 -+ 0. We then say that the two norms 11 and 11 are compatible. Let @ be a linear space, (11 1, , n = 1,2,...} a sequence of norms on @, satisfying condition (1.3.2), and such that any two norms 11 * Ilm and 11 11% are compatible. Then @, with the metric topology defined by (11 ) ,1 in accordance with (1.3.1), is called a countably normed space. Obviously, countably normed spaces form a special class of countably pseudonormed spaces. Now, construct the completion Q1of @ with respect to the norm 11 * ( I . Consider those elements T E Q1 for which there exists a sequence {vn} C @ such that

-

-

-

-

(1.3.3)

Denote the totality of such elements cp by Q2; clearly, G2 is a linear subspace of G1. For any rp E Q 2 , choose a sequence {y,} C @ satisfying (1.3.3). Then, (1.3.3) implies that limm,n-too I II vn 112 - 11 vm112 I = 0. We define (1.3.4) II v 112 = II Vn 112.

kz

The value of 11 p (I2 is independent of the choice of the sequence I n fact, suppose that {#,> C @ is another sequence such that

{v,}. (1.3.5)

1.3. Countably Hilbert Spaces and Rigged Hilbert Spaces

389

Then, by (1.3.3) and (1.3.5), the sequence {I/Jn - Q I ~ }is fundamental with respect to 11 1 1 2 , and hence, by condition (1.3.2), also fundamental with respect to 11 * Ill , moreover, limn+m11 t,hn - Q I Ill~ = 0. Since 11 * /I2 and 11 \I1 are compatible, it follows that limnjro I(I/Jn - yn = 0. Therefore, Iim, jm I( t,h, (I2 = limn+m[I Q I I(2~ . I n particular, when QI E @, we may choose qn = QI, n = 1 , 2,..., so that the norm defined by (1.3.4) coincides with the original norm 11 QI Furthermore, if QI E Q2 and {yn> C @ satisfies (1.3.3), then

-

II v - Vn 112 = iz I1 Vm - v n I12

9

hence, (I. 3.6)

lim /I F - yn /I2 = 0. n-tm

This shows that @ is dense in Q2 . Moreover, we assert that Qj2 is complete with respect to the norm /I 1 1 2 . I n fact, suppose that {I/Jn} C Q2 is fundamental with respect to 11 )I2. Since @ is dense in G2, we may choose elements Q I E~ @ such that

-

II $n

- Vn

-

1

112

< pt

3

n = 1, 2 9 . a .

-

(1.3.7)

Then, {yn> is also fundamental with respect to (1 (I2, and hence fundamental with respect to 11 * ( I l . Consequently, there exists a QI E Q1 such that (1.3.3) holds. But this means that QI E Q 2 , and it follows as before that (1.3.6) holds. Hence, in view of (1.3.7), we get lim

n-m

II v - 4 n 112 = 0,

which proves that Q2 is complete. Thus, O2 is a completion of @ with respect to the norm (1 1 1 2 . By repetition of the preceding arguments, we arrive at the following result.

-

Lemma 1.3.1. Let @ be a countably normed space, with norm sequence {I1 l l n , n = 1,2,...}. Then, a completion Qjn of @ with respect to (1 [In can be chosen for every n in such a way that

-

-

3 a23 ... 3

Lemma 1.3.2.

3

**.

3 @.

@ is complete if and only if 0)

Qn.

Q, =

(1.3.8)

n=l

PROOF. Assume that @ is complete. Choose any QI E

(-)El. Since @

390

APPENDIX

I.

TOPOLOGICAL GROUPS AND LINEAR SPACES

is dense in @" with respect to the norm an element cpn E @ such that

there exists, for every n,

1

Iln < ;'

II 'p Then, for 1, m

11 - I,l

% I

> n, we have

Hence, the sequence (vl}is fundamental with respect to 11 * 1, for every n, that is, {vl} is a fundamental sequence in @. Therefore, there exists a $ E @ such that

:;\

I1

*

- 'pz Iln = 0,

from which it follows that # = v, and hence v E @. Thus, (1.3.8) holds. Conversely, assume that (1.3.8) holds. If {vm} is a fundamental sequence in @, then {y,J is also a fundamental sequence in @,, (with respect to 11 * ) ,1 for every n. By the completeness of @", there exists rp(") E @" such that (1.3.9) lim 11 'prn - qdn) \In = 0. m+ w

But

@n+l

C Qn , hence q+mf1)

(1 , p + U

- P(")IIn

E

@"

, moreover,

< II ~m - v'~+')Iln + II ~ r -n v ' ~ ) < II ~ r -n v'~+')IIn+l + II v m - Iln (In

v(")

+

0.

Therefore, rp(") = v("+l),n = 1, 2,,.., so that all the elements y(") are @", and (1.3.8) identical and may be denoted by v. Since q E holds, we have v E @. Moreover, by (1.3.9), the sequence {vm} converges to v in @. Thus, @ is complete. ] n, then Qn, is a subspace of @". Thus, regarding anand Qrn If m as Banach spaces with norms (In and 11 * /Irn, respectively, we may define a linear operator Trim, from Qrn to o n ,by

n,"=,

-

Trim : 'p -+

E

'p,

>

Qrn.

It follows from (1.3.2) that T,m (m n) is bounded; we call it the imbedding operator from Gm to @, . A complete countably normed space will be called a countably Banach

1.3. Countably Hilbert Spaces and Rigged Hilbert Spaces

39 1

space. If all the norms 11 * [ I n which define a countably normed space @ are induced by inner products (9, + ) n , that is,

then we call @ a countably inner product space. A complete countably inner product space is called a countably Hilbert space.

Example 1.3.1. Let A denote the finite closed interval [a, b] on the real line. Let K ( d ) denote the totality of infinitely differentiable complex-valued functions y ( x ) on the real line which vanish outside of d. Obviously, K ( d ) forms a linear space with respect to the ordinary linear operations. For any rp, E K ( d ) , we define

+

(q,

=

v=o

I

pl'Y'(t)

p ( t ) dt,

fl

= 1, 2

,...,

(I.3.10)

A

where yt0) means just rp itself. We write 1) y \ I n = [(rp, rp)n]1/2. Clearly, K ( d ) , with the sequence of norms (11 n = 1, 2, ...}, forms a countably inner product space. Consider those functions rp, defined on the real line, which vanish outside of A , have an absolutely continuous derivative of order n - 1, and such that rpcn) € L 2 ( d ) .Denote the totality of such functions p by &(A). Define an inner product on K n ( d )by formula (1.3.10). Clearly, &(A), with the ordinary linear operations and the inner product (1.3.10), forms a Hilbert space. In fact, Kn(d)is just the completion of K(A) with respect to the norm 11 ]In . Moreover, K,(d) 3 &(A) 3 ... 3 &(A) 3 -.* and

-

-

m

W) =

Kn(4 n=l

consequently, K ( d ) is a countably Hilbert space. 2O

Spaces of Continuous Linear Functionals

Let @ be a countably normed space, with norm sequence (11 * 1 I n , n = 1,2, ...I, and, for every a, let @% be the completion of @ with respect to the norm 11 * ) I n . Then On is, of course, a Banach space; we denote its conjugate space by Qnt, and the norm in Qnt by

If rn 2 n, any element of Qnt, when restricted to Qrn,is also continuous

392

APPENDIX

I.

TOPOLOGICAL GROUPS AND LINEAR SPACES

with respect to the norm 11 * [Irn, and similarly, when restricted to @, is continuous with respect to the topology of @. Thus, denoting by Gt the linear space consisting of all continuous linear functionals on @, we have

moreover, we assert that

(I.3.12) In fact, if F E Qt, then there is a positive number E such that p E @, 0) < E implies I F(p)l < 1. Choose no sufficiently large, so that Zz=n,+l(1/2m)< ~ / 2 and , choose a positive number 6 < 1, such that a/( 1 6) < c/2. Then, when I( p llnO < 6, we have p(p,

+

<

whence I F(p)( < 1. Therefore, I F(p)I (1/6) 11 p llRO for all p E @, and hence F can be extended to a bounded linear functional F E @La . This proves (1.3.12). Now, let Qi be a countably inner product space, with the topology defined by the sequence of inner products ((p, +)%, n = 1,2,...). Let (p, $) be another inner product on @, which is continuous with respect to the topology of @, in other words, there exists a positive number E, such that, if rp, $ E @ , p(p, 0) < E, p($,O) < E , then I(p, $)I < 1. It then follows from the preceding discussion that there is a positive number 6, and a natural number n o , such that [(p, $)I < 1 whenever 11 p l n O < 6, 11 $ l n o < 6. Consequently,

I(v, $11

< M II P

lln,

II # IIn,

9

P>$ E @,

(I.3.13)

where M = 1/a2. Forming the completion of @ with respect to the new inner product (p, $), we obtain a Hilbert space H 3 @. We may, in the usual manner, identify H with its own conjugate space Ht. The restriction to Qi of any element ( of H t (= H) is a linear functional on @, defined explicitly by p -+ (9,5). Moreover, by (1.3.13),

hence

8 : p -+(9,0,p E @,

is a continuous linear functional on @,

1.3. Countably Hilbert Spaces and Rigged Hilbert Spaces in fact, 5 E ,@; of spaces

. Thus, we obtain

393

an imbedding of H in Qt. The triple

@CHC@+

(1.3.14)

is called a rigged Hilbert space.

Example 1.3.2. Define an inner product on K(A) by J A

T h e completion of K(A) with respect to (rp, #) is just La(A),and K(A)t is the space of generalized functions on A. Thus, we obtain the rigged Hilbert space K(A) C L 2 ( A )C Kt(d). m

Let K = (Jn=l K(A,), A , = [-n, n]. We specify that a sequence {rp,} converges to rp in K, provided that, for some K, {rp,} C K(d,), rp E K(A,), and that {rp,} converges to 'p in the topology of K(A,) (as defined in Example 1.3.1). Accordingly, we say that a functional f on K is continuous provided that f(rpn) +f(rp) whenever {rp,} converges to rp in the above sense. I t can be proved that K is the exact union of the sequence of spaces {K(d,),n = 1,2,...} (see Definition 4.3.5).

Example 1.3.3.

APPENDIX

BACKGROUND MATERIAL ON FUNCTIONAL ANALYSIS IN HILBERT SPACES

§11.1. Operators of Hilbert-Schmidt Type, Nuclear Operators, and Equivalence Operators

lo Basic Properties of Hilbert-Schmidt Operators and Nuclear Operators I n this subsection, we restrict our considerations to Hilbert spaces. Let H and G be two Hilbert spaces, T a completely continuous linear operator from H to G. Let T* denote the adjoint of T. Then T*T is a completely continuous self-adjoint linear operator from H to H, moreover, for any cp E H,

that is, T*T is a positive operator. According to the spectral resolution theorem for completely continuous self-adjoint operators, there is an 394

11.1. Hilbert-Schmidt-Type, Nuclear, and Equivalence Operators

395

orthonormal system of eigenvectors {en) of T*T, with corresponding eigenvalues An2 > 0, such that, for any 'p E H, (11.1.1)

Let g , = (l/A,) Ten . Then, 1

( g m 9 gn) = -(Ten U r n

, Te,)

1 XnXm

= -( T * T e n

, em)

=

,

is the Kronecker delta, that is, {g,} is an orthonormal system where in G. Now, for any 'p E H, there is a vector u 1{en)such that

Since u

1{en),u I T*Tu, that is, (Tu,Tu) = 0. Hence,

(11.1.2)

where A,

> 0 and limn+mA,

= 0.

Definition 11.1 .I. If, in (11.1.l), we have C,"cl An2 < co, then the operator T is said to be of Hilbert-Schmidt type (written briefly as H-S type); if Z:-l A, < co,then T is said to be a nuclear operator. Obviously, every nuclear operator is of H-S type, but an operator of H-S type is not necessarily nuclear. Any continuous linear operator of finite rank (i.e., having a finite-dimensional range) is, of course, nuclear. Lemma 11.1.1. Let T be a bounded linear operator from a Hilbert space H to a Hilbert space G. Then T is of H-S type if and only if T * T is nuclear. PROOF. The necessity of the condition is obvious. Conversely, if T * T is nuclear, then there exist numbers A, > 0, C An2 < 00, and an orthonormal system {en} in H , such that (11.1.1) holds. Hence, (11.1.2) also holds. It then follows that T is completely continuous and is of H-S type. ] Lemma 11.1.2. Let T be a completely continuous linear operator from a Hilbert space H to a Hilbert space G. Then, T is of H-S type if

396

APPENDIX

11.

FUNCTIONAL ANALYSIS I N HILBERT SPACES

and only if there exists a positive number 1M such that, for any orthonormal system {p),} in H , (11.1.3)

PROOF. Assume that T is of H-S type. Then, there exist orthonormal systems {e,} and {gy} in H and G, respectively, such that (11.1.2) holds. By (11.1.2) we have, for any p) E H , (11.1.4)

Let (cppz} be an arbitrary orthonormal system in H. Then, by the Bessel inequality,

C

I(vn >

ev)12

< It eu t12

(11.1.5)

= 1.

n

Using (11.1.4) and (11.1.5), we get

c II

Tvn

(I2

C

=

n

v

n

I(vn

ev)12


Thus, choosing M = CAy2 < 00, we obtain (11.1.3). Conversely, suppose there exists an M < co such that (11.1.3) holds for every orthonormal system {rpy} in H . Form the decomposition (11.1.2), and choose p), = e, . Then, since Tev= Xug, , it follows from (11.1.3) that < co. ]

xAy2

Remark. Actually, it suffices to assume that T is a bounded linear operator, and that (11.1.3) holds for every orthonormal system, in order to deduce that T is completely continuous, and, consequently, of H-S type.

Lemma 11.1.3. If H , G are Hilbert spaces and T is a H-S type operator from H to G, then T* is a H-S type operator from G to H . PROOF. First, since T is completely continuous, T* is also completely continuous. Now, let {yGn} be any orthonormal system in G. Choose a complete orthonormal system {p),} in H. Then, by the Parseval equality and the Bessel inequality, we get n

n

v

11.1. Hilbert-Schmidt- Type, Nuclear, and Equivalence Operators

397

By Lemma 11.1.2, there is a positive number M such that (11.1.3) holds, hence,

c I1 T*+n (I2 < M .

Again applying Lemma 11.1.2, it follows at once that T* is of H-S type. 1 Lemma 11.1.4. The totality of H-S type operators, from a fixed Hilbert space H to a fixed Hilbert space G, forms a linear space with respect to addition and scalar multiplication. If T is a H-S type operator from H to G, B a bounded linear operator from a Hilbert space H l to H , and C a bounded linear operator from G to a Hilbert space G I , then the operators TB and CT are of H-S type. PROOF. Using Lemma 11.1.2, it is easily seen that the H-S type operators from H to G form a linear space. Next, suppose that the operator T , from H to G, is of H-S type, and that the linear operator C, from G to G 1 , is bounded. Given any orthonormal system ( y v } in H , we have, by (11.1.3),

c II CTV, 112 < I1 c 112 c I1 T," (I2 < I1 c (I2

Jf-

Again, by Lemma 11.1.2, it follows that CT is of H-S type. Finally, if B is a bounded linear operator from H l to H , then, since T* is of H-S type, it follows, by the result just proved, that B*T* is of H-S type. Using Lemma 11.1.3, we conclude that TB = (B*T*)* is of H-S type. ] In the present book, we make use of the following sufficient condition for a bounded self-adjoint operator to be of H-S type.

Theorem 11.1.5. Let B be a bounded self-adjoint operator in a Hilbert space H . Suppose there exists a positive number K < CO, such that, for any finite set of vectors v1 ,..., qn in H satisfying the conditions ( T k ,TI) = a , , k, = 1,2 ,... 72, (I I. 1.6) 9

(Bljlk, T I )

= 0,

k # 4 k,

=

1,2,*.., n,

(11.1.7)

the inequality

c n

k=l

(BTk 7 4d2


(11.1.8)

always holds. Then, B is of H-S type. PROOF. (i) We shall first prove that, if A, belongs to the spectrum of B, and A, # 0, then A, is an isolated point of the spectrum. We may

398

APPENDIX

11. FUNCTIONAL

ANALYSIS I N HILBERT SPACES

< <

assume that A, > 0. Let { E A ,a X b} be the resolution of the identity corresponding to B. If A, were not an isolated point, there would exist a sequence of points {An}, in the spectrum of B, converging to A,. For definiteness, we assume that the sequence {An) is monotonic increasing, and that A, > A0/2. Choose any sequence of numbers (pn) such that A0

- < A,

2

< p1 < A, < p2 <

**'

< A, < pn < " * < A,.

Since every An belongs to the spectrum of B, we have E., - E,,, # 0, n = I , 2, ... . Choose vectors vn E - E J H , 1) vn = 1. Then, for any n, the vectors 7, ,...,7n satisfy conditions (11.1.6) and (11.1.7) [because the subspaces (Ew,+l - E J H are mutually orthogonal and invariant under B]. On the other hand,

if

(11.1.9)

From (11.1.8) and (11.1.9), we get

But, since n is arbitrary, this is impossible. Hence, we conclude that every nonzero point of the spectrum of B is isolated, and is therefore an eigenvalue of B. (ii) The eigenspace HA,, corresponding to any nonzero eigenvalue )Lo of B, is finite dimensional. For otherwise, there would exist infinitely many mutually orthogonal unit vectors {qk} in HAo;for every n, the vectors v1 ,..., vn would satisfy (11.1.6) and (11.1.7) (since Bnk= Aoqk). But then (11.1.8) becomes

<

do2 K ,

n

=

1, 2,...,

which is impossible. I t follows from (i) and (ii) that B is completely continuous. (iii) Let pl , p 2 ,..., pn ,... denote the totality of nonzero eigenvalues of B (repeated according to their multiplicity), and vl, qa ,..., vn ,... a corresponding orthonormal system of eigenvectors. Thus, (11.1.10)

11.1. Hilbert-Schmidt- Type, Nuclear, and Equivalence Operators

399

moreover, for any n, the vectors q 1 ,..., qn satisfy conditions (11.1.6) and (11.1.7). Hence, by virtue of (11.1.8), n

c

pk2

k=l

<

(11.1.11)

for any n. Comparing (11.1.10) with (11.1.2), we conclude from (11.1.11) that B is of H-S type. ] Repeating the line of reasoning used in the proof of Theorem 11.1.5, we can also obtain the following result.

Theorem 11.1.6. Let B be a bounded self-adjoint operator in a Hilbert space H. Suppose there exists a positive number K < 00, such that, for any finite set of vectors q1 ,...,qn satisfying conditions (11.1.6) and (11.1.7), the inequality

holds. Then B is a nuclear operator. Remark. In Theorem 11.1.6 (Theorem II.1.5), one need only assume that inequality (11.1.12) [inequality (11.1.8)] holds for every finite set of vectors ql ,..., qn belonging to some dense linear subspace of H and satisfying conditions (11.1.6), (11.1.7); the conclusion of the Theorem still remains valid. Let B be a nuclear operator from H to G, and write m

/I

111

=

1

9

?k)l,

k=l

where the supremum is taken over all the orthonormal systems { f k } and {qk} in H and G, respectively. I t is easily verified that 11 B < 03. T h e number 11 B /I1 is called the trace norm of B. The following theorem gives another important relation between H-S type operators and nuclear operators.

Theorem 11.1.7. nuclear operator.

The product of two H-S type operators is a

PROOF. Let H I , H , , H3 be Hilbert spaces and B : H , -+ H , , C : Hl H , operators of H-S type. Then T = BC is a completely continuous operator from Hl to H , . By (11.1.2), there exist orthonormal --f

400

APPENDIX

11.

FUNCTIONAL ANALYSIS IN HILBERT SPACES

systems of vectors (eu> in HI and {g,) in H 3 , and numbers A, such that

> 0,

TP,== C A ~ ( P , , en)gn

We must prove that n=l

In fact, An

= (Ten 3 gn) = (BCen 3 gn) = (Cen > B*gn)

d MI Cen 11'

+ II B*gn 112)*

By Lemma 11.1.3, B* is of H-S type, hence, by Lemma 11.1.2, we have

1 < 31 (T II Cen + 1II B*gn An

112

112)

< 00,

as required. 3 Remark. It can also be proved that any nuclear operator can be resolved into the product of two H-S type operators. Next, we give another useful sufficient condition for an operator to be nuclear.

Theorem 11.1.8. Let T be a linear operator from a Hilbert space H to a Hilbert space G. If there exist families of vectors {h,} in H and {#,} in G, such that T has the representation

then T is a nuclear operator. PROOF. In view of (11.1.13), it may be assumed that {h,) and {#,} are countable. Form the operator

Then T, is a continuous linear operator of finite rank, and is therefore completely continuous. Moreover, since

1) c m

II(T- Tn)v 11

=

u=n+l

1 u*12+1 c 11

(v, h v ) +u <

m

hv II II 4, II II v 11,

11.1. Hilbert-Schmidt- Type, Nuclear, and Equivalence Operators

401

it follows by (11.1.13) that

Hence, T is completely continuous. Representing T in the form (11.1.2), and using the Bessel inequality, we get

Therefore, T is a nuclear operator. ] 20 Equivalence Operators

T h e concept of an equivalence operator, as defined below, was first introduced by Feldman [l] in connection with the equivalence of Gaussian measures.

Definition 11.1.2. Let H , K be Hilbert spaces and A a linear operator from H to K, having the following properties. (i) A is a homeomorphism from H onto K. (ii) A*A - I is a H-S type operator on H (here, I denotes the identity operator in H ) . Then A is said to be an equivalence operator (from H onto K). Note that condition (ii) in the above definition may be replaced by - I is a H-S type operator on H . (ii)’ (A*A)1/2 In fact, if A is an equivalence operator, then A*A - I = T is a selfadjoint H-S type operator. Let T be represented in the form

Tv

=

C An(v, en) en,

C An2 < 00,

An

> -1,

where {en) is an orthonormal system in H . Then, one easily calculates that ((A*A)1/2- I)p, =

But (I

C((1 + An)1/2

- l)(v, en) en

.

+ hn)1l2 - 1 is a real number, moreover C ((1 + hn)1/2-

<

\

402

11. FUNCTIONAL

APPENDIX

ANALYSIS I N HILBERT SPACES

C An2 < 00. write S

=

Hence, (ii)' holds. Conversely, assume that (ii)' holds, and (A*A)lI2- I . Then A*A- I

= (I

+S)2-I

= 2s

+

5'2,

which shows that (ii) holds.

Theorem 11.1.9. T h e inverse and adjoint of an equivalence operator are equivalence operators; the product of two equivalence operators is an equivalence operator. PROOF. Let A be an equivalence operator from H onto K , and B an equivalence operator from G onto H . Obviously, A B satisfies condition (i) in the above definition. Now, set A*A

=I

+ T,

Q

= (B*B)l" = I

+ S.

By conditions (ii) and (ii)', both T and S are of H-S type. Suppose that x E G and Qx = 0. Then [I Bx ]I2 = (B*Bx, x) = (Q2x, x) = 0, but since B-l exists, it follows that x = 0. This shows that -1 is not an eigenvalue of S . Therefore, since S is completely continuous, -1 is a regular point of S , so that ( I S)-l is defined and bounded on the entire space G. Hence the operator V = BQ-l is one-to-one from G onto H , moreover, for any x E G,

+

( V X ,V X )= (BQ-lx, BQ-'x)

= (B*BQ-'x,

Q-'x)

=

(x, x).

Therefore, V is a unitary operator from G onto H , and so we have

+ V*TV)Q = I + (2s + V*TV + SV*TV + V*TVS + S2 + SV*TVS).

(AB)*(AB)= QV*A*AVQ

= Q(I

By virtue of Lemma 11.1.4, the above expression shows that (AB)*(AB)- I is of H-S type, that is, A B is an equivalence operator. The other assertions of the theorem can be proved in a similar fashion. ] 3O

Nuclear Spaces

Definition 11.1.3. Let @ be a countably Hilbert space, with the sequence of inner products {(*, s ) ~ } . Let @% be the completion of @ with respect to (., *), , and, for m 3 n, let T," be the imbedding operator from Qm to O n .Suppose that, for every n, there is an m 2 n such that Trim is a nuclear operator. Then @ is called a nuclear space. Since the product of two H-S type operators is a nuclear operator (see Theorem 11.1.7), it is easily seen that a countably Hilbert space @

11.1. Hilbert-Schmidt- Type, Nuclear, and Equivalence Operators

403

is a nuclear space if and only if, for every n, there is an m 2 n such that Trim : !Dm +. !Dn is of H-S type. Furthermore, since the product of a bounded linear operator and a nuclear (H-S type) operator is again a nuclear (H-S type) operator, it is easily seen that a necessary and sufficient condition for a countably Hilbert space to be nuclear is the existence of a sequence of natural numbers n1 < n2 <

-.. < nk <

such that every T ; y : @nr+l +. Qnk is a nuclear (or H-S type) operator.

Example 11.1.1. K ( d ) (see Example 1.3.1) is a nuclear space. I n fact, we shall prove that the imbedding operator T,n+l:K,+,(d)

--f

Kn(d)

is of H-S type for every n. T o simplify notation, we shall suppose that d = [0, 2n-1. Let K,’(A) denote the totality of complex-valued functions on A which have an absolutely continuous derivative of order n - 1 and a quadratically integrable derivative of order n (defined almost everywhere), Then &’(A) forms a Hilbert space with respect to the ordinary linear operations and the inner product -),, moreover, &(A) is a closed linear subspace of Kn’(A).Let P, denote the projection operator from K,‘(d) onto K,(A), A, the imbedding operator from K;+,(A) to &’(A), and C,+, the imbedding operator from &+,(A) to K;+,(A). Clearly, TE+’= P,A,C,,, . (a,

Since C,,, and P, are bounded linear operators, we need only prove that A, is of H-S type, whence it will follow, in view of Lemma 11.1.4, that T,”+’ is of H-S type. Consider the orthonormal system

in KA+l(A),and the orthonormal system

in K,’(A). Let y

E

K;+,(A);expansion of y in a Fourier series shows that (11.1.14)

404

APPENDIX

11. FUNCTIONAL

ANALYSIS IN HILBERT SPACES

where n+l

(11.1.15)

In fact, (11.1.14) and (11.1.15) follow from the expansion

and the formula

which is derived using integration by parts. From (11,1.15), it is easily verified that CAm2< 00. Therefore, A, is of H-S type. ] 511.2. Tensor Products of Hilbert Spaces

The definition of a tensor product of finite-dimensional spaces can likewise be extended to the infinite-dimensional case. I n the present book, we shall be concerned only with tensor products of inner product spaces (see von Neumann 121). Definition 11.2.1. Let $jl ,..., 5, be vector spaces (e.g., over the complex numbers). Consider those functionals @(fl ,...,fn), f,E By, v = 1, 2,..., n, which satisfy the conditions (i) @( ..., af, ,...) = I%@(...,f v ,...), a any complex number, and (ii) @( ...,f, g, ,...) = @( ...,f,,...) @( ...,g, ,...), for every index v = 1, 2,,.., n. T h e totality of such functionals forms a linear space with respect to the usual operations of addition and multiplication by scalars; denote this linear space by @ 5,. Next, we further assume that each 8,is an inner product space. Given any set of vectors fy0 E $ , , v = 1, 2,..., n, we denote the functional @(fl ,...,f,)= (fyO,fv)by the symbol or, alternatively, fi0 @ . * The totality of functionals of the form

+

+

n;',

nL1of:,

nL1 ofno.

1

n

(11.2.1)

constitute a linear space, which we denote by other element of the form (11.2. I), say

n:!,

.

@ 5, Given any

11.2. Tensor Products of Hilbert Spaces

405

we define

Lemma 11.2.1. T h e number (@, Y ) defined by (11.2.3) depends only upon the functionals @ and Y, and is independent of the particular representations (11.2.1) and (11.2.2). Moreover, (@, Y ) is an inner 05 , . product on PROOF. Choose any two representations of @. Subtracting one from the other, we obtain another representation of the form (11.2.1), but in this case the left-hand side 0 = 0. Hence, to prove that (11.2.3) is independent of the representation of @, it suffices to show that the right-hand side of (11.2.3) is zero whenever the functional @ = 0. But (11.2.3) may be written as

nizl

(11.2.4)

which vanishes whenever @ = 0. Thus, (11.2.3) is indeed independent of the representation of @. Noting the fact that (@, W ) = (Y,@), it follows at once that (11.2.3) is also independent of the representation (11.2.2) of Y. It is obvious that (@, Y )depends linearly upon @. Next, we prove that (@, @) 3 0 for all @. Observe that, for any finite set of complex numbers

t1 !..’, 51

9

9

that is, the matrix ((f P9 fY 9))p,q=1.2 ,...,1 is nonnegative definite. Therefore, this matrix can be expressed as a sum of 1 matrices1 ( “ J ~ ~ o L 1 ~ ~ ) p , p = 1 , 2 , , . . , z , s = 1, 2,..., I, whence we have Y

(11.2.5) Let X = (x,,&,.~-~.~.....~ be any nonnegative definite 1 x 1 matrix. Then there exists an I x 1 unitary matrix U = (uD,J such that X = UAU*, where A is a diagonal matrix whose diagonal entries are the eigenvalues A, ,..., Al(Ay > 0) of X . Consequently, X is the sum of I matrices of rank one, that is,

.....I ,

(~~~.Y(~”)””~zz,.v~~v~l’*)),,.~~l.a

” = 1, 2,*..,1.

406

11. FUNCTIONAL

APPENDIX

ANALYSIS IN HILBERT SPACES

From (11.2.5), we deduce at once that the Schwarz inequality

w’z

I(@,Y9l d ((a,@)(K

n:El

is valid. Consequently, if @ E @ sj, , (@, @) = 0, then (@, for every Y E @ $5,. I n particular, choosing Y = f“E s j v 9 we get

n:E,

,

fI O f u )= 0,

= (@,

@(fl ,.*.,fn)

Y)= 0

nl10 fu

v=l

hence @ = 0. Thus, we have proved that (@, Y) is an inner product. ] Throughout the following, @ 6, will be regarded as an inner product space with respect to the inner product (11.2.3), and we shall write

ni:l

I/ @ 11

=

(@,

a)l’a*

It is easily verified that

Moreover, since

for any @ E

nLEl @ sj, , it follows that

Lemma 11.2.2, in

Let {GS},s = 1, 2,... be a fundamental sequence

niEl @ sj, . Then, there exists a functionaI @ nr=, E

,...,fn)

@(f1

= lim r+m @,(fl

for every n-tuple of vectorsf, E 43, , v PROOF. By (11.2.6), we have

=

,...,f n )

I , 2,..., n.

Consequently, for any fixed n-tuplef, ,...,fn , lims+m@,(f1 ,...,fn) exists; denote this limit by @(fl ,...,fn). It is easily verified that the functional @(fi

nL G3 1 nr-, @ sju be the totality of functionals @ in nr=l

,...,fn) E

Let

sju ‘

407

11.2. Tensor Products of Hilbert Spaces

satisfy the following condition: there exists a fundamental sequence {OS} In in @ sjY such that

nV=,

for every n-tuple f v E sj, , v = 1, 2, ..., n. When this condition is satisfied, we say that {QS} is a defining sequence for @. If Y E @ sj, , and {Ys} is a defining sequence for Y, we define the inner product of @ and Y by

n,”=,

(@, Y)

=

lim (as, Y,).

n-m

(11.2.8)

I t is easily verified that the above limit exists, is independent of the choice of defining sequences for @ and Y, and does, in fact, define an inner product on @ sj,. Moreover, from Lemma 11.2.2, the reader can easily deduce the following result.

n;=,

n,”El

Theorem 11.2.3. @ sj, forms a Hilbert space with respect to the inner product (11.2.8). I n fact, If=, Q sjv is a completion of the inner product space

I-IS Q 43,.

n;=, Q sj, is called the tensor

Definition 11.2.2. T h e Hilbert space product of the inner product spaces sj, ,..., 8,.

Example 11.2.1. Let 52, = (G,, 23, , Pv),v = 1, 2,..., n, be a family be the Hilbert space formed of measure spaces. For every Y, let L2(QV) by the totality of quadratically integrable functions on 52” with the usual linear operations and the inner product

Let 52 = (G, 23, P ) be the product of the measure spaces 52, ,..., 52, . Then, any @ €L2(52)may be regarded as a functional @(fl ,...,f,), f, E L ~ ( Q , v) ,= 1, 2 ,...,n, namely, W

l

,...,fn)

=

I

~@(W)fi(%)

**..fn(Wn)W W ) ,

R

w = (wl

,..., wJ.

n;=,

In accordance with this identification, @ f, is just the function -..f,(w,) in L2(52). It is easily seen that the tensor product @ L2(QV) is, in this case, simply L2(52).

fl(wl)

n:=,

408

APPENDIX

11. FUNCTIONAL

ANALYSIS IN HILBERT SPACES

We now consider tensor products of identical factors $jl = = 5, =8. n In this case, we write &(,) in place of @ 8,; the elements of 5 j ( l E ) are called tensors of ordev n over 5. Let Zn denote the symmetric group of degree n, that is, the group of all permutations of the symbols 1,2, ..., n. If T E Z, , we denote the result of applying the permutation T to the symbols 1,..., n by ~ ( 1,..., ) rr(n), respectively. Now, form a 0 5,, as follows: if linear operator V ( " ) ( T ) in

nv=l

n:El

then

c I

V'"'(7r)aJ=

n

@f3?,.

p=1 v = l

ni!l

It is easily seen that V(,)(T)is a unitary operator from @ 9, onto itself. Since @ &, is dense in $j(,), V n ) ( v can ) be uniquely extended to a unitary operator from & ( l Z ) onto itself. Furthermore, the correspondence rr -+ V(,)(T-~) is a unitary representation of the symmetric group in the Hilbert space (for the definition of a unitary representation, see 511.3). Consider the bounded linear operator

n:cl

in

&(").

Actually, S, is a projection operator. I n fact, since V(,)(T)*=

P ) ( T we - ~have ), 1 s,* = c V(")(,-1) = s, . n! ~

,{TT' I T' E Z},

On the other hand, for any T E Z, Consequently, V'"'(7r)s,

=

I

= {T'T T' E Z },

1 -1 V(n)(7rT')= s, = SnV(n)(7r), nI ,,,

= 2,

.

(11.2.9)

Therefore, since Zn is of order n ! , we obtain s , 2

=

J&z

V'"'(77)s, = s,

I

n

Formula (11.2.9) also shows that every vector of M , = Sn$P), the range of S,, is invariant with respect to all the operators V n ) ( r r ) ,

11.3. Unitary Representations of Groups

409

. The elements of M , are called symmetric tensors of order n, and the projection S, is called the symmetrization operator. M , , being a closed subspace of a Hilbert space $j("), is itself a Hilbert space with respect to the inner product (@, Y).

rr E Z,

Example 11.2.2. If, in Example 11.2.1, we take Q, = Q, = = Q,, then the symmetric tensors of order n are just the symmetric functions f ( w ) in L2(sZ),that is, those which satisfy

for all permutations rr E En .

Lemma 11.2.4. Let {g,} be a complete orthonormal system in the Hilbert space $j. Then,2 '!

)"'S,(gp

(k,!k,! ... kl!

@

... @g!'),

0 < k,,

1

1 k, = n v=l

(11.2.10)

forms a complete orthonormal system in the corresponding space M , of symmetric tensors of order n. PROOF. Clearly, (11.2.10) is an orthonormal system; we need only prove that it is complete. Suppose that g E M , is orthogonal to all of the vectors (11.2.10). Then

But every vector in $(), is a limit of linear combinations of vectors of the form g$ @ *.- @ g t t , moreover, S,g = g. Consequently, g is orthogonal to the whole of a(,), that is, g = 0. This shows that (11.2.10) is complete. ] 511.3. Unitary Representations of Groups

lo General Concepts Definition IL3.1. Let 6 be a group, H a Hilbert space. Suppose that, to every element g E 8 , there corresponds a unitary operator U ( g ) * Here,:g

represents gl 0 ... 0g, kl IBCtOrS

.

410

APPENDIX

11. FUNCTIONAL

ANALYSIS I N HILBERT SPACES

on H , and that, for any g , ,g , E 6, we have U ( g l ) U ( g , ) Then the correspondence

=

U(g,g,).

is called a unitary representation of the group 6 in the Hilbert space H. If we let U denote the totality of unitary operators in H , then a unitary representation U of 6 in H is simply a homomorphism of 6 into U. If there exists no nontrivial closed linear subspace of H which is invariant under all the operators U(g), g E 8,then the representation U is said to be irreducible. Clearly, U is irreducible if and only if { U ( g ) I g E G}' (see $2.3, 2O) consists of just the operators X I , where X is a number and I is the identity operator in H . Now, suppose that 6 is a topological group, and U is a unitary representation of 6 in a Hilbert space H. If, for every fixed pair of vectors T, ht, E H , the function (

wvv, *I

is continuous on 6, then the representation U is said to be weQkly continuous. I n other words, U is a weakly continuous representation provided that, when U is given the weak topology (see $2.3, lo), the correspondence U : 6 -+ U is a continuous homomorphism. If, for every fixed q~ E H , the correspondence h + U(h)v,

hE8

is a continuous mapping from 8 to H (relative to the norm of H ) , then the representation U is said to be strongly continuous. I n other words, the representation U is strongly continuous provided that U is a continuous mapping from 6 to U when the latter is given the strong topology (see $2.3, lo). Obviously, any strongly continuous unitary representation is also weakly continuous. Actually, the converse is also true.

Lemma 11.3.1. A weakly continuous unitary representation is also strongly continuous. PROOF. Let U be a weakly continuous unitary representation of a topological group 6 in a Hilbert space H . Using the definition of a unitary representation, one readily calculates that, for any q E H , go,g E 8,

11.3. Unitary Representations of Groups

41 1

Let E be an arbitrary positive number. Since U is weakly continuous, there exists a neighborhood V of the identity e in 8,such that

for every h e V . But ( U ( e ) y ,y) = ( y , y), hence, if g E ~ , V(which is a neighborhood of go), it follows from (11.3.1) and (11.3.2) that

Therefore, U is strongly continuous. ] Let U , :g -+ U,(g), g E G , k = 1, 2, be unitary representations of a group G in the respective Hilbert spaces H I and H , . If there exists a unitary operator Q from HI onto H , such that U,(g) = QU,(g) Q-I, g E G , then U , and U , are said to be unitarily equivalent. 2 O One-parameter Groups of Unitary Operators

Let H be a Hilbert space, and { U ( t ) , -co unitary operators in H such that U(t1

+ tz)

=

U(t1) U(t,),

-CO

< t < co} a family of

< t , , t z < CO, U(0) = I. (11.3.3)

Then, { U ( t ) , - co < t < a} is called a one-parameter group3 of unitary operators in H . In other words, saying that (U(t ),--a3 < t < a>is a one-parameter group of unitary operators in H means that the correspondence t --t U ( t )is a unitary representation of R (the additive group of real numbers) in H . If this representation is strongly (or weakly) continuous, then we say that the one-parameter group of unitary operators { U ( t ) , -a < t < a}is strongZy (or weakZy) continuous. If, for every fixed pair of vectors y , $ E H , (U(t)T, $1,

--co


is a Lebesgue measurable function of t, then ( U ( t ) , -co < t < co} is said to be weakly measurable. Clearly, any weakly continuous oneparameter group of unitary operators is weakly measurable. Conversely, we have the following result.

Lemma 11.3.2. In a separable Hilbert space, any weakly measurable one-parameter group of unitary operators is weakly continuous. The term “parameter” in this designation refers, of course, to the real variable

t , -a

< t < co.

412

APPENDIX

11.

FUNCTIONAL ANALYSIS IN HILBERT SPACES

This result is due to von Neumann. Because the proof is rather lengthy and involves techniques not closely related to the subject of the present book, it will not be given here. A proof may be found in, for example, Guan Zhao-zhi [2]. For the proof of the following lemma, see, for example, Riesz and Sz.-Nagy [l].

Lemma 11.3.3. Let { U ( t ) ,- 00 < t < a}be a strongly continuous one-parameter group of unitary operators in a Hilbert space. Then, there exists a resolution of the identity { P A ,-a < h < 00} such that .m

T h e result of Lemma 11.3.3 can be further elaborated, as follows. We define a linear operator A in H: the domain of A is to be

and, when x E D(A),

I

m

Ax

=

hdP,x.

-W

We then write U ( t ) = &A,

(11.3.4)

and call A the infinitesimal generator of the one-parameter group of unitary operators { U ( t ) ,-00 < t < a}.Actually, the right-hand side of (11.3.4) can be interpreted as (itA(P, - P-,))k e i t A = lim exp(itA(P, - L,)) = lim n+ m n+ m k! k=O

1

Moreover, it can also be proved that, when x E D(A), Ax

= lim t-0

U(t) - I

or, written more concisely, A = - -IUd( t ) I i dt

it

x,

. t=O

NOTES AND REFERENCES TO THE LITERATURE

Chapter I

$1.1. Some of this material is common knowledge. Regarding 3" and 5", see Segal [3]. T h e notion of a pseudo-a-finite measure (in 6") was introduced, for convenience, by the author. Regarding Lemmas 1.1.13, 1.1.14 and Corollary 1.1.15, see Naimark [l]. Lemma 1.1.16 (7") is due to Prohorov (see Gel'fand and Vilenkin Ill). Regarding 8", see Doob [l]. $1.2.

See Segal [I].

$1.3. The generalization of the classical theorem of Kolmogorov [l], in terms of projective limits, is due to the author. $1.4. These results are due to Kakutani [ 13, except for Lemma 1.4.2, which is due to the author. Chapter II

$2.1. For the basic ideas and properties of topological algebras, see Naimark [l] or Xia Dao-Xing [2]. Regarding the representation of 413

414

NOTES AND REFERENCES TO THE LITERATURE

symmetric multiplicative linear functionals as points of an infinite product space (Lemma 2.1.1), see Xia Dao-Xing [l]. 52.2. These results were proved for normed algebras by Gel’fand and Naimark, but the discussion given here applies to a still more general context. Lemma 2.2.1, Corollary 2.2.2, Lemma 2.2.3, and Theorem 2.2.4 are due to the author (see Xia Dao-Xing [2]); the methods used are entirely different from those of Gel’fand and Naimark. I n particular, the use of the Kolmogorov Theorem to study positive functionals, and to relate the theory of positive functionals more intimately with measure theory, is due to the author (see Xia Dao-Xing [l]). $2.3. Some of the concepts introduced here are due to von Neumann; see von Neumann [2], [3], [4], [5], [6], [7], [8], or Naimark ([l], Chapter 8). $2.4.

See Segal [l], [2]. Chapter 111

53.1. Lemma 3.1.3, for the special case where G is a finite-dimensional space, can be found in Gel’fand and Vilenkin [l]. Theorem 3.1.13 was first proved by Yan Shao-zong [l], using a method due to Girsanov and Mityagin [l]. Lemma 3.1.31 was also proved by Yan Shao-zong [l], using a different method. Theorem 3.1.32, concerning ergodic measures, was first proved by Segal [5] (although he stated the result in a somewhat different form). All the other results in $3.1 are due to the author; some of these may be found in Xia Dao-Xing [5],[9], while some are published here for the first time.

The basic notions concerning characters, and certain results for locally compact groups, are well known (see, e.g., Naimark [l]). Aside from these, the theory in $3.2 is due to the author; some of the results are published here for the first time, others may be found in Xia Dao-Xing [5]-[9]. 53.2.

53.3. Lemma 3.3.3 and Theorem 3.3.4 (concerning positive definite functions on discrete groups), as well as Example 3.3.2 and Lemma 3.3.9 (concerning locally compact groups) are all well known (see, e.g., Naimark [l]). The remaining results are due to the author, and were previously published for the case where there exists a finite quasiinvariant measure (see Xia Dao-Xing [6]). 53.4. The theory of Fourier transforms on locally compact groups is well known (see, e.g., Naimark [l]). Theorem 3.4.8 (for Kth order

Notes and References to the Literature

41 5

cyclic measures) has been discussed by Ou Rui-sen [2]. T h e remaining results in this section are due to the author. Regarding other research along these lines, see Ou Rui-sen [2] and Shen Hai-yu [2]. Chapter I V

$4.1. Except for Example 4.1.2, all the results in this section are due to the author, and are published here for the first time. 54.2. As regards linear functionals, the basic concepts and propositions given here are well known. Regarding quasi-linear functionals, the results in 2” and 3” are due to the author (some may be found in Xia Dao-Xing [S]). Lemmas 4.2.16 and 4.2.17 are due to Erohin (see Minlos [l]). Theorem4.2.18 is also due to the author, but Corollary 4.2.19 was first proved by Shen Hai-yu, while Corollary 4.2.20 was first proved by Mityagin [I], using a geometrical argument. Example 4.2.1 was first given by Chen Jing-jian in his university graduation thesis. Yan Shao-zong [3] used the L2-Fourier transform to solve a certain type of functional variational equation, similar to that which occurs in quantum field theory. 54.3. The notion of a cylinder measure was introduced by Gel’fand (see Gel’fand and Vilenkin [I]). Lemma 4.3.7 is due to Mityagin [l], Corollary 4.3.9 to Erohin (see Minlos [I]), and Theorem 4.3.12 to the author (see Xia Dao-Xing [3]). For another approach to the representation of continuous positive-definite functions, see Xia Dao-Xing [3], Ou Rui-sen [I], and Shen Hai-yu [l]. Concerning the material in 5”, see Gel’fand and Vilenkin [l]. T h e notion of a generalized stochastic process was first formulated by Gel’fand (see Gel’fand and Vilenkin [l]); concerning the results in 60, see also Xia Dao-Xing [3] and Yan Shao-zong [2]. Chapter V

$5.1. Concerning the general properties of Gaussian measures, see Doob [l]. Theorem 5.1.4 is due to the author; Feldman [2] attempted to give an example of an ergodic measure not equivalent to any Gaussian measure, but the proof of Lemma 6 in Feldman’s paper is open to question (because the assertion that Ck-+ 0 lacks adequate justification), and so the general validity of his example remains uncertain. More extensive discussions of Wiener processes may be found in Doob [l] and Paley and Wiener [l]; Koval’cik [I] gives a fairly comprehensive bibliography on the subject. 55.2. The fundamental criterion for the equivalence of Gaussian measures (Theorem 5.2.5) was first obtained by Feldman [l] and

416

NOTES AND REFERENCES TO THE LITERATURE

Hhjek [I]. The proof given here by the author is a modification of that given by Rozanov [l]; it avoids the use of relatively complicated devices such as entropy, and so 0n.l Concerning Example 5.2.1, see Xia Dao-Xing [4]. A number of survey type articles on the equivalence of Gaussian measures may be found in the proceedings of the Symposium on Time Series Analysis, Brown University, 1962 (Edited by M. Rosenblatt), Wiley, New York, 1963. $5.3. The notion of a standard Gaussian measure was introduced, for convenience, by the author. Theorem 5.3.5 summarizes results obtained by Minlos (see Gel'fand and Vilenkin [l]) and Mityagin [l], but some parts of the present proof were devised by the author. Example 5.3.1 is due to Ou Rui-sen [l] and Chen Jing-jian. Theorem 5.3.7 is due to the author. As for the elementary properties of Wiener measure, these are, of course, well-known. $5.4. The general results presented in this subsection are written here for the first time in their present form, but are not essentially new; they originate in certain results of Cameron and Martin [11 and Segal [4] concerning Wiener and Gaussian measures. Chapter VI $6.1. Theorem 6.1.1 is due to Wielandt [l]; Theorems 6.1.3, 6.1.4 and Corollaries 6.1.5, 6.1.6 are due to von Neumann [I]. The remaining results of this section originated in the work of Segal [3], [6]. $6.2. Regarding lo and 3", see Segal [6], [7] and [8]. Theorem 6.2.6, as stated here, is due to the author.

$6.3. Lemma 6.3.4 was obtained by Umemura [I], using Bernstein's theorem; the proof given here is due to the author. Theorem 6.3.5 was first obtained by Segal [8]; again, the proof given here is due to the author.

The remaining results in $6.2 and $6.3 are similar to those obtained by Segal ([4], [5], [6], [7], [S]), using the notion of a weak distribution, but here, they are formulated in terms of quasi-invariant measures. Gel'fand used quasi-invariant measures on conjugate spaces of nuclear spaces to construct a special class of unitary representations of the commutation relations (see Gel'fand and Vilenkin [l]). Translator's note: For a very elementary and easily comprehensible proof, see Brody El].

BIBLIOGRAPHY

Brody, E. J.’ [l] An Elementary Proof of the Gaussian Dichotomy Theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 20 (1971), 217-226. Brush, S. G. [l] Functional Integrals and Statistical Physics, Reo. Modern Phys. 33 (1961), 79-92. Cameron, R. H., and Martin, W. T. [l] The Orthogonal Development of Nonlinear Functionals in Series of FourierHermite Functionals, Ann. of. Math. 48 (1947), 385-392. [2] Fourier-Wiener Transforms of Functionals belonging to L, over the Space C , Duke Math. J. 14 (1947), 99-107. Chen, Jian-gong [l] Real Function Theory (in Chinese), Kexue Chubanshe, Peking, 1958. Doob, J. L. [l] “Stochastic Processes.” Wiley, New York, 1953. Feldman, J. [l] Equivalence and Perpendicularity of Gaussian Processes, Pacific J. Math. 8 (1958), 699-707. Gel’fand, I. M. [l] Some Problems in Analysis and Differential Equations (in Russian), Uspehi Mat.

’ Translator’snote: Reference added by translator. 417

418

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Nauk. 14 (1959), No. 3 (87), 3-19. [English transl.: Aner. Math. SOC.Traml., Ser. 2, 26 (1963), 201-219.1 Gel’fand, I. M., and Vilenkin, N. Ya. [I] “Generalized Functions,” Vol. 4 (in Russian). Moscow, 1961. English translation: Academic Press, New York, 1964. Girsanov, I. V., and Mityagin, B. S. [I] Quasi-invariant Measures in Linear Topological Spaces (in Russian), Nauln. Dokl. Vysf. skoly Fiz.-Mat. Nauki 2 (1959), 5-9. Guan Zhao-zhi [I] “Essentials of the Theory of Topological Spaces” (in Chinese), Kexue Chubanshe, Peking, 1958. [2] “Lectures on Functional Analysis” (in Chinese), Gaodeng Jiaoyu Chubanshe, Peking, 1958. Hhjek, J. [I] On a Property of Normal Distributions of an Arbitrary Stochastic Process (in Russian), Czechoslovak Math. J. 8 (1958), 610-61 8. [English transl.: “Selected translations in Mathematical Statistics and Probability,” pp. 245-252. American Mathematical Society, Providence, Rhode Island, 1961.] Halmos, P. R. [I] “Measure Theory.” Van Nostrand, Princeton, New Jersey, 1950. Kakutani, S. [I] On Equivalence of Infinite Product Measures, Ann. of Math. 49 (1948), 214-224. Kolmogorov, A. N. [I] “Foundations of Probability Theory” (in Russian), Moscow-Leningrad, 1936. [English transl.: Chelsea, New York, 1950.1 Koval’Eik, I. M. [I] Thewiener Integral(inRussian), UspehiMat.Nauk. 18(1963),No. 1 (I09), 97-134. Mityagin, B. S. [l] A Remark on Quasi-invariant Measures (in Russian), Uspehi Mat. Nauk. 16 (1961), NO. 5 (lo]), 191-193. Naimark, M. A. [I] “Normed Rings” (in Russian), Moscow, 1958. [English transl.: Noordhoff, Groningen, 1959.1 Ou Rui-sen [I] Measures on the Spaces 1” (in Chinese), Fudan Daxue Xuebao (Journal of Fudun University) 8 (1963), 39-46. [2] Measures on Banach Spaces (in Chinese), Doctoral dissertation, Fudan University, 1964. Paley, R. E. A. C., and Wiener, N. [I] Fourier Transforms in the Complex Domain, Amer. Math. SOC.Colloq. F’ubl. 19 (1934), Chap. 9. Riesz, F., and Sz.-Nagy, B. [I] “LeGons d’analyse fonctionnelle,” Budapset, 1952. [English transl.: “Functional Analysis,” Ungar, New York, 1955.1 Rozanov, Yu. A. [l] On the Density of one Gaussian Measure with Respect to Another (in Russian), Teor. Verojatnost. i Primenen. 7 (1962), 84-89. Sazonov, V. V. [I] On Characteristic Functionals (in Russian), in “Proceedings of the Sixth AllUnion Conference on Probability Theory and Mathematical Statistics, pp. 455-

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462, Vilnius, 1960. Gosudarstv. Izdat. PolitiEesk i Naufn. Lit. Litovsk. S.S.R., Vilnius, 1962. Segal, I. E. Equivalences of Measure Spaces, Amer. J. Math. 73 (1951), 275-313. Decompositions of Operator Algebras 11, Mem. Amer. Math. SOC.9 (1951), 1-66. Abstract Probability Spaces and a Theorem of Kolmogoroff, Amer. J. Math. 76 (1954), 721-732. Tensor Algebras over Hilbert Spaces, I, Trans. Amer. Math. SOC.81 (1956), 106-134. Distributions in Hilbert Spaces and Canonical Systems of Operators, Trans. Amer. Math. SOC.88 (1958), 12-41. Foundations of the Theory of Dynamical Systems of Infinitely Many Degrees of Freedom, I, Kgl. Danske. Videnskab. Selskab. Mat.-Fys. Medd. 31 (1959), No. 12, 1-33. Foundations of the Theory of Dynamical Systems of Infinitely Many Degrees of Freedom, 11, Canadian J. Math. 13 (1961), 1-18. Foundations of the Dynamics of Infinite Systems, 111: Mathematical Characterization of the Physical Vacuum for a Linear Bose-Einstein Field, Illinois J. Math. 6 (1962), 500-523. Shen Hai-yu [I] Measures on the Spaces L’ (in Chinese), Fudan Daxue Xuebao (Journal of Fudan University) 8 (1 963), 205-220. [2] Measures on Banach Spaces and Groups (in Chinese), Doctoral dissertation, Fudan University, 1964. Umemura, Y. [I] Rotationally Invariant Measures in the Dual Space of a Nuclear Space, Proc. Japan &ad. 38 (1962), 15-17. von Neumann, J. [I] Die Eindeutigkeit der Schrodingerschen Operatoren, Math. Ann. 104 (1928), 570-578. [2] On a Certain Topology for Rings of Operators, Ann. Math. 37 (1936), 111-115. [3] (with F. J. Murray) On Rings of Operators, Ann. Math. 37 (1936), 116-229. [4] On Rings of Operators 11, Trans. Amer. Math. SOC.41 (1937), 208-248. [5] On Rings of Operators 111, Ann. of Math. 41 (1940), 94-161. [6] On Some Algebraical Propertiesof Operator Rings, Ann. ojMath. 44( 1943),709-71 5. [7] On Rings of Operators IV, Ann. of Math. 44 (1943), 716-808. [8] On Rings of Operators. Reduction Theory, Ann. Math. 50 (1949), 401485. Wielandt, H. [I] Uber die Unbeschranktheit der Operatoren der Quantenmechanik, Math. Ann. 121 (1949), 21. Xia Dao-xing (Hsia Tao-hsing, Sya Do-Sin) [l] Positive Functionals on Algebras (in Russian), Dokl. Akad. Nauk SSSR. 121 (1958), 233-235. [2] On Semi-normed Rings with Involution (in Russian), Dokl. Akad. Nauk SSSR. 124 (1959), 1223-1225; Izo. Akad. Nauk SSSR. Ser. Mat. 23 (1959), 509-528. [3] Measures on Functional Spaces (in Chinese), Fudan Daxue Xuebao (Journal of Fudan University) 7 (1962), 121-131, 8 (1963), 261-262. English trans].: Sci. Sinica 12 (1963), 479-494. [4] On the Equivalence of a Class of Gaussian Measures (in Chinese), Fudan Dame Xuebao (Journal of Fudan University) 8 (1 963), 15 1-1 61.

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On Linear Topological Spaces with Quasi-invariant Measures (in Chinese), Fudan Daxue Xuebao (Journal of Fudan University) 8 (1963), 273-277. [6] Positive Definite Functions on Commutative Topological Groups with Quasiinvariant Measures (in Chinese), S h w e Xuebao 14 (1964), 340-352. English transl.: Chinese Math.-Acta 5 (1964), 369-381. [7] On Dual Quasi-invariant Measures (in Chinese), Shuxue Xuebao 14 (1964), 680-688. English transl.: Chinese Math.-Acta 6 (1965), 70-78. [8] On Some Properties of Quasi-invariant Measures (in Chinese), S h w e Xuebao 15 (1965), 81-90. English transl.: Chinese Math.-Acta 6 (1965), 382-392. [9] On Quasi-invariant Measures in Linear Spaces (unpublished). Xia Dao-Xing, W u Zhuo-ren, and Yan Shao-zong [l] “Theory of Functions of a Real Variable and Essentials of Functional Analysis” (in Chinese). Shanghai K e w e Jishu Chubanshe, Shanghai, 1963. Yan Shao-zong [l] A Note Concerning Quasi-invariant Measures (unpublished). [2] On Generalized Random Processes (in Chinese), Fudan Daxue Xuebao (Journal of Fudan University) 9 (1964). [3] On a Class of Functional Variational Equations (unpublished).

[5]

INDEX

A Adjoint function, 207 Algebra, 50 Banach, 51 commutative, 50 concrete Weyl, 345, 364 multiplication, 81, 82 norrned, 50 operator, 59 contraction of, 79 over La(@, 126 weakly closed, 74 seminormed, 50 countably, 50 symmetric, 53 symmetric tensor, 370 von Neumann, 74 w*-,74

B Bore1 cylinder, 31, 159, 171, 248

Bound lower, 24 greatest, 24 upper, 24 least, 24 Brody, E. J., 416 Brush, S. G., 103

C Cameron, R. H., 416 Canonical Gaussian measure space, 284 projective limit, 33, 35 system, 354 unitary representation, 353, 354 Center, 78 Character, 135 induced, 140 Characteristic function, 18, 160, 235 functional, 277, 366 421

422

INDEX

Chen Jing-jian, 415, 416 Commutation relations Heisenberg, 338 von Neumann form, 340 weakly continuous unitary representation of, 338 Weyl, 338 Complexification, 355 Conjugate, 52 space, 233 Consistent family of projections, 31, 34 family of measures, 31 Continuous cylinder measure, 259 linear functional, 51 measure, 162 Convolution, 168 Correlation number, 18, 244 Cyclic element, 83, 207, 366 family, 207 measure, 185 measure space strongly, 207 strongly Rth order, 207 subspace, 83 Cylinder measure, 31, 255 continuous, 259 integral with respect to, 256 open, 34 weakly, 263

D Defining sequence, 407 Determining set, 6 joint, 41 Direct sum, 10, 75 Discriminator, 303 Doob, J. L.,247, 413, 415 Dual algebraic, 135, 232 measure space, 191, 254 topological group, 138

E Equivalent, 24, 30 unitarily, 79, 357, 411

Exact union, 274, 393

F Factor, 78 Feldman, J., 401, 415 Fourier-Gauss transform, 329 Fourier-Hermite coefficient, 332 Fourier transform, 191 L1-, 183 La-, 191, 254 Function adjoint, 207 characteristic, 18, 160, 235 convex, 382 homogeneous, 382 distribution, 18 joint, 18 measurable, 2, 3, 4 positive-definite, 164 pseudocontinuous, 256 quasi-characteristic, 160, 161, 240 Functional characteristic, 277, 366 continuous linear, 51 convex, 382 correlation, 244, 245 Hermite polynomial, 332 measurable linear, 234 multiplicative linear, 53 positive, 52 quasi-continuous, 214 quasi-linear, 236 symmetric, 52 Fundamental sequence, 51

G Gel’fand, I. M., 104, 274, 386, 413, 414, 415, 416 Girsanov, I. V., 414 Gradient transformation, 352 Group associated with one-particle system, 353 character, 135 dual topological, 138 maximal quasi-invariant left translation, 107 multiplication, 187 one parameter, of unitary operators, 41 1

INDEX

strongly continuous, 410, 41 1 weakly continuous, 410, 41 1 weakly measurable, 41 1 spectral, 165 transformation, 126 of La(Q), 126 quasi-invariant, 106 Guan Zhao-zhi, 69, 144, 150, 412

transformation, 105 weakly, 411 Measure, 30, see also Measure space continuous, 162 cyclic, 185 cylinder, 31, 255 extension of, 2 Gaussian, 281, 283 invariant, 217 H normal, 186 restriction of, 3 Hhjek, J., 416 ring, 30 Halmos, P. R., 1, 2, 3, 9, 30, 33, 36, 69, rotationally invariant, 379 84, 85, 105, 107, 127, 128, 132, 176, spectral, 98 183. 255 Measure space canonical Gaussian, 284 I dual, 191, 254 ergodic, 131 Infinitesimal generator, 412 weakly, 131 Invariant subspace, 78 invariant, 106, 217 Involution, 52, 355 left, 107 Irreducible, 343, 361 kth order cyclic, 185 Lebesgue type, 225 K linear, 234 Kakutani, S., 314, 413 localizable, 1, 29 distance, 42 locally finite, 12 inner product, 42 normal, 186 Kolmogorov, A. N., 413 pseudo-a-finite, 12 Koval’Eik, I. M., 415 quasi-continuous, 214 quasi-invariant, 105, 106 L left (right), 106, 107 standard, 244 Left quasi-invariant element, 107 strongly, 106 Linear weakly, 106 basis, 54 regular (outer, inner), 16 space of measurable transformations, 214 sample probability, 40 Lower semicontinuous, 385 spectral, 98 standard Gaussian, 312 M strongly cyclic, 207 Martin, W. T., 416 strongly Rth order cyclic, 207 Mathematical expectation, 18 totally o-finite, 1 Maximal quasi-invariant Wiener, 290 left translation group, 107 Minlos, R. A., 415 linear subspace, 216 Mityagin, B. S., 414, 415, 416 Mean value, 18, 244 Measurable N function, 2, 3, 4 linear functional, 234 Naimark, M. A., 15, 50, 65, 413, 414 ring, 30 Natural ordering, 32 space, linear, 234 Natural projections, 33

423

424

INDEX

Nondegenerate pair, 364 Norm compatible, 388 trace, 399 Null set, 3 0 Observables, 364 Operator algebra, 59 annihilation, 371 creation, 371 equivalence, 401 Hilbert-Schmidt type, 395 imbedding, 390 multiplication, 81 nuclear, 395 ring, 59 Schrodinger, 339 symmetrization, 409 Ou Rui-sen, 415, 416

P Paley, R. E. A. C., 415 Partition, 10, 171 Probability distribution, 18 joint, 18 Process Gaussian, 283 standard, 312 sample, 40, 277 Wiener, 290, 291 Projections consistent family of, 31, 34 natural, 33 Projective limit, 31, 32, 34 canonical, 33, 35 Projectively complete, 34 Pseudometric, 381 homogeneous, 382 k-, 120 left-invariant, 381 Pseudonorm, 382 k-, 120

Q Quasi-character, 140 similar, 140

Quasiconvex pair of functions, 116 Quasi-invariant set, 130

R Radon-Nikodym derivative, 90 Random variable, 17 Representation irreducible, 410 Schrodinger, 339 unitarily equivalent, 41 1 unitary, 410 canonical, 353, 354 Riesz, F., 346, 412 Ring group, 168 measurable, 30 measure, 30 operator, 59 maximal commutative weakly closed, 80 weakly closed, 74 Rosenblatt, M., 416 Rozanov, Yu.A., 416

S Sample probability measure space, 40 probability space, 277 process, 40, 277 space, 277 Schrodinger operators, 339 Schrodinger representation, 339 Schwartz inequality, 53 Segal, I. E., 81, 104,214,413,414,416 Self-conjugate, 52 Semimartingale, 19 Seminorm, 50 Separated, 215 Sequential completeness, 51 Shen Hai-yu, 415 Space conjugate, 233 countably Banach, 390 Hilbert, 391 inner product, 391 normed, 388 pseudonormed, 387

425

INDEX

of linear functionals which distinguishes points, 249 linear metric, 383 linear pseudometric, 383 nuclear, 402 pseudometric, 382 rigged Hilbert, 393 sample, 277 spectral, 165 linear, 262 state vector, 356 union, 274 Spectral group, 165 linear space, 262 measure space, 98 countably decomposable, 99 normal, 99 space, 165 Standard Gaussian measure space, 312 Gaussian process, 312 quasi-invariant measure space, 244 Stochastic process, 40, see also Process generalized, 276 linear, 276 Strong neighborhood, 72 System canonical, 354, 356 infinitesimal, 355 conventional free-field, 376 Fock-Cook, 371 group associated with, 353 single-particle, 353 Sz.-Nagy, B., 346, 412

determined by sequence of pseudonorms, 387 induced by convex function, 382 k-, 120 /L-, 126, 147, 239 p-weak, 147 strong, 72, 145, 238 146 uniform, 73 weak, 56, 72, 145, 238

u-,

U Umemura, Y., 379, 416 Uniform multiplicity, 81 Unit, 50 Upper semicontinuous, 384, 385

V Vacuum state vector, 376 Variance, 18 Vilenkin, N. Ya., 104, 274, 386, 413, 414, 415, 416 von Neumann, J., 404, 414, 416

W Weak Bore1 set, 159, 248 Weak neighborhood, 72 Wielandt, H., 416 Wiener, N., 415 X

Xia Dao-Xing, 260, 413, 414, 415, 416

T Tensor of order n, 408 product, 407 symmetric, 409 Topology admissable, 111

Y Yan Shao-zong, 414, 415

z Zhao Zhong-zhe. 247

Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Paul A. Smith and Samuel Eilenberg Columbia University, N e w York

1 : ARNOLD SOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures on Theoretical Physics, Volume V I ) 2 : REINHOLD BAER.Linear Algebra and Projective Geometry. 1952 BUSEMANN A N D PAUL KELLY.Projective Geometry and Projective 3 : HERBERT Metrics. 1953 4 : STEFANBERchiAN A N D M. SCHIFFER. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. 1953 5 : RALPHPHILIPBOAS,JR. Entire Functions. 1954 BUSEMANN. The Geometry of Geodesics. 1955 6: HERBERT 7 : CLAUDE CHEVALLEY. Fundamental Concepts of Algebra. 1956 8: SZE-TSEN Hu. Homotopy Theory. 1959 Solution of Equations and Systems of Equations. Second 9: A. M. OSTROWSKI. Edition. 1966 10 : J. D I E U D O N N Treatise ~. on Analysis. Volume I, Foundations of Modern Analysis, enlarged and corrected printing, 1969, Volume 11, 1970. Volume 111, in preparation Curvature and Homology. 1962. 11 : S. I. GOLDBERC. 12 : SICURDUR HELCASON. Differential Geometry and Symmetric Spaces. 1962 13 : T. H. HILDEBRANDT. Introduction to the Theory of Integration. 1963. ABHYANKAR. Local Analytic Geometry. 1964 14 : SHREERAM 15 : RICHARD L. BISHOPA N D RICHARD J. CRITTENDEN. Geometry of Manifolds. 1964 16: STEVEN A. GAAL.Point Set Topology. 1964 MITCHELL.Theory of Categories. 1965 17 : BARRY P. MORSE.A Theory of Sets. 1965 18: ANTHONY 19 : GUSTAVE CHOQUET. Topology. 1966 A N D I. R. SHAFAREVICH. Number Theory. 1966 20 : Z. I. BOREVICH A N D J U A N JORCE SCHAFFER. Linear Differential Equations 21 : J o s t LUIS MASSERA and Function Spaces. 1966 22 : RICHARD D. SCHAFER. An Introduction to Nonassociative Alegbras. 1966 23: MARTINEICHLER.Introduction to the Theory of Algebraic Numbers and Functions. 1966 ABHYANKAR. Resolution of Singularities of Embedded Algebraic 24 : SHREERAM Surfaces. 1966

25 : 26: 27 : 28 : 29 : 30 : 31 : 32 : 33: 34: 35 : 36 : 37 : 38: 39 : 40 : 41 : 42: 43: 44 : 45 : 46 : 47 :

48 : 49 :

FRANGOIS TREVES. Topological Vector Spaces, Distributions, and Kernels. 1967 PETER D. LAXA N D RALPHS. PHILLIPS. Scattering Theory. 1967. OYSTEINORE.The Four Color Problem. 1967 MAURICEHEINS.Complex Function Theory. 1968 R. M. BLUMENTHAL A K D R. K. GETOOR. Markov Processes and Potential Theory. 1968 L. J. MORDELL. Diophantine Equations. 1969 J. BARKLEY ROSSER.Simplified Independence Proofs : Boolean Valued Models of Set Theory. 1969 WILLIAM F. DONOCHUE, JR. Distrihutions and Fourier Transforms. 1969 S. CAIRNS.Critical Point Theory in Global MARSTON MORSEA N D STEWART Analysis and Differential Topology. 1969 EDWIN WEISS.Cohomology of Groups. 1969 HANSFREUDENTHAL A N D H. DE VRIES. Linear Lie Groups. 1969 LASZLO FUCHS.Infinite Abelian Groups : Volume I, 1970. Volume 11, in preparation KEIONAGAMI.Dimension Theory. 1970 PETER L. DUREN.Theory of HP Spaces. 1970. Borm PAREIGIS.Categories and Functors. 1970 PAULL. BUTZERA N D ROLFJ. NESSEL.Fourier Analysis and Approximation : Volume 1, One-Dimensional Theory. 1971 EDUARD PRUGOVEEKI. Quantum Mechanics in Hilbert Space. 1971 An Introduction to Transform Theory. 1971 D. V. WIDDER: MAXD. LARSEN A N D PAUL J. MCCARTHY. Multiplicative Theory of Ideals. 1971 ERNST-AUGUST BEHRENS. Ring Theory. 1972 MORRISNEWMAN. Integral Matrices. 1972 GLEN E. BREDON. Introduction to Compact Transformation Groups. 1972 WERNER GREUB,STEFHENHALPFBIN,A N D RAYVANSTONE. Connections, Curvature, and Cohomology: Volume I, De Rham Cohomology of Manifolds and Vector Bundles. 1972 XIA DAO-XING.Measure and Integration Theory of Infinite-Dimensional Spaces : Abstract Harmonic Analysis. 1972 RONALD G. DOUGLAS. Banach Algebra Techniques in Operator Theory. 1972 112 preparation E. R. KOLCHIN.Differential Algebra and Algebraic Groups T. BENNYRUSHING.Topological Embeddings WILLARD MILLER,JR. Symmetry Groups and Their Applications

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