Symmetric Hilbert Spaces and Related Topics: Infinitely Divisible Positive Definite Functions. Continuous Products and Tensor Products. Gaussian and ... Processes (Lecture Notes in Mathematics, 261) 3540058036, 9783540058038

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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Zurich

261 Alain Guichardet Faculte des Sciences de Poitiers, France

Symmetric Hilbert Spaces and Related Topics Infinitely Divisible Positive Definite Functions Continuous Products and Tensor Products Gaussian and Poisson ian Stochastic Processes

Springer-Verlag Berlin· Heidelberg· New York 1972

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Zurich

261 Alain Guichardet Faculte des Sciences de Poitiers, France

Symmetric Hilbert Spaces and Related Topics Infinitely Divisible Positive Definite Functions Continuous Products and Tensor Products Gaussian and Poisson ian Stochastic Processes

Springer-Verlag Berlin· Heidelberg· New York 1972

AMS Subject Classifications (1970): 46C10, 46M05, 60G15, 81A20

ISBN 3-540-05803-6 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05803-6 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.

© by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card Number 72-76390. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

Table of Content

Introduction .

3 3

Chapter 1. The symmetric measure space of a measure space. • 1.1. The symmetric set of a set. • . . . . • . 1.2. The functor S on the category of sets. • • . 1.3. The symmetric Borel space of a Borel space. 1.4. The symmetric measure space of a measure space. 1.5. Application to linear processes and factored probability spaces. . . . . . . . • . . • •

12

Chapter 2. The symmetric Hilbert space of a Hilbert space. ••

18

2.1. Definitions and general properties • . . . 2.2. The unitary operators UA,b,c and the 2.3. Relation with sywtietric measure spaces

Po:.

18

22 28

Chapter 3. Positive definite functions of type (S) • . 3.1. Positive definite functions. • • . • • • 3.2. Positive definite functions of type (S). Definitions and first properties . • . . 3.3. The case of commutative locally compact groups

30 30

35 38

Chapter 4. Conditionally positive definite functions and infinitely divisible positive definite functions.

47

4.1. Conditionally positive definite matrices and kerne Is. . • . . • . . . • . . • . . • 4.2. Conditionally positive definite functions on groups • • . . . . . • • . • . • . • • . . • 4.3. Infinitely divisible positive definite functions Chapter

5. Boolean Algebras of tensor decompositions of Hilbert space . . .

. . .

. . . . • .

. • .

a

Chapter 6. Factorizable positive definite functions on current groups. • • •.• 6.1. Definitions. • . .•. 6.2. Results. . . • . . •.. 6.3. Study of other current groups. Chapter 7. Gaussian measures on topological vector spaces •• 7.1. Definitions and general properties 7.2. Relation with symmetric Hilbert spaces. . 7.3. Examples of Gaussian measures . . 7.4. The Wiener transform • . . • . . 7.5. Equivalence of Gaussian measures . . .

4 6 7

47 50 56 62 80 80 82

90

93 93

96

• 100 103 105

IV 7.6. Quasi-invariance of Gaussian measures with respect to translations. . . • • • . • • • • • 7.7. Ergodicity of Gaussian measures with respect to rotations. • Chapter 8. Continuous products. • . Introduction. • . . Continuous products of complex numbers. Continuous tensor products of Hilbert spaces.. Continuous tensor products of symmetric Hilbert spaces. . . . . . • • • • • . 8.5. Continuous products of measure spaces. A. Infinite tensor products of Hilbert spaces • • Definitions . . . . . . . . • . . Associativity and commutativity . • Decomposable vectors . . . . Tensor products of operators

Appendix B. Cohomology of eroups. .

120 121 126 133 136 148 148 149 150 153 154

B.1. General definitions. • . . B.2. Study of Z1 in the case of a unitary representation.. Appendix C. Boolean algebras • • • • • C .1- General properties • • . . . C.2. and 0.3. Multiplicative measures • . . .

115 120

8.1. 8.2. 8.3. 8.4.

Appendix A.1. A.2. A.3. A.4.

112

154 154 157

functions

Appendix D. Restricted products of sets and measures.

157 160 162 163

D.1. Restricted products of sets and Borel spaces • D.2. Measure spaces . . . . D.3. Restricted products of measure spaces • . • .

163 165 166

Appendix E. Gaussian and Poissonian measures and processes. E.1. Gaussian measures on Rn• . E.2. Poissonian measures on R E.3. Stochastic processes. . •

168

Appendix F. Canonical commutation relations

172

168 169 169

Appendix G. Measures on topological vector spaces •

174

G.1. General properties • • . . . . . • • • • G.2. Relation with linear stochastic processes.

174 177

Appendix H. Conditional expectations and martingale theorem..

179

H.1. Conditional expectations. • . • • • • H.2. The martingale theorem. • . . • • • • • • H.3. Hellinger integrals and equivalence of measures.

179 180 180

Appendix I. Desintegration of continuous cocycles • . • •

182

v Notation index Terminological index . Bibliography •

• 188

• • • • 190 • • 194

INTRODUCTION

As was recognized by J.M.Cook ln 1953 the notion of symmetric (or exponential) Hilbert space, which is quite similar to that of the symmetric algebra of a vector space, is fundamental to a construction maQe in 1932 by the Russian physicist V.A.Fock in order to provide a more rigorous foundation for Quantum Field Theory. This construction, which is now calleQ the Fock representation of the canonical commutation relations, is presented in ) 2.2 and Appendix F. More recently the Japanese physicist H.Araki supplied a new impetus to the theory of symmetric Hilbert spaces, showing that it is related to infinltely divisible positive definite functions on groups and provides a new method for constructing all these functions on a given group. These ideas were further developed by K.R. Parthasarathy, K.Schmidt and R.F.Streater ; we present them in chapters 3 and 4. Chapter 5 is devoted to another deep result of Araki concerning what we call Boolean algecras of tensor decompositions of a Hilbert space, Which shows that symmetric Hilbert spaces play to some extent a universal role in the theory of continuous tensor proQucts of Hilbert spaces. This result is applied in chapt.b to the determination of the so called factorizable positive definite functions

OG

some groups which can be considerea as continucus

2

products of groups. From another direction J.Neveu and independently G.Kallianpur established recently a link between symmetric Hilbert spaces and Gausbian stochastic processes, giving a new interpretation of the works of Wiener, Ito, Cameron and Martin about the so called Wiener stochastic process. We present these ideas in chapt.7 where we have felt it useful to maKe a systematic exposition of the theory of Gaussian measures on topological vector spaces. There is also a relation between symmetric Hilbert spaces and Poissonian processes, but a less direct one, by means of the more elementary notion of symmetric set of a set which we introduce in chapt.l. Finally in chapt.8 we discuss various possible notions of continuous products which are potentially useful but which are not as yet quite precise, and we try to determine the directions in which one can search for precise definitions.

After writing these notes we learned that K.R. Parthasarathy and K. Schmidt gave at the same time in London a lecture on a similar subject entitled "Positive definite kernels, continuous tensor products and central limit theorems of Probability Theory" •

Chapter 1. THE SYMMETRIC MEASURE SPACE OF A MEASURE SPACE

In this chapter we introduce several elementary notions which are rather similar to that of symmetric Hilbert space. } 1.1. The symmetric set of a set

a set ; for each n s: 1,2, .... we denote by SnX the n quotient set of X by the symmetric group Gn acting on x" by perLet X

be

mutations ; SoX is by defini tion reduced to a single point w Definition 1.1. the set-theoretical sum (or disjoint union)

.

n::o

SnX

will be called symmetric set of the set X and denoted by SX. For every n-uple ( xl"" the class of (Xl'"

n in X we denote by

xn

cl( Xl'··X n)

X in SnX ; then we define a composition law n)

in SX by cl (Xl' •• x n) + cl (y l' •. YP )

:=

cl (xl' .• x n' yI' .. Yp)

•• X = cl(x l, .. x n) w = Cl(X l,·· Xn) ; n) l, SX becomes a commutative monoid with unit element GV, the so called w

cl(x

free commutative monoid with unit generated by X. Identifying SIX with X we can also write cl (Xl' .• Xn) ::. The canonical injection X

Xl

..• + x n . SX

will sometimes be denoted by

We can also describe the monoid SX ir. the following way be the set of all mappings

X _

N such that

N(x)

= ()

let

(X)

B

almost

4

everywhere (i.e. except for a finite number of x) ; N(X) is a commutative monoid for pointwise addition; there is an isomorphism of SX onto H(X) which carries each element

into the

function N where N(x) is the number of indices that

xj

= x.

j::

1, ... n

such

We shall refer to the first and the second descrip-

tion respectively as the S-pi¢ture and the N-picture.

i

1.2. The functor S on the category of sets

The monoid 3X has the following universal property : if G is a

.-...

CMU (commutative monoid with unit) every mapping u" :

tends to a unique morphism {;.(xl -t •••

T

Xn) )

SX

= U(X1 )

t-

G T

u : X

G ex-

defined by

}

u(X n )

= 0

(Ll)

Therefore 3 can be considered as a functor of the category

CMU : if we have two sets X and Y

sets into the category and a mapping

SX

3Y

of

f: X

Y there exists a unique morphism

Sf:

such that the diagramm X

where f is an arbitrary bounded Borel function on SX. We also have the n-th convolution power of r considered as t' n a measure on the monoid SX, and Sr.:: elft' == the convolution

nl

exponential of Sv

. If

X

r.

If

t'

'='

0, S r: '"

moreover

set' or v):::

is reduced to a single point a and r Uan

be identified with !i and

Sf'

with the Poisson measure

= 1, 60

SX can

L (n!) -1

n.,.o

Sr-'+

J. n

Properties of St'. \i)

Suppose that,... is non-atomic

let

be a Borel total order

8

1. 3 ; then if we identify Sn X

relation on X and define X' n as in with X,n, t'n is identified with

n xn) in X such that

subset of all n-uples (Xl"" n i::/= j ; X - Y

r

is f'6 n_ negligible since

is a symmetric Borel auoae t of the subae t s

In fact let Y be the

c (A ..... Y " ""

X' n)

x",

r@n(A"Y)

# xj

for

is non-atomic ; if A

is the disjoint union of

A 1'"1 Y

where 6'

xi

Gn ; then

runs over n!

:;:;

n! f'@n{Ar, X' n) ana on the other hand

(ii)

I f u is a Borel mapping

where Su is the mapping in

j

1. 2, u( r

)

and.

X _

SX

Sue S r-

SY

)

we have

Y

S(u{,..»

Su(S)-'-)

-.=.

associated with u as shown

are the images of rand

s»

respec-

tively under u and Su (easy verification). (iii) For every

e

f

1

L (X,r)

-< Sf< , t >

H

,f

1

belongs to

L ( SX,S,....)

and (1. 6)

exp( < r: ,f > ).

In fact for every n, f is rn-measurable and bO

9 Q

=

(nl)-l jlftXl)+ •. +f(Xn)/ dr(xl) •.. dt'(x n)

£,'

(nl)-l

J

(n !) -1 n 00

n:::l :::

e

which proves that

f

«

(.­tf(xl)/t" •• ­rlf(xn)j) dr(x1)···cli'(X n ) .

n­l)! ) -1

/Ir ll

JI

r

II n­l

r: • if I ;:. . 1/,.. 1/ n­l

-c,

containing

all P-negligible sets and making measurable all functions

X(n) B

'

B E A , we have to prove the following Proposition 1.1. For every We already know that inclusion. '/; A and S( T;\!!)

SeA x g)

A Ed

we have

A

« '1 and we must prove the converse A A are inverse images under the projection

of two Borel structures on

S(A'" R) ; the

first one is staLdard, the other one is less fine ; it is sufficient to prove that the second one is generated by a sequence of subsets which moreover separate the points (see [9], App. B}. To do this we can suppose

A

T ; then

et

is generated by a sequence

Al,A 2, ••. and we can suppose that the set [An}contains T and is closed for finite unions and complementation; then it is known

16

(see e.g.[17Jlemma 6.1) that (l is the smallest set containing

1

and closed for countable increaSino/nions and complementation ; but if Am, A ,xi n) converges simply to xin) , and on mi the other hand we have x(n) T­Am

=

This proves that the functions xin)

» ; now take a I

generate

m

countable basis of open subsets U in R ; the sUbsets p V

m,n,p

generate

'D' .

=

Let us now prove that they separate the points of

S(TxR).

Let N,N' two distinct elements in a finite set

in T SUch that

o

N( t,a) ::

since

there exists

S(Tx R)

if

t # to, ••• tq ;

N' we can suppose that there exists N(to,ao)

take m such that

:j=.

8

0

such that

N'(to,ao)

Am contains to but not t l, •• t q

n such that

L

a

Wn(a) N(to,a)

Z.

1­'

a

there exiJlts p such that

L a

wn(a) N(to,a)

e

Z. (.IJn(a) N'(to,a) 4 a

then by (1.12) we have

U

p Up

I

W

(a) N (to,a) n

there exists

17

X(n)(N) A

-=-

X(n)(N') A

z:

m

m

o»

a

n

(a) N(to,a)

1- w n {a)

N/(t ,a) 0

a

'

f/

i.e. N E

V

m,n,p

N

f

e

V

m,n,p

U P U

P

Chapter 2.

THE

HILBERT SPACE OF A HILBERT SPACE

In this Chapter we introduce the symmetric Hilbert spaces as well as the vectors EXP a

and the unitary operators

UA,b,c

which

will playa fundamental role throughout this lecture.

J

2.1. Definitions and general properties

the n-th Hilbert tensor power tric group S

G

n

6n

H&n

= R or

£ we can form

= 1,2, .•

; the symme-

K

Given a Hilbert space H over the field where

n

acts by unitary operators

U

s,n

in

for each

we have

the operator

P

(n!}-l

n

U

s EGn

s,n

is an orthogonal projec-

tion, the image of which is denoted by SnH, the set of all elements in H8 n Which are invariant by all U , and called the n-th s,n 1 symmetric power of H. We set S H H. One gets

=

SnH in the following way : let

an orthonormal basis of

e 1,e 2,··

be an orthonormal basis of H (we suppose H separable for the sake of simplici ty); for any sequence of positive integers SUCh that

"i

+

n

2 + ...

(n!}l/2 (n

1

!

n

2

=.

n1 , n 2,···

n , the element

! ...

}-1/2

belongs to Snn ; and these elements constitute an orthonormal basis of SnH•

19

Definition 2.1. The

SH of H is the Hilbert

snH , that is the set of all sequences

1. if

and (x I y)

=.

L: (x

n

x ,,2 < 00 n

x

= (xn )

with

the scalar product is given by

;

I y n) •

The space SH is sometimes called "exponential space of H" (see

[2) and [25]) ; we shall give in example 3.1 a new interpretation of SH (for a real H) as the real Hilbert space associated with a kernel on H. Defini tion 2.2. For every x

-e.

SH

(x ) n

EXP a

such tnat { 1

z:

x

EXP a

a

li

H we denote by

n

::

( n :,)-1/2 a rS> n , that is

, a , (2!)-1/2

ae a

, (3!) -1/2 a'1'" An

EXP a/EXP x) 1

e

(ai' x)

;

0;

consider f as a mapping of the real vector space H into the real vector space K ; its p-th derivative at a point x is given by (p)

fx

(bl, .• bp)

""

= 2.. A,

1

e

(ai' x )

(alb

... (a,lb )

ll)I P

21

in particular for each

and will be identically zero

beR

we

shall have f

lp)

vp

:;; :2

(b, •• b)

o

i

: 0,1, .. n-l

A.1. 's

this is a linear system with respect to the

which admits a

non trivial solution; its determinant {of Vandermonde type) must be zero, and there exist two distinct indices i and j such that (a-a.lb)

(a.Jb) , Le. J

.

0 ; the vectors

J

1.

are not null and every vector b is orthogonal to at least one of them, which is impossible by part a) of the proof. c) We finally prove that the

are total in SR, i.e. that

EXP a

the closed linear sUbspace L generated by them is equal to SR. Since SnH is generated by tensors of the form cient to prove that each

=

a

belongs to L. For

EXP ta ; we nave

we have

f(n)(t)

a

18

n

Example 2.1. The space SK, with phic to

L

2

(E, v ;K)

E"

z

0:0

Yt

where v is the reduced Gaussian measure on

1

en

into

(n!)-t h

(nl)-l a n h {x ) n

ax-a

the function x_

f(t)

KE E or C, is canonically isomor-

=

e

E.

form an orthonormal form an orthonor-

mal basis of SK, there exists an isomorphism

00

E se t

L.

In fact by App.E.l the functions basis of 1 2(R, Y ;K) ; since the elements

which carries every

t '(

since

a

Land

a ill' n, it is suffi-

n 2/2

SK

1

2

(E, Y

and every EXP a

;

K)

into

22

Proposition

2.'. Suppose H is the Hilbert sum of a family of Hil-

; denote by u i the element EXP U of SHi' h(u) There exists a unique isomorphism T; SH __ SHi carrying iEI EXP a into a i for every element a = (a i) E H with a i z: O oert spaces

a.e.

(For the notion of infinite Lensor product of Hilbert spaces, see App.A). Consider (EXP

a;. £a i), b::.

alEXP b)

e

(b

i

)

with

2(a

(a I b)

::

o

a.e.; we have

ne

(a./ b. )

a i '" b i

e

il

b ) i

-=

then the proposition follows from the fact that the elements EXP a are total in SR and the elements ® EXP a Remark 2.0. Denote by E

c

htu )

i

are total in @

SRi

the complexification of a real vector

space E ; then SH c is canonically isomorphic to (88) c • In fact consider an element x -to iy in Hc with x , y (8 ; developing the element

(x.riy)@n

we get first a number of members containing

an even power of y and secondly a number of members containing an odd power of y ; denote respectively by a and it the sum of these members ; then there ( x-s- . ). EXP

a

with /\ E £.

=£. -t.O}

;f-

and

23

a

E

H ; we recall that

implies ,\ '" A', a:. a'.

EXP a : A'EXP at

Definltion 2.3. We denote by

t

H the group of all unitary operators U in SH such that U and U- l preserve HI globally. Our next

aim is to determine the general form of these operators. For any Hilbert space H, 1£(H)

will denote the group of all unitary opera-

tors in H. First each unitary operator A in H extends in a natural way to a unitary operator iu SH, namely (I,A,A call it

2

, •.. ,A

@

n

, ... ); we

UA,O,1 ; we th us h ave UA,O,l (EXP x )

and therefore UA,u,1

UA,U,l

=

(2.1)

EXP Ax

belongs to

tH

moreover the mapping

A

is a morphism of groups. for each x and y in H we

Secondly consider a vector b in H have

setting

x = Y

exp (-tllbIl 2

(;J Ib») EXP (x ... b)

exp (-tllbIl 2

(ylb)) EXP (y+ b)

we have (EXP x I EXP y) ;

(x I Y)

since the elements

EXP x

and X (b being fixed) are total in SH,

there exists a unique unitary operator in SH carrying each EXP x into the corresponding X ; we call it U

l,b,l (EXP x )

=

U

I,b,l

exp(-tllb Il2 - (x l b)

and we have EXP(x-+ b)

(2.2)

24

and moreover

U

UI , b , 1

vne easily checks that

exp{i Im(blb'»

UI,b+b',l'

c !l. (the group of all complex numbers having mo-

dulus one), every

b 10 H and every

U

::.

A,b,c

U

=

UI " b' 1

Now for every

clearly

belongs to

I,b,l

U

A,b,c

we set

U

l,b,l UA,O,l

C

belongs to

A,b,c (EXP x )

A 'U.(H)

j

by (2.1) and (2.2) we have

c exp(­lIbI/ 2/2 ­ (Axf b)

=

(2.4)

b)

in particular A,b,c (EXP 0)

U

c exp(-I/bll 2 /2) EXP b

:::

moreover one easily checks that U U A,b,c A',b',c'

I

Lemma 2.1. The mapping into

U

:::

AA',b+AbT,cc'exp(i Im(bIAb'»

(A,b,c)

r----,lIo

U

A,b,c

from 1L(H)

(2.6) x H x

!l.

H is bijective.

(2.,) A,b,c = UA" b' ,c' j c::: c' ; then (2.4) implies Ax+b::: A'x+b

Proof of the injectivity : suppose implies

b::. b'

and

for every x, i.e. A

U

= A'.

Proof of the surjectivity. Let U be an arbitrary element of U(EXP 0)

is of the form /\ EXP(­b)

it

with /\ c;c.£

5H ;

, b 4E H ; multiply­

ing U by a scalar of modulus one we can suppose ).. > 0 ; we then have U

l,b,l (U(EXP 0»

= ),

exp(­llbI12/2 + /Ibfl2) EXP 0

25

since U is unitary and).. exp(llbU 2/2 ) »

0

we have A exp(1J bll 2 /2)

Ul,b,l U=U' we have U' 5 and U'(EXPO) = H EXP u . We are thus led to prove our assertion in the case where

=1

setting

= EXP

U(EXP 0)

EXP(Tx) 1

0 • For every x in H, U(EXP x) is of the form

where T is a bijection of H onto itself

=

(EXP x I EXP 0)

= /\ (EXP Tx ) EXP

=

(U(EXP x ) I U(EXP

we have

0»

0)

for x,y in H

= =

(U(EXP x ) I U(EXP s)

(EXP x I EXP y) (EXP Tx J EXP Ty)

(Tx I Ty)

=

He (Tx I Ty)

e

(x I y)

mod

(TxITy) (2.7)

2Tri

He (x I y)

=

Let us consider H as a real Hilbert space with scalar product He (x I y) ; T is a bijection of H which

0 and the scalar

products; therefore it is an E-linear and orthogonal operator. Now by (2.7) for any real /\ we have I Ty) - (xly» which implies

=

(Tx I T ,\ y) - (x lA y)

(TxITy) - (xly)

H, T is a

=0

E

; for the complex structure of

preserving 0 and the scalar products ; there-

fore it is a £-linear and unitary operator. Finally Theorem 2.1. The set

1L(H) X H x

g

U

= UT,O,l

•

is a group for the following

law: (A,b,c) (A',b',c')

=

(AA',b+Ab',cc' exp Ci Im(bIAb'»)

(2.8)

26

with neutral element (1,0,1)

j

it is a topological group for the

product topology of the strong topology on U(H), the strong topology on H and the ordinary topology on U

A, o, c

g.

The mapping

is a bicontinuous isomorphism of U(H) x H

)
" (

J

)J

but (b g lw) does not make sense in general and the function

41

b)

that c is continuous, or, equivalently because, of a), that c g tends to 1 when g tends to e. We can suppose that g remains in a

compact C ; choose V compact such that (3.7) holds; take an '1> O. There exists a compact

K,) V

.A

such that

v (G - K)

il

using

(3.8) we have

>-

If( - 1)2.

1 - i J(g, X ».dv(-x)1

(l+h(g, X

».dv( 1\ } I + if

I

K-V

+

hG-K (ex .s

I

> -l-iJ(g,;ddy / ;

the first integral tends to 0 since h is bounded and the integral

fa

t-l,) 2

too since on

J(g,1}

dr(

K-V, v

the second one tends to

tends to 0

oX )

is bounded and

-l

Q

as well as

converge uniformly to 0 ; the third one is less than

multiplied by a con,tant. We have thus proved Suppose that A does not contain the trivial representat ion ; then for every l-cocycle b or equivalently for every w

-

in H, the function of the function

Im(b IA

(g,g')

g

I----'J>

g

g

b

s ,)

is the coboudary

1m !«x,g > - 1 - i J(g,

X

}).dv(X) ;

therefore (3.5') has a solutlon, given by (3.9) ; the corresponding

If

is given by '(g)

= ex p

[!

> - l-i J(g,X ».dV(-X)J

Conversely if v is a positive measure on

J1< X,g.>

- 1)2. d v( X )

(3.10 )

A

G -ft} such that

is finite for every g and tends to 0

when g tends to e , the function !.f defined by (3.10) is a c.p.d.L

42

of type (8) : one has only to take for equivalent to

v , say

function such that

fA

:::

=-

II (1)

r i/

where

a normalized measure

r

is a strictly posi tive 2 H:: L (G,I"'),

1 , and to set

LV

==

r-t , '«: mulUplication operator by the function b

g = f'unc t i.on

Case 2.

It

It

.e > -1)

'( (x

)-t ;

c

g

as in (3.9).

We now suppose that A is trivial.

In that case a l-cocycle b is nothing but a ism of G into H and the corresponding C -1 - i Jlg,X».dV(X)] (3.12)

where Xo is a character of G ; Q is a continuous positive quadratic form

null if G is a union of compact subgroups) ;

Y is a po s i tive Radar, measure on

II _lI2.d'll(;\')

A

G - lE 3 such that the integral

is finite for every

g

e G and tends to

li

when g tends to e ; and finally J is defined as in lemma 3.2 and is null if G is a union of compact subgroups. Moreover such a measure v is boundea on the complementary of every neighbourhood of E.

45

Example 3.2. For function

G

X.

= Sn

the function J can be replaced by the

g. X /(1-J.,,2)

where

L g.

g.'X:::

j

J

X.

X2 , which has the same effect. On the other hand J

ten as

J

and

;x:2 =

v can be wri t-

where 6' is an arbi trary

dv (it )

finite positive measure without mass at 0

j

then (3.12) can be

rewritten as

where Xo is an element of gn, Q a positive quadratic form and 6

a

finite positive measure without mass at O. We thUS see that our p.d.f. is the product of 3 p.d.f.: a character, a Gaussian p.d.f. and a Poissonian p.d.f.(see App.E). Example 3.3. For

G;::.

, theorem 3.2 does not apply but we can

proceed as follows : for the trivial part of the representation A, bg can be written as b g :;: g} where H j then (b Ib ,) :;: g g 2 gg' is real, 01 is equal to 1, c is a character X-.. and !f (g) 11 X(g) exp(_kg 2 )

where k is an arbitrary positive number. For the

(J

non-trivial part we can take

g.F(X)

where F is as in the figure. Finally we have a formula quite analogous to (3.12) :

r (g) =

eXP[i g X C

Example 3.4. For

kit j I G

=

2

there exist c.p.d.f. of type (S) of a

different kind, for instance (setting

-

I - i J (g,:x ) ) . d \I (jd

as already noticeo. in lemma 3.4 , Im(b"lb") g g'

a

;

is null; therefore

is a real character of G and we have

=

He

::

-i1bgIl2/2 - i[o(g) + 1m !(ex,g> -1 - i J(g,J

- 1 - i J(g,X».dv(>:).

Theorem 4.2. Let G be a separable commutative locally compact group; the continuous hermitian conditionally positive definite functions on G null at e are exactly the functions .,.-(g)

=

i J(g) - Q(g)

t

1 G

.e '>

- 1 - i

J(g,x».d\l(%)

(4.4)

55

0

where

is a real character of G and Q, v ,J are as in tho 3. 2.

(See[ 34], th.4.4.)

There remains only to be proved that for each

0 ' Q and

v ,

formula (4.4) defines a c.h.c.p.d.L null at e , which is trivial.

G = En

Example 4.1. For

to what has been said in ex-

ample 3.2, (4.4) can be =

i

o

- Q(g) +

tnis formula

sometimes referred to as "Levy-Khinchin formula".

Example 4.2. If U

1S

a union of compact

if G is a p-adic group t'(g)

gp)

(for instance

(4.4) takes the simpler form

)a((-l).dv(A).

'"

Remark 4.3. If tne c.h.e.p.d.f. of th.4.2 is real, it is equal to its real part and formula (4.4) becomes 't"(g)

::::

- Q(g)

= where

v'

= v!2.

Q(g)

-t"

(

J-C

-1,

G

Re( - 1) .dr'(;t) I..o(x,g> - 112. d Y' (;;r)

This formula can be proved by a more direct method

(see [20}) ; but one can hope that the present method can be extended to other groups than those which are considered here.

56

4.3. Infinitely divisible positive definite functions Definition 4.4. A continuous positive definite function

on a

topological group G is said to be lnfinitely divisible if for every positive integer n there exists a continuous positive definite function w satlsfying

w

n

'f.

=

Example 4.3. If 'it is a continuou.s hermi tian conditionally posi'+' tive definite function, e is an i.d.c.p.d.f. (cf.prop.4.3). We shall prove later

a partial converse of this result . ."..

We now consider triples (H,U, S)

where H is a Hilbert space,

U a (continuous unitary) representation of G in H and clic vector for U ; we recall that two triples (H,U,

ft)

f)

a unit cyand (H' ,U ',

are called equivalent if there exists an isomorphism of H onto

H' carrying U in to U' and

f in to

'..::.

> •

A triple (H,U,

Definltion

is infinitely divisible is for

every positive integer n there exists a triple (Hl,U that (H, U,E) is equivolent to (K , U1IP nlK ' z:) 1 c 1 osen- SUbspace

0

f

HIa9 n

't"Qiln )

generated by the elements

(It should be noted that for it is included in

such l, where K is the

n > 1

K is distinct from

since

Sn Hl). Then a c.p.d.f. is infinitely divisible

iff the associated triple does so.

I

Pr OPOSi tI On 4.b. If (p is an infinitely divisible continuous positive definite function, the set of all g such that

I is a (open and closed) SUbgroup. (See [33J, lemma 4.2.)

*

0

57 Ite

can suppose 0

a finite partiLion 1. =

such that It is clear that

o

1 ; g is 6 -multiplicative (same

reasoning as in lemma S.2) and therefore non increasing; then f is positive and non decreasing; moreover we can apply part b) of the proof of prop.C.l with

f

replaced by g ;

becomes our f and

(S.l/is proved. The last assertion of our statement follows from lemma C.2.

70

Proof of theorem 5.1. \ie choose a set K with a biJectlon

a)

U- 1 ( CJ ) . By lemma 5.1 with every

the U element of K as

we set

iff

x'

o

have b)

=

ti'

x' ,.. x(J

0 ; clearly

a Y&!,

K

fi

satisfying (i) and (ii) with x::: a We define the subsets KJ by a & K.t)

- w&

lJ

is a non decreasing family and we

=K .

'" Lo], K

Ko

a ( K

(x )

we can xaaacc i at e a family Va

HO • We define

K

V:

1L

We now define a function

(a,b)

(alb)

with the notations of lemma 5.2 where

x

= Da

by

(alb) =

= Vb.

, y

We have

the following properties (UaIDb) (ai b)

:::

::.

()

(x,,}Y6)

=1

(ala)

:;t

0

(a I a)

=0

if

=0

a

since

implies

x::: w

x

= "-;;' ,

a

, h(e) ::: O.

for

1 (recall that

(x (x y ) a

a;:: 0

since

( x Ii I wt »

::: 1

v implies

het) -

= 11 via 1/ 2 ).

(xIx)::: 1 ,

x=w (alb)

(immediate)

a E K , b e Ky '

==:> (a [b ) ::. 0

a

I')

.=

(alb) :: 0 of the form

since for every

=

/12

Vb E y

e

@

:.

(x k jYj ) "1 11

:::

( x if

1

K{)

w, B

I

wI'/

1

1',

we have

and

G'

(x jj IY," ) 2'2

) (tv 7

2

I Y1)

2

)

:::

1 .

a EKe' • In fact for every we have

e

O

YE H

71

e

1

(a I b)

(x/y)

o ::: then for every

(xeIYO) (X ' {,Jel tl

=

(xli

Z

:::

zi}:iI'

O

6 H

Z t7'

= O

since H is total in H this implies X

e = wa '

a

to

K

a

, )

IX

\ ()

o

a

-v.)

:::.

0 ,

I

We think that the present construction of the scalar product in K is simpler than Araki's; but unfortunately we have not been able to simplify the construction of the linear structure given in c). c)

We now endow K with a linear structure, i.e. we define b

X

8

,

e .£ , a, b E K • Set

for ol, X

I

d

,

' '7

y'

X

= Ua

a +

, y = Ub and define

as in part a).

c.l) We first prove tnat the element

'l:'"' __

of H has a limit in H where

."

(8 )

i Le I

runs over the set of all

finite partitions of 11 • We have to prove that for every there exists a ti tion

$ finer

partition than

II z_

-

'-to

't we have "

S

(5.4)

E.

max (;I x 1/2

;:.

> ()

r such that for every finite par-

. Set

For the sake of simplicity we take k

.-

r-

/I xII

4)01/1-

(5.5)

choose £' such that k[exp(t l I

/Ixl14 log !Ix 11 2

k[exp(c'/t(1 4 ,l x Il 4 log I/x 1(2

- 11J - 1]

f 2/ 4

(5.6)

e 2/4

(5.7)

72

and small enough in order that

t.'

t

/Ix ,,2

e

t

- 1 -

t

t

2

(5. 7' )

•

Taking the notations of lemma 5.3 with

=

f(8)

since

- nx /I

I

1 , /I x (} II

Ilx II

-2

=

t)

I

.

• II

it

we have

x II 2/1/ xd It 2

x 0 ;1

!' II X II

we have

f{B)

5.3 there exists a finite partition of

By lemma

such that

ViE I ; we shall wri te

instead of Z,:i

X

uJ

oi '

Setting

x'

e,1. '

1.

r.r

Ai

v-\ '

= {Y i )

xi we have

Z,

1.

=@Zl'

Let

d be a fini te parti tion of 11

fi

ne r than

7': :1 :

where i runs over I , J runs over some set J, , and iJ. 1.

1.

:=

(&,

,)

1.,J

"

J

e,1.,J, ;

we have

= W.

1.

X. 1.

x!

1.

x. 1.

::

@cJ. ,

::

@

W. 1

j

l.J

x

j s

iJ j

( w ij + x • ) ij

Cit

j

w,l.J.

where u i is the sum of those terms in s&l (UJ i j + x 1j ) j tain exactly one ral ones. Then

x1 j

which con-

and vi the sum of those which contain seve-

73

= Wi

zi

'*" ...._J x 1' @(w .. +r,(xi · )

lJ

j

J

where w. is the sum of those terms in

which

1

contain several z

't

= s:

Set

u

=

Xij Z.

j

i

W (w

i

I (w i +

:::

1

i

Z.

::

lj

i +

.:(

i

1

1

(w i + K u.1 + wi)

i

c< u i ).

(5.10)

u. +r.lv.)

We shall prove that

II Z'1

- u II

,s

/2

(5.11)

II

- u II

-s t /2

(5.12)

f;.

which will imply (5.4). We have by (?e)

(w ijlXi )

I,xi

x tl '

-=

u.J J

'> x

-:

w8 .

ei

,

1

a cco ru i.ng to lemma 5.1 we can write H

x

:.

=

H

h

0'

X

a'

h

e \&l

(kJt)

) i

iEl (

® hI

x

Hr}.

°i w

whence

a.= 0 .

e. ,

KtJ is the Hil-

Ko . 'so It is enough to prove that

, a

0

f:}

partitlon

bert sum of the a

x .1.

a

:.

0

78

f)

Finally we must construct the isomorphisms

For

a,b

f

CPg; H(j-+ S KG •

K we have

(Ua)Ub)

(EXP a j EXP b)

;

since the elements Ua (resp. EXP a) are total in H (resp. S K

, we can consider F( S) as a SUbgroup of the product group F

(S. ) 1

ic I

n

which contains the restricted product

if I

the definition see lemma 3.1). The restriction of an a subset S is denoted by

f

( s)

• The elements Xg,S

I

F

(S,)

(for

1

f

F

to

defined by

t E S

if

if not generate the group F. We now consider a positive definite function note by

negligible if

we de-

F(S) ; we say that S is If-

its restriction to

cp(:)

r on F ;

is identical to 1. We assume the following

condi t i ons

If (t) =

(i)

1

'f is factorizable,

(ii) L.i

3i

where

S, S.1 '

L /q z and

(S.) 1

(S)

f) If

1

E

(f

(f

a , e. for every countable partition

n

, and for every

(S,)

1

l)

i

(S.) 1

t

t

must be the co boundary

- 1 - i J(g,?:».dyt(;t)

which we

; this implies that it is both symmetric and anti-

symmetric, hence null, and ).., t is a real character. We finally obtain ;0-

of

h (EXP 0)

®

S(I! e ) J

..; a

(a) E' F J

an isomorphism T

J

h (1)

2

@L(R.,M·,R) J

J

and finally by App.D.3 an isomorphism

-

/ J -

a

2 J

12)

98

carrying each element

f j (where

@

for almost all j) into the function We thus get an isomorphism T of every

RlP

a

with

n

a

X-

3H

r----'lI>'

onto

f ..: 1

L2(E',,.,. ;!!)

J

carrying

into the function

F

::

exp(

and by continuity this still holds for

a

H.

J

and

n f /.x j ) •

a. 2/2)

j

exp(

2 L (R j , t' j ;!!)

f j

-

J

J

Proof of the last assertion: for any real numbers

-)/ a,,2/ 2)

u l'· •• Uk

we

have

=

L

n=o

(nl)-t.(u +••• lal

n!

= all the other hand

=

+..• ukak - (uiilalll2 K

= n

=

- u

2

J

Ii

1" . . .

a II 2/ 2 ) J

II n, 1 1/ a II J u J (llJ.1) - • h n . ( j=l J J J lC

n

a J III a j 1/ )

99

then our assertion follows by identiiying the coefficients of "i

ul

nk

••. uk

in both expressions of

Corollary 7.1. If

e

l,e 2,

T(EXP(ulal + ••• ukak

».

•.. is an orthonormal basis of H, the

elements

n

J-=l

constitute an orthonormal basis of

L2 ( E' , r

when

(n

l,n2,

••• )

runs over all sequences of positive integers Which are zero almost everyWhere. In fact the elements

form an orthonormal basis of SH (see

j 2.1). QED

By remark 2.0 we car, aSSE,lI't that SH

c

phlC to

is canonically isomor-

which is the complexification of

if)

is)

the followlng theorem gives further information about this isomorphism. Theorem 7.2. The isomorphism T of theorem 7.1 extends to an iso-

SH c

morphism a

into

a

and

L

2

EXP a

if)

into

carrying, for every exp( a - r( a )/2).

a

in He '

100 rv

a

'V

'V

"1- i a a l + i a 2 ' a l and a 2 E H). 2 if l the required The proof is quite similar to that of th.7.1

is defined as

a

isomorphism is the tensor product of the various isomorphisms 2 L ( R , fA j;£)

s(£ e j )

j

described in example 2.1.

Remark 7.2. The preceding results, as well as many subsequent ones, can be transposed to the general situation of linear stochashc processes by using App.G.2. Suppose for instance that we are given a Gaussian linear process X on E with a probability space (0, d.,F) with continuous characteristic f'unc t i ona.l , and denote by

'1

the sub- 6 -algebra of Q d ef'Lr.ed by the functions

Xx ' x E E. Tnen th.7.2 yields an isomorphism of SH c onto the space L2(D,'D,F 11.».

§

7.3. Examples of Gaussian measures

Example 7.1. We take functions on

R+

E

=

j) (R

= [0,+00[

. _R), the space of all real

-+ '

which have compact supports and

are infini tely differentiable on ]

0,

+

00 [

and infinl tely diffe-

rentiable on the right at 0 ; we endow it with the usual, inducWe consider the quadratic

tive limit topology of the space form Q on E defined by Q(x)

=

and the associated symmetric bilinear form

o

f(t) x(t) dt • It can be proved that

(see e.g. [4] ) ; therefore we can asso-

R a function J s defined,. -almost every-

Js (f)

= f(s)

f:>

f

; we thus get a sto-

Js ; moreover

R : s

oelongs to

Y

one can prove that

and that the covariance of our stochas-

=

tic process is given by

Min(s,t) [the reader can

easily make a formal verificatlon of this fact by approximating

Js

by a sequence of functions belonging to

E}.

This stochastic

process is called Wiener process because it has been introduced by

in his mathematical description of the Brownian motion. Now for every x in E we set

Ax

= x'

(the derivative of x) ;

A is a biJective mapplng on E and we have Q(Ax)

= . :x(u} 2

du o therefore A extends to an isomorphism of

onto H, the

completion of E for Q. Applying th.7.l we obtain an isomorphism 2 2 of S(L (R+ ; R» onto L ( E " fA ;R) ; its restriction to the subspace Sn(L 2 (R..- ; R» is called "n-th stochastic Wiener integral" and the image of this subspace is callen "n-th homogeneous Wiener chaos" ; the decomposition of an arbitrary element of 1 2(E' ,r- ;R} on the basis descriOed in cor.7.l is referred to as the "CameronMartin decomposi tion" (cL [24] and [30J

7.3).

102

Example 7.2. We take

=

E

r

del' a Gaussian measure

R

; then E'

=

I R

and we consi-

on E' which is invariant uno e r the shift

v

U::: (u ) E E' ; its FouA : (Au) n = n e -Q/2 must be invariant under the analogous tran-

transformation riel' transform

sformation in E, hence Q must be of the form

=

Q(x)

L

a

p,q

p-q

x

P

x

q

where a is a real positive definite function on

• Let

V

be the

measure on I such that with every fx(t)

=

x n

(x n) EE the function f x on I c we associate int v r Irx 12 ) and the xn e ; then we have Q(x) :::

=

2 to an isomorphism of Hc onto L x extends Applying th.7.2 we get an isomorphism

mapping

x

I---'J;l>

f

cr, v ).

carrying every sufficiently regular function h on I into the fun(u )

:L'x

u

where the x n 's are the Fourier coefficients of h. Let V denote the multiplication opet rator in L2 (I,v) b yie ; then it is easily seen that L carries ction on E' : u

the operator U V,O,l 2(E', U in L t') : (u

n

:

(I,V,V

r) (u )

n

n

, ... )

into the shift operator this result can be used

to study ergodicity, mixing,etc. properties of U (see [42]).

103

7.4. The Wiener transform Let

r

be a Gaussian measure on E' •

We begin with the case where

JA

-

is the reduced 2(R,f Gaussian measure. We denote by T the i somorphi smt, Sf L

a)

E

::

R and

described in example 2.1 and by W' the unitary operator in W'

we have

::

U

::

i,O,l

W' (EXF a)

EXP ia

Sf

(l,i,i 2 , ••• )

f . We set

";I a

tnis is a unitary operator in

called Wiener transform

in [40]. We also have

W

where Z is the isomorphism

2 L ( R)

2 L ( R,

aefined by

(Zf)(x)

and F is the usual Fourier transform :

=

(Ff)(y)

t

- ""

e i xy/ 2.f(x).dx

[ To prove this it is sufficient to verify that Z-lF Z T EXP a

\j a f

f

W T EXF a

=

and this is a straightforward computa-

tion] •

Moreover W carries every function eiax-a2/2

and every h

n

into

inh

n

(this is clear from the first

definition of W) ; every function x n into every function

h (x2- t ) n

into

(i2-t)n

1

n

hn and n x ; this can be seen

from the following formal computation (Which could be made rigorous ) :

104 00

(n!)-l W(x n)

4n:o

:::

W(

L

an 2- n/2 (n!)-l

= w(e ax/ Y2 ) 2

2

.r:

ea /4.w(eax/J2 - a /4) :.

e

a

2/4

• e

iax/J2

f

2 a /4

2 +a /2 e i ax/ V 2

whence

::.

in h (x/V2). On the other hand n

Lan (n1)-1 W(h (x/ {2 » n

=

1'( Lan (nl)-l b

=

W(eax/V2 - a /2)

=

w(e-a2/4. e ax/ i2

=

e

-

2 -a /4

2

• e

-

a

(x/ V2 »

n

2/4)

2 i ax/ V"2 + a / 4

e i ax/ Y2 in 2- n/2 (n!)-l an x n

::.

b)

f

We now pass to the general case. We call

what was called isomorphism (EXP 0) S£

T, W',

j

'

Tj

,

Wj' Wj

in part a); by th.7.2 we have an 2(E', T: SHe --...". L J-') we can identify SHe with and

I'

1 2 ( E' , r-)

wi th

U) 1 2 (R,

we can consider the unitary operators and we have again called Wiener transform. Note that

W'

::

.

0

Aj

>

y

-1

moreover

J

for almost every j, say for

Aj " 1

'f J

whence

=

and denote by

(A j +l ) - t

rj

2

L 1\ j

j ?' m

.

1


-

""

2 -2 - 1 . (6+ 2 ) ) J J 1

'l

J'"

Y 2J (J J +-2 )-2

(7.5)

110

Taking

F

=

g

f

j

we see by (7.4) that 4(,,·+1)(r·+ 2)-2 vJ

which implies that

J

fj is bounded by some constant A

then (7.5)

implies

If a finite dimensional sUbspace F

is included in another one F 2, K onto S2Fl ; therefore KF F2 1 KF has a limit K in S2H• Finally take x in E and F F g x 2) l we have l is the orthogonal proJecGion of

=

y(kc+t) )

in E such that

Proof of (i). For every n there exist

CP (xi ,X j

)

t

'='

ciated with Q

where 0 ; moreover every invariant normalized

f' c defined by

measure can be written in a unique manner as v is a no rma.lazed posi tl ve measure on

where

g +.

In fact the ergodic invariant normalized measures are the extreme points iI, the set of all invariant normalized measures ; and if E is nuclear every c.p.a.f. is the Fourier transform of a'measure. Remark 7.6. In particular every Gaussian measure

r

on E' is In-

variant and ergodic under the group of all continuous linear transformations in E preserving the covarlance or r ; in the case where E and Q(x) L x 2 , this fact also follows

=

n

from [21]: since identical,

r

is of the form

e

r«

where the J-O e1

we have

t

2

tx-ax /2 dx

-00

for every integer

f

i-

CO

n

= 0

we have

2

x 2n e- x /2 ax

-0.0

f

2 x 2n-l e -x /2 ax

+00

:::

O.

_'l

The Hermite polynomlals h n are defined by

a > 0

169

2

e

ux-u /2 n""o

h

n

(x)

=

2 e X / 2 (_l)n

e -x

2

12 1 2(R,

(n )-t h

form an orthonormal oasis of n where y is the reduced Gaussian measure on R. the functions

§

Y )

E.2. Poissonian measures on R A prooaoili ty measure

f

R is called Poissonian {or mixed

on

if its characteristic function nas the form

= where Y

exp

rlR

J

i k t - 1) dV(k) +

a finite positive measure on

(E.l)

R and a a real numOer.

If V lS concentrated on a single point, say v

= >. dk

'

t' a s saa d

pure Poissonian ; tnen