Lecture notes in mathematics

Eckmann 486 ~erban Str&til& Dan Voiculescu Representations of AF-Algebras and of the Group U oo r Springer-Verlag Berlin... Representations of iF-al~ebras and of the 6roup Lecture

Edited by ,~ Dold and B Eckmann

486

~erban Str&til&

Dan Voiculescu

Representations of AF-Algebras and of the Group U (oo)

r

Springer-Verlag

Berlin Heidelberg-NewYork 1975

Representations of iF-al~ebras and of the 6roup

(Lecture notes in mathematics ; 486)

Bibliography: p

Includes indexes

i Operator algebras 2 Representations of alge-

bras 3 Locally compact groups 4 Representations of

groups I Voiculescu~ Dan-~-irgil, 1949- joint authoz

II Title III Series: Lecture notes in mathematics

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I N T R O D U C T I O N

Unitary representations of the group of all unitary opera-

tors on an infinite dimensional Hilbert space endowed with the

StTong-operator topology have been studied b y I.E.Segsl ([30]) in

connection with quantum physics I n [ 2 ~ ] A.A.Kirillov classified

all irreducible unitary representations of the group of those unl-

t a r y operators which are congruent to the identity operator modulo

compact operators , endowed with the norm-topology Also , in [ 2 ~

the representation problem for the unitary group U(oo) , together

w i t h the assertion that

n e d

The group U(oo)

U(OO) is not a type I group , is mentio-

, well known to topologists , is in a cer-

tain sense a smallest ~ f i n i t e dimensional unitary group , being

for instance a dense subgroup of the "classical" Banach-Lie groups

of unitary operators associated to the Schatten - v o n Neumann

classes of compact operators ([~8 S) Also , the restriction of

representations from U(n+~) to U(n) has several nice features

which make the study of the representations of U ( ~ ) somewhat

easier than that of the analogous groups S U ( ~ ) , 0(oo) , S O ( ~ ) ,

S p ( ~ )

Th.~ study of factor representations of the non locally

compact group U(OO) required some associated C ~- algebra The

C*- algebra we associated to a direct limit of compact separable

groups , G = lira G n , has the property that its factor repre-

s e n t a t i o n s c o r r e s p o n d either to f a c t o r r e p r e s e n t a t i o n s of G e e ,

or to f a c t o r r e p r e s e n t a t i o n s of some G n a n d , since the d i s t i n c -

t i o n is e a s y b e t w e e n these two c l a s s e s , it is of e f f e c t i v e use

t a t i o n s f o r e a c h p r i m i t i v e ideal are the d i r e c t l i m i t s of irredu-

c i b l e r e p r e s e n t a t i o n s of the U ( n ) ' s , b u t there are m a n y other

i r r e d u c i b l e r e p r e s e n t a t i o n s 9

U s i n g the m e t h o d s of C h a p t e r I , we s t u d y ( C h a p t e r IV)

c e r t a i n c l a s s of f a c t o r r e p r e s e n t a t i o n s of U(oo) w h i c h r e s t r i c t e d

t o the U ( n ) ' s c o n t a i n o n l y irreducible r e p r e s e n t a t i o n s in a n t i -

s ~ e t r i c tensors This yields in particular an 4nfinity of non-

equivalent type III factor representations , the modular group

in the sense of Tcmita's theory (~32]) with respect to a certain

cyclic and separating vector having a natural group interpretation

Analogous results are to be expected for other types of tensors

The study of certain infinite tensor products (Chapter V)

gives rise to a class of type I I ~ factor representations As in

the classical theory for U(n) , the ccmmutant is generated by a

representation of a permutation group In fact it is the regular

representation of the ~nfinite prmutation group S(oo) which

generates the hyperfimite type II~ factor Other examples of

type lloo factor representations are given in w 2 of Chapter V

Type II~ factor representations of U(oo) were studied

in (E3@],E35 ]) and the results of the present work were announced

in ( 38]

Concluding , from the point of view of this approach ,

the representation problem for U(oo) seems to be of the same

kind as that of the infinite anticommutation relations , though

"combinatoriall~' more complicated Of course , a more group -

theoretical approach to the representations of U ( ~ ) would be

of much imterest

Thamks are due to our colleague Dr H.Moscovici for drawing

our attention on E2~S and for useful discussions

The authors would like to express their gratitude to Mrs

S a n d a S t r ~ t i l ~ f o r her k i n d h e l p in t y p i n g the m a n u s c r i p t

The g r o u p U ( ~ ) is the d i r e c t l i m i t of the u n i t a r y g r o u p s

U ( ~ ) c U(2) c c U(n) c , e n d o w e d w i t h the d i r e c t l i m i t

t o p o l o g y L e t H be a c o m p l e x s e p a r a b l e H i l b e r t space a n d [ e n l

a n o r t h o n o r m a l b a s i s T h e n U ( o o ) c a n be r e a l i z e d as the g r o u p

of u n i t a r y o p e r a t o r s V on H s u c h that Ve n = e n e x c e p t i n g

o n l y a f i n i t e n u m b e r of i n d i c e s n S i m i l a r l y , we c o n s i d e r GL(oo) the d i r e c t l i m i t of the GL(n) ' s

B y U & ( ~ o ) we denote the g r o u p of u n i t a r i e s V on H s u c h

t h a t V - I be n u c l e a r , e n d o w e d w i t h the t o p o l o g y d e r i v e d f r o m the m e t r i c d ( V ' , V " ) = Tr(IV' - V" I ) A l s o , b y U(H) a n d GL(H)

w e d e n o t e all u n i t a r y , r e s p e c t i v e l y all i n v e r t i b l e , o p e r a t o r s on the H i l b e r t space H

The bibliography listed at the end contains, besides references to works directly used, also references to works we felt related to our subject We apologize for possible omissions

C O N T E N T S

C H A P T E R I O n t h e s t r u c t u r e of A F - a l ~ e b r a s a n d t h e i r

r e p r e s e n t a t i o n s I

w I D i a g o n a l i z a t i o n of A F - a l g e b r a s 3

w 2 I d e a l s i n A F - a l g e b r a s 20

w 3 9 S o m e r e p r e s e n t a t i o n s of A F - a l g e b r a s 31

C H A P T E R II T h e C * - a l g e b r a a s s o c i a t e d to a d i r e c t l i m i t o f c o m p a c t ~ 57

w I T h e L - a l g e b r a a s s o c i a t e d t o a d i r e c t l i m i t of c o m p a c t g r o u p s 87

w 2 T h e A F - a l g e b r a a s s o c i a t e d t o a d i r e c t l i m i t of c o m p a c t g r o u p s a n d its d i a g o n a l i s a t i o n 62

C H A P T E R III T h e p r i m i t i v e i d e a l s of A ( U ( o o ) ) 81

w I T h e p r i m i t i v e s p e c t r u m of A ( U ( | )) 81

w 2 D i r e c t l i m i t s of i r r e d u c i b l e r e p r e s e n t a t i o n s 93

C H A P T E R I V T y p e I I I f a c t o r ,rep,resentations o f U ( o o ) i n a n t i s v m m e t r i c t e n s o r s 97

C H A P T E R V S o m e t y p e IIco f a c t o r , r e P r e s e n t a t i o n s of U ( o 0 ) 9 127 w 1 , I n f i n i t e t e n s o r p r o d u c t r e p r e s e n t a t i o n s , 127

w 2 , O t h e r t y p e IIoo f a c t o r r e p r e s e n t a t i o n s ,., 146

A P P E N D I X : I r r e d u c i b l e , r e p r e s e n t a t i 0 n ~ of U ( n ) , 155

N O T A T I O N I N D E X , ~ 160

S U B J E C T I N D E X , 164

B I B L I O G R A P H Y o~ 166

Our a p p r o a c h to the r e p r e s e n t a t i o n p r o b l e m of the u n i t a r y

g r o u p U ( ~ ) r e q u i r e d some other d e v e l o p m e n t s , also w e l l k n o w n

f o r the UH~ - a l g e b r a of c a n o n i c a l a n t i c o m m u t a t i o n r e l a t i o n s

C h a p t e r I is an e x p o s i t i o n of the r e s u l t s we h a v e o b t a i n e d in

t h i s d i r e c t i o n , t r e a t e d in the general c o n t e x t of AF - algebras

We shall use the b o o k s of J D i ~ n i e r (~ 6 ],[ T ]) as r e f e -

r e n c e s for the c o n c e p t s a n d r e s u l t s of operator a l g e b r a s

If MT , M 2 , are subsets of the C * - algebra A ,

the l i n e a r m a n i f o l d (resp the c l o s e d l i n e a r m a n i f o l d ) s p a n n e d

b y ~ _ ~ M n A l s o , for a n y s u b s e t M of A , we shall denote

r a t h e r t e d i o u s v e r i f i c a t i o n s

A n a p p r o x i m a t e l y f i n i t e d i m e n s i o n a l C ~- a l g e b r a ( a b r e v i a -

t e d A F - a l g e b r a ) is a C - a l g e b r a A such that there e x i s t s

a n a s c e n d i n g s e q u e n c e l & n } n >Io of f i n i t e d i m e n s i o n a l C ~- sub-

a l g e b r a s in A w i t h

e l e m e n t s of A , r e l a t e d to a suitable " system of m a t r i x units

f o r the d i a g o n a l i z a t i o n of A w i t h r e s p e c t to C " , such that

T h e r e f o r e , in p r o v i n g the inductive step , we m a y assume

t h a t An+ ~ a n d A n are b o t h f a c t o r s But t h e n it is c l e a r t h a t

An+ ~ = A n ~ ( A ~ ~ An+~) , Cn+~ = C n ~ D n + ~

a n d , since C n (resp Dn+~) is a m a s a , in A n (resp in

A'n ~ An+h ) ' it f o l l o w s o b v i o u s l y that Cn+ ~ is a m a s a , in

An+ ~ 9

(iii) The e q u a l i t y we have to prove is obvious for k = o

A s s u m i n g t h a t it is true for a f i x e d k , we get

C n + k + ~ = ~ C n + k , D n + k + T ~

= < C n , ~ (~ Cn+ k , D n + k + [ )

c C n + k + ~

w h i c h p r o v e s the d e s i r e d e q u a l i t y b y i n d u c t i o n on k

c o m p l e t n e s s a n d in order to e s t a b l i s h some n o t a t i o n s , we shall

p r o v e it We m a y suppose that A is a f a c t o r T h e n {qil

i I

is a c o m p l e t e set of m u t u a l l y e r t h o g o n a l a n d e q u i v a l e n t m i n i m a l

projections of A n , thus , for a fixed index i o ~ I n , we can

find partial isometries v i ~ A such that

( ~ ) v~vi = qi o ' viv~ = qi ' vi = viqio qivi ; i a I n i then an arbitrary element x E A n is of the form

F

(2) x = i , j ~ i n ~ viv~ ' lij ~ ~ "

If ~ : A n ~ C n is a conditional expectation , then

~ ( v i v j ) ~ ( q i ( v i v j ) q j ) = q i q j ~ ( v i v ~ ) = Jij qi ' thus

~ ( X ) = ~ ~ i j ~ ( v i v ~ )

l,J Moreover ,

I.~.3 Now denote by ~pj~ j ~ j n the minimal projections

of Dn+ i Since Cn+ ~ = ~ C n , D n + ~ , it follows that the mini-

mal projections of Cn+ i are the non-zero qipj , i m I n , j EJn

Since Pj ~ Dn+ i C A ~ , for each x E A n we have

Pn+~l An = Pn I A~

a s in the p r o o f of I ~ ~ ( i i )

(ii) S i n c e P n + k ( X ) = P n ( X ) = x for x ~ C n , we have

P ( x ) = x for x ~ k_# C n a n d , b y the c o n t i n u i t y of P , w e

n:o infer P(x) = x f o r all x g C Thus , P : A ~ C is a

P r o o f (i) A g a i n , we shall assume t h a t A is a f a c t o r

a n d use the n o t a t i o n s i n t r o d u c e d in S e c t i o n I.~.2 We d e f i n e

II%(Yk - yll : II (Yk- y II IIy - yll

lira IIQn(Yk) - Yll = 0 But Q ~ ( y k ) E A~ a A n + k , hence

(ii) By C o r o l l a r y I.~.5 we have P ( A ~ N An+ k ) = A ~ N Cn+ k

a n d using (i) we obtain

Proof By Lemma I.I.6 it is sufficient to prove the

e q u a l i t y of the statement only for x ~ A n and y m A~ ~ An+k "

Thus , f i x n >i o , k >i c , denote b y (qil i m i n the

m i n i m a l p r o j e c t i o n s of C n and b y IPJ) J ~ Jn,k the minimal

p r o j e c t i o n s of A~ ~ Cn+ k By Lemma I.~.~.(iii) it follows that the non-zero qip j , i g I n , j E Jn,k ' are the minimal p r o j e c - tions of Cn+ k We define

P n + k / n (z) = 7 pjzpj ~ (A~ ~ Cn+k)' for all z ~ A

Pn+k/n(Xy) = x Pn+k/n(y) for all

As in Sections I.~.2.,I.~.3 we see that Pn+k/n

conditional expectation of A~ f% An+ k with respect to

and

Pn+k/nl A ~ A n k : Pn+kl A~ f] A~+ k 9

Moreover , for any z ~ A ,

z)) = P n ( ~ p j z p j ) Pn(Pn+k/n (

It can be proved that

= <~A~ ~ C n + k , A~+ k (ACn+k+ ~ )

I.~.8 In this s e c t i o n we shall d e t e r m i n e suitable systems

k ~ K n there is a system of m a t r i x u n i t s for the f a c t o r ~ w i t h

r e s p e c t to the m a s a ~ ~ C n , that is a set

S u c h a system is c o m p l e t e l y d e t e r m i n e d once we c h o o s e an index

P R O P O S I T I O N The systems ~eij ; i,j ~ I n ,

m a t r i x units for A w i t h r e s p e c t t o C n c a n be c h o s e n such

t h a t , f o r e v e r y n >i o , the f o l l o w i n s a s s e r t i o n h o l d s :

^(n+~)

_(n) is a sum of some ~rs

re(n)

Proof We proceed by induction Let ~ iK12 J be the

be some system of matrix units of A~ (] Am+ T with respect to

~ u : C B c i ~ u*cu E C The kernel of this homomorphism is easily seen to be

~ C : ~ 1 ~ n ~ C

n : o

r (n) k k ~ n )

For given systems of matrix units ~eij ; i,j E I n ,

satisfying condition ($) of I.&.8., we shall construct a sub-

Let U n be the subgroup of Q J [ n consisting of all

direct product of % A N C by U Moreover , U and P awe iso-

morphic and since

A = l.m.(UnC n) = l.m.(CnU n)

A = c l m ( U C ) = c I m ( C U )

I.~.~O We now denote by ~-L the Gelfand spectrum of the

commutative C*- algebra C Then I ~ is a compact topological

space , C ~- C(IO_) and we m a y view P as a group of homeomor-

phisms of ~ Thus , we obtain a topological dynamical system

(.0_, P)

associated to the given AF - algebra A

Consider the Hilbert space ~ 2 ( ~ ) with orthonormal basis

b t ~ ~ } and denote by ( I- ) the scalar product

Each function f e C(~'I) defines a "multiplication opera-

Thus , U n C U n + ~ and putting

U = O U

n=o

it follows that U is a subgroup of q~ and

and that ~ n is the semi-direct product of ~ n f~ Cn b y U n

f : C(il) , and V ~ , ~ P

THEOREM Given an arbitrary AF - algebra A there exist

a) g m a s a C in A ,

b) ~ conditional expectation P o_~f A with respect t_~o C ,

c) a subgroup U of the unitar~ ~roup of A ,

such that

(i) u* C u = C for all u ~ U ,

(ii) P(u*xu) = u*P(x)u for all u ~ U , x ~ A ,

(iii) A = c.l.m.(UC) = c.l.m.(OU)

Moreover , !e~ ~ be She Gelfand ~ t r u m of C and

b_~e the ~roup o_~fhomeomorphisms o _ ~ f ~ induced b y U Then there

in the p r e c e d i n g Sections It r e m a i n s to c o n s t r u c t a * - isomor-

p h i s m b e t w e e n A and Ẵ)-, P ) w i t h the stated p r o p e r t y

F i x n ~ o denote b y ~ ( n ) the image of uĩ) g U n C U

a n d , identifying an element of A with its image under ~(n) and

a point of ~ w i t h the c o r r e s p o n d i n g vector in ~2(k~) , we have

h a v i n g the d e s i r e d p r o p e r t y

Q.ẸD

Note that , although the concrete C*- algebra Ăk~L, r )

a n d its conditional e x p e c t a t i o n w i t h respect to C ( ~ ) d e p e n d

o n l y on the choice of the minimal projections , the c o n s t r u c t e d

9 - isomorphism b e t w e e n A and A ( ~ , P ) is e s s e n t i a l l y b a s e d

on a suitable choice of the complete systems of matrix units

I ~ ~ For later use , we shall give a convenient

description of ~ as well as of the action of ~ on ~

The Gelfand spectrum ~ - n of the commutative C *- algebra

can be identified with the finite set ~q(~)~

i E I n the minimal projections of C n The map ~ n+~ ~ ~ n cor-

responding to the inclusion map C > Cn+ T associates to

every minimal projection of Cn+ ~ the unique minimal projection

of C n containing it Since the C*- algebra C is the dlwect

limit of the C*- algebras C n following the inclusion maps ,

it follows that ~ can be identified with the topological inverse

limit of the discrete spaces ~-~n following the maps

-CAn+ ~ ~ Kln 9

Therefore , the points of ~ can be represented as

sequences

t =

where q(~) is a minimal projection of C n , for all n >i o

A point t ~ ~ is adherent to a set ~ c ~'~ if and

only if , for each n >i o , there exists s a oo such that

In this s e c t i o n we s t u d y the c l o s e d two s i d e d ideals of an

A F - a l g e b r a A and we i n t e r p r e t the r e s u l t s in t e r m s of the t o p o -

T h e n for a n y c l o s e d two s i d e d ideal J o f A w e h a v e

n = o

P r o o f The c a n o n i c a l * - h o m o m o r p h i s m s A n / J N A n ~ A/J are injective a n d t h e r e f o r e isometric F o r any x e J there is

a s e q u e n c e x n ~ A n w i t h lim llXn - xll = 0 It f o l l o w s t h a t lim Iix~/Jii = o , h e n c e lim llXn/JN Anl I = 0 T h u s there is a

s e q u e n c e Y n ~ J O A n such that lim llXn - Yn~ = 0 and this

a r g u m e n t s (~ 6 ]) , we shall prove the f o l l o w i n g

L E N ~ A Le ~t I b_~e ~ U - stable c l o s e d ideal o f C The _~n

J(I) = I x ~ A ; P ( x * x ) ~ I 3 i_~s ~ c l o s e d t w o s i d e d ideal of A

P r o o f C l e a r l y , J(1) is a c l o s e d s u b s e t of A

Since (x + y) (x + y) ~ 2 ( x * x + y~y) , f r o m x , y ~ J(I)

w e infer

P ( ( z + y) (x + y ) ) ~ 2(P(x~x) + P ( y y)) ~ I ,

R e m a r k t h a t the c o r r e s p o n d e n c e s J ~ w Ij a n d I ~ J(I) are i n c r e a s i n g w i t h r e s p e c t to i n c l u s i o n

1.2.5 C O R O L L A R Y Let J~ and J2 be c l o s e d two s i d e d

i d e a l s of A T h e n

J~ = J 2 r J~ ~ C = J2 n C

P r o o f F o l l o w s o b v i o u s l y f r o m T h e o r e m 1.2.@

Q E D 1.2.6 C O R O L L A R Y Let J b e ~ c l o s e d two s i d e d ideal o_~f A T h e n

J = the c l o s e d two s i d e d ideal of A ~ e n e r a t e d b_~ J ~ C

P r o o f D e n o t e b y J~ the c l o s e d two s i d e d ideal of A

g e n e r a t e d b y J • C C l e a r l y , J~ ~ J S i n c e

J ~ O C c J ~ C c J ~ O C , the e q u a l i t y J = J[ f o l l o w s f r o m 1.2.5

The proof will be given in Section 1.2.[0

Remark that Theorem 1.2.4 further implies

This correspondence is decreasing with respect to inclusion

The closed ideal I of C is P - stable if and only if

the subset ~ I of iO_ is P - stable Owing to Theorem 1.2.$

it follows that

J ; ~ ~176 C

is a decreasing one-to-one correspondence between the closed two

sided ideals of A and the ~ - stable closed subsets of ~

A closed two sided ideal J of A is called primitive

if it is the kernel of an irreducible representation of A It is

known ( [ 5 S ) that J is primitive if and only if , for any closed

two sided ideals J~ and J2 of A , the following implication

holds :

o_~f _O_ c o i n c i d e w i t h the c l o s u r e s of the [1 - orbits

The set o J j N C a s s o c i a t e d to a c l o s e d two sided ideal J

of A h a s a simple d e s c r i p t i o n in the terms e x p l a i n e d in S e c t i o n

I ~ ~ N a m e l y , bet ~[ be a r e p r e s e n t a t i o n of A w i t h kernel J

T h e n the set o O j O C c o n s i s t s of all p o i n t s t e n h a v i n g the

p r o p e r t y

P r o o f I n d e e d , suppose the c o n t r a r y h o l d s Then , for

e v e r y k > n , there exist m u t u a l l y orthogonal c e n t r a l p r o j e c t i o n s

are p o s i t i v e o p e r a t o r s c o n t a i n e d in the c e n t e r of the v o n N e u m a n n

f a c t o r g e n e r a t e d b y T[(A) in L(H) , t h e r e f o r e t h e y are scalar

o p e r a t o r s

P~ = X~ , P2 : h a ; ~ ' ~ 2 #- [ 0 , ~ )

N o w f r o m (~) we ~ f e r ~ + ~ 2 = ~ , w h i l e (2) implies t h a t

~ = ~ 2 = 0 This contradiction proves the Lemma

Q.E.D

(i) 7[(p(~) p(n)) ~ o for all n > ~ ;

there exists k >/ n such that UI (p(k)q) ~ 0

Proof Indeed , let us write the set

~_~ ~q ; q is a minimal projection of C n and ~[(q) # 0 I n={

as a sequence le{ , e 2 , , ej , "''I Owing tb Lemma ~ , we

find by induction a sequence Ip(kj)l of minimal central projec-

Clearly , this sequence can be refined up to a sequence

having the stated properties

~p(n) 1

Q.E.D

sequence ~p(n) 1 as in Lemma 2 The condition (i) satisfied b y

This m e a n s that

the n o t a t i o n b e i n g as in S e c t i o n I % ~ Therefore , p(n) is the

c e n t r a l support of the m i n i m a l p r o j e c t i o n ~(n) in A n Since

t o qI(p (n)) ~ 0 , it f o l l o w s that

(n)~

T [ ( q to- ~ 0 for all n ~ ~

Thus , t o s co and c o n s e q u e n t l y r ( t o ) c oo

N o w consider t ~ co and f i x n ~ ~ The c o n d i t i o n (ii)

s a t i s f i e d b y the p(n) , s shows that there exists k n ~ n such

that

(kn)q(

Therefore , there is a minimal p r o j e c t i o n r

c e n t r a l support p in Akm such that

1 2 ~ The p r i m i t i v e s p e c t r u m Prim(A) of A is the set

of all p r i m i t i v e ideals of A e n d o w e d w i t h the h u l l - k e r n e l topo-

l o g y The p r e c e d i n g r e s u l t s s h o w that Prim(A) c a n be i d e n t i f i e d

w i t h the set of all c l o s u r e s of ~ - o r b i t s D e f i n i n g an e q u i v a -

l e n c e r e l a t i o n " N " on ~ b y

t~ N t 2 < ~ p ( t & ) = ~ ( t 2) ,

it can be easily verified that Prim(A) is homeomorphic with

the quotient space ~ / ~ endowed with the quotient topology

1.2.~2 In his approach to AF - algebras based on diagrams,

0.Bratteli has also studied the closed two sided ideals Instead

of considering the intersections of the ideals w i t h the m.a.s.a

C , O.Bratteli considers the intersections w i t h the smaller sbelian

subalgebra generated b y the centers of the A n ' s , the results

b e i n g quite similar (see K ~ S, 3.3 , 3.8 and ~ S , 5.~.) 9 His

approach is particularly well adapted for problems such as the

determination of all topological spaces which are spectra of AF -

algebras ( s e e ~ S, 4.2 a n d S 3 S)

w 3 Some representations ~ AF - algebras

We consider an AF - algebra A = ~ A n T together with

n=o the m.a.s.a C , the conditional expectation P : A ) C and

the group U as in w 9 Let ( ~ , ~ ) be the associated topo-

logical dynamical system and ~ the sigma-algebra of Borel sub-

sets of ~ 9 In this section we shall study two kinds of repre-

sentations of A , $I~ and ~ , associated with ~u_ quasi-

invariant measures ~ on the Borel space ( ~ , ~ ) 9

A positive measure on ~ will always mean a positive

is a positive measure of mass & , i.e 5 ( ~ ) = ~ Two positive

null-sets

Then ~ is ~ - invariant (resp [~- &uasi-invariant) if ~ = (resp ~ equivalent to ~ ) for all ~ E P The positive measure

is ~ - ergodic if the only ~ - invariant elements of ~ , 5 ) are the scalars 9

Then ~ can be regarded as a state of the commutative C*- algebra

C -~ C(~-2) and therefore

bert space H 5 and a cyclic unit vector ~ r H~ for ~ p such that

the bicommutant notation , T[~(A)" 9