Lecture notes in mathematics

Lecture Notes in
Mathematics
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


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Authors
Dr. Serban-Valentin Str&til&
Dr. Dan-Virgil Voiculescu
Academie de la Republique
Socialiste de Roumanie
Institut de Math@matique
Calea Grivitei 21
Bucuresti 12
Roumania

Library of Congress Cataloging in Publication Data

Stratila, Serban-Valentin~ 1943 Representations of iF-al~ebras and of the 6roup

(Lecture notes in mathematics ; 486)
Bibliography: p.
Includes indexes.
i. Operator algebras. 2. Representations of algebras. 3. Locally compact groups. 4. Representations of
groups. I. Voiculescu~ Dan-~-irgil, 1949joint authoz
II. Title. III. Series: Lecture notes in mathematics
(Berlin); 486.
QA3~
no. 486 [QA326] 510'.8s [512'.55] 7~-26896

A M S Subject Classifications (1970): 22D10, 2 2 D 2 5 , 46 L05, 4 6 L 1 0

ISBN 3-540-07403-1
ISBN 0-387-07403-1

Springer-Verlag Berlin 9 Heidelberg 9 N e w Y o r k
Springer-Verlag N e w York 9 Heidelberg 9 Berlin

This work is subject to copyright. All rights are reserved, whether the whole
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~ by Springer-Verlag Berlin - Heidelberg 1975
Printed in Germany
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

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INTRODUCTION

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 by I.E.Segsl ([30])
connection with quantum physics . I n [ 2 ~ ]
all irreducible unitary representations

A.A.Kirillov

in

classified

of the group of those unl-

tary operators which are congruent to the identity operator modulo
compact operators

, endowed with the norm-topology

the representation problem for the unitary group
with the assertion that

U(OO)

. Also , in [ 2 ~
U(oo)

, together

is not a type I group , is mentio-

ned .
The group

U(oo)

, well known to topologists

tain sense a smallest ~ f i n i t e

, is in a cer-

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
easier than that of the analogous groups
Sp(~)

of

U(~)

SU(~)

somewhat

, 0(oo) , S O ( ~ )

.

Th.~ study of factor representations of the
compact group

U(OO)

required some associated

non

locally

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-

,


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IV
sentations correspond either to factor representations of
or to factor representations

of some

G n and , since the distinc-

tion is easy between these two classes
This

C*- algebra is an

of finite-dimensional

algebras

. For the

, it is of effective use .

AF - algebra

C~- subalgebras

c e d and studied b y O.Bratteli ([~])

Gee ,

.

, i.e. a direct limit

AF - algebras

, introdu-

, are a generalization of UHF -

UHF - algebra of the canonical anticommutation

relations of mathematical

physics there is the general method of

L.Garding and A.Wightman ([12S) for studying factor representations
and , in particular

, the cross-product construction which yields

factor representations

in standard form . So we had to give an

extension of this method to

AF - algebras (Chapter I) . For

U(~)

this amounts to a certain desintegration of the representations
w i t h respect to a commutative

C - algebra

, the spectrum of which

is an ~nfinite analog of the set of indices for the Gelfand - Zeitlin b a s i s ([37])

9 For

U(oO)

in this frame-work

classification of the primitive
bra

, a complete

ideals of the associated

, in terms of a upper signature and a lower signature

possible (Chapter I I I ) .

O*- alge, is

Simple examples of irreducible represen-

tations for each primitive ideal are the direct limits of irreducible representations

of the

irreducible representations

U(n)'s

, but there are m a n y other

9

Using the methods of Chapter I , we study (Chapter IV)
c e r t a i n class of factor representations of
to the

U(n)'s

U(oo) w h i c h restricted

contain only irreducible representations

in anti-


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v
s~etric

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 ~
the classical theory for

factor representations

. As in

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

Type II~ factor representations

of

w 2

U(oo)

of Chapter V
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
"combinatoriall~'

more complicated

. Of course

theoretical approach to the representations of

, though

, a more group 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.


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Vl
Sanda Str~til~ for her kind help in typing the manuscript

The group
U(~) c U(2) c
topology

U(~)

is the direct limit of the unitary groups

... c U(n) c

. Let

an orthonormal

H

. Then

of unitary operators

V

o n l y a finite number

that

U&(~o)

V - I

the metric
we denote

be nuclear

space

U(n),

Appendix

space and [ e n l

can be realized

such that

Ve n = e n

n . Similarly

as the group
excepting

, we consider

GL(oo)

' s .

the group of unitaries

V

on

H

such

, endowed with the topology derived from
- V" I ) . Also

, respectively

, by

U(H)

all invertible

, wo - topology means weak-operator

and

GL(H)

, operators

on

strong-operator

topology and

topology.

it might be useful for the reader to have at h a n d

certain classical
of

H

Hilbert

H .

so - topology means

Since

separable

U(oo)

GL(n)

= Tr(IV'

all unitary

As usual

on

we denote

d(V',V")

the Hilbert

, endowed with the direct limit

of indices

the direct limit of the
By

...

be a complex

basis

.

facts concerning

especially

the irreducible

in view of Chapters

about these representations.

representations

IV and V, there

is an


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vMI

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.

Bucharest, March 12 th 1975.

The Authors.


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CONTENTS
CHAPTER

I

. O n the s t r u c t u r e
representations

w I . Diagonalization

of AF - a l ~ e b r a s a n d t h e i r
...........................

of AF - a l g e b r a s

w 2 . I d e a l s in AF - a l g e b r a s
w 3 9 Some r e p r e s e n t a t i o n s
CHAPTER

I

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

3

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

20

of AF - a l g e b r a s

..........

31

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
of c o m p a c t ~

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

57

w I . The L - 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
of c o m p a c t g r o u p s
w 2 . The AF - 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

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
CHAPTER

III. The p r i m i t i v e

w I . The p r i m i t i v e

87

.....

62

..........

81

)) . . . . . . . . . . . . .

81

i d e a l s of A ( U ( o o ) )

s p e c t r u m of A ( U ( |

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

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

97

C H A P T E R IV . Type III f a c t o r ,rep,resentations o f U ( o o )
in a n t i s v m m e t r i c
CHAPTER V

tensors

. Some t y p e IIco f a c t o r ,rePresentations
of U(o0 ) . . . . . . . . . . . .9. . . . . . . . . . . . . . . . . . . . .

w 1 , Infinite tensor product representations
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
APPENDIX
NOTATION

...... ,.

127

...... ,.,

146

: I r r e d u c i b l e ,representati0n ~ of U ( n )
INDEX

SUBJECT INDEX
BIBLIOGRAPHY

127

..... ,.

155

...................................... ,~

160

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , ....

164

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o~

166


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CHAPTER I

ON THE STRUCTURE OF

AF - ALGEBRAS

AND THEIR REPRESENTATIONS

The uniformly hyperfinite

C*- algebras (UHF - algebras)

,

w h i c h appeared in connection with some problems of theoretical
physics

, were extensively studied , important results concerning

their structure and their representations being obtained b y
J. Gl~mm ([15]) and R. Powers ([Z4])

. They are a particular case

of approximately finite dimensional

C ~- algebras (AF - algebras)

c o n s i d e r e d b y O.Bratteli ([ i ]) , who also extended to this more
general situation some of the results of J. Gl~mm and R. Powers

.

Our approach to the representation problem of the unitary
group

U(~)

for the

required some other developments

, also well known

UH~ - algebra of canonical anticommutation relations

.

Chapter I is an exposition of the results we have obtained in
this direction

, treated in the general context of

AF - algebras.

We shall use the books of J. Di~nier (~ 6 ],[ T ]) as references for the concepts and results of operator algebras

If

MT , M 2 , ...

are subsets of the

.

C*- algebra

A ,

then we shall denote b y

< M~
the smallest
l.m.(M~

, M 2 , ...>

or

C - subalgebra of
, M2

, ...

)

A

(reap.

containing
c.l.m.(M~

~_~ M n
n
, M2

and b y

, -..

))


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2
the linear m a n i f o l d

(resp.

by

~_~

n

Mn

. Also

by

M'

the commutant

, for any subset

M'
A maximal
C*- algebra
that

A

the closed linear manifold)

of

=

M

{xE

abelian

in

A , we shall denote

A :

subal~ebra

(~)

y ~ M}

(abreviated

C ~- subalgebra

.

m.a.s.a.)
C

of

A

of a

such

C' = C .

to a

expectation

C*- subalgebra

such that

B

in

A

~

#) P ( x ) ~ P ( x )
5) P(yxz)

IIxll

~

J. Tomiyama

= yP(x)z

onto

projection

([33])

A

with respect
P

: A

B

for all

x ~ A , x ~ 0

for all

x e A

for all

x 9 A , y,z ~ B

of

A

of norm one of

A

An approximately

~B

;

sequence

algebras

A

with

;

;
.

with respect to
onto

B

B . Conversely
of norm one

. In what follows we

only in order to avoid some

.

finite

is a

an ascending

expectation

of J. Tomiyama

tedious verifications

in

x ~ A

expectation

is a conditional

AF - algebra)

for all

has proved that any projection

shall use the result
rather

P(x*x)

, a conditional

is a (linear)

A

C - algebra

is a linear mapping

3) P(x) >~ 0

Obviously

of a

:

2) llP(x)il

ted

of

A ; xy = yx

is an abelian

A conditional

of

M

spanned

dimensional

C - algebra

l & n } n >Io

A

C ~- algebra

(abrevia-

such that there exists

of finite

dimensional

C ~- sub-

,


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A

=

~ n~o
An~=

We shall suppose that

Ao

( =

is one dimensional

stands for the identity element of
For

~) n = o A~

C*- algebras

A

obvious (star) isomorphism

and

B ,

A .
A

~

B

Diagonalization s

Given an arbitrary

will denote some

, in which case corresponding elements

will sometimes be denoted b y the same symbol

w ~

, A o = C.~ , where

.

AF - algebras

AF - algebra

A
n=o
we shall construct a
tion

P

of

elements of

A

m.a.s.a.

C

with respect to

in
C

and a group

A , related to a suitable

for the diagonalization of
A

=

A

A , a conditional
U

expecta-

of unitary

" system of matrix units

with respect to

C " , such that

c.l.m.(UC)

I.~.i. We define b y induction an ascending sequence
of abelian

C ~- subalgebras

C o = Ao
where

Dn+ ~

;

in

A :

Cn+ & = ( C n , O n + ~

is an arbitrary

LEM~,~A . .For al__!l n ~ o

{Cn}

m.a.s.a,

and all

in

,

n $ o

A~ ~ An+ &

k ~o

we have

.

,


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(i)

Cn

(ii)

A~

(iii)

is a

projection

of

a n d we have

a)

pz = p

x ~ An+ ~ ~
An+ ~

pA n

pC n

is a

, there

. If

in

b)

m.a.s.a,
y l

y e An

An

is c l e a r

. If

in
>

is o b v i o u s

h a v e p r o v e d that

jections

of

Cn

p

so we suppose

.

is a m i n i m a l
,

PAn+ ~

central
is a f a c t o r

is a

projection

commutes

in

An

with

of

An

such that
of

zA n

pC n , t h e n

zy e C n , since

Cn

is a

py = p(zy) c pC n .
in

(PAn)' ~

(PAn+ ~)

to the c e n t e r

with

PCn+ ~ =

of

.

A~ ~ An+~

~pC n , PDn+~

.
.

.
a)

, b)

, c)

px ~ Cn+ ~

. Since

~

we

infer that

for a n y m i n i m a l

is a f i n i t e

An+ ~ , it f o l l o w s

Therefore

z

commutes

that

belongs

homomorphism

is an i s o m o r p h i s m

py

m.a.s.a,
p

.

is a , -

, thus

. It f o l l o w s

px ~ PAn+ ~

An+~

Cn+ ~

pA n

py

and if

with

, since

If f r o m

of

it for

n = o

p ~ Dn+ ~ C Cn+ ~

is a c e n t r a l

PDn+ ~

c)

;

:

commutes

m.a.s.a,

p

is o b v i o u s for

Cn+
~'

, then

A~ N A n + k

.

and such that the above map

zy ~ A n

This

in

and we prove

, since the map
pA n

;

An

, A~ f~ O n + k >

Cn

Consider

This

m.a.s.a,

. (i) The c l a i m

it is true for

onto

is a


Proof

onto

in

Cn+ k

On+ k =

Indeed

m,a.s.a,

central

sum of m i n i m a l

that

, we m a y assume

px ~ PCn+ ~ , t h e n we

that

x ~ Cn+ i
An+ ~

projection

central

pro-

.
is a f a c t o r

. With


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this assumption

minimal central projection of
is a factor
b o t h in

, qAn+~q

An

A~l] An+~ , we have

a')

qC n

is a

b')

qDn+ !

c')

qx ~ qAn+~q
a')

An . Since

tions of

An+ ~

and

An+ ~

=

and , since

qA n

in

, c')

qx E Cn+ ~

q E C n C Cn+ ~
q

isa
,

qAn

is central

.

(qAn)' ~ (qAn+~q)

.

qCn+ ~ = < q C n , q D n + T ~

we infer that

9

qx e qCn+ ~ , then

for any minimal central projection

is a finite sum of minimal central projecx c Cn+ ~

.

, in proving the inductive step , we m a y assume
An

are both factors

An~(A~

~ An+~)

C n (resp. Dn+~)

,

is a

. But then it is clear that

Cn+~

m.a.s.a,

A'n ~ An+h ) ' it follows obviously that
An+ ~

q

:

commutes with

, b')

~

in

m.a.s.a,

A n , it follows that

Therefore
that

m.a.s.a,

is a

we have proved that
of

A n , then

is also a factor and , since

and in

If from

q

' ~ . If
x E An+ ~ ~ Cn+

, consider again

Cn+ ~

=

Cn~Dn+

in

~

A n (resp. in

is a

m.a.s.a,

in

9
(iii)

The equality we have to prove is obvious for

Assuming that it is true for a fixed
Cn+k+ ~

=

~Cn+ k , Dn+k+T~

=


c

k , we get

(~ Cn+ k , D n + k + [ )

Cn+k+ ~

w h i c h proves the desired equality b y induction on

k

.

k = o .


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(ii)

Let

E

be an abelian

subalgebra

such that

A~ FI Cn+ k C E C A~ n An+ k
Then

c~+ k =
a

n cn+k~ c < C n , E> C An§

Since

Cn+ k

hence

E = A~ ~ Cn+ k . T h u s

in

is

m.a.s.a,

in

An+ k , i t

A~ ~ Cn+ k

follows

is

that

indeed

a

E C Cn+ k

m.a.s.a.

~ ~ An§
Q.E.D.

I.~.2. Denote b y
C n . For each

x s A

{qi}

is a projection
conditional

In fact

Cn

projections

2
qi x qi
imI n

"

and that the map

Pn

of norm one of

is a

of

A

A

onto

in

: x ~

C n' . Thus

with respect

m.a.s.s,

of

to

C n!

Pn

; Pn(x)
is a

9

A n , we have

P(An)

= Cn .

,

PnlAn
i s the unique conditional
a n d it is faithful

x ~ An
Of c o u r s e
completness
prove

=

Pn(X) ~ C~

expectation

Since

the minimal

we define
Pn(x)

It is clear that

i ~ In

, this

is

:

An

~

expectation

of

An

with respect t o

Cn

, i.e.

,

Pn(X~X)

a well

= 0

~

known fact

it . We m a y suppose

that

set of m u t u a l l y

An

x = 0

. However

and in order to establish

is a complete

Cn

9
, for

some notations

is a factor

erthogonal

the

sake

of

, we shall

. Then

{qil

i

and equivalent minimal

In


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projections of

A n , thus , for a fixed index

find partial isometries
(~)

v~vi = qi o

'

vi ~ A

viv~ = qi

then an arbitrary element
(2)
If

x
~ : An

~

Cn

~(vivj)

such that
'

x E An

F
i,j~in

=

io ~ I n , we can

~

vi = viqio

qivi

;

i a In i

is of the form
viv~

'

lij ~ ~

"

is a conditional expectation , then

~(qi(vivj)qj ) = qiqj~(viv~)

= Jij qi

'

thus
~(X) = ~

l,J

@ iiq

~ij~(viv~)

= ~

qixqi = Pn(X) 9
I

Moreover ,
x*x = ~ ( ~
i,j

and therefore

k

Pn(X*x)

~ki ~kj)ViV~

= o "--> ~ k h

I.~.3. Now denote by
of

Dn+ i . Since

Cn+ i

Pj ~ Dn+ i C A ~
Pn+~(x)

, for each

x = 0

the minimal projections
, it follows that the mini-

are the non-zero
x E An

qipj

, im In , j EJn.

we have

=

,~
i ~ In, j ~ Jn

qiPJXqiPJ

=

~
qixqi
i cl n

~__
Pj
J ~Jn

=

~,
qixqi
iEl n

=

Pn I A~

Therefore
Pn+~l An

k,~ ~ >

~pj~ j ~jn

Cn+ ~ = ~C n , D n + ~

mal projections of
Since

= 0 , (u

=

=

=

Pn (x)


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a n d so

Pn§

: PnlAn

Oonsequently

, for

foran

x 9 k.~ A n

n>~o,k>o

we m a y d e f i n e

n=o

P(x)

=

Pn(X)

if

x E An

.

Then

P ~ 1 7 An ~ x :

~ P(~)e U

n=o

is

a projection

on

n=o

of norm one

.

We define
C

=

~ k~Cn>

( =

n=o

a n d we denote

k._#C n )
n=o

again by
P

: A

the u n i q u e b o u n d e d l i n e a r

~- C
extension

of

P

: k . J An

>

n=o

PROPOSITION
a_

m.a.s~a,

in

(i)

A~

and

C

P

with respec t to

: A

,
>

C

s u c h that

C

O =

=

Pn o p

. (i) C o n s i d e r

x ~ C'

Pn(X)

= x . Thus

Cn 9

A ,

A' ~ C

is

n ~ o ,
~C n , A~6~C~

n ~o
=

.

expectstion

of

A

,
p

. There

llm llXn - xll = 0 . F o r each

and therefore

in

is a c o n d i t i o n a l

and , f o r each
P ~ Pn

Proof

m.a.s.a,

, for e a c h

O 6~A n = Cn
(ii)

is a

~

n=o

is a sequence

n > o we h s v e

xn~

An

x g C~

,

llPn(~n) - ~II = llPn(~n- ~)ll ~ ll~n- ~II
It f o l l o w s

that

lim lJPn(Xn) - xJJ = 0

and

, since

Pn(Xn) a C n ,


www.pdfgrip.com

9
we get

xq: C . Hence
Since

contains
that

C l] A n

Cn

B y Lemma

Cn

is a

=

(ii)
P(x) = x

of

projection

A~

C

Pn+k(X)

for all

of

Finally

A

An

which

A n , it is clear


implies that
in

C =~C n , A~C>.
A n' follows now

.

= Pn(X)

= x

for

x ~ C n , we have

and , by the continuity

A

onto

with respect

, for any

P(Pn(X))

C

x g C . Thus

of norm one of

expectation

in

is s m.a.s.a,

I.~.~.(ii)

x ~ k_# C n
n:o

P(x) = x

of

we have

< C n , A n' / ~ C n + k ~

Since

for

A .

subalgebre
m.a.s.a,

k > o . This obviously

as in the proof

in

.

The fact that

. ere

m.a.s.e,

is an abelian

I.~.i.(iii)

Cn+ k

infer

is a

and since

C ~ An = Cn

for every

C

x ~ A

to

, P

: A

of
~

P , we
C

is a

C , that is , a conditional
C .

we hsve

=

P ( ~
qixqi )
i ~ In

=

~
qiP(x)
i e In

=

P(x)

,

e~(P(x) ) =

i ~ In qie(x)qi

=

i ~ In qiP(x)

=

P(x)

,

qll

are the min

al projections of

.
Q.E.D.

I.~.#.
unitary

We now consider

el~ments

u E An

U*CU = c

, ~n

is a group

for any

~n

consisting

of all

with the property
u *

Clearly

the set

Cn

. If

c s A~ 6]C

u

=

Cn

.

u e 14 n , then
9 Since

U~Cn u = C n

C = < C n , A~C~ C > ,

and
we get


www.pdfgrip.com

I0

(3)

u~ C u

Then

, from

Cn+ ~ = C ~ A n + ~

=

we

ql n

C

.

infer

C

~n+~

:

~ n

U~Cn+~U

= Cn+ ~

. Thus

,

"

We define

Then

is a group

satisfies

the r e l a t i o n

PROPOSITION

A

Proof

. (i) It suffices

to avoid
. Let

~qil

satisfying

see that

is clear

and any

=

c.l.m.(C~)

x ~ A

,

u

~ LL

.

u E ~

, we assume

be the minimal

the r e l a t i o n s

it suffices

A

9

to show that

, denote b y

Qvi~

(1)

that

projections
i ~ In

. Owing

An

of C n

the partial

to r e l a t i o n

(2)

to show that

v i iv*

B u t this

for all

complications

i ~ In

I.~.2.

of

c.l.m.(~C)

u~P(x)u

notational

, as in S e c t i o n

isometries

=

=

P(u*xu)

is a factor

we

. (i)

elements

9

(ii)

In order

and

of u n i t a r y
(5)

"

n=o

i,j m I n

~nCn

.

, since
viv j

=

uijq j

where
uij = i - (v i - vj)(v.~ - v~) s ~ n

(ii)

It Is e n o u g h

to prove

the claim

'

qj ~ Cn

only for

"

x ~ ~
n=o

Since

iX

=

~
n=o

~n

' we m a y suppose

that there

exists

n >i o

An

.


www.pdfgrip.com

11

such that

x ~ An

and

u a 9/n . Define

P'(y)
Since

P(u*yu) e C n

easy to check that
of

An

=

and
P'

w i t h respect

uP(u*yu)u*

U*CnU

: An

to

for

= C n , we have
~

Cn

y cAn

.

P'(y) a C n . It is

is a conditional

C n . B y Section I.~.2.

expectation

it follows

that

P' = P n l An = P I An 9 Thus
UP(u~xu)u e = P(X)

9
Q.E.D.

I.~.5. 00ROLLARY

. If

Proof

u ~ S/n

l.[.4.(ii)
w i t h any
and

. For any

,
u

, since

u*P(x)u
E ~n

x e ~

we have

= P(u'xu)

" But

P(x)

, then

= P(x)

P(x) E A~

.

u*xu = x , thus

. Hence

P(x)

, by

commutes

obviously commutes with any

A n = l.m.( ~/n0n ) , it follows

that

c e Cn

P(x) ~ A~

.

Q.E.D.

I.~.6. LEMMA

. Fo__~rany

(li)

Proof

A~C

. (i) Again

and use the notations

n ~ o

=

we have

:

< ~
A~(~Cn+k~
k=o

, we shall assume that

introduced

=

J

in Section

vi v

An

is a factor

I.~.2. We define

for all

A

i~l n
Then
fact

Om(x)
, Qm

commutes with all

: A

Consider

)

A~

vlv ~

, hence

is a conditional

y ~ A~ . There

Qn(x) E A~

expectation

is a sequence

. In

9

Yk ~ An+k

such

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12

that

lim IIYk - YII" : 0 . Since
k-~

y e ~A~ , we have

~"
"~nty)
: y .

Thus

II%(Yk - yll : II (Yk- yII
and

lira IIQn(Yk) - Yll = 0 . But

Q~(yk ) E A~ a A n + k

(ii) By Corollary I.~.5. we have
and using

, hence

P(A~ N An+ k ) = A ~ N Cn+ k

(i) we obtain
P(A~)

Therefore

IIy - yll

, for every

= c ~ A~N

Cn+ k >

C ,

Q.E.D.
I.~.7. PROPOSITION

. If

P(xz)

x ~ An
=

and

P(x)P(y)

y ~ A~ , then
.

Proof . By Lemma I.I.6. it is sufficient to prove the
equality of the statement only for
Thus , fix

n >i o , k >i c , denote by

minimal projections of
projections of
the non-zero
tions of

Cn

and by

and

y m A~ ~ An+k "

(qil i m i n

IPJ) J ~ Jn,k

the

the minimal

A~ ~ Cn+ k . By Lemma I.~.~.(iii) it follows that
qip j

, i g I n , j E Jn,k ' are the minimal projec-

Cn+ k . We define

Pn+k/n (z) = 7
pjzpj
J r Jn,k
Then

x ~ An

Pn+k/n : A

in particular

~ (A~ ~ Cn+k)'

(A~ ~ Cn+k)'

for all

z ~ A .

is a conditional expectation,

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13

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

for all

As in Sections I.~.2.,I.~.3. we see that
conditional expectation of

A~ f% An+ k

x~A

n , y~A

.

is the unique

Pn+k/n

with respect to

A~' f% Cn+ k

and
Pn+k/nl A ~ A n . k

:

Pn+kl A~ f] A~+ k

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

=~---qiPjxpjqi
i,~

Moreover , for any

9

z ~ A ,
=

Pn+k(Z)

therefore
Pn o Pn+k/n = Pn+k/n ~ Pn = Pn+k
Thus , for

x E A n and
P(xy)

"

y g A~ (~An+ k , we have
=

Pn+k(xY)

=

Pn(Pn+k/n(XY) )

=

Pn(X Pn+k/n(Y))

=

Pn (x)Pn+k/n(y)

=

Pn+k (x)Pn+k(y)

=

:

P(x)P(y)

Q.E.D.
It can be proved that
A~ ~ Cn+k+ ~

=

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

=

Pn+k+T/n ] An+k

which implies
Pn+k/n I An+k

Thus , we may define a map
P

P~/n

/n(X) = Pn+k/n(X)

:

A

if

>

(A~nC)'

x E An+ k

This map is a conditional expectation , P ~ / n I A~
P

=

Po~/n o Pn

=

Pn ~ P ~/n

9

by


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14
I.~.8. In this section we shall determine suitable systems
of matrix units for the finite-dimensional

C - algebras

An

.

Cons ider

kGr~
the decomposition of

~q(~)I
k
~
J i cI n
k ~ Kn

An in factor components

the minimal projections of

~

and denote b y
(hC n . For each

there is a system of matrix units for the factor

respect to the

m.a.s.a.
I (n)
eij

~

~

with

~ C n , that is a set

k}
; i,j ~ I n

consisting of partial isometries

(n) , q(~)

q(~)

eij
such that

e(n) e(n)
~jr (n)
ij
=
eis

(n)*
^(n)
eij
= ~ji

'

"

Such a system is completely determined once we choose an index
i o ~ Ik
n

v i = ~(n)
~ii ~ , i r I nk , since

and the partial isometries
e(n)
~
lj = viv j

k
i,j ~ I n

,

.

The whole system of matrix units
~ e (n)
ij
is a linear basis for
then

; i,j E Ink ' k ~ E ~ }

A n . If

k
i,j 6 In

,

h
r,s ~ In

and

h ~ k ,

k~Knl

of

e(.n)e (n) = 0
ij rs
PROPOSITION

. The systems

r (n) ; i,j ~ I nk ,
~eij

m a t r i x units for

An

that

n >i o , the followins assertion holds

(4)

, for every

with respect t o

_(n)
each ~ij

Cn

is a sum of some

can be chosen such

^(n+~)

~rs

:


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15

Proof . We proceed by induction . Let

be some system of matrix units of

and the non-zero

e(~)fjj

A~ (] Am+ T

containing the

with respect to
Cn+ ~

are partial isometries between such

projections . Moreover , for every
= ~

i~,i 2

,

~(n) f .
~i~i2 ~j

j

I^(n+~)1
~rs
j

Now we may take as

1

are the minimal projections of

(n) f
ei~i 2 J~S 2

e(n)
I~i2

be the

w. .e.ot o

system of matrix units for

Dn+ ~ . The non-zero

re(n)
~ iK12 J

"

any system of matrix units of

An+ ~

e (n) f
i~i2 J~J2 's .
Q.E.D.

I.~.9. There is a homomorphism of the group
group

~

of

~ - automorphisms of

corresponding

~ - automorphism
~u

C , namely , for

~u E P

: C B c i

~

~

onto a
u ~ ~

, the

is

u*cu E C

.

The kernel of this homomorphism is easily seen to be
~

C

:

~1~n~

Cn

n:o

For given systems of matrix units
satisfying condition
group

U

of

3J[

Un

~

C

~0~ be the semi-direct product of
by

U .

be the subgroup of

QJ[n consisting of all

=
U

where , for each
Un

_(n)
7
~k
k ~ K n i g In

k ~ Kn , ~ k

is generated by the

k~n )

of I.&.8., we shall construct a sub-

such that

its normal subgroup
Let

($)

k

r~eij
(n) ; i,j E In ,

ei'~k(i)

is some permutation of

Ink . Thus,


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16
u(n)
ij
with

=

~ _ =(n)
(n)
e(n)
(n)
~li - ejj + ij + eji

k
i,j ~ I n , k ~ K n

. Remark that

e(n)

~(n)~(n)

ij

=

~lj

~(n),(n)

~ j

It is easily verified that

=

Un

~ i ~ij

9

is the set of all

u ~ n

of

the form

(n)
eij

~ij
kaE

and that

n

~0,~

' ~ij ~

for a l l

i,j

i,j ~ In~

~n

is the semi-direct product of ~ n f~ Cn

Thus , U n C U n + ~

U

direct product of

Un .

and putting
U

it follows that

by

=

O
Un
n=o

is a subgroup of

%AN C

by

q~

and

U . Moreover

~I

, U

and

is the semiP

awe iso-

morphic and since
An

=

l.m.(UnC n)

=

l.m.(CnU n)

A

=

c.l.m.(UC)

=

c.I.m.(CU)

we have

I.~.~O. We now denote by ~-L
commutative
space , C

C*- algebra

C . Then

the Gelfand spectrum of the
I~

~- C(IO_) and we may view

phisms of ~

.

is a compact topological
P

as a group of homeomor-

. Thus , we obtain a topological dynamical system

(.0_, P)
associated to the given

AF - algebra

Consider the Hilbert space

b

t ~ ~}

and denote by

Each function

A .

~2(~)

(. I- )

with orthonormal basis

the scalar product

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


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17
tot"

Tf

on

~2(~-~_) by
Tf(h)

=

h c ~2(C2)

fh

On the other hand , each element
operator"

V~

on

~2(~)

~ ~P

.

defines a "permutation

by

Let us denote by

C)

A(~,
the

C*- algebra generated in

f : C(il)

,

and

V~ , ~

P

L(~2(~))

m.a.s.a.

C

in

U

AF - algebra

A

there exist

A ,

b) ~ conditional expectation
c) a subgroup

Tf ,

.

THEOREM . Given an arbitrary
a) g

by the operators

P

o_~f A

with respect t_~o C ,

of the unitar~ ~roup of

A ,

for all

u ~ U

,

for all

u ~ U

,

such that
(i)

u* C u

(ii)

=

P(u*xu)

(iii)

A

=

Moreover

C
=

u*P(x)u

c.l.m.(UC)
, !e~ ~

=

*

,

c.l.m.(OU)

be She Gelfand ~ t r u m

b_~e the ~roup o_~fhomeomorphisms o _ ~ f ~
i_g a

x ~ A

induced by

of

C

and

U . Then there

- isomorphism
A

~

A(~,~)

(t)

=

(xtlt)

such that
P(x)

,

t~il

;

xeA

.

Proof . The first part of the Theorem was already proved