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
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
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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-
,
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-
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.
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
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.
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
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
, -..
))
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-
,
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
.
,
(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
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 .
(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
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
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)
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 ,
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
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
.
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
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
=
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,
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
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
:
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,
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-
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
18 in the preceding Sections phism between Fix
A
and
n ~ o
under the map ponding to
'
U
A(~)-, P )
denote by ~ ~
q(~) ~ C n C
T ( n3)
=
. It remains to construct a * with the stated property ~(n)
Tf(n)
the image of
Ij
and by C
- isomor-
f(~)
.
ui~) g
the function on ~
UnC
U
corres-
and put v (ij n)
'
=
V (n)
j
nk , k ~ Kn .
;
i,jr
;
k ~n i,jgI n , k
-ij
Then the correspondence e(n) ij
=
, (n)~(n) ~ ~ij ~ j '
~ i
v(n)T(n) ij j
has a unique linear extension up to an isometric
~(n) : An
~
- homomorphism
~ A(~I,P)
and , identifying an element of a point of
*
,
An
with its image under
with the corresponding Pn(X) (t)
=
(xt ~ t)
vector ,
in
t ~ ~'~
~2(k~) ;
~(n) and , we have
x ~ An .
Since the systems of matrix units were chosen with property (4)
, it follows that ~(n+~) I An
We thus obtain an isometric A(~V_, P )
*
=
having the desired property
~
. oo 4 ~ An n:o - isomorphism
- homomorphism
which clearly extends to a
A
~(n)
*
of
into
A(~I,P)
. Q.E.D.
Note that , although the concrete and its conditional
C*- algebra
expectation with respect to
only on the choice of the minimal projections 9 - isomorphism between
A
and
A(~,P)
C(~)
A(k~L, r ) depend
, the constructed
is essentially based
19
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
The Gelfand spectrum Cn
~-n
C n . The map
responding to the inclusion map every minimal projection of of
Cn
Cn
Cn+ ~
containing it . Since the
limit of the
C*- algebras
it follows that
~
Cn
~ Kln Therefore
~q(~)~ ~
> Cn+ T
C *- algebra of all
iE In
n+~
.
~ ~n
cor-
associates to
the unique minimal projection C*- algebra
C
is the dlwect
following the inclusion maps ,
can be identified with the topological
limit of the discrete spaces
-CAn+ ~
on ~
of the commutative
can be identified with the finite set
the minimal projections of
~
~-~n
inverse
following the maps
9 , the points of ~
can be represented as
sequences t where
=
q(~)
is a minimal projection of
A point
t ~ ~
only if , for each
C n , for all
is adherent to a set
n >i o , there exists q(~)
=
~
s a oo
n >i o .
c ~'~
if and
such that
q(n)s
remark that the last equality implies :
Finally , for any ~u(t)
c ~
u ~ ~i
is determined by
o~k~
all
and any
t E ~
,
the point
.
2O q(n.)
I.~.~2. means
0.Bratteli
embeddings
useful for constructing . For instance
commutative
has studied
. These diagrams
, 0.Bratteli
([@])
is the center
The diagonalisation
,
AF - algebras b y
the factor components
approximately
C*- algebra
n~o
ordered set called the diagram
, which reflects
and their partial
u.q(~) u
([i])
of a certain partially
AF - algebra
bras
=
of
of the
of the An
' s
are particularly
finite dimensional
C*- alge-
proved that any separable of some
AF - algebras
AF - algebra
presented
here
. is
c l o s e l y patterned
after a similar method first used in the study
of the
anticommutation
canonical
physics
by
L.G~rding
w 2
relations
and A.Wightman
Ideals
in
([~])
of mathematical .
AF - ~igebr@s
In this section we study the closed two sided ideals of an AF - algebra logical
A
and we interpret
dynamical
the results
system associated
in terms of the topo-
to a diagonalisation
1.2.T. We begin with a known result ( [ i S , LEMMA sequence
of
. Let
A
b e s_ C - algebra and
C*- subalgebras
such that
n=o
An
3.T.)
of
A .
:
be an ascending
2~ T h e n for any closed two sided ideal J
=
J J
of
A
we have
n)
.
n=o
Proof
. The canonical
are injective a sequence
and therefore
xn ~ An
with
Yn ~ J O A n
entails
isometric
. For any
x e J
~ A/J there
lim llXn/JN Anl I = 0 . Thus there
such that
lira ~x - Ynll = 0 n->~
1.2.2. N o w let
An/JNAn
is
lim llXn - xll = 0 . It follows that
lim Iix~/Jii = o , hence sequence
* - homomorphisms
lim llXn - Yn~ = 0 n-~
is a
and this
. Q.E.D.
A = < ~_J A n >
be an
AF - algebra with
n=o
moa.s.a.
C =
~
Cn)
, conditional
expectation
P
: A
~ C
n=o
and group ponding
~
constructed
group of If
J
as in
* - automorphisms
=
~
the cortes-
C . A , then
J ~ C
~ - stable closed ideal of
arguments
of
is any closed two sided ideal of Ij
is a
w ~ . Denote b y
C . Conversely
, using standard
(~ 6 ]) , we shall prove the following
LEN~A
. Le__~t I
b_~e~
U-
stable closed
J(I)
=
I x ~ A ; P(x*x)
i_~s ~ closed two sided ideal of Proof Since
. Clearly
, J(1)
ideal of
C . The___~n
~ I3
A . is a closed subset
(x + y) (x + y) ~ 2(x*x + y~y)
of
, from
A . x , y
~
we infer P((z + y) (x + y ) ) ~
2(P(x~x) + P(y y))
~
I
,
J(I)
22
thus
P ( ( x + y ) * ( x + y)) Since
we
(ax)~(ax)
~ ~<
I , i.e.
x + y
llaU2 x*x , from
e J(1)
.
x e J(1)
and
a r A
infer
P((ax)*(ax)) thus
P((ax)~(ax))
e
I
ilall 2 P ( ~ x )
~ ,
i.e.
ax
In order to show that suffices Then
to c o n s i d e r
, by
P(x"x)
u*P(x*x)u
~ I
a I
, .
, a e A
u ~ ~
> xa
~ J(1)
, it
, c ~ C (see I . ~ . # . ( i ) ) .
, we have
P((xa)*(xa)) Since
with
I
J(I)
x E J(1)
a = uc
I.~.4.(ii)
e
~
and
= P(c"u~x*xuc) I
is
and t h e r e f o r e
~-
= c~c u * P ( x * x ) u
stable
P((xa)*(xa))
, it f o l l o w s 6 I , i.e.
. that xa
~ J(I)
.
Q.E.D. 1.2.3. algebra
Let
J
be a c l o s e d
A
and
LEMMA
.
x ~ J
P~oof
.
Clearly
P(x)
=
of the
AF -
x E A . Then
>
x e J Since
two sided ideal
~
P(x)~
, for each ~
lim Pn(X) n-~
J
.
n >I o
Pn(x) e J
we have .
, the Lemma f o l l o w s
. Q.E.D.
1.2.4. She
THEOREM
AF - ~ I g e b r a
A
. F o r an 2 c l o s e d two s i d e d
J
of
we have
Thus , J ~ I j correspondences
ideal
between
~nd
I.~
> J(I)
are in, v e r s e t o one a n o t h e r
the set of all c l o s e d two s i d e d ideals
J
23
o_~f A
and the set of all
Proof P(x*x)
. If
~ J g~C
P - stable c l o s e d
x ~ J , then
= Ij , i.e.
Conversely
j
, by
x*x e J
x a J(Ij)
1.2.%.
ideals
hence
, by
I
of
C
1.2.3.
.
,
.
a n d 1.2.2.
, we have
<<_/~nAn">
=
n=o
I%--0
p and therefore
it s u f f i c e s
{~An
JnAn Since central
J•
An
projection
to show that
; Pn(,X*x)~ (JnAn) nOn~,
is a two s i d e d p
of
An
x e An Then
x = px + ([ - p)x
PAnB Pn(X*X)
*
ideal of
such that with
Pn(X*X)
.
A n , there
J ~ A n = pA n
is a
. Consider
e J
and
= Pn(PX*X
+ (~ - p)x*x)
= pPn(X*X)
+ (T - p ) P n ( X * X )
whenc e Pn((('i
using
I.~.2.
we
-
p)x)*(('l
-
p)x))
= 0
.
infer that (~
-
p)x
= 0
and therefore x
The e q u a l i t y closed
ideal
of
I
= px ~ pan
=
Ij(1)
= J n An
is o b v i o u s for a n y
U
- stable
C . Q.E.D.
24
Remark that the correspondences are increasing w i t h respect
1.2.5. COROLLARY ideals of
J .~ w Ij
to inclusion
. Let
J~
and
I ~
J(I)
.
and
J2
J~ ~
C
be closed two sided
A . Then J~
Proof
=
J2r
. Follows
=
J2 n C
obviously from Theorem
.
1.2.@. Q.E.D.
1.2.6. COROLLARY
. Let
J
be ~
closed two sided ideal
o_~f A . Then J
=
the closed two sided ideal of
Proof
. Denote b y
generated b y
J~
J • C . Clearly J~O
the equality
J = J[
A
~enerated b_~
J ~ C .
the closed two sided ideal of
A
, J~ ~ J . Since
C c J~
C cJ~O
follows from
C
,
1.2.5. Q.E.D.
1.2.7. COROLLARY is faithful
. The conditional
expectati0n
, P(x*x)
>
P
: A
9 C
: x
Proof
~ A
= 0
. Follows from Theorem
1.2.~.
x
= 0
.
applied to
J = 0 . Q.E.D.
1.2.8. algebra
A
THEOREM
there
. For any primitive
is a maximal
ideal
I
ideal o_~f C
J
of the
such that
AF -
25
J ~ C
=
~ u m II
u~ Iu
The proof will be given in Section Remark that J
Theorem =
1.2.4.
der
[~
further implies
{x g A ; P(x'x)~
1.2.9. Denote by ~
1.2.[0.
u'Iu,
(~) u ~ I~ 1
the Gelfand spectrum of
as a group of homeomorphisms
of
C
9
and consi-
-(7- .
There is a one-to-one correspondence between the closed ideals
I
of
C
-~ C ( ~ )
and the closed subsets
oo
of
A'I
,
which is given by c~ I
=
{t ( ~ s
This correspondence
f(t) = 0 , (~) f ~ I ]
is decreasing with respect to inclusion .
The closed ideal the subset
~ I
of
I
iO_
of
is
C
P-
is
P-
stable if and only if
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 known ( [ 5 S )
that
two sided ideals holds
:
J J~
.
A
. It is
is primitive
if and only if , for any closed
and
A , the following implication
J2
of
26 J A
=
J[N
J2
> either
~ - stable closed subset
c ible if , for any of
_O_
J
oo
=
J~
of i~_
or
J
, the following implication holds >either
J2
will be called
~ - stable closed subsets
co = ooiU co 2
=
uoi
" [1_ irredu-
and
u02
:
co = c o i
or
cO=
6o 2
Thus , the correspondence J ~ carries the primitive
>
~176 C
ideals of
A
reducible closed subsets of ~ i
.
Let us denote by
the
by
P(t)
its closure
THEOREM A
~(t)
P-
orbit of
t
, P-
E ~
it-
and
. Then Theorem 1.2.8. rephrases as follows
. For any primitive
there is a point
C - stable
onto the
t o E ~-[ ~
c
ideal
J
of the
AF - al~ebra
such that =
P ( t o)
9
This entails the following property of the topological dynamical system
COROLLARY o_~f _O_
A
. The
:
P
-
stable
P-
irreducible closed subsets
coincide with the closures of the
The set of
(i~,P)
oJjNC
[1 - orbits .
associated to a closed two sided ideal
J
has a simple description in the terms explained in Section
I.~.~.
Namely , bet
Then the set property
oOjOC
~[
be a representation of
consists of all points
ten
A
with kernel J. having the
27 q l ( q ( ~ ))
~
0
for all
n>/o
.
1.2.~0. For the proof of Theorem 1.2.8. we need two Lemmas. Let space
H
U~
be a factor representation of
LEnA
~ . Let UT(eL)
~hen there exist
J
e~ , e 2 ~
0
.
b e pro~ections o f
,
k >~ n
U T ( e 2)
Proof every
0
such that
.
and a minimal central projection
~
0
,
~[(pe 2)
~
0
p
of
.
. Indeed , suppose the contrary holds
. Then , for
k > n , there exist m u t u a l l y orthogonal central projections , p(k)
of
Ak
with + p(k)
(2)
~
An
such that ~[(pe~)
p(k)
on the Hilbert
such that ker UT =
Ak
A
7[(p(k)e~)
=
0
Since the unit ball of that the sequences Denote b y
P~ ' P2
, L(H)
:
,
7[(p(~)e2 ) is
=
wo - compact
{7[(p(k))},{~l(p(k))}
are
0 . , we may assume
wo - convergent
their corresponding limits
. Then
.
P~ ' P2
are positive operators contained in the center of the von Neumann factor generated b y
T[(A)
in
L(H)
,
P2
, therefore they are scalar
operators P~ N o w from
(~)
= X~.
we ~ f e r
~
:
+ ~2 = ~
ha
;
, while
~ (2)
' ~2
#- [ 0 , ~ ) .
implies that
28 ~
= ~2 = 0 . This contradiction proves the Lemma . Q.E.D. LEPTA 2 . There is a_ sequence
projections
p(n)
of
An
for all
for an,y minimal projection
there exists
k >/ n
of minimal central
with the properties
(i) 7[(p(~)... p(n)) ~ o (ii)
~p(n)}
such that
q
n>~
of
Cn
; with
~[(q) ~ 0
UI (p(k)q) ~ 0 .
Proof . Indeed , let us write the set ~_~ ~q ; q n={ as a sequence
is a minimal projection of C n and
le{ , e2 , ... , ej , "''I
find by induction a sequence
~[(q) # 0 I
. Owing tb Lemma ~ , we
Ip(kj)l of minimal central projec-
(kj) tions
p
of
Akj
such that
kj ~ kj+~ SI (p(kj)p(kj_~)
(k~)
...p T[(p(kj)ej)
~
)
~
o
,
0
Clearly , this sequence can be refined up to a sequence
~p(n)1
having the stated properties . Q.E.D. Proof of Theorem 1.2~8. Put sequence
~p(n) 1
the
, s
point
p(n)
t o ~ ~'~
co = ooj(~C
and choose a
as in Lemma 2 . The condition (i) satisfied by
and the compacity of ~-~
entail the existence of
such that P(n)(t o)
~
0
for all
n ~
.
29 This means that
q(n) to
~
p(n)
for all
n ~
the notation being as in Section I . % . ~ . central
,
Therefore
support of the minimal projection
~(n) to
, p(n)
in
is the
A n . Since
qI(p (n)) ~ 0 , it follows that
. (n)~
T [ ( q to- ~ 0 Thus , t o s co
for all
and consequently
Now consider satisfied by the
t ~ co
p(n)
, s
n ~ ~
r(to)
and fix
c
.
oo
.
n ~ ~ . The condition
shows that there exists
(ii)
kn~
n
such
Ckn
with
that
p Therefore central
support
q to
support
p
in
Akm
o
r
~
0
is also a minimal projection
p
in
(kn)
of
u r '[lk~
Akn , there exists
,
of
such that
(kn)q(~)
Ckn
with central
Such that
.
r Thus
/
, there is a minimal projection
r Since
(kn)q(~)
=
u
q to
u
.
u* q(~) to u .<
q(~)
.
On the other hand ,
u* q (kn) to Since
q(~)
and
it follows that
u* q(n) u to
u,~ q(n) u u
~<
to
are both minimal projections
in
An
,
3o
q(~)
=
u* q(n) u to
We have proved that
, for each
s s p (t o )
n > &
, there exists
such that =
This means that
t e p ( t o) .
Therefore
,
co w h i c h proves Theorem
=
r ( t o)
1.2.9.
and its equivalent
form
, Theorem
1.2.8. Q.E .D. The above proof shows that the kernel sentation
of the
this result
AF - algebra
A
is a primitive
is kno~n for all separable
On the other hand
of any factor repreideal
C*- algebras
, the same proof
, but
([ 5 ]) .
shows that any primitive
oo
ideal of the
AF - algebra
A = ~_~
An~
is the kernel
of a
representations
of
n=o
direct limit representation the
An
' s .
1.2.~.
The primitive
of all primitive logy
of irreducible
ideals of
A
. The preceding results
w i t h the set of all closures lence relation
" N " t~ N
on
spectrum
~
t2 <
Prim(A)
of
A
is the set
endowed with the hull-kernel show that of
~-
Prim(A)
orbits
can be identified
. Defining
by ~
p(t&)
=
topo-
~ ( t 2)
,
an equiva-
31 it can be easily verified that the quotient space
~/~
Prim(A)
is homeomorphic with
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 with the
m.a.s.a.
C , O.Bratteli considers the intersections with the smaller sbelian subalgebra generated by the centers of the
A n ' s , the results
being 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 algebras ( s e e ~
w 3
S, 4.2. a n d S 3 S)
.
Some representations ~
We consider an
AF -
AF - algebra
AF - algebras
A = ~
AnT
together with
n=o the
m.a.s.a.
the group
U
C , the conditional expectation as in
w 9 . Let
logical dynamical system and sets of ~
(~,~) ~
P : A
) C
and
be the associated topo-
the sigma-algebra of Borel sub-
9 In this section we shall study two kinds of repre-
sentations of
A , $I~
invariant measures
~
and
~
, associated with
on the Borel space
(~,~)
~u_ quasi9
32 A positive measure on ~
will always mean a positive
regular Borel measure on i-~ . A probability measure is a positive measure of mass measures
~ , ~
on ~-~
are equivalent if
continuous with respect to with respect to
~
,
& , i.e. 5 ( ~ )
~
and
~
that is if
~
~
on
= ~ . Two positive ~
is absolutely
is absolutely continuous and
$
have the same
null-sets . For a positive measure of ~
onto ~
Then ~
is
(resp. ~ is
~
on
we shall denote by
~-
invariant (resp.
are the scalars
~
and a homeomorphism the transform of ~ by ~
[~- &uasi-invariant) if
equivalent to ~ ) for all
~ - ergodic if the only
~
~-
Then
~
~
=
~ E P . The positive measure invariant elements of ~ ,
5)
9
1.3.~. Th e construction of the representations Let
~
.
be a
p-
~[~
.
quasi-invariant probability measure on ~
can be regarded as a state of the commutative
.
C*- algebra
C -~ C(~-2) and therefore
is a state of
A . The
struction associates to bert space
H5
Gels ~
(abreviated GNS) cona representation
and a cyclic unit vector ~(x)
=
~
qT5 of r H~
(~[~(x)~F I ~ )
For the yon Neumann algebra generated by ql~(A) the bicommutant notation , T[~(A)" Since
~
is
P-
A
for ~ p , in
on a Hilsuch that
x e A L(Hs)
. we use
9
quasi-invariant , its support
~
is a
33 ~-
stable closed subset of ~-~ . Then it follows from J~
=
Clearly , ~
on
~~
A .
as a state of
and it is is
that
I x ~ A ; P(x*x) (t) = 0 , (~) t g ~ - ~ 5 1
is a closed two sided ideal of
~5=
w 2
~-
C
is faithful
if and only if
invariant if and only if the measure
~-invariant.
1.3.2. Let us recall that a state ~(xy) PROPOSITION there exists a
=
. A state
~
~(yx)
~
of ,
A
is central if
x , y
~ A
.
of
A
is central if and onl2 if
~ - invariant state
~
of
C
such that
T In this case
~ = T
Proof riant
. If ~
. Moreover
IC " is central
, for fixed
minimal projections of ~(Pn(X))
, then
~I C
is clearly
n ~ o , denoting b y
(q~
P-
inva-
iEI n
C n , we have
= T(~ qixqi ) = T ( 2 xq i) = T ( x ) i g In i g In
,
x g A .
C
and
Hence
Conversely
, for
~
a
P-
= ~ o p , we shall prove that x = u~c~
, y = u2c 2
T(xY)
the
with
invariant state of
~
is central
. Indeed , for
u~ , u 2 ~ U , c~ , c 2 ~ C , we have
=
~(P(u~c~u2c2u~u~))
~(u~P(c~u2eRu~)u~) :
=
~(P(c~u2c2u~))
=
:
~(P(uac2u~c~))
= T (~)
~(c~P(u2c2u~))
=
34 This ends the proof since
c.l.m.(UC)
= A . Q.E.D.
1.3.3. We shall prove that the representation standard , more precisely we have PROPOSITION measure on
~
Proof
. Le__~t ~
. Then
~
. B y the
Suppose
x
~ UTp(A)"
density theorem
~uasi-invariant probability
construction
~ ~
is separating
is such that
xk g A
is cyclic . For
A
SI~(A)"
we have
Kaplansky
there is a norm-boun-
such that ~IV~(xk)
converges
x . Hence
(4_) k->lim~~oP(x~kxk)(t) d~(t) -- k-~limII~rl~(xk) ~ pIi 2 : IIx ~tll 2 To prove that
x = 0
it will be enough to show that
T[ in a total subset of c.l.m.(UC)
.
, so all
x g A
x ~ p = 0 . By the
and the separability of
ded sequence of elements strongly to
P-
beg
~
is
:
is c2clic and s0parating for
GNS
we have to prove is that
71~
H~. Thus , ~ ~
x~
being cyclic and
=0.
= 0 A
for =
, it will be sufficient to prove that
= o
for all
ugU
, ceC
,
for
uEU
, c~C
,
i ogC
.
that is
II
(Xk<,C)l
II
:
o
all
k--~ ~
or equivalently
( (2)
lim~ (c*c)(t)
(u"P(X~k)U))(t) d~A(t) = 0
Thus we must prove that
(&)
~ (2)
; UEU
and , since
~
is
35 - quasi-invariant , this will follow from the following more general fact : "If
{fk~
a uniformly bounded sequence of positive
is
measurable functions on
_eL
, f ~ L~9 (I~, ~)
a probability measure on _0_ with respect to
~
absolutely continuous
, then
lim k ( t ) d~(t) = 0 k - ~ ~/hf
) k lim[ ~ J ~ f(t)fk(t ) dg(t) = 0" h = ~d9
To verify this assertion , let derivative of ~
and
with respect to ~
. Then
be the Radon-Nikodym h ~ L~(I~,~)
, h ~ 0
and we have to show that lim I h(t)f(t)fk(t) d~(t) = 0 k@~ Consider
En = Then
I t ~-k*~ ; h ( t ) ~ n t
( lin |
so for given
~ ~ 0
h(t) d~(t)
there is
no
=
0
such that
h(t) d~(t) ~ En o On the other hand , by the hypothesis there is for
k >~ k~
~ofk (t) dt~(t) 4 Hence for
k ~ka
we have :
2noM
k~ ~ ~
such that ,
36
I
5 h(t)f(t)fk(t) d~t(t)" =
+
h(t)f(t)fk(t) d~t)
En o +
I h(t)f(t)fk(t) d~(t) I-LTM Eno n~
+
4
M2
I fk(t) d~(t) ~L
5 l"k'- E
noM ~
+
h(t) d~A(t)
no
+ M2 *
2noM
2-~
= s
"
This ends the proof of the Proposition . Q.E.D.
1.3.4. PROPOSITION . Let
~
be a
C-
~uasi-invariant
probability measure on ~'L . Then we have ker~5 Proof . By Proposition x a ker~---~
P(x*x)
3.3.
T[~(x)~ } I
and since
=
J~
. we have
= 0 <
> ~(x*x)
P(x*x)(t) d~(t) = 0
is a continuous function on l~-
x ~ ker~5(----> e-----~
= 0
it follows
P(x~x)(t) = 0 , (~) t m ~ L ~ x
~ a~.
Q.E.D.
1.3.5. The Gelfand spectrum of via
Gelfand isomorphisms
C
~- C(L~L)
UT~(C)
is
~L~
. Hence ,
, UIs(C ) ~- C(I~)
, the
37 restriction of
~
to
C
c(~l)
The s t a t e of
C(~)
corresponds to
~ c ~
~
c l~l~
corresponding to
~ c(~l~)
~
can be f a c t o r e d through
t h i s homomorphism , the corresponding s t a t e of restriction element
of the measure
~(C)
B ~[~(c)
to i t s support
~- f ~ C ( ~ )
(~(c)~1 Moreover
~
~)
=
~5(C)"
C(-(~ r) ~
being the
. For any
we have
Ixlf(t)
d~(t)
, we know that the vector state
faithful normal state of
.
x ~. ~ (x ~ I ~ )
is a
.
It follows then from known results ( ~ g ] ,
Prop. ~ , w 7 ,
Chap. I ) that : The Gelfand isomorphism
~(C)
~- C(A~5)
has a unique
extension to a normal isomorphism
(3)
~T~(c)" -- T ~ ( ~ , ~ ) such that , for
~5(C)" B c
(c~l~)
z
~- T,~(-~,i~) f ~ L~(AO-, ~ )
= I~f(t) did(t)
, we have
9
The following result is also well known If
E
is a faithful normal semifinite trace on
L~(~, ~)
then there is a unique sigma-finite positive measure (4)
on
~L
, equivalent to
~
z(f) for any ~ ( # L , ~ )
, such that
=
I'3Lf(t) d~(t)
, f>10
.
38 1.3.6. PROPOSITION . Let probability measures on
~& ' ~2
be
~-
quasi-invariant
A~- . Then the representations ~ &
are unitarily equivalent if and only if the measures ~
'~2
, ~2
are
e ~u ivalent . Proof . Since ~T~& , ~ 2
are equivalent , they have the
same kernel , so that , by Proposition 1.3.4. , ~ Moreover , there is a normal isomorphism
= ~-
P',_
~2 (C)" ~_ q~2(C)"
~
"
which extends the isomorphism
rJ'['[4.~.(C) B qj'[~,.t(c)
,',, c]'[~.2(c) E. ~J[F2(C)
That is , there is a normal isomorphism
L~(
,,J.,~,'1_.)
equal to the identity on the equivalence of ~ i
C(~&)
dM 2
= ~
Then there is a sequence converges in
L&(A~L,~&)
a) {Cn}
=
and ~ 2
Conversely , suppose ~
h
"" L
(_.Q")...~2,~.. 2 ) 0(~2)
" , ~2
are equivalent . Consider
~ L ~(..o,~.~.)
,
h >t0
.
cn ~ C = C(L~L) , c n>1 0 , such that {Cn 2] to
h
and therefore
is a Cauchy sequence in
b) fy(t)h(t)
. This easily yields
L2(i~ ,~&)
;
d~(t) = n~lim ~3f(t)Cn(t)2 d~{(t), (V) f ~C(~l).
Next we have II~ and from in
H
(Cn)~ a)
. Put
- qTD ( C m ) ~ U 2
= fxTlCn(t) - Cm(t)I 2 d~(t)
it follows that {~T~ (Cn) ~ ~ I is a Cauchy sequence ~
=
lp i m ~
(Cn) ~
E
H
. In view of
b) ,
39 for all
x E A
we have
:
(~:'la-i (x)'~l I~) =
(~j'[ ~.~.(CnXCn)~,l[ ~p~.)
n-~oolim
:
lira f P(x)(t)Cn(t) 2 d~r n ~ co i_O_
:
#
=
(P(x)(t)h(t)
d~[(t)
~xt
= (P(x)(t)
d~2(t)
=
ill
hence
: Thus
, there is a unique
isometry
( x ) ~111 V
of
v(~[~jx)~2) = ~(x)~, Clearly
, V
is intertwisting for
T[~2
into
H52
and
x ~ A
.
q[~i
.
Since the same kind of argument shows that equivalent to a subrepresentation
of q l ~ 2
Hp~ such that
~[
is also
, the SchrSder -
Bernstein type theorem gives us the desired result
. Q.E.D.
Let us emphasize that , the representations standard
, two of them are quasi-equivalent
are unitarily equivalent
on
~
1.3.7. Let
~
. For each
n ~ o
nal expectation
q[~
being
if and only if they
.
be a
r~- quasi-invariant
probability measure
there is a strongly continuous conditio-
40
Pn : T[~(A)"
q[p(A)"
T(~.(Cn)'('l
>
defined b y
i ~I n where
{qi] i s n
are the minimal
P~n(q'[~.(x)) and
, since
of
= q'[~ ( P n ( X ) )
C n . Clearly
,
x e A
,
X
,
x e ~I,.(A)"
,
p o Pn = P ' we have
B y the strong continuity
Moreover
projections
, for any
of
x ~ A
Pn~
EA
.
we infer
.
w@ have 2
= ('-J~p(:Pn(X)*PD(x))~pI ~p) *p n(X)) = (~ ~ P ) ( P n ( X ) ~4 ([t ~ e ) ( P n ( X * X ) ) = ~t(P(x*x) ) 2
= I1%(x) and
, again b y the strong continuity
of
Pn~
,
t
By Lemma 1.2.3.
, there
P~ : Tgp(A)
is a projection
x
~
~K~(A)"
of norm one
> ~(c)
such that
(7)
Pl~(i1171,(x))
= qI~(P(x))
,
x e A
.
P~(~(X))
=
,
x g An
,
Since (8)
P~n(q'[~(x))
.
41
it follows from
(6)
that n=o
Hence , for any
T' E q[~(A)' , we have
n=o
Since by 1.3.3. show that
P~
~[~A)'%~
is dense in
H ~ , the preceding results
is strongly continuous on bounded subsets of
~[~(n~J_o= An ) . Using the Kaplansky density theorem , we can extend P~
up to a linear map
pF : IT~(A)'
) q]~(C)
strongly continuous on bounded subsets . It follows that
P~
is
a projection of norm one and also a normal map . Thus we have (see also[ 6 ], Th. 2 , w 4 , Ch.I) : P~
is a ultraweakly and ultrastrongly continuous
(9) conditional exptctation of ~[~A)" with respect to g[~(C)". Owing to the relation (8) and to the continuity of
P~
and
P~
,
it follows that
(~o)
=
,
.
n.-~
Then we have also (~)
,
where cular (~2)
P~(x)
is regarded as an element of
Le~
x ~ q]~(A)
~)
. In parti-
,
P~
The conditional expectation
is faithful .
Also , clearly , we have (~3)
P~(u*xu)
=
u*P~(x)u
,
x ~(A)"
,
u ~(u)
.
42 1.3.8. PROPOSITION probability measure
~(A)"
on i~t
~
be _a
. Then
~ - guasi-invariant
~[~(C)"
i_~s a_ m.a.s.a,
in
. Proof
we have
. Let
9 Consider
x s 7[~(A)" N q[~(C)'
Pn~(X) = x , for each
and , since
~
n >i o . By
is separating
,
. Since
(~0)
x aT[~(Cn)'
we infer
x = P~(x) e ~[~(C)"
. Q.E.D.
1.3.9. PROPOSITION probabilit,y measure if and onl.y if Proof p e ~[~(C)" p 0
~
on i9_ is
be _a [~- 9uasi-invariant
. Then
T[~
qT~
is a factor representation
is a factor representation
L~ (i~, ~ )
~ ~[~(A)" (] (~[~(UC))' or
~
[~- erg~odic .
. Suppose ~-
. Let
be a :
~-
and let
invariant projection
~[~(A)" (~ T[~(A)'
. Then
and hence is either
~ . Conversely
tral projection and clearly
p
, suppose P
is
~
E ~[~(A)" ~-
is
P-
. By 1.3.8.
invariant
ergodic and consider a cen,
p E ~(C
. Thus , p
),, -~ T~(XI,~)
is either
0
or
~
Q.E.D. 1.3.~0. PROPOSITION probability measure o_~n ~ i if and only if lit,y measure A
~
o_~n ~ -
. Let
~
be _a
. The representation
i_~sequivalent to some . Moreover
i_~s quasi-equivalent
[~- quasi-invarlant ~[~
is finite
[~- invariant probabi-
, ever~ finite representation
t__ooa representation
~[~
9
o_~f
.
43 Proof . Suppose let = r
~
is a finite representation of
i~ be a normal faithful finite trace on . Then the representation
representation state
Z o~
there is a that
of
A
Z oqT
the measures
GNS
~
and
Gonversely
~
~
~(q)
=
is central
construction for the
, b y Proposition 1.3.2.
invariant probability measure
w o7[ = 9 o p . If
with
and
is quasi-equivalent to the
obtained via the
. Because ~-
~
~(A)"
A
is some
~
~)
on
~L
such
, the equivalence
of
follows from Proposition 1.3.6.
, if
~
on ~'~ , equivalent to
~
, then ~
and ~ 9
1.3.6. Moreover
being central
, ~ e
, ~9
is a
for the yon Neumann algebra
~-
~J~(A)"
invariant probability measure are equivalent b y is a trace-vector
. Q.E.D.
1.3.r
PROPOSITION
. Let
~
b_s a_ ~ -
quasi-invariant
P r o b a b i l i t y measure on ~-~ . The representation finit__~e if and onl,y if ~-
~
~I~
is semi-
i_gs equivalent to some sigma-finite
invariant positive measure on ~)_ . Proof
~(A)"
. Let
~
be a normal semifinite faithful trace on
. We shall prove that the restriction of
is semifinite
iz to " ~ ( C ) "
. Thus , for any
y ~[~(C)"
with
y ~ 0
,
y ~ 0
,
we must prove the existence of z Since
~ q[~(C)"
with
0 % z ~ y
-t is semifinite and faithful
and
0 ~12(z) ~ + o~
, there is
.
44 x
~ ~[~(A)"
Moreover , since
~
with
0 ~< x ~
and
0 ~(x)
~ + o~
is a separating vector , we have
.
xI/2~ % 0
and therefore
Consider x n = P~n(X) ~ ~ ( C n ) ' Because of the well known properties of the trace ([ 6 ], Prop. ~ , w 6 , Chap. I) , we have -d(xn) = Z ( T i n ~ ( q ig
i) x ~ ( q i
)) : Z ( ~ x~(qi))= i~I n
Z(x)
.
By the relation (5) from 1.3.7. , we get (Xn~l~)
=
(x~l ~)
~
0
9
Note also that 0~<Xn4Y Thus , there exists a weak cluster point {Xn}
z
of the sequence
and we have
O.
% y
z ~ f'] ~ ( C n ) '
: ~[~(c)"
n=o z(z)
G
lira ~ f ~ ( X n )
=
z(x)
<
+ o~
,
n - ~
where we have used the fact that
~[~(C)"
and the weak lower semi-continuity of The existence of a measure
~
on
-~
of the assertion
~-
from
m.a.s.a.
(I.3.8.)
.
invariant sigma-finite positive
equivalent to (4)
Z
is a
1.3.5.
~
is now an easy consequence applied to the restriction
45 of
z
to
Z~(c)"
=
Conversely
L~(fl,~)
, let
tive measure on ~-L
~
.
be a sigma-finite
, equivalent to
~
[~ - invariant posi-
. For
x e ~(A)"
, x >i 0,
we define ~(x)
=
(JfLP~(x)(t) dg(t)
which is correctly defined since with respect to That
Z
~
W
is absolutely continuous
. Thus we get a
weight
Z
on
(q~(A)") +
@
is faithful and normal follows from the corresponding
properties of
P~ (see 1.3.7., (9) and ([2)) , ~
continuous with respect to 9 a direct consequence To prove that
Consider
ej t g
. Also , that
of the fact that ~
~
-g
being absolutely is
semifinite
is sigma-finite
is
.
is a trace , we must show that z(x'x)
=
projections
in
ej P~(x*ejx) t P~(x*~x)
z(xx*)
and
,
~[~(C)"
with
x ~ ql~(A)"
.
Y(e~) < +oo. Then d
ej P~(xejx *) ? P~(xx*)
and therefore ~ (x'x)
=
-~(xx ~)
=
lim
j-~
lira
j-~
I
ej P~(x*ejx) d9
I
ej P~(xejx*) d9
Hence it will be sufficient to prove that
(~5)
for all
~oej P~(xejy) d9
x , y
=
I~Lej P~(yejx) d~>
~ ~f[~(A)" and all
j
,
.
Let us now show that it is sufficient to prove
(~-5) only
46 for
x , y a ~Tw(A) . Indeed , for
x , y ~ ~I~(A)"
by Kaplansky's density theorem , two sequences strongly convergent to
x , y
~5(C)"
that is
@j = ej d9
=
@j(P~(xe jy))
9j(P~(YkejXk) ) =
9j(P~(yejx))
is strongly
are normal posi-
,
,
k~lira ~ Ioej P~(xkejYk)d~
= %Oe J P~(xejy) d~
lira Ii~Lej P~(YkejXk)d~ k-~
= Io_ej PW(yejx) d$
Now we may further restrict the proof of when
P~
~-- L~ (A~, 5) , we have
lira $ j(P~(Xke jYk) ) k@o@ lim k~
{Xk] ' {Ykl C ~ ( A )
respectively . Since
continuous on bounded sets and since tive functionals on
we can find ,
x = u~c~ , y = u2c 2
But then , by the !s
~-
P~(xejy)d~
with
u~ , u 2 ~ l ~ ( U )
invariance of $
(~5)
, c~ , c 2 ~ ( C ) .
, we have
= I
u~ P~(c~eju2c2eju~) u~ d~
= I
P~(e~eju2c2eju~) d~
=
to the case
l l,t. ej Pg(u2c2eju,lc~) d~
= I~k ej P~(yejx) d~ Q.E.D. The preceding Proposition shows that all semifinite representations
51~
can be obtained by choosing , for any given sigma-
47 finite
~-
ty measure
invariant positive measure ~ , ~
an equivalent probabili-
'I[~.a\,, j
. Then the trace on
is given by formula
(~4) 9 Note also that the above proof shows that the restriction of the trace to
ql~(C)"
is semifinite . It follows that , in the
factor case , ~ [ ~
is discrete (i.e. type I) if and only if
or equivalently
, is discrete (i.e. completely atomic) .
~
~
,
1.3.~2. Summing up the results of the preceding Sections , we obtain the following Theorem . THEOREM . Consider an conditional expectation dynamical system
P : A
(A~_,[~) . Let
probability measure on - ~ with cyclic vector the state
~oP
AF - algebra
~~
of
a
~-
C ,
be a_ P -
quasi-invariant
be the representation of GNS
A
construction t__oo
A . Then : ~~
is (cyclic and) separating
q~(A)".
is a m.a.s.a, =
with m.a.s.a.
and associated topological
associated by the
for th__~eyon Neumann algebra
3) ker ~
~
and qT~
~) qf[~ is standard , i.e.
2) qT~(C)"
~ C
A
in qT~(A)".
I x ~ A ; P(x*x)(t) = 0 , (~) t c s u p p o r t o_~f~}.
4) qT~ (A)"
is a factor
<-->
~
is
5) gT~ (A)"
is finite
(~>
~
-is equivalent to
invariant probability measure on ~
finite representation of
A
[~- ergodic .
. Moreover , every
is u a ~ - e ~ u i v a l e n t
to some
q~.
48 6) ~[~( ~
is semifinite
<~>
~
is equivalent to
~ - invariant sigma-finite positive measure on _CL. 7) T [ ~
' q [ ~ 2 are e~uivalent <---> ~
Moreover , concern in~ the t.ype of cas~
(i.e.
when
(i) ~ [ ~ card(supp~)
~
c[[~ , in the factor
is ergodic) we have :
is of type In = n
, ~ 2 are equivalent .
;
<------> ~
is discrete
and
<~>
is discrete
and
n ~ ~ .
(ii) ~f[~ is of type Io~
~
card(supp ~) = eo . (iii) ~[~ is of type II~ non-discrete
r
~
i_ss equivalent to
F~- invariant probability measure on _CL
(iv) T[~ is of type I I ~ a non-discrete
~-
~
>
~
9
i_~s equivalent to
invariant sigma-finite infinite positive
measure on i-)- . (v)
T[~ is of t.ype III
t_~o any sigma-finite
P-
<
group
P
is not equivalent
~
~ (~
]).
2.5.) has given a necessary and
sufficient condition that 8 state of an state . This result was extended by AF - algebras :
.
, then the
non-measurable with respect to
1.3.~3. R.Powers ( E 2 ~ ] ,
i~_
is not equivalent to any
P - invariant positive measure on ~
is said to be
the case of
~
invariant positive measure on
In the last case , i.e. if sigma-finite
>
UHF - algebra be a factor
O.Bratteli (~ i ~ , #.#.) to
49
-Let of
A = <~ An> n=o
be an
A . The representation
GNS
construction
if for every
for all
x e A
Y E A~
g[~
there is
associsted to
n $ o
A = < ~J A ~ n=o
be an
' 502
quasi-equivalent
such that
AF - algebra and ~
by the
GNS
7[ T @
construction
if and only if for every
asso, are
~ > 0 there
x ~ A~ .
riant probability measures on in
~
bability measure o_~n i~i f e C(i~4)
g ~ C(~-)
i~-
. By
Pn
~ op
in order
[~- quasi-inva-
we shall denote the
.
1.3.~4. PROPOSITION
riant
:
such that
to obtain ergodicity and equivalence criteria for
every
and
, ~o 2
' g[~2
We shall use these results in the case of states
Un
if and only
, have shown
~T
image of
by the
AF - algebras
ciated to
for all
~
UH~~ - algebras
A . The representations
n ~ o
a state
2.7.) for
states of
is
~
.
O.Bratteli (~ i ~ , ~.5.) for
(~7)
algebra and
is a factor representation
The same way , R.Powers ( [ 2 # ] ,
Let
AF-
. Let
~
b.~ea_
. Then
~
i_~s [~ - -erfi0dic if an__ddonly if
t h-ere i__ss n >i o
we have
p-
~uasi-invariant
pro-
such that for all Pn- inva__.._-
80
Proof . As we have seen , if
~l~
~
is
P - ergodic if and only
is a factor representation . Hence , by
(~6) ,
~
is
- ergodic if and only if the following holds : for levery
x~A
there is
r(P(xy)) for all
-
n ~ o I suih that
r(P(x))~(P(y))
~<
II
~(y)
y s A~
We must prove that this is equivalent to f of every
I
(~9)
[for all
That An' ~ C
is
x ~ C
~n - invariant
(~8) ~-~ (19) the
set
there is
of all
establish
Thus , for
For
that
x g Am
y E A~ C A~
P(x)P(y)
('18)
Pn -
we have
y E C .
invariant
(~9)
holds
choose
such that
follows immediately from the fact that
Conversely , suppose to
n >i o
for
n > m
elements
of
C
9
holds . Clearly it is sufficient x
in the
dense
such that for
P(y) a A' g]C n
set
z ~ A~NC
(by I.~.5.) and
(by I.i.7.) . It follows that
i
U An 9 n=o
we have
P(xy) =
51
Q.E.D. 1.3.15. From the other result , (~7) , of O. Bratteli
we obtain in a similar way
PROPOSITION
. Let
bility measures on
~_
~i
' ~2
. The_._~n~
be
~ ~ 0
fo~ all
f ~ C(II) .
P n - Invariant
P - quasi-invariant probs~2
there is
are e~uivalent if
n ~ o
1.3.~6. The irreducible representation s Let for every
~
be a
such that
~5
.
P- quasi-invariant measure on ~ -
f ~ C(i~) , the operator
and
:
and
and only if , fo r ever,y
R. Powers
~5(Tf)
on
. Consider,
L2(i~,~)
defi-
ned by ~(Tf) and , for every
h
=
fh
~mP
; h~L2(l~,~)
, the operator
~(V~)
on
L 2 ( ~ , ~)
defined by (f~(~)
Then
Tf ---*~(Tf)
presentation seen
h) (t)
~
=
k~-
, V~
of the
h(#-[(t))
~9~(V:) C*- algebra
, t ~i~!
h~L2(l~l,~).
oan be e~tended to a * A(lq, P )
,
as it is easily
,
Composing this
* - representation
of
re-
A(AD-, [')
with the
52
-
isomorphism of
A
onto
* - representation of
A(l~i, P )
(see I.~.[0.)
, we get s
A , which will be still denoted by
~5
This notation makes sense only in the presence of a fixed morphism of units for
A
onto
A(~,~)
* - isomorphisms (that is , dif-
ferent systems of matrix units) may yield different
f~ is irreducible Indeed , L~(i-l, ~ ) ~(A)
is a
~
<
>
m.a.s.a,
in
is the set of
Let us also remark that
r~I
C
we have ~
and
A
~
and
A(i~, P )
:
i_.ss r -
ergodic
L(L2(I~,~))
and the corn-
invariant elements of ~(I-[, ~). does not depend on the choice
of the system of matrix units which implements the between
* - represen-
A .
Concerning the irreducibility of
mutant of
* - iso-
(that is , a fixed system of matrix
A ) , since different
tations of
.
~ - isomorphism
and therefore the equivalence of
, even obtained from different
* - isomcrphisms
~i
, entails
2 the equivalence of the measures Conversely
, if ~
multiplication by between
~
and
(d~[/d~2)~2
and ~ 2
~[ ~2
and
~2
"
are equivalent
, then the
implements an unitary equivalence
regarded as representations
of
therefore
, between the corresponding representations
obtained
via
the same
* - isomorphism
A
=
A(~, ~) of
A(~, ~)
A
,
.
1.3.17. In this Section a general method of constructing -
representations
of
A(i~, P )
outlined . Unfortunately
and hence also of
A , will be
, it is of limited use for our purposes
and
53 since it is difficult to decide when such a representation factor representation For
is a
.
f ~ C(I-L) and
~
P
let us denote by
f~
C(~)
the function defined by f~(t) Then , if
~
=
f(r
is a representation of
space
H , we have that
~ ~
of
and
is a
r
Moreover
,
f~
~(Tf)
t ~ ~
A(~,~)
>~(V~)
.
on some Hilbert
is a unitary representation
* - representation of
C(A~-) .
, we have
f(vt)f(Tf)r
=
T h i s i s w h a t i s known a s a c o v a r i a n t example[9], and , in case example [ 4 S ] , -To
Def. 2 ;[3~], H
3.~o
is separable
Prop.
3.5.)
(Tf )
representation
( see for
) of the dynamical system
, the following
(~,~)
i s known ( s e e f o r
:
give a representation
~
of the dynamical system ( ~ , ~ )
is equivalent to give : (i) a
~ - quasi-invariant measure
~
on
~-
(or rather an equivalence class of such measures); (ii) a 5 - measurable field of Hilbert spaces t .' ~
Ht
(iii) for each
over ~ r~
~
;
, a measurable field of Hilbert
space isemorphisms
~,t
: H~(t)
~
Ht
such that
Then
~(Tf)
is the multiplication operator by
f
and
;
54
(~(V,),)(t)
for any
~
d : (~)
J~
Ht
y2 ~@_~,t~(~-~(t))
dp(t)
Let us remark that :
In order that the preceding representation of the dynamical system should yield a representation of
A(~h, P ) , it is
necessary and sufficient that the following additional requirement be satisfied : (**)
T~,t = T#',t
for all
t~O-
such that ~(t) = ~'(t).
It is also easily seen that I A necessary condition for the factoriality of the above representation is the ergodicity of that
dim H t
~
and the requirement
be almost everywhere constant .
For instance , in the case of the one dimensional trivial field of Hilbert spaces over and
t E ~-~
~
and
~,t
, we get the representations
= I ~
of
for all
~ ~ P
1.3.~6.
Further , in the above general context , The equivalence of two such factor representations entails the equivalence of the corresponding measures on the equality of the numbers
dim H t
I~
and
.
Moreover , in view of the special nature of our group
~
,
an infinite algorithm can always be given for finding the solutions
55
of
(*)
and
(~)
, as in the case of the canonical anticommuta-
tlon relations ( ~ Z ~
; see a l s o ~ )
.
1.3.~8. Most of what has been presented in this Chapter has its roots in the study of the representations of the canonical anticommutation relations and of the associated
UHP - algebra
and topological dynamical system . In this case
~
by a smaller group
~o
freely acting on ~-~
presence of a measure
~
Then the
A ( ~ ,Po)
of
C*- algebra
C(~)
by
Po
on
~
can be replaced
and which , in the
, has the same "full group" as ~ . is isomorphic to the cross-product
and results similar to Theorem 1.3.~2. are
well known . The representations in fact ,
via
~
we have considered correspond
the isomorphism
product construction" of systems (A~, ~ , P )
A
z
A(17.,~ ) , to the "cross-
W. Krieger ( ~ Z ~ ) for the dynamical
.
Namely , given an arbitrary dynamical system even if
~
does not act freely , W~ Krieger has constructed a
standard yon Neumann algebra ~o(~)
in
( ~ , ~, P )
~(~)
~ (~)
together with a
and has described the type of
m.a.s.a.
~(~)
in a
manner completely similar to t~at in Theorem 1.3.12. A detailed exposition of W. Krieger's construction can be found in the book of A. Guichardet (~4~,Chap. VII) , where it is also pointed out that there is a unique conditional expectation of ~ respect to
~o(~)
9
(~)
with
,
56 The construction of W. Krieger shows that
~ (~)
is gene-
rated by a covariant representation of the dynamical system ( ~ , ~ ) . This extends to a via
the
tion of
* - representation of
* - isomorphism
A
A(A~,~)
~- A ( I ~ , P )
, to a
and therefore , * - representa-
A . It can be shown that this representation is unitarily
equivalent to the representation corresponds to with respect to
~I~
in such a way that
~I~(C) ~ , the conditional expectation of ~o(5)
corresponds to
P~
~ oP
~
(~)
and the state of
associated to a certain cyclic separating vector for out by W. Krieger corresponds to
~o(~)
~(~)
A
pointed
9
Our choice of an exposition where W. Krieger's construction does not explicitely appear was motivated by the fact that once I-L and
P
are fixed , the representation does not depend on the
isomorphism chosen between
A
and
A(~, P)
(that is , on the
systems of matrix units) . Also , to make our exposition more selfcontained , we had to reprove in this frame-work some known results in the case of W. Krieger's construction .
CHAPTER II
THE
C*- ALGEBRA ASSOCIATED TO A DIRECT LIMIT OF CO[~PACT GROUPS
Let us consider a sequence
[e]
=
GO c
G{ c ... c G n c
of separable compact groups such that each in
Gn+ ~
Gn
of Haar measure zero . Let further
limit of the groups
is a closed subgroup Go@
of involutive Banach algebras
M(Gn)
, the completion of which is an
is a closed ideal in
~(n)
M . The group algebra
M(G n) , hence
~ L~(Gk)
:
is an involutive subalgebra of
M(Gn)
.
Gonsider the involutive Banach subalgebra
The
L = L(G~)
of
defined b y L
=
L(Go o)
measure-theoretic
that for
~ k 6 LK(G k)
=
=
k ~ L(n ) n={L
assumption ,
,
define a direct limit
involutive Bsnach algebra we shall denote b y
M
denote the direct
G n , endowed with the direct limit topology
II.%.~. The measure algebras
L[(Gn )
...
Gn+ ~ c
k = ~,
C
M
made at the beginning insures ..., n,
IL(n)= zk=llkll L{<%)
we have
58 Hence
is closed in
L(n )
sum of the
L~(Gk)'
element
s
L
s
~
,
and it is the topological
k = l,...,n
. It
follows
that
direct each
can be uniquely re~resented b_~ a series
and
For further use we also consider
L%(G k)
L(n)= L
k> n Then
L (n)
algebra
is a closed two-sided ideal of
L / L (n)
is isomorphic to
The algebra
L = L(G~)
factor representations
of
G~
factor representations
of
L
of
G
or of some
11.1.2. Let of
G~
L(n )
L
and the quotient
.
can be used in the study of the . In fact, as we shall see, the correspond to factor representations
Gn .
f
be a continuous unitary representation
. Then we can associate a representation
as follows. The restrictions
~n
sentations of the measure algebras sentation of
of
~
to
Gn
~
of
L
define repre-
M(G n) . These yield a repre-
M . Finally, restricting this representation to
we get the representation
L ,
~-[~ we were looking for.
For completness we must also record a second kind of representations of of
Gn
L . For
fn
a continuous unitary representation
we get a representation of
M(G n)
and, b y restriction,
59 a representation
of
L(n ) . Since
this yields a representation
L(n )
~n
of
is isomorphic
to
L //L (n),
L .
II.~.3. Before going any further let us fix for each
n
a
sequence
such
that
u(n)
J Then
[u~n)I~a~
is an approximate
unit for
L (n-i)
, i.e
,
j--~ We shall use repeatedly continuous
the following remark.
unitary representation
If
~ T
is
its extension to the measure group algebra and if a sequence
~n
of measures converges w e a k l y * converges
to a measure
~
, then
in the weak operator topology to ~ ( ~ )
II.{.4. Eow let There are two oases a). ~ Since
of a compact group and
is a
~
.
be s factor representation
of
G .
: there is
n ~ ~T
such that
L[(Gn+%) contains an approximate
q~(L (n)) = 0 . Let
-c (~n)
nor
~
~ ( L (n)) : 0 . We may view
T[ (L~(Gn+~))
unit for
be the smallest
n ~ Z
~-~ as a representation
= 0 .
L (n) , we have such that of
60
L(no)-- L / L (n~ since
~
Since
.
is factorial,
L~(
Gno
)
is an ideal of
(no)
,
it follows that
ST kuj
L( )
and
no)
converges
in the strong operator topology to the identity operator. Let no and
be the representation Tno
of
Gno
be its extension to
corresponding
M(Gno)
. For any
toT[ I L~(Gno) ~ ~ L(no)
we
have
~no(~)
: wo -.lim ~n (~ * uj(n~ O --~ao
0
f
= w o - lira ~ ( ~
*
j --~o
k
~n~ uj
: w e - lira T[(~)T[(uj
)
(n o )
j --~
) :T~(~)
This means that
'17 b). Suppose Since
L (n-~)
=
T[~n ~
~-~ (L~(Gn)) ~ 0
is an ideal of
9 (n)) follows that ~T (uj
L
for all and since
~l .
G-C is factorial,
it
converges in the strong operator topology
to the identity operator and so ~TIL~(G n) ~n ' Zn
nr
be the representations
of
is non degenerate. Let
G n , M(Gn)
respectively
,
corresponding to 'I"[I L~(Gn ) . We shall prove that
(~-)
-gn+~
[ M(Gn)
= ~n
"
=
'
Clearly , this implies
fn+~ IGn
fn
which allows us to define a continuous representation such that
~
of
G~
61 T[
The following computation
=
T[f
9
, with ~ E M(G n) , establishes
(1) :
(n))
Zn(~) = wo-lim Zn( ~ ~ uj 3.~oa
= wo-I im ~[ (~ * u~ n) ) 9 (n+~.)))
= wo-i im
= wo-I im (wo-lim T[(i~oo ~ ~ uj(n) ~ ui' (n+{))) j~
~n+~(~ -u~n)~"
= wo-I im (wo-~im i-~
9 (n.~))) "i
oc,
= wo-lim~ =oTn~l(~ * uj(n))
= Zn+~(W) II.~.5. Summing up the preceding discussion
, we obtain
the following THEOREMS. ~ factor representatipn of representation
some
Gn
q~[~n where
, nC~{~}
~n
L(Go~)
is alwa2s
is a factor representation
o~f
.
Thus , there is a canonical one-to-one correspondence betwee n She factor representations of of the factor representations
of the
L(G~) Gn' ~
and the disjoint union (n ~ ~ ~ [ ~
) ~
It is clear that this correspondence preserves the yon ~eumann al~ebra generated b.y the representations
and also the equi-
valence o_~frepresentations. Since the yon ~eumann algebra generated by a factor representation of a compact group is finite-dimensional represent atipn of
L(G~)
, an~ factor
which generates an infinite-dimensional
62 y o n Neumann algebra
(tvpes Io=, II , I I I )
t._o _a factor representation
w 2
~
of the direct limit group
AF - algebra associated G~P~G~
g~_~
and ! ~
Since we are concerned with involutlve
Banach algebra
envelopping see
L(G~)
C*- algebra
, A(G~ )
is an
corresponds
outlined
II.2.~.
* - representations
A = A(G~)
of
L(G~)
envelopping algebra
C*- algebra
X . For
LEMMA
. Let
b_~e a n increasing bras there
such that is a
,
X
~
simple remark
k~ Xn n=~
of finite =
. As we shall
of
involutive
A
qTn
the
Banach
becomes much easier
. Banach algebra
dimensional
of
X
and let [Xnl
involutive
X . Suppose moreover
- representation
some of the
task to determine
of a given
be an involutive
sequence
the
I o
L , the determination
b y using the following
to consider
AF - algebra
it is a difficult (~7])
of the
and the aim of this section
in Chapter
In general
.
d!Gg~G%!G~ion
will be to carry out for this particular constructions
G~o
to a direct limit of
, it is natural
AF - algebra
automatically
subalge-
that for each n e
whose restriction
to
~
i_~s faithful
. Then
The proof
II.2.2.
X
is the direct limit of the
is obvious
For
X = L
' ~ .
, so we omit it .
we shall construct
a special
sequence
Xn
63 First some notations. In order to avoid notational complications in the sequel we shall denote the convolution as a usual multiplication
: a * b
Let
Gn
=
a b
.
be the dual of the separable compact group
G n , i.e. the
set of equivalence classes of irreducible unitary representations A
of
G n . For
fn s Gn Xfn
we denote by : Gn
>
its character , by d~n its dimension and by
= 9(fn(e)
~n
the corresponding conjugate representa-
tion . Then Pfn
:
d
fn X { n
~
L~( Gn)
is a central projection (i.e. selfadjoint idempotent) in
M(G n) o
L(n)
"
We write A
(fn ~ G n , fm e Gm , n < m )
fn4 fm if
~n
appears in the restriction of f n < fm
<
>
~m
to
P~n P~'m ~
Gn
.
Then
we
have
0
and P~n Pfm Pfm = I ~ n ~
n , fn<~m
fn 6 ~n
We denote by
the multiplicity of
fn
in the restriction of
~n+1
to
Gn
9
64 We can find a sequence
~ n ~ n=O,~,2,.., n
& n c
of finite sets
A
[-J Gk
k=o
enjoing the following properties (•
A
CA
n
n+~ A
(ii)
~-~ A n n=o
(iii)
=
~j ~ ~j+~
[-~ Gk k=o
fj+~
,
A n
"===~
fj E A n
Then we can take Xn
=
S Bfj fj ~ n n
,
where Bfj = pfj L~(Gj) = pfj M(Gj) = pfj L( j) The Xn
Bfj ' s
are finite-dimensional
involutive subalgebras , so
is finite-dimensional and selfadjoint . Moreover , for
j ~
we have B~j Bfk C so
Xn
M(Gj) Pfk M(Gk) C
Pfk M( Gk)
=
Bfk
is also a subslgebra . By the Peter-Weyl theorem is dense in
L~(Gj)
hence
n=o is dense in Gn
on
j : o ~j ~ ~j Bfj
X = L . Consider
An
the regular representation of
L2(Gn ) . Then the extension of k n
faithful and so , taking
~n
to
M(G n) O
Xn
=Slkn (the representation of
is L
k
65 corresponding to ~ n condition
by the construction in II.~.2.) , the last
o f Lemma I I . 2 . 1 .
is also
Therefore the envelopping limit of the s ional , An
C ~- algebras is equal to
its canonical image in
An = ~ Xn
is an
For any element
C ~- algebra . Since
A = L Xn
is the direct
is finite-dimen-
as involutive algebras and we have
=
A In particular , A
satisfied.
AF - algebra. x ~ L
we shall denote by the same symbol
L = A .
11.2.3. For further investigations it is necessary to decompose the algebras
Xn
into factors .
We define the selfadjoint idempotents p~n) =
> fj
P~J
s
L ~ ( G j ) ~ Xn
EAn(] 8j
and q(n)
~j : p~j
(i
_(n)]
s
- pj+~,
M(Gj.~
The condition (iii) satisfied by the
~n
) #]
' s
xn
~
~j
implies
~(n) while condition (i) implies p(n) j S ince
4
~(n+T) ~j
9
Pfj E L~(Gj) ' Pyj Pj+i-(n)E L~(Gj+~) and L~(Gj)~ LI(Gj+~) = [01
it follows that q(n)
~
0
for each
~j E A n
66
LENA..The decomposition of
Xn
into factors is
q(n) ~j Bfj
xn : 2 ~j~A n Proof. If q(n) fj Pfk
j < k (&
=
Pfj
= P~'j
we have (n)) - Pj+~
Pfk
pj+~.,
-
,,
~j§ ~ ~j+~ h n"'
Pf'j+~. Pfk
Therefore j < k
~
q(n) ~j Pfk
=
0
q(n) q(n) =
0
.
q(n) q(n) =
0
.
=
Pfk
q(n) fj
and in particular j ~ k
~
rj
rk
It is obvioms that
rj rj
are mutually orthogonal. Then we have
Henc e
~j J ~j E a n
7
q(n) fj
=
fj t a n
P~j (& - PJ+~
j = o
~j ~ A n N G j
)-
4
n
:
(n))
7
_(n))
j=o n
=
-
.
3=0 =
p~n)
o(n) -
=
~
.
~n+~
~'j+"L)
=
0 .
6y Moreover , by their definition ,
q(n) ),j
M(Gj)
j , we have
hence for
x g B~k q(n) x
On the other hand for
, k ~
:
x _(n)
~j
x ~ B~k
q(n)
,
k ~
q(n)
x q(n)
are central elements in
, we have
j
x
=
0
,
q(n)
=
0
.
q~n)~j x ~ q(n)~j B~j
=
=
x
Hence the map B~j ~ x,
~
q(n)~j Xn
is a non-zero involutive homomorphism of the factor q~n)
B
fj
, so that
is itself a factor . Q.E.D. Consequently , the decomposition into factors of the finite-
dimensional
C*- algebra
An
An : ~
is
q(n)
B~j
~ ~n i.e.
are the minimal central projections in
An .
II.2.4. Now we construct a maximal system of minimal pro jections in
A~
and a corresponding system of matrix units for A n .
The idea behind our construction is that of the Zeltlin basis , i.e. first we write the representation
Gelfand ~n+~ I Gn
68
as a sum of irreducible representations of
G n , then we split
again these representations into irreducible representations of Gn_ i
and so on . Since
G O = {e~ , at the end we find one-dimensio-
nal spaces . However , this will be done algebraically in Fix the representations
An .
~n s Gn ' ~n+i ~ Gn+~ ' ~n < ~n+~ "
Then P~n B~n+~ Pfn = (P~n Pfn+~ ) B~n+~ (P~n P~n+{ ) C B~n+~ is a factor of type
I(d~n[~n+~:fn])
B~n~ x t
. Since the map
~ P~n+~ x ~ P~n+~ Bfn
is a non-zero involutive homomorphism , it follows that is a factor of type
Id
Pfn+~ Bfn
. Moreover ,
~n P~n+~ B{n
=
Hence the commutant of of type
~
P~n+~ P~n B~n P~n Pfn+1 P~n+~ Bfn
in
9
Pfn Bfn+T PFn
P~n Bfn+~ Pfn
is a factor
l[~n+~:~n] " We choose a maximal system
p(_ n ~ k
,
~_ ~< k
4 [fn+~:fn]
of mutually orthogonal minimal projections in (Pfn+~ Bfn)' ~ (Pfn Bfn+~ Pfn ) Of course ,
=
k=~
P fn Pf n+~
,
69 A
We denote by
~o
S(~n) ' ~n g Gn ' the set of all symbols
kj
k~.~~.
"'"
~ ~j-~
kn ~ "'"
~ fj
~ fn-~
" fn
where A
~j ~ Gj and we define n
= ~
P
k.
Then {P~I~ ~ S(~n) is a maximal system of mutually orthogonal minimal projections in
Bfn
9
Indeed , this is obvious for
n = o . Suppose this is true
A
for all
~n g Gn
A
, with
n ~ ~
fixed , and consider ~n+{ ~ Gn+~ "
Then [P~ Pfn+~I ~ s S(~n) is a maximal set of minimal projections in
fn+
~ ~ k
P~n+~ B~n . Since
~ [fn+~:fn]
is a maximal set of minimal projections in the commutant of P~n+~ B#n
P~
in
Pfn Bfn+~ Pfn
fn+
, it follows'that
~ a S(fn ) ' ~ ~< k
is a meximal set of minimal projections in
4 [fn+~:fn] P~n Bfn+~ P~n" Hence
7o
is a msximal set of minimal projections in For symbols
~ ~ S(fn) k) fn+~
~
Bfn+~
and i ~ k ~ [fn+~:fn]
. we consider the
in S(~n+~) " We have
2
P(~ k) ~n+ 0 ~ S(fn)
=
Pfn P~n+~ "
, r ~ k 4[fn+r
11.2.5. We choose now by induction a system of matrix units
[E%,~] ~,~ ~ S(fn ) A
in
B~n . Suppose this was done for all ^
fixed, and consider
~n s Gn ' with
n ~
~n+~. E Gn+~. 9 Then
is a system of matrix units in
so that
[P(~n
~E
P
k
J
is a system of matrix units in the involutive subalgebra
7
P
k
(~n__~fn.~)
71 of
B~n+[ . We choose an arbitrary completion of this system to a
system of matrix units in
Bfn+[ .
The systems of matrix units constructed as above have the property that each Pfn+r E~,~
;
~C ,~ a S(9n)
is expressed with respect to the linear base [E~,gl ~,@ ES(~n+%) of
as a linear combination with coefficients only
Bfn+[
0
and ~.
Namely
[ fn+r '~ n7 J
('i)
Pfn+% Ea('~
k
II.2.6. Denote by tions of
= ~." ( ~ k ~'n+'l)'C/~k'~n+~)
Perm S(fn)
S(~n) 9 For any ~ s
the set of all the permuta-
S(fn )
~"
we define
~' ~'(~)
n
=r ~ S ( f n )
U o_ = Let us fix in
9n ~( 9n+~
S(fn+%) k
+ (~. - pfn)
and
s
Xn
~ 6 Perm S(~n) . Since any symbol
is of the form
> fn+~
we can define -s (o( -6(
V
with
~ 6
S(~n)
~ ~ Perm S(fn+[) k ~ fn+~ ) = ~(
by k ) ~n+~
k >fn+S. ) =~.(~)k_.~fn+ [
~ ~ s(f n)
9
72 (2)
Pfn+~. U~r = Pfn+a_ U Z
Indeed ,
Pfn+[uz =
= Pfn+~
/3 g
E{~,r(~) S(fn+, 1 )
P~n+~~ ~ S(fn) E
+ Pfn+~~
k
\ E
= PYn+~ ?
'q'(~) + PYn+~-/
E S(fn)
= Pfn+~ V~
= PPn+~ V~
E~6'< e
s(f~)
' ' ' 'J'nK JPn+~.'fn~fn
Pfn+~ ? "
+ P~n+~ (~ - Pfn )
= Pfn+~ U~ II.2.7. Concerning the AF - algebra A we first remark that
I q(n)I is a maximal system of minimal projections in
73
q(n)
and we denote by
C n the corresponding
Since for every
(n)
P~ q]~j
~ (S(fj)
=
m.a.s.a.
in
An
9
we have
- Pj+~.
J
~
~j+{ E ~nl'~Gj+~ and since obviously
P-~j+I = it follows that
Cn
V-
/
Pl6
is generated by
~P~I ~ a S ( f j ) , Let us denote by
o ~ Dn+
~j ~ ~ n
"
the abelian subalgebra of
An+ ~
generated by
[P(~ k-~-~+') 1
~j ~ ~j+~ ~ A n + ~ ,
'j+~
~n,
~k,
[~+~:~
It is clear that C n+ ~
:
o Dn+~
C
9
We assert that
Indeed , consider
A'n C I ~ + , j .
~i E A n , x ~ Bfi
and
fj ~
~j+i E A n + i
fj+~ ~ /kn , ~ G k ~ [fj+i:~j] " Because of the condition (iii)
74 satisfied b y the A n ' s , we cannot have if
j+~ ~
i
we have
P~j+~ Pfi
P(rj. k,~j+ On the other hand,
x p~ pf
if
= p
:
0
~i " Therefore
and so , o
~.
i < j+~
pf
=
fj+~ ~
we have
x E P
,
P
c (Prj+~. B~),
and so
Putting Dn+ ~ it follows that
=
Cn+ ~ (~ A~ (~An+ ~
Dn+ ~
is a m.a.s.a,
in
A~ (~ ~n+~
and
0n+~ :
n=o
is a m.a.s.s,
in
A
and we have the corresponding conditional
expectation P of
A
:
A
with respect to
~ C
C .
.
,
75 11.2.8. Denote by ~ pectively . Of c o u r s e , ~
n n
, ~-~
the spectra of
C n , C res-
can be identified with the discrete
set of all the minimal projections in
C n , hence with the
discrete set of all the symbols
where
~ E S(~j) The map
Cn+i
~
E a n
~-~n+i
the inclusion in
,
. canonically corresponding to
~ n
C n C Cn+ ~
associates to every minimal projection
the unique minimal projection in
Cn
containing it .
This means that the above map associates to the symbol kj_
the symbol ~i-~
fo where
i 4 j
is maximal with the property ~i 6 A n
The spectrum ~-~ of the spaces ~
n
9
of
C
t
=
is the topological inverse limit
and of the maps ~ n + ~
condition (ii) satisfied by the that the set ~
ki> ~ i a ~ n
. . .
i n ' s
m~n
" Owing to
, it is easy to see
consists of all the s,ymbols
(~o(t) k[(t) ~
~r
>
...
) fn_r
& 4 n
< no(t)6
where
~n(t)
s
Gn
~ v{+oo}
kn(t)>
fn(t)
> ...)
76
~< kn(t) %
[~n (t) : ~n_%(t)]
thus, no(t ) E ~ ~
[+co} indicates how many groups
in the symbol of
t E~
; Gn
are involved
.
Bearing in mind the very definition of the inverse limit topology and the special nature of the maps
~n+i
finds the following description of the topology on Let be
co C ~
and
t ~ t
> ~n ~
' one
.
. Then
E
uO
if and onlE if any of the following conditions holds :
(i) (ii)
t
~ ;
no(t) = +co and for every m C ~ there is s E ~ O
fn(S)
= ~n(t)
,
kn(S) =
such that
kn(t)
for an,y n .< m ; (iii)
no(t) < +oo and the set
[~no(t)(s)
;
s ~ u O , ~n(S) : ~n(t) , ~ ( s )
= ~(t)
, (~) n < no(t)]
is infinite.
11.2.9. Now we remark that, for each
fj is a system of matrix units in
is a system of matrix units in
~j g ~ n
'
g s( q(n) B
fj &'
hence
An . Using the relation
(~)
it is
easy to verify that each element of the system of matrix units in An
is written as a linear combination with coefficients only
0
?7
and
~
in the linear base of
An+~
given by its considered system
o f matrix units. As in the case of general
AF - algebras we denote by
the group of unitary elements in system of matrix units in
An
An
Un
associated with the above
and we have
U n C Un+ i which allows us to consider the group U =
~_~
Un C A
n=o
The group
Un
q(n) U
Yj
is generated by the elements
~- + (~ - qt~jn))
Fix such an unitary in
; ~
Un
i ~j+~
Perm S(~j)
,
,
i = C,...,m E Z~n(~ Gj+~
using (2) we find permutations -~iE Perm s(~i+i)
,
i = i,...,m
,
i = {,...,m
such that U
It follows that Pj+i U~F =
n
and denote by
the elements of the set
P i
fjaA
=
P i
U
m
ZV i = 9
P i
U o-
fJ+~
m
i = i
~J+~ Uzi
m
~-
P i
u~.
. By
78
_(n)
= Put
w
=
~j+~
9 Then we have
...
p(n) ~+~ U~ =
U
Ur~ "'" UTm
(n) pj+~ W
_(n)~ W = _(n) ~j+~ W + (~ - ~j+~.
,
: pfj U~ + (i - pfj)
U -~ w ,
and also similar formulas with
W-~ . Therefore
_(n) U-~- =
(n) W-~= (~ - Pfj ) Pj+l (i - Pfj
- Pffj
_(n)
and so U ~ W -~ = (p~ U + (~ - p~j))(p~n) W-~ + (~ _ ~J+~JJ~(n)~
J =
(n) + p ) o~J+~
p ~ p(n) + (~
+~ =
q(n) U
-
p(n)
('i _ ~(n)~ U w +
(~ _ p(n))
j+~ - pfj
J+~
+ (~ _ q(n))
The above discusion shows that the group
U
i_~sgenerated by the
elements Uw
;
S(fn)
Q- E Perm
,
~n ~ ~n
We know from the general case of an in Chapter I ~ ~
that the group u C u~ = C
and that
U
induces a group
U
n s ~ .
AF - algebra treated
invsriantes the m.a.s.a. C , i.e.
for every [~
'
u ~ U
of homeomorphisms of ~ .
From the preceding remarks we infer the following concrete description of
P
. Fix
n ~ ~
and define a transformation
,
~n ~ Gn
and
~-a Perm S(~n)
79
t
o__.f.f._~
as follows
~(t) if
--f(n
, fn' ~)
:
(~r(fo(t) k~(t),... kn(t)) fn(t))
=
t E ~
,
no(t)>
n
and
~n(t)=
~n
'
kn+~(t)
-f
i(t)
and
f(t) = t in the contrary case. Then formations
P
is the group generated b_~ the trans-
: A
~(n
, fn
,o-)
, o-~eerm
S(fn)
, ~n g G n
Thus, we have fulfiled our task of describing ~=~(G~)
and
a direct limit
~=
Gc~
~(G~o )
for the
of compact groups
Gn
Gn
the numbers
and
in Chapter I w ~
the choice of the for all
~n
E ~,#
' s
A(Go=)
of
. In concrete situations, ~(Go~)
[ ~n+~ : ~ n~
is not unique
and
P(G=@)
"
Let us recall that the isomorphism between established
n e
suitable
AF - algebra
all we need to know in order to construct are the sets
,
A
and
A(-(I,[ I )
, it does depend on
(even in the case when
[~n+i:~n ] ~
' ~n+~ ) "
II.2.~O. For the applications we have in mind we must explicitate when the representation given representation
of
Go=
of
A
corresponding
is a representation ~
be done by using the special
associated to T[~
Thus space
H
, let
and
~
~ o p
be a representation
of
G~
~)-
correspon-
ding , as in Chapter I w 3 , to a measure state
on
Sl~
to a
. This will 9
on a Hilbert
~ ~ H , II~II = ~ ' a cyclic vector for
~
. Denote
)
80 by
T[
the representation
T[f
of
L
associated to
~
as in
Section II.[.2. and use the same symbol for the corresponding representation of of
L[(G n)
corresponding to
riant measure
of
A
A = L . Then ST
~
IL[( Gn)
is the representation
g I Gn . Consider a
~-
quasi-inva-
on i~i.
In order that q'[
be equivalent to
associated to
and
~-[
T[~
be equal to
and the state ~ o P , it is
necessary and sufficient that
~(P(x))
=
for all
x c L[(G n)
and each
B~n
and
n e ~ . Since the
B~n 's
span
L~(Gn)
is spanned by the elements of the form x
=
~g * P~n
it is sufficient to verify
'
(3)
g ~ Gn
'
only for these
x .
On the other hand , we have
~(p~x p~ )
~(P(x)) = /k-~ S(~n)
:y
~(P~) TrB~n(P~X P~ ) ~ s(fn)
Therefore , in order that
T[ -~ ~
and
it is necessar,Y and sufficient that
(pD TrB rn for all
A
g c G n , ~n E Gn
and
n e~
.
*
* P2
CHAPTER
III
THE PRIMITIVE
We consider now a special situation
, namely the group
discussion groups
are completely
the classical
Chapter we shall the
U(oo)
w ~
described
A(U(~))
~q_t-~_~
U(n)
and a
"lower
B y the classical
of
U(oo)
to
. In this
the primitive
ideals of
Q~
A(U(oo))
AF - algebra A
signature"
A
description
=
A(U(oo))
is determined b y an .
theory of Hermann Weyl U(n)
, the
can be identified w i t h
set of all the signatures m~ >i m 2 ~
where
, it seems difficult
ideal of
of the unitary group
the discrete
of signatures b y
is to give a complete
of the
see , a primitive
III.~.T. dual
spectrum
signature"
of the unitary
.
The primitive
of the primitive
in terms
representations
The aim of this section
"upper
representations
study in some detail
C ~- algebra
As we shall
which caused the preceding
theory of Hermann Weyl
the irreducible
A(U(oo))
case of the above general
. While the irreducible
U(n)
classify
IDEALS OF
mi
Z
,
i = ~
Then
, for
... ~
mn
, 2 , ... , n
.
""
n-i
) r
A U(n-i)
)
u(n)
A
fn
:
>
...
.
82 we have
if and only if m (n) j_~ >/ m_(n-~_) j_~ ~/ m(n) j
9
,
~< j ~< n
snd , in this case , [~n : ~n-~
=
~
Taking into account these classical facts and the results of Section
II.2.8.
, it follows that the points of the space
are the symbols t
=
{(m(~)(t)) ~ ~ j ~ n ]
~ ~ n~
no(t )
where
~ no(t ) ~
(n)
~(n-~)(t ) ~
Therefore , s point of ~ where
a
~ b
means
s 4
+ oo
m(~)(t)
.
looks like the picture below , b .
83
r m~ ) <
l\ m(~)( m~ ) <
m~ )
/\
m (~} (
m~) <
m~ )
m~ # <
m~)
/\/\/\/\ 9 ~
mC~'~)(
,,, (
m(~ - )<
/\/\
=z
,
.
<
"
~
9 (
m~<,
'
,,
J.
9
" <
/\ /\/\ ~-~
~'--,l.-c""O
/
.4 \.4 \.4 \ / \ . <
\.4 \.
O@e@@@@@B@@@@@@@~@@@@o@@O@e@@ee@~@@@@@@@~@@@@@@@@~@@@@@@O@@@@@@I
~
The description of the topology on ~ of the transformation group Sections with
II.2.8. and
no(t ) = + oo
~
and the description
= ~ (U(~))
follow obviously from
II.2.9. Roughly speaking , a point
is adherent to a set c o c ~
one can find a point in co
t ~ ~-~
if for every "line"
with the same "beginning" until that
line . Also , the generators of the group
~
change these begin-
nings among themselves , leaving fixed the rest of the picture. III.~.2. We recall from Chapter I w 3 that the primitive ideals of
A
the orbits of
correspond in a canonical way to the closures of ~
. In
Lemma III.~.3.
we shall determine these
sets . First , some notations ; for we define
te~
and for
~ 4 j < no(t)
84
Lj(t)
=
sup I m(3)(t)
Mj(t)
=
(n) t ) ; j ~< n < no(t) 1 ~ 2 inf [ mn_j+z(
; j ~, n < no(t) ~ g
~ v ~+oo} {---}
, .
These definitions can be easy visualised on the picture of t. Since m (n-~ j_~)(t) >11 m(~)(t ) it follows that L[(t) >2 L2(t) >i ... >i ~j(t)>1 N~_(t) 4 If
no(t)
... ~ Mj(t) ~< ...
M2(t)
< + c~ , we have
+~ while if
...
> Lj(t) no(t)
(no(t)-i) m j (t)
=
=
+o~
=
Mno(t)_ j (t)
>
- oo
, we have
Lj(t) >/ m(3)(t) >i Mn_j+~(t)
,
that is Lj(t) III.~.3. L E n A the closure of the
2).
for every
tog ~
t o - orbit of [~
~). I_~f no(to)
itg ~
Mk(t)
. Consider
Lj = Lj(t o)
cO =
~
< + oo
,
.
, denote b_z uo = [~(t o)
in ~
, Mj = Mj(t o)
j,k ~ ~
an d put
; ~ ~ j < no(t o )
9
then
; no(t) = no(t o ) , m (no(t)-~) j (t) = Lj ; ~.<J<no(t) ] .
I_~f n o ( t o )
co (~ It g n ;
=
+ ~o
no(t)= + ~ }
,
=
85
; ~j~n~+~ OO n
r, oCt
~-~
<
I,
=
; no(t ) < + oo, Lj ~ m(~)(t) >i Mn_j+ ~ if
L~
M~
=
+
.
oo
Proof. The first statement is obvious. In this case we have co
= P(t
o
)
.
In order to prove the second statement , denote
e
= {t ~ ;
no(t)=
+oo,
Lj~
~ (t o)
C
m(~.)(t)) Mn_j+ ~ ; @ 4 j ~ n ( + ~
I.
Obviously , (~)
@
9
We shall show that (2)
by proving , for any
sE ~
the following assertion
, any
t ~ ~(to)
m(~)(t) k
~d
i 4 i ~ k ~ n-i In order to prove
Af(s;n,j)
induction as follows
and any
, which we denote by
there exists
~Pr ever2
n a ~
m(~)(s)
i
satisfying k=n
we fix
Af(s;n,j)
, i~ s ~
i~
j
and we proceed by
:
(i)
Af(s;[,~)
(ii)
Af(s;n,j)
,
such that
=
or
i~j~n
is true , ).Af(s;n,j+~)
, ~ ~ j < n ,
86 (iii)
Af(s;n,n)
Let us prove A).
m(~)(t O) ~
~_- Af(s;n+r
9
(i) . There are two possibilities : m(~)(s)
or
B).
m(~)(t O) ~
m(~)(s)
First we suppose that A). Since
s ~ @
m(~)(t o) >/ m(~)(s)
we have
m(~)(s) ~
ME = inf Im(n)(to) ; n ~ ~ I
so there exists a unique
We define
m(h+~)(t h+~ " o) 4
and
Im(~)(s)
Af(s;%,{)
'
such that
h ~
m (h)( h " t o ) '~ m(~ )( s)
Then
9
if
i=k<~h
if
i~k
m(~)(s)
or
k>h
holds with
t
=
i ) ~ ~ i_< k
~ ~k<+~
sinc e
and it is obvious that there exists t
=
~r ~
with
~ ( t o) ~ F ( t o)
.
Next , suppose that B). Since
s c @
m(~)(to) ~ m(~)(s)
we have m(~)(s) _< L~ =
so there exists a unique m(h)(t o) ~ We define
m(~)(s)
h >~ and
sup /m(~n)(to) ; n ~
~}
such that
t o) >"
)(s)
,
87
Af(s;~,~)
i = ~
and
k ~ h
,
m(k)(to )
if
i ~ ~
or
k ~ h
.
t C
~ ( t o)
satisfying the
holds with
Now we prove assertion
if
I
i(~)_Then
m(~)(s)
(ii) . Choose
Af(s;n,j)
(n) t) ~ A). mj+[(
. Again , there are two possibilities
(n) mj+[(s)
B). m~n)(t) ~
or
:
mj+%. (n)fs)
and we begin with the first one , so we suppose that _(n) t) ~ A). ~j+~( We continue
(n)Is) mien.
in two steps .
(IA) We show that there exists
h ~
0
such that the
followin 6 statement is true StA(S;h)
: ~.(n+h-[)t j+h ~t) ~ m(n)f j+~ ~
+ 9
and
m(n+h)f§ j + h + ~ . / ~ m-(n)t j + ~ s) .
Suppose the contrary holds . Then m( n+h-[)r (n)ts ) + [ j+h ~ t~J >i mj+[. Since
t ~ P(to)
there exists
m(n+h-~)t +~ = j+h ~/ By the definition of
Mn_ j
h o >~ 0
for every such that
.(n+h-4)t " j+h ~t o) for every
On the other hand , s E 6) , hence ~
This is a contradiction
h >/ h o.
we have
Mn_ j = inf m(n+h-[)(to) (n)Is) + ha~ j+h >/ mj+%~
Mn_ j
h >~ 0 .
(n)Cs)
mj+~. .
(II A) We show b/ induction on
h ~
0
tha t
.
88
StA(S;h)
~
Af(s;n,j+~)
StA(s;0)
It is obvious that have proved that
~
StA(S;h-~ )
.
Af(s;n,j+~)
. Suppose we
> Af(s;n,j+~)
. From
StA(S;h)
we Infer I m(n+h)(§ ~ m(n+h-i)(§ j+h ~ >i j+h ~
(3)
~(n+h)c§ ~ ~j+h+~.~, ~ Since
t
satisfies
,
_(n)(s) mj+~
Af(s;n,j)
m(n+h-2) (~ ~j ~ j+h
(4)
(n)(s mj+~L" ) + PL
~
we have
~< m(n-i j+T)(t) = _(n-i)~ ~, j+{, s ) ~< m j(n) + ~ ( s) .
We define
m(k)
m(k)(t)
if
k ~ n+h-~
or if
(n) mj+~(
if
k = n+h-~
and
s
)
in% [m (k-~)(~ i-g ~ Using
(3)
and
(#)
t
by
if
[ ~ i ~ k
=
Af(s;n,j)
k = n+h-i and i > j+h .
~(t) and
~
[ ~ k < +~ ~
~
such that
r ( t o)
StA(S,h-~ )
t' , The induction hypothesis in
Af(s;n,j+[)
i < j+h ,
we see that
and it is obvious that there exists
Then both
and
i = j+h
' ~~(k+~)t i ~t) ]
t' =
t'
k = n+h-~
are satisfied (IIA)
replacing
insures
that
holds .
Next , suppose that B) 9 m(n)(t~ j+~, ,
~
m(n)(s) j+~,
We proceed again in two steps . (I B) We show that there exists
h ~
0
sucb th@t the
89 following statement is true (n)(s) - ~ StB(S;h) : ~~(n+h-~)( j+[
~(n+h)(~ (n)(s). ~ j+~ ~ j ~ mj+~.
and
s ~ e , in the contrary case we would obtain the
Indeed , since
following contradiction : m j(n)cs) +~
=
Q Lj+~
sup ~ ( n + h - ~ ) r +
j+~
"
hgZ =
~0"
sup ,.(n+h-[)(§ ,,, j+[ ~
(n) s ) mj+~(
6
he~; (II B) We show by induction on StB( s ;h) Indeed , from
StB(S;h)
>
4
(5)
satisfies
.(n+h-[)(~ ~ j+[ ~J 4
Af(s;n,j)
.
(n) s) - [
mj+[(
,
we have
(n) s m(~-~)(t) = m(~-T)(s) >/ mj+~().
m(j+h-2)(t) >
(6)
Af(s;n, j+~)
_(n)(s)
.(n+h)(+~
t
that
we infer
I .(n+h)t ~ j+2 <~J
and since
h ~ 0
Putting
H(k)
m(k)(t)
if
k ~ n+h-[
or if
= Im%n)(s )
if
k n+h-%i =
and
inf [m(k_~)(t) and using
(5)
and
m(k+[)(t) 1
(6)
k = n+h-[
and
i<j+~
,
i = j+[ if
k = n+h-[ and i > j+~ ,
we obtain
t' = [(~(k))~ ~ i ~< k } ~ ~ k < +oo which satisfies both
Af(s;n,j)
induction hypothesis in The proof of
(IIB)
(iii)
and
StB(S;h-~) . Hence the
implies that
Af(s;n, j+~)
is similar and we omit it .
holds.
90 We continue the proof of the Lemma . By (~) and (2) we have
(to> a @
c Veto5
and so
(7)
uO
But
=
is obviously closed in
~ [t~
; no(t) = + o ~ }
with
respect to the relative topology , thus a~)
/A
[tEn
; no(t)=
+~]
=
~
,
which proves the first part of the second statement of the Lemma. Consider now Therefore
s ~
, s s e
with
no(S ) < + o o .
Then
s ~
e
.
if and only if the set of all symbols (m(~ ~s))(t)) ~ 4 J ~ no<S)
with t ~ e
and
m(k)(t) = m(k)(s)
is an infinite set . Hence ~ sup [ m (~
s
-
for
~ oo =
O
[ ~ j ~ k < no(S) implies
m (n~ no(S ) (t) ; t e
I
and so L~I -
Conversely s ~
, if
with
L~ - M~ = +oo
no(s) < + o~
Lj ~ m(~)(s) ~ belongs to
O
M{
= ~o
=
+ ~o
then it is clear that any
point
and Mn_j+ ~
for all
~ ~ j~
n ~no(S)
.
This completes the proof . Q.E .D. III.~.#. The next Lemma answers a natural converse question.
91
LEMMA . For any siren
Lj ~ Z ~ [ + ~ }
, Mj~-~}
(j g ~)
such that +~
>i L~ >i L 2 >/ ... ~ Lj ~
there exists a ~oint
to~_~
... ~ Mj ~ ... ~ M 2 ~ M ~
with
Lj = Lj(to)
,
~-~
no(t o ) = + ~
such that
Mj = Mj(to)
,
j ~ ~
.
Proof . We distinguish three different situations : !). Suppose there is Then the point
tog~
a ~ ~
with
inf Lj ~ a ~ sup Mj j~ j~
we are looking for can be defined as
follows inf {Lj
,
a+n-j I
if
~ ~
j ~
n
if
n+~ ~ j 4 2n
if
~ ~< j ~< n+~_
if
n+2 ~ j%2n+~_
m(2n)(to ) sup {M2n_j+~
m(2n+~_) J
(to)
linf
Then the point
inf Lj = - ~ j~
toe ~
Then the point
=
sup M
j~IN
to ~ ~
a+n-j+2} , so
Mj : _Qo
for all
j ~ ~
.
inf {Lj
J
= +~o
, so
' --
n-j} L
J
,
= +o9
~ ~ j ~< n ~ + ~ . for all
..........
j ~ ~ .
we are looking for can be defined by o)
Since
,
we are looking for can be defined by
m(j)(to) 3). SuPpose that
a+n-j+~}
, a+n-j+~}
{Lj
= . s u p ~M2n_j+2
2). Suppose that
,
:
sup
'
"< J
< +
inf Lj >i sup Mj , these three cases cover all the j~ j~
possible situations and the Lemma is proved . Q.E.D.
"
92
It is obvious that for any given integers L[ > L 2 )
... ~
L n _%
,
no <
+oo
,
0
there is a point
to~
, with
no(t o) = n o , such that
(no-i) m j (to) = Lj
for all
~ ~ j < no
.
III.%.5. Thus , taking into account Theorem 1.2.9 . , Theorem II.[.5.
, the remarks in Section III.[.[. and the preceding
Lemmas , we obtain the following THEOREM . The primitive spectrum of the
C*- algebra A(U(oO))
can be identified with the set of all the s,Tmb0!s
where either
no(~) = + ~
and , for all
~ ~ j < +oo we have
Lj( ) o__rr no( t ) ~ ~
Mj( )
and for all
~ ~ j < no(W) ' we have
~ Mno_J(I) = Lj(~) ~ Lj+~(~) Namely , if ~ g0m~
U(k))
= Nno_J_~(~)
is a factor representation of
, then the kernel of ~ f
Lj9 = sup {sup I m(~); n ~ j}} where the first
sup
,
U(~o)
(o_.rro_~f
corresponds to the s2mbo ! Mj = inf {inf { mn_j+i, (n) .. n
,
and the first
inf
J}I
are taken over all
A
signatures
(m(~),
The points
... , m(~ )) E U(n)
which appear ~in
~ ~ Prim(A(U(oo)))
with
to factor representations of no( E ) = n o 6 ~
U(oo)
~ I U(n) 9
no(~) = +oo correspond
, while the points
correspond to factor representations of
~
with
U(no-%).
93 It is easy to describe the topology on the space Prim(A(U(~)) ) Thus , consider
E C Prim(A(U(oo)))
.
and
~o
E
Prim(A(U(o~)))
.
Then we have
if and only if
C i)
an.y of the following conditions holds :
~o ~ E
(ii)
no(~o ) = +oo
(lii)
no(~o
.an%,for all sup~Lj(~)
; and , for all
) = no ~
~ ~ j 4 + ~,
+ oo ,
9 ~ j < no ,
; ~ ~ E} >~ L j ( ~ o ) = Mno_~(~o)>~ ~ f ~ M n o _ J ( ~ ) ;
~E}
For example , the one point set ~ I c P r i m ( A ( U ( ~ ) ) ) , '
for
. where
all
~,Q
j
is everywhere dense .
w 2
D_ir_ect limits of irreducible representations
The direct limits of irreducible representations of the U(n) 's provide us with a wide class of irreducible representations of U(oo) . In particular , every primitive ideal of corresponds to sentation .
U(oo)
A(U(oO)) which
is the kernel of at least one such repre-
94 Among these representations there are also the representations considered by I.E. Segai (~3oS) and A.A.Kirillov ( [ ~ ] )
Consider a point Thus
t
~
= E'~ (U(oo))
~2~
... ~ f n <
with
.
no(t)
,
t where
=
A ~ n E U(n)
(?~
,
n s
III.2.~. Let
5
.
be a completely atomic
probability measure concentrated on the for all
~g P
"'" )
~-
~ - orbit
quasi-invariant ~(t)
. Then ,
we have > o
and , for all Borel sets
B C ~-~
we have
s g ~(t) N Clearly , ~
B
is ergodic and therefore the representation ~5
is irreducible (see 1.3.~6.) . Moreover , the kernel of
~
corresponds to
~ (t) (see
|
z.2.9.)
.
III.2.2. On the other hand , since
~n ~
~n+~
, there are
isometric embeddings in : Hpn ~
Hpn+i
such that (~n+~l U(n)) ~ in
=
in ~ ~n
"
95 Moreover , since
[~n+~:~n ]
a scalar factor of module
~'s
~ , the
In'S
are unique up to
~ .
On the completion following the
=
Ht
of the direct limit of the
there is a natural representation of
H~n ' s U(~o) .
It is easy to see that the representations corresponding to two different choices of the
in'S
are unitarily equivalent . There-
fore , we may denote this direct limit representation by
III.2.3. Any two representations Hilbert spaces
H (j)
~(J)
of
U(oo)
~t
"
on
, j = !,2 , such that the subspaces
n=o
P~n
are one dimensional and cyclic , are unitarily equivalent . Indeed , if
then She functions of positive type determined by on
U(oo)
~(~)
and
~(2)
are equal , as can be easily seen considering the
restrictions to the various
U(n)
.
III.2.4. From the above remark we infer : Th e representations
~
and
~t
are unitarily equivalent.
In particular , The representation
~t
primitive ideal of
A(U(oo))
o~f ~ -
.
orb it
~(~)
i_gs irreducible and the associated corresponds to the closure
96 Moreover
,
Two representations if and only if
III.2.5. groups Gn ' s
t'
~t
and
~t'
= ~(t)
In the general
for some
irreducible
such a representation
concentrated
P
be unitarily
on the corresponding
.
of compact
representations
representations
one can choose
that the representation
~
case of direct limits
, the direct limits of irreducible are still
are unitaril 2 e~uivalent
of
of the
G~o . Also
, for
a system of matrix units such equivalent
U - orbit
.
to
~
with
CHAPTER IV
TYPE
III
FACTOR REPRESENTATIONS OF
U(oo)
IN ANTISY~(ETRIC TENSORS
We shall study some representations of restrictions to the
U(n)'s
U(o~)
whose
contain only irreducible represen-
tations in antisymmetric tensors , i.e.
representations with
signatures of the form (~,...,~,0,...,0)
IV.~. The notations and the results contained in Section III.~.~.
will be used without any further reference.
Consider the set (f~4 with
~n ~ U(n)
co c ~
f2 4
.-.
consisting of all symbols < fn ~
"'" )
of the form
fn :
{,~, . . . . 4 , o , . . . . . . . . '9) kn-times (n-kn)-times
Clearly , ~o can be identified with the set of all sequences {knl n ~
of positive integers enjoing the properties
k~ ~ {o,~
,
~+~
- ~
~ {o,~
.
The map
allow u_~st_~o identif,y
~o
with the ~roduct set
{o,~1 ~~
.
It is easy to see that by t.his identification
the topology of
COrresponds t_gothe product topology of discrete topologies
__on {0,~ ~~
~o
98 The set
~
is a ~ -
orbit whose closure corresponds
(see III.~.5.) to the upper signature the bower signature mations in
r
to
[Lj = ~ ; j ~ ~} and to
{Mj = 0 ; j m ~ ] . Restricting the transfor~o
we get a transformation group
[~
on ~o .
In order to describe this group in the identification =
,.o
we fix
n E ~
and
a
to,
permutation
}
~
of the set
~ 0 , ~ n such that
n
~-(~176
= (~i'''''~n)
and we define a transformation
~
~n
on
~n,~(~i,...,~n,~n+i,...)
snd
~o
~
)
"
consists of all trans-
~n,~
set {0,~} n
which preserves the sum of the components of the
i_~ss_ permutation of the
.
We remark that the set with
~
'
by
formations
elements
n e ~
= i~__~,=~i
(r162
=
Then it is easy to see that the ~roup where
n
I:~i
n = no
is a subgroup of C~o
We consider on
~
~ where each
~no,~ o of all transformations
5n
nr 0
9
= (O,i] H~
=
the product measures
n~__~~ n
is a probability measure on {0,~}
:
pCO
~ p(O) ~
,
~
,
:
0
~P
:
,
n
It is obvious that any such measure
~
on
~
is
~n,~
99
quasi-invariant
.
The representations we shall study are the representations ~,
~
of
A(U(oO))
considerations the
~
which correspond to
~
by the general
of Chapter I w 3 9 Therefore we are interested in
- ergodiclty of ~
, in the
~-
measurability of
~o
and in the equivalence of two measures of the above type .
IV.2. Concerning the ergodiclty , the main instrument is Proposition 1.3.1@, which was obtained as a consequence of the Powers-Bratteli theorem . Using this Proposition that the measure
~
is
for every
, it follows
~o~- ergodic if and only if f ~ C(~)
and every
such that for every
~r,~-
s ~0
invariant
there is r ~ g E C(oO)
we have
It easy to see that it suffices to verify this condition only for functions
f ~ C(~o)
of the form
j=~ ~ i if (0j)n f((~S)S=~)
=
n
if where
n
;
,
(~j)j=~ = (~j)j=l
n g ~ , ( ~ j ) j n E ~O,K~ n Moreover
n
(~j)j=[
are arbitrary but fixed .
, we may suppose that the function
depends only on a finite (but non-fixed) number
g E C(uO)
N ~ ~
of compo-
nents of its argument . Thus , consider ~r,~-
n ~r
< N . Since the function
invariant ' there is a function
~
such that
g
is
100
~((ps) ~)
= T(/~+...+/~r ; #r+~ , ... , ~N) , ( # ~ ) j : ~ o .
Let us denote Dk
=
(~j)j=~
~0
;
~j = k
,
0 ~ k ~ r
,
Dk Then
j:~ n
N.~ (to) k:O
fg d ~
( ~ j )N-r j:~
{o,~I N-r
=
N-r ~ {0,~iN-r k:o (tj)j:1 and so
,
the relation (~) becomes
(~)~=~g(o'~N-~lj:Ipr+~--~ ~(k'~''"
- ~
z_-~ i "I I
This last relation is implied by the following
(2)
~(D~)-~(Dk)~:~p
i
J
one :
~
Putting n
ko
= ~
Dk(J,h ) we have
~i
=
(~i)~_~ ~ ~o
;
#i = k
,
j ~ h
,
101 n
5(Dk)
=
~ 5(Dj({,n)) ~(Dk_j(n+{ir)) J=o
~(D{)
(~i) = ~(Dk-ko(n+i'r)) !=I p i
Thus , in order to prove the r ~ -
ergodicity of ~
, it is
sufficient to show that
l*m tliPi
],I:~_~__olt<(Dk_ko
Since n
8=0 it is sufficient to show that (3)
lim
~=ol~Dk(n+~,r)) -~(Dk+{(n+~,r))l
=
0
IV.3. For technical reasons we consider the power series
p(~) : ]Z o k zk km~
with
~ k~2
ickl <
+
and we define
It is easy to see that
(5)
il'll is s norm and
IIzn P(z)II
We remark that
~(Dk(n+{,r))
= IIP(z)I
"
is exactly the coefficient
in the development r
=
and that for any
I C N
we have
5
Ck zk
ck
: O,
102
(6)
Finally , the sufficient condition dicity of
~
is rewritten as follows
(7)
lira
(~ - z)
(3)
for the
~
- ergo-
:
(p(O) + z p ( ) )
=
0
.
i
t h e n the measure Proof
~
is
~
- ergodic
.
. B y the assumption in the statement we distinguish
the following three cases
:
A). there is a sequence
lira _(o)
S-~
=
Pis
n < i[~
p(o)
i~
n ~ i~<
o
i2 <
...
such that
i s s~_~_ P-(~) = +
IS
IV.5. Case A).
such that
is
'
p(,l) :
"
For graphical convenience we put
p(O)
=
A
,
~
_ p(O)
=
~
Then we have
l~
(,~ - z)
~<
~im
(p() + z p())
ii
( ~ - z)
m ~ ~ IJ
;
O)
0
C). there is a sequence
S->9@
such that
n < iK < i 2 < ...
-uis
S"~
...
0 < p(O)< a_
J
B). there is a sequence
lira _(o)=
i2 <
=
(p(O) + z ~(~) ) is ~is
L1
.
103
lim
i-~ (~.-z) I I
~< m 2 -~ ~
~
~
is *z
s=m~
) ~is
m .-~aQ
Since
lim
9m
(7) it is sufficient
= 0 , in order to prove
to
m - ~
show that (8)
g c k
l~
kk~m-k]~
m-*~
C mk
where
~
stands for the binomial
Now fix A(m) whose
,
6 >
0
,
are the integers k
respectively
(&+2E,+~) There
0
,
the sets ,
D(m)
,
E(m)
,
m - k
, (~[,&+a]
a , b , c >
0
and
, (['$~,~] N o6
~
~ & , , (~f'$'/~,['$-~]
such that for
c a r d A(m)
<
a m
,
card F(m)
<
a m
,
card B(m)
>
b m
,
card E(m)
>
b m
,
card C(m)
>
c m
,
card D(m)
>
c m
.
, for any
k[ ~ A(m)
Cm~ k~ m - ~ also
, for any
r
k& ~ F(m)
(~-
,
k , 0 % k $ m , such that the quotient
m
we have
Moreover
F(m)
to the intervals
, (&+a,]L+2g]
exist
C(m)
=
k+'~
coefficient
and consider
B(m)
elements
belongs
- -A- m - k I
m
and any
k 2 E C(m)
we have
+ 6 )-bm C m!<2 kk2 ~-~2
and any
k2 E D(m)
;
we have
[o,~] >
No
104
Thus k ~k m-k Cm @
k g A(m}~F(m~
9 m-k
k ~ C(m) Cm ~< 2 -ca (~ + a ) - b m
k e D(m)
m
.
On the other hsnd , for -
k ~ B(m)~C(m)~D(m)~E(m) -~- } - - g ~
..< ~
we hsve
.
m
S~ce
Cmk kk ~m-k
=
~ , it follows that
k ~ A(m)L~F(~) ~ m
k a g(m)~ C(m) ~JD(m) dE(m) k~g
~2
k Cm kk ~m-k zm-k U
k kk ~m-k A(m) ~F(m') Cm
k~B(m)~C(m)~D(m)oE(m) # ~ (~ +~ )-bm
m +
~- k ~ 2s
.
Hence
8 n d the measure
~
on co
IV.6. Case B ). (~ - z ) ~ i=n+~
is ~
- ergodic
We remark that
[I
(~
-
§
z-~ i=n+~
105
: (~ - z) Therefore
Case B).
IV.7.
meduces to
qaseC)~
LEPTA ~. Let
lim
Case C).
We shall need the following two lemmas .
-(n)
~(~),
~)~(~) j=~ ~
(z ~( ) + ~())
... , AN(n)
= k > 0
and '
be positive such that
lim n ~
max ~j~N(n)
~(~) = a
0 .
,Then
lim
(~ +
n->=~
)z) -
: 0
=
Proof of Lemma ~ . There is
n o ~ IN such that for
n ~ no
we have
<
~(n)
2~
and
'k (~)
max
< ("i + 2~) -]"
Then the following inequalities are easily verified :
ll~li J : ~ e~(~)z g
(~ -
(~ -
~(n)
(n)
~ j=~
(~+~jz)
_
(~+
(~())2)-~
_
lJ
Ji .e 2~"
U =,z I~ - 2~ whence
~jmax~N(n) ~(~))-~
-
~] "e2~
.
106
=
II
n-~
J
=
j
"l.
=
0
.
,].
Since
a lim
z~
e
the Lemma follows
LE~FI~:A 2 .
-
e
=
0
.
lira
e -~,,II:: - . >
e)~Z ,ill =
0
Proof of Lemma 2 . The proof is based on arguments similar to those used in Case A) to prove (8) , so we shall be brief in details
. Thus , fix
A(~)
,
0 < s < :/2
B()~)
,
C()~)
whose elements are the positive
and consider the sets ,
D(k)
integers
, k
E(~)
,
F(aA)
such that the quotient
belongs respectively to the intervals [~+2~,+~) There exist
, [Ii+~,:+2&) , [_~,,l_.+&) , [ ~r, a _ ) a > 0
and
card B ( k ) >
card c(k) ~
~o ~ 0
, [ ~ ,~_~ )
sdch that for
~.
)~ ~)~o
ak
.
carriE(E) > a k
,
o
,
card O(k) > 0
.
, (O,T~ ] we have
Then
,) k ~ A(X~-, F(~,) and so
j = 0
k~C(k)~D(X)
107
~-~
k' A(~)~F(~)
k:
Finally , lira
A--~
e -)'II([ - z) ekZll
k
e B(k)~C(X)~.,D(A)~.,E(~)
~-
'1 -
~: 2 ~ and this proves the Lemma .
Let us now return to the proof of the Proposition in
Case C). By Lemma 2
there is
Owing to the assumption in Case C)
~>0
such that
<
9
~
and to Lemma
indic e s n < j~ < J2 < "'" < Jp such that e-~e~Z
_
~.(p~O)+ S='1_
z p(~)) Os
We infer
iim
(~ - z)
r ~->~
(p(~) + z p()) =
(~ - z) "1--[-(p (~ si=l~
Js
+ z p
))
~ ,
IV.~.
we find
108
II(~ - z) (e-% e ~z
2&+E
-
s= ~ ( P ~ ~
z P~)))II§
- z)e-~zll
=3~.
Hence lira r -->=~ and the measure
~
(~ - z)
(p()
+ z
=
0
= on ~
is
~-
ergodic .
This ends the proof of Proposition IV.4.
IV.8. Concerning the PROPOSITION limit points
~-
measurability of
If the sequence
0 < p ~ p' <
~
{P(m j m=
, then the group
~
we have
has two different r~
is non-measu-
rable with respect t_~o ~ . Proof . By the assumption we can find a sequence iK < i 2 ~
... ~ in ~ ...
such that llm
_(o)
k-~=
Pi2k
=
p
and
lira
k-~==
We consider the transformations
p(o)
=
p,
12k-~ defined as follows
~k e ~
:
where a)
if
for every ~2k+2s_ ~
~ ~ s ~ 2 k-i
we put
= ~2k+2 s
b) in the contrary case , let
so
be the smallest
s
with
109
~ s <~ 2 kwl
and
~ ~2k+2s_[
?j
= ~j
k+2so_T
~2k+2s ~
; then we put ~
for all
s J ~ ~ \ 12k+2so-~ ' 2k+2so I
= <2k+2so
= ~2k+2 s -21 0
Let us further denote 8
=
sup {_(o) ~(o) + _(~) pi~)kl kg~ Pi2k-T ~'i2k Pi2k-~
p(O) M
=
sup
i2k-~
max
p(.~-) z2k
12k_I Pi2k
: ~f k~
p(~)
_(o)
i2k_~ ~i2k
, 'o(o)..... ~ ( ~ ) i2k_i =i2k
fp(.~)
lip( o ) -(~) ~I i2k-~ Pi2k minl[p'(K') _(o) ~ i2k_~ ~i2k
p(O)
12k~I
' Ip(O)
i2k
p(.~_')' -
i2k-~
12k
The assumptions insure that 0 < 0 < ~
,
0 <M<
+oo
f k
= {~6uo;
,
~ >
0
.
Denoting
we see that 2 k-~
Now , for
%~
A k
we have
d~ and for ~ o \
Ak
we have
~-(A
k) <
+Oo,
I9
110
,s.~.
(~) g
~ ~i2k-~
/p (,~-)
] ~'p(,'l_) k ~ ~/V/pl(o I)
i2k
I I --( 0 )
[ i2k-~ Pi2k
3
_(o) t p("l ) ; k E Ilt'I "
[ i2k-1
i2k
Hence
(9)
M
(~o)
~---~'k)-%I 4
,
M
lim k~ These facts enable us to end the proof with an argument
similar to that of
L.Puk~nszky ([~7]).
Suppose there is a sigma-finite on ~o , equivalent with ~
,~
- invariant measure Q
and consider d?
f
There exists a Borel set is bounded on
d~
:
F C ~o
9
with
~(F) ~
such that
f
F . Let us denote if
~'(~) Since
f(~k(~) )
k=[
= If(~)o otherwise (~)
=
f(~)
,
~-
it follows that
~
0
F
F
dp-/
almost everywhere ,
111
By the relation
(~0)
we infer
On the other hand , owing to relation nature of the transformations
for any function
~
The relations
~
~k
L~(~O,
(~)
, we infer that
~-)
and
(9) and to the special
.
(~2)
are obviously contradictory,
thus the Proposition is proved . Q.E.D. IV.9. Finally , consider two product measures on c~
defined by the sequences
ip(n~
)}
and
~
and
~(~)~ {9 (0) n,~nJ
respectively . Some simple necessary and sufficient conditions for the equivalence of ~
and ~
are known ( ~ ] , [ ~ g ] )
We reproduce here the result of Let
a , b
9
V.Golodets ([~@]) .
be real numbers such that
O
and put
l
Then the measures ~
and ~
are equivalent if and onl 2 if the
followin~ conditions are simultaneousl F satisfied :
112
(P(~n) - na(~))2 ~ _(o)~(~)
T-
(i)
/ 6
~
n e Io{'~I{
na
n
<
+ o~
,
J~ n
~',I o
n e ~'.I{
%(iii)
/ n ~ ~I
~ n
+
/ n ~ ~\I&
~
IV.{O. By the considerations in II , any measure
~
on
<
Chapter I w 3
+ c~
.
and
Chapter
~O defined as above by a sequence
{P(~) ' P(~)I which satisfies the conditions of Prepositions IV.4. and of
IV.8. U(~)
gives rise to a factor representation of type I I I , . 0n the other hand , if the sequence
{p(~) , p(~ )]
satisfies only the condition of P~oposition
IV.~. then we have
an irreducible representation
. In what follows
~
of
U(~)
w_~e shall construct concrete realisations of these types of representations . Thus , fix a sequence
{p(~) , p(~)]
such that
o < p(~)'~ < and put
Denote by
~
the corresponding product measure on ~
IV.~T. Consider a separable Hilbert space
H
.
with an
113
orthonormal basis
~enl
. We define Hn
We shall identify elements
T ~ L(H)
, GL(~,r
GL(H)
and
.
with the set of invertible
such that
and
U(H)
r ej
GL(n,~)
T ej Thus
n ~
=
=
U(~)
ej
for
j ~
n
.
can be identified with subgroups of
, respectively
.
We shall use the following bounded linear operators on H : T n ~ GL(n,~)
;
T n ej
= ~qj ej
, j = ~.,2,~
;
An E GL(n,~)
;
A n ej
= p(~)ej
, j = ~,2,...,n
;
;
A ej
= p(~)ej
, j ~
It is clear that (I -
An) Tn2
(I - A) + A V n
=
(I - A m ) + AnV n , for all
IV.~2. We denote by of
Hn
Yn
=
Xn
"/~"
k th
exterior power
k
~ k=o
AIIn
n
Yn
the
and we consider the Hilbert spaces n
On
k A-- H n
V n ~ L(H n) 9
=
~ k=
k AHn
k ~
AHn
9
o
we consider also the exterior multiplication ,and on
Xn
we consider
"/~ " ,but given by the rule
a multiplication
, denoted by
, denoted again by
114
( a i ~ b i) A ( a j @ b j )
=
( a i A aj)(D(bi/kbj)
i
j
ai , b i ~ AHn
,
aa , bn ~ A H n
With these conventions we define the vectors E)n , ~ n
E
Yn
'
' ~n
s
Xn
as follows : rl
II
a = ~
k = o
9 e I~
A..
.
A
e ik
l~<...&i k
n
"~n = A
(qP(~
+ PV~-~'eJ ) :
n
=VP(g)'''P(n~ ~ . . k=o
=
k
k
n
n
~n
~qi~"'qJ ei~A--.Aei .
A (~| j;~.
+ ej|
Z
F
(ei~^"""^eik) | (ei~A"""Aeik)
k=o i~<.. o~ ik
n
j=~ n
= qP(~)" "P(~
of
~..
IV.~3. There is a natural irreducible representation k GL(n,~) on /~H n , namely that with signature (~,...4,o, . . . . . . . ,o) k-times (n-k)-times
Hence we have the representations n
.
115
n
~-n = where
I nk
k
~=
k
Jn | In
of
GLCn,r
on
stands for the trivial representation
Xn of
, k GL(n,C) on / ~ H n
We shall denote b_~ the same s.vEbol th__~erestriction of g representation from
GL(n,C)
ts
U(n)
and its extension to the measure
algebra. It is easy to verify that representation sentation
~n
Jn
and that ~ n
•
n
is a cyclic vector for the
is a cyclic vector for the repre-
" Since
~n(Tn) en = Z T Tnei~"" "^Tnei k k=o i&<...~i k
,
n ~n ( Tn)2 n =
Z F
(Tnei~"""ATneik) ~ (ei~A'''Aeik)
k=o i&<...ai k
=
n
it follows that ~n
is a cyclic vector for the representation
~n
~n
is a cyclic vector for the representation
Q-n
IV.&4. By the way we have obtained
Jn(~n-~)~n
nC C% For any
V ~ GL(n,C)
=
det ( I - An) 1/2 e n
,
=
det (I - An )I/2 ~ n
"
we have
116
9 9 < i k (Vei~ . . . A V e i k ) ~
ei~A ... A e ik)I
n
)
I
(eiA...Aeik)@(eiA...Aeik) k_-~oi~<...
n
(Vei A "--AVeik I ei A "''Aeik ) k=o i~<...
= ~Xn (~n(V)) where TrYn denotes the natural trace on L(Yn) . This shows that
v ,
~
(~n(v)~nl ~n )
is a central function on GL(n,~) and that if V B
V =
is diagonal ,
9
"X n we have (~n(v)~n~ ~n ) = A Y Ai[''" ~i k k=0 i[~...
= ~
({ + Ai )
= det (I + V) Sinc e V ~
~
det (I + V)
is also a central function on GL(n,C) , it follows that (~n(V)~n ~ In ) = det (I + V) Therefore , for all V ~ GL(n,~) we have
for all V s GL(n,Z) .
117
({rn(V) ~n I ~n)
IV.&5. Since
det (I - An)
(~'n(TnV Tn) ~ n I ~n )
=
det (I - An)
det (I + TnV T n)
=
det (I - An)
det (I + Tn2 V)
=
det ((I - An)
+ AnY)
=
d e t ((I - A) + A V)
C_ Hn+ [
Hn
Yzc
=
rn+~
, we may consider
~+~
,
x~c
'
In : Xn
The isometric linear maps Jn : Yn
>
Yn+[
Xn+ ~
defined by
Jn(~)
=
7 A ( ~(o) n+~"
In(~)
l_(o) ~.@~ = ~ A ( ,VPn+~."
~-
+
~ n ~
en+~)
+ "
' ~
en+~.@en+~.) , ~
{ Yn
'
{Xn,
allow us to consider Y = the Hilbert space direct limit of the Yn'S following the Jn'S , X = the Hilbert space direct limit of the Xn'S following the In'S . Since
Jn~n
Vo For all
= ~n+i
and
: li~ ?n ~ Y U
~ U(n) C. U(n+{)
~n+~ (U) ~ Jn
=
Jn~ Jn(U)
In~n
= ~o
P
~n+~ =
' we may define
lira )
~s n
e
X
, respectively
.
=
I n o ~rn(U).
we have ,
o-n+{(U ) o In
Therefore we get the following representations of and
x
U(=o)
on
Y
:
=
the direct limit of the representations Jn '
=
the direct limit of the representations~r n .
We shall denote by the same symbol the corresponding ~epresentations
118
of
A(U(OO))
. We remark that
~o is a cyclic vector for the representation
~
,
~o
~
.
is a cyclic vector for the representation
IV.~6. Our aim is to show that to
Tiff and that
~
a- is unitarily equivalent
is unitarily equivalent to
to a suitable system of matrix units
~
corresponding
E ~,/3 " Only the first veri-
fication will be done in full detail
.
Thus , for the equivalence of the representations rI[~ , using the results obtained in
Section
II.2.~O.
W
and
, we have
to verify that :
~ S(r
~n
A
for all
g g U(n) , ~n g U(n) If
~n
and
n ~ ~ .
does not correspond to s signature of the form (~, .... ,~,0, ........ ,0)
then we have r
=
o
, ~n(p~n)
=
o
and
p(p~n )
:
0
!
so the above equality is trivially satisfied . Suppose now that
~n
corresponds to the signature
(9' .... '9 ,0, ........ ,0) kn-tlmes (n-kn)-tlme s Them
on
~n
~
,
0 ~< k n ~
n
is (equivalent to) the natural representation of
Hn
, hence
~n
defines a
*-
isomorphism
9
U(n)
119
f n ' Ben ---'--'>
r,(/~
&)
Consider
~. = (~&4, f2 ~" ""4~ fn-&Z, fn ) ~- S(~'n) where
fj
Define
=
4,9, ........ ,o,) kj-times ( j-kj)-times
(~j)j=n E {0,&} n
s~=
,
(,~, . . . .
~ ~ j ~ n
.
, & 4 j~ n
,
such that
~s
=
kj
and denote by ~ i& < i2 <
... < ikn ~ n
those indices for which ~
=
&
,
m = &,2,...,k n
.
The reader should keep in mind the correspondences
We denote e~
=
el& 9 /k e i 2 A
... /k elk n ~
Hn .
It is easy to verify that ~n(p~)
=
the orthogonal projection of
A
H kn
Since
l, ~ e ~ S ( ~ n )
~rB (x)
fn
onto
is an orthonormal basis in - n / ~ [
= ~r fn(X) = 7
~S(fn)
(~'n(x) e~l e~)
~ e~.
n ' we have
,
x e Bfn .
120
f
•
On the other hand , owing to the identification oo = ~0,~
~(P~} = a=a_ .I I p J =
pc
we find
q%
We proceed to verify (&#) . By the above we have
J
rn =
p(
_
~ ~ s(j~n)
~s~ qisI (fn(g)%1 e~)"
Since ~(p~n ) is the orthogonal projection of Xn typic component of type ~n
of ~n ' we have
~n(P~)~n =j~_~Jl. ~
/~q~s/(ei~ A'"
-
i~<...
~
"Aeikn
onto the ise-
)|
~
i ) kn
henc e
(~n(~g" P~n) ~nl ~n) = (On(g) ~n(P~n) ~n =~
p(
fn(g) e~| e~
s
I ~n(P~n) ~n ) =
q~is~ e | eI
Therefore th_eerepresentations ~ and SI~ are unitaril2 e~uivalent. Let us also indicate how for ~ and ~5 we can proceed
121
similarly
. First we specify the system o f matrix units
means of which we identify
A
cient to do this only for
and
~ ,#
E %,~
by
A(ID_, P ) . It will be suffi-
~ S(~n) , n s ~
, where
~n
is
of the form ....
,%,9,
........
,
kn-t ime s (n-kn)-times We choose the
E
~n(E~,#
's
such that
) e# = e~
; ~ , # ~ S(fn)
,
0 % kn ~ n < + ~
Then , restricting again the verifications to the prove that
~
and ~
a~e unitarily equivalent
the ~tates dstermined by
~
.
B~n'S , one can by showing that
and respeg$1ve!y by the yector
L2( ,
)
are equal .
IV.~7. Taking into account IV.4.
and the last result in
conclusion
Section 1.3.~6.
Section IV.T6.
Y
, we derive a first
:
THEOREM . The representation space
, Proposition
J
of
U(oo)
on the Hilbert
constructed as above is irreducible provided that :
n='l.
IV.~8. Concerning the representation G-, we first remark that the function of positive type on has s simple expression
U(oo)
associated to ~ and
. Indeed , from relation (~3) we ~ f e r
(~(U) ~ I ~)
=
det ((I - A) + AU)
;
U s U(~176 9
122
Thus , the function ~A(U) is of positive type on sentation of
U(~)
=
U(~o)
aj
=
p(~) ,
The function T E L(H)
and , denoting by
~[A
~_
~-
=
ajej
;
j E ~ ~ T
, we have
L(H)
is defined by
0 < aj ~ 9
dot T
=
I + N
It follows that the function it is easy to see that
the repre-
,
j ~ ~
,
.
T~
of the form
~
U ~ U(oo)
=qT~
A ~ GL(=~ ,~) C A ej
where
;
associated to ~TA
Recall that here
det ((I - A) + AU)
YA
~
is defined for all operators , with
N
nuclear operator
is defined on
is continuous on
.
U~(~).Moreover
U~(o=)
,
with respect
to the topology defined by the metric
d(U,,U") Since type on
U(Oo)
is dense in
U~(~)
=
~
[U'-u"[
U~(~)
, ~
is a function of positive
Ui(~)
qT A
extends to a representation
, denoted also by
Using again the density of
~-[A ' which is associated to U(~)
ven Neumann algebras generated by
in
U~(==)
V
STA(U(~)) and by qTA(Ua(~))
be a unitary operator on iv
,u~(~)~u;
is an automorphism of
U~(o@)
H . Then
-~ v * u v ~ and
~
we obtain that the
are equal. Let
.
.
Thus , the representation of
, U,,U" ~ u ~ ( ~ )
u~(~)
.
123
(~A o iv)(U ) Therefore
~V~AV
= ~V.Av(U )
U
;
~ U~(~o)
is also a function of positive type on
and its associated representation
~/V*AV
.
U~(o@)
coincides with ~ A o ~ .
Since the yon Neumann algebras generated by ~V,Av(U~(~o)) and by
~IA(U~(oo))
tations
UIV~AV
are the same , it follows that the represen-
and
~A
are both of the same type (without
being necessarily equivalent)
.
Thus , for any injective operator diagonable with respect to function
~/A
some
A ~ L(H) , 0 ~ A ~ I ,
orthonormal basis of
H
, the
is of positive type . By the spectral theorem one
can easily check that any operator
A ~ L(H) , 0 ~ A
~I
, is
the norm limit of a sequence of diagonable injective operators A n ~ L(H) , 0 ~ A n ~ PROPOSITION
I . It follows that . For any operator
A
E L(H) , 0 ~ A ~ I ,
the io____~n func t : U~(oo) B U!
r
det ((I - A) + AU) E
i_~s continuous and of positive type on Denote again by ciated to
~A
UT(oo)
the representation
~A
of
U~(~)
I~A . The following problem naturally arises
PROBLEM . Under which conditions on tion
.
factorial
A
asso-
:
is the representa-
? Which is the type of th__~ecorresponding
f act0r ? When are tw O such representations equivalent ? As we remarked above , the representations V
~ U(H) , are of the same type . Therefore
S y A and SKV,AV ,
, th__~etype of
~l A
124
depends only o_nnthe spectral properties of
A . However , it seems
difficult to decide when
are equivalent ~
qIA
and
UTV,AV
IV.~9. The considerations concerning the representation ~allow us to give a partial answer to the above problem. Indeed , taking into account
Theorem 1.3.~2.,Proposition IV.4.,Proposition
IV.8. and the first conclusion in Section IV.~6.,we obtain the following TKEOR]~M . The representatio_n space
X
(r of
U(o~)
on the Hilbert
constructed as above is factorial provided that
L
sin ip(n0) , p(:n)~
=
+ co
and the corresponding factor is of type III provided that the sequence ip(~
has two different limit points in (0,~) .
Note that necessary and sufficient conditions for the equivalence of tw~o such representations are contained in Section IV.9. COROLLARY . Let eigenyalues
A
{aj~ jg ~
~ L(H) be a_ dia~onable operator with
suc___~hthat
0
< a3 <
Then the representation
qT A
~
of
, U(Oo)
j E ~ . as associated to the function
o_~f Positive type ~A(U)
=
det ((I - A) + AU)
is factorial Provided that
,
U
E
U(oo)
,
125
j
d
7
min l a j ,
~-
aj}
=
+ co
= ,'1_.
and the 9~
factor is of ~
III provided tha t
the sequence ~aj} has two different limit points in (0,~)
@
The last condition in the Corollary means that the essential spectrum of
A
has (at least) two different points in (0,~)
IV.20. If
.
p(~) = aj = a = constant , then the correspon-
ding product measure representations
~A
~ ~
on co ~
is
~l~
C~
- invariant , so the
are of type
II~
and they
correspond to the character U Type
~
~
det ((~ - a) + aU)
I I ~ - f a c t o r representations of
,
U(~o)
U ~ U(oo).
were studied
in (~3@] ,~35S) , where also other classes of such representations were obtained . It would be interesting for instance to extend also the construction of type
II~ - factor representations in
symmetric tensors , as was done here for antisymmetric ones ~
IV.2~.
Finally , let us mention that the modular automor-
Pmhism group {~t} = (~(U(oO)))',
t~m
o_~fthe vo__~nNeumann algebra
( Aej = sjej , O ~ a j c T
cyclic separating vector Namely , with automorphisms of
st(V)
~o
(UIA(U(oo)))" =
) corresponding to the
(see 532J ) has a group interpretation
B = A(I - A) -i , consider the following
U(~o ) : =
B it
V
B -it
,
V ~ U(o~)
;
t E
126
Then
: ~-t(~IA(V)) = UKA(St(V))
,
V ~ U(~)
;
t s ~
9
This can be verified by reduction to the case of
((~:A IU(n)) Ix~)" C :.(xn) The reduction is possible since of the operator adjoint
(~IA(U(~)))"~o
Xn
~d
tnEX z
.
is invariant for the closure X~o~
~
X*~o
and for its
, which is the analogous operator with respect to the
c ommutant .
CHAPTER V
SOME, TYPE
II C O
FACTOR REPRESENTATIONS OF
U(o@)
In this section we shall show that some natural representa$ions of
U(oo)
on the infinite tensor products of the underlying
Hilber~ space are factor representations
of type
IIQ@ and we shall
study the equivalence of different such representations
as well as
their commutation properties with the corresponding natural representations of the group
S(oo)
of finite permutations of
~ .
This section does not depend on the rest of the work
V.~.~. Consider a separable Hilbert space normal basis
~en~
n~N
H
.
with an ortho-
' define n
Hn
=
and , as usually , identify operators and
T E L(H)
U(oo)
respectively
~=
~ ej
GL(n,C)
,
na~
,
with the set of invertible
acting identically on
H ~ H n . Then
can be identified with subgroups of
GL(H)
GL(~,~)
and
U(H) ,
.
Consider also the Hilbert spaces : m
~n
=
~
Hn
=
Hn~''"
~Hn~
; n,m ~
m-times There are natural representations
~ nm
and respectively of the symmetric group space
~m
such that
and ~ nm S(m)
of the group on the Hilbert
U(n)
128
m
)
; ~'''''~m
~ Hn
'
U ~ U(n)
; ~'''"~m
e Hn
, o- ~ S(m)
,
m
Denote by ~n
m
and
Nn
the von Neumann algebras generated in ~m(u(n))
and
L(~)
by
jm(s(m))
respectively . Let us recall the following classical result (~$6]) THEOREM (Hermann Weyl) . The yon Neumann algebr a commutant of the yon Neumann algebra
~, If ~
=
' "'" ' ~ m ~ Hn
~m
~
in
m
Nn
is the
L ( ~ m) :
.
are linear independent vectors ,
then the vectors
G )m :~
are linear i n d e p e n d e n t i n
[~n~
' ~ ~ ~(~}
~s
:
71( j)
. Since
Nm n
,
o- E S(m)
,
is linearly spaned by
' i~ ~ollows that
m ~j
is a separating vector for
Nn
that is Y ~ Nmn
,
y~j~__~ OI =j
~
y
=
0
But a vector is separating for a yon Neumann algebra if and only if it is cyclic for the commutant . Thus , by the above theorem we see that
129
m
is a cyclic vector for
~
.
V.~.2. We fix a strictly increasing sequence
K
of positive
,
n ~
integers
we define the Hilbert spaces n H (n) = ~ H
= H~
... @ H
,
n~
~
n-times and the linear isometric maps
In :
~
;
~
~ekn+~
.
We consider the Hilbert space direct limit
of the
H (n) 's
following the
The Hilbert space
~K
In
Ts
.
is nothing but von Neumann's
infinite tensor product of a sequence of copies of
H
along the
sequence of vectors ek& , ek2 , ... , ekn , ...
~
H
Let us recall that for any sequence of vectors ~
' ~2
' "'" ' ~ n
' "'"
~
H
such that
there corresponds a "decomposable vector"
n->~
"
e k n + ~ @ ekn+2 |
E
130
which depends linearly on each
J
9 If
j:[
~j
is another de-
composable vector , then
= I T < jr j) j=~.
where the infinite product is absolutely convergent . The set of decomposable vectors is total in
Let of S(m)
~
S(o~)
~K.
be the discrete group of finite permutations
, that is,the obvious direct limit of the symmetric groups . There are natural unitary representations and
of the groups
U(~176 and respectively
=
\j=~
S(~)
U~j
j=~
~-~(j)
on
~K
such that
,
U
~ U(o~)
,
,
r
s(~)
,
for any decomposable vector
V.~.5. Before going any further we shall prove a simple lemme on the commutants of von Neumann algebras . For a v o n mann algebra if
N C L(~)
p c M' (resp. p ~ M)
Mp C L ( p ~ )
we denote by
M'
Neu-
its commutant , Also ,
is a projection , then we denote
by
the induced (resp. the reduced) yon Neumann algebra.
We recall that (M')p
LEMMA . Lg~
=
(Mp)'
M , N C_ L ( ~ )
.
be yon Neumann al~ebras such
131
that
N C M' . Suppose there are (i) a_~n increasing sequence
o_~f M
which generates
~ Mnl
M , i.e. -
which generates
:
(k~
--
[Nnl
N , i.e. -
Mn)" ;
n=~
(ii) a~n increasing sequence o_~f N
M
o_~fvo__~nNeumann subalgebras
N
o_~fvon Neumann suba!sebras =
(~
-
Nn)"
;
n=~
(iii) a_~nincreasing sequence of projections
Pn ~ Mn'f'h Nn' ;
such that Pn ~
~
and
((Mn)Pn)' = (Nn)Pn
for each
n .
Then M !
:
N
9
Proof . We have to prove that der
x ~ M'
and
y E N' . For each
X ~
Mnl
M' C N" = n
N . Thus , consi-
we have
Y ~ N n'
so , by our assumptions , Pn x Pn g ((Mn)Pn)' = (Nn)p n
'
Pn y Pn
~
((Nn)Pn)'
It follows that Pn x Pn y Pn Taking the limit when
=
n --~ =
S inc e
y E N'
for each
Pn y Pn x Pn
n
, we get
yx
.
was arbitrary , we obtain
x ~ N" . Q.E.D.
V.~.4. We denote by M K
and
NK
the von Neumann algebras generated in ~X(U(~))
L ( ~ K)
and 9 K ( S ( ~ ) )
by
132 respectively. We shall prove THEOREM . .The yon Neumann algebra of the yon Neumann algebra
MK
in
=
N K
(MK) '
N~
is the commutant
L(~ K ) : .
Proof . Clearly , (M~)'D N K . For each Mn
and
and
denote by
Nn
the von Neumann algebras generated in ~K (U(kn))
n
L ( ~ ~)
by
"~K (S(n))
respectively . Then , obviously , M n
and
Nn
fulfil the condi-
tions (i) and (ii) in Lemma V.~.3. On the other hand we have
, and
~
~ k n C q~kn+~
n--~
is an invarlant subspace for both the representations
~xlU(k ~)
and ~XlS(n ) . Moreover , it is clear that the represen-
tation ~K IU(kn)
induces on ~ ~
the representation
~KJS(n)
the representation
induces on
~
~
and
the representation Z-A
Denote by
Pn
the orthogonal projection of ~ K onto
Then we have Pn ~
a_
'
Pn E Mn'/h N n'
and
(Mn)Pn : thus , by
,
(Nn)Pn = N ~
Hermann Weyl ' s theorem , ((Mn)Pn) '
=
(Nn)Pn
,
~
.
133
Therefore , all the assumptions in Lemma V.~.3. are fulfiled end the Theorem follows . Q.E.D.
V.~.5. THEOREM . The representation is a factor representation of type Proof . We shall show that that
M~
is a type
~K
~of
U(cc)
o_~n ~
IIo@ . N~
is a type
II~
factor and
II~@ factor .
By the remark at the end of Section V.~.~. it follows that ~ekj
E ~ ~
is a cyclic vector foro@ ~o
is a cyclic vector for tlng vector for
MK
=
~ ~S(~)
'
n~
~ . Therefore
j=~ekj ~ ~ Since
N K = (MK) '
~o
is a separa-
N~ .
On the other hand , for any
where
~Ukn
9 S(O@)
we have
if
~=
E
if
~
~
is the identity permutation . It follows that
the vector state
is a faithful finite normal trace on the von Neumann algebra
N~ .
Let us recall that the von Neumann algebra generated by the left regular representation of
S(OO)
is the hyperfinite
type II~ factor and that the canonical trace on this factor induces the same function of positive type on does (see ~ 8
S(~)
as the above ~
]) . It follows that the representation of
S(~)
134
induced by
9K
~NK tO~]
on the invariant subspace
o~ ~
is
unltarily equivalent to the left regular representation of Denote by
p
the orthogonal projection of ~ K
S(oo).
onto ~N K ~o ]
Then p
E
(N K)'
MK
=
and the preceding discussion shows that (NK)p
is a type
II~
factor
The central support of the projection
p
is the same as
the central support of the orthogonal projection of
~K
onto
the subspace :
:
thus
p
is a faithful projection in
w
,
(NK) ' . It is known that any
induction by a faithful projection is an isomorphism (see[ 6 ~) , in particular the canonical map NK
~~
(N~)p
is an isomorphism . Therefore , N K it follows that MK
MK
is s type is a type
II~ II
factor
Since
(M~) ' = N K
factor 9 In order to show that
is i~finite we shall construct an infinite family of mutually
orthogonal and equivalent projections
{qn~
in
M
Consider the vectors ~n
=
ek~@''" @ e k n _ ~ | ekn| ekn| ekn+2| ekn+3| "'" ~
and denote by Then
qn
the orthogonal projection of
onto
~ IN K ~n ~.
135 q=
~
It is clear that for any
(N K ) '
=
M K
n ~ m
and for any
~m
for
(z, ~ ~ S(oo)
we
have
t hu s qn Finally , define
_L
Un ~
U(~)
r
n~m
by
ekn+ ~
j=k
n
ekn+ 2 U n ej
and ~-n ~ S ( ~ )
ek n
if
3 = kn+ z
ej
otherwise
by = ~
b
o-n
...
n-T
n
n+~
n+2
n+3
n+~
....~
9 ..
n-~
n+2
n+~
n
n+3
n+4
/
M K
with
Then
so
(Un) qn It follows that
~(Un)
initial projection
qz
= qn
qn+i
is a partial isometry in
and final projection
qn+~
"
Thus , qn
N
qn+~
which completes the proof of the Theorem . Q.E.D.
136
V.&.6. For any permutation
~- ( S ( ~ )
we can define a
K U~ ~ U(c'~) by
unitary operator K
U ~ ekj
= ek(j)
U~ e k
:
for each
ek
~
j
k ~ ~
~ ~
,
{kj}
.
:
Then
is a subgroup of
U(~@)
. Of ceurse
isomorphic as abstract groups
and
, ~_K(~)
S(~)
are
.
Since K
"~(~'-~') ~o
= it follows that
As we have seen , the von Neumann algebra
i_~s gene-
(N~)D
rated b y
~(s(~)) I Moreover
, its commutant
[~ ~]
, the yon Neumann al~ebra
(MK)p
i_~s gene-
rated by
This can be seen as form
~n
orthono~al
basis
w i t h the Hilbert space
follows. in
translation b y
w
representation of representation
, for every S(~)
[~ ~o] "
[2(S(~))
comes the left translation b y
The vectors
(Y
Thus
{'~w(~-)~0 ;~'~S(~176
we ~ay
i~ent~y
in such a way that and
~W(U~)
~ ~$
~(~)
be-
becomes the right
~ E S(oo) . Since the left regular
is the commutant of its right regular
, our assertion is apparent
.
137
V.[.7. The following theorem shows that the representations ~,
obtained from essentially different sequences
K
are non-equi-
valent . THEOREM 9 Le__~t K = [ ~ }
,
K' = [ ~ }
- - ~ two strictly be
increasin6 sequences of positive integers . Then the representations 5,
an d
5K. of
there exists
U(~)
no ~
are unitari!y equivalent if and onlE if
S
such that =
~
for all
n ~ nO
.
Proof . If this condition is satisfied , then it is obvious that the representations
~"
Suppose now that operators
~
Un ~ U ( ~ )
and
n > m ,
~' ~K,
are identical . are equivalent . Define the
by
U n ek
If
and
L
-e~
if
k
=
kn
,
ek
if
k
~
~
.
we have ~K (Un) ~
=
-~
for each 1
C ~
,
C ~K
SO
for each n-~
Since
k~ ~m m=~
~K
=
, it follows that
km
nllm ~ (Un) ~
=
- ~
for each
~
~
and
~
9 Kt
By the equivalence of the representations nl~im~ K~ ( U n ) ~ But it is clear that
=
-7o'
where
~
~o' =
~
we infer ek~ c ~ '
.
138
r ~, Therefore
~ , ..., ~ , ...}- - > r
, there exists
mo ~ ~
such that
I~I n>,~o ~
[~I n~ ~
B y a dual argument we find
m o' g ~
[knl n ~ m ~ Changing
, if necessary
, the number c
~ ~o,
~ Choose
n o >/ m o
"
mo g 9
{k~
we m a y suppose that
n>m~
n o' > m o'
c
k ~'
such that
.
[kn~ n ~
On the other hand , there exists
9
such that
c
Iknl n ~ m o
~o' : d
" such that
{kn~ n ~ mo
9
= kno . Then it is easy to see that
I~} n ~ n~
=
Ik~
=
no
n ~ no
It remains to show that n oi
Suppose the contrary holds , for instance n ot choose
k
a ~
, l~l =
i
n >
m
,
r
r
E
~
,
Vn
~ ~
V n ek If
+
, such that
kr and define the operators
no
=
U(~~
by
[~e k
if
[
if
ek
~ k 4 k k>k
n
,
m
.
we have
(~K(Vn)~[ ~)
=
~n
H~
2
for each
~ ~
~or each
~
~
C
~
,
so
(ff'(v n) ~ i ~ ) nl~'m~
9n
llkll2
~.
.
139
By the equivalence of the representations
(~.~'(Vn) ~' I ~ ' )
(~)
2
xn
:
On the other hand , if
n > m
~x
and ~
Kt
we infer
~'~'
for each
~ g
for each
~' g ~ C _ ~
for each
~, E
we have
(~'(Vn) ~'l ~') = ),n+r ll~'ll 2
K',
SO
(2)
=
krll~ll2
k~
for
nlim C~'~'(Vn) ~n ~' l '~' )
KNots that
k m ~ km_ r
subspaces of
~K.
=
(~)
large , so
~
o
9 m ~k~
as
and this implies
~' Comparing
m
K,
and
=
(2)
~~mm--~l
.]
we get
and this is a contradiction . Q.E.D. V.~.8. For every
U e U(==)
(~x(U) ~o I ~o )
=
we have
~(Uekj
I ekj)
Therefore , the function
(3)
= ]-F
j-Ji (Uekj
is of ppsitive type o_~n U(o~) U(~)
and
~"
,
ekj
)
;
u
eu(~)
is the representation of
associated to ~K 9 We shall show that
to the metric of
U~(~)
~K
i_~suniformly continuous with respect
,
d(U', U")
=
Tr IU' - U" I
,
U' , U" e U ~ ( ~ )
140
Since this metric is both left and right invariant and since
~K
is of positive type ( recall that this implies
I~ ~C~'~
- ~c~"~l ~
~
~
I~- ~'~'~-~"~I
,
see ~ ], ~3.~.7. ) it is sufficient to prove the continuity of ~ in the identity
I E U(oo) . First remark that for any normal
operator
H
T
on
and for any vector
-~
Thus , we obtain
~ a H
we have
((Re T)&, ~)
+
l((Im T)~, ~) I
(IRe TI ~, ~)
+
( Im TI ~, ~)
j•=
(Uek.0 I ekj )
.TT~ + l(~e~j1%) - ~
- ~
0 =
=
exp
~
exp
(2
It follows that ~
)
~ I((U - I)ekj I ekj)l
~l (=IlU~ - j
ekjl ekj
-
-
~i
"l
extends to a uniformly continuous
function of positive type on UT(~), denoted also by ~.Consequently, the representation ~
of
U~(oo) , denoted again by ~
U(oa ) .
extends to a representation
o~f
141
It is easy to see that formula that for decomposable vectors
=
U
j
Since generated in algebra
MK
g
~
~ j E~K
and
U(oo)
by
generated by
factor representation of
V.~.9. Let
V
~K(U~(~))
j
=
=
U
we have
j
~K(u(~))
. Hence
~K
is a_ t.YPe II~o
U~(oo ) . Moreover , th_~eequivalence of
factor representation of
is a type
II~ factor representation of
It is clear that
H . Then
. Therefore
II
U~(~)
U(~)
~K o iv I U(oo)
and
.
is the natural represen-
on the Hilbert space infinite tensor product
of a sequence of copies of
H
along the sequence of vectors
(Vek( , Vek2 , ... , Vekn , ...) Thus , we obtain the following result . Consider an arbitrary orthonormal system oC
=
(a~
,
a2
,
and define the Hilbert space
...
,
an
~K
reduces to their e~uivalence
be a unitary operator on
U~(~)
.
is equal to the yon Neumann
is a type
U(c~)
U ~ U~(~)
and
U(oo) .
is an automorphism of
tation of
and
U~(~)
U~(o~) , the yen Neumann algebra
an__~d ~ Ki as representations o_~f U&(oo) s_~srepresentations of
extends to
(U)
is dense in
L ( ~ K)
(3)
,
...)
oC in
H
,
142
as the
~finite
the
sequence
of
U(oo)
tensor
product
~c . T h e r e
on
is
of
a sequence
a natural
of
copies
of
H
along
representation
such that
for all decomposable vectors
=
J
"
THEOREM . For any orthonormal system ~ = ~an~ (. H , the representation of type
II~
5~ of
U(~)
on
is a factor representation
.
This extends Theorem V.~.5. Theorem V.~.#. , namely
the commutant of
natural representation of Note that
~
There is also an extension of
S(oo)
~
is ~enerated by the
on
is the representation of
U(o~)
associated
to the function of positive type
y~(u) =
.[ [(Uaja aj)
; u ~ u(o= )
V.~.~O. Now the following problem arises : Given two orthonormal systems
~ = lan~ , ~ = ~bn~
in
H,
find necessar E and sufficient conditions in order that t.he representations
~
and
~
be equivalent .
143
Theorem V.~.7. contains the answer to this problem
in a
particular case . In general , if there is an arbitrary permutation of such that b n = e n a~(n)
for suitable
then the functions of positive type the representations
~
and ~
~n ~ C ~
and
, l~nl = ~
~,
are equal , so
are equivalent.
On the other hand , suppose there exists an operator U
a U~(~@)
such that bn
=
Ua n
9
Then the Hilbert spaces
~
sentations
are identical 9
5~
and ~
and
~
coincide , so the repre-
A reasonable conjecture might be that The representations
~
are unitarilF e~uivelent
an d ~
if and onl 2 if there exist permutation an operator
~ U
of
N
~ U~(~o)
such that U b n = @n ar
for suitable
One
Z
, lenl = [ .
A weak result in this direction is the following
PROPOSITION . Let < : [an~ , ~ = [bn~ be orthonormal systems in u(oo)
H . Suppose that the representation s are 9quivalent . Then there are finite sets
and a bijective map
and /
144
~ :~\F~
~ ~\F
such that lira
lib n
-
a~(n)ll
en
n - ~
Proof . Let
= o
~Cn~
for
suitable
~
C
, tenl
be an orthonormal basis o f H
contains the orthonormal system
=
an
{an}
= ~
.
which
such that
Ck~
is a strictly increasing sequence of positive integers. Define the operators ~C
Un
ck
=
by
Un 6 U~(~o) if
k=k n
~
~k~
[ ck
As in the proof of Theorem V.~.7. we see that for each lira (~(Un) ~ ,~)
=
~ g ~%
-I1~112
n - ~
Since
~
is equivalent to ~ ,
we obtain -i
,
that is (4)
lira ~ n-~ j
(Un bjl bj)
=
-i
~j
=
="1.
Write bj
=
~ j ck k
where
(bj |Ck)
= "1.
Then the relation (4) becomes (5)
lira n - ~
Since
(~ - 2 =
) =
-~
.
145
it follows that for each
n ~ 9
there is at most one index
j a
such that Owing to relation integer
no
(5)
2
<
0
we infer the existence of a positive
with the following property
for every
9
n ~ no
:
there is a unique
-
2
<
j(n) ~ ~
0
such that
.
Since
using again the relation
(5)
we obtain
n ~
12
j(n)
) =
-i
Since =
k=~ the map n 6 ~ is injective
; n >/ nol 5 n
. The relation
(6)
)
rewrites
lim l(bj(n) l a n ) n-~2o@
By a dual argument we find {n E ~
; n >imo} ~ n
j(n)~
i
=
mo E ~ >
and an injective map
i(n) ~ ~
such that ~
l<~n,ai
From these it follows easily ~hat for sufficiently large n s ~
we have
146
i(j(n))
=
n
Therefore there are finite sets
F~C
bijective map (namely a restriction of ~- : ~ ' F ~
~ , ~ C ~
n
>
and a
i(n) )
~ - F
such that
n < ~. l(b~I a~Cn )) Define
, for
n e ~\F~
=
{
9
,
8n
(bnl a~(n)) = l(bnl a~(n))l
Then 0
< (b n I @n a~(n))
r).-IP
and clearly
I~
llbn - e n a~(n) ll =
0 Q.E.D.
Suppose that %
and~
are orthonormal bases in
order to prove the above conjecture
H . In
in this case , it would be
sufficient to show that c ard F
:
card F p
and
(b n 18 n ar
] /~ +
n =~ where the notation is as in the preceding Proposition w 2
Other ~E~9
II
~9G~K
.
representations
In this section we shall briefly sketch the construction of another kind of type
II~
factor representations of
U(~)
.
147
The notations and the results from Chapters I , II , I I I will be freely used . We shall often identify a representation
fn
E U(n)
with its
signature
stands for the projection
corresponds
p~n E A = A(U(@@))
to the signature
V.2.~. Let
. For example ,
co C ~
where
~n
E U(n)
( m l , . . . , m n) .
be the closure of
ponding to the upper signature the lower signature
p(m~,...,mn )
(co, &
P-
orbit corres-
, 0 , 0 , ... )
and to
(0 , 0 , 0 , ..~ ) . The points of
~o
are
the symbols t such that
=
(~i(t) ~
~j(t) E U(j)
~2(t) ~
... ~ ~j(t) ~
... )
is either of the form
(m(t),~,o,
.
.
.
.
.
.
.
,o)
( j-2)-times or of the form (m(t) ,~0,0, ....... ,0t) ( j-~ )-times Let further
co'c ~
consist of those points
= +o@ such that for large enough ture of
a
~-
~j(t)
is of the form
j (depending (m,~,O,...,O)
t a co on
with
t)
no(t) =
the signs-
. Clearly
,
~A& ~
is
invariant Borel subset of co . Let us also denote
)
=
{t
,jt)
:
j :
.
Consider now a system of positive real numbers
such that (~)
C(mn)
=
~p~m
c(n+'i) P
;
m,n e ~
.
148
For a given system of numbers there is a u n i q u e on
uo
[~- i n v ~ i a n t
[c (n)} satisfying
, sigma-finite
(~) ,
Borel measure
such that :
(i)
~ ( e ( f [ < . . . < f n ))
(ii)
~(,,o \ ~ ' )
Moreover , any
U-
=
C (n) m
=
0
if
9n = (m,~,O,...,o)
,
.
invariant Borel measure
~
on
~o
, such that
is of the same kind , i.e. defined as above by a system of numbers ~c'(n)I satisfying the conditions
o ~c'(n).<
(~)
and such that
c (n)
;
On the other hand , consider the closure of corresponding to the upper signature the lower signature
(~,
(0 , O , 0 , ... )
consisting of those points
t ~ ~O~
m,n
~
~ - orbit
0 , 0 , ... )
and the subset
with
~
.
~O~
and to cO~ ofuo~
no(t ) = + o~. Consider
also
(f~'"~gn)
=
t~;
~j(t) -- 9j
Then for every system unique (i~) (ii[)
~-
satisfying
j --~,...,n
9
(~) there is a
invariant , finite Borel measure ~; on uO~ such that 9(e~ (~...<~n))
=
(n)~ cm+
~(~0{\
=
0
Also , if ~)~ is a
~0[) ~-
if
fn = (m,O,O, 9 ..,0)
.
.< ~'
%
0
9' also corresponds to a system of numbers ~c '(n)}
the conditions (~)
,
invariant Borel measure on o0[ such that 0
then
[o (n)}
for
and such that
satisfying
149
c (n)
o 4c'(~ )
These remarks show that,for a system of positive
fc(mn)t s a t i s f y i n g godic ~-
if
and o n l y i f
ergodic
t h e measure
~ on co i t
t h e o t h e r measure
~
on
on
Now a f i n i t e
~
P-
invariant
yields a type
II~
it
P-
er-
defines is
sentations
U(oo)
of
the restrictions
II~
of U(~)
factor repre-
of which to the
only according to representations
(m,O,O,...,O)
ergodic Borel
factor representation
, such measures arise from type
U(n)
' s
of signatures
.
The character responding
~o~
, ~-
and conversely
decompose
def~J~esis
.
V.2.2. measure
(&),
numbers
of the type
II~
factor representation
cor-
to "~ is then given by :
X
IU(n) = 4y
(n) $~(m,O,....... ,0)
Cm+~
m = o
(n-~)-times
It is known that the functions V ; on
U(=o)
of
U(~)
>
are characters (see[
])
det ((~ - b)(I - bV) -i) of type
and for
V
II~
~
U(n)
,
O~b<~
,
factor representations we have
o@
det ((~ - b)(l - bV) -~)
=
~ m = o
bm(~ - b)n ~(m,O, ....... ,01)(V} (n-~)-times
The last equality can be easily established by computations maximal torus of
U(n)
on a
.
Thus , _t~.~. measure ~
on co r
to th e system
150
c (n) = m i gs ~ - ersodic
~
indeed , but it is not finite
7
. For
...4 ~n-~ 4 (p,~, ....... ,~))) (n-~)-times
q>~P
O
on ~o defined by these numbers
This can be proved as follows ~(e(~<
,
.
V.2.3. The measure ! g Sin~ma-finite
bm-~(& - b) n
q>~P
.
p ~ ~
we have
=
(n-~)-times
bq-i(~ - b)n+~
=
c(n)m
n-times
bP-~(& - b)n
Suppose now we have proved ~(e(~&4...
4 (P,~, ....... '9 )))
~
k bP-~(& - b) n
(n-%)-times for all above
p ~ &
and
n ~&
. Then , using the same equality
, we have
~( e(~r
. . .
< fn-~ < (p,o,
. . . . . . .
,9)) )
( n-K )-times
bP-~(~ _ b) n
= Thus
k bq-%(~ - b) n+%
+
(k + &) bP-i(~ - b) n
9
, we must have ----
( n-~ )-time
s
+
~
9
as
151
V.2.4. By the general theory , the ~ ant , sigma-finite measure
~
tor representation
U(~)
~[~ of
on ~o C ~
ergodic , P - invari-
defines a type I I ~
fac-
. What singles this factor repre-
sentation out up to quasi-equivalence ? The answer is the following: Given a semifinite representation
~
of
U(~)
, i_~ti_~s
~uasi-e~uivalent to qTp if and o n l $ i f
(2)
Tr (~(P(m,~,O,~ . . . . . . .
,0) ).
= d(m,~,O,. . . . . . . .
(n-2)-times (3) m,n
,0) bm-~(T - b)=
(n-2)-~imes
P(m,~,O, ....... ,0) ) (n-2)-times
:
z
.
What we have to show is that two representations satisfying these conditions are quasi-equlvalent . This will follow ~rom usual arguments about (unimodular) Hilbert algebras if we make the following remarks . The condition (3) implies that
Z f(B(m,~,p ' ....... ,0)) m , n (n-2)-times is weakly dense in
(~(U(~)))"
and , since ~(B(m,~,O,...,O ))
are finite dimensional factors , the trace is completely determined on the above finite-dimensional subspace by the condition (2) . Moreover . . . . . . .
m , n is a
* - subalgebra of
the ~ -
subalgebra
,q))
(n-2)-times (~(U(~o))) '' and it is ~ - isomorphic with
152
~'-
B(m,~,p, ....... ,q)
m , n of
A
=
A(U(co))
(n-2)-times
.
V.2.5. With these preparations we can now exhibit another realization of the representations Let
~
.
be the natural representation
underlying Hilbert space Consider also U(co )
~
H
92
of
U(o~)
with orthonormal basis
a type
II~
on its
lenl n~ IN
factor representation
" of
of character det ((~ - b)(I - bU) -~)
in standard form on a Hilbert space Then on
(~(U(~))
K
|
and
~ g K
is a type
a trace vector. Yloo factor acting
H @ K , the trace being defined by the weight @
= Let further tions
~
and ~ 2
~3
b-~n~=
~en|
~
be the tensor product of the representa -
:
The projections ~3(p(m,~,O, ....... ,0) )
( n-2 )-times are finite projections
~3(p(m,r
in
'.......,0)) a (~r (n-2)-times
and thus
(~(U(~o))|
~2(U(~o)))"
. Indeed ,
| @2(u(n)))"
153
~ (~}3(P(m,T,O, . . . . . . .
,9) ))
(n-2)-times
=
b -~"
n ~__ eOej|
(~3(P(m,~, 9, . . . . . . .
j='l.
,0 )))
(n-2)-times
since HI~)K
= (Hn@K) {t} ((H{3Hn)~} K)
and the representation of equivalent to
921U(n)
U(n)
on
( H e H n) @ K
, being quasi -
, contains only signatures of the form
(m,0,0,...,O) . Now , we have b -~ / ~) J = ~ ej|
o ~ 3 IU(n )
b-~X (~, o , . . . . . . . ,9) (n-~)-times
-zf2
I u(n)
bm(~ - b)n ~((m,0, ....... ,O) =
,o, ........ ,o)
(n-~)-times
=
b-~(~
- b)n~(~,o,
=
m = 0
. . . . . . . ,o)
(n-~ )-times +
(n-~ )-times +
b-~ / ~
b m ( ~ - b)nE~((m+~,O , ..... ,0t) +$((m,~,O, ..... ,0)]
m = ~
(n-~)-times
(n-2)-times
where we have used the known fact that ~(~,0,0,...,0) Xr for
m ~ ~ 9 Hence we have
:~(m+~,0,0,...,0)
+~(m,~,0,...,0)
154
~~
=
....... ,9) )) (n-2)-times
bm-~(~ - b)n d(m,~_,O,....... ,0) " (n-2)-times
Consider p(n) = ~
~,3(p(m,~.,O,. . . . . . .
,0))
(n-2 )-time s Then we have
p(2) 4 p(3) ~ p(4)<
....
and P(n)(H @ K)
is an invariant subspace for
~3(U(n)) .
It follows that L
=
V n
P(n)(H @ K) =
2
is an invariant subspace for
f3(U(~))
,
It is now easy to see that the restriction of ~3 satisfies the conditions
(2)
and
Thus , the restriction of representation of
U(~)
~3
to
(3) 9 to
L
is a t3~pe II
factor
, quasi-equivalent to S ~ .
V.2.6. The problem which naturally arises in connection with these type
II~o factor representations of
U(~)
the tensor products of irreducible representations of sidered by
I.Segal (KsoS)
and
factor representations ([3~])
A.Kirillov ([~])
is to study U(oo)
con-
and of type
II~
APPENDIX
: IRREDUCIBLE REPRESENTATIONS OF U(n)
As we already mentioned (III.~.~.),
there is a bijection
between equivalence classes of irreducible representations of U(n) and decreasing
n-tuples m~
m2~
...~ mn
of integers (the "signatures"
of irreducible representations).
Below we shall describe for each signature m~
m2~
...~ mn
an irreducible representation of
U(n)
in the corresponding equi-
valence class. In order to avoid mixed tensors, we first consider only positive signatures (i.e.
mn~
0) and then we indicate a way
to obtain also representations for the remaining signatures,
star-
ting from the positive ones. For a positive signature m~ $ m 2 ~
~
~ mn~
0
consider the following diagram (Young's diagram)
:
Ill the rows of which have lenght
m~ , m 2 , ... , m n
insert in squares the numbers ~ , 2 , 3 , . . . , m }
respectively and
,m = m~ + m 2 + ... + m n ,
filling first the first column, then the second column and so on. For instance,
if n = 3 the signature
(4,3,~)
yields
:
156
r
Consider ~
,
2
,
S(m)
... , m I
14161
1
the group of permutations of the set
and
P , Q
the subgroups consisting of those
permutations which conserve the rows of the Young diagram , respectively its columns (horizontal and vertical permutations). Denote by ~(~) the sign of the permutation Let acts and
Hn ~
be the
~ ~ S(m).
n-dlmenslonal Hilbert space on which
be the natural representation of
U(n)
=
U(n)
on
. m-times
Consider also the representation ~
~[(~)Qj__~Jl
=
of
S(m)
on
j~_-1(j);~'~
and define the linear map
~m
~ Hn
R : ~mn
~ ~m
by
commutes with the ~(g) , g ~ U(n)
and
such that :
'
0- E S(m)
#
( p , c ! ) E P x (~ Then
R
invariant subspace for the ~(g) The restriction of cible representation of signature
is an
, g E U(n) .
~ to the subspace
U(n)
R ( ~ m)
R(~mn )
is an irredu-
in the class corresponding to the
(m~,m 2, ... ,mn) .
For a general signature
m~ >~ m 2 ~ let
~(m~,...,m~)
...~
mn
be a representation in the class of the signature
157
m~ ~ m ~ where
m~ = mj - m n
~ ... ~ m n' > 0
and then consider the representation
U(n) ~ g
r
~
(det(g))mnf(m~,...,m~)(g)
@
This is a representation in the class corresponding to (m~,m2,...,mn). Let us consider a few particular cases : a) Suppose Then the subspace
(m~,...,m n) = (m,O,...,O). R(~)
~ nm
of
is just the space of symmetric
tensors, i.e. the space of those
~nm
such that
for every b) Suppose
Then
R(~)
(m%,..~
That is,
R(~)
c) Suppose
~S(m)
.
= (~, .... ,~,0, ........ ,0) . k-times (n-k)-tlmes
is the subspace of these ~(~)~
~
= ~ (~)~
~ E ~
such that
for every
~ E S(k) .
is the space of antisymmetrlc tensers of degree k. (m~,...,m n) = (d,...,d) .
Then the corresponding representation is one-dimensional U(n) B g
.....
~
z
(det(g)) d .
A fundamental result concerning irreducible representations of
U(n) Let
is m~
the character formula . m 2 ~ ... ~ m n
be a signature. Then
158
m~+(n-~) z~
m~+(n-~) z2
9
m~+(n-~) zn
z~ 2+(n-2)
m2+ (n-2) z2
9
m2+(n-2) zn
mn z~
mn z2
.
Zn mn
m
(z i - zj) l<j is a polinomial in
z~ , ... , zn , z~ ~, ... , Zn~, we shall denote by
~(m~,...,mn)(Z~'''''Zn) Consider
~(m~,...,mn)
"
an irreducible representation of
U(n)
cor-
responding to the signature (m~, ...,mn). Then the character formula can be written as follows :
f(m~,...,mn)(g) where
~
, ... , ~ n
= ~(m~, ... ,mn)(t~' "" "'~n)
are the eigenvalues of
g ~ U(n)
Let us mention that the decomposition of the restriction of an irreducible representation of representations of
U(n)
U(n+~)
to
U(n)
into irreducible
can be easily obtained from the preceeding
fundamental formula. The corresponding result has already been recalled in Section III.~.i. Standard references for the preceding results are ~36S , [37] 9 This being an appendix, the authors apologize for having to omit the natural justification of the correspondence between irre-
159
ducible representations and signatures (highest weights of irreducible representations of reductive Lie algebras).
160
NOTATION INDEX , S , ~ , ~ are respectively the set of positive integers, the set of all integers, the set of real numbers, the set of complex numbers. H
denotes a separable Hilbert space with a fixed orthonormal
basis {en~ and scalar product
(-I.). L(H) is the algebra of all
bounded linear operators on H. For a compact space i-h , C(i9_) is the algebra of all complex continuous functions on iO_ . For a locally compact group G, M(G) is the convolution algebra of all bounded complex regular Borel measures on G and L{(G) is the ideal of absolutely continuous measures with respect to the Haar measure on G. The convolution is denoted either by "*",or simply by juxtaposition if no confusion arises. ~= stands for the Dirac measure concentrated at the point g 6 G. The notations U(n) , U(oo) , U~(oo) , U(H) are explained in the introduction.
CHAPTER I
; GL(n) , GL(oo)
page
, GL(H)
page
An
Z
~n
49
A
3
[-~
15
Cn
3
~-n
19
0
6
~
16
Pn
6
Pn+k/n
13
P
6
PCO/n
13
II n
9
A(~l,~ )
17
II
lo
~2(fl)
17
un
~
Tf
(f~ c(fl))
17
u
16
v~
(t ~
17
f~
(fE C(~-),~ "~[~)
~u ~ P ~(t)
53
~0
32
i~
~
32
(t6]'[)26
~
32
(u ~ %t) , C(t)
C )
161
q(n)
(t e l'~.)
19
~'[~
32
Ij , J(I)
21
J~
33
lu.~, cO I
25
f~
51
Pn~
40
P~"
40-4}
u (n) •
i6
> I.m.(M~,M2,...)
Prlm(A)
CHAPTER II
i
M'
2
i
e .I .m. (M~[,M 2 , ...)
1
A ~-- B
3
so
page
page
G n , Ga)
57
~n { f m
63
L = L(G(D )
57
A = A(Goo )
62
[fn+~ "fn] s(f=)
63 69
_~ =~'~(GoD )
79
71
V = V'(G~ )
79
Perm S(fn) A n
M
57
Xn = A n
L(n ) , L (n)
57-58
S-If
58-59.
, T[~n
~n
63
64 64,67
B~j
64
p(n) j q(n)
65
fj
65
Vg,U~ ((r~Perm S(~n))71 d~n
63
~n(t)
(t ~ 9_)
~n
63
kn(t)
(t ~ ~ )
Pfn
63
no(t)
(t ~ _O.)
68
P(gnk--) fn+~)
P
k~
(fo {Eg,jg}~,~ g S(jOn) 70-7i
[ j
'
75:6 75-76 75-76
~kn~ ~n ) >
"'"
je~
69
162
Note. - In Chapter II the convolution (see p. 63).
is denoted by juxtaposition
For any element x ~ L, we denote by the same symbol its canonical image in the envelopping C~-algebra A = ~ (see p. 65 ). -
- no(t) is a positive integer or the symbol +co, depending on t gi~- , which indicates how many groups G are involved in the n concrete description of t g / 9 - (see p. 75-76).
CHAPTER
IIl
A = A(U(co))
page 81
page Lj(t),Mj(t)
(t s
n
84
})
p =p(U(oo))
CHAPTER IV
83
(for ~ g Prim(A(U(oo)))
page
page
oo
97
A(m), ...,F(m)
103
[~
9s
A(~), ...,FO,)
1o6
~n,O-
98
p , p'
108
rn, uo
98
Hn
113
~n ' ~
98
Tn
113
98
A~
:13
A
113
99
Yn t X n
113
ko
lOO
en , ~ n
ll4
Ok(J,h)
:oo
~n ' ~n
1:4
~
::4
p(O) , p(~n) !
Dk , Dk n , r , N
ioo
i = p(O),~ = :_p(O) :o2
J
,:
Note.
:::
- ~he n o :
II " ~L (ll::(z)IU is d e f i n e d
~A
:22
on page :o:.
C km stands for the binomial coefficient. - The sign /~ stands for the exterior product (see p. 113 ).
-
163
CHAPTER V
page
page
S(m)
z27
~K
13~
S( CO )
130
~
142
Hn
127
~oc
142
T~ ~ nm ' V nm ~ N mn
~c~) F.K ,
~,',
M K i, N ~
127
~
i,~
147
128
L~O~.,u..)~
148
~
~ ~sso. ~o {O~m ~}
~
129
~ asso. to
148
13o
y~
is2
131
~2
152
133
/__'(|)
c(~ )
152
i~6
Note. ~ In Chapter V w 2 we replace an irreducible representation ~n E U(n) by its signature (m~,...,m n) in notations such as
~fn ' dfn ' P~n ' B & - The sign ~
stands for the tensor product.
184 SUBJECT INDEX
I). The following terms are used with their usual meaning, as in the monograph of J. Di~nier ([7]) : Approximate unit C~-algebra Center of an algebra Central state Character of a representation Commutant Conjugate representation Cyclic vector Dimension of a representation Dual of a compact group Envelopping C*-algebra Equivalence of projections Factor Factor representation Faithful state Faithful trace Function of positive type Gelfand .spectrum of a commutative C*-algebra Gelfaud-Naimark-Segal (GNS) construction Induced von Neumann algebra Involutive Banach algebra Irreducible representation ,-Isomorphism Kaplansky density theorem Multiplicity yon Neumannalgebra
von Neumann density theorem Normal isomorphism Normal state Normal trace Peter-Weyl theorem Primitive ideal Primitive spectrum Reduced von Neumann algebra Regular representation ,-Representation Semifinlte trace Separating vector Strong (operator) topology State Tensor product of representations Types of von Neumann algebras : - I n , Ioo , II~
, IIco , I I I
- discrete / continuous semifinite / purely infinite finite / properly infinite Types of representations Uniformly hyperfinite (UHF) algebra Ultrastrong topology Ultraweak topology Unitary equivalence of representations Unitary representation Weak (operator) topology
Not___~e. A unitary representation of a topological group is assumed to be continuous and its type is defined to be the type of the generated yon Neumenn algebra.
165
II). The following terms are defined and/or explained in the present work : AF-algebra (approximately finite-dlmensional C*-algebra) Approximate unit Central state C ovariant representation Conditional expectation Equivalent measures Ergodic measure GNS (Gelfand-Naimark-Segal) construction Invariant measure Irreducible subset Krieger construction Lower signature of ~ ~ Prim(A(U(oo))
M~(~) ~ M2(~) ~ . . .
~ M~(~) ~ . . .
m.a.s.a. (maximal abelian subalgebra) Non-measurable group of transformations Positive measure Powers-Bratteli theorem Primitive ideal Primitive spectrum of a C*-algebrs Probab il ity measure Projection of norm one Quasi-invariant measure Signature of an irreducible representation of U(n) System of m a t r : L x units Topological dynamical system associated to an AF-algebra Upper signature of ~ r Prim(A(U(co )) L~(~) ~< L2(~) ~< ....< Ln(~) .< ... Young diagram
page 2 59 33 53 2 32 32 32 32 26 55
92 2 48 32 48,49 25 3o B2 2 32 155 14
16 92 155
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