Lecture Notes in Mathematics Edited by ,~ Dold and B. Eckmann
486 ~erban Str&til& Dan Voiculescu
Representations of AF...
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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
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. ~ by Springer-Verlag Berlin - Heidelberg 1975 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
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
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 .
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
(Xk10
.
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 ~~(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{
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 , fn2 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 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=kh
holds with
t
=
i ) ~ ~ i_< k
~ ~k~ 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+[
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+~ ,
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
=~
)z) -
: 0
=
Proof of Lemma ~ . There is
n o ~ IN such that for
n ~ no
we have
0
such that
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
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
imo} ~ n
j(n)~
i
=
mo E ~ >
and an injective map
i(n) ~ ~
such that ~
l 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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