Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
731 Yoshiomi Nakagami Masamichi Takesaki
Duality for Crossed Products of von Neumann Algebras
~{~
.~_ Springer-Verlag Berlin Heidelberg New York 19 7 9
Authors Yoshiomi Nakagami Department of Mathematics Yokohama City University Yokohama Japan Masamichi Takesaki Department of Mathematics University of California Los Angeles, CA 9 0 0 2 4 U.S.A.
AMS Subject Classifications (1970): 46 L10 ISBN 3 - 5 4 0 - 0 9 5 2 2 - 5 Springer-Verlag Berlin Heidelberg NewYork ISBN 0 - 3 8 7 - 0 9 5 2 2 - 5 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging ~nPublicationData Nakagami,Yoshiomi,1940Dualityfor crossed products of yon Neumannalgebras. (Lecture notes in mathematics: 731) Bibliography: p. includes index. 1. Von Neumannalgebras--Crossedproducts. 2. Dualitytheory (Mathematics) I. Takesaki,Masamichi,1933- 11.Title. III. Senes: Lecture notes in Mathematics(Berhn) 731. OA3.L28 no. 731 [QA326] 510'.8s [512'.55] 79-17038 ISBN 0-387-09522-5 Thts work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publishe~ @ by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
INTRODUCT ION
The recent develol~uent in the theory of operator algebras showed the importance of the study of automorphism groups of yon Neumann algebras and their crossed products.
The main tool here is duality theory for locally compact groups. Let
•
be a yon Neumann algebra equipped with a continuous action
locally compact group
G.
For a unitary representation
be the ~-weakly closed subspace of ators ~(U
T
from
® V)
• ~(~)
~U
into
for any pair
where
~
~.
~
of
G~
2
of a
let
~G(U)
spanned by the range of all intertwining oper-
It is easily seen that
U~V
~U~u)
~(U)~(V)
of unitary representations of
means the cOnjugate representation of
basis for the entire duality mechanism.
U.
is contained in
G~
and that
~(U)*
=
This simple fact is the
At this point~ one ~hould recall the form-
ulation of the Tannaka-Tatsuuma duality theorem. In spite of the above simple basis~ the absence of the dual group in the noncommutative case forces us to employ the notationally (if not mathematically) complicated Hopf-von Neumann algebra approach to the duality principle.
It should
however be pointed out that the Hopf - yon Neumann algebra approach simply means a systematic usage of the unitary This operator tion table.
WG
In this sense~
be overestimated. that a non-zero
WG
on
L2(G × G)
given by
(WG~)(s~t) = ~(s~ts).
is nothing else but the operator version of the group multiplicaWG
is a very natural object whose importance can not
For example~ the Tannaka-Tatsuuma duality theorem simply asserts x ~ £(L2(G))
is of the form
regular representation~ if and only if
x = p(t),
where
p
is the right
W~(x ® 1)W G = x @ x.
When the crossed product of an operator algebra was introduced by Turumaru~ [76]~ Suzuki~ CE]]~ Nakamura-Takeda, Zeller-Meier~
[51,52]~ Doplicher-Kastler-Robinson~
[20] and
[79]~ it was considered as a method to construct a new algebra from a
given ccvariant system~ although Doplicher-Kastler-Robinson's work was directed more toward
the construction of covariant representations.
Thus it was hoped to add more
new examples as it was the case for Murray and yon Neumann in the group measure space construction.
In the course of the structure analysis of factors of type III, it
was recognized [12] that the study of crossed products is indeed the study of a special class of perturbations of an action 1-cocycles. point algebra = G
~
on
More precisely~ the crossed product ~
in the von Ne~m~ann algebra
® Ad(k(s)),
where
k
~
~ x
by means of integrable G
is precisely the fixed
~ = ~ ~ £(L2(G))
under the new action
is the left regular representation.
With this obsers s vation, Connes and Takesaki viewed the theory of crossed products as the study of the perturbed action by the regular representation~
[14]; thus they proposed the
comparison theory of 1-cocycles as a special application of the Murray - yon Neumann dimension theory for yon Netmm~_u algebras. In this setting, the duality principle for non-commutative groups comes into play in a natural fashion as pointed out above. abelian.
If
~
is a "good" action of
G
on
Suppose at the moment that ~,
so that for each
p c G
G
is
one can
iV
chOos~ a unitary
u
such that
~s(U) = (s,p)u,
p
generate
If we drop the commutativity assumption
~.
on the fixed point algebra
then this unitary
an action of
replaced by something else. together with the
ated by
P(G),
A(G)
,
~G
T~
and
~
g~ven by
u's
from
G,
u
gives rise to
together with then
@
~G(f)(s,t)
= ~st);
~
Q(G)
action of the "dual" will be given as a co-action action of a ring of representations.
and 6
~. of
G
Neumarm algebra
formulated
Theorems 1.2.5 and 1.2.7, are proved there.
the integrability
of 1-cocycles.
The equivalence
tween closed normal subgroups
of
is established in Chapter VII.
of our theory, G
theory
the Galois type correspondence
and certain yon Neumann algebras
We must point out that the restriction
should be lifted through an application
Banach *-algebra bundles,
spectral
of an action, dominant actions and the comparison
As an application
ality for subgroups
in Chapter
in ~4 in Chapter IV.
In this paper, we present the dualized version of the Arveson-Connes analysis,
acts
as well as a Roberts
The crossed product of a v o n
of co-actions and Roberts actions is established
G
The precise meaning of an
by an action of the "dual", a co-action and a Roberts action~is IVand duality theorems~
gener-
At any
is "good", then the "dual" of G
L~=(G)
the second
the predual of the von Neumann algebra
is generated by the "dual" of
~.~
should be
One is the algebra
[ 27,2~ ; the third is the ring of unitary representations.
rate, it will be shown that if the action on
or these
There are a few candidates.
co-mul~i~.[c~tion
is the Fourier algebra
T~;
[30]~ and its dualized version.
be-
containing
~
of the norm-
of Fell's theory of We shall treat this some-
where else. The present notes have grown out of an attempt to give an expository unified account of the present stage of the theory of crossed products for the International Conference
on C*-algebras
Marseille,
June 1977.
is particularly
and their Applications
In theoretical
to Theoretical
physics, the analysis
Physics,
CNRS,
of the fixed point algebra
relevant to the theory of gauge groups and/or the reconstruction
the field algebra out of the observable algebra.
In this respect,
of
the material pre-
sented in Chapter VII as well as those related to Roberts actions are relevant for the reader motivated by physics. cerning C*-algebras area.
It should, however,
is more needed in theoretical
be mentioned that a theory con-
physics.
It is indeed a very active
The authors hope that the present notes will set a platform for the further
develol~nent. The present notes are written in expository style, while Chapters III, IV and V are partially new.
The references are cited at the end of each section.
The authors would like to express their sincere gratitude to Prof. D. Kastler and his colleagues at CNRS, Marseille, while this work was prepared.
for their warm hospitality extended to them
COIYl E E T S
Chapter
i. A c t i o n ,
co-action
§ I. D u a ! i L y
(Abel Jan ease) . . . . . . . . . . . . . . . . . . . . . . .
2
§ 2. 'DuaiiLy for c r o s s e d
pz'oducLs
(General
case) . . . . . . . . . . . . . . . . . . . . . . .
4
- Tatsuuma
duality .....................
14
action
§ i. S u p p l e m e n t a r y
IT. E l e m e n t a r y
and T a n n a k a
formulas ............................................
!crope:'ties of c r o s s e d
§ i. Fixed p o i n z s
30
§ 3.
integrabil[ty
an(l d o m i n a n c e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
valued weights ...........................................
B6
and o_Derai~or w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
:n~ef4rab,e a c t ! o n s
§ 4. Domi_ua~ t a c t i o n s
}.7
and c o - a c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
and co-:~,ctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
IV. Spec tr Lain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § I. The C o n n e s
spectr'±m oF co-act, ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6? 6~
§ 2. Spec~,rt~v of acZ'oi:s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
§ 3. The c e n t e r of a c r o s s e d
{5
§ L. C o - a c n i o n s
V. P e r t u r b a t i o n
§ 2. D o m i n a n x § j. A c t i o n
V].
product
and F,~) . . . . . . . . . . . . . . . . . . . . . . . . . .
and [{obert ae~,ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§ 1. C o m p a r i s o n
Chapter
21 24
§ 2. i n t e g r a b i l J t y
Chapter
20
p,~oducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§ I. O p e r a t o r
of c r o s s e d
19
§ i{. Com,mutan~,s o f c r o s s e d p_~'oaucT,s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C hap L e r I17.
Chapt er
i:roducvs . . . . . . . . . . . . . . . . . . . . . . . . .
i_n crosse,'i F:'oduets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§ ]'. Charac%er-za-$ion
Cha.nt~r
I
products
§ 3. R o b e r t s
C h a p t er
ann d u a l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
for c r o s s e d
of act,ions and c o - a c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . of' l - c o c y c l e s
of a c t i o n
and c o - a c t i o n . . . . . . . . . . . . . . . . . .
l-cocycles ...............................................
of G on ~ h e c o h o m o ! o g y
,'8
88 -~9 92
space ...............................
97
101
Relative
eommutan~
of c r o s s e d p r o d u c t s . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§ I. R e l a t i v e
co~nutant
~heorem ........................................
102
§ 2. Stabil.ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
Vll. A p p l i c a u i o n s
theory .....................................
111
and c r o s s e d product, s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
§ 1. S u u g r o u p s
§ 2. SubaJ.gebras
to G a ] o i s
in c r o s s e d ioroducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
§ 3. O a l o J s
correspondences ............................................
119
§ 4. G a l o i s
correspondences
125
(I]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix .......................................................................
129
References .....................................................................
136
L I S T OF SYMBOLS
IN = The set of n a t u r a l numbers,
[1,2,...
}.
= The ring of integers. Q
= The rational number field.
~R = The real number field. C
= The c o m p l e x n u m b e r field.
~+~SR+:
The non-negative
~,2,
...
:
~,~,
... :
Hilbert
parts.
spaces.
Subspaees.
~Ti,~,
... :
Vectors in a Hilbert space.
al~S2~
...
Vectors in a fixed orthogonal n o ~ n a l i z e d basis.
G :
:
A l o c a l l y c~mpact group.
L2(G)
= The Hilbert
space of a l l square integrable
right invariant Haar measure L~(G)
ds
= The a b e l i a n yon N e u m a n n a l g e b r a
ca
functions w i t h respect to a
G.
of a l l e s s e n t i a l l y b o u n d e d functions
w i t h respect to the Haar m e a s u r e acting on £(~)~£(~)~ ~,~,P,~,
...
:
... :
~,C~
:
von N e t m m n n algebras.
:
e,f,...,p,q,
The center of ...
:
Unless otherwise
stated,
l~, acts on
9, and
2.
A b e l i a n von N e u m a n n
....
on
b y multiplication.
The algebra of all bounded operators.
acts on G,8, ....
L2(G)
algebras.
~
and
h
respectively.
~ V ~ = (~ U ~)"-
Projections.
Aut(~)
= The group of automorphisms
(*-preserving)
of
Aut(~/~)
= The group of automorphisms
of
the vo~ Ne1~nann subalgebra
of
•
pointwise
•
leaving
fixed.
= The i d e n t i t y automorphism. = The symmetry reflection: T~Tr =
Traces.
w~,~,
e ~ G
x @ y -~ y @ x.
...
:
r,s,t~
:
Linear functionals,
states or weights.
The unit. ...
:
Elements in
G.
The right regular representation
of
G.
0(') ~(.)
= =
The left regular representation
pt(x)
=
p(t)
x p(t)*
for each
x ~ £(L2(G)),
t c G.
kt(x)
=
k(t)
x X(t)*
for each
x e £(L2(G)),
t e G.
of
G.
f * g(t) = / f ( t s - 1 ) g ( s ) d s V
G f~(t)
= f(t -I)
;
;
~.
VII
fb(t) = A(t)f(t -I) ; f~(t) = f(t -I) ,
f~(t) = A(t)f(t -I) •
~(G) = [p(t) : t ~ G}" '(G) = IX(t) : t ~ G}" = ~(G)' A(.) = The modular function of A(G):
The Fourier algebra of (g~.
~,5,~
f)(t), t c G,
..- : Actions of
,%,~, ...
G
: Co-actions of G
G. G, which is identified with
i.e.
on a v o n G
on
R(G).
Neumann algebra:
The action of
cz6 :
The isomorphismof
%G :
The co-action of
G
on
,%(G) with
,%~ :
The co-action of
G
on
,~(G)' with respect to
L'~'(G) with
L~(G)into
(~G)t = Pt; (~G f)(s't) = f(st).
L~(G)@L"~(G)
k, ~ G, ) t = X t-i : (,q~f)(s~t) = f(ts).
with
,%G(a(t)) = 0(t) @ p(t).
VG~(s,t ) = ~(st,t),WG~(s,t ) = ~(s,ts) V ~( s, t)
C~'~'~...
:
= A(t)~'~(t-!s,t),W~(s,t) The actions of
G
9,(G)'
such that
on a v o n
:
The co-actions of
ge:
The action of
G
on
The co-action of
G
G
~ L2(G x G) .
Netunann algebra with respect to
on a yon Neumann algebra
~P'e with on
defined by:
~e
~'(G):
oct'.
( ~ ' ~ ) ° ~, = ( ~ ) ~$=[x~,~:~(x) = x ~ l } ; U ~ = [ ~ e:
5~(k(t)) =
;
= A(s)~(s,s-lt),~
(~' ~ ~) oa' = ( ~ % ) 8",g'~...
=
( 5 ~ g ) o ?-=(&gBC} ) ° .P,.
k(t) @ X(t), ef. Chapter I, ~ ;~. 9, VG,WG,V6_,W ~ : The tmitaries on L [ G × G) = L2(G) @ L2(G)
L
~f,g(0(t))
( C ~ & ) o.,~=(g~C~G) o,~.
on a yon Net~nann algebra:
~G :
~
by
A(G) = L2(G) ~ * L2(G).
with respect to
~'(G):
o ~, ~:~,(~) = ~ , l } . ,~e(xe) = c~(X)e@l
with
for
e£.T~.
~;e(ye) = ?'~(Y)e@l for
e e :.d.
ze W n.~ = {x ~ ~, : o(x*x) < ~o} ; m~ = n~n o.
f~.:,%}
or
[.,,0,%,,;~}:
~he
a~S
cc~struet~on:
(,~.,~.(x)¢(~)lqo(Z))
= (~lz)¢.
=
C~(z*xy), x £ ~, y,z c nCl). Wr(resp- w~):
The right (resp. left) representation
of a right (resp. left) Hilbert
algebra. A
= The modular operator.
J~ = The modular tmitary involution. ~O = The modular automorphism. ×
G = The crossed product of
~
by
G
with respect to
~.
×~.G = The crossed product of
~
by
G
with respect to
~'.
×6 G = The crossed product of
h
by
G
with respect to
5.
xs,G = The crossed product of
~
by
G
with respect to
5'.
a = The dual of
5:
~(y) = Adl@w~(y ~ i)
for
y e ~ ×
G.
VIII
~' = The dual of
~ = The d u a l
~':
of
5:
5' = The dual of
~'(y) = Adl~w~(y ~ i)
~)(x) = A d l ~ v ~ ( X ® 1)
5':
o ~
and
for
,~'(x) = Adl~v~(X ~ l)
~ = (L @ q) ° (~ @ ~,) and
= Adl~v~
for
y ~ • × , G.
x ,. ~ x~ G. for
x ~ TI x~ ,G.
~t : Ct v a,
(~t = O~t .2 k t -
= Adl@WG ° ~ . p = The action of
G
on
~3 = The co-action of
G
.~(,~.)' with respect to on
C(G) = The set of continuous functions on
G.
C (G) = The set of continuous functions in Y(G) = The set of continuous functions in supp(c):
The support of which
v . ~LG,pG.
~
~ e A(G),
d't
L~(G)
defined by:
@(G), ,~ T (G)
9G(P(f)) = f(e) ;
K,J:
;
#•~(f) =/'f(t)d't
is the left invariant Haar measure
The wights on
where
vanishing at ~. wanishing outside a compact set.
which is the closure of the smallest set outside
~G(f) =,j~f(t)dt where
C(G) C(G)
vanishes.
The weights on
I ,,I. ~G,~G.
Q '(G) : f~(x) = Adl~v~(X ~ i).
5(h)' : a(y) = Adl~4G(y ~ i).
A(G)~
~(t)dt.
defined by:
@~(k(f)) = f<e)
for
A(G)+ ,
is the set of positive definite functions in
The operators on
L2(G)
defined by: I.
l'
(Kf)(t) = A(t)2f(t -I) ; G:
(Jf)(t) = A(t)Nf(t -I) .
The dual group of an abelian locally compact group equivalence on
~2
classes of irreducible
(continuous)
G
: Unitary representations
[W,~w} = The unitary representation
or the set of(unitary)
End(~) = The set of endomorphisms [P,~I] = The Roberts actions of
of
G
G
of
G,
on
~w"
[~
}.
p~q~
... ~ G.
(Definition 1.3.1).
~. (Definition 1.3.2).
~G(Wl,W2) = The set of intertwiners
of
~G(Pl~P2) = The set of intertwiners
of
= The set of all Hilbert spaces
D~ = The endomorphism of
of
conjugate to
of
~,
corresponding to v
= The ring of unitary representations
~(~)
G
unitary representations
spaces.
X p '.X q ,... : Normalized characters of ~,~w],
A(G).
Wl Pl ~
~ : p~(a)x = xa
and
w2
in
~.
and
P2
in
End(~).
in for
~
such that x ~ ~
and
~t(~) = ~. for all a ~ h.
t.
IX
"~ × Q = The crossed product of ~I by with respect to {p,ll] , (DefinitionlV.4.3). P gc~ = The .~-valued weight on ~, given by 0 there exists I((Yj -xj)~j I .~lj)I< t for
and
Since
fc(s)N~(t) =
is a strongly dense x. a £ such that j o
*
sub-
il((yj -xj) ~ i)(i ~ WG)(~ j @ fo)ll < .a
j = I,,2. Therefore
]((SCK(Y j) -yj)~j1~]j)[ < I(Sq~K(Yj -xj)~jlT, j)I + l((:~qjK(Xj) -xj)~j ['ij)] + I((xj
-yj)~j1-~j)1
< a,jh] ,ll]j.!+ I((~ .K(Xi) -xj)~j I-',j)1 + ~ . Thus it remains to show t~mt the second term converges to Adl~ W (z ° @ i) = z ° ® ~(r)
for
of th~ form wn~.k=lZk
zk = Yk ~ fkP(rk)'
with
z ° = y ~ f~(r),
0.
Since
5(Zo) =
it follows that, for any
z e £o
n
(6(~K(Z)gJ I 'i J)
k=l
(zk~j ] T,j)~K(~k) ~ (z~j 1',j) •
Q.E.D.
11
Lemma P.9.
(i)
If
&(x) = Ad
~} (x @ i) l@W G
for
x { £(.9 ~ L2(G)),
then
x e [5 (x), :~ *. A(G) ,~ }((O)]" •
(ii)
y~'a
If
%
is a co-action of
~(~)
and
n~(~)
G
on
~efine~by
"a,
ther: the set of all
0
~l. Let
~, ,~ 6 ~
there exists a
I((~(x)
by Len~na 2.8. S > 0
If
z
and
f, g ~ ~(G).
.~, £ A(G) q ~(G)
-x)({
commutes with
For any
such that
,F f) lz*(~, ~ ~))1 < ~
~ (x) ~D
for all
~ .~ A(G) N }{(G), then for any
we have
which implies (ii)
xz = zx.
If
y ~ N,
then
$(y) ~ h ~ [{(G). Since
( ( L ® %G) ( t ) ( y ) ) ,
'.o @ ,$) : /(L, :9 ?'G)(~'(y))
,
it follows that
~'.,(y) ~ { ',,)(?(y)) : ~ ~ A(G) n ~(G) ]"
Q.E.D
Len~na 2.10. ~%-@ £(L2(G)) = ~(~)' V (C ® g(L2(G))). Proof.
It is clear that
8(~) V (C ® £(L2(G))) c ~
the reversed inclusion, we shall show that [ ~ (C @ £(L2(G))) ' = 8(U)' N (~(~) (9 C),
where
~ £(L2(G)).
To conclude
(L2(G))) ' = ~' '~ C ~ ~(n) ~ is the Halbert space on which
acts. For each
If
~ ~ A(G), set
x ® i e 5(~)' O (£(R) ® C),
~ A(G)~
then we have, for any
y e ~, e e £(q).
and
12
(~(y),~) : (~ (y),~) : (~(~),~x® ~> = ((x ® 1 ) 5 ( y ) , ~ ®
in
%
h;
%(y)x.
(y) =
hence
~) = ( 5 ( y ) , x ~ ®
~)
(~(y),x~) = (~(y)~,~) ,
~.
so that
~) = (5(y)(x ® 1 ) , m ®
But by Lemma 2. 9.li,
kY): Y e h,~ e A(G)}
is total
x g N'.
Q.E.D.
Proof of Theorem 2.7. (2.19)
Let
K
denote the unitary on
L2(G)
defined by
K~(s) = A(s)i/2g(s -I) •
We have then Kp(s)K = k(s)
We define a map (2.20)
If
~
of
,
h ~ £(L2(G))
YO,(s)K = p ( s )
into
.
~ ~ £(L2(G)) @ £(L2(G)) as follows:
~ x ) = (1 @ 1 ® K)(1 ® WG)($, ® l.)(x)(l @ Wg)(l ® 1 ® K)
y e ~
and
x = 8(y),
then
(8 ® L ) ( x ) = (~ ® 5G) ° 5 ( y )
by ( 2 . 5 ) ,
hence (2.2)
entails that (2.21)
~(5(y)) = (i ® i ® K)(5(y) ® 1)(l ® i ® K) = 5(y) ~ 1 .
Then3 by
dfrect computation, we have rT(I ® f) = i ® X(f) ,
(2.22)
By the previous lemma, C ® g'(G); thus (N ×5 G) x~
G
~
~ ~ g(L 2 (G))
~(1 ® X(r)) = 1 ® 1 ® 0(r) .
is generated by
maps the generators of
5(U), C ® L~(G)
N ~ £(L2(G))
~nd
onto those of
by Proposition 2.4.ii.
It remains to be shown that
5 • w = (~ ®
L) ° ~.
From (2.8), it follows
that o ~(x)=~(x)~l,
~
(~×~G) ;
and trivially ~(1 ® 1 ~ s(r)) = 1 ® 1 ® P(r) ® o(r) . On the other hand, we have, by direct computations
[
~(5(y)) = 8(y) e 1 ,
J
(2.27)
g(lef)=lef~l
Y~h;
,
- g(l ® k(r)) = i @ k(r) ® o(r)
Thus,
~
intertwines
~
and
~.
f ~ T?(G) ; ,
r C G .
Q.E.D.
~3
NOTES
The definition of a co-action, Definition 2.3, and the construction of the crossed product
N ×~ G
were given independently by Landstad [42,43], N a k a ~ m i
46] and Str~tila-Voiculescu - Zsid6 [59~60].
Dual co-actions and d ~ l
Proposition 2.4, were introduced independently in [,13, 6,~0] theorems for crossed p r ~ u c t s ,
Lemma 2.10.
and the duality
Theorems 2. 5 and 2.7, were proved there.
presented here are taken from [k71 which resembles [60].
[h5,
actions,
The ~ y
Here we take an idea due to Van Hceswijck to [77].
The proofs
to the proof is in On the other hand
Landstad, [42], prepared Theorem II.2.1.(ii) in order to prove Theorem 2.7. results of this section were generalized to the Eac algebra context [22,26].
Various
14
~3-
Roberts action and Tannaka-Tatsuuma duality. In this section~ we shall discuss the duality for the "automorphism actions" of
a locally compact group on a yon Neumann algebra through a formalism given by Roberts. In order to avoid unnecessary complications~ we consider only compact group~ in this section~ while this restriction can be lifted ~,~ithout serious difficulties if one really needs to do so. Definition 3.1. rin~ if
i) Wl e ~
A collection c W
trivial representation of each
.~. e Z
For each
Let
End(~)
We leave the general ease to the reader.
and ~ ~
~i' '~2 ~ ~'
of unitary representations of
Zl ® ~2 c ~
of
falls in
e
G
for every pair
belongs to
R.
-i,~2 c Z;
is called a ii)
The
If the conjugate representation
again, then the ring
9
is said to be self-ad~oint.
we denote
be the set of all *-endomorphisms of
anq the idc-nt.'Ly prcsurv'ng for endomorphisms. (3.2)
G
~.
Here we assume the normality
For each
el, P2 e End(~)~
o~(p2,.%) = [a e ~ : aPl(y ) = o~(y)a , y e ~
we write
.
We then have the following relations among these sets:
OgG(I"r3,TT2)~G(I~2,~I ) c JG(~3,,.~l)
~G(~2,~l ) ® ~a(~,~i)
(3.3)
I °0~(~2'°11)~ L~(~,Ol
;
= Ja(~ ~ ~.S,,.-1 ® ~i) ;
~(~ " °~'~." ~i) ;
) ~ ~a(~ .o,o I
o),
and
(oe,ol)~l(~(oe,Ol))
c.~a(~_
~e'~z
°z) ;
(~.~) L,~2(J~ ( P2 ,oI))~(P2,, Pl, ) Definition 3.2. on
N
is a
A Roberts action
is a composition
=
~~(o 2, °
{p,O
[0 ,-iWl,W2 : ~ )~ l ,~ ~
o-weakly continuous linear map of
o2'P 1,
°Ol )
of a ring (I 9 ] ,
~G!Wl,~2)
where into
~
of representations of O p~ c-.End(N)
and
:i~l,~2
~(D,~I, DW))~ such that
15
i) i~)
for every
°~'l@ ~2 = °~i ° P~2 '
•
a e ~%(~_p.,~l) iii)
a' e gC(n2,T ~ Tl ) ~
and
,,~l.~(a)*= n 2,~a~).
v) for e v e r y
,_
.(i) = i ;
,
iv)
-%£~,%(a'))
,(a ~ a') ~ ~l..~,.l(a) %1
h~2Z '~~ , ~l~ w I
a~ Ja(~i,~2) ;
",_,.(ah _(b) =,,~i, (ab) i "~ ~, '% %
a e ~(~l,~_o)
and
b e ~(q2,~3).
Before giving an important example of Roberts
action, we need a few prepara-
tions concerning Hilbert spaces in a yon Neumann algebra. Definition space
~
of
~
3.3-
A Hilbert space i_~n a yon Neumann algebra
i)
For every
x,y e ~, ~×x
x ii)
and
y*x
~
as the inner product
aR ~ {0}
whenever
of a unitary.
algebra is not interesting. algebras. A noz~alized tern {~i : i ~ l , . . . , d ] is chosen, the map:
of
a ~ 0, a e ~ . ~
with norm one is an isometry.
is finite, then every Hilbert space in
the scalar multiples
(x~y)
y;
It is easy to see that every element of if
is a closed sub-
is a scalar multiple of the identity;
hence one can consider
Hence,
~
with the following properties:
~
is one dimensional and
So a Hilbert space in a finite yon Neumann
Thus, we must consider properly infinite yon Nc~nann
orthogonal basis of a Hilbert
space
@
in
N
is then s sys-
of isometrics with orthogonal ranges and ~ = i ~i~i = 1 . Once it d < x ::~ ~ i = l Uxju : -. i is an endcmorphism of ~ and does not depend
on the choice of a basis; hence we denote it by
p~.
One can characterize
p9
by
the equality:
(3.5)
.o~(a)~ = x a ,
x ~ ~,
It is easy to check that for Hilbert spaces closure of the linear subspace spanned by
a ~n
~l
x~×, x e ~l
and and
~2
in
~,
Y e ~2'
the is
G-weak
16
economically identified with
pR(U) =
n h;
£(~)'
£(~,2,~j_); hence o~ ( ~ ) ~ ( ~ )
n=
An important feature of Hilbert spaces in Hilbert spaces in y e ~,,.
~
is that for any pair
h, the closed subspace spanned by the products
is that the product
Let
Moreover, we have
.
is naturally identified with the tensor product
abstract.
~(~)
£(R2,RI) ~ ~.
hence
@l,R2
xy~ x g RI
~i ® R2"
RIR ,, is a concrete object sitting in
~
[~;,G,~}
be a covariant system with
~
while
R
[~t : t e G}.
If
~ e %(~),
~
is
We denote by
globally invariant under
then we have, x,y e R,
(~t(x)'~t(y))
Hence the restriction of
@ R2 1
properly infinite. ~
and
Here the point
In the following situation, this point becomes clearer.
the collection of all Hilbert spaces in
of
to
~
=
~t(Y)'X~t(X) : Gt(~'~X)
=
~t((x!y)l) = (xly) .
is a unitary representation {~,~}.
Then
%(~)
G
on
denote this representation by
~
tion of representations
which is~ in turn~ a ring in the sense that
of G
or
of
~.
We
turns out to be a co3~Lee
(3.6) where
wI
and
are isometries in
w2
It is not hard to see that
~
with
WlW ~ + w2w ~ = i.
0 R, R e %(,T.), leaves
~
globally invariant, and
also by (3.5) that
~G(O~R2,otR1)
C A
a,(p~ ,o R ) 2
C
'i
m°t .
We then set
I
Pa,R(x) = p~(X) , x
(3.7)
%t%,~,~l(a)
~ ~,~, ~ ~
%/m.) •
= a , a ~ oga(~ ,O~l),,.
RI, %. ~ ~(r,O
A straightforward calculation shows that
,
a Roberts
~°~R' I~2'~RI
action of
~(~)
on
D~.
~
•
: ~,~i,~2 e % ( ~ ) }
is indeed
We now have the following Tannaka duality theorem in our context: Theorem tion of
3.4. Assume that
[~,~}
G
is compact.
If every irreducible subrepresenta-
is equivalent to some representation
in
~(~),
then each
17
e Aut(~./~~) form
dr
lesving every member
for some
~ e ~(~)
globally invariant must be of the
r e G. 5)
It should be pointed out that the above theorem can be generalized to a locally compact group, if we assume that representation of
G.
~(~)
contains a member equivalent to the regular
Thus, the Tatsuuma duality theorem in our context remains
valid also. Proof.
For each
a,b ~ ~
~ ~ ~ (~)
we set
q,b(t) = a*~t(b) , and denote by C (G) is a
the set of all such functions.
* - subalgebra of
C(G), because
Now, we define a map
U
of
~ (~)
C (G)
It is easy to check that
C (G)
is a self-adjoint ring.
into itself by
Uq,b = f(a),b which is well defined because dt
ll'~fa,b,l~ = ~(a)*f~t(b b*)dt o(a)
_- ~(a*f~t(b b*) at a)
= ~(J]fa,bIL~l) -- llfa,bLi~ • It also follows that
U
is an isometry.
Moreover,
U
is multiplicative.
Indeed~
U(q,bfo, d) = U qe,bd = f~(ac),bd = %(a),b%(e),d " Since U0 t = pt U on
Ca(G ) .
on
C (G),
Lenm~ 3.5 below tells us that
U = kr
for some
r e G
Therefore ~(a)*~s(b ) = a*~_l (b) = Sr(a)*~s(b) r s
for all
a,b ~ ~, ~ ¢ ~ ( ~ ) .
Therefore
~(x) = ~rCx) for all
x £ ,~, ~ ~ ~4c~(.~) and for all
5) fixed.
Aut(~,/~%) means the group of all aatomorphisms of
x £ ~cY. •
leaving
'~ pointwise
18
If
[~,~}
for some
is an irreducible subrepresentation
~ e ~(~)
by assumption.
and an orthonormal basis
of
[Vl, ...,v~
of
~
w =~a.v.* JJ
.
Then
w e ~
and
~
then
[&,~] ~ [~,~]
[al,...,ad]
of
such that
~r(ak) = ~ j }~._a.; ~r(Vk) = Ejk j k V ~ ~t
[~,~],
Then there exists a basis
=~.
j •
Therefore,
if
a e ~,
then
c(wa) = Wc(a) = W&r(a ) = ~r(Wa) , so
q = ~r
total in
on ~.
~.
Since
G
Therefore,
Lemma $-5.
Let
G
is compact, the collection of these spaces
~ = ~r
on
is
~.
Q.E.D.
be a compact group,
globally iuvariant under
~
Pt' t e G.
If
A
q
and
B
be
a *-subalgebras
is an isomorphism of
A
of
onto
B
C(G) such
that
then ~ in A.
is of the form
~roof.
Let
By assumption
~
and
(ii),
multiplication
X
~
r
~
i.) ~ o pt = Ct o ~ ,
t e G ;
i)
f e A
II±'ll2 = I'~TIIo
for some
'
r e G,
hence oreserves the adjoint operation
be the closures of
A
is extended to a unitary
representation of C(G)
and
B
in
of
~
onto
U
on Lg(G).
L2(G) ~.
respectively. Let
We then have, for each
7
be the
f ~A, g e B,
u~(f)u-½ = ~(~f)g Thus, the unitary U
gives rise to an isomorphism of the uniform closure of
onto the uniform closure of
~(B) i~.
However, the map: f e A ~ ~(f) l~ ~ £ ~ )
extended to a faithful representation the faithfulness Furthermore, w([A])
~.
-([A]~)) I~ of
of the extended representation
onto the closure of
[A]~
onto
[A]~
[B]
at some point in for every
[A]~
G,
~ B ) I~(= ~([B]~)I~)
in
is
C(G),
where ~]
~.
~(A) I~,
which is
gives rise to an isomorphism
~. [A].
is given by evaluating each function in
we can find an element
r' e G
such that
(~f)(e) =
f e [A] . We then have
C~f)Ct) = Cpt • ~f)Ce) = PtfCr ') = f(r't), Thus, putting
A
is indeed given by restricting the space
which extends of
of
follows from the fact that
Thus, the isomorphism of the closure of
Noticing that every character
f(r')
of the closure
the extended representation
to
~ A ) I~
r = r '-I,
we get
~ = k
f e A,
t ~ G .
as required.
Q.E .D.
r
NOTES The materials p r e s e n t e d
in this
section
are mainly taken from [55]-
See a l s o
[3]
19 §4.
Supplementary formulas. We shall give definitions of an action, a co-action and a crossed product with
respect to
g(G)':
An action into
~'
~ ~ L~(G)
of
G
with respect to
(~'* ~) o ~' =(~ ~ )
T where (~G
is an isomorphism of
= Adv~.(f ,e, 1)).
isomorphism of
~
L~(G)
A co-action
into
(~.~)
~ ~ ~(G)' (~' ~ ~)
where (~
~
5~
(Y)
is an iscmorphism
: ~w&(Y
~(G)'
is an isomorphism of
satisfying
(~.i)
(~f
on
~ 1)).
~e
the yon Neumann algebras The dual co-action
(~')^
of
5'
of
,
L~(G) .~, T~(G) G
o_n_n ~
with
/ I ~(~Gf)(s,t) =f(ts),
with respect to
~(G)'
• ~, = (~ ~ ~)
into
pr~ucts
~'(~) v (C ~ ~(G)')
o ~,
,
~,(G)' ~ ~,(G)' with 6~(r) ~ m~,
and
and the dual action
G
and
~ x~,
~
: ~(r){ ~(r),
are demned
are defined by
(o,)^(y) = Aa~w~(y ~ l) ,
y ~ ~ x,
(~.~)
(~')^(x) = Adl~v~(X ~ Z) ,
x ~- ~ x~, G .
Now, we shall list the associativity conditions for
WG, W~_, V G
(w~ ~ i)(~ ® o)(w~ ~ i) : ((, ~ ~)(w~ ~ l ) ) ( w ~ =
l)
(, ~ ~>G)(w~) -
(~.6)
(wG ~ l)(l ~, ~)(w G ~ l) = ( L e 5G) (W G) "
(~.7)
(w~ ~ 1)(L ~ ~)(W~ • ' ~ l) : ((~ ~ o)(w~ ~ l))(w~ ~ i) =
~.8)
as
5'(b) V (C ~ L~(G], respectively.
(5') ^
(~.~)
(~.5)
is an
satisfying
~(G)'
crossed
into
o~'
(L ~ 8~)(w~) . -t 7~ (L ~ ~o)(wG ) •
1)(L ~ ~)(v e ~ l) : (L ~. ~G)(va) -
(4.9)
(v G .
(~.10)
(V~ ~ 1)(L ~ ~)(V~ ~ l) : ( ,. ~ ~G) (v~) .
(~. li)
(V~ ~ 1)(~ ~ ~)(V~ ® l) : ( ~ ~ % ) (v G) •
(~.~)
(V~*~ ~)(~ ~ ~)(V~* ~ l) : ( L ~ ~ ) (v~*) .
G ;
and
V~:
CHAPTER II.
ELEMENTARY
Introduction.
PROPERTIES OF CROSSED PRODUCTS.
The image of the original algebra in the crossed product
is characterized
as the fixed point subalgebra under the dual action (resp. the
dual co-action),
Theorem 1.1, in §l.
crossed product is characterized
Combining this with the duality theorem,
as the fixed point subalgebra
product of the original algebra with
£(L2(G))
under the tensor product of the
original (resp. co-)action and the regular (resp.co-)aetion, Section 2 is devoted to a characterization characterization
the
of the tensor
Theorem 1.2.
of a dual(resp,
co-)action.
The
sho~-d be viewed as a sort of an im.primJtivJty theorem.
In ~3, we consider
the com~utant of the crossed product oy a closed subgroup
which is shown to be the fixed point subalgebra of the crossed product of the commutant of the original algebra under the action (or co-action) group.
of the quotient
Here, since we do not treat the Banach algebra bundle of Fell,
have to restrict ourselves
to normal subgroups
for actions.
[ 3 0], we
21
#l.
Fixed ~oints in crossed products. Given an action G of G on [~ or a co-action 5 of G or U5 the fixed, point subal6ebra of ~ or ~, i.e.,
on
~
we denote by
;
m~ = { x ~ m : ~ ( ~ ) =x®l] ~ ={x~. : ~(x) = x ® ~ ]
.
With these notations, we can characterize the location of the original von Neumann algebra (or more precisely the image under the action or the co-action)
in the
crossed product as follows: ^
Theorem 1.1.
(a)
(~ x~ G)~ = ~(~).
(b) (n x~ a)e Proof.
(a)
8(h).
It is clear that
~(~)
is contained in
(~ x~ G) ~.
We have only
to show the reverse inclusion. Since action
~
{~}~
and
~ ® Pt = Adl~o(t)'
is implemented by a unitary representation
u(t)xu(t)* given by
{~(~)~L ® p]
for
x ¢ ~.
Here we identify
(u~)(t) = u(t)~(t)
for
u
Then
~'
is an action of
I'll' ~ £ ( I ' . 2 ( G ) ) Applying
G
is generated by
~ c $ ® L2(G).
Ad
on
~'
~'(~')
Then
of
G
such that
Jt(x) =
£($) @ L~(G)
J(~) = u(~ ® C)u*.
We put
x ~ m'.
with respect to and
u
with the unitary in
~ ' ( x ) = ~*(x ® 1)u ,
~' @ L~(G)
we may assume that the
C ® L~(G).
R'(G).
By Lemma 1.2.6,
Therefore
= C~'(ll1') V (C ® L~(G)) V (C ® fC'(G))
.
to the both sides and considering the commutants, we have U
(l.1)
~(m): (m' ® C)' n (c ® ~ ( G ) ) ' Now suppose that
y a ~xc~ G
to the right hand side of (1.1). commutes with
m' ® C y ® 1
~' ® C
and
and u(t)®k(t)
and
n { u ( t ) e X(t): t ~ G}' ~(y) = y ® i.
It is straight forward to see that
u(t) ® X(t) for all
We want to show
t ~ G.
for all Since
t a G.
Thus
y
&(y) = ( l @ W G ~ ( y @
(i ® w~(C ® C ® f(a))(1 ® WG)
Since~ by Lemma 1.2.6 (C ® L~(G)) V W~(C ® L~(G))WG = f ( G )
it follows
that
y
commutes with
C ® L~(G).
~ f(G)
,
~ ×
belongs G
commutes with
confutes with
=d
y
1)(l®WG),
22 ^
(b)
We have only to show that
~5(~), g ® 5G} a unitary
w
and
(U ×5 G)5 ~
5G(X) = WG(X ® I)WG,
in ~(R) ~ R(G)
ccntained in 5(~).
we may assu~ae that
5
Since
[U,5] T
is implemented by
such that
5(y)=w×(y®i)w
,
y~
~,
(1.2) (. ® i ) ( ~ Let
K
be a unitary on
and
~' = ( l ® ~ ) w ( i ® K ) .
L2(G)
®~)(w
®z)
defined by
Since
: (~ ® 5G)(w) .
(K~)(t) = a(t)i/2~(t -I)
~(n®C)~c~.~(~O(S))
and
for
{ ~ s2(e)
w ~ ~(R)~(O),
it follows that ~..,(m c c)~ × c ~, ~ R(c) .
Here we set
5'(x) = w'(x ® i)w '×, Then
5'
is an isomorphism of
w' ~ ~(R) ~ R'(G)
A'
Therefore,
® ~)(w' ~i)
~,' is a co-action of is generated by
~' ~ ~(-L2(G)) Applying
~' @ ~(G)'
and
w'
satisfies
and (~,' ® l ) ( L
n' @~(T.-~(G))
into
x ,: ~'
Adw*(l ® K)
G
on
5'(n')
= (L ® 5 ~ ) ( w , )
.
~P.' with respect to and
C ®£(L2(G)).
:: ,~;'(n') v Cc ® I f ( G ) )
R'(G).
By LemmaI.2.10,
Therefore
v (C ® ~(G))
.
to the both sides and considering the commutants, we have
~(n) = (~' ~ C)' P, (w*(C @ ~(G))w)' n (c @R,'(G))' Suppose that with
~' ® C.
Y ~ ~ ×5 G
Since
(1.2)
and
~(y) = y ® i.
It is clear that
~ x5 G
eommules
implies
( i ® wo)(w* ® 1)(~, '~ o-)(w × ® ! ) ~ (~,~ ® ! ) ( 1
~ We_) ,
we have
A~wG((w~(i for Thus
f e L~(G). w*(1 ® f)w
® f)w) ® i) = (~*(~ ® f)w) ® i
It is known that
WG(X ® I)WG = x ® i
commutes with
C ® T:~(G). Therefore,
w*(C ® IJ~(G))w. Thus
y
commutes with
v
( l ® VG)(y ® l)(.l® VG) , y ® i
C ® C ® R'(G) Therefore,
y
commutes with
and
71' ~ C
and
if and only if ~ xs,) G
x a L~(G).
con~uutes with
w*(C ® L~(G))w.
Since
~(y) =
commutes with
( i ® V~)(C ® C @R'(G))(I ® VG)*
C ® R'(G).
Q.E.D.
23
Combining Theorem i.i and the duality theorem for crossed products, we have a characterization
of crossed products.
Theorem 1.2.
(a)
• ×~ G = ( ~
(b) ~ Proof.
(a)
£(LP(G))) ~.
~ = (~ ~ Z(L2(G))) g
~$~kir~ use of ~he isomorphism
v
of
~L ~ £(L2(G))
onto
(m ×a G) ×a a obtained in (m.2.12), ~e h~ve
~(m ~ Z(L2(G))) ~) ~ ((m ~ G) ×a °)~ by the duality theorem for action. 1.1.
On the other hand, (b)
~
~
The right hand side is
on
~ ×(~ G
Making use of the isomorphism
obtained in (1.2.20),
bj/ ~
(1.2.8),
of
~(~, x
G)
by Theorem
(I.2 • ~ ~nd (I.~.13).
~ @ £(L2(G))
onto
(~ ×5 G) x^ G 5
we have
~ ( ~ ~ Z(T2(G))) ?~) = ((n ×~ G) ×~ G) ~ by the duality theorem for co-action. Theorem 1.1.
On the other hand,
?~ = n
The right hand side is on
%% ×5 G
5(h ×5 G)
by (I.2.10),
by
(I.2.11), (1.2.21)
and ( I . 2 . P 2 ) .
Q.E.D. NOTES
The fixed points in the crossed product, Theorem i.i, are characterized in [4P~43,46,60].
The presented proof is taken from [46].
the crossed product as the fixed point subalgehra, f)Igerness [15,16 ]
and by ~3,47~60] .
Kae algebra context in [26].
The characterizations
of
Theorem i.~, were obtained by
Theorems ].1 and ]..2 are generalized
to the
24
§2.
Characterization of crossed products. When one studies an action or a co-action of a locally compact group
G
on a
von Neumann algebra, often it so happens that the action or the co-action is already dual to another co-action or action. one is dual to something else.
Thus, we need to know e x a c t l y ~ h e n the given
The following gives a convenient characterization
for those actions and co-actions. Theorem 2.1.
Let
8
be a co-action of
G
on
N.
The following three condi-
tions are equivalent: i)
There exist
avon
Neumann algebra
•
and an action
~
of
G
on
such that
{~ ×~ G,
{~,~1 ~ ii)
There exists a unitary representation
(2.1)
u
of
There exists a unitary
(2.~)
u
in
G
in
h
such that
t c G ;
5(u(t)) = u(t) ® p(t) , iii)
h ~ L~(G)
such that
(u ~ 1)(~ ~ ~ ) ( u ® l ) = (L ® ~G)(U) ;
~(u) = (u ~ Z)(l ~ w C) ,
(2.3) ~here
&} ;
~:
(~G~)
o(~0.
If any of these three conditions holds, then
~
is generated by
~5
and
u(t), t ~ G. Proof. (ii)~
(i) :> (ii): (iii):
Trivial.
Given a unitary representation
(2.1), we shall use the same symbol (u~)(t) = u(t)~(t)
for
u
u
~ ~ R @ L2(G).
Then
= ~(t,s),
G
in
h
satisfying
~ ~ L~(G)
u(st) = u(s)u(t)
Furthermore, (2.1) means precisely (2.]3). N~meZy, if U~(s,t)
of
for the unitary in
defined by
means (2.2).
~ ~ ~ ® L2(G × G)
and
then
~(u)E(s,t) = U ( 5
@ g)(u)(U~)(s,t)
: 8(u(s))(~)(t,s)
= (5 ® ~)(u)(UE)(t,s)
= (u(s) ~ ~ ( s ) ) ~ ( t , s )
= u(s)(U g)(ts,s) : u(s)~(s,ts) : (u ® l)(l ® WG)~(s,t ) .
(iii)=~ (i): isomorphism of
h5
For each into
x c R 5,
~ ~ L~(G).
we set
c~(x)= u(x g l)u*.
By (2.3), we have, for each
Then
(~ is an
x ~ ~5
25
= (u ® 1 ) ( 1 e WG)[(L ~ = )
o (~ e ,~)(~ e 1)](1 ® W~)(u× ®Z)
= (u e, i)(1 e, WG)[(~ ~ ~)(~(~) ® 1)](i e WG)(u* ~l) = (u ® 1)(l ~ wa)[(, ® ~)(x ~ iL~(~×~))](~ ® W~)(u* ® l) = (u~)(~)(u*®~)=~(~)el
.
Therefore, a(x) falls in ~5 ~ L~(G), so that (z maps D.5 into ~5~L~(G). By (2.2), G. satisfies the associativity condition, (I..°_.4); hence it is an action of G on ~5. Next, we want to show that h is generated by ~5 and u(G). By Lemma 1.2,10, the yon Neumann algebra By Theorem 1.2,
R = ~ ~ £(L2(G))
~ ~8 G = ~ .
by (2.3), it follows that u(C ®9(G)')u "×'. Since
~
Since
?I X5 G
is an isomorphism of
is also generated by
u(l®X(r))u*=u(r)~X(r),
Finally, we shall show that
and
C ® g(G)'
[~,~}
~-~ = h5 @ ~(L2(G))
onto
[~,5"}
and
~% must be generated by ~5 and u(G).
{~,5} T [R5 x
G,~}.
For each
x e R5
we have
u~(x)u* = u(x * 1)u* = a(x)
(2.4)
L u~(u(t))u
Therefore,
~d u o ~
is ~
~5
and
u(G)",
× = ~(u(t)
isomorphism Aduez
on
is generated by
Ad u
o (5 ¢ . ~ )
e ~(t))u ~ = L ~ ~(t)
~=~Vu(~)" o~ =d
onto
.
~ 0 .
By ( 2 . ~ ) ,
we have
oAd u o
which yields our assertion.
Q.E.D.
The du~l version of the above theorem is the following: Theorem 2.2. are equivalent : i)
Let
There exist
~
be am action of
G
a von Neumann algebra
on
~.
The following three conditions
~% and a co-action
5
of
G
such that
ii)
(2.5)
%
iii)
(2.6) (2.7)
There e x i s t s a
o~=~oX
*-isomorphism
t
w
of
L'~(G)
(or ~ o ~ = ( ~ )
There exists a unitary
w
in
into
oX)
• @ B(G)
•
such t h a t
on S~(a) .
such that
(wX ®l)(~ ~ ~)(w* ~i) = (~ ® ~o)(w*) ~(w*)= (w*~.1)(1~va) In the case (ii),
(or ( % ® l ) ( w *) =w*(1~p(t)))
~r, is generated by
~
and
~(L~(G)).
.
on
26
Proof.
(i) ~# (ii) : Clear.
(ii) => (iii): We may assume that unitary representation L~(G).
u
of
G
~
on
is standard and
~.
~
is implemented by a
Then (2.5) implies
Adu(t) ~ . . . .
By Mackey's imprimitivity theorem, there exist a Hilbert space
isometry
U
of
~ ® L2(G)
onto
~(f) -- u ( i ~ Therefore
U(g ® L~(~))u-i=
)~
on
and an
,9 such that
O u -i ~
~
~(~)
and
u(i~
~,(O)u -I
~t ° Adu = Adu ° (~ ~ ~t )
on
~ ~ ~(~).
Therefore we may assume that = R ~. Li~(O)
,
~(f) : i(~ Z f
for some unitary representation Now, we define a unitary
v
of
G
on
,
u(t) = v(t) 'g k(t)
.~,
w in £(~) ~ ( G )
for
by
I~W
associativity condition (I.[l.p), w*
satisfies (2.6).
and
w ~ ~ ~ ~(G).
w(L'~(G)) c ~,
it follows that
A
(i @, X(t))~u(t)~ - £(R)~L~(G). o.
Since
Since
W*G
satisfies the C~ L ~ ( G ) ~ ) ~ ( G )
for
f ~ L (G),
I~WG(
Since
G~ { l) = (W 0 ~, 1)(1 ~ V~) ,
d(~)(V~l)(W
the condition (2.7) is obtained. (iii)
> (ii)
Put
w~ = (i @ K)w
where (K~)(t) -- A(t)i/2{~t-i " (2.7) implies
(2.8)
)
for
and g
6
@ ~ K ) = [w K @
it follows that
Ad
o ~ =~
o Ad
wK6~I ~o (f) :{ ~ , x
~)(f) =
LP(G)
wK(i ® f)w K
Then
.
~0(f)
m ~ £(L2(G)).
Since
l)(i Z VG) , Since
~
= T, x
wK•
.~
G
by Theorem i o
o.
The associativity condition (2.6) i~iplies that (i ~ W c ) ( w Since
{i)(~
~*~G = (K @ I)WG(K ~ i),
® c)(w e l )
= (w @ i ) ( i
~ %)
.
we h~ve
(i e wa)(~ K ~ I)(L ~ ~)(w e i) = (w K ® l)(1 e wa) , and hence
Adl~(%(f) Since
~z(m) = ( m x
G) d
~ i) ~ T0(f ) ® i
by Theorem i.i,
therefore there exists an isomorphism (~ o n.
Since
and hence that
(,~,@ g) o ~z = (~ ®,ZG)
we have ~
of
,~o(f) ~ on
~(z) = w * ( y .
by
~
~)w .
It follows from (2.7) that o ~(y) = ~ o
~dw.(y~) o ~(y ~ l)
= Adw*x i ° AdI@'V G : ~(y) ~ ~ .
5(y) e F£~ ~ £ ( L 2 ( G ) ) .
Therefore into
~
hence
~ ~(G). 5
Moreover,
w
satisfies
is a co-action of
G
on
By iemma 1 . 2 ~ ~ ~ = ~. ×
G.
~erefore
~
onto
(hi) there
condition (2.6)
~C~
and
~' ×c~ G
and
=d
cxisz
C ® L~(G).
Adw~ "
of
By Theorem 1 . 2 ,
~,~}
onto
[~,~}.
~(C ® C(~))~K K a H i l b e r t space
R
and an i s o m o r p h i s m
w
of
[~ such that ~ = R ® L2(G)
Since
the associativity
i s g e n e r a t e d by
i s generated by ~
i s an i s o m o r p h i s m o f
~.
C o n d i t i o n ( 2 . 8 ) g i v e s an i s o m o r p h i s m
According to L~(G)
w e ~ ~ g(G), 5
Since
,
~(f) = I R $
WK(C~® L~(G))wK c C~® C ( G ) Finally we shall show that
f ,
~ L~(G), •
{~L,c~} ~ { ~
k •
G ° ~ = (.~® ~ ) °
is generated by ×5 G,g}.
If
T~
Y ~ ~c~
ana ~ ( # ( G ) ) and
.
f ~ #(G),
then (2.9)
w.*~y)w.. = ~K~(y ® i)~K = ~(y)
Therefore (2.~),
Ad
*
o a
is an iscmorphism of
*@1
on
~CL and
•
onto
[~a ×8 G.
Since, by (2.9) and
wK ° (~
®
~)
~(L~(G)).
° c~ =
.% ° A d
.
o C~
Q.E.D.
Here we g e n e r a l i z e t h e c o n c e p t o f t h e d u a l a c t i o n and t h e d u a l c o - a c t i o n defined in
P r o p o s i t i o n I . 9.4
as follows:
28
Definition 2.~. action of (b) G
on
G
on
(a)
~,
A co-action
~,
if
An action ~ of G on ~
if
[~,~} ~ [n x5 G, g}
5
of
G
on
[~,5] ~ [~ X~ G, &~
~
is said to be dual or the dual
for some
In,5}.
is said to be dual or the dual co-action of
for some
{~,~}.
In the rest of this section, we assume that a bijection between the set of all unitaries
G
w
in
is discrete. ~ ~ g(G)
Then there exists
satisfying the
associativity condition
(w ® I)(L ~ ~)(w ~ i) : (~ ~ ~G)(~) and the set of all partitions
{e(t) : t c G]
(2.11)
w =
of the identity in
by the relation
~ e(t) ® 9(t) . t~G
Thus our Theorems 2.1 and 2.2 are stated in terms of these partitions. Theorem 2.4. w c h ~ g(G)
(a)
If
5
is a co-action of
G
on
~
implemented by a unitary
satisfying the associativity condition so that
5(y) = Adw.(y ® 1),
then the follcwing twc conditions are equivalent: (i)
5
(ii)
is dual;
there exists a unitary representation
u(t)e(r) : e(rt-1)u(t) (b)
If
~
for all
u
of
G
in
h
such that
r,t.
is an action of
G
on
~,
then the following two conditions are
equivalent: (i) (ii)
~
is dual;
there exists a strictly wandering projection
[~t(e) : t c G} Proof.
(a)
is a partition of the identity such that
which is equivalent to (b)
u(t)e(r) ® p(z') =
~ s
Condition (2.6) is with (2.11).
i.e.
= 0
for
t ~ s.
which is equivalent to
for all
t,r.
equivalent to the existence of a partition Condition (2. 7)
~t(e(r)) @ p(r -I) =
r.
~,
e(s)u(t) @ o(st) ,
u(t)e(r) = e(rt-l)u(t)
r
some
for
~t(e~s(e)
Condition (2.1) is equivalent to 2 r
[e(t) : t c G]
e c ~
~t(e(r)) = e(tr).
~ s
is equivalent to e(s) ® D(s-lt) .
Thus we have only to set
e = e(r)
for
Q.E.D.
29
NOTES
The equivalence of (i) of (i)
and
(ii)
in
and (ii) in Theorem 2.1 is due to Landstad [42]; that
Theorem 2.2 is due to Landstad [~3], Nakagami [46] and
Str~til~ - Voieulescu - Zsid6 Kac algebra context in [26].
[60].
Theorems 2.1 and 2.2
are generalized to the
30
§3.
Commutants of crossed products° In this section we shall define an action
and a co-action
a
of
G
3
on the commutant of
of
G
5(N)
on the commutant of
~(~)
in order to show an imprimi-
tlvity theorem, which will be applied to the commutants of crossed products.
The associativity (I.2.11) for
~
and
(I.2.5) for
5
gives us
Ad(~va)((~(rn) ~ C) a (~.~.~ ~)(r~ ~, S(G)) ; Ad( ,~W~ ) (S(rO.~ C) a (S ® L)(r~ ~ ,~(G)) Since
VG s ~(G) ~ L~(G)
and
WG .-: L~(G) ~ a(G)
,
we have
(~.]-)
A~(l~V~)(c~(,~)' ~, c) ~ ~(r~), ~ if(G)
(3.2)
Ad(l~G)(,%r0'
Besides,
VG
and
WG
Definition 3.1. ~(G)'
e c) c ~(n)' ~ ~(a)
satisfy the associativity conditions (I.4.i!) and (I.L.6). We define an action
and a co-action
.
e
of
G
on
Ad~v~ (~
B
of
G
on
~(~0'
(_~.3)
~(~) =
~ l) ,
x ~ ~(,~,),
(3.4)
a(y) = Ad]igWG (y ® l) ,
y E B(~)'
For a closed subgroup
H
of
G
with respect to
5(h)' by
we denote
~ ( ~ \ (0 = L~(G) P, X(~)' #(G/H) = IF(O) n p(~)' mx a H
=c~(m) V ( C ~ p ( H ) " )
"rl x 5 ( H \ G )
= 5(1t) V (C ® £ ~ ° ( H \ G ) )
.
Now, we shall show an imprimitivity theorem: Theorem 3.2. (~. x~ H)'
(3.6)
(a)
(Assume that
is an action of
G
(r~ x
II is noznal).
with respect to
~(G)'
The restriction and
H)' : (rp, x~G), v (C ® S ( G / H ) )
.
of
~
to
31
(b)
The restriction of
s
to
(~ x 5 (H\ G))'
is a co-action of
(3.7)
(~ x5 G)' = ((h x 5 (H\G))') ~
(3.8)
(n x~ (H\ Q))' = (a x5 G)' v (C ® ~(~)") Proof.
(a)
Put
~o = (B Xa H)'
Since
H
G
and
,~ ~ g(~) @ L~(Q),
is normal and
we have r~H
AdI~VG(I ® p(r) @ f ) ~ C ® t)(H)" (~ L~(G) ,
Combining this with (3.1), we have of
~
to
~
is an action of
G
~(~P.o) c ~L° ~, L~(G). with respect to
Therefore the restriction
Q(G)'
O
~e~t we s ~ Z ( ( ~ X 5 H)') ~. then
show (3.5).
Since
x @ 1 = Adl®~#(x ® l ) vG
(~
VO ~ ~(G) ~ L~(G),
It suffices to show the reverse inclusion.
c
belongs to the commutant of
x~ ~)'
x ~ (~(r~)') ~,
Tf
e~(G)
for
(C ® @(G)) V V~(C @ @(G))V G = R(G) @ ~%(G)
(3.9) Therefore
x E O~(1~)' p~ (C ® ~ ( G ) ) '
= (m Xo~ G ) '
,
Finally we shall show (3.6).
.
, )r~
((~ x G H) )P c (c~(l~.) As
~
Thus
c (m xo G)'
commutes with
C ® p(H)"
O
V~(I ® p(r))V G = p(r) ® p(r), 8(~o) is contained in isomorphism of
L~(G/H)
onto
is the canonical msp of
G
£~(G/H) on
~
is an action of
G/H
Since
(3.10)
C ® £®(G/H) c ~o' ~
~o
~o ~ ~ ( G / H ) .
¢(f) = f ° ~
for
f
and as Let ~ ~e an ~ if(G/H), where
Here we set o ~(x)
x e mo
,
such that
~. = 8t t
with
t = tH°
is an isomorphism of
L~(G/H)
into
m°
,(f) = 1 ~ ( f )
on
with
G/H.
~(~) = (L ~ ~ @ ~-i) Then
@~(G),
We set
,
~. °-(f) = ~t(l e ( f t
o~))= ~
satisfying
if) ,
Therefore, by Theorem 2.2, ~b is generated by (~o)~ and T~f(G/H)). Since (~o)~ = (~o)~ and w(L~(G/H)) = C ®~(G/H), ~o is generated by (~ ×~ G)' and
32
(h)
~t
no= (nx 5(H\G))'
WG ~_ f'.~'(G) ~r £ ( G ) ,
Since
Adl(~WG(C ® £~°(H\G) g C) = C ® ~ ( H \
a ( h o ) c No i 9(G).
Combining this with (3.2), we have a
to
n
is a co-action of
o
~t
we s h ~ l
G) ® C .
Therefore the restriction of
G.
show ( 3 . 7 ) .
Since
W~ ~ s~(a) ~ ~(G),
(~x~ G)' ~ ( ( ~
it suffices to show the reverse inclusion.
If
A d l ~ G ( y ~ i)
C g l~(G) ~ L~(C),
belongs to the ccmmutant of
Y • (B(h)') S,
then
( ~ G ) ) ' ) ~,
y ® ! =
fcr
(3 .ll) Therefore~
y c S(n)
n (C ® L~(G)) ' = (n xg, G)'
( B X ~i]] ( H X a ) ) ' )
Finally we shall show (3.8). contained in n ~ 0(H)". '9(DH(r)) = p(r~,
where
Let OH
Thus,
~ ~ (IS(~) ' )~" C ( ~ X s G) '
As
n°
commutes with
~,_~ be an isomorphism of
C @ £(H\
G), t(no)
oH(H) '' onto
p(H)"
is the right regular representation of
H
on
is with
L-@[).
Here we set ~H(y)
Then
aH
= (L ~ & ® ,~.-1) o a ( y )
= (L S L ~ ~-l)
y ~ n
£ ~ X(H) c n
is a co-action of H on ~ . Since o
~H(Z ~ X(r))
,
o
and
o
o ~:(1 ® >,(r))
(3.12) = (1, S 1, ® 9 - 1 ) ( 1 ~ k ( r )
it follows from Theorem 2.1 that (no)sH = (~o)a,
no
Corollary 3.3.
(h x 5 G)'
(a)
[~ is dual on
The action
(~(~.)')#
The co-action
s
is dual on
(3.14)
(~(n)')~
Proof. (b)
is generated by
is generated by
(3.13)
(b)
n°
(a)
By (3.10),
By (3.12),
@ 0(r))
a
~
= (,~. ×
5('a)'
,,y
G)'
and
= (t~ x s a)'
is dual on
is dual on
and
5(n)'.
a(~)'.
= 1 @ ,~,(r) ® DH(r)
(~o)sH C ~ X(H). Q'(T0,
and
and
,
C @ ).(H). Since Q.E.D.
33 Corollary 3.4.
(a)
If
~
u
is implemented by a unitary
in Z~)~L~(G)
satisfying the associativity condition
(u ~
such that
1)(L ~ o)(u
®
l)
=
(L ~ %)(u)
~(x) = u(x $ l)u* , then
(3.15
(~ x~ G)' = (~'~ ¢) v u(C ® ~(G)')u* b)
If
5
is implemented by a unitary
w
in
£(~) ~ ~(G)
satisfying the
as soci ativity condition ( w * ~ l)(L e ~)(w* ® l) = (~ @ ~G)(W*) such that
5(y) = w*(y @ l)w,
(3.~6)
(~ ×~ @ '
Proof.
(a)
= (~' ® C) V w*(C e Z(G))w
By the associativity condition,
According to Theorem 1.2, is the intersection of Therefore
then
(~ ×a G)'
• ×~ G
is
~ @ £(L2(G))
%(y) = y ® 1
where
v~f) = w*(l ® f)w
(3.17)
~' @ C
f "~ L~(G).
y ¢ ~(L~(G)) '
,
The associativity condition implies that
(i ® WG)(L ® ~)(w* ® l) = (w* &¢ i)(i ® WG)(W ® l)
~(y) = Ad(w~l)(~Wp(~l)(y y e I-KL~(G)) ' implies
G
azld u(C ® ~(G)')u*.
so that
Therefore
~t = ~t ® k t ~ × ~
u(t) ® k(t), t ~ G.
we have only to show that, for each
if and only if for
Since
with the commutant of
is generated by
(b) By virtue of Theorem 1.2, y ~ ~ ~ £(L2(@),
u ( l ® k(r))u* = u(r) ® k(r).
(~ @ £(L2(G))) ~.
~ i)
~(y) = y ® i.
Conversely, the associativity condition implies that (l ® W p ( w * ® 1)(L @ ~)(w* ® l) = ( w * @ l ) ( l * W p and hence that Ad]it~G(~(f ) ® i) = ~(f) ® i •
,
34 Therefore, by (3.14), we have for each
f e L~(G)
~(f) ~ (N ×5 G)'
and hence
~(f)
commutes with
The same type of formula as (3.17) is obtained from the associativity condition on
(5.18)
u:
(1 @ V~)(L ~ ~ ) ( u ~, l ) = (u ~ 1)(1 ~ V~)(u* ~ l )
NOTES
The action
~ , Definition 3.1, is introduced oy Landstad, [43].
The
impr~itivity theorem, Theorem 5.P and Corollary ~.~.a, "~ are due to Nakagami [4'7]. Corollary ~.4 is due to [16,~3,43,47].
y
Q.E.D.
by Theorem 1.2.
C}~PTER llI. IIgfEGRABILITYAND DOMINANCE
Introduction.
As in the case of actions, [ 14 ], integrable co-actions play
a crucial role in the analysis of the crossed product.
To do this, we shall prepare
in §i elementary properties of the operator valued weight associated with the Plancherel weight on
~ ,
the
CG on ~
~(G)
of
G
to
DG and for a co-action
integral with respect to semi-finiteness of
as well as the Haar measure
-valued weight 5
of
CG "
G
5
£(~) ~ ~(G) -
~, the
~"-valued weight
in §2.
~
C (G)
For an action
Making use of
on
~
g5
is then the
(resp. 5) is defined by the ~5~ we shall show there
on a s t a n ~ r d von Neumann alge0ra
implemented by a representation of unitary in
on
~G "
is defined as the integral with respect
The integrability of
g~ (resp. g~)
that an integrable co-action
g
[~,~J
is
which is identified with a certain
this identification is discussed in Appendix.
It is,
however, conjectured that the implementability for a standard von Neumann algebra holds without the integrability assumption for 5. In §3, integrab!e actions (resp. co-actions) are characterized as reduced actions (resp. co-actions) of the second dual actions (resp. co-action), which is also equivalent to the point spectrum property of all reduced actions (resp. coactions), (3.1) and (3.4).
In ~ ,
it will be s h o ~ that among integrable actions (resp. co-actions) there
is a unique, up to equivalence, largest one, which is dual and of infinite mul~iplicity.
Such an action (resp. a co-action) is called dominant.
36
§i.
Operator valued wei6hts Let
~G
(resp. ~6) be the faithful, semi-finite, normal weight on
L*(G)
by integration with respect to a right (resp. left) invariant Haar measure. malize
bG
and
b~
so that
Given an action
a
given
We nor-
b~(f) = bG(Af), f e K(G).
of
G
on
~
we set, for each
f e Ll(G)3
(~f(x),~} = (a(x),~ ® f} =f(at(x),~)f(t)dt G
(l.1) G
Let
F = {g e ~(G) : 0 S g S & } .
ing net in
~ + . We define an
(1.2)
If
x e B+,
then
~-valued weight
[ag(X) : g e F}
8
Ca(x ) = sup{~g(X) : g e F} ,
if the right hand side exists in
%
~.
is an increas-
by
x e ~+ ,
Since we have
° af = a ( s ) %
f ,
f e~(O)
;
"S
A(S)ksF = F ,
ga(x)
falls in
~
if it exists.
As in the case of numerical valued weights, we
consider q~ = {x e ~ : ~g(X*X) ,
It then follows that
q~
g e F,
is a left ideal of
~.
is bounded}
Put
x-
~a = q~qa "
It is easily seen that be denoted by
8
is extended to a linear map of
(1.3)
~%(f) = ~6(f)l , f e L~(~) n ~l(a,d's)
Proposition 1.1.
(1.4)
~-' into
~ a . As a special case3 we have
e(x i ) ~ e ( x )
if xi ~ x
in ~+ ;
h a,
which will
37
(1.~)
=(i(~)):~(x)¢l,
(1.6)
~,~_~=
(1.7) the m a p : x
~
~
and i (a~b)=~L~(~)b, a,b~m~, ~ ~=
e .~ - ~ % ( a xb) e ,~
~s
~-weakly continuous for each
a,b 6 % .
The proof is very similar to that for the corresponding properties of weights, so we leave it to the reader. We
now want to dualize the above construction of
first dualize the notion of Haar measure. on
~(G).
We need the Tomita algebra which gives rise to
Tomita algebra structure in
~(G)
f*g(t) = Jf
(~i~f)(t)
[
We then have
(1.9)
We introduce a
= -,(t)
f,,~),
f~(t) = f(t -1) ;
= ~ c : (flg)
fr'(t) = A(t)f(t--'-~-U~ - ) ,
f e
=
f(t)g-eU
dt
•
K(G) ;
I '~(f) = ,,.~(~Lf)
where
k
and
= o(~f ~) ,
p are of course the left and right regular representations of
(1.10)
,,,(f) =
ff(s)X(s)d~
, o(f)=ff(s)p(s)~
G
~r(~(G))
of the Tomita algebra
yon Neumann algebra
QI(K(G)).
~(G)
and get a full right Hilbert algebra 9/' with
K(G)(or
~I)
and
.
@ (G)
is the right yon Neumann algebra
and its conmutant
9'(G)
is the left
Thus, we set
~ = ~(G)'
algebra
G
G
It then follows that our von Neumann algebra
(x u )
R(G).
as follows:
f(ts-1)g(s)ds ,
~(f)
with
~ . For this, we must
So we shall define the Plancherel weight
~(91') = ~'(G).
and
~' = ~(C-)"
9/ with
@r(9/) = ~(G)
and a full left Hilbert
The modular unitary involution
is given by:
j~(s) = A(s)~ ~(s -I) , ~ c L2(G)
J
associated
38
The canonical weight
(1./2)
$C
~r(~I) = ~(G)
on
*G(rrr(g)*~r(f))
where
e
is the unit of
G.
is then given by:
= (fig)=
(fX'g~)(e) = ( g ~ *
We shall call
~G
f)(e)
,
the Plancherel weight on
e(o).
The corresponding modular automorphism group is then given by
(1.13)
ct(x) : A-itx A it ,
x e ~(G)
.
Set
(l.l~)
= [¢? e ~(G)x+ :(l+a)C0 < *G
for some
s = ~
> O} .
We then have
(1.~)
CG(X) = sup[~(x)
Suppose now we have a co-action each {8
~ ¢ ~(G). = A(G), : ~ e y]
G
on
Of course,
g6(x)
have~ for each
~ e ~
8
on
~.
It then follows that
, x ~
does not necessarily exist.
x e ~+
By (I.2.18), we define~ for
We set
~(x) =snp{~ (x) : ~ ]
only for those
A.
a linear transformation
is upward directed.
(1.1~)
8 of
: :~ e ~} .
So we mean that
g6(x)
is defined
such that the right hand side of (1.16) exists.
We then
,
(1.17)
= (8(x),~ ~ %> ,
x ~%+
As before we set
(i.13)
q5 = Ix e ~ : gs(x*x) exists}
We then extend
g5
to a linear map
have the following formula Lemma
1.2.
g8
d~(x) =,c(x)l, We identify first
which are of the form:
~G).
into
~.
As a special case, we
of
G
on
xe ~=m~G
~(G)~
.
with the algebra
A(G)
of functions on
G
g ~ x. f, f~g e L2(G), under ~he correspondence:
~f,g ~ ~(G). g ~ * f ~ A(G). ~i E K(G)
P8
P5 = qsq5 "
which requires a little bit of proving:
For the co-actlon & = %
(1.19) Proof.
of
and
is a net such that
It follows from the construction of
~ (~i) ~ ( ~ i ) z i
then
~
~G
that if
(x) = (x~iI~i) I ~G(X)
39
for every put
x • ~(G)+.
g = f~.
Suppose
~ e A(G)+
~s a state.
For any
f • A(G)+
we
Then
= • Since
f,g • A(G)+,
(1.20)
we have
p(f) ~ 0
and
p(g) ~ 0
and hence
(~p ® CG ) o 5(p(f)) = g(e) : f(e) : @G(p(f))
Now let
{qt : t c ~
associated with
SG"
•
be the modular automorphism group of
g(G)
We then have
(1.21)
Gt(P(s))
= A ( s ) ;It p ( s )
,
s ~. G,
t
• ~
o~
.
;
hence
(1.22)
8
Thus, if we set so that if
¢
and
~ c A(G)
oG
t : (=t ®L)
* = (*G ® ~) ° 5 ¢G
(L®%)
o~=
for a state
commute in the sense of
is a state.
G-weakly dense part, see [80,81].
with
{,T] • ~
is dense in
LI(G),
g(G).
for any normal state
~
(1.23)
then
, o ~t = ¢'
We claim now that
Since the set of all
the set of all
O({ ~ ~ )
Therefore, we finally get
on
g(G).
{ * "i with
o 8 ,
Q .E .D.
which means precisely (I.19).
Proof.
In the general case, we have
For each
x c ~,
{,~ c
CG = (~G ® ~) ° 5
By linearity, we have
~(z), G = (¢~ ~ ~)
Corollary 1. 3 .
* = CG
To this end, we need only to check the equality
on a
is G-weakly dense in
~ • A(G),
[80].
~ e N:
and
we have
4O
= ((~
s 5~)
o ~(x),~
®,
=
sc> = ( z , ~ > ( ~ ( x ) , ~
o
(l,~)<e~(x),~)
which means by definition that Proposition 1.4.
®
.s. %),
(z.:b)
= (e~(x) ~ z , ~ , ~) ,
g~(x) e 5 .
With a co-action
gs(sup xi) = sup gg(xi)
,
Q.E.D.
5
of
G
on
~A~ we have the following:
for any hounded increasing net [xi} in ~+ ;
,~ 5~'n ~ ~-_ ~.~ ;
(1.2~)
~.~(~xh) = a~.~(x)b ,
(1.27)
(z.2~)
a,h ~ n "~~ x e
p5 ;
the map : x e ,~ ~ gg(a xa) is q-weakly continuous for every
Proof .
The normality, (1.25), of
(1.26) and (1.27):
If
a ~ ~
g5
and
a E q8 .
follows directly from its construction x ~ ~
~len we have for each
~ ~ ~
= = (~(x*x),(a~*) hence if
x ¢ qs~
then we have xa ~ q5
;
and
--- ¢ ~ ( x * x ) , a ~ *
*..~'c_> = <ee(a"~ ~a),~> ;
thus (1.P6) and (1.~7) both foZlo~. Claim (1.2~) is routine.
Q .E .D.
41
We state here a formula which has been implicitly used, while the proof itself is routine:
(1.29)
8q~ = 80p8~ , ~,¢ e A(G) . ..... 1.5.
If
{¢i]
is a net in
A(G) D E(G)+
approximate identity in the Banach algebra
(1.30)
such that
[Z~i }
is a bounded
LI(G), then
l{mj'SP(S)*~i(X)l ® p(S)dS
8(x) =
G
in the G-weak topology for every
x
of the form:
x= 5(y) , y~u, Proof.
If
~,~ ~ A(G) ~ ~(G),
s e G ~ ~(p(s) ¢) e A(G)
®~A(a) nx(G) .
then the
has a compact support
A(G)-valued function: K = (supp ¢)-l(supp ~),
so that the
function:
is continuous and supported by
7s~ G For each
f ¢ A(G),
K.
Hence the following integral makes sense:
(p(s)*~ i
)(y) @ o(s)es = ~ . .
we compute
(p(t),J # f(s)~(p(s)*¢)~> =7
F g(t-ls)]f(s)i P A(t) ds dt ~G
= rb(~_:) = ~G(~) . Thus (1.2) yields the conclusion. (1.34):
The proof is similar to that for (1.32).
(1.35):
For a
y ~ h,
~ ~ h.
and
h,k ~ L2(G)
we compute:
G
G
G Recalling that
5(y) c R ~ ( G ) ,
we compute the following in
~(C).x. = A(G):
44
o@%((%"~2 @ O(.~(z)(1 @ o(h)))) by (e.~_8) ,
--- -~o,@,~G(~(~)(! @ p(h))) = w no @ , z @ p(h))
1 1 (Z~ ® l)w*(A 7 ® i) = w*
which gives
a x,y ~ q,~
For any
(w*(x @ v(f)*)lY ~ = (,,~ ® ¢ G ) ( ( y *
and
f,g c ~(G)
.o(g)
),o®,}G
® 1)5(x)
(l
~ p(f#
on
,
a q, @ ,
.
we have
* g)))
by
= (.~, ~ %)(~(y*)(x @ 0(~ * ~)))
(2.12)
,
by (2.19)
,
= (,~@ ~,,G)(~(y*)(l@ ~(#))(x @ p(7))) = (.~, @ % ) ( ( o ~ ~, (x) = (w*(y*@
@ p ( ~ ,--f ) ) ~ ( y -x-)( 1 ® ~ ( ~ ) ) )
p(gV))lo:_~i(x*) @ s(f,,))a,®¢ G l
= (w*(~'j;aj@ Cp(g)*)I/~ a~x @ cp(f)*)~@% I
c)(x @ o(f)*)).@%
= ((~] @ ~)w*(a~- @ z)(a @ c)(y ~ p(g)*)l(% = (x @ o(f)*l(a:,, @ C)w*(a @ c)(y ~ p(g )* )),@,,
by (2.26) .
YG
Thus
w*
satisfies
(2.25)
and so
w
is a unitary.
Using the same computation
as
(2.26) we have
1 w(A~:
Since
1 @ 1 ) w e = A g~, ~ 1
both sides are self-adjoint
operators
on
a q~ @ n~,O .
with the same core
qa ® ~
,
CG
(2.21-).
we have
Q.E.D.
NOTES
The importance is taken from [ 60].
of the integraole The integraOility
action was first pointed
out in [14]. Lemma 2.5
for the Kac algebra version was given in [ 25].
54
§3.
Inte~rable actions and co-actions. In this section we shall give a characterization
actions, assuming the proper infiniteness Theorem 3.1.
If
~G
of integrable actions and co-
of the fixed points algebras.
is properly infinite, the following three conditions are
equivalent: (i)
G
(ii)
is integrable; For any non zero projection
such that
x = fxf
f ~ T~ ~ C, there exists a non zero
x ~
and
(~.l)
(leV~)~(x)
=x®l,
or equivalently (~t @ L)(x) = (i ® X(L))*x
(iii)
[ ~ ' ~ ~ [~'~]e
Proof.
(i) :~ (ii):
z ¢ q~
with
Suppose
~
z ® i = f(z @ !)f.
(3.2) for
for some projection
Then
e
in
is integrable.
For a~Lv g { K(G)
(x{)(s) = A(s)~s(Z)- L ~ { ~ $ ® L2(G).
;
fxf=x.
, =
Since
j
z {0,
~'~-~. Then there exists a non zero we set
g(t){(t)dt we have x ~ 0 .
Since
Ilxmil2= jIl~s(Z)~ 2A(~)d~ = II~(z*z)ll II~!i2 ~ i~(z*=)ll IIg,l~ I1~I1~ , x
i s bounded on
(3.2),
then
~ ® L2(G).
[x,x' ® i] = 0
((1 ® v~(x)~)(s,t)
If
we r e p l a c e
and so
~
x ~ ~.
by
( x ' ® 1)~
with
x'
~ ~'
in
Since
: A(t)~(~(x)~)(t-ls,t) ~t)½~t(x)~(t-ls,t) £
i"
by (3.2) ,
A( t)GA( t-ls ) ~ t (~t_is (z) ) J g( r)~( r ,t)dr A( s)~Gs( z)fg( r)~( r,t)dr ((x ® 1)~)(s,t)
we have
x = fxf
(ii)----~(iii): set of all defined by
x e ~
,
satisfying (3.1). We use the
2 x 2
satisfying (3.1).
matrix method due to Connes. Let
a
be an action of
G
on
Let
~
~ ® F2
be the
55
(3.3)
"/to : ¢ x l l
Xl2~
\x21
(
~--t(XlzL)
x22/ ~ >
~t(x]A~)(1 N ) ' ( t ) ) * ~
X(t)),,~t(Xpl)
(I N
,~t(x22)
/
"
Therefore, i f x e J, then x @ e21 ~ (~ ® F o) ~. Then condition ( i i ) implies that the c e n t r a l support of 1 @ e l l in (~ @ F2)~ is majorized by the c e n t r a l support of I @ e22 in (~ ® F2)Y. Since i~,.~ is properly i n f i n i t e and ~ @ C ® e22 is contained in ((~ @ F 2 ) Y ) I ~ e 2 2 , i ® e22 Noreover,
l®ell
Therefore, w e J
is
(~. ® F2)~ ,
there exists an isometry
by (3-3)(iii)
ble on
c-finite in
Put
: (i):
~ .
e = ww*
Since
Thus
~
~
w e ~,
Then
•
is also properly infinite in
e e
~(L'-(G)),
~
~
{~(x] = {i~.~}l®p~
(~ is integrable on
~
and
by (iii).
--
--i
1 ® p e: ~'~and hence
(~
in
(~ @ F.2)~o
w ® e21 e (i~.® F2)7
implies
and (iii) is proved.
is integrable on
is integrable on
then
i ® ell < i ® e22
because
.) e in
so
(~i® F2)~.
®
p
e ~ ~-~'~,~e If
p
is also integra-
is a minimal projection -
is integrable on
~'l®p"
,~..
Since Q.E.D.
Next we shall consider the dual version of the above theorem. Theorem 3.2.
If
~%5
is properly infinZte~ the following three conditions are
equivalent : (i)
5
(ii)
is integrable; For any non zero projection
such that
y = fyf
there exists a non zero
y c
( 1 @ WG)~(y ) : y ® 1 .
(3.k) [~,5-'}
(iii) Proof. such that xj ~ q5
~ [~'~}e
(i)=>(ii): z = 5 (z)
and
for
some p r o j e c t i o n
Suppose that
for some
aj s n~ G
(7.b) where
f £ ~%'~"® C
and
e
~x
we define an operator
y
Thereex~stsanonzero
a non zero
d~.,
•
z~q5
For any
~G
by
(~
;!
71 I x
X"
~ "t~"
. . . . . . .
1Ill2
~Jz > O,
" '~®?O--
II/
< ~
z
il (i):
This follows from the similar arguments to those in the proof
(iii) ~ (i) in Theorem 3.1.
Q.E.D.
According to Theorems 3.1 and 3.2, we have a stronger result than (1.32) and (1.34). Corollary 3.3 (a) (b)
If
Proof. ~}
5 (a)
~ ~'~]e
~(L2(G)).
Then
If
G
is integrable~
is integrable, The case where
~
is properly infinite:
for some projection i ® p c
the above equivalence.
g(~(pG) = ~ .
g5(%5 ) :: h5.
e c ~.
Let
p
Since
I
is integrabl6,
be a minimal projcetion~ in
, which corresponds to a projection Therefore
G
f c ~G
through
58
Since
~ is dual,
~
= ~,(~)
preserves this equality. The general case:
Put
is an integrable action of ~(~) of
by (1.3~,).
~ = • @ F G
o~
~
and and
= ,~o~. Take a minimal projection
~. (b)
Since
f c ~.~, the induction
if
of
Thus (a) is obtained° ~ ~- (L ~ a) o (a ® ~)
~8
is properly infinite.
p ~ F
on
~.
Then
Therefore
and consider the induction
~l®p
Then (a) is proved. Similarly as above.
Q.E.D.
NOTES
In Theorem 3.1, the equivalence (i) (iii) due to [47 ]. [47 ]
In Theorem 2.2. the implifications
and the implication (i)
> (ii) is new.
is due to [i~4] and the rest is (ii)
~ (iii) ~> (i)
are proved in
59
§ 4.
Dominant actions and co-actions. In this section we shall introduce two new concepts "semi-dual" and "dominant"
for actions and co-actions, and show the implications : "Dominant" :=~ "Dual" = ~ "Semi-Dual" =-> "Intregrable" Definition 4.1.
An action
there exists a unitary
G
(resp. co-action 5)
v e • @ ~(G)'(resp.
(4.1)
is said to be semi-dual, if
u ~ ~1 ~ L~(G))
such that
~(v) = (v @ i)(i ®V'G)
(4.2)
(resp.
~-(u) = (u ~ i)(i ~ WG) )
It should be noted that if Moreover, if
~
or
~
t~ (resp.
'5) is dual, then it is semi-dual.
is semi-dual, then it is integrable by Theorems 3.1and3.2.
Condition (4.2) is equivalent to the condition Chat for each exists a unitary
u(t) 6 ~
such that
t ~ G
there
5(u(t)) = u(t) ® p(t).
From the above definition the following corollary is immediate by considering the fixed point subalgebras. Corollary 4.2.
If
and
~
(4-3)
m x
G ~- ,~i ~ @2(L2(G))
;
(~..4)
'n x 5 G % r ~ ~ ~(:B2(G))
.
Definition 4.3.
~
An action
are semi-dual, ~hen
(~ (resp. co-action
5)
is said to be dominant
if (i)
Q
(resp.
5)
(ii)
~
(resp.
I%5)
is dual; is properly infinite
In the above definition,
(4-5)
condition (ii) implies
{g,,,(~} g {~.,~--} (resp.
If we combine this with
(4.6) a
[~,?)} T ~ , ~ } )
•
condition (i), then
[~,,C~] g f~,~} (resp.
f o r any dominant
.
(resp.
[~t,~} 7 {~,~)})
5).
Now, we shall give an analogous, but a stronger result as Theorem 11.2.1 and I1.2.2. Theorem 4.4 four
If
~,~
(resp.
~15)
conditions are equivalent for
is properly infinite then the following
C~(resp.
5):
60
(i)
It is semi-dual;
(ii)
It is dual;
(iii) It is dominant; (iv)
There exists a unitary
w
K
~[ ~ ~ ~(G)' (resp.
G(w K) = (w K X 1)(L ® V~) , i . e .
u ~ N ~L~(G))
(o t @ L)(W K) = wK(I @ k ( t ) )
such that
,
(reap. ~(u) = (u ® 1)(L S WQ)) Proof. Thus
(i) => (ii):
and
5
Combining (4.5) and (h.1) (resp. (4.2)), we have (4.6).
are dual.
(ii) ~-> (iii)
and
(ii)~(iv):
(iv)~
(i):
obvious.
There exists a unitary
w c ~ @ p(G) (resp.
u (~
~ L~(G))
such that ((~t @ L)(w*) : w*(l @ p(t))
by ( 1 1 . 2 . 7 ) ( r e s p . and
(II.2.3)).
wK = (i ® K ) w * ( l
Here, we put
® K).
w K ~ ~,~ ~(G)'
Example. dual.
Let
: £(L2(G))
[(u) = (u ® i)(i @ WG)) /O
(K~)(t)
-1
= A(t)l/~(t
)
e L2(G)
for
Then (Gt @ ~ ) ( w ~
which implies
(resp.
and
= w~l
~(w K)
@ X(t))
,
- (w K ~ l)(l ® V~) .
Q.E.D.
We shall give an example of an action which is semi-dual but not
G
he a locally compact abelJan group and
G
its dual group.
Put
and
(u(~)g)(t) = g ( t - s )
s,t ~ o
(v(p)~)(t) = (t,p)~(t)
for
~ ~ L2(G).
Let
G
p ~ G
Then the commutation relation
be the action of
G × G
on
~
defined
u(s)v(p) = (s,p)v(p)u(s)
holds.
by
c~(s,p)(X) = Adu(s)v(p)(X) . Then
G(s,p)(U(t)v(q) ) = ((s,p),(q,t))u(t)v(q)
other hand,
~G = C.
by Theorem II.2.2.
Therefore, if
and hence
o: is dual,
is semi-dual.
On the
~. must be isomorphic to
L~(GxG)
This is impossible for a non trivial
G
G.
The situation will be clearer by introducing the concept of regular extensions. Definition 4.5. t -e G t
Let
(~(.) : t ( G -+o:t e Aut(~)
a homomorphism up to inner automorphisms.
(s,t) ~ G x G -~ u(s,t) ~ •
of unitaries satisfying
G s o G t = Adu(s,t)
° Gst
he a Borel mapping
For each Borel family
with
61
the von Ne~m~nn algebra
• x~, u G
generated by
(~(X)~)(t) = ~t(x)~(t)
is called a regular extension of If we define a map
~
on
,
~
~(~)
and
pU(a),
(pU(r)~)(t) = u(t,r)~(tr)
by
G
~x~, u G
with respect to
~
where
,
and
u.
by
a(y) = Ad I g W ~ (y g i) ,
then
a
is a co-action of
G
on
~ Xo,uG
by direct computation.
The relation
between a semi-dual co-action and a regular extension is given by the following, which is also an extension of Thcorem 11.9.1. Theorem 4.6.
Let
5
be a co-action of
G
on
and
u.
N.
The following three
conditions are equivalent: (i)
[N,5} % [~ ×
uG,a]
for some~,a
(ii) There exists a Borel map
t ~ G -Tu(t) e N
with unitary values such that
5(u(t)) = u(t) @ p(t) , t ~ G. (iii) There exists a unitary
u
in
N ~ L~(G)
such that
~(u) = (u ~ ~ ( l ~ W G ) .
We leave the proof to the reader.
NOTES
~e
general theory of dominant actions is developed in [14].
The part of this
section dealing with an action should be viewed as a supplement to [14; Chapter III]. A
characterization of semi-dual actions is adapted from
[50].
CHAPTER IV.
SPECTRUM.
Introduction.
The Arveson-Connes
theory of the spectrum of actions of an
aDelian group was very successful and played a vital role in the structure analysis of a factor of type I17. In this chapter, we shall try to generalize to the noncommutative Since a co-action for instance,
~
is in essence an action of a commutative
it is natural to expect that the dualization
theory would Oe smoother than the non-commutative §i that this is indeed the case. spectrum
F(~)
group of
G.
on
are introduced For a dual
C x~G ,
G .
by amending
p(8)
V. 2.9
Of Theorem V.~.o.
sp(8)
of
We then prove that F ($)
It will be seen i~ 5
and the Connes
is a closed sub-
is shown to be the kernel of the restriction of
An example
This misoehavior
object~ A(G)
of the Arveson-Connes
generalization.
Na~Lely, the spectrum
there.
~ , F(~)
Theorem 1.5.
subgroup of
and
this analysis
case including the dualization.
of
shows that
P(~)
F(~)
can fail to be a normal
will be corrected in the next chapter
to the normalized Connes spectrum
2n($),
see Definition
Section P is devoted to a non-commutative version of the Arveson-Connes in the campact case.
Here we employ 7{oocrts' apparatus.
abelian case is then replaced by G, where Msp(~)
~(V)
~I3~(V) is defined by (2.8).
The eigensubspaee
for a unitary representation Making use of
is defined and serves as a non-commutative
~V),
theory in the
[V, ~V }
the monoidal
of
spectrum
version of the spectrum of
for an abelin group. Section 3,
the connection
between
F($)
and the center of
This relation will become sharper in Theorem V.2.6. ×~G
is not satisfactory
yet.
N ×sG
is given.
The analysis of the center of
We do need further effort in this area.
Section 4 is devoted to the analysis of the relation between co-actions and Rooerts actions. Theorem 11.8.
It will be seen there that these two things are indeed equivalent~
63
§i.
The Connes s~ectrt~u Following Connes'
of co-actions.
idea,
[12], we shall introduce the essential
co-action named the Connes spectrum, recall the duality If
is non zero, then the following
x = p(t)
(ii) (iii)
for some
are equivalent:
5G(X ) = x ® x, i.e., <x,~¢) = (x,~)<x,~>
for all
~,~ ~ A(G).
W~(x @ I)W G = x ® x.
A(G)
and vice versa.
the hull of a closed ideal
of
supp(x) = ~
p(t)
The support
in
A(G)
supp(x)
[g ~ A(G) : 5G, (x) = 0].
(SG,o(x),o)
Of course,
three conditions
t e G.
It is known that the annihilator ideal of
First of all, we
theorem for locally compact groups:
x .z g(G)
(i)
and show some properties.
spectrum of a
= (SG(X),~
if and only if
x = 0.
®
is the regular maximal
of
x < @(G)
Here
~>
5G, ~
is defined as
is defined by
~ ~. A(G)
,
.
The only if part is a Tauberian
theorem. Let
C j , ~ ~ ~.
be a linear map of
) : x Adu(t) , whose
is called the dual action of the Roberts action
{P,9}
and
82 Lemma 4.5 Let g, then
~
be the dual of a Roberts action
[p,.q]
of
g
on
~.
If
[~,~# ~
(i) {~,v(~Q]~ ~(~ x ~) (ii) { ~ , ~ ~ {~,v(%O} Proof (i) Let [~,~a} be a trivial representation of G with dim ~ = 1 and a 0 a normalized vector with ~ ~ = Ca 0 . Let [Cl,..,Sn] and [~l .... '%} be " the corresponding orthonormal bases of ~ and ~.~ respectively. Take a ~ ~ G ( ~ ® ~,~)
and
~ ~ JG(~ ® ~,~) aC
Since
=~7
o
(aaol ~j ® ak) = 5j,k,
j
so that
®cj
-" = ~ s ®~ ' aC'o + j j •
it follows that
(l ®a*)(~lQ(%~)
=(l ~ a * ) E ~ : j ® ~ j ~ , : ~ o J
Using our assumption (1T @ a*)(a (9 1D
[v,~
: i
: [~ @ ~ , %
and hence
o~(T~i~,~(a)*)q~@.~,~(~) = i .
(4.17)
From the above assumption, ~ ~w
we have
and
b c JG(,W' ,w),
V(Co) = i.
Since
V(b~) = p(~],w, ~(b))V(~ )
for
it follows that
p(~_
(a)) = p(r,_
=~
(a))V(~o) ~ V(a~o)
V(Zj ® aj) =~V(~j)V(~j)
,
and hence, by Le~na 4.4,
E V(aj)V(aj)* = ~ V(~j)p(r,_
= ~
P(Pw(h_
(a))*V(%)
(a))*)V(~j)V(~j)
= p(%(rL_
(a))~)v(~-%)
= p(%(r~_
(a))%,~,~(a)) = l
TT~L
by (4.12) ,
by (4.Z7)
6
83
Therefore ~ ~
V(~_~
by
is a Hilbert space in
(4.16),
(ii)
it follows that
~ x
g.
{~,V(~}
____~t(V(~)) = V(~(t)~)
Since
for
~ ~^(~ x R). P P
It is clear from ~^
V(9) Pt(V(g)) = V(~) V(w(t)g) : ( ~ t ) g l g ) l
for
~i
Q.E.D.
~ ~rr" G
In the rest of this section we assume that
is compact and
dt
the normal-
ized Haar measure. Proposition 4.6. Proof. that, if
~)P = p(~) . P By (4.16) it suffices to show that
[,~,~
(~ x
~ ~
is irreducible,
~J
or
~
'
(n x
then
P
~)P c p(n).
First we recall
~(i) :
Let
U(~,)
such that (1.2) holds.
from this easily.
~ - .,"~C~)' '~, ~ ×op)^-
We consider the restriction
of the
h'
dual co-ac~,ion ~ K
of
G
contained for every
to
such that
@
.
e ~ K •
Let
in the interior of t c K
be a p r o j e c t i o n
L
L
such that
and
e ~ L.
clt(q) q = 0
/
from the fact that
¢~(x (t))(l ~ p ( t ) ) d t
•
= [0]
all
0
f e A(G)
G
t e L .
such that
such that
for every
subset is
K
f(t) = i
x c ~(K).
Let
q
We then have
x ¢ ~,,x O. o
(1.3)
Hence,
If
for every compact
subset of
~f(x) - x for
q~f(x)q
This follows
9'~K)
be a compact
and supp ( f ) c L~ then
(1.3)
x :
We claim that
holds for every
x
of the form
^
for every
x e o~(K),
we have
~G xq - q~f(x)q - 0 .
But condition (ii) ~(K)
= {0]. (i)
says
Hence, ~o = ma([e]), = ~ D )
@ (iii)
u(N)
> 0
.
Since
N
%m~ler the projection
to the second
cross-section
theorem,
such that
~ H %.'
t 6 L]
= i.
Thus
•
HVt
[e] ] - N •
P:t%'=D',tl component
T ,N
there exists a measurable
-[e]. %,
P:
is the image of' the BOT'el set
[(t,'y) ~ G X
h
for every
: Set [~e
Suppose
= O
that V [q :(Tt(q)q
We then define
a
=e Y
e] c o x
r
is analytic. C-valued for
By the von Neumann
function:
~ ~ N
~ c N ~ h
and an operator
6 G u
on
104
~
r,2(G)
as follows:
(u{)(%,,t)
: {(,,,th
-i t
) ~,,
where we consider the central decomposition
' @ ~ ~ @ L2(G)
of
'
¢-:
/~ ~ ~v)d.(o~) It is then easy to check that
u ~ o(~)'
O(~!×oG)
but
u / q(~). Q.E.D.
A n important
consequence
of the theorem is that the freeness of the action of
each individual group element of property,
Consider
G~n
can be constructed
; l~ )
action on
Then consider any non-transitive G = G1 × G~n;l~),
[p,u]
F = Pl x l~n ,
transformation
.
l~n
action
-~ of
and if
o
equipped with the Lebesgue measure t'ree action
I~ = u I × m
Let
If
G
.
{?i' 7'1' GI}"
m
Set
We then get a n o n ~ r a n s i t i v e acts freely on
property.
be an abelian yon Neumann algebra equipped with an
Auto(C ) = [o < Ant (Q) : o-,:.~ t - .~'tc, t e G]
is faithful, then the action
.,~ satisfies
the conditions
is ergodic on (% in T h e o r e m 1.1
[C~., c~} •
Proof.
Let [f,u]bc~ t h e G - m e a s u r e
T h e o r e m i.i. action of
W e note that
G .
space considered
H = Auty(O)
acts on
then
follows
~l(p - N) = 0
.
[_~]
in the proof (iii)
that
N
For
each
is
invariant
compact
set
!e]
} •
under
H; h e n c e
either
K
G
e ~ K ~ set
in
with
~ (ii) of
also and commutes w i t h the
Set
N = {7 s ~:HT/ It
A more precise
group such that each single group element
1.2.
G •
case.
co,mutant
as follows:
ergodic group and
the relative
the descrete
But it does not have the relative commutant
Proposition
for
does not ~aarantee
which shows a sharp contrast with
example of this phenomena
ergodic
G
u(N)
= 0
or
105
rK It follows that ~(I'K) = i
or
FZ
=
{To
r:Kn
H
is open and H-invariant.
FK = ~"
If
s ~ e,
~ e FV(s).
Hence
[V(si) : i = 1,2,...,n}
of
YV(s)
K,
PK~
of
fK
y e r
s
such that
F.
D , we have K
of
~(F-N)
~
and
s? ~ ~,
y ~ V(s)~,
so
which
Choosing a finite covering
DV(s I) U ... UV(Sn) = i ~ I D V ( s i )
sequence of compact subsets of
is dense in
such that
we have
for every compact subset
Since
V(s)
is dense in
By the Baire property of
-N = Qn=lFN •
Hence either
then there exists
that there exists a compact neighborhood means that
= ~ 3.
G G
rK / ~ ,
with
so that
e ~ K .
such that
If
~(rK) = i
[Kn]
G - [e] =
= lim ~(~K ) = i,
n
N
"
is an increa~ing
Un=IKn
' then
we have
must be null.
Q.E.D.
n
Corollary 1.3.
If
with the faithfulness
G
is abelian, then the ergodicity of
of the restriction of
G
to
C~
G
on
C~
together
yields the conditions in
Theorem 1.1.
Denote by Za(G,h(C~) ) a e Z (G,h(C~))
we set
the set of all unitary l-eocycles in
6a = Ada* ~
$a(~(x)) = ~(x)
Theorem 1.4. (i)
G.
a 6 B(G,h(%))(i.e.
Proof. (i)
Suppose that
Adb(t)((~(x)) = (x(x)
satisfies (1.1), t -> b(t)
that
.
then
is a bijective isomorphism of
a ,-, i)
if and only if
B ¢ Aut(~ x
G/~(m)).
Z (G,h(~))
onto
a
we have
for
x ~ ~, b(t)
is a unitary 1-cocycle in
(ii)
That
~a e Int(~ ×
~a = Adu
b(t) = ,~(a(t))
for some unitary
u ~ (x(D.)' ,O (~ ×
G)
r £ G.
Since
satisfies (1.1),
this is equivalent to
v £ C~
and
~
for some
Since CL a(t) £ C~.
it follows
~ = !3a.
is equivalent to
v ~r(V*) = a(r).
and
(~([~)' ,O (In x~ G).
b(st) = b(s)(L ® ps)(b(t)),
Z (G,CD)~ with
G), i.e.
G).
.
belongs to
b(t) 6 cx(C~) and so
is strongly ccntinuous and
5 a ~ Int(~ ×
We set
b(r) : ~(~ ~ O(r))(1 ~ ~(r))*
Since
For each
G/~(~)).
(ii)
Since
Z (G,%).
Then, by direct computation,
Ba(l ~ p(r)) = (Z(a(r))(l ®p(r))
li' (~ satisfies (l.l),
The above map :a - ~ a
Aut(m ×
,
x
u
in
D.x
u(1 ® p(r))u* = (~(a(r))(L @ P(r)) u = ~(v)
G for
for some Q.E.D.
106
Proposition
i.').
(i)
If
o
is integraDlc
(1.4)
(~':')' (ii)
If
G
Proof.
is
aeelian
(i)
Let
and
";.
..D _ ,, o ~..
is
and satisfies
(i.i),
then
r ~ , cl;: • dual,
then
(1.4.)
for some faithful
implies
normal
(1.1).
state
~
on
~ .
~x
As
~
is integra~le,
'~ ° Q t = rp . Let
J
o
is a faithful,
We may assume
oe the modular
that
~
unitary involution
u(t)q~(x)~_(o~(x)) .. .~ is square integrable.
for
x e ~ f
for all unitaries
.
is
2.~.
"t : :~t @ L.
~(~ ® 5 C ) ( 5 ( y ) )
If
of
P = T X~ H. An action
not.
~T.~ £(L2(G)).
the fixed points
.
of
G
cn
,
h ~
l~f G
~>1 ® f , Z ' l
defined
by t h e map:
be a yon Neumann subalgebra cf
h ~
G
x }>Adlc~(xg'l containi~
)
5(h).
is a factor, then the follcwjn6 two conditions are equivalent:
(i)
~
(ii)
~ = ~ ~
is globally
Proof.
5d
(~\~)
invarJant.
for some elosed
s~b~ro~p
~
o:°
G.
Without any loss of generality we mekv assume that
(i) ~ (ii): Let
~
be the co-action of
Then, by Corollary II.3. 3 there exist isomorphism
w
of
5(h)'
onto
G
,~n 5(h)'
an action
(h /T> C)' ×~ C
(~ of
G
such that
h
is standard.
defined by (II.~3.4). ('n
(h X 5 G)'
and an
(~ o 17 = (w ~ L) ° £.!
We want to consider the foll,'>w~r~ correspondences:
~-~
Since P ~(G).
~
is globally
Using the property
contained .in ~' ~ ( G ) . is globally
5d
(~ invar~ant.
and obtain that
~'
--~
w(?' )
invariant,
Adlc{W~
W G ~= L'~(G) @ e ( G ) , G
This means that S~nce
(~ ~
9' G)'
® C)
is contained in
we find ~hat
is globally
A d l ~ / (~' ® C)
,, invarian~, or
~s
w(~')
is a facto-r, we can apply Theorem 2.1
117
~(~,)= ( ~ % o ) ' ~ H
for some closed subgroup
cf
G.
Remembering the property
(11.2.4) cf
~,
we
have ~(y) = ~(y),
~ :~ (~ %
c)'
( ~ ) ' ~ ~,'× c
[a i
7
Since
~x)
: y(x),
u ~ N(y(~')),
then
l x ~ ~'
[,s i c
I e- >
I
~
I O.
Therefore xe i = u i l g
and hence
x = ~ xe. e 6~.
~.=H Q P~
Consequently,
is =-weakly dense in
~P. . As
(xei)ei e ~H H
q O~ ~ ~.
Thus it remains to show that
is finite and
g (~t(x))=
A(t)-ig=(x),
N
each element Ct~ ~ % ( x ) is
~-weakly dense in
wi~h
x ~ ~belongs
,,h. Since
H
to ~ = ~ ~.
is finite, the set of all
Since ~
is du~l,
--~tsH ~t (x)
~=
with
U
x ~ ~
is
dense in
~H
=-weakly dense in m ~ T
= {~t~t(y)
: y ~ ~}
T h u s ,~.~: n ~ is c-weakly
.
Q.E.D. NOTES
The concept of full group is defined for an abelian yon Neumann algebra by Dye [21] and generalized to a general von Neumann algebra by Haga and Takeda [37] as in Definition 4.1 and by Connes [12], independently.
Theorems 4.2 and 4.3 are obtained
in [21] for abelian yon Neurmann algebras and in [37] for general yon Neumann algebras.
Theorem 4.5 is obtained by Aubert
[5] for a finite group
G.
APPENDIX
To a unitary representation
[u,$}
ponds bijectively a non-degenerate
*
of
G
on a Hilbert space
representation
n
of
~
LI(G)
there corres-
on
~
in such a
way that w(f) = / f(t)u(t)dt
If
G
is abelian, then
LI(G) = ~A(G)~ -I.
,
f ~ Ll(G)
.
Therefore, a unitary representation
the dual grOup
G
algebra
The aim of this appendix is to discuss such a representation
A(G).
corresponds to a non-degenerate
*
representation
of
of the Fourier for
general locally compact groups. Theorem A.l. degenerate
*
(a)
There exists a bijection between the set of all non-
representations
u 6 £($) @ L ~ ( G )
[w,~}
of
LI(G)
and the set of all unitaries
with the associativity condition:
(A.I)
(u @ i)(~ @ q)(u ® i) : (L @ ~G)(U)
,
which is determined by the relation: (A.2)
(,(f),~> = (u,
w
of
w
:n 6 £ ( R ) . .
satisfies (A.I) is equivalent with
(u%)(t) = U(t)~(t)
and
~
to
~:
For' each
~ 6 A(G)
for
the map:
turns out to be a bounded Hermitian form.
~(~)
in
is a linear map of
for ~,~ ~A(a)
[U,~}
,
Thus (a) is a known result stated in the above.
then a bounded operator
Clearly,
~ 6 A(G)
~(@.,
£(E)
A(G)
There exists
such that
into
£(~).
According to (A.3), we have,
130
= <w ® 1,((~ ® ~)(w ~ 1))(,.~ ~. ~ ~ ¢)>
(A.5) = = <w,,~(~) ® ¢> = •
Therefore,
w
is multiplicative
Next we shall show t ~ t e A(G).
and
w
w(A(G))
is a
*
is abelian.
representation,
~(~)
namely,
= ~(¢*
for
Since
=
= <x~(~),.~>
and <w(x ~ i),,~ ® ~> ~-
it follows unitary,
that,
[x ® l,w]
w(A(G))'
= 0
is equivalent
to
x ~ w(A(G))'.
is closed under the adjoint operation.
turns out to be an abelian von Neumann algebra. by
,
Since
Therefore,
w
is
G = w(A(G))"
Here we denote the dual m a p of
w*:
Since
w
is multiplicative,
character of
This mean
A(G).
that
if
~
is a character
Since these characters
w ( ~ ) * = ~(~)
Since
that
w
w(A(G))~
is unitary,
= 0
~,
then
w*,~, is also a
we have
•
F i n a l l y we shall show that the Suppose
of
are self-adjoint,
for
*
representation
~ < R.
~ ~ f = 0
For any
for any
w
is non-degenerate.
,, ~. R, f , g { L2(G),
f c L2(G).
Therefore,
we have
~ = O, i.e.
w
is
is non-degenerate. The correspondence tation of
@~ ~(G),
A(G).
[6?].
Since
of
~
to
w:
~ = w(A(G))"
Therefore
Let
{~,~}
is abelian,
be a non-degenerate ~t follows that
~ ~
*
represen~(G) =
131
"~ E £(A(G),G) = ((i. ~)y A(G))* = 0, @'k R(G) : CI @ ~'(G) Here we denote the element in
G ~ @(G)
corresponding to
w
by
.
w.
Then we have
(A.4):
= < w , ~ > Making use of the multiplicativity
, ~A(a)
of
w
get the associativity condition (A.3). Now, the representation Since, for any
*
representatioh
of
~,lj c ~
of
A(G)
w
(A.5),
we
is unitary.
is extended uniquely to a
which is denoted by the same symbol
f,g E ~(G)
and
and the similar computation as It remains to show that
[w,R]
C (G) = C*(A(G)),
, ~(R)..
*
In,R].
,
: (,w(g ~ * f),.~ .). =,l'(w(Itlf),:~'~,q)g-V~dt we have
(A.6)
(w(~
f))(t) : w(xtlf){
, locally
a.e. in
t .
Therefore w ~ ((~f)[~(,,eg))
:/._( w().l- lf){ Ii w(k ; lg)Ti)dt
(A.7)
=,f (,~(ktl(gf)),{~{,,,>dt . For each
,& c ~(R).,
tional~ that is
the map:
f ~ C (G) -> (~,(f),{~.',> is a bounded linear func-
w~6~ is a Radon measure on
./ <w(X
G:
(f,wnem) = (w(f),'~>.
~f)),~,, )dt = /~/ (
Therefore
s)dt
(A.8) = (f I ~) /'d~*-~,T~(s) , where the last equality is obtained by Fubini Theorem, for hand,
w
is non-degenerate.
f c C (G), 0 < f < i can choose
f
so that
contains the identity. f z i
~f ~ ~ G ) .
So the weak closure of the set of
implies
Since
w(f) i i.
w
is a
*
~(f)
On the other with
representation~
we
Therefore, the first equality of
(A.8) gives us
(A.9)
({ I h) = , / d ~ *',~{,.,(s) .
Combining (A.7), (A.8) and (A.9), have the similar equality as (A.6):
we find that
w
is an isometry.
As for
w* we
132
(w*(~i ~ g))(t) : ~((~[ig)~)~ in place of it.
Repeating the same argument as above, we find that
w*
is an
isometry.
Q.E.D.
Example 1.
If
[w,~} = [WG,L2(G)},
ing non-degenerate multiplication
*
representation
operators on
then w satisfies (A.3). The correspon*o [~,L-(G)} of A(G) is given as the
L2(G): ~(~0)~ = o~ ,
Indeed,
~ -c L2(G)
.
(~(:Of,g),~,,i) = (WG(~ @ f) I "I @ g) = ((g# * f)~ ! ~I,)
Example 2.
If
G
identity such that given by
is discrete, there exists a partition
w = ~t~G e(t) ~ p(t)
w(~) = ~ t ~ G
[e(t) : t ~ G}
of the
The corresponding representation
~
is
o(t)e(t).
For any unitary representations £(~j) ~ L~(G)
"
uj
of
G
on
~j,
or unitaries
uj
in
satisfying (A.I), we consider a representation: o
t
of i
u
G
on 2 u .
.
~i ® ~o, ~ Then
~ ul(t) ~ U'-(t)
which is also a unitary in
£(~i ~ ~ )
@ L=(G)
denoted by
ui . u 2 = (i~u2)(L ~ ) ( u I ~ I )
(A. i0)
: ((~ ~ ~)(ui ~ l))(i ~ u ~) on
~i ® '~2 ® L2(G)"
representation
In = (~'Vl~'l
~ ~ A(G)
where
*
*.
,
(A.11)
U
u I .~ u 2
2•
is the unitary corresponding to
~
by (A.4).
,
A(G) R1 ®~2
133
Proof.
Put
is a unitary in
wI = w l '
w2 = w,2
and
£(~1 ~ ~2 ) ~ L®(G)"
(~¢~)
o (LCL~)
w = ((L ® ~)(w I ® 1 ) ) ( 1 ® w2).
Then
Since
o (L¢~a
~ ~) = (~ C L ~ G )
o (L®~)
,
it follows from (A.3) that
: ( ~ )
o (L~o)((w
=(~e~)
° (~)('l
i®i~i)(~®~O(w ~i)
i~i~i))
.
Again, by (A.3), we have
(i~w~i)(~)(l~w~l) Therefore
w
=(~G)(i~w~)
satisfies the associativity condition:
(w ® l ) ( ~ @ ~ ~ ~)(w @ l ) = ((C ~
® ~)(w I ~
l ® l))(l
• ((~®~)
®w 2 @ i)
o ( ~ O ( w
iei~l))(~o)(l~w~,l)
: ((~ ~ ~ e ~) o (~ ~ ~)(w I ® i))(~ ® ~ ~ ~)(i ® wn) =(~®~)(w) Now, for any
. ~ e A(G)
and
wj ~ £(~j).,
(~ ' ' ~ l * W~2> = J7
~(st)
we have
d.~ l(s) d.~2(t)
=,'/'<Wz,* z ~ pt~)d'2 *~ 2(t) =
i/"(1 ® p ( t ) , ( ~
1
where the third equality follows from (ptm)(s) Since
(~l @ ~)Wl
~j e A(G),
= 3 ( l , ~ j ) , ' /" (p(t),o.j >(in " *~J ~_ o (t ) J~
~,
(w2,~ 2 @ ~j)
J>_3 -: (I ~ w2,
v, a~ ,~j ® Co S %0j) _
J>_3 :.
Therefore,
P
' ~t ,
: which implies (A.13).
Q.E.D.
In the above theorem if representations of
A(G) (s I
on
sj(j = 1,2) ~j ~
s2)%~*
are one dimensional non-degenerate
then they are characters of
A(G)
and
f) = ,!~Sl
