Duality for Crossed Products of Von Neumann Algebras

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 <~(x),~0 = <~(x),~.'~ %>. g6 = The

N~-valued weight 'on

q~ = [x~ ~ :801(x*x) q5 = [Y ~ ~ :86(Y'Y) Ker c ~ H: m ×

exists], exists],

= It ~ G :&t = 1

A closed subgroup of

on

~I given by ~

= q~q~*. b5 = q~q6"

&,)

G.

OLH = ~(,m,) v ( c ® ~ 4 H ) " ) . ~ ( i ' \ G) = L'~°(G) 0 X(H)' ~ ( G / t t ) = L~(G) n p ( i ) ' t~ x~ (~\ a) = ~s('n) v (c ~ ~(H\ a)).

<%(y),~>

~ <~(y),:,~ ~ CO~.

CHAPTER I. ACTION7

Introduction.

CO-ACTION AND DUALITY.

This chapter is devoted to the formulation of the duality for

crossed products of yon Neumann algebras

involving non-commutative

groups.

the duality theorem for abelian auto-

To do this~ we must reformulate

morphism groups.

In

automorphism

§17 the usual duality theorem for crossed products

involving

only abelian groups is presented without proof together with its consequence structure theory of von Neumann algebras of type III. the duality principle

In §27 we shall review first

for abelian groups to pave the way for noncommutative

We then formulate the duality theorem for crossed products groups.

Here we take the Hopf-von Neumann algebra

principle.

Namely,

G

Neumann algebra

on a v o n

into

~ ~ L~(G)

phism of action

such that

L~(G) 5

of

showing that a continuous

into G

= (~ $ 5G) o 5 ,

on

~

corresponds

~

where

~(G) of

J

of a locally compact group

uniquely to an isomorphism

as an isomorphism of

o ~,

where

~G

(GGf)(s,t) =f(st), ~

into

~@g(G)

p

G

g(G) @ g(G)

5G(P(S))

and

5G

theorem~ Theorems 2.5 and 2.77 as the non-commutative

of

we introduce a cosuch that

is the isomorphism of

= 0(s) ~ p(S), s c G.

w

is the isomor-

is the yon Neumann algebra generated

regular representation such that

action

given by

and non-

approach to the duality

(~ ® ~) o ~ = (~ ® ~G)

L~(G) ~ L ~ ( G )

groups.

incolving non-commutative

We present a proof which takes care of the both cases~ abelian

abelian.

in the

(5®~)

o5

by the right

~(G)

into

We then prove the duality version of the usual duality

theorem mentioned above. Section 3 is devoted to another approach to the duality principle due to Roberts which follows more closely the spirit of the Tannaka-Tatsuuma theorem referring directly to a ring of representations. Hilbert

spaces in

merit of unitaries

Here, the notion of

a yon Neumann algebra plays a crucial role~ which is a replacein the case of abelian groups.

We shall see in the subsequent

sections that the Hopf-von Neumann algebra approach

is convenient

the crossed product while the Roberts approach has an advantage automorphism reader.

duality

groups over the former one.

in constructing

in the analysis of

Section 4 is merely for convenience

of the

§i.

Duality for crossed products r (Abelian case) We begin with discussion of a duality for crossed products involving only

abelian groups first. Let

G

be a locally compact abelian group with a Haar measure

use the additive symbol for the group product. of

G

with the Placherel measure

Neumann algebra by

~Xoz G,

[~}~

dp.

We denote by

Given an action l)

the crossed 2roduct of

is constructed as follows:

~

by

G

s

~

ds~ where we

the dual group

of

G

on a yon

which will be denoted

A representation

~

of

~

on

D ® L2(G)

is given by (i.i)

~(x)~(t) = Gt(x)~(t), x e ~,

a unitary representation

u

(1.2) Then

of

G

on

t e G,

~ ® L2(G)

~ e ~@

L2(G) ;

is then given by:

u(s)~(t) : ~(s + t), s,t e G . ~

u(G).

G

is the yon Neumann algebra on

~ ® L2(G)

Next~ we construct a unitary representation

v

generated by of

G

on

~(~)

~ ® L2(G)

and by the

following (1.3)

(v(p)~)(s)

: ~s,p~ ~(s),

s ~ G,

p ~ 8

.

We then have v(p)~(x)v(p)*

: ~(x),

~ ~ ~,

p ~ 8

(]..~) v(p)u(s)v(p)*

Hence the automorphism of X

G

phism on

= (s,p>u(s),

£(~ ® L2(G~

s ~ G, p ~ 8 .

induced by

v(p)

leaves the generators of

inv~riant up to multiple by scalars~ so that it gives rise to an of

~ x~ G,

~ X

G.

which will be denoted by

automor-

Thus we obtain an action

~

of

We shall call it the dual action.

Theorem l.l

In the above situation,

(m x

G) x~ G ~ ~ g £(L2(G)) .

The isomorphism carries the action the action ~

~p.

of

G

on

~

~ @ £(L2(G))

of

G

on

(~ ×

G) ×^ G

dual to

~

into

given by

i) An action of a locally compact group G on a yon Neumann algebra • means a homomorp?:Jsm a of G into Aut(~) such that s ~ G , ~ (x) ~ ~ is G-weakly con tinuous for each x e ~. The composition [~,G~G} or [ ~ } will often be called a covariant system. The topology in Aut(hl) should however be considered as the point-norm convergence topology in the predual under the transposed action.

where

~(s)

ks

means the inner automorphism of

2 ~(L-(G))

induced by the representation

of a on ~2(~):

(~Xs)~)(t)--~(t-s) ,s,t ~ G ,

~ ~L2(G) .

The proof will be given in the next section. the modular automorphism group~

Applying the above theorem to

we obtain the following structure theorem for

factors of type IlI. Theorem

1.2

If

~

is a properly infinite yon Neumann algebra, then there

exists a unique properly infinite but semi-finite yon Neumann algebra

N

equipped

with a one parameter automorphism group [St] and a faithful 3 semi-finite 3 normal -t traceT such that T o ~ = e T and ~ N ×~]P. If ~ is a factor, then ~ is ergodie on the center admit a multiple of

% L~(]R)

of

N.

If

~

is of type

III, then

{%,el

with translation as a direct s ~ m ~ u d and

does not N

must be

of type I I . We leave the detail to the original paper analysis

I~1

and subsequent structure

[ 14].

NOTES

Historically, the duality theorem for crossed product, Theorem l.l, was discovered through the structure analysis of a factor of type III~ Theorem 1.2, [14~ 69].

Related references:

[2~12~14~67,68,69].

§2.

DualitF for crossed products.

(General case)

In this section, we shall discuss a general duality theorem for crossed products involving non-abelian groups.

Since we do not have a dual group for a non-

abelian group, we must reformulate the duality theorem for abelian groups without making use of the dual group before we move to the non-abelian case. Suppose

G

is an abelian locally compact group.

you Neumann algebra on

L2(G)

Let

~(G)

denote the

generated by the regular representation

p

of

G,

where ~(s)~(t) = ~(s + t) ,

Denoting by

3

s,t c G , ~ ~ L2(G) .

the Fourier transform of

L2(G)

onto

L2(~),

we have

2ff_~(G)3-1 = ~(G), .~.(G)3-I = L~'(~) ,

where we c o n s i d e r L2(G))

L~(G) with and

L~(G) ( r e s p . L~(G))

acting by multiplication.

G

R(G) via the Fourier transform.

in terms-of

L*(G)

approach tells us that Namely,

L~(~

and

L~(G)

R(G)

Q(G). and

= f(s

We then see that other.

g(G)

+ t),

does also the co-multiplication

~a(o(s))

~L~(G), ~G}

L2(G) ( r e s p .

we identify

We then formulate the duality of

G

carry the structures dual to each other. ~G

which is defined by:

f ~ L~(G)

,

s,t

¢ G ;

8G:

= p(s)~o(s)

and

G,

Indeed, the Hopf-von Neumann algebra

carries the co-multiplication ~G(f)(s,t)

and

as a yon Neumann a l g e b r a on

Since we want to eliminate

,

{~(G), 8G]

s ~G

.

serve as the dual system of the

At this stage~ we see that the dual group

G does not appear in an exc!iciL

form. Let us review what we have done in the above. means that we translated the group structure of L~(G)

together with the isomorphism

SG'

L~(G) ~ L~(G);

and that of the dual group

of

~(G)@~(G).

~(G)

the same

into

G

into the yon Neumann algebra

the co-multiplication, of G

into

~ e both sysbe~

@(G)

~L'~(G),~G]

commutative di~ram: i

Indeed, the above procedure

!

7 neuL--~neueu

L~(G)

into

with the isomorphism and

[~(G),6G}

8G

satisfy

We now remove the commutativity

assumption from G, i.e.

G

is now a locally

compact (not necessarily abelian) group with a right Haar measure

ds.

(resp. k)

on

let

be the right (resp. left) regular representation of

~(G) = p(G)" (resp. ~'(G) = X(G)"). L2(G) ® L2(G)

on

I

=

L2(G

X

G)

as

G

We define unitary operators

Let L2(G), and VG

and

WG

follows:

VG~(s,t ) = ~(st,t), ~ ~ L2(G

x

G), s,t g G ;

(~.l) WG~(s,t )

For each

f g L~(G)

and

~(s,ts) .

x e ~(G),

set

"aG(f ) = VG(f ® 1)~ = A~G(f @ l) ;

I

(2.2)

• %(x) : ~(~ ® I)W o : A%a* (~ s l)

.

We then have

®O-a

a=

(~a)

.a o

(2.3) (% ® ~ ) "

~a=(Le%)

o%-

We must now formulate the notion of an action of in terms of

~L (G),aG}.

ventional sense is given. with certain properties. s ¢ G ~ as(X ) ~ ~

on

Suppose that an action

G

a

on a yon Neumann algebra of

G

on

~

in the

con-

This means that we have a map : (x,s) ~ ~ × G~as(X ) ~ If we fix an

G,

x ~ ~,

then we get an

which is, in turn, an element

~(x)

H-valued function: of

~@L~(G).

Namely, we have a map

~a :x ~ ~ ~ ~a (x) c ~ @ L'~(G) from ~ into ~ @ L~(G). The isomorphism property of each a s reflects to the isomorphism property of ~ , i.e. is an isomorphism of s ¢ G ~ a s ~ Aut(~)

~

into

~ @ L~(G). The homomorphism property of the map:

is translated to the commutativity of the diagram: TT

r~

~

:

~ ~

T,°°(G) I

1

We then discover that the isomorphism of the crossed product as in (1.1).

~

has already appeared in the construction

In order to complete or to start our program,

we must, however, prove the following first:

Proposition 2.1.

A normal 2) isomorphism

~

of

•

into

~ ~ L~(G)

satisfying

the equality:

(2.~) gives rise to an action Proof.

~

of

G

on

~

First we shall prove that

Making use of the duality of for each

f ~ Ll(G)

~

and

with ~(~)

~

= ~.

is invariant under

I~ ® ~ ;t ¢ G~.

~,,, we define a linear transformation

~f

on

as follows:

<~f(x),~)=<~(~)s~®f), Noticing that for each

fsg ¢ LI(G), h

~ ,

~ %

LI~(G)~

c

(h,f*g) = (~G(h),f @ g) = (pg(h),f) 3) , we have (~ ° ~f(x), ~

g) = ( ~ f ( X ) s % ( ~

g))

= <~(~),~.(~® g) ® f> = <(~ ® ~) • ~(=),~ ~ g ® f)

=((L®~G)

o ~(~),~®g®f)

= ((~ ® pf) ° ~(~)s ~ ~ g> hence we get under

~ o ~f = (~ ® pf) o ~, f e Ll(G).

[L ® Pf : f e LI(G)).

fi(s)ds

~(~)

[fi }

in

converges to the Dirac measure at a given point

converges to

Ps

for s 6 G.

We then define

~f = ~f,

Thus,

If we choose a net

in an appropriate sense, so that

f e Ll(G)s

Us = - i

we get

tified with an isomorphism

action of

G

on

action of

G

on

s

~.

~

by

~)s

[0fi }

is invariant under

~

of

~

into

~

of

~ @ L~(G)

G

~,

~

may be iden(2.4).

We shall

and call it an GG

is indeed an

G

with respect to

which will be denoted by

~ ×

s(~) and ~

C ® e(G)

is

(or simply the crossed

G.

2)

We consider throughout only normal maps for von Neumann algebras.

3)

Pg = I g(t)P t d t . G

~s

p.

The yon Neumann algebra generated by by

on

satisfying

In this respects the co-raultiplication

~

t ®

Since we have Q.E.D.

for an action and the isomorphism

called the crossed product of product of

~(~)

such that then

~ = ~ .

L'=(G) induced by

Definition 2.2.

s e G,

• (~ ~ °s ) o Us s ~ G.

Therefore, we have established that an action

use the same symbol

is globally i n w r i a n t LI(G)

We are now ready to define an action of the "dual" of algebra

~.

We should simply replace

commutative diagram for Definition 2.3. U @ g(G)

~.

L~(G)

A co-action of

(~®L)

product of

~

by

G

G

Proposition 2.4. ~,

by

5G

in the

~

is an ~somorphism

5

of

~

into

®~G) o ~ .

8(~) 5

and

C ®L~(G)

is called the crossed

(or simply the crossed product of

~

by

~ X 5 G.

(Dual co-action and action),

i)

Given an action

~

of

G

if we set

~(Y) = Ad(I~'~)(Y~'D ® I ) ,

(2.6)

~

is a co-action of ii)

G

on

~ ×

5

of

Given a co-action

(2.7) then

on

o ~=(~

with respect to

which will be denoted by

then

~G

Namely~ we have the following:

The yon Neumann algebra generated by

on

and

on ~ yon Neumann

such that

(2.~)

~),

by ~(G)

G

y e m×~ G ,

G, G

which will be c"8/_led dual to

on

%(x) = Ad(l®V~)(x @ 1), ~

is an action of

Proof

i) Since

G

on

~ ×5 G~

N

C{°

first, if we set

x e ~ X5 G ,4)

which will be called dual to

W G e L~(G) g R(G), 1 ®,W~

and

~(~(~)) = ~ ( x ) ® l e

G)~C

~(x) ® l, x e m,

5°

commute,

so that

(2.8)

(mx

.

Since we have W~(p(r) ® Z)W G = ~(~) ® p(r),

r ~ G ,

we get

(2.9) Equality (2.5) is also seen by checking the generators

ii) co~ute;

For each

We have

V G £ @'(G) ~ L~(G),

so that

~(~)

and

p(G)

5(x) ® l, x ~ ~,

and

hence

f e L~(G)~

we have

x(f) : ~(f ® 1)v~* , 4)

For the definition of

V6,

see the list of symbols.

of

~ × ~ G.

1 ® VG

),(f)(s,t)

where

-- f(t-ls),

so that

@.n) X

But

is nothing but

x5 G

into

w~

with the notation in Proposition 2.1.

Thus,

5

m~ps

(~ x 5 G) ~ L~(G). Equality (2.:-) can be checked by looking at the

generators separately.

Q.E.D.

Theorem 2.ft. (Dual~ty for actions).

i)

~

is an action of

(mx a G) x ~ o ~ 5 £'( m ~ ( 0 ) )

ii) action

If

~

the seccnd dual action of

G

~

of

C

on

£(L2(G)) defined

on ~ @

on

~, then

•

(F X

by

G

G) ×^ G

~t = at ® kt

is conjugate to the under the above

isomorphism.

on

Lemma 2.(~.

m ~ ~(o)

Proof. Let

C (G,.~rO denote the C~--algebra of all continuous h.-valued functions

G

a(m)

(c ~ f'(o)).

v

vanishing at infinity with respect to the norm topology in

C (G,~)

is naturally identified with the

any two distinct points tions

=

f

and

~

s,t

in

G

and

C -tensor product x,y e ~,

with compact support such that

f(t) = g(s) = 0.

Let

a = ~-l(x)

and

~.

Of course,

C (G) @~. ~.

For

we choose two continuous funcf(s) = g(t) = i

b = a~l(y).

Set

and

~(a)(l S t ) + a(b)(l $ g)

z e a(~) V (C @ L~(G)). We then have

~(s)

= %(a)f(s)

z(t)

=

y

+ %(h)~(s)

= %(a)

= ~ ;

.

Therefore, the partition of unity shows that every element of

a(m) v (C ® ~'(C)).

a~proximate~ by

Proof of Theorem 2.5.

a(m), C ® g(G)

and

By definition,

o ~(~).= Q(~L) ~ C, ~(C ® ~(G)) = C g' %G(~(C)) ~ £(L2(G)) ~ £(L2(G))~

is well-

Q.e.D.

Thus, our assertion follows.

By the previous lemma,

"C ® L"(G).

C (G,~)

~ ~ £(L2(G))

(~ ×a G) ×& G and

is generated by is generated by

C ~ C ® L'~)(G). in the algebra

we set

~(x) = (l~Vo)~(a ~ ~)(~)(i ~ vc), ~ ~ m ~ Z(L2(G)) •

(2.12)

We %hen have

f

xe~;

V(~(x))= )(x) ® 1 , ~(1

® o(r)) ~(1

® f)

= 1 ~ p(r) = 1 ~1

~

~ o(r) r ,

,

r e G ; f ~ f'(c)

.

Thus, we get S

£

G

•

we

q(m @ £(L2(G)~ = (m X

G) X~ G • Since

f

for each

xc~;

~ o ~(X) = ~(x) ~ i ,

(2.14)

~(l ® o(r)) = 1 ® p(r) ® I ,

~(i®f) = i ~ Applying

(~s ® ks) o ~ = ~

have

~ ® t

~(f) ,

to the above, we see that

Proof of Theorem

i.i.

Suppose

r~G;

G

W

.

f e D~(G)

intertwines

is abelian.

nihg of this section together with Proposition 2.1

~

and

~.

Q.E.D.

The discussion at the beginand Theorem

2.5 yields the

desired conclusion.

Q.E.D.

In order to put Theorem 2.5 in the form s3amnetric to the next result, we express the action

~ of G

on

~ @ £(L2(G))

in the following formula, which is

directly checked by looking at the generators: (2.15)

.~x) = Ad(l ® V ~ ) ° (L @ ~) ° ( ~

Theorem 2.7.

(Duality for co-actions)

If

~)(x), x c ~ 9 £(L2(G)). 5

is a co-action of

G

on

B,

then (2.16)

8(y) = nd(l ® WG) O (~ ~ G)o(8 ® g)(y),

is a co-action of

G

on

N ~ ~(L2(G))

y e ~ @ £(L2(G)) .

and

{(~ x 8 a) ×g G,~} ~ ~ ~ ~(L2(a)),g} . The map

8

defined by 5(y) = AdI~w~ (y ® i) , y e £(R ~ L2(G))

is a co-action of the co-action

G

5G.

Lemma 2.8.

on

£(R ® L2(G)),

We shall use this

(i)

If

S

converges to (ii) E > 0

and

1

If

for each

in the following lemma.

[$K : K c S}

in

A(G) n W(G)

G

ordered by

such that

and

~j ~ E(G,~)

there exists a

for

j = 1,2,

~ c A(G) n ~(G)

then for any

such that

I((5~(yj) -Yj)~j I qj)l < a , where

5~

is defined by

SK(t)

t e G.

yj e £(R @ L2(G))

qj e R ® L2(G)

5

is the set of all compact subsets of

set inclusion, then there is a net

agrees with

whose restriction to Q(G)

(5$(y),w) = <6(y),~ ® $>

for

!~ e £(R ® L2(G))..

10 Proof.

(i)

Let

fO

be an element in

fo(t) For each

K ~ ~

_> 0

and

~(G)

with

/'fo(S)ds

= 1 .

we set

f fK(t) = J

fo(st)ds

and

fD

":K =

K Then

0 ~ fK(t) ~ i

tends to

G~

(ii)

and

%K ~ A(G) q Y(G).

q~K(t) comverges to

The functions

w(G × G,~) supported by on Ky~ then

~

.

fo,~K

fK(t)

Since

converges to

1

as

K

i. j = 1 ~o

(i @ WG)(~ j ~ fo)

for

K 1 × K2

K. < ~.

for some

If

are elements in h = i

h ~ W(G)+.. satisfies

I

1)(1 ® WG)(gj @ fo)

(~5~)K(yj)~j I ~j) = ((yj

(1 ® WG)(~ij ® fK ))

= ((yj ® i)(i @ WG)(~ j @ fo)

(! ~ i ~ h)(l @ WG)(,ij ® fK ))

and hence ](5.~.K(yj)~j[r,j) I ~ :l(yj ~ i)(i ~ WG)(g j ~ fo)' l[h'l J",jll , o L--norms.

where the right hand side are

Let

£

be the set of all finite linear O

combinations of elements in

£(~) @ C, C × L~(G)

and

C X,~(G).

f~:s(g)~(st) and (fo(s))* = 0s_I (f)~:(s)*, the set £o algebra of £(~ @ L2(G)). For any ~: > 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

<~(~),'~>

:<~(v),~o~>

5 (y)

for

with

,~'~.

is

z ~ £(~ ~ L2(G))

and

~P o-weakly dense in Proof. any

(i)

~ > 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))

<(~ ~ ~) o ~;(y), ,.~~. ¢ ~ ¢

, ~ ® '$ @ ~)

= (~(y), ~.~,~ ,~, ~) ~ ~)

: <~ (y)~- ~.('~ ,~ '0> = <~(':' ( 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) ~
o ,'~, it follows that

~. such that

~, = O

~ ° ~'t = ( ~ g 0t) ° &

27

a ° a t ° W = (C ® Dt ) ° ~ ° W = (~ ® ~t ) ° ~o = w°

L~(G).

on

Thus

at o ~ = ,~ o X t

(hi) and ( i i i ) ~

kt

°

on

= ~

o

w

kt

o

T]~(G).

(i): We set the map

~> 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

<~(r}(f)), ~ ® ~i > = • 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),~

®,

= <~(x),~ ~ (, .~ ~.o) =

(1.25)

%>

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

~ ~ ~

<~(a×x*xa),~ e ¢~> = <(a × e 1)~(x×~)(a ¢ Z),~ ~ ~G> = (~(x*x),(a~*) hence if

x ¢ qs~

then we have xa ~ q5


~ ,o> ;

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
G

f

¢o(t),(ts-1)f(s)ds

G

G

= ~(t)J' A(S) ~ (S) f(s-lt)~ G

~(t)XA~(f) (t) with

~ e ~t.

= m(t)J ,~(s- )f(st)ds = ~(t)fa(s)t(s)(Xsf)(t)ds G = (p(t),~O(kA~(f)))

;

42

(z i,

G

-.j G

= ~/5(y), ~ e,][" f(s)e.o(O(S)×l~i)ds) C

:/y),~

a mX~,~: i

"a," i

~lence ~.~e get, for every

<~i,~>

=

sc that Q.E.D.

Proposition 1.6. a) If co-action

~

(1.31)

of

G

on

~ ×

C ,.~mCG c b~

~

is an action of

G

on

~,

then we have, for the dual

C, the following:

and

~^(i x ) ~

= ~'G(X)l,

x emir G ;

^

If

5

is a co-action of

du~l action

b)

~

of

G

on

(1.33) cc(L~(c-) OLl(G))c~g 0-.3~)

(1.35)

G

on

~,

then we have the following for the

h x~, C:

~d ~(! ~:)= •(g)1, ~ ~LI(G) OTf(O) ;

Lg@~,) = ( , x~ c-)~"

if

g ~ X(C),

then with

~ = g~" x. g ~ A(G) ,

PJ~((1 ~ g)-×~(y)(l ® ~)) G

=

~; o ~

OD,

Cp

y ~n •

43

Proof

(1.31):

By definition, the restriction of

with the canonical co-action

5G,

i.e.,

~

to

C @~(O)

~(i ® x) = i ® 5G(X),

coincides

x ~ ~(G).

Hence

our assertion follows from (1.19). (1.32): x e m,~o,

By (1.24),

then

g~(~) c

a(l ® x) ~ p~

(~ X~ G)~.

If

a e (N XG G)~

by (1.26) and (1.3!).

and

We then have

g~(a(1 ® x)) = a ~ ( 1 ® x) = ~G(x)a . Thus, choosing

x e

so that

CG(X) # 0,

we conclude

a

g~(p~).

~

Hence

(~x~ G)s ~i~(~). ~c O

(1.33):

Recalling

(1.2.11),

we have for each

=

~ e n~

and

f e L~(G),

®~®f~A>

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

G = /(p(r)ks(g)hlks(g)k)Z~s)ds G =//g(s-ltr)h(tr)g(s-lt)k(t)A(s)dt

ds

G×G =~g(str)h(tr)g(st)k(t)ds dt GXG =/(/

gV(rs-l)g(s)ds)h(tr)k(t)dt

G×G = / ( @ ~ . g)(r)h(tr)k~dt = ~(r)(p(r)hlk) • G Thus, we get

/

~ks(g)h,ks(g)kA(s)as = ,~n,k

"

G

Therefore~ we finally obtain the following: (~((i ® g*)5(y)(1 ~ g)),(~ e ~h,k )

= (~(y),~® ,~h,k> = <~ o ~ ( y ) , ~

~ ~h,k >

Q.E.D.

Let [~,%,~]~ and [N~,$~,~¢} be the GNS representation of with n and $ on ~. Let respect to faithful semi-finite normal weights (resp. .¢) be the set of x c ~ with ~0(x*x)< (resp. ¢(x*x) < ~). We denote by S¢,~ the closure of the conjugate linear map: ~(x) ~n~,(x ), x ~ n O n~. Denote the polar decomposition by s~,~ = j¢,~ ~!/2 ,~, . Then J~,~ is a conjugate linear isometry of Dq0 onto ~ singular positive self-adjoint operator on %.

and Z~¢,~ is a non-

Proposition 1.7. Let ~ be a faithful semi-finite normal weight on ~. and = ~ o s-1 o ~ . Then

45

(i)

is a faithful semi-finite norm~%l weight on

~

(ii>

if

~

=d

~

unitary representation of u(J

G

canonically implementi~gz ~,

then

u

is the

J~ = (Jq0 ~ J)u@=

®J). Proof.

Assume that

~

n = g(O,~P,)nqf 6) Then n 0

is standard,

i.e.,

turns out to be

(x.

~),' s ) . x~(s)

[ ~ , $ ] = [i~0(~0,~0, }.

an

~(x)

,[x :r~ % (y( r - l ~ ~ ) ~"-= n(s~

s (~(s-1)) × .

=/;~(x(s))(1

',~ K ~ ) )

~x

~J

Then

~
is

a-weakly dense in

into

~ ® L2(G)

by

n:

G.

x ~ n ~,

m ×~

G

by

n~.

We denote the linear map of

x ~.,

s ~

~:

n

a .

is a left Hilbert aJ.gebra with respect to

n(x)n(y) and dense in

into

n 0 n~

Js ,

(~(~))(s) = n ~ ( ~ ( s ) ) , ~(n N n~)

We s e t

involutive algebra with respect to

We denote the involution preserving isomorphism of

Then

~ ×~ G;

~ e the modular unitar~ involutions and

~ @ L2(G).

Define

= n(x . Y), ~

and

2

:~(x)~ = n(x ~)

by

m(~]-itA it I

t c ~

by the uniqueness of standard

Since J~,.~ = J

Form

.

and so, for

(~,(x ~ ) )( s ) = a( s )i/2u (s ) ,4,,n~(x~(s-i )) -1 o

= n(s) /~u(s) Jq~n~(~-s~X(S)*)) A(s)-i/2J~n =n(~)

-i

_~(s)*) ~°g~S

o

/ ~ j s q~~co0~l - , c? ~cc," ' x '~ s ) )

= (ff~/2n(~))(s) it follows that algebra 6)

~ / JI12 A

'q(n n n#). K(G,~) = support.

,

is the closure of tile involution

The canonical weight The space of

~

on

~X~ G

#

of the left Hilbert

is given by

~-walued continuous functions on

G

with compact

46

:

Therefore,

S¢ =' 7 ~ I/2

and hence

~((~ .~ ~)(e)) J~ = 7

or

9(-

: ~ o ~-i

J~ m = ~"

o eS(~(~ ) ~(~))

.

Q.E.D.

NOTES

The g e n e r a l t h e o r y o f o p e r a t o r v a l u e d w e i g h t was d e v e l o p e d by Haagerup~ [ 3 3 ] . On the other hand, operator valued weights associated with an action or a co-action are treated i n d e p e n d e n t l y i n [11,,34,1!2,!~6,60] .

47

§2.

Integrability and operator

w.

We begin ~¢ith the definition of integrability: Definition 2.1.

An action

is said to be integrable,

in §i is semi-finite, namely, For example,

~G

~

(resp. co-action

5)

of

if the operator valued weight

and

Pe

5G

(resp.

p5 )

is

g~

G

on

~

(resp.

o-weakly dense in

are both integrable.

(resp.

gS) •

~)

defined (resp.

if).

By virtue of Proposition 1.6 a

dual co-action and a dual action are integrable. Lemma 2.2. with any

(a) If C is integrable and + ~o ~ ~*' then we have

(2.1)

(~ ® g)(O~x)) = ~(X)UG(g) ,

(b) + ,J 6 ~., o

If

5

is integrable and

x ~ ~o '

,,~i= (*~o ° g5 (or

(~o ® ~G ) ° O),

g ~ LZ(G)+

(~o ® '~fG) ° 5),

with any

then

(2.P_)

(',~® ~)(5(y))

Proof.

(a)

= '~(y) ~(1)

,

Y : PS'

~ e A(G)+

Our assertion follows from the fact that: e~(~s(X))

(b)

~ = :~o ° gC (or

We may assume that

= a(s)-~(x)

y e p~.

for

x ~ ~

.

Then we have

(~ ® ~)(5(y)) = ~(5(y)) = <~5(5(y)),(~o > =

sup< ~(5(y)) ,~o > ~ sup< 5o(5~(y)) ,~o > CeF

teF

= <~(~5(y)),~ ° ®

~> by

= <e~(y) ® 1,~ o ® ~> =(0®~)(y®1)

(1.24),

.

Q.E.D.

We now present the main result of this section: Theorem 2. 3. a unitary

w

in

If

~

is standard and

£(R) @ ~(G)

5(z) =

(w*®

Proof. integrable, @ ~(G)

Let

~ = '~o ° g5

'~ @ SG

~,

is integrable on

then there exists

such that

(2.3)

(2.~)

5

w*(z ® Z)w ,

1)(L ® q ) ( w * ®

l) = (L @ 5G)(W*)

.

~ on ~. Since 5 is o semi-finite, normal weight on ~ @ 9(G). Since

for a faithful normal state

is a faithful

is standard, it is assumed to act on the

L2-cQmpletion

of

.~ ® , '~G"

48 For any

xj e u., and ~

yj ~ u.. ~ "G

we have

2

,~- xj ~ yj.[,~G :

~*~. s y~y~)

(~,, ~ ~,~)(~(x~.~j))

:E

= (~, ~ ,~.s)(E(~ ~ ~)

b~ (~.2) ,

*~,(x~xj)(1 e yj))

= IIE~(x])(~ ~ ~)il,~a

•

Therefore we have n 5(xj)(l ~ yj) ~. ,,, , ~';'G

j=l We define an isometry

w*

(2.6)

on

w%,~

for

x. c u , J '~"

yj ~ ,. '~G

L2(n ~ 9(G),'~ ~ '~G) = ~ ~ L2(G)

(j:l .... ,n) .

by

,,(x e y) : -,~:~,,,(,~,<x)(1~ y)) .

For the moment we assume that

w

is unitary, which will be proved later in a series

of lemmas. For any

z £ ~l, we have w*(z ~ l)~.,~@,,~(5(x)(l ~ y))

, ( zx = w* ~ ] : n ~ G

~

y)

= ~L,~,~,G(~(zx)( i ~ Y)) = S(zh~.%(S(x)(1

By the assumption on ® L2(G). Now, if

w,

the set of all

~ y)) .

5(x)(l @ y), x -c ,~, y e "~G

Thus (2.3) is obtained. f,g e LI(G) n L2(G),

then

8a(,,o(fb)(1 ~ ~,(gb) ~ (p ® p)((w$~ ® ~ ) ) ~ ) and so

(2.7)

=

~.c ~ ¢o(( o ~ p)(@~(f ® ~)~e~)

= %%®,.a(~G(o(r 5)(J- .~ o(g ~))) .

is total in

49

Therefore, if

x 6 ",0 and

a,b 6 n, , then with ~G

Y = '~ ® "G ® ~'G we have

(w* ® 1)(L ® o)(w* ® l)riy((x ® 5G(a))(l ® i ® b)) = (w* ® l)qw([(L ® ~) ° (5 ® g)(x @ i)}(i ® 8G(a))(l ® i @ b)) = q%,(((5 @ I-)5(x))(l ® ,~G(a))(i @ i ® b))

= r~((~ ~ ~o)(~(x)(i~ a))(i ~ I ~ h)) = (i ® WG)qy( (

® g)(x ® i)(i ® a ® b))

by (2.7)

= (1 ® Wg)(w* ® 1)qy(x ® a ® b)

= (z ~ w~)(w* ~ 1)(z ® w~)~W((x ® ~o(a))(z ® 1 ® h)) , Q.E.D.

which entails (2.4) Before going into our lemmas, we note the following facts:

(2.8)

So( ,,,,~G) c ,,VG® '¢'G"

(2.9)

SG(.,~o)(l ®.~,G) c U~,GS,,:G ;

(2.i0)

.¢G(~(f)~p(g)*)

= ~(~(g~

. f))

.

Indeed, (2.8) and (2.9) were proved in the above proof.

Equality (2.10)

follows from the calculat~ on:

%(p(f)p(s)~)(~)*)

= (flo(s)g)

= (g~ * f ) ( s -z)

(2.ii) =(~(s)*(g ~ * f))(e)

Len~ua 2.4.

If

'~ = "~o °

(2.i2) for all

for some faithful

+

~='o ~ ~*'

.

then

(,4 ~ CG)(5(a)*(b ® p(h))) = (~ @ CG)((a*@ l)5(b)(i@ p(hv))) a,b ~ "9

Proof.

(2.13)

g5

= ~o(p(s)o(.~ ~ * f ) )

and

For any

(,~

f,g,h e A(G) O K(G),

we have

,@G)(SG(p(f))(p(g) ® o(h))) = (wlG ® ,$G)((p(f) ® i)BG(p(g))(l®p(h")))

by direct computation. hl,h 2 ~ L2(G).

h ~ A(G) ? ~(G) .

Put

Since

h c A(G),

it is of the fore

Y = ~o ® '~G ~ ~IfG" For each

x,y ~ ~,

h~ . h I

for scme

we have, by (2.10),

50

(Y ® o(g) ® ~(h2)*l(~ ® ~)(x

®

~(f))*(l @ i

®

D(hl)*)) #

= ~((~ ® S~)(x ® ~(f))(y ® ~(g) ® ~(h2~. hi))) (2.14)

= T((x ® ~(f) ® i)(~ @ 5G)(y ® ~(g))(l @ i @ ~(h I . ~2)))

= ((~, ® ~ ) ( y

® ~(g))(1

® 1 ® ~(~l)*)l(x

® ~ ( f ) ® ~(~2))*),t,

,

where the second equality follows from (2.13) and v

(2.15) Since

(h~. hi)" = hi . ~2 K(G) ~ L2(G),

it follows that

y ® p(g) ® P(h2)* c n# Since

and

~(hl)*

, p(~2)* ~ .,,, ~'G

5G(p(g))(l ® ~(~i)* ) ~ ,¢G@~IG , it follows that

(~ ® ~G)(y ® p(g))(l ® I ® p ( l l ) * ) We fix these elements.

~ .~

Then (2.14) is extended uniquely to

as a bounded linear functional of Replacing ( x ® p(f))* by 5(a),

x ® D(f). we have

Since

L2(~ @ R(G~,w ° ® ~G )

a e uS, we have

5(a) ¢ n o®@G.

( y ® p(g) ® P(h2)*l (t) ® #)(5(a))(1 @ 1 ® P(hl)*))~f (2.16)

= ((~ ® ~o)(y ® p(g))(l ® i ® p(~i)*)lS(a) ~ ~(~2)*)~ Similarly,

P(h2)* , p(~l)*~

5(a)(1 ® P(hl)* ) ~ q~ ®'@G' o

u

and 5(a) ~ P(h2) '~G it follows that

c~ uy. Since

(~ ~ O(~(a))(l ® i ® P(hi)*)~ "~ . Therefore, (2.16) is ex~ended uniquely to L2(n @ ~(G), ~o @ @G ) as a bounded linear functional of y ® p(g). Since b e qS' we have ~(b) ~ "Wo@~G" Replacing y ® p(g) by 5(b), we have (5(b) @ P(h2)*l(5 ® ~)(5(a))(l @ 1 @ P(hl)*)) T •

(2.17) = ((s ® O(s(b))(i Since

a,b c qs'

® i ~ p ( ~ i ) * ) l S ( a ) ® P(h2) )y

each argument belongs to

uy.

Therefore, (2.17) gives our desired

result (2.12) by using (2.10) and (2.15). Len~na 2.5. If w is a faithful, semi-finite~ normal weight on then

(2.18)

$ ° ~

= ( ~ t ® ~) ° ~ "

Q.E.D. satisfying (2.12),

51

Proof.

P~ virtue of ~he KMS condition,

there exists a function [~ e C : 0 < I m

If

a

~ < ~

F

for any

x,y ~ .~

® V. and f~ A ( G ) 0

(G),

continuous on and holcmorphie in the strip

such that

F ( t ) = (w®,$G)( ~

( x @ p ( A - l f ~ ) ) 5(y)) ;

F(t + i) = (~ @ ¢G)(5(y)~t

(x @ p(A-If ~)))

is an analytic

element in

(2.z9)

g(G)

with respect to

CG(ab) = CG(t~ CG_ (a))

~

CG

,

then

.

Since

¢ (2.20)

dte(p(f))

= p(A ~-:; f) ,

t c ]R ,

it follows that w F(t) = (w ® ¢G)((ot(x ) ® 1 ) 5 ( y ) ( l ®

~G v o t (o(f))))

~G = (~ ® C G ) ( 6 ( ~ ( x ) ) ( Y ~ % t ( p ( f ) ) ) )

(2.2Z)

by

(2.12)

= (~ ® %)((~ ® o G)(s(~(x)) (y ® p ( f ) ) ) ) , and

f))))

by

(2.12)

(2.22)

= (~ ® %)((y ® p(f))(~ ® ot%(~(o~(x)))) • If

z ( n ,

that

then

6(z)

llZn-'5(z)'IJ®$G

is approximated and

by a sequence

llZn*-5(z)* j@,$.G

ccnverge to

and (2.22), there exists a sequence of functions in the strip:

0
~ < 1

[Zn}

G

n

in

0.

Therefore,

~G

~a Gn(t + i) = (~ @ ,~G)(Zn(L ® o t )(5(~t(x)))) Further we set

(2.23)

G(t) = (.~ ® CG)((b ~ o ¢G t ) (~(o~(x)))S(~))

j

LG ( t

such oy (2.21)

continuous on and holomorphic

such that

Gn(t ) = (~ ® SG)((L ® a t ) ( 5 ( ~ t ( x ) ) ) z n)

f-

"~ @ "~G

+ i) = ('., ® SG)(6(z)(L ® atG)(6(gf(x))))-~

.

52

Since [Gn(t)-G(t)I

< iIZn-~,~(z)iI - -

~

;*~(ot(x))iI.;@,iI G

~I'G

[Gn(t + i) - G(t + i)l --< Iz* n -

it follows that G(t + i))

Gn(t ) (resp.

Gn(t + i))

on each compact subset of JR.

is a holomorphic extension of symbol

~ ~ ' II'?(Ct(X))-)(-II~j~I~C_

G~

such that

G

n

G

converges uniformly to

G(t) (resp.

The three lines theorem implies that there

to the strip

0 < Im ~ < l,

converges unifoz~ly to

G

denoted by the same

on each compact subset of the

same strip. Finally we shall go into the proof of (2.18). under

& @ G~ G

by (2.20)~ there is an action

~

(~ ® ~t )(5(y)) = }(yt(y)) Since

'2 = ('0 @ ~G ) ° 5~

define

~t

Then

on

5(~)

by

is an action

of

,, is

]R

on

y

invariant.

As

~(~)

of IR ,

on

is globally invariant ~

such that

y ~ n •

Hence

~

commutes with

c ~.

We

~(~q). Indeed,

,i,

.f

"G ) ° 5 o qs+t(y I~' Cs+t(~'~(y)) = (g ® qs+t ) = (i ® q ] G )

o 5 ° a '~ s ° (~t ° ° - t ( Y ) )

= Os(5(~t o ~t(y))) = ~s(Gt(~'~(y)))

and the map: indicates ~(~).

t ~ ~t(5(y))__

that

~

is the

Consequently,

is ~-weakly KMS

continuous

automorphism

of

(~ @ G t j ° 5 o G t = ( c t ~ c

for each

t ) o 5 on

The following lemma verifies the assumption on Le~ma 2.6.

If

w

y £ ~.

~ ® %G ~5(~),

w

~

Therefore,~

i.e.

(2.23) on

~t =~Jt $~G

and (2.18)holds on

~ Q.E.D.

in the proof of Theorem 2.3:

is a coisometry defined by (2°6), then

w

is a unitary

satisfying

(2.24)

w(4~ ® l) w* = :L ~ 1

(2.2~)

w*(J®®C) =(J ®c)w,

where 9,

~ and

and C

Proof.

J

is the conjugation: Let

a

If

z ~ q j,

are the modular

then

qa

operator

f £ L2(G)

be the set of analytic

and the modular ~

unitary

involution

£ L2(G). elements

in

with respect to

G uJ"

for

53

i

=

(~ @ 3_),~:o@,~G(.~,(oi/2(z))(1 @ p(h))) ±

= (~@

(2.26)

J-)>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~ = <~(z*z),,~> 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

(~
, ~ aj) = x~, z ,$fG(a aj)'i.,:~

!,~ is a faithful, semJ.-finJte,

~.

~ is integrable.

0cA(G) 3]{(G).

r~,.,¢,k(~.x j ~-" .~

in

normal weight

"~'G

(,,,i ° ~ ,)G) o 5

on

, ~% for some

+

faithful

~o ~ 'ft.. Since

,I

,'. .. E ".+,,.l,,l*,:j

<~(z*z),; ~q0) _< LzllOi(~ ~ )%.) for some

'.,~t.,),,x i''7''

~ 1 ) ! L oJe

" 7'C,(- • ,,d

< and r:er,cc,,oy

II z + (~**:~.j~. 1;

I!2

(X

(~'<'s . . . . )'

j

,,.

~'., ~.(,:i*~. ~(<" ' * , ;s '" , j )x j ) ~ " '-"..,,'.~,,... '

~',.(,~*'-~ ,; :<. ;.x.K )*~':z ~..,u*,,.).~.,. ,; <, d )x]'

2 < :lJl: -

":G(d~aJ)5(z)(xJ ~

,

>

;!

71 I x

X"

~ "t~"

. . . . . . .

1Ill2

~Jz > O,

" '~®?O--

II/

< ~

z

il<]VtjI

-¢]~

~u e

lldl12

'iJ,,

CG

iI

'

Xx.

a

~

a. "2

8'['*'@'1+'G

we have

56

Since y

q5

is dense in

h

and

z ~ 0~ 5 ( z ) ( x ®

i)/

0

for some

x

q~ .

(I

Therefore

is a non zero bounded opermto_r. In our proof we may assume that

identified with the L2-completion

5(z)(y'x @ l) for all

.

of

It remains to show that

y

is standard.

"~@'@G'

Therefore

and that

~ $ L2(G)

N' ® C

is

acts on it.

Since

(y, @ 1)5(z)(x ~ l)

y' c N' , (3.5) implies that

0 e A(G) ~ ~ G ) ,

N

[y,y' ~ i] = 0,

satisfies (3.4).

it follows from Lemma 1.5

that

and hence

Since

y

~I

~ "

z = 5 (z) for P ~-weak limit of

5(z)

is the

/50(r)'iI; (z) ® D(r)dr . For any

b e "'@G we set

(3.6)

F(r,@) : g((.%p(r).~,(z)x) @I i)(i @ p(r) ® b) .

Then F(rp@) = ~($,p(r).L(z) @ i)~(l ® p(r))~(x ® i)(l @ i ® b) = ~(Sp(r)*~(z ) ® p(r))~(x @ i)(i ® i ® b) , so that lira £F(r,,$,)dr : ~(5(z))~(x @ i)(i ® i ~ b)

(3.7)

'~ = (~ @ "%G)(Z(z))~,(x ® i)(i ® i ® b) ,

where the last equality follows from standard and

5

is integrable,

8

a o 5G = %G"

satisfying (2.3) and (2.4) by Theorem 2.3.

(3,8) Put

Since

N

is assumed to be

w

is implemented by a unitary

in z(~){~(a)

Then

~(y) = A(~ i~OXW*(~l)(y ® l) • Y = ~ @ '~'G® '~G" Since, for any

x e q5

and

a,b c "~G'

we have

((~ ® ~)(w* ~ l))(y ~ I)((~ ® o)(w ~ i)h~(~(x ® i)(i ® a ~ h)) = ((~ ~ ~)(w* ~ i))(y ~ I)TI~(x ~ a ~ h)

by (2.6),

: ~G(d%)((~ ® ~)(w* ~ l))~W((~(z ) ~ l)(x ~ i ~ h)) by (3.P) , : ,~G(d*a)((& ® ~)(w* ~ i)) lira £~(((~p(r).¢(z)x) ~ i)(I ~ p(r) ~ h))dr : ~'a(d*a) ~im / W(~((~ (r).¢(~)~) ® 1)(~ ® ~(~) ~ b)) dr

57 i

= CG(d*a) lira i ~Iy(F(r~Ij))dr ¢ J

by (3.6)

= '@'G(d*a)~iy((L ® ~3G)(5(z))~(x ® i)(i ® i ® b))

by (3.7),

= '$G(d*a)(1 ~ W~)~,~((~(z) ® 1)Z(x ® 1)(i ® i ® b)) by (2.7), = ( 1 ~ W~) (y ~ l) T~ (~( x ~I i) ( 1 ~ ~ ~ b)) ,

by (3.5),

which gives (3.4). (ii) ~ (iii): = Ad I ® W G

Let

J

be the set of all

~ , we have a co-action

o

{

(3.9)

Therefore ~ x i j

=Ad

o (~®~

®~)

~

of

o

(~

y ¢ ~

with (3.4).

G

~ ® F2

on

defined by

~)

V

v = (~ ® L @ ~)(1 ® 1 ® 1 ® ell + 1 ® W G @ e22 ) .

® eij e (~ ® F2)~

xll ~

if and only if

, x12 ¢ ~ , x21 ~ J

and

x22 E

Condition (ii) implies that the central support of is majorized by the central support of properly infinite,

Since

1 ® 1 @ e22

i ® i ® e22

in

.

i ® 1 ® ell in (~®F2)~ (~ ® F2)~. Since ~5 is

is also properly infinite in

(~ ~ F2)~.

1 @ 1 @ ell is G-finite in (~ @ F2)C ~ 1 ® 1 ® ell < l ® l ® e22 in Thus there exists an isometry u in J. Put e = uu*. Then e c-~5

Since

(~ @ F2)~. and (iii) is

established. (iii) => (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

<~Jy),¢

y ~ ~

with respect to

Definition

i.i.

is the hull of

~

c A(G)

It is clear that

5.

m

y : 5

SPs(y )

of

E

of

G

~ ~ A(G)

is considered as the spectrum

Y ~ h

~ A(G) : 5 (y) = 0}. -p

with -r.espect to The spectrum

5

sp(5)

is the of

= 0], is the closure of the union of all

Now, we shall prepare some elementary properties subset

defined by

In other words:

= ~

sp(5)

~(G)

supp(~ (y)), ~ ~ ~.

The spectrum

hull of the closed ideal

into

= <~(y),:~, ~ ~> ,

The closure of the union of all of

h

of spectrums.

sps(y), y ~ N. For each closed

we denote

(1.1)

h5(E)

= [y ~ ~ : sps(y

) C E] ,

which is a G-weakly closed linear subspace by Lemma 1.2.ii below.

Lemma 1.2. of

ft ~ a:~(t) (ii)

(i)

sP5(5 ( ( y ) ) c sps(y) ,q supp(q0),

where

supp(o)

is the closure

/ 0}.

sP5(y ) C E

if and only if

neighborhood of E. " ( i i i ) sps(y* ) -= sps(y) -1.

5 (y) = 0

for

.~ ~ A(G)

vanishing on some

64

(iv) U

of

t ~ sp(a)

if and only ~f

ha(U) ~ [0]

for all compact neighborhoods

t.

(v) (vi)

If

E

If

or

E

F

is compact,

ha(E)'~5(F)

then

is a closed subset of

G,

C h S(E["). 5 PE

then the projections

and

5 qE

given by

v{supp(xx >) : x <-~a(~.)] are central projections of Proof. Therefore

Put

my c m z

If Since

(i)

%0¢ = 0

and hence

y~,

EU~

Conversely, if such that

(E).

then

empty by (i), we have

E

5¢(y) = 0,

(iii)

Since Let

%0(t) ~ 0, mt

then

5¢(z) = 5%0(5~,(y)) = 0.

SPa(Z ) c_ sps(y ). ~ ~ A(G)

such that

it follows that

For any neighborhood is disjoint from

and

~(t) ~ 0.

we have

denote the ~deai

,~,(t) ~ 0

and

%0~ = O.

t / spa(z ) . Thus (i) is proved. U

of

supp(%0).

5 (y) = 0. %0 t / E, there exists a neighborhood

%0 vanishes on EU

%0(Y) = 0.

C

If

implies ,~.%a~(z)= O,

Suppose

%0 vanishes on

.

z = 5 (y)..

t / supp(%0), there exists a

(ii)

m

h a.

e

and

%0 c A(G),

Since

sps(a%0(y))

U

e

of

and

if is

%0 ._ A'G)

F_,'omcur assumption it follows [,:te,

t / sp~(y).,, Therefore

spa(y ) c E.

[p(t)]"

t c spa(y*),

of

A(G).

If

then

( m ) x = m , and (mr)~ = % - i we have % c m 1 and hence Y xy t. sp$(y ) c spa(y)-l. Changing the role of y and y , we spa(y ) c spa(y × )-i .

m, .

Since

t yl c spa(y ) . Thus have

(iv) Then %

U

Suppose that

~5(U~ ~ [0]

for all compact neighborhoods

has a non empty intersection with some

belongs to the closure of union of Conversely, suppose that

there exists a 5~(y) ~ 0

%0 • A(G)

for some

y e h.

spa(y ) with

t (sp(5).

such that Since

sps(y ) with

U

y 6 ~l, and hence to

For any compact neighborhood

%0(t) ~ 0

and

spa(a%0(y)) c u

of

t.

y : ~1. Therefore

supp(~) c U.

Since

sp(5).

U

of

t

t ( sp(a)~

by (i), it follows that

a (y) • ~a(u). %0

(v)

We may assume by

y £ ~la(F). For any

(iii) that

(~ (~×_

<$®(xy),%0>

and

F

%0 (- A(G)

= <6(x)a(y),

is compact.

Suppose that

we have

® ~ %0>

= limjF (5(x), a(r)×~(y)~ o(r)~> dr

where

~(r,~) = a

D(r)×-~(y)w. Therefore, ~ (xy) =

lim,.~¢ (r,#)(xJp(r) dr

.

x (~5(E)

and

65

Since sPSG(z~(r)) = sPSG(z)r and SPSG(¢)(r,$)(x)) c sps(x),

it follows that

sPSG(¢~r,~)(x)D(r)) c sP5(x)r . If

U is a symmetric compact neighborhood of

vanishes on the complement of

UF

for all

~

e,

then the map : r ~ 5:(r).×¢(y )

with

supp(¢) c U.

Therefore

sp5 (¢ (xy)) c sps(x)UF

G Since

U (vi)

is arbitrary, the left hand side is contained in We may assume that there exists a unitary

the co-action ~% such that

As in Appendix,

W

belongs to

w(C (G))'

If

Thus

W e £(R) @ g(G)

there exists a *-representation

(TT(90)~Iq) = (W,m~,q®%0),

follows that belongs to

5.

EF.

~,7 (= ~.

r
xy ~ ~IS(EF). which implements

w of

C (G)

on

From the proof of Theorem A.I, it We claim that

then for any

P~E (resp.

a ~ ~5(E)

and

qS) q0 e A(G)

we have (a~)~IT)

= (~(~)~I~7) = < W , ~ , a ~

@ ¢

= <W,®~, a ® ~ >

< ( a ® l)W,m~,~ i ® ~> = <~(a~,m~, 7 ® ~> If

%o = ~f,g,

then (aw(%o)~[7) = (W~(a.)(~ ® f ) l q

On the other hand, for any

where

¢ = wf, h.

6 ~

This means that

(a~(~,)~[ Thus,

ar<%o)~ = 0

r<%0) leaves Since

for every

(i -pE)~ 5

7) = o ,

Hence we get

~ ~_ ~ . so that

~%O)~

: w * (p~5 ® l ) W

( l - P E.)5~

Hence

PE5 { w(A(G))'

: PE8 ~ i

follows from the fact that

5 p _~ = q=.

is°a twosided module over h5.

belongs to

we have

under the inner automorphisms induced by unitaries of both in the center of

•

we have

is variant, which means that

5(p~) 5 q~

h c L2(G),

8(a)([ ® f) = 0.

a~hS(E),

W~_ ~ ( A ( G ) ) " $ ~ ( G ) ,

The assertion for

and

®g)

p °and ~1 . Hence

both invoriant PE

and

qE

are Q.E.D.

66 ,< For each projection restriction ~15,

then

of

~5 to

~e"

i.~.

where

e

Theorem

i.~.

Proof.

Since

Since

in

~'

we define a co-action

It then follows

that if

f

,%e

of

G

on

is the central

The Connes

spectrum

-(5)

runs over all non-zero The Connes f(5)

~'(~)-i : "(?')

of

,% ~s the intersection

projections

spectrum

d(5)

in

by (iii)

is a closed subgroup

to be in

F(~)

of

F(~:)sp(,~e)csp(~>e)

it is necessary

~nd suffSe~ent ~t, 5e e c: PU = e

that for ar~ compact reighbcrhood U of t and a non-zero p_.'cjeetion ~e and qu = e . Hence for any s ,-sp(~,e) and a compact neighborhood V of te

h[']e('J:) o hte(b-)]q" Hence C ~ O s p ( ~ e ) ~ o . h a v e t s ', s p ( ~ e ) . ~neorem i.,~.

Since

[U'.;]

If

5

is dual,

Fp.) = I t .q ~ C- : ~t : ~

(ii)

r(~)

h

(i)

for ne±~hhorhoods or

ts,

~..~e

on

(~ x~. .~],

and

"

Suppose that

dueulity of crossed

.

then

= r(~).

satisfyir~,

s we have

Q.E.D.

(i)

Proof.

( '0 ~ { 0 ]

form~ a t a s i s

G.

to show tne groap property.

of Le.~a 1.q~ we have only to show

For a t oC

of all

~If;.

is clearly closed ~ it suffices

e:• h 5 .

for all projections

onto

h as tile e support e in

s ~ ; e) = sp(?~f).

Definition sp(se);

e

~% is duaJL.

5 ow=(w@L)o(~, products

There exists

where

~

for action te]ls

"N5.

~

of

]I5 X , G

~qen the

us that

C. ~] --. r'n'~ @ ~ ( L ; " ( G ) ) ,

{h %

an isom(Pphism

Js an action c,f C- on

.

Q]

Therefore

[1.2)

Ker ~

We then define

an action

G = Ker ~

~ %X~

B

of

G

on

~ %5~£

~

u(t) = ~,(i @ p(t)).

=

.

If

e

(1.3)

3 t = Adu(t).

° ~t

"

Since

is a non zero central projection

enS({t])e

Consequently~ and only if

~

S

= ~

5s = L

e ~ C 5 (ii)

"

5(u(t)) in

"r~r,

= u(t) ~ 0(t), then

5t(e ) =

and

u(t) eu( t)*

tion

Then

Ch5

by:

B t ° (w ° G) = (w ° &) Here we set

= Ker (%

Since

= e~3:u(t)e :-e~%~"#t(e)u(t)

on the center of on

Ch~,,~,

if and only if ,5 is dual,

h ×- G O

if' and only if s c r(~)

•

if and only if eC,s(e], / 0

G

S

= L

on

C ~,~ "fl"

by (1.3).

~,,~"~ = {~,~}

if

for all non zero projec-

by Theorem II.2.1.

Therefore

F(~) =

67

~(~) = U(,~)

by the duality for crossed product.

It remains to show that

I"(,5) =

r(~). o y 6 ~1, z (. £( L2 (G)), "~i 6 I%., uJo,..c £(L'-(G)).

For any

and

".o 6 A(G),

we have

<(~)~o(y e z), '~'l~ "~2> = (s (y) ~ ~ , <~z e w,°> " Thus

5

: 0

is equivalent to

sp(5 e) ~= sp(~ e ~ l )

for

(~)%0 : 0.

e ~ %`5.

It should be noted that, if For general 5~ P(~)

Thus 5

Since

%~

: an`5 @ C,

it follows that

F(%) = r(,~-~, ....

is dual, then

Q.E.D.

F(,5)

is not necessarily normal and so

is a normal subgroup of

G.

F(5) / F(~)), unlike the

abelian case, as seen in the following: Examl~le. ,5 of

5G

to

sp(5) = H.

on

H

is a non-normal closed subgroup of

p(H) "

=

is an integrable co-action of

On the other hand,

generated by 1 ® p(t)

If N

p(t) @ p(t), t 6 H,

and

AdV ( l ® f ) = l ® f

A ×5 G, r ¢ i'(~) Corollary 1.6.

(i)

%xsG C

(ii)

C~X~

(iii)

i(~)

Proof.

f(~) = ker and

~1`5'

=

c, r(`5)

=

~) [

Ch

is eauivalent to

C=C~

r- tU = tH

~(G/H).

Gsince

for every

is

5r = ~ ® ,\r

t e G.

The following three (resp. two) conditions are equivalent: ~5(~%)(resp. C~,IX G c C~,0< G

@(,~i)). C

~(~,)).

G.

The equivalence

(i) ~-~ii):

then the restriction Since

= ~ tHt -I. Indeed, ~ >< G ~×~,G t~G `5 C ® L~(G) ° Since Ad v (o(t) @ o(t)) =

, we have

G c %(~)(resp. =

G, G.

The case of

of (ii) and (iii) is immediate from Theorem 1.5. 5.

Put

.,h = N ×8 G

and

~ = 5.

According to the

duality for crossed product we have only to show

Since

C~, = % ® C,

for any

x (: ~

we have

x ~ m~<-~x~l,:

~

,

wY~ich implies (l.i.-.). The case of any

~.

The proof can be done similarly as above by noticing that for

y c % y ~ ~`5 <=~ y ® i ~ ~

because

,

AdlfgWG o (L ~ o) o (5 ® L)(y @ i) = y ~ i ® i.

Q.E.D.

68

NOTES

The spectral analysis for which had considerable of factors of type III. Connes spectrum

~

in the abelian case was developed by Arveson

impact on the subsequent development The spectrum of

I'(~) was introduce~

~

G

[47].

[46] .

The

by Connes [12] and used to classify factors of

in the abelian case was determined

to Nakagami

of the structure theory

was introduced by Nakagami

type III into those of type 111X, 0 < k < i. x

[4],

in

The relation between [14].

U(~)

and

Theorems 1.4 and 1.5 are due

69

§2.

Spectrum of actions. If

G

is abelian, the spectrum of an action of

spectrum of a co-action

5 = AdI@s

° ~

of

G

on

~

where

coincides with the S

is the Fourier

unclear yet.

We shall try to clarify the situation in this section.

LI(G),

be the enveloping

G

•

L2(G)

C*(G)

L2(G).- However, if

on

transform of

Let

onto

G

is non abelian, the situation is

C*-algebra of the involutive Banach algebra

and

W*(G) the universal enveloping von Neumann algebra of the secand dual of C*(G). Now, for each

f ~ Ll(G)

we define

Ll(G).

m

of all

Since the

f ~ LI(G)

satisfying

~-weak closure of

ideal, it is of the form Definition 2.1.

m

.

~f ~ 0 in

is a closed two sided ideal of

W*(G)

is also a closed two sided

for some central projection

W*(G) es

The spectrum

namely,

a'f by

~f(x) =/"f(t)~t(x)dt The set

C*(G),

sp(~)

e

w~(~).

in

r'(~)

and the essential spectrtm

are

defined by (2.1)

sp(G) = i - e

; S

(2.2)

F(s) = inf[sp(se) :e c a s, e % O}

To see the situation more closely, we shall assume throughout the rest of this section that

G

is compact.

Let

G

be the equivalence classes of all irreducible ^

unitary representations p ~ G

of

the corresponding

G, ×p, p a G

the normalized

central projection in f / Xp(t)p(t)dt

Ep =

character of

G

and

Ep,

W(G):

.

G

Then

C~(G) = ~ p ~

CEp.

Prol~osition 2.2.

If

[U,~ U]

is a r e p r e s e n t a t i v e

of

p ~ G,

then the f o l l o w -

ing four c o n d i t i o n s are e q u i v a l e n t : (i)

E p -< sp(~). sf / 0 for some

(ii) (iii)

(s t ~ a)(X) = X(I @ U(t))

(iv)

[U,~u}

Proof. condition all

f e EpL2(G).

(i) ~ (i)

f ~ EpL2(G)

for some non zero

is equivalent to a subrepresentation (ii):

As

E

[~,~}.

is a minimal projection in the center of

P is equivalent to if and only if

X ~ ~ ~ £(~U ). of

Ep(l- e ) % O. EpL'~(G) c m ,

It is clear that

if and only if

~,(G),

&f = 0

Ep < es.

for

Therefore,

(i) is equivalent to (ii). (ii) ~,~ (iii): Suppose that

~f/ 0

Let

d = dim ~U

for some

and

[EI,...,s d}

f ~ E~2(0).

an orthonormal basis of

The~e exists an

x ~ L~ ~ t h

~U"

~f(x)/O.

70 Therefore ,.

(2.3)

is non zero for some ({I~){,

j,k.

Let

Tz

denote the operator on

~U

defined by

T~,~=

and

(2.4) Then

(U( s)sj ISk) -as(X) ds

Xjk ~

X :-j'~',kxjk 'Z'T~sj'ak X % O, X ~ ~ ~ £(~)

and

(2.5)

(a t ® L)(X) : X(1 ~ U(t)) .

by direct computation. ( i i i ) ~ ~ii): (iii).

Suppose that a non zero

satisfies condition

x ~ m ~ ~(%,)

Then it is of the form

(2.6)

X =j,k )~ Xjk ~ Taj'~:k

by fixing an orthonormal basis

[al,... ;~:d} of

(2.7)

at( Xj k) = ~

~j. Condition (iii) implies

U(t)~kla~)Xj~

and hence Xjk =/.!U(t)gkl~k )-at(Xjk)dt . Since

X / 0, Xjk / 0

for some

j,k,

which implies (~i).

(iii)--~ (iv): Using the same notations as in the proof of fix some map of

j ~U

with

Xjk / 0.

Put

to the subspace of

a k = Xjk ~

for

spanned by

k - l,...,d.

(iii):> (ii),

Let

[a k: k=l,...,d]

V

we

be a linear

such that

VE k = a k.

Then (2.7) implies atVS k =~ (iv)=~ (iii): Suppose that [~,~].

Let

V

iU(t)SklS~)VS ~ = VU(t)£ k . [U,~ U]

be the intertwiner of

is carried. Let {£1 ..... Sd} k = 1,...,d. Then

~U

is equivalent to a subrepresentation of onto the subspace of

be an orthononnal basis of

~U

~

on which

U

and

a k = Vg k

for

~t(ak) = VU(t)s k ::~ (U(t)Skla~)a ~ •

Put Xjk = a k for all j,k. x(l~u(t)) a~d X ~ 0 .

Then

×

defined by (?.6) satisfies ((~t @ L)(X) = Q.E.D.

71

For each irreducible unitary representation (2.8)

[U,~ U}

Proposition 2-3.

Assume that

unitary representation of

G,

we denote:

~a

is properly inifinite.

There exists a representation

(ii)

There exists a unitary in

Proof.

( i ) ~ (ii):

with

corresponds to

s.

If

[U,$ U]

is a

Let

[a,R}

in

M (~)

equivalent to

[U,~}.

ha(U).

Suppose that

dim ~o = dim R.

orthonormal bases of

[U,~u}.

then the following two conditions are equivalent:

(i)

~

~a,R} % [U,~u}.

Let

~o

Iv I ..... Vd}, [w I ..... Wd}

~o, ~ and ~U'

respectively.

and

be a Hilbert space [~i ..... ad }

We may assume that each

be wj

in the above equivalence:

(2.9)

(U(t)gj I gk) = wkat(wj) . Fow, we define an isometry

(2.1o) Then

G

ma(U) = Ix ~ m ~ £(~u) : (a t @ g)(x) = x(l ,2.u(t)), t ~ G} ,

which plays the role of eigenspace associated with

in

of

V

V~ = ~ V-] ~ j

® sj = ~ v j ~ j .

of

~

onto

~ @ ~U:

vj~ @ gj ,

~ ~ ~ .

Here we set W = VZw.v.* V-1 J J

Then

W

is a unitary and

~ ® ~U:

W = ~ vj*wk ® T~j ~ k ~ ~. @ £(.~U) • Using (2.9), we have

(a t ® L)(W) .... vJt(w~) $ TSj,g ~

= E vj*WkWkat(w~) ~ T~j,~ = ~ vjw k ® T

* j ,U(t) ~k

= W(I ® U(t)) . (ii)----~(i): space in

~

with

Suppose that

W

dim ~o = dim %

is a unitary in and let

.~.~(U). Let

Iv I ..... v d}

and

~o

be a Hilbert

[Sl .... 'gd}

Y2

are orthonormal bases of onto

~ g ~

by

~o

(2.10).

and

~U~

w.

Then

[w I ..... Wd]

respectively.

Define an isometry

V

of

Put : v-lw

V v..

is a basis of a Hilbert space

2

in

~ (~).

Indeed, since

(~t e ~) ° Ad V : Ad V ° qt

I~:k)VkVj

V-I(1 ® U(t))V = ~ (U(t)aj j,k by direct computation,

it follows that

~t(wj) = ~t(V-~TV)vj

= (V-~V)

(V-I(I ~ U(t))V)vj

(2.11) =~k (U(t)gj ]gk)V-Iwv~< Moreover,

(2.11) shows that

The monoidal spectrum that

~(U)

=,~,U(t)ajk I gk)Wk •

[U,~j} ~ [~,R]. Msp(~)

Q.E.D.

due to Roberts is the set of all

contains a unitary associate(~ with a representative

The following theorem ~dll give a sufficient condition for

Proposition 2.4. ergodic subgroup

g

Assume that of

Aut(~)

~+

[U,~ U}

such

of

p.

Msp(~) = [p ~ G:E p-< sp(~)].

is properly infinite.

com~uting with

p ~ G

~t' t ~ G,

If there exists an then

^

pProof.

By Proposition 2.2,

fore, it suffices to show that we set

~ = ~ ® ~

z2

~(U)

of

belong to

on

Ep <_ sp(~) Ep _< sp(~)

k ~ £(~U).

~.

l~Jt PU = P,t~ £(~U),

~(U)

p e Msp(~).

Then the left support

are both invariant under

C ® £(~).

is equivalent to implies

Since

g

zI

ThereT ¢ g

and the right support

is ergodic on

~'t = (~t ® L

% [0}.

For each

and

~,

zI

and z 2

~t = +t g AdU(t)"

Since

(e U) ,~: (u)(, u ) : m~(u) , zI

and

z2

Adl~u(t)(z2)

are central in

(J U)

= ~t(z2) = z.o ~ an~

and U

infinite, we can choose a unitary in Proposition 2.5. [p e G : E p-< sp(~)}

Assume that and

~

(+JTT)~, respectively.

is irreducible,

z 9_ = i.

Thus, z I = i. Since

~.~ is properly

.,~+(U). ,,~

Q.E.D.

is properly infinite.

is faithful, then

Since

~

is dominant.

If

Msp(~) =

73

Proof.

As shown in Theorem 1.3.4,

the set

C (G)

of all functions

fx,y

defined by fx,y(t) = x ~ t ( y ) , is a *-subalgebra of fx,y(t) = fx,y(e) f, t c H}.

C(G).

for all

Moreover, if

Let

t e H,

C(G)"

= C(G).

representative of

p

and

then

~t = L

R c ~(~). ~t = ~

on

on

~.

Then

t e G

A ~ ~(~). Since

for all

p e G.

[~,q}

Thus

~

~

p e G,

Since

is faithful, %(~)

H = ~e} contains a

is dominant.

is a Roberts action of

is a representative of

such that

Ca(G ) - = ~fe C(G) :k~f)-

The Peter-Weyl theorem then implies that

Proposition 2. 5 . If [~,~} 6 ~ (~)

,

be the closed subgroup of

x,y ~ ~

Msp(a) ={p 6 G : Ep ~ sp(~)}, and so

H

x,y ~ R, R ( ~ ( ~ )

Q.E.D.

~(~)

on

h

and

then the following two conditions are

equivalent. (i)

Ep ~ f(~).

(ii) epR(e ) ~ 0 Proof.

for all non zero projection

e ~ C

Condition (i) is equivalent to

(e ® 1),~.~(U)(e ® l) % 0

for all

e £ %.~,

where

Proposition 2.6. generated by

~

Lemma 2.7. tion

[~,D } Proof.

of If

If

and If

{U,~} ~ [c~,R}. ~

is dual and

Q.E.D. ~

is properly infinite, then

~

G

is

is semi-dual, then for any irreducible unitary representa-

there exists a unitary in

~

•

~(,T0 .

~.~(~).

is semi-dual, there exists a unitary

(s t ® t)(w) = w(1 ® p(t)).

If

G, there exists a projection

[w,~w} e

in

w

in

~ @ R(G)

such that

is an irreducible unitary representation of

R(G)'

such that

[.w,~ } = u[p,L2(G)]e u* for some isometry is a unitary in

u

of

eL (G)

~. @ £ ( ~ )

onto

and

Proof of Proposition 2.6.

% . Since

(~t ~ ~)(v) = v(1 ® ~(t)). Combining Proposition 2.3 and L e m m

any irreducible unitary representation in

~(,~.). Let

bases in (2.12)

%U

and

[a I ..... gd] ~,

e { ~(G)', v = (i ~ u)w(l ~ u*)

and

[U,,%U]

Iv I ..... Vd]

respectively.

If

of

2.7, we have for

an equivalent element

[~,~]

be the corresponding orthonormal

x c ,%,

,F(U(t)gJ I gk)-~t(x) d t =

G

Q.E.D.

then 9~ g~(xvj)v k "

74

Let

p

be an element in

G

associated with

~U,~I~.

The set of functions

(U(t)~ 1 ])

does not depend on the choice of a representative of

denoted by

f~

nom

dense in

• ~(G)

The linear span of such functions with by Stone-Weierstrass theorem.

combinations of elements of the fo~u (2.12). and

]::~; x

Ccnsequently,

~ (m).

p,

~,~i ~ ~

f~

,(t)=

which will be and

p ~ ~,

is

is approximated by linear ~

is generated by

~

Q.E.D. NOTES •

I'L

i

Propositions 2.6 and 2 8 are due to Roberts [ppi and Propositions 2.2, 2.4 and 2.5 are taken from [~j].

75

§5.

The center of a crossed proauct and

:'(5).

In this section we shall give a necessary and sufficient condition for a crossed product to be a factor.

The following proposition gives us a sufficient condition

for the fixed point algebra to be a factor. Proposition 5.1.

If

~(resp. 5)

is dual, then the following two conditions

are equivalent : (i)

% × G C &(,~,), ( r e s p .

(ii)

C

Proof. [~,5}.

F(5) = G);

, (resp. C ~, C % ) .

C ~

The case of

[~L,~] : Since

~

is dual,

[,~.,s~ % [h ×5 G~ ,'q}

for some

By means of the duality theorem for crossed product and Theorem II.l.l, it

suffices to show the equivalence:

% ~ c ~ n ×~,~, a

(~.1)~

~<~

?~(%) ~ % × ~

^

Since

% ® C

is elementwise

(3.1) is equivalent to

%

~ invarianfl, the condition in the left

~ C c 5(~),

(:~.,o)

~(~)

hand s i d e o f

which is equivalent to

c ('n x s c-)' . p~

Indeed,

% @ C c 5(~)

% C "n",

is equivalent to

(~)(z~l) =z~l~l, Moreover, all

C~c

z e %

~&

and

for

z~C~.

is equivalent to (3.2), for (3.2) implies f ¢ L~(G).

[5(z),l @ f] = 0

for

Condition (3.9) is clearly equivalent to the condition

on the right hand side of (3.1). The case of

[~,5] : Since

['~,5} ~ [~. x(~ G,~}

for some

[,Tt,~.

By the duality

theorem for crossed product and Theorem II .1.1, it suffices to show the equivalence:

%, e c ~ ,~. ×

(~.3)

G <-~ ~(c~) ~ %,× G .

This is done as a similar argument as in the above case. of (3.3) is equivalent to equivalent to

~(C~) c (~ ×

~

® C c ~(m), G)'

That is, the left hand side

which is equivalent to

~

c~,

which is

w~ich is equivalent to the right hand side of (3.3) Q.E.D.

Corollary 3.2. i~(5) = G). tion

e

in

Proof.

Assume that

If

~ (resp. 5)

~

(resp. ~5).

C

c%,

is dual, so is

or e

% ~ G C ~(m). (rest. 5 e)

' rssp.

C~',?~ C Ch

or

for each central projec-

By Theorem II .2.1 (resp. 11.2.2) and Proposition 3.1.

Q.E.D.

78

Theorem 5.3-

of

(i)

• x

(ii)

C~G

The following two conditions are equivalent:

G (resp. h ×~ G)

is a factor.

C ~(m) (resp. 2(5) = G)

and

~(resp. 5)

is ergodic on the center

• (resp. h). Proof.

The case of

~(z) = z @ i

for

[~,~}.

z e C~,

( i i ) ~ (i) : Since

(i) -> (ii):

then

~

= ~ ×

~

on

It is clear that

~(z) commutes with G

~ ×

.C~ G c ~(~).

G. and hence

If

z = i.

by Theorem II .1.2, it suffices to show that

is a factor. The ergodicity of

Of(x)

Moreover,

C~x G c ~ ( ~ )

The case of for

( i i ) ~ (i): on

Ch

~

[h,5}.

z e Ch,

i)

.

x G 1

on

e,~"&(X

C~,

,R'

C~.V~GC ~(~) C_~C~

5(z)

commutes with

by Corollary 1.6. ergo

~

on

Since~ ~

ioity of

on

%.

Because,

by Corollary i. 6.

.~

CBx~G C ~(h). and so

is a factor. That

Since

Here we use

is a factor.

h x5 G

~

C~.

on

by Proposition 3.1.

we fin~ that

It suffices to show that

to

x @ i ~ 1

=

(i)=~ (ii) : It is clear that

then

is equivalent to that of

C ~ x ~ G C ~(~)

~

is equivalent to

the above ergodiclty " " of

z ® i

is equivalent to that of

is equivalent to

~ -~ C ~ G c ~(~)

is dual,

=

%

~

g(z) = z

The ergodicity of

CO><5G c %(h)

is dual,

If

z = i. 5

is equivalent to

~ G ~(~]"

%

is eauivalent, r o f cto

Q.E.~ Theorem 3.4. is integrable and Proof.

Assume that ~

~

(resp.

h 5)

G c ~(~)(resp. F(5) = G),

The case of

[~,~}:

Since

~

is properly infinite. then

~ (resp. 5)

If

~ (resp.

is dominant.

is properly infinite and

~

is inte-

grable, we have

(3.4)

fm,~} ~ ~,~} F ~'g}e

for some projection C~× G c ~(~) dua~ and infinite,

f <

e

in

@.

Let

f

be the central support of

e

in

~.

That

is equivalent to that CI~x G c ~(~) by Corollary 1.6. Since ~ is ~f is central, ~ is also dual by Corollary 3.2.- Since ~ is properly and

~f

are also properly infinite.

On the other hand, since tion with central support

f,

~

is c-finite and

it follows that

~

is dominant on

~

e

e ~ f

f~'~}e ~ f~'~ f ' which implies that

Therefore,

by (3.4).

~f

is dominant on

is a properly infinite projecin

~f.

Therefore

77

The ease of

{'a,,q :

Since

is propez'ly infinite and

$

is integrable, we

have

(3.5)

for some p r o j e c t i o n

e

in

'

•

-

-

•

.

The r e s t of t~le pz'oof ~s the same as the a b o v e c a s e Q.E.D. NOTES

A n e c e s s a r y and sufficient c o n d i t i o n , T h e o r e m ~.[j, for a crossed p r o d u c t to be a factor was o b t a i n e d in terms of

P(~)

by

Connes-Takesaki

and 0y N a k a g ~ n i for the non-a,oelian case [47]. and [47] for

5 .

!14]

for the a D e l i a n case

T h e o r e m 3.' is due to r 14] for

o.

78

§4.

Co-actions and Roberts actions. In this section we shall discuss the relation between a Roberts action and

a co-action.

We indicate by

[Wr,~r}

a representation of

equivalent to the right regular representation of Given a rirg

g

of representaticns of

and the relation between them.

C

G

which is q~asi-

G.

we consider two objects

~,

vg

Now we set

(4.!) Then

~

(a,)~,w,c~

For each irreducible representation operator

a

is a unitary representation of

represented by a matrix

vg(~)

on

(4.2)

~9

•

~.

on

£(~)

Each element of

is

Condition (1.3.~) implies that

~,~

~ R

and for each

~ e ~

we define az

by

v~(~)~¢,,w,

~~

"

Then it satisfies

,~(~)ii

: llg;I ,

I"T

(4.3) v~(~ e q) = v (~,)~(~) In particular, if

II~II = l,

(4.4)

then

vg(~)

,

I I

Now suppose a Roberts action

[D,~}

'~TT' "

is an isometry.

~g(t)v~(~) = v~(TT(t)~)~(t)

is given.

E

of'

,

Moreover

t -c G .

on a yon Neumann algebra

We set

considering each

0

as a representation of

trivial representation~

p~

~

is an isomorphism of

on

R ~

= R. into

As

~

contains

~ @ ~(~).

Condition

(1.3.2) implies

(b

,)~%(~)'~

h

, ~

Jn(%,%,)

fora11

~,~'~

.

Conditions (iii) and (iv) in the definition of Roberts actions g~ve us a *-homomorphism

~I of

w)~(G)' into

p~(~)'

such that

79

(4.5) If

~((%,,)) at,w,

= (:I_T,~,(%,=,))

is an isometry,

so is

, %,~, ~ ~G(~, ~') •

Ti~,~,(a,)

ri~,w (i) = 1.

since

Next we want to construct a Hilbert space on which we realize the crossed product of

N

by

~

with respect to the Roberts action

set of all bounded linear operators

[p,h}.

@

of

~9

to

~

to

~

by

¢(w),

Let

F(p,~)

be the

satisfying

(4.6) If we denote the restriction of

(4.7)

¢

9w, n,(aw,~,)¢(w')

Assume

that

~

has an element

representation

of

with

= 0(t).

@(~,r(t))

G.

[~r~r}

Then there Let

= ¢(~)a

then it is written by

,w,

quasi-equivalent

exists

F0(P,~ )

an isomorphism

be the set of all

to the right ~

of

~r(G)"

$ ~ F(p,~)

regular onto

~(G)

with

ii~ii~ = @G(@(¢(~r)*~(Wr))) < ~ ,

(4.8) where

SG

is the Plancherel weight on

Len~na 4.1. Proof. p(t).

If

Let

0

$(~r) = 0,

~(G).

then

¢ = 0.

be the isomorphism of

Wr(G)"

onto

g(G)

with

O(Wr(t))=

Since

~('r ® l)a : ~ % ® l , . r ( a ) ® ( = r) : 0 , for any trivial representation

i.

is a unitary in

with

we have

a ~ JG(.r e i,~ r)

¢(Wr ® l) = O.

If

[~.,~j e ~

•

L~(G) @ £(.~#

and

U

211

(U~)(~) = w(t)~(t)

for

~ c L (G) ® $~ ,

then (4.9)

Ad(o@g)-l(U)(Wr(t)

and hence

@(TTr @ T~) = O.

@(w)b we have

¢(W) = 0.

Since

= 9.~,~r®.~(b)@(.~. r ® ~) Since

[w,~#

From this lemma, we ~ o w We denote the completion by Lenm~a 4.2. and

e ~,

Zf

where

@ l) = Wr(t) $ w(t)

~) e

,

is arbitrary in

that

1

b e OG(,~,w r @ w) , ~, ¢ = 0.

JJ defined by (4.8)

Q.E.D. is a norm on

Fo(p,~).

L2(p,R).

F(p,~),

then

p(y)¢, ¢v9,(~) e F(p,~)

for

y e n

and

80

(p(y)@)(w) = pw(y)D(~)

Proof.

any

a ~

Since

~G(~,W')

pg(y)h(b) = 9(b)p~(y)

and

~

b ~ ~(G)',

it follows that, for

= hr, ~,(a),

~(p(y)@)(w')

Similarly,

for

•

- ~p~,(y)$(~')

= p~(y)~$(w')

= pw(y)<~(,w)a

= (~(y)¢)(w)a

since for any (a@l)~=,lw,,,g~,,v,~_(a (a ® 1 S

@i

= ~p~,(i ) ,

.

), a= ~i.,,v,(a)

and

~ = 9w, w(l ),

a c ~G(U",:')

by condition (ii) in the definition of Roberts action~ it follows that

ove(~)a~,

= ¢(~" s r,)(a ® i )(~, e

~)

Q.E.D. Therefore we can define a crossed product as the followirg: Definition h.3.

Assume that

regular representation

of

G.

~

Let

has an element quasi-equivalent p(y)

and

V(~)

be operators on

to the right L2(p,~)

defined by (~,.lO)

for

(~(y)¢)(~) = p~(y)O(=),

@ c ~(p,~).

action

[P,'I}

The crossed product of

y ~ ~ ,

~

by

R

with respect to a Roberts

is the yon Neumann algebra generated by

which will be denoted by

h ×

p(~)

and

V(~w),~ e @,

g.

It is immediate from (4.10) and (4 .ii) that

(4.~)

v(~)~(y) Lermna 4.4.

If

[~,~

(4.13) l~roof. isomorphism

=

~ ~

~(~Jy))v(~) and

,

~/] ~ ~.,

~ ~ %

.

then

v(~0*v(~) = (~ l~)l • Since

r,r

~

wr(G)"

of

a c ~G (~r @ ~,~,r)

is quasi-equivalent onto

to

(~r ® ~)(G)"

is an isometry and

e = aa

Wr @ w with

-i

,

by (4.9),

there exists an

~(,~r(t))= ~ r ( t ) @

then

v
If

81

e(~(Wr(t)))e = Ada(~r(t))

If

•

f = Ti~r@~ ,wr(a)*i~r@W,Wr (a), then Ada(¢(Wr)*@(~r) ) = Ada(¢(~Tr)*f¢(%))

= e(®(%@~)*+(%e~))e Since and

a

is an arbitrary isometry in

@(Wr@W)*¢(~r®W)

JG(~r@~,Wr)

~ (~r®~)(G)",

and since

¢(TTr)*¢(~,r)

~ ,Wr(G)"

it follows that

(~( ¢( TTr)*¢(Wr) ) = ~( TTr ~ ~) *¢( TTr ® TT) If

¢, Y C Fo(P,~ )

and

~,rl E ~TT'

(v(~)®Iv(~)~)

then

= ,G(~(~(~)%(%®~)%(%®~)v~(~)))

(~.~) -X- , -~ = ~G(O(V~(VI) (~('w.(i~ r) @(Wr))VR(~))) where

@

is an isomorphism of

On the other hand, if

~r(G)"

f ~ A(G)

onto and

vR(9)*~(Wr(f))v~(~)~ = v~(q)*~(Wr(f))(~®

R(G)

with

~ ~ ~r'

then

where

g(t) = f(t)(~t)~{9 ).

(glu)(®l~)

@(Wr(t))= p(t).

~)

: v~(9)*J'f(t)(~r(t)~@~t)~)dt

= %(g)~

,

= Ff(t)(~t)~ITi)~r(t)~r dt J

, Therefore, the right hand side of (4.14) is equal to

by (I~.8). Thus (4.13) is proved.

Now we define a unitary representation

U

Q.E.D. of

U(t)¢ = SwR(t) -I ,

(4.15)

G

2

on

L (p,R)

e Fo(P,~)

by

.

Then, it is immediate that U(t)p(y) = p(y)U(t) ,

(4.~6)

U(t)V(~) = V(~t)~)U(t) Therefore,

~ ×p R

restriction to denoted by

~ .

h ×

y~h, ,

is globally invariant under the action P

R

t ->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

~
is a trivial representation,

where

dt

is the normalized Haar measure.

Therefore

Now, we denote by

g

the normal expectation of

~ ×

~

onto

(~ x

P

P

g(x) =10t(x)dtd-- ' x e h X

P

~)P: P

g .

The set of elements of the form

x =~

p(yj)V(~j)

yj e ~ , ~j ~ ~ . J

is

a-weakly dense in

~ X

g

and

P ~(x) = 2

p(yj)V

~(~)

~(t)

.

Q.E .D.

Lemma 4.7.

If

~(~,)

contains a representative

is implemented by a unitary representation u(x ® l)u* Proof. Cv I .... ~Vn}

for

x c ~,

then

~

c ~:u(t)~

Choose a representative

~,~p}

be an orthonormal basis of

~p.

u

of

of

p

G

on

= ~ , t c G} c ~%(~I) Put

of

for all ~

p ~ G

such that

is cyclic for p

for each

and ~(x) =

~. p c G.

Let

84

~

E

is the central projection in

E

*

=yu(t)d~

P E°

Then

n

= n ! ~ v.e~,(v..)u(t)dt ~ j=l J ~ J

. and

u(G)"

Eo~ = [~ag9 :u(t)~

= ~ , t ~ G}.

~p Since

..

.fr

k

E_ = n ~ vj t i utt)dt ) V. = n ~ ~, j <,.' J P it follows that

E

Up

~ ~ (T~o$)-.

Since

~

~E

p

vm. ~. , J o,I

= 1, Eo$

is cyclic for

~. Q.E.D.

We are now ready to prove the equivalence between the crossed product by a coaction and that by a Roberts action. Theorem 4.8. (i)

If

5

Assume that

G

i s compact and

is a co-action of

an element quasi-equivalent a Roberts action

[p,-i}

G

on

~,

to the right regular

of

g

on

~

h

is properly ~ i n i t e .

there uxi:t

a r~ng.

representation

of

~

containing

G

and

such that ^

(ii)

If

R

is a ring containing

ular representation

of

exists a co-action

5

Proof. If we define

(i)

and

of

Put

[p,q}

G

G

[P,I} on

~

~(,~.,~,(b))

in

= ~

= b ,

then it is a Roberts action of w

~ (~)

~

on

~.

such that

,

is properly infinite,

to the right regular representation

Therefore we can construct a crossed product

(4.~m)

~

~ ~%(~)

ring.

,

is a dual action,

(a t ® g)(w*) = w*(l ® p(t))

quasi-equivalent

!

there

,

~,

Since

Since

stant

~,

is a self-adjoint

~ ~(m)

b ~ JG(~,~,)

~ (~)

m ~ ~(G) ~

(8(y)),

Theorem 11.2.2.

Let

to the right regon

satisfying (4.18).

[~,~} = [~ ×5 G , ~3]. Then

5(n~(y))

exists a unitary

~

by

(4.19) (4.20)

an element quasi-equivalent is a Roserts action of

~ ×

of

~,

G

by Proposition

where

R = .~ (~).

P be the projection ~f ~ @ LP(G) onto 2 @ C! where o function. We shall identify ~ with 2 @ CI. Then

= ~ ( y ) E ° == y

,

by

there exists an element in

E

~oS(y)

there

y c n ,

i

~(~)

2.3.

is a con-

85

(4.22)

Eob = bE o = "'1~, ~(b)

Now,

for any

~ c R ~

b e ~G(W',,~)

L2(G)-- we define

¢_ %

by

¢~ (w)a = Eoa ~ , a e ~I~ '

Then

¢~

is a bounded linear operator of ~ ® D w

b e ~G(W',~),

.

-r e

to

f~

R

.

and

#~ e Fo(0,R ).

then 0~(w')ba = Eobag = ~i~, w(b)Eoa~- = "iw,,w(b)¢g(w)a

hy (~+.e2).

Indeed, if

Moreover, if

~,Ti e R,

f.g c: L2(G)

az~d a,b £ ~

,

, then r

(9:i ~ g(%)*¢~ ~ f(~r)~ Ib) = (b%o~(~

~ f) l~i ~ g)

= ;(b*(l ® k(t))a(~ ® f)lq ® g)dt =/b*~t(a)(~ITi)(X(t)flg)dt

,

an~ hence +]®g(1~r) ¢~®f(~r)

= (~IT~)Wr(~O~) ,

qc = ~f,g ,

which implies that =

,

ll~ef!l e

9+

'+'G(e(~®f(~r)

Therefore we have an isometry for ~ 6 R @ L2(G).

W

of

.o

¢~®f(~r)))

R ® L2(G )

: iI~ ® f,i o L'(p,~)

into

such tlmt

Next we shall show by usimg (4.21) and (h.pp) that

(L.23)

wS(y) = p(y)W ,

(4.24)

w~ =v(~)w

(4.25)

If

~ e R @ L2(G)

,

W(I @ k(t)) : U(t)W ,

and

b e ~,

then

y ~ h , ~

~

, ~

t e G .

¢5(y)~(~,)b = Eobg(y)~ = E°P~,(6(y))b~w, =

p~ (5(y))Eob~ = 5(p~(y))Eobg = p~(Y)Eobg = p~(y)0g(r,)b, TT

~ ,

and so

"

WS(y)~ = % ( y ) ~

= o(y)~

= p(y)W~ .

86

If

~ ~_ R @ L2(G)

and

b ~ ,~. ,,

Wa~

If ~ e R ® L2(G) and ®g&[i(t) = (i~R(t)-l)(~)~,

b

then

~a~

=

¢~vg(a)b,, = @~ba =: Eoba~ : ~,a~(~')b,

¢~vR(a)

=

~, then and so

( II

W ( 1 @ )~(t))~

V(a)$~

=

:I

V(a)W~

= I~;(l®~.(t))

~ :

.

- Eob(l ~ X(t))~

~(Zi~X(t))~(~)b

@~7~(t) -I

and so

= U(t)W~

= EeSt(b)~

=

.

Using (L.2~); (!4.24) and (4.25), we s ~ l l show that W is an isometry of ~ LP(G) onto L 2 (p,~). If ~ £ L2(p,~) and ¢ = U(t)~ (= @ ~ ( t ) -i) for all t ~ G,

then

{w,~J

~(.)

:I 0

~or a n

irreduci~o

is a trivial representation,

some unitary 9~ = C,~)(u )

uv, ~ ~ and

by (4.19).

~'w' = CS(u ,);

some non zero

~ 6 C

for a trivial

w.

and

Then

then

If

and non t r i b a l

~

c :~(h) axed hence

{w,Dw}

then

and

,

i"

~

~

w'

for

tl~t is,

,~,(u ,u~)

Here wc set

and

~

0~(Y) = u yu

is of the fore

by (*~.20).

Indeed, if

~ ~.

are trivial,

~-7',~.I.w,}

a e ~G(VT',~)

"~. _(a) = mu ,u* ,'~ = ~ .~ u~®

{~,~j

for

~ : ~(v.)5(u ) 1~

are trivial, then

and

(~').~(u,) ~,u~

-i

Cu*~ @1 = ~ -i ~I=, ~(a)~( ~)I~(U ) : ~-l~( ~' )a~(~ ) : ~(~')~(u

Consequently,

we have

~ e WEo ~

O

Since ~

contains an element quasi-equivalent

so does

~^(~l x o

tative of

p

p

~)

for all

by Lemma 4.5. p ~ G

isometry of

~ ® L2(G)

onto

by Propos.~tion 2.6 and

(ii)

w ,: ~}, Since

then

Ad W

h ×

~

h x5 G

s

p

by Lemma 'l.7.

~)

of

contains a represenThus, the right

This means that

h ×

~

is generated by 5s generated by

gives the equSvalence

W

is an

5(h) p(h)

and and

{g~ : irreducible [V(~.) : 5rre-

(4.18).

{£ has an element quas_~-equz'.valent to the right regular representa-

tion, we can consider the crossed p-roduct By Lemma 4.5,

]~^(YI x

L2(0,~).0

Finally, if we notice that

ducib!e

to the right regular representation

Therefore

by Proposition ,°.3 and Lemma 2.7.

hand side of (!:.26) ~s cyclic for'

6 ~}

and so

WE ~ = [~ t L2(p,~) :U(t)0 = ~,t ~. G} .

(4.26)

G,

(D = R ® L2(G))

,) •

h ×

g.

Here we set

M~(~.0 has also an element quasl-equivalent

{m,(~} :

[h

×

to the right regular

g, B].

87

representation.

S~nce

w s ~.~ £(L2(G))

with

is properly ir,~'inJte, there exists a unitary

]q

(~t @ g)(w*) = w*(l ® ~(t))

(~ is a dual action, for

by Proposition P. 3 . Therefore,

f,~.,.~] ~ ['..~..'~}: [,,~'~.,~: ~ .

Q.E.D.

From this theorem we know that if we consider the following correspondences :

then

{h /%1 C, ~!] -~ {h x 5

G, g2 ].

Namely, there is an isomorphism

'~ ×~, G onto ~ x},) G with _F(~ 6~ L) ° ^[~i __ $,P ° 7. Since 'i ? we ,may set w I = (=, 2 L)(I ~ WG) and w 2 :: 1 @ W G. Then .

w~(1 (~ p(t)).

Since

It follows that

(L ~ L {~ ¢~G)(W~W2).

{~,~,l]

(,'-.:2) ~ o v : ~ o (t'~l)t~ we have

w~w 2 E Z2(~) ~ 9 ( G )

~

Therefore

and

~

of

1 @WG

~{ (~I x,, G) ~ Q ( G ) ~ • ( ~ * ~, (<:'.2)t .~ L),- 2, r

~

~

:

((:}l)t .~. L)(w{): Wl(l ~ P(t))-

(w~w 2 ~ I)(L @ %G ~ L)(w~w2) =

['[1~.51} arid [h~:".2] are outer conjugate:

{~z(~), L e ~a]

~

{'~2(~),Adw{~~ ~ f~2(~), L e ~,G~ ~ ~

[~,~2]

In the same way, if we consider [h,~z,f%,,k] ] (ii)~ [h,G,5] (i I {U,ao.{%,,]o]] then

[h ×DI % ,

is realized in

8]] ~ [" ×0 •

R 2,

a 9, ~9].

Since

2

pl(n)

the eovariance equivalence of

corresponds to [h,{01~r[] ]

in some sense will be obtained.

NOTES The materials of this section are adapted from [55].

and

~2(~)

and

{h,{D2~2]]

~l

C_HAPrER V. PERTURBATIONS

Introduction. flow of weights

OF ACTIONS AITD CO-ACTIONS.

This chapter is concerned with ths non-commutative

of Connes-Takesaki

[14 ].

Since the "dual" of a non-commutative

is not a group~ an action of a non-colmnutative a non-commutative l-cocycles G

G

goes well.

on

G

Contrary to this; the dual of the "dual" of

~l gives rise to an action of

~

to the center

C.~%G

G

is the given

duality theorem says ; thus a co-action G

itself on the cohomology space, i.e.

an action on an abelian yon Neumann algebra, which is isomorphic of the dual action

group

does not give rise to an action of

group on the cohomology space, even though the comparison theory of

itself as the Tannaka-Stinespring-Tatsuuma

,% of

version of the

to the restriction

of the crossed product

discussion here is adapted from the flow 6f weights,

~ ×,;, G.

The

[i:'] ~ without major change.

Of

special interest would be ~aeorem 2.6 which says that the normalized Connes spectrum Fn(,%)

is inv~riant under perturbation

kernel of the action of

G

by l-cocyeles;

on the cohomolog~v space.

hence it is precisely the

89

§i.

Comparison

of 1-cocycles

of action and co-action.

In this section we consider a fixed action

~ .

The dual version

Given a strongly values

the set of all perturbed

continuous

map

for a co-action : t ~ G ~ v(t)

actions

by l-cocycles

of

is also discussed.

e ~

with partial

isometry

satisfying v(s) ~s(V(t))

= v(st)

(i.l)

v( s -i) o o]l(v(s)*) , we define

a

an operator

in

~ ~ Ln(G)

by

(a~)(t)

= v(t)~(t)

for

~ c $ ~ L2(G).

Then (a ~, 1)(c~ Z ~)(a)

~- (L 'm ~G)(a)

(1.2) aa* = e a 09 i , a*a = ci(ea) where

ea

is the left support

Proposition there exists

i.i.

If

a strongly

Proof.

of

a

v(t)

.

is a partial

continuous

map:

isometry

t e G

+v(t)

in

~ ~L°:'(O)

~ ~

satisfying

satisfying

(1.2),

(i.i).

By (i.~) we have

(,y~ ~ ) ( a ) and hence

v(t)

,

o{ a(t))=

:(a~)i)(l~VG)(a®l)(i~VG)~

a*( ~ @ .ot )(a)

- 2Zl(a*( ~ @ 0t)(a)).

Then

locally almost t ~v(t)

everywhere.

is a strongly

Put

continuous

map with (i.i). Q.E.D.

Definition isometrics partial

a

1.2. in

isometrics

(l-cocyeles) Let

~ ~ L~(G) b

in

Z (@~IT.) () (1.2).

be the set of all partial

satisfying

N ~ ~,(G)

Let

Z,~(G, ~)

be the set of all

such that

(b ~ i)(~, ~ L)(b) : ( L g, ~0)(~); bb* = e b ® i , b*o for some projection Since

8(eo)

e o e h"

a(t)~t(ea)a(t)*

reduced yon Neumann

=

= e a , we consider

algebra

~e

defined

the action

a O, of

G

by

a a ~ x) = Ad a o ~(x)

Similarly,

we consider

the co-action

b~

of

,

G

x ~

on

e

a

•~e~ defined

by

on the

9O

b~'(y) = Ad b o ~(y) ,

We denote the fixed point subalgebras of o

oy

~

a

and

~

O

~'e

Definition i.~. "~ Z~:(G,'n)) o

~resp.

~)

(Cohomologous)

~ec

with respect to

a~

and

are e~uivalent

class of l-cocycles.

We say that

and w-rite

a

and

a ~- o , i f

there

b

in

ZG(G,T)

exists

an element

c

in

such that

b~(c*~1)a~(c)

,

~

(resp. b = (e* ~ l)a~(e), Moreover,

and

, respectively.

Next we shall consider the cohomologous

(resp.

Y e Geb

a < b

if

a ~ (e ~ l)b

(c~l)~o~)

a - (c g l)bS(c~)).

for some projection

e

in

b

(resp. b ) .

The following proposition is crucial for this section, which deduces the comparison problem of l-cocycles into the comparison problem of projections

in the

sense of Murray and von Neumann. Proposition 1.4. a,b ¢ Z ( G ~ )

(a)

we define

(1.3)

(G,~)

a < b (resp. a ~ b)

(1.4)

and

For any

<~t = °'t ® L •

is equivalent to (resp. e a ~ ell ~ e o ® e22 )

~c . (b)

Let

and

~ = n ® F2

c e ZN(G,~)

~ ~ (L ~ c) ° (~ ~ ~)"

For any

a,b e Z~(G,~)

we

by

(1.5)

c = (L ~ J)(a ~ ell ~ b ® e~2 ) . a ~ b (resp. Therefore~

a ~ b)

if

is equivalent

e a ® ell

(= e a ~ ell + eb ® e22 )

in

[aj}j= 1

by

1.a,

we d e f i n e

is a sequence in then

g

a projection

Ca(b ) ® ell : (Central support of

mutually orthogonal,

--c

in

b)

a ~ b .

Makine use of P r o p o s i t i o n ~:~resp. 'a8)

to (1.4)

and e b ® e27 have the same central support --c ~ (resp. ~c) and a (resp. h a) and b(resp,

are properly infinite, then

If

Y2 by

e a ® ell ~ e b ® e22

define

Then

~ = ~

c(s) = a(s) ® ell ~ b(s) ~ e22

Then

in

Let c ~ ~

Za. 3

Ca(b )

eb ® e22

in

Z (G,~),, (resp. Z~(G,~)),, such that is also a

cz ~a (b) =s~2~aj (b)

an~

1.-coeycle

,

~(Zaj) =VCb(aj)

.

in the center

~C)(e a ® ell). [ea.} J

is

of

91

Let

If

a

[vj : j c I~ ] be an orthonormal basis of a Hilbert space in

is a 1-cocycle~

(1.6)

~, ( r e s p .

~1).

then

a=D

(~j~l~(vj)*

j=1 v

is also a l-cocycle

such that

Proposition 1.5. equivalence classes of

The map

a .~ ~

ca

b e Zs(G,~)

and

b

is properly

infinite.

is an ordered isomorphism of the set of all of infinite multiplicity with

the set of all projections in the center of of

v

l ~ a ( r e s p . I~a )

b < ~

onto

H a ~ where the infinite multiplicity b is defined by the proper infiniteness of

92 §2.

Dominant l-eocycles In the following we call an element

semi-dual, dual or dominant according as

a

in

Zo(G,~)(res p. Zs(G,b)) integraole,

a~)(resP.a~ ) has the corresponding

property. The following proposition is a restatement of the definition of a semi-dual action and co-action. Proposition 2.1. (a)

If a unitary

(2.i)

a e Zo(G,~ )

and

is a semi-dual , then

(~ ~ ~)(a ® i) -- (i ~ v~)(~

®

~)(a ® i)

Z_(G,~) . (b)

If a unitary

(2.2) in

~,~= ~,~ £(LP(G)) (resp. ~ = ~% ,~,£(L2(G)))

®~)'(o®~(resp. ~=(~ ~o)o(5®,)).

G=(,

in

Let

b e Z~(G,~)

is semi-/ual, then

(L ® ~ ) ( b ® i ) - - ( l ® W G ) (

L ®o)(b®l)

Z_(G,~). 5 Proof.

(a)

If

a e Zci(G,~) ,

then

( t ~) J)( a g i) 6 Z_(G,~).

Since

-v

(2.~)

(V~ ® i)(~ ® ~)(i ~ V~) =- Adi®vG(V ~ ® i),

we have

I®V~

follows that

~ Z_(G,~)

As

a ® i®

i commutes with

(I ® V~)( ~ ® ~)(a g i) c Z (G,~.). Since

(o ® ~ ® a

v

in

k ~g(G)'

such that

it

is semi-dual , there

_

exists a unitary

I &)(l ®VG),

~

96

(i 10 V~)(a:i)(v ) : v

$ i.

Therefore

(v ~ i)((i ® V~)(L ® o)(a ® 1))~ (v~ = ( L ® ~)(a ® i) and hence (2.1) follows. (b)

If

b c Zs(G,b),

(2.t) we have

then

(& ® o)(b ® l) e Z (G,~).

Since

(W G ® i)(~ ® a) (i ® WG) = A d I ® w ~ (W G ® i), i ® W G e Z(,,G,~).

follows that

As

b ~ i ® i

commutes with

(i ® W G ) (L ® ~)(b ® i) e Z (G,~).

exists a unitary Therefore

u

in

~l ~ L~(G) ' suchSthat

Since

(8 ® L ® L)(I ® WG) , b

is semi-dual , there

(i ® WG)(oS)-(u* ) = u* ~ i .

(u ® l)((i® WG)(L ® o)(b ® l))g(u*)

(~. ® ~)(b @ i)

it

93

and so (P.2) follows.

Q.E.D.

Proposition ,°.2.

(a)

If

a

is a unitary in

Z (G,~.),

then

i e v~ -- (1 ® v~)(, ~ ~)(a ~ ~)

(2.~)

in z_ (~,~). (b)

If

b

is a unitary in

(~.~) in

then

l ~ w G-- (z ~ WG)(~ ~ ~)(~ ~ ~)

Z (G, ~). Proof.

by

Zs(G,~) ,

(a)

Since

Proposition 2.1,

it

i (9 V6 ~ Z ~ G,~.,) and (1 ,~ v~)(L ~ ~)(a ® l) o r e m a i n s t o show ( 2 . 5 ) . By d i r e c t c o m p u t a t i o n

Adl.~V,~a

e Z_(a,[)

® i) - (~ $ a) ° (t_ ® ~G)(a ).

a e Z (G,~) , it follows that

Since

Ca* ~ i)((1 ~ v~)(L ~ ~)(a ~ l))[la) : ( a * e Z)(1 ~ V~)((~ ® o) o (~ ~ O~G)(a)) : i ,~ V ~

(b)

Since

by (i.~),

.

1 ® W G e Z (G,~)

and

(i ® WG)(& ~ o)(b ® i) e Z (G,~)

Proposition 2.17 it remains to show (2.6).

by

By direct computation,

Adl,w~(b ® i) : (~ ® o) o (~ ~ ~a)(b). Since

b e Z~ (G ~) ,

it follows that (b*@ i)((i $ WG)(~ ® c)(b @ l))~(b) = i ® W G. Q.E.D.

This proposition

shows that

(2.7)

(~)~°

42.8)

(bS) No Ad b = Adb$1 Proposition 2.3

a

and

b

Let

are dominant,

Ada = A d a @ l

a

then

and

b

a ~- b .

° ~

on

~,,,

° ~

on

E.

be unitaries in

z (a,~) (resp. z~(G,~)).

If

94

Proof.

(a)

Propositions

Since

a

and

b

are dominant,

they are semi-dual

.

Combining

~.i and 2.2, we find that

1~%' ~-(,®~)(a~l) in

Z(~(G,~).

Therefore

a - (~ ® (~)(a

in

ZF (G,~).

Therefore~

oy Proposition

e

where

p

~,,~ F 2

=

properly

and

infinite,

Proposition

(b)

and

[ - (~ ~ j)(b @ i) are equivalent

i.:~,

% ell ~ e

,~, e ,:,, .....

in

is given by (i.]3) for

c

a

9

C

and

b.

1.4 implies

is given b y ( l . ~ ) f o r

Theorem 2.4 There

and

~b

are

a ~- b .

By the same argument,

(i)

~,La

Since

~ ell ~ e ~ ,~. e ,: .... i ~ e D ~ e;.);.;. _ _

e

~ ell ~ e

.~. e.~

in

{c ,

-

and c a -- b.

;

we nave

e a ,~ ell ~ e

Thus,

9~ i)

Let

[ and

T. (resp.

is a dominant

where

.9 : .~ ® F 2

b

o .

h)

Since

a

and

be propoerly

unitary

a

in

h

are properly

infinite, Q.E.D.

infinite.

Zr(G,}.) (resp. Z~G,~)).,

which

is unique up to equivalence. (ii) An element

b e Zr~(G,~ ) (resp.

Z5(G,N))

is integrable

if and only if

b
Proof. orthonormal

(i)

The case of [~,o]:

basis [vj : j c ~ I

Then

(= Z(vj ~ 1)~(vj)*). Therefore

we may assume

ally isomorphic

to

exists a dominant

a

that

[~,~] unitary

. in

If

~

of a Hilbert is a l-cocycle ~o

is properly space in unitary

is properly

Since

~ . and

infinite.

i @ V~ e Z-(G,~)

Z?~(G,~,).

infinite,

~a

a =

is properly

Therefore

and

By Proposition

Put

we can choose an

~ = Adlgv.

infinite.

[~,~]

is spati-

O ~ ,

there

P.3 it ms unlque up to

equivalence. The case of is properly a dominant

[11,~]: By using

infinite. unitary

in

Since z~(G,~).

the same device

i ® W G e Zg(G,~) By Proposition

as adore we may assume

and

that

5 ~I

~ = AdI~.W G o g , there exists

2.3 it

is unlque up to equivalence.

95

(ii)

It is immediate from Theorems Ill.3.1andlll.3.2

• ~(resp. NS)

By virtue of Theorem 2•4 and the ex~nple in §rV.l , l-cocycle perturbation, spectrum

,

for we may assume that

is properly infinite by the same device as above.

F(5)

contrast to the abelian case•

F(5)

Q.E.D. is not stable under

This defect of the Connes

is restored as follows:

Definition 2.5• ,~GtF(5)t-i

The normalized Connes spectrum

Fn(5 )

is the intersection

which is the largest normal subgroup contained in

Theorem 2.6.

The normalized Connes spectrum

n (5)

F(5).

of a co-action

va~iant under the perturbation by any unitary 1 - cocycle in Zs(G,N )

5

is in-

and is equal to

r(~). Proof.

By our favorite

2 x 2-matrix arguments, our assertion is equivalent to

the claim that if two projections l'n(Se ) = i'n(Sf).

Let

u

e

and

f

in

~5

be a partial isometry in

with

and

be an arbitrary compact neighborhood of

be compact neighborhoods be a function in We then have

of

A(G)

a*a ~ ~e

t

and

t-lst

such tb~t

and

aa* s

.

t

Set

and eI =

supp ~ c

eI

belongs to

eI

V[supp(x*x)

: x ~ ~'~

and

K.

s.

then uu* = f.

sps(flu ) Let

K

KVK -I c U. Set

a=5

and V Let

(flu)~0.

~/ ~ ~A(G) supp( ~ (a)*5~, (a) ).

~C

By Lemma 2.7 below,

N~

be a point in

respectively such that

~(t) = i

~f]

Let

u*u = e

fl ~a 5

U

fl ! f"

N

~ake a non-zaroprojection s e Fn(Se ) . Let

with

are equivalent in

e

~

Since

t-lst ~ 7'(5 i) e1

(V)} :-~/[supp(xx*) : x ~ ~%

(V)} = e I

e1 so that we can find an

x~ n

(v), ~l' '~'2~ A(O)

such that

fl 0 ~ 5,#l(a) x 5 o(a)*,_ c- ~5

Hence

U 0 sp(5

that is, of

fl)

J ~;

s ~ u(5f).

Fn('6e).

so

Thus

By symmetry,

Len~aa 2.'7. For any

s ~

(U) .

sp(~fl)

for every non-zero projection

Fn(Se ) c ?(sf); we have x 6 N,

~fl (KVK -1) c

so

Fn(Se ) c Fn(5f )

n(~Se) = i'n(&f) . we have

~/[supp(5< (x)*F (x)):~ ~- A(G)} = p ~ F

~f

fl ~ "n'" ,

by the normality O.E.D.

96

Proof.

We may assume that

(1"(C.)~1'~')

f~ is implemented

= k' Wrr~'.~.' ~ ~ "~ ®,:@)

5(a)

=: w~(a e Z)w ,-x

¢

Y = [5 (x) :~ ~ A(G)}.

w such tb~t IT.

,i A ( C - )

~: ~

,

~,r,

;

;

I-

a r(A(G))" ~Q(G)

w

Let

:,

by a unitary

Then

Y

.

is a subspace

of

~I invariant

under

5@,

~9

e A(G).

Suppose

p~ ~ 0, ~ ~ .~.

~ = ,,f~g c A(G),

For any

-., ~z R

and

y -- Y,

we

have

\]~' ~,Y~l !

= \W ,,~,= . y @ q¢) :_ \~y II @ l)w1~,.~,., ~ ~)

,b

For any

~ ~ ~

and

h ~: L'-(G),

we have

(5(y)(t ~f)

[ c ~h)

= <Xy),~,C (~,(y)

'

=(~ (y)~lt)

¢ ~'

fl,h

~

=0

) with

,'.....

;

v

5 ( y ) ( ~ ,~ f ) = 0 ;

thus

N a m e l y we have

s o that

p~(q0)~ = 0.

( i - i0)~ invariant.

Hence

(Yw(O)@ I ',) = 0

Hence (p$

i)

p

and and

f o r every

q ~ ~;

TT((2) commute because w

commute,

which means

hence w(o.)

leaves

the

5(p) = w ~ ( p @ l ) W =

p ® I.

Q.E.D.

Corollary F(b) = G

yw(.~)~ =0.

2.7

If

if and only if

i'(5)

is no~nal,

!'(~) = G.

then

f(~) = U(~).

In particular,

97

§3- Action of

G

on the cohomolo~F space.

In this section we assume the proper infiniteness

for

~

and shall only give

a sketch how to make an action on the cohomology space which relates to the Connes spectrum

~(5) .

For convenience we simply denote 2

Let

D = ~ ~ £(~ (Zs))

£(~(z~)).

W e define a

(3.1)

and

lea,b: a,b e ZSI

1-cocycle

u

~)(

u = (~ ®

Z5(G,h )

by

Z 5.

the canonical matrix units in

Z~a,O) (T = (~ ~ ~) ° (~ ® L))

in

by

~ a ® eaa ) . a ~Z~

Then, the map

(a ® q) o (5 ® L))

(= AduO U

is also a co-action of For each in

G

on

a c Zs(G,h )

~ . We denote by

we denote by

ph ( a)

9

the center of

D

the support projection of e a ~ e aa

9 .

Proposition 3.1. projections

in

(i)

The map

Ph

transforms

Zs(G,a )

o-finite

9.

(ii)

ph(a) = ph([) •

(iii)

ph (a) ~ p.n(b)

if and only if

a < ~ .

v

Proof.

onto all

(ii)

That

aj = (vj ® l ) a S ( v j ) *

v

ph (a) ~ ph (a) is clear.

Since

a = 2 aj

with

by (1.6), v

P~(~) =~b %(a)~ %b =~L~b(V%(aj))j ® e~b<_ pn(a) (iii)

That

cb(a ) <_ Cb(b ) ,

(i)

ph(a) ~ Ph(b) if and only if

Let [e. : i e 11

if and only if ~ < b

by

.

p~(a) <_ ph(b) ,

if and only if

Proposition 1.5.

be a family of mutually orthogonal non-zero projections

1 a

in

p

with

q-finite

,

e. ~ pb(a) . For each i z I is countable.

Conversely,

form

p~(a)

c I, e (e a ~ eaa ) ~ 0 .

we shall show that each

f o r some

(3.1)~s Z e a ,~_ eaa

and

a e Z~(G,h). e <_ s~(u),

q-finite projection

Since the left it follows that

support

Since

e

s~(u)

in

h

is

is of the

~ of

u

defined by

98

e ~ Since

e

is

V

[Pn(a) : a c z~(o,n)}.

there exists a sequence

J-finite,

[aj : j c ~ ]

in

Z&(G,h )

such

that

e <_ V Take a l-cocycle

h 6 Z~(G,~)

[Pn(aj) : J 6 ~;}.

such that

e <_ p.n(b). Choose a l-cocycle

a. < b

for all

j e ~

.

Then

a c Zs(G,~I) such that

e(e b ~ ebb) :: Cb(S, ) ~ ebb • Now,

e

p~(b)

and of

are ooth projections in

p~(a)

D-- dominated by the central support

and

eb @ ebb

e(e b @ ebb) = p,fl(a)(cb ~ ebb) • Consequently,

Proposition 3.2 the center of e e Cho If

b

If

onto

In particular, a

Q.E .D

e = p~(a).

b -c Z&(G,~), @d

with

therc exists uniquely an isomorphism

d

p~(b)

such that

Pb(e) = p~((e @ i) b)

Po

of

for

p0(eo(a)) = ph(a)p~(0).

is a 1-eocyele in

Z~(G,~), then

(1 ~ o(r))a is also a 1-eocycle in

z~(0,~). Definition 3.3

A homomorphism

~

of

G

into

Aut(@)

is defined Oy

~tP~(a ) - p~((l ~ ~(t))a) for all

a e Zs(G,~ ).

Theorem 3.L

The following three conditions are equivalent for

(i)

a

(ii)

a < b

(iii)

The map

Proof.

is integrable. for some (or all) dominant t ~ ~tP~(a )

is

D e Z~(G,n). ,,>

o-strongly continuous.

(i) ~ > (ii) : By Theorem 2.4.

a ~ z~,(G,n) :

99

(ii) ~(iii) : Since is properly infinite. and some projection

ph(a) = ph(~)

Then e

by Proposition 3.1, we may a s s ~ e

a ~ (e ~ l)b

in the center of

for some dominant l-cocycle b

.

Since b

and hence there exists a unitary representation b~( u( t) ) :-u( t) ~ p(t) .

Since

a ~ (e ~ l)b

u

and

of

that

a

b e Zs(G,~)

is dominant, it is dual G

in

h

such that

(i ~ o(t~)a~ (i ~ p(t~ (e @ l)b,

it follows that ph(a) : pb((e 9 1)b)

(3.2)

p~(1Z

Because, since

o(t))a)

c b* ~ Z ~(G,h), b"

p~((u(t)eu(t)*® l)b) .

we find that

b~(U( t ))c5 (u(t)) ~ (: Z,b(G,'n) by direct computation.

Since

(1 2 o(t))c we have

(u(t)* % i) b,5(u(t)) c,%(u(t)~ 5(u(t)),

(I ~ p(t))c ~_ bS(u(t~c[~(u(t)~ . Here we replace

c

by

(e q l)b

and

obtain

b~(u(t))(e @ l)b$(u (t)~ =

(u(t)eu(t)*

®

1)b

.

Therefore (3.2) i~ true. On the other hand, we have an isomorphism ~d (d : p~(b))

such that

Pb

Pb(e) = p,n((e ~ l)o i

of the center of by

Proposition 3.2.

~

b

onto Therefore

the map

p~(a) = pb (e) ~ t p n is

J-strongly continuous. (iii)

and

(&) = pb (u(t)eu(t)*)

~ (ii) :

f =V[3t(e

Let

) : t ,c Go].

Go

be a dense countable subgroup of

By assumption, each

3t (e) for some sequence It n n f = V [~t (e) : t e G] and it is we have

f = p~(b)

: n e ~]

in

c-finite and

for some dominant

.~t(e)

G .

Let e ~ p~(a)

is a ~-strong limit of

G o . Therefore, 3

b e Zs(G,~).

invariant. Since

By Proposition 3.1

e < f < p~(b) ,

100

a 4 ~ < b

by Proposition ~.i. Q.E.D.

Corollary 3.5. a e Zs(G,h )

where

such that

d = p~%(a) and

Connes spectrum

Pn(5 )

If

~

"- p~'operly infinite, there exists a dual is

[~k,~a~] T [~ ×,~,.G, ~]j and

~ ~d

is the restriction of

is precisely the ker~nel of

~

to

~d"

Thus the normalized

~d.

NOTES The contents of this chapter are generalizations of the flow of weights on a factor of type III in [i4].

CPAPrER Vl RELATIVE COMMUTANT OF CROSSED PRODUCTS.

Introduction.

The relative

commutant

of the original algebra in the crossed

product plays an important role in the study of the crossed product and/or the fixed point algebra.

Section i is devoted tc the analysis

of the condition under which

the relative commutant is contained in the original algebra. the relative commutant property, of

G.

But~ in the continuous

case, the freeness

case,

of each individual group element

does not yield (i.i) as in the example following Theorem i.i. criteria for a stronger relative

In the discrete

(I.i), is equivalent to the freeness of the action

commutant property,

i.e.

Theorem i.i gives a

a necessary and suffi-

cient condition for the relative commutant of the center of the original algebra to be the original algebra itself. very primitive.

Generally speaking,

the results in this area are

We need a greater effort to understand the relative commutant

property to claim, anything definite. In §2~ we treat the stability of actions following Connes-Takesaki~ Again we know very little about it.

[ik].

102

§i.

Relative

colr~utant theorem.

It is known that,

if

is descrete,

G

the following

two conditions

are

equivalent : (i) x

=

ot

is free on

~,~ for each

Indeed,

for

all y

implies

~<~.)' p (~ xoG) c ~.(~,.). p( t )) c ,-~rc)' 0 (~×,~,G),

~_~t~O,,~xt) ( l *

xto:t(y ) = y x t continuous

for all

y ~ F..

However~

thc above equivalence

Theorem 1.1

The following

conditions

(i)

-~C~)'

n ~ ×,o)

= ~);

(ii)

For any compact

subset

K

of

for

q ~ p

such that

q < ~,.

If

on ~

[F,~;]

L~(F,[~)

such that for every

rio

which does not contain the unit p c C~

for every

is a standard

is isomorphic

are equivalent:

there exists a non-zero

and

f~t(q) q = 0

(iii)

~

G,

e ~ G , and any non-zero projection

projection

in

does not hold for

o<%).

cX ~.)' n (~.x. ~ ) -

G

and only if

in this section is mainly concerned with

(1.1)

element

if

group.

Our interest

N

= yx

o)~ (ii)

of

t / e (mot(Y)

G-measure

space such that the canonical action

[c~,G,~],

7 c ~ - N ,

t c K;

then there exists a B o m e l null set

the stabilizer

subgroup

H

of

7

7 reduces

to

[e]

Proof. separable Let

F

.

(iii)

> (ii) : Let

C ~ -suoalgebra

of

be the spectrum of

that ~zt(a)(~) = a(t-l~), measure on

F

H

C~ A •

be a globally ~

jn~ariant

Then

C

lim !Io~t(x) - xll = 0 for every x c A. t-~e acts on ~ continuously in such a way Furthe-~more,

to a faithful normal

state on

is quasi invariant under the action of

be identified with the action of

o - w e e k l y dense

such that

a £ A, t c G, ~ e I~ .

corresponding

that the measure

A

G

on

L:~(P,~).

Set

let C~,G

~i

be the Radon

It then follows and

~ C D ,G,c~]

can

103

r K = [~ e F : t ~ By condition (iii) action of

G

~ u(p - 2K) - 0

on

U .

neighborhood

Ut

of

neighborhood

V(t)

For each ~ t

of

K.

set

Set

for every t £ K

PK

q e PK

is open in

and

t ~ K

tU t 0 U t = @

such that

open covering of' the compact ~V(tl),...,V(tn) }

and

such that

of

/ ~

p

b y the continuity of the

there exists a compact

Then

we can choose a

V (t)U t Q U t = @ .

K,

Then

IV(t) : t e K}

is an

so that wc can choose a finite subcovering

U(T)~n

Ut..

such that

i= i

( ~. 2)

,

•

]

Then

U(~)

is a neighborhood

of

l

KU(V) n u(~) = ~ .

Thus every point Condition (ii)

~ ~ PK

admits a neighborhood

ii) now follows >(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<,n

Let

U

.

semi-finite,normal

acts standardly and

Since

u

~

~ ~e an isometry

weight on m such that P L- completion of ~

on the

a unitary

in

is integrable, of

~

into

£(gO)~ L~(G) the map:

gO ~ L2(G)

(U~) (t) - u(t)~, Then we have,

for any

(l.5)

x e £(~)

u~=

Since

and

u(G)")

defined

by

~ c gO •

t e G,

~(~xl)~*u,

J((,T!~) ' n ~.)J -- ( m v

definedby

t->u(t)~,g~n

uu(t)

Q p,'

=(l~,~.,(t))u.

and

(o(~.) V (C ® 9(G)") p u(~' ~9 C)u*

(~ ~p) a ~Xr,0' = ~dcr ~)

Dy assumption U

(i.i),

is isometry, (ii)

of

G

into

y e: (TO.), Q ~,~ ,

JyJ c C~, and hence

We may assume

on ~®

if

~

that

implementing

L2(G)

~

then

UJyJ c %(C~.)U = U C ~

by (1.5).

Since

Let

representation

y ~ ~.

is standard.

canonically

,,z , If,P4]

uf=~ - u ( ~ ¢ f),

~ ~ ~ ,

u .

be a unitary We define

a map

~

of

by

f c r.2(G).

Then we have

(1.6)

u(x ~ l)u*Uf = u~x ,

First,

(1.7) where

we notice

that,

if

x ~ ~(D).

y c ,~(,~.)' ;~ (~T.×..G) ,

then

~*g yuf J ~ (~.2:)' p ~ (- c~) , J

is the modular unitary involution

for

,~.

Indeed,

since

y e C~(~0' , (1.6)

107

implies that because, if

U*g yUf e ~'

y c -G ~ X,C'

Since

x ¢ T' ~, u(G)', then

, (1.6) implies

UgyUf ~ m V u(G)",

Ufx = u(x ® l)u* Uf ~ (x ® l)Uf

and hence

[u~ yUf, x] = o . We now show (i.i).

By Theorem II. 3.P. (a) we have only to show that y ~ C~(D)' n ~:×.G) (/

By assumption,

G

is a belian and

there exists a unitary

Vp ~ ~

V'p = JVpJ ~ ~'.

Ju(t)J = u(t),

Since

(i.8) If

Therefore for each

c~(Vp) = (t, p) Vp.

we have

p ~

Put

u(t)v~u(t)* - (t,p)v~

and

~ f) = (v~ ~ ~)u( ~ ~ pf).

then

n (~,X G) ,

(~l®

y c (C ~ L'~(G)) '

is dual.

such that

u(~

y e ~.)'

~

>

p)u(~ ® f)lu(n ~ g)) - (yu(~ ~ pf)lu(~ ~ g))

oy(1.8),

: (y(V,p ® 1)*u(v~ ~ 1)(~ ~ f)lu(~, g)) : (~(v~

~ f)l(v~ ~ ~)u(~ ~ g))

= ( ~ ( v ~ ~ f)lu(v~ ~ ~g))

b~l.8)

= (yufv lU gV ) = (yuf~!U~g~) = ((i ~ p)yu(~ ~ f)lu(~l ~ g)). Thus

[y, i ® p] = 0

Proposition

for all

1.6.

(i)

p E G

If

and hence

o

y e (C @ L~(G)) '.

Q.E.D.

is integrable and satisfies the relative

commutant property:

(1.9)

~(~0' n('~x

G) ca~.x G'

then

(1.10) (ii) Proof.

(~;~)' n ~,= c If

G is abelian, (i)

Proposition 1.5.i.

t h e n (1.10)

il:lplics ( 1 . 9 ) .

We use the same device and notations as in the proof of If

y c(b..°;)' r,. LT. ,

be the complex conjugation operator on

then

L2(G).

UJyJ ~ CM.,x G U Then

by (1.5).

Let C

108

UJu(t)J = ( J ~ Therefore

JyJ

(ii)

commutes with

C)(I®

Ju(t)J ,

o(t))(J ~ C)U = ( i ®

and hence

p(t))U.

y e ~

We use the same notations as in the proof of Proposition 1.5. (ii).

First we notice that (l.ll)

(i®

(1.12)

(u(r) ® 7\ (r)) Uf = ~ ( r ) f

If

y e a(~)' n (F~× G), then O,

~(r)) Uf

Uo(r)fU(r )

JU* yUfJ e ~:) by g

"

(1.7).

Since

Ju(t)J = u(t), this

implies that (1.13) Since

Ug y Uf c ~ . y c !:~)', it suffices to show that

y e (C ® 9(G)) '.

By (1.11)

and

(i.13)

we have

Ug y(1 ® o(r))U~._ = Up(r).g(1 @ p ( r ) ) y U p ( r ) f. Since

y e ~x G

and

G

is abelian,

(l.ll)

implies

U~(r).g(1 ® D(r))YU.(r) f : U g ( l ~ o(r)]yUf.

Therefore

y ¢ (C ~ It(G))'.

Q.E.D.

It should be noted that (1.10) is true for any integrable modular automorphisms, [14]. NOTES

The equivalence (i) ~

(ii) in Theorem 1.1 was obtained independently in [14,57].

Theorem 1.4 is taken from [14].

Propositions 1.5 and 1.6 are due to Paschke, [54].

109

§2. StaOility In this section we shall discuss the stability of actions or co-actions under l-coeycle perturbations.

Definition 2.1.

An action

:). (resp. co-action

all unitary l-cocycles are equivalent, there exists a unitary

h)

is said to be staole, if

bh&b is, for each unitary 1-cocycle

c e ~(resp. ~) such that

a

a = (c* ® i) o (c) (resp.

(e* ~ 1)~:(e)). Lemma 2.2.

If

q

satisfies (i.i), then (i.i) holds for both of the follow-

ing: for each unitary

a c Z,),(G,k);

(ii)

on

or

Proof.

(i)

(i)

a~

some unitary

~ ~ F n (n e ~

~) given by

We may assume that

u e £(~) ~ L=(G)

~

with

is standard and

a~(~,,) v (~ ® .~ ( 0 ) )

(ii)

a(X

:){x) - u(x ® l)u*

for

(u ~ I) (g ® G) (u ~ l) = ;,~G(u). Since

(a~<,~.)) ' = ( A d a u ( ~ ® ~ ) ) '

it follows that

~t = ~t ® L •

satisfies (l.1) on

- a .~.(~)'a*

-: a(~,: ~ p ) a * , ~ .

Since

(:){ ~.) @ Fn)' q (( "~(~,,) ~ F n) v (~ ¢ ~(O) ~ .. )) c "~ ~.) @ Fn (~ ~ ~") (( :'(~) @ Fn)' : ~(~ ~ ~n)' it follows that

(~ satisfies (l.l) on

~,

F n. Q.E.D. a

Pro~osition 2.'~ all unitaries

Proof. c(t) = i ® (i.i) on

If

~

satisfies (l.l).and

a e Z (G,(,, ~), then

Take a unitary ell + a(t) ~ e22

~ ® F2 .

~

a e Zo(G,#~)., We set and

~t ~ ~t ® L •

~ - ~®

Then, by Lemma 2.2,

-C

D

coincides with

central supports of

~c

for

Thus

eI = i ~

unitary

i ~ ejj in

~ ,

in

the fact

j = i~2.

ej (~ C ~

ell + i ~ e22 = e 2 , namely,

central support in

F o, -C

Therefore, Proposition i.'~ implies that

which implies that the center of

i ® ell ~ i ® e22

is properly infinite for

,,, is stable.

-e

~

a e Z (G,~) ,

As we assumed that i ® ell

and

(m)' -C

C~ ~ ~ . Then

Let

ej e ~

e~

satisfies

P ~ = C~ , i~

be the

O • •

Since

implies that i ® ell + i @ e22 < ej. i ~ ell ~

a

and

i ~ e22

have the s~ne

is groperly infinite for every

i ~ e~,,/~, are properly infinite in

~c.

110

Consequently,

i % ell ~ 1 ~ e22

Corollary 2.71.

If

G

in

~c ,

namely,

is finite and

~;t

a ~ i.

is free for each

t / e , then

o

is stable. a

Proof.

~e

case where

N

is properly infinite.

properly infinite for all unitary t / e ,

o' satisfies (i.i).

The case where

D

a e Z

(G,D).

Therefore,

is finite.

Since

Since

~

= ~ ~ F

and

satisfies (i.i) by Lenzna 2.~, and hence the center of

trace on on

~.

T~' • Then the restriction Therefore,

Thus ea ~ ell ~ eb ® e~2 ~

C g

for any projections e ~ f

a

C of

in

is finite,

to e

O ~ ~(~

and

is the

Let

~

~

Then O ~

by

De the center valued

is also the center valued trace f

~T ¢~-> e ~ f

for all unitaries

.

is

2.~.

"t : :~t @ L.

~<7

~

is free for all

,~ is stable by Proposition

Let

Proposition i.~, which implies that

~t

G

in

in

a,. b

.in

,,

~.~ . Z. y(G,~.). .

Conseouently,

D.

Q.E.D.

EOTES

The stability of an action played an important role in the structure analysis of a factor of type Ill, [14].

IIowever= we have to admit that the general study

of the stability of an action is in a very primitive one should study also the approximate ular automorphism action of ]R by Connes-St~rmer

[13].

stability

stage.

of actions.

on a factor of type III I

The materials

The authors feel that For example~

is approximately

presented here are taken from [14].

the modstable

CI-IAPI~ER VII APPLICATIONS

Introduction

TO GALOIS THEORY

Galois theory here means the theory concerning intermediate

Neumann sublagebras

yon

between the entire yon Neuamnn algebra and the fixed point sub-

algebra under a prescribed action (resp. co-action) related to the analysis

of

of the algebra of observables

the quantum field theory of Doplicher-Haag-Roberts,

G.

The theory is intimately

in the axiomatic

[ 17].

approach to

It should, however,

be

pointed out that since our theory is concerned with von Neumann algebras, but not C*-algebras~

we have still a long way to go in order to build a theory applicable

directly to physics.

Nevertheless,

the study in the von Neumann algebra context

would provide a way to go further. In ~l, we shall study the crossed product by a closed subgroup homogeneous

space

H~G

according to an action or a co-action.

tell how one can recover a given closed subgroup subalgebras,

i.e.

H

is determined by

• x

Section 2 is devoted to the association yon Neumann subalgebras

H

of

G

H

or the right

Theorems

1.1 and 1.2

by the crossed product

H (resp. ~ x 5 (H\G)). of closed subgroups

H

to intermediate

in the crossed product under a natural regularity condition.

In ~3, under regularity conditions we show Galois type correspondences closed subgroups and intermediate The regularity conditions

subalgebras

between

for semi-dual actions and co-actions.

are fullfilled when one has a sufficiently

phism group commuting with a given action of a compact group.

large automor-

Hence~ if the situation

arises from physics, and the action is given by the gauge group, then the space translation will provide the regularity condition. mechanisms, be in with

we shall show that an automorphism

~(G)

for a compact

c~(G),

Theorem 3.8.

G

if

~

~

Making use of all the duality leaving

The last section is devoted to the full group analysis provides a correspondence automorphism

~

pointwise

commutes with a large subgroup

between certain subgroups,

~

of Dye-Choda,

called full groups~

group of the crossed product and intermediate

subalgebras.

fixed must commuting

which also of the

112

§!.

Sub6roups and crossed products Let

×~ H

H

be a closed subgroup of

and

~ ><5 (H\G)

K

with

~ x5 (H~G).

O

I

I ~,x~H

H

[e}

will turn out to be the or the set of

t e G

xH

]%><5 O

i

I

H ~

(:Z(I]'P,) , or

H

This indicates the one side of Galois type corres-

H ~

{e}

Then

~(~ x H) c h ~ p(K)"

J~,X~ G

G

We denote by

We shall give a characterization of

in Theorems 1.1 and 1.2.

smallest closed subgroup with ~t = & on pondences:

G.

~x 5

(H\o)

5(]%)

O

the right or the left regular representation of

on L2(H). Theorem i.i.

~ = ~ ~

G.

Then

~,XG H = [y • n : (9(y) -: n @ p(H)"] ,= n 0 (C ~ £~(G/H))'.

(ii)

H

Proof. in

Let

(i)

is the smallest closed subgroup (i)

K

satisfying

By (I.2.8) and (I.2.9) it is clear that

(~(~ X~ H) c ~l @ p(K)".

~(~ xG H)

is contained

n @ p(H)". Next we shall show that

~(y) • h ,~ p,(H)" implies

y • (C ® f(G/H))'

For

this it suffices to show that

(1.z)

f(S/H) ® C ~ (Adw (C ® ~XH)')) v (c ® ~(T2(G))) .

Indeed, y

y ® i

commutes with

must commute with

C @ £~(G/H)

(~(f) c £~(G/H) @ L*(G),

IF~ill = 1

Since

where

by (i.i).

and

C ® C ® £(L2(G))

l~ow, if

(c~f)(s,t) = f(ts).

f .< ~(G/H) O C(G), For any

g,h c K(G)

f c D(H)' If

and

(7

~d (f -i )(1 ~ g -1 )d r

(x~(f) = Ad W (i ® f)~

~ • L2(G x G),

(i ~ h)

Fg,h

belongs to the right hand side

thegn

r ~f(r-ls)

is continuous and bounded, if

g(r-l)dr

converges to the Dirac

f ® (A-lh).

f ® (A-lh)

Making

i,

F converges weakly tc g,h belongs to the right hand side of (i.I).

.

measure at the unit, then to

then with

r

(Pg,h~)(s't) =(]~ f(r-ls)g(r -l)dr) A(t)-lh(t)~(s,t) Since

and hence

we set

~,h =

of (i.i).

AdI@WG(~I' ~ p(H)')

we have the inclusion (i.i) for

~(G/H) n c(G)

is weakly dense in

f(G/H) O C(G)

~(G/H),

Therefore

A-lh

in place of

we have (i.i).

converge weakly ~(G/H).

As

113

m X~ H = ~ n (c ® £ ~ ( G / H ) ) ' .

Finally we shall show that

y ~ ~ n (c ® ~ ( G / H ) ) ~. Here we may assume that be the modular unitary involution of

•

and

~

~x~

Suppose that

is standard. G,

Let

respectively.

J~

and

It then

follows from Proposition III.1.7 that

= (Jm ® J)u* = ~ (J~ ® J) , where

u

is the canonical unitary implementation of

7(c ® A ( G / H ) ) ' 7 and hence

)y7 c h' n (c ® ~(~\o)),.

for induced representations, (~.XH

H)'

onto

[ 65].

~.

= (C ~ ~(H\o))'

Here we apply the m a t t n e r - M a c k e y ' s

There exists a natural isomorphism

~' 0 (C ® ~ ( H \ G ) ) ' ,~(x' ® l H )

~(~

~H

is the restriction of

X H H)'),

theorem

of

such that

= x' ® 1 G ,

x' "- ~' ,

~u(t)®~(t)):u(t)®x(t) where

Then we have

(~ to

which is generated by

H.

,

Therefore,

~.' ® C

and

t ~ H, ~y7

belongs to

u(t) ® X(t),

t c H.

Since

7(u(t) ® X(t))7 : i ® p(t) , it follows that (ii)

y

belongs to

~ ×(~ H. Q.E.D.

It is cleam from the first equality in (i).

Theorem 1.2.

Let

(~o):

~ = ~ ~

G.

(i)

~

(ii)

H = it ," G : ~t(x) = x,

Then

{x ~ ~ : ~ t ( x ) = x ,

t ~ ~]:~n(c~x(H))'

x e )I ×t~ (I{kG)] •

Proof'. (i) it sufI'iccs to eonsi<~cr the co_mmut,ant of (!1.3.8).

(ii)

Then

K

Let

K

be the set of

is a closed subgroup of

by (i), we have

H = K.

t < G

G

such that

containing

~t(x)=

H.

Since

x

for

x e h ~

~ x8 ( H \ G ) c h

(H~G).

x8 (K\G) Q.E.D.

114

NOTES

A Galois theory of yon Neumann algebras are initiated

by Nakamura and Takeda

[51] and Suzuki [ 61] for discrete groups. Theorems 1.1 and 1.2 are obtained by Takesaki [ 69] for abelian locally ccmpac~ grups and Nakagami [ 47] for non abelian groups.

115

§2.

Subal6ebras in crossed products Let

9

X5 G)

be a v o n

containing

of the form case

H

~×~

Neumann subalgebra of a crossed product G(~)

H

or

(resp.

5(h)).

~ X 5 (H\ G)

is determined by

.9

and

.9

~×~

G

(resp.

We shall give a condition for

for some closed subgroup is determined by this

H

H.

of

~

G.

to be In this

Therefore, we have

the other side of Galois type correspondences:

I oE_LbH I

I

I .9 ~ I

1 {e}

a(~.)

,

l

~,(h)

G

For the s~ke of a technical reason we shall assume that factors in Theorems 2.1 and 2.2.

I >H

~

and

~X 5 G

are

In Theorem 2.3 we replace this assumption by

other conditions. Theorem 2.1. If

Let

.9 be a von Neumann subalgebra of

containing

a(~).

is a factor, then the following two conditions are equivalent: (i)

.9

(ii)

.9 = ~, xcz H

Proof.

is globally

fying

~

invamiant.

for some closed subgroup

H

cf

G.

(ii) ~ (i): It is clear.

(i) --~ (ii): Let

h = N X~ G

a(9) c h ~ p(H)" .

and

Since

H

the smallest closed subgroup of

&(y) e "a ~ ~(H)"

coincides with the closed subgroup generated by 9

~L ×~ G

is contained in the set of all

y (: h

with

by Theorem i.i.

Let

~

be the isomorphism of

~(C~(x)) = qH(x)

and

~(i ~ ~(r)) = i ® ~H(r).

(2.1)

is equivalent to sp&(y)

for all

H

onto

~. x H H

satis-

sp&(y) c H, H

y ~ .9.

£](y) e N @ ~(H)", i,e. • x

G

Therefore

.~(y) e ,T,x H satisfying

Then

~(~) : ~(~) : ~ ×j H

Here we set

5 = (~H)^.

Since

N. is a factor,

Now we claim that ~(.9) x 5 H the ergodicity of z @ lH

for

5

is a factor.

on the center of

z a w(C~).

n(9)

F(5) = H

by Theorem IV.l.5.

To this end, we need only to check

by Theorem IV.'J.3.

Suppose that

5(z) =

Tile inclusion relations (2.i) imply that

Cfi(~h)

:

~ ( . 9 ) 5 (~ X H H ) 5 = c~H(E',) • CJ

~,

Since r<.9)8,

z

is a factor, so is

~(p)5.

Since

must be a scaAar operator.

On the other hand, by (2.1), we have

~Co) 5

is contained in the center of

116

The d u a l i t y onto

for crossed

~ ~ £(L2(H))

product

gives

us a n i s o m o r p h i s m

as in the proof of Theorem 1.2.7.

~ @ L'~'(H) by Lemma 1.2.6.

~'

of

Then

(~' ×~H H) X~ H

~,(~H(~) X5 H) =

Therefore,

~ L~(H) C 17'(17(~) X5 H) c ~'~ ~ £ ( L 2 ( G ) ) Since

w(e) x~ S

is a factor, it must be

( ~ × H H) X5 H. hene~e

Considering

5(y) ® l

~,

This means

we h a v e

~9)

,-<,9) X 5 H =

= ~. × H H

~

and

(x of

G

on

1 @ k(f),

h x5 G

carries the generators

respeetSvely,

i.e.

g

leaves

On the contrary~ we want to consider a co-action

makes

C ®L~(G)

fix and

5(~)

which agree with a ec-action Theorem 2.2. h x5 G

Let

~

5(y)

and

and Q.E.D.

i ® f

to

5 ( h ) f i x e d but C@L'~(G) .d :~" of G on h ×=~2 G w~ich

move:

5 ( y ) }->(~ ® 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)'


~erefore,

~'

is generated

(h ~

G)'

and

9 = (h XS, G) n (c ~ X(H))' = ~l ~, (H\G) (ii) ~ (i): The commutant of C ® X(H).

s(y) = y ® 1

Since

it follows that contained in

~'

.

in other words,

by Theorem 1.2. ~s generated by

(h XsG )' and

m~nd 8 ( l ~ X ( r ) )

= l®X(r)®p(r),

y -~ (h ><5 G ) '

£

Since

C ~ ~(H),

~ = h X 6 (H\G)

for

is globally

.~' ~ ~(G).

r ~G

invariant.

Therefcre,

WC_ (I.~(G) ~ ~(G),

~

A d l ~ ~ (~' ® C)

is g l o b ~ l yG

5d

is

invariant. Q .E .D.

For a discrete group

G

we have a more general result.

Theorem 2. 3 . Assume that of

~x(% G

Aut(~×~

containing

G)

C

~(~).

is discrete.

If

~

Let

P

be a v o n

Neumann subalgebra

is free and there exists a subgroup

S

of

such that is globally

S

~nvariant;

(a)

~(~)

(b)

S

is ergcd~c on the center of

(c)

S

is trivial on

~(m);

C ®~(C),

then the following two conditions are equivalent: (i)

9

is globally

~

inva~'iant and of the form

faithful normal expectation (ii)

~ = ~x~

Proof. Let

etH

H

g(~x~

G)

for some

g.

for some subgroup

(ii) =>(i): The

S

of

invmriance of F(G)

be the projection Jn

H

G. m x

with suppcrt

~s clear from (a) and (c).

H tH.

Let

g

be a faithful

normal expectation defined by g(y) = Since

~ (i ® etH)Y(l ® etH ), tH ~G/H

etHPr(etH ) = etH

~ ( ~ ~) =

~

for

e

in

~

~

is free,

hence

° y = v

e c ~(C~). o g.

etHPr(etH ) = 0

for

r ~ H,

Since

g(l ® P(t)) :~ (1 ® p(t))e

by Lemma 2.4 below.

S~nce

~

n p ~(m)'

n~,~

@:

e' and

and

we have

H.

(i) ~ (ii): Since projection

r ~ H

y (~ m XG G .

Since S

is

P

is globally

a trivial

on

~ C @~(G)~

for some central

is free, we have

~(c m)

i nvariant we

(1 ~ p(t))e = (1 ~ ~(t))~(e),

and have

~ ~ S

is free, we have

118

Therefore the ergodicity of set of

t £ G

with

Z

on

G(C~)

implies

g(l ® ~)(t)) =. i ® p(t).

e = 0

Then

H

or

i.

Let

H

is a subgroup of

be the G

and

= g ( ~ ×~ G) = • x~ H. Lemma 2.4. g

Let

~

Q.E.D. be a v o n

Ne~mann subalgebra of

a faithful normal expectation of

~

onto

h.

If

u

~

with

h' ~ • c h

is a unitary in

~

and with

.×.

~

= ~,

then

g(u) = ue = eu

for some centr~l projection

e

in

h.

-v

Proof. u g(u)

Since

c %

g(u)x = g(uxu~u) = uxu g ( u )

~(u)~(~)~(u)

Therefore

g(u)

=

e E %

and

~

is a partial ~sometry. e = S(u)

Then

for

all

x c h,

we h a v e

and h e n c e

~ ( u ) = ue = e u ,

~(u)

for

=

We set = ~.(~ S ( u ) )

u~(u)x

= ~ ~(u)

~= x u ~ ( u )

.

for

x ~ ~.

Q.E.D.

NOTES

Theorem 2.1 is obtained by [691 for abe!tan locally compact groups and [47] for non abelian ones. H. Choda [ 6 ] .

Theorem 2.2 is obtained in [47].

Theorem 2.3 is obtained by

119

§~.

Oalois correspondences M~ir~2, u s e

of the results obtained

correspondence. we denote by

in §!ii and 2, we shall zive a Galois

For a w)n Neumann algebra

£(~1,92)

91

rand its von Neuma~n subaigebra

the set cf all w)n Neumann subai6ebras

~

of

91

92

con-

e raining

~2"

Theorem 3.1. bijective

If

h5

correspondence

of globally

5

is a factor and

b

is

semi-dual,

then there exists

between the set cf closed subgroups

invariant yon Neumann subalgebras

H

of

G

and the set

~ ~ £(h,hS):

(5.1)

where

K

runs over all closed subgroups of

Proof. 5

The case where

is dominant

on

h.

~

G.

is propez'ly infinite:

~j virtue of Theorem III.4.4~

Combini$~ '±~eorems i.i and 2.i, we have the desired corres-

pondence.

Th~gene~ Then

~

ease:

: h~ ~ F

P~t

~=hE~,

where

~(H)

and

H(~)

H = H(F(H))

equivalent to then

are defined

[(~) ~ [ ~ .(H)", and hence

Corolla~y :3.2. Assume that

Proof.

factor, remaindez'

oz'

~" ~ £ ( L 2 ( G ) ) ~ h x b O.

clcsed subgroups sub~l~ebras

If

,%(Z of

G H

~. is a factor and on of

(z

x ~ ?(H(~)),

: 9.

Q.E.D.

If

?" is integrable

By Theorem IV.'J.4,

Since

&~ez'efore,

of the proof is the same as in Theo:r'em "j.l.

Theo_,'em "3.3.

,~ F ,,

ffs

Y(5) = Q h5

and

t,

is

h

is a

is a factor.

J s semi-dual~

and the set of g.loba]ly

9 <: ~(~,,,~p(Z) in such a way that

5(z

The

Q.E.D. then there exist a

[P. and a biject:ive eor_"espondence between the set of G

an~

as in Theorem "~.i.

is properly infinite:

is a factor by Theorem 1V.7.3.

.,~(H) =?(H)

= ~ .~ o ( H ) "

There~ore, if

'N"% is a factor.

h $~

Therefore,

Since

5(~)

9(H(9))

"n.

Therefore

,

~(~) = H ( [ ) . Thus,

The case where

h X~, G

cc-action

(5.2)

x : e.

Since

then there exists the same cor-~espondence

dominant on

is se~£-dual.

similarly as ([~.l).

we have

h

~

9 : ~(H(@))

and

= H ( ~ ( H ) ~ F o) = H ( ~ P ( H ) ) .

x @ 1 ." ~ ( H ( ~ ) )

F(5) = C,

o (s ~ ~) ~_d N = ~ L .

is a properly infinite factor and H = H(~(H))

we h a v e

[= ( ~ )

invax.iant von Neumann

120

Procf.

The case where

~

is properly infinite:

Since

~

is dominant,

Theorems 1.2 and 2.2 give us the desired correspondence. The general case: is a factor,

~

Put

~ = m~_F~,

is s~mi-dual and

~

H = H(~(H)) where

~(H)

and

on

P(H).

~t = ~ hence

H(~) Since

and hence

follows that

and

If

for

If

Therefore

t ~ H(~(H)),

x c 9.

~. fis a factor.

then there exists a co-action

~t = ~

x e ~(H(9)),

t (H(9).

and hence

Assume that

~ = P @ F . Then

it follows that

H(~(H)) = H.

~t(x ® i) = x ® I

Corollar~ 3.4.

and

is properly infinite.

o = 9(H(~)) ,

~(H) = ~(H) ~ F . Thus

x @ i e ~(H(~)) = ~

C ~ . ~ G c(~(~)'

@ &

~ F

are defined similarly as (3.2).

t 6 H(~(H)) = H.

t 6 H(P)

~t=~t

= ~

If

5(~ of

then

Since Thus,

then ~(H)

and

~t(x) = x

on

for

H(9) = H(~),

it

9(H(9)) = ~.

Q.E.D.

(~ is integrable and G

on

m

and the same

correspondence as in Theorem 3.3. Proof.

By Theorems 3.3 and IV.3.4.

Lemma 3.5.

Assume that

by a unitary representation

is discrete.

u,

(~ is a free action implemented

There exists a faithful normal expectation of

(ii)

There exists an action

onto

~,~' xy G

such that

v

of

G

on

,~x) = y(x)

(~)'

onto

~'.

IT.' and an isomorphism

for

x ~. ~'

and

-

of

w(u(t)) = 1 @ p ( t )

t. Proof.

for

Put

x c ~'.

vt(x ) = u(t)xu(t) ×

Since

~

is free~

onto

~.'

and

~,' and

¢ = ~ o g. (m~) ',

Let

fcr

so is

(P be a faithful normal state on

of

If

then the following, two conditions are equivalent:

(i)

(~)' for

G

Q.E.D.

~',

x ~ T.'. Then ~.

g

g(u(t)) = 0

Since,

and

{w~,%,~%,}

for

x(t), y ( t ) ~

(~u(t)*x(t)l~u(s)*y(s)) = , \t s ¢

of

v : ~ ~ ~,u(s) s

where

fs

computation

is a funetionin we

have

%

onto

y(s) ~

~2(a)

Let (~(~)'

be GNS representations ~',

(sfg(u(st-l))x(t)] : /

~t,s v

t ~ e.

y(S)×u(st-l)x(t

= there exists an isometry

for

a faithful normal expectation of

[~,$@,g~}

respectively.

Thus

g(u(t)*)~t(x ) = xg(u(t)*)

,~ ® ~2(G)

defined by

%(y(s))~fs

with support

~s}

and

' llfsll2=l.

By direct

121

v%(x)v ~ , ~ ( u ( r ) ) v -1 : 1 ® ~(r)

Setting

w = (wQ ® ~)-i ~ Ad v o wQ,

When

G

~s discrete and

expectation of

(~)'

the spatial equivalence Theorem 3.6. expectation of Aut(~)

~

onto

~

c ~

.

we have the desired result.

~s sea,-dual,

~'.

Q.E.D.

then there exists a faithful normal

Indeed, we have only to consider the eommutant of

[~,~} ~ [~,~}.

Assume that

(~)'

,

onto

G

is discrete and there exists a faithful normal

~'.

If

~x is free and there is a subgroup

C~;

and

g

satisfying

(i)

g

(ii)

y(u(t)) = u(t)

is ergodic on

for all

~ ~ g

and

t ~ G,

then there exists a bijective correspondence between the set of subgroups and the set of globally

g

invariant yon Neumann subalgebras

faithful normal expectation of Proof. (~)' onto

Since

onto

~

(~P~)' onto

~'

G

such that

can apply Theorem 2.3 to

~(~') = v(m')

[~,~].

I 9

and

st = L

Since

~(K(~)) = ~

This implies ~' ×~ H

on

~

P'

and an isomorphism

~' = ~' V u(H(P))"

Z' X

G

onto

of Since w(~')

K

Y

~-~

I

i ® D(t) c v(e'),

we have

by Theorem 2. 3 , we have and so

~ = P(H(P)).

t 6 H(~(H))

H(9) = K(r
w(P') = ~' >< H(P).

Since

~(H)

is equivalent to

corresponds to

1 ® c(t) e ~' xy H.

H(P(H)) = H.

Q.E.D.

Up to this point, we often use the following correspondences:

(a)

[~}

by (II.3.4), 5(h)'

= {h X5 G, $]

and

£

then there exists an action

onto

(h X5 G)' ×y G

we

K

Remark. If

(~)'

I

I~(~') = ~' ×

I

if and only if (~ = ~ P ' ) ) ,

w

~(u(t)) = i ® o(t), t ~ G.

I ~

I

in the diagram, we have

This shows that

G

We consider the diagram:

I ~

I Since

of with a

is free and there exists a faithful normal expectation of

~',t::ere e~ist an action ~ of G on ~'

~' x

H

~ ~ 2(~)

in such a way as (3.2).

is free and there exists a faithful normal expectation of

of

of

such that

the co-action of y

of

G

on

(R ~

G

on G)'

5(R)'

defined

and an isomorphism

122

H

1%

I

h x 5 (H\G)

=

( h ><8 ( H \ G ) ) '

~

t5t~~ Here, gll

g12

a discrete (b)

I

~(h)

~

;%

{~(~)',~}

are bounded for a compact

If

{h,5] = [~ x~ G, ~}

and

of a(~)'

onto

(m×5 e)' ×~ G

t~=

r~~ = g.. ij ~nd

G,

~

then there exists a co-action

g31

{(~%G)'x

c, ; ]

and

g21

and

g22

.

Further,

are bounded for

G.

(II.3.3),

Here,

~

are operator valued faithful semi-finite normal weights.

g.. and

=

(1l ><5 G ) ' X H 'f

~

the action of ~

of

G

on

G

on

~(~)'

(~ X~ G)'

such that

iT.x~c~ ~

(mx~a)'

~

~((mx~G)')

iT, X(:z H ,-->

(lb. X(::z H)'

~

(iT, XQ; G)' x~ (H\G)

~

{(~-, xa o ) '

~(rO

,---~

{c~(r~),,~]

x~ ~, ~::}

are operator valued faithful semi-finite normal weights. g32

a discrete

defined by

and an isomorphism

are bounded for a compact

G,

and

g41

and

g42

Further,

are bounded for

G.

Theorem 3.7ergodic subgroup

Assume that S

of

G

Aut(~)

is compact.

If

commuting with

~

is faithful and there is an

at,

t ~ G,

then there exists a

bijective correspondence of (3.2) 0etween the set of closed normal suDgroups G and the set of globally

Proof. IV.2.5,

The case where

~

~,

It is clear that

we may assume that

is globally ~

and

with

iT.(~= ~

of

Put

invariant.

L=(G)

into

~L~(G)).

(~t ° ~ = ~ ° kt'

The general case:

S

•

9

Since

by Theorem 1.2.

G/H(£)

@

H

of

P ~Q(~,~).

By means of Proposition

~ c @(H(@)).

H(P) = {e}

(~ is faithful] on

(~ and

exists an isomorphism

•

~1 (~ is properly infinite:

~(H(£)).

generated by into

invariant yon Ncumann suolagebras

iT.. Therefo-~,e, H = H(P(H))

From this assumption, @

~

is considered as a faithful action of

fixed points are

because

and

(~ is dominant on

suffices to show that normal,

~

on

Since

It H(£)

P(H(P)).

and must show that

and hence

is

Since the 9 = ~.

(~ is dominant on

P,

By means of Theorem II.2.2, there with w

(~t ° ~ = ~ ° Xt

and

9

is also an isomorphism of

is generated by

~

~ = ~,@ F , ~t : O~t ® ~'

and

r~L~(G)).

'~ = @ ~ ~

and

is L~(G)

Thus

9 = ~.

123

= {T ® T' : T ~ 8,

T' c Aut(F )].

Then

= ~(~(c))

and

~(~(~)) .

~ =

Q .E .D.

The rest of the proof is the same as that of Theorem 3.3. Theorem ~.8. of

Aut(~)

Assume that

is of the form Proof.

G

commuting with ~ = ~

is compact.

~t,

t ~ G,

for some

r

The case where

~

a < Aut(~L/~ ~)

commuting with

r ( G.

is properly infinite:

IV.2.2 and IV.2.4, every irreducible some representation in

If there is an ergodic subgroup

then

subrepresentation

By virtue of Propositions of

[~}

is equivalent to

)~G(~). Therefore, Theorem 1.3.4 tells us that we need only

to check

Let

for all

~(~)=~

(3.3) (vj : j = l, .--,d}

HAlbert space in basis.

~

be an orthonormaZ basis of

with

dim ~0 = d

and

w

is an isometry of

V(t) =

onto

and

•

is considered as by

O0(x ) = ~ ujxuj

R0

into itself.

o0(~ ) ~ (R0,R0), and

Since

P,~= p0(~) @ (R0'~0)"

(R0,.~0) (R0,R0)

Since

; oo

{T0 : T e ~]

defined

is the subspace of those elements in

and

T e 8

mapping

h(z, it follows that

q = q ® we define

T0

and

L- by

0 ~% is ergodic on

is equivalent to the fact that is ergodic on

x ~ ~. ,

is the endomorphism of

f~,

The proof of (3-3) is now obtained as follows. c(R) m R

Indeed,

.

kjk ~ £,

is contained in

For each

0 m = c0(m ) $ (R0,~0).

X

[V,R 0] = w*{(Z,~}w.

where

at = at ® ~

sion

bc a

is a lineaz' span of elements of the form

kjkUjXU k = o0(x ) F ) j k U j U k

p0(~).

~0

its orthonormal

~ Vk(Zt(vj)uku j . j,k

(~t(w) =D(Zt(vj)u 3 = wV(t)

Since each element in

on

R e ~M~(m)" Let

[u. : j = l,..-.d]

×

and

R0

(3.4)

on

.

We set d w = ~ vju; j=l

Then

~<:~(~,)

~0(~),

so is

[~0

Since

w×C(w) ((R0,~0).

it suffices to check

: T

<

= wR 0,

$]

on

the inclu-

But, since

124 m

To this end we set

a

-X

= 7(w)w

.

Since

C~t(a~. ) = ~.(w.F(t))7(t)x~ * = a T by (3.4), it follows from the a s s u m p t i o n

on

~

that

~(aT) = aT,

which implies

(3.5). The general = IT v T'

case:

: "[ r ~,

We set T' ~ A u t ( F

~,, = i~..~ P , )}.

Then

c~t = c~t ~ ~, ~ = C~

~ = d ® ~

for some

r.

and

Thus~

~ = c~ .

r

f

Q .E .D.

NOTES

A Galois correspondence Connes and Takesaki ones. M

•

groups

(Theorem 5.i) is obtained

Choda

[4 ]

.

[6].

A Galois

Kastler,

correspondence

A Galois

tained by Kishimoto

by

[14] for abelian locally compact groups and ['17] for non abelian

Theorem 3.3 is obtained in [!-7]. Lemma ~.5 is obtained b y Connes

by H. Choda

Araki~

for continuous

[41]

for discrete

correspondence

groups

for compact groups

by a slightly different method.

Takesaki and Haag [~.].

,)

[12] and

/

(Theorem _~.o) is obtained (Theorem 3.7) is oo-

Theorem 3.8 is obtained by

125

§4.

Galois correspondences

(II)

In this final section we shall give a different

sort of Galois correspondence.

In the first half of this section we introduce the concept of a fall group of automorphisms

of

~

and get a Galois type ccrrespondence

will be called the Dye correspondence.

finer than Theorem 2. 3 , which

The s e ~ nd half is an introduction to a

new approach to the Galois correspondence. In what follows we assume that faithful action of of

G

on

~t

: t ~ G}

1.

Dye correspondence For any

e = e(O,Y)

Aut(~)

~,

~.

G

is a (countable) discrete group and

We occasionally

identify

and use the same symbol

y e Aut(~)

Definition 4.1.

G

a

with the subgroup

for them.

there exists the largest central projection

satisfying that the automorphism

O ~ Aut(~)

G

~

For a faithful action

(v-l~) e ~

of

is inner on

G

on

~

~e"

the set of all

satisfying sup e(~,~t) = 1

teG is called the f~ll group of full if

For each C~

~

such that

O e [G]

G,

which is denoted by

[G].

A group

G

is called

G = [G]. there exists a partition

[~tl(e(t))

: t c G}

{e(t)

: t e G}

of the identity in

is also a partition of the identity.

Then

if and only if

(4.1)

~(x) =

for some unitary

v ~ ~.

Therefore,

u ~ N(~(m)),

where

N(~(m))

of unitaries

u

~

in

Theorem 4.2. correspondence

G

) ,

~ ( [G]

with

of

of

G

)

If

~ c ~(~ ×~ G, ~(~)) onto

=

~u

~=Ad

in

m~

U

o~

G,

for some

i.e. the set

~

~s free, there exists a bijective H

of

[G]

and the set of

with a faithful normal conditional

~:

i-~H(O) = [y c Aut(m) H ~D(H)

~(~)

u~(~)u × = ~(~).

(Dye correspondence

~

x c

if and only if

is the normalizer

between the set of fall subgroups

von Neumann subalgebras expectation

D e(t~t(vxv tcG

~ N(~(m))

:~

o y = Ad

: ~

o ~,

o y = Ad

o ~,

u ~ N(~(m)) y

a H}"

Q ~} ;

.

U

Proof.

See [37].

In order to translate the above theorem into the Galois type correspondence, we choose a suitable Hilbert

space

~

so that the action

~

~s implemented by a

126

unitary representation

of

u

G

on

D

with

(~t = Adu(t]

on

~,.. Then

~',

defined by (4.2)

O~(x)

is an action of

G

cn

~,.'. l,'or e~ch 6'(x)

is an automorphism and call the

= u(t)xu(t)

of

=

x (. ,~"

~ - [G]

of the form (4.1) we define

x'_. e(t)O~(x), t~G

8'

by

x ~ 17,~'

,~'. Denote the set of a&l such

C-full group determined by

Theorem 4. 3 .

×,

~'

with

8 ({ [G]

by

[G]c

G.

Assume that

(i)

(~

is implemented

by a unitary

representation

(ii)

(~

is free;

(iii)

there exJ sts a faithfu], normal conditional

u

of

G;

and expectation

of

(~.~)' onto

~'. If

~'

is defined by (4.q) and

Aut(~J), subgroups

HC

cf

[C] C

is the full grcup of correspondence

[(~ : t ~ G}

cf

(~,~)' onto

in

between the set c'.f C-full

and the set of vcn lleumann sub.-~.ehras

a faithful normal expectation

(4

[G]

then there ex_~sts a bJjective

? ~ p(,~.,~(~) with

~':

.~) H C i->~,. 0 [u ~ ii(~') : Ad U e- HC}'

where

N(~')

is the normalizer of

Proof. action

Jn

Accordi~z to our assumptions

y (~ (~')

such that

~'

of

(~,,.~)' (j), (JJ) and (i~!), Lemma 3.5 gives an

G

on

~,'

and an isc,morpbism

..~(x) = ~(x),

x-

~.'

r~nd -(u(t)) = i ® o(t),

free and there exists a faithful normal expectation

~

of

of

(~.~)' onto t ~ G.

~.' ×

G

~,' ×

Since ontc

~

r~°')

G is from

w

our assumption on

(~

and

o,

we can apply Theorem 4.2 to

~',~}.

Now, we

consider the diagram:

~?~> ( ~ ) ' ~ ~,'× c

[a i

7

Since

~x)

: y(x),

u ~ N(y(~')),

then

l x ~ ~'

[,s i c

I e- >

I

~

I <----~.

H

HC

I and

~N(~'))

~ o @ = Ad u o ~

Theorem 4.2 gives the correspondence

= N(~(~')),

is equivalent to between

~'

and

Jf

0 c Aut(m')

~ = Ad H:

i w- (u)

.

and Therefore,

127

H~-e~"~,u

N(~.') : Ad u ~ H~"

~

~'~,[ad u ~' which ~mplies Remark.

If

not necessarily

if

8

,

~

Q .E .D.

is free, then there

of

G

exists

surjective.

The converse

Moreover,

inclusion

of the identity

of the identity.

an ~njection:

into the set of full subgroups ~ ~

in

cf

C17,, such that

v c ~

H ~ [HI

[G].

H = 9([H]).

is shown as follows.

is of the form (4.1) for some unitary

t ~ H]

n~'}

our result.

set of subgroups

is clear.

• u ~N(~,)

,

Indeed, That

from the

Th~s injection

~ ×5 H ~ O([H])

~ ~ [HI

if and only

and some partition

[~l(e(t))

: t c H]

is

[e(t)

:

is also a partition

Setting U =

(1 .~. p(t

~.

l(e(t))v)

,

tcH

we have -i

u

Ad

o (% = (~ o ~.

unitary

Since

subaAgebras Aubert

in

then

and so u ' = u~(z) for some O(~l) o r , , ~ . ~usthe of intermediate

G, (~(~,)) than that given in Theorem

von Neumann

2. 3 .

ccrrespondenee 4.4. of

Let

subalgebra

9

be ~in~te!y

wandering,

~,

~

{~t(e~)

: t ~ G]

~r = L

on

(iii)

the set of 4.5.

If

9

Since

system

~

is a partit~on

r c G ~

~r(ei)

~.

For each yon Neumann

cf the identity

with

~

r

: L

of the identity ~ ci

~n

P

subgroups

H

wandering

of

G

9

is said to

i ~ I;

i g I;

and

is finite.

a bijective

correspondence

of (i~.2)

and the set of yon Neumann

subalgebras

partition.

is dual, we may assume that

~h,5].

for each

for some

on

is dual, there exists

with a finitely

Proof.

cn

[e. : i c I] ].

whenever

the set of finite

E ~(~,~)

O

jf

(ii)

Theorem

be an action of

a partition

(i)

covariant

Ad a o ~ = (% o ~, ,'

u-lu ' <~(CI~) wh~chmeans

gives rise to a finer classification

~(~. x

Definitien

between

u' c N((~(~,,)) and

is free,

~

:~enee u' ~ m % H ,

z c%.

Dye correspondence

2.

If

U

u' ~ ~(m').

~he correspcndence

[~,~} = [U X 5 G, g}

H(9(H))

= H

fcr some

is a consequence

of

Theorem 1.2. Next we shall show show the reverse

~(H(@))

inclusion.

= ~.

Since

g

be the

Let

# c @(H(~)) ~-valued

is clear,

weight

:(Ill .1.2) : + a

(X)

--

~

t~G

A(t)~t(x ) ,

X

4

~

•

on

~

it suffices defined

by

to

128

Let 0

{e i : i £ I} for all

i ~ I

be a finitely wandering partition. by (ii).

Therefore,

for any

If ~!{

x 6

t { + O p~

H(@), and

then ~t(ei)e i = + ~ 6 ~.,

(g (xei)e i, '~} = < ~ a(t)~t(xei)ej/O) t~G = ( _~ A(t)~t(xei)ei,~!) t£H Since

V ~t(ei) = i

by (i) and

H

= (x

~ a(t)~t(ei)ei,u) t~H

•

is finite by (iii),

~' ,5(t)~t(ei)e i = uiei t~H for some

~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,<~ @ f) ,

(b)

f ~ Ll(G)

,

w e £(~)..

There exists a bijection between the set of all non-degenerate

sentations [~,R]

of

A(G)

and the set of all unitaries

*

w ~ £(R) ~ @(G)

reprewith the

associativity c erudition (A.3)

(w @ 1)(~ @ ~)(w @ l) = (L @ ~G)(W)

by the relation (A.4)

<,~(%0)/~) = <w,~ ~ 0~> ,

Proof.

(a)

That a unitary

u c £(~.) $ L~(G)

to the existence of a unitary representation ¢ F~ ® L2(G). (b)

The correspondence

(~,TI) e R × R -~ (w,~,r~ @ ~0>

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) = <.(~) ® 1 , w ( ~ ~ ~)> = <w,,~(~) ® ¢> = <~(~),'~(~)> = <~(~o),~("0,~'> •

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 ~ l)w,:~ ® ~> =

<~(~),~>

= <x~(~),.~>

and <w(x ~ i),,~ ® ~> ~- <~(~)x,~>

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 <~ 3 of Chapter I we use the notation

t -~ ul(t) ~ u2(t)

Definition A.2. Rj

for

j = 1,2.

instead of

uI

w. be non-degenerate J The product ~l * w2 of w I

Let

for

Theorem on

Rj,

and

satisfies

A.3.

then

(A.~)

and

.

~2

of

A(G)

on

is defined by

*,

v2'~'2)

If

wl * ~2

~j(j

= 1,2)

are non-degenerate

is a non-degenerate

*

*

representations

representation

of

A(G)

of on

that

= ((~)

(Wl ~ i ) ) ( i ~ 2 )

,

(~l * ~2 )(g~ * f) : [ ~ ( o ~ l g ~ ~ ~2(~[if) dt , f,g ~ ~ o ) w

for the

~.j ~ £(Rj) * •

w i.~2

(A.13)

representations

and

(~i * ~2 (co) '~i ® "~'2> = (~'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),

=
is of the form ~-J~3 wj .~ ~j

it follows that

(p(s),p(t)~) for some

. ~.J [ £(~1).

and

134 *, 1 * ,~2':*~2> = ,i" ~ ( i ® p(t),,~'j ~ ~Cj)~ 2 2 ( t) (%0,Wl.~. " J_>3 *

-

"

*,J

>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,

<((L ® o)(w I ® 1))(i ® w p ,% e '~2 ® °>

Wl * 'w2 is a non-degenerate

It remains to s~-ow (A.13).

*

For any

representation of

f,g <: K(G)

r......

and

Ffi

f,g# e K(G),

with (A.12). we have

O( rs)di:~1( r)d~%( s)

= jj./ g~rt-l)f(ts)dt

Since

A(G)

,:j e £(~j).

d,~ul(r)dw~*'2(s)

.

we use Fubini theorem and

<(~1 * ~2 )(g~ * ±')'<% ~ :'>P

'<,~z( -1 % g~,%><~2(xtlf),%> ~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
=

~ -i g~slt )f(ts2)dt = ( g ~ *

f)(sls2)

•

135

T h e o r ~ A.4. w 6 h ~ ~(G)

If

5

is a co-action of

with (A.3) and

corresponding to

w,

w

G

on

is the non-degenerate

*

representation of

A(G)

then 5
Proof.

implemented by a unitary

It is clear by (A.6).

<~ = :~f,g> .

Q.E.D.

REFERENCES

Ira]

Araki, H.; Some properties of modular conjugation operator of von Neumann algebras and a non con~uutative Radon-Nikodym theorem with a chain rule, Pacific J. Math, 50(1974), 309-35h.

[2]o Araki, H. ; Structure of some von Nemmann algebras with isolated discrete modular spectrum, Publ. RIMS, Kyoto Univ., 911973), l-lk.

~ ~

%i~

[3]

Araki, H.~ Kastler, D., Takesaki, }$ti'and Haag, R.~ Extension of KMS states and chemical potential, Commun. Ymth. Phys., 53(1977), 97-134.

[4]

Arveson, W.B.; On groups of automorphisms of operator algebras, J. Functional Analysis, 15(1974), 217-2),3.

[5]

Aubert, P.L.; Th$orie de Galois pour uric W*-algSbre, Comment. Math. Helvetici,

[6]

Choda, H. ; A Galois correspondence in a yon N e u ~ n n algebra, Preprint of Osaka Kyoiku Univ., 1977.

[7]

Choda, H.; Correspondences between yon Neumann algebras and discrete automorphism groups, The 2nd Japan-US Seminar on C*-algebras and its Applications to Physics, 1977.

[8]

Choda, M.; Shift automorphism groups of von Neumann algebras, Proc. Japan Acad., 50(1974), 470-475.

[9]

Choda, M. ; Normal expectations and crossed products of yon Neumann algebras, Proc. Japan Acad., 50(1974), 738-7L2.

[10]

Choda, M. ; The fixed algebra of a yon Neumann algebra under an automorphism group, T6hoku Math. J., 28(1976), 227-234.

[]_7_]

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