Lecture notes in mathematics

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
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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

đ V)

ã ~(~)

~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 ~ ã ì , 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
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) đ f> = <(~ đ ~) ã ~(=),~ ~ 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

~ ì

s(~) and
~

C đ 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 đ I ) ,

(2.6)

~

is a co-action of
ii)

G

on

~ ì

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(lđV~)(x @ 1),
~

is an action of

Proof

i) Since

G

on

~ ì5 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

~ ì ~ G.

1 đ 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

ã

(F X

by

G

G) ì^ G

~t = at đ 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)) ã

(2.12)

We %hen have

f

xe~;

V(~(x))= )(x) đ 1 ,
~(1

đ o(r))

~(1

đ 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), ,.~~. Â ~ Â

, ~ đ '$ @ ~)

= (~(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 = (~ đ

L) ~.

From (2.8), it follows

that
o ~(x)=~(x)~l,

~

(~ì~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