Actions of Discrete Amenable Groups on von Neumann Algebras

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1138 I IIII I

Adrian Ocneanu

Actions of Discrete Amenable Groups on von Neumann Algebras

Springer-Verlag Berlin Heidelberg New York Tokyo

Author

Adrian Ocneanu Department of Mathematics, University of California Berkeley, California 94720, USA

Mathematics Subject Classification (1980): 20 F 29, 46 L 40, 46 L 55 ISBN 3-540-15663-t Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15663-1 Springer-Verlag N e w York Heidelberg Berlin Tokyo

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically these of translating, 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 "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

SU~RY. groups

We study

classification of a m e n a b l e A main

up to outer

groups

result

hyperfinite

my a c t i v i t y were ments

The

in

in the proof, I would

Zeeman

Benjamin results

in

generosity

Craig

and accuracy.

conjugacy

up,

as well

at the U n i v e r s i t y Foia~,

~erban

Romania,

of Warwick,

England.

as my

Pimsner,

Str~til~

and

improve-

Mihai

Evans,

during

as several

as well

Zoia Ceau~escu,

from David

obtained

for their

Klaus

INCREST Sorin Popa, support.

Schmidt

and

at Warwick.

Weiss was very kind

to send me a d e s c r i p t i o n

of the

to their publication.

to V a u g h a n

with which

My wife Deborah

done

support

factors.

group on the

fellow at I N C R E S T - B u c h a r e s t ,

Arsene,

[36] p r i o r

I am grateful

up to outer

of this paper were m a i n l y

The w r i t i n g

and e s p e c i a l l y

further

Christopher

were

the

of the actions

II h y p e r f i n i t e

of an a m e n a b l e

like to thank C i p r i a n

Grigore

Dan Voiculescu, I rec e i v e d

[34].

conjugacy

amenable

We give

II 1 factor.

results

as a r e s e a r c h

announced

colleagues

type

of d i s c r e t e

algebras.

on the type

is the u n i c i t y

of the free a c t i o n

ACKNOWLEDGMENTS.

the actions

on factor yon N e u m a n n

Jones

he h e l p e d make

Ileana g e n e r o u s l y typed and edited

for useful

discussions

the results

and for the

in this p a p e r

known.

h e l p e d me carry on my work. the m a n u s c r i p t

with

remarkable

skill

TABLE

OF

CONTENTS

Introduction

. . . . . . . . . . . . . . . . .

I

Chapter

1

Main

5

Chapter

2

Invariants

Chapter

3

Amenable

Chapter

4

The

Chapter

5

Ultraproduct

Chapter

6

The

Chapter

7

Cohomology

Chapter

8

Model

Action

Splitting

Chapter

9

Model

Action

Isomorphism

References Notation Subject

Results

. . . . . . . . . .

and

Classification

Groups

Model

Rohlin

. . . . . . . . .

16

. . . . . . . .

23

Action

Algebras Theorem

. . . . . .

31

. . . . . . .

41

. . . . . .

59

. . . . .

77

Vanishing

....

. . . . . . . . . . . . . . . . . . Index

Index

8

95 112

. . . . . . . . . . . . . . . .

114

. . . . . . . . . . . . . . . .

115

INTRODUCTION

In this p a p e r we study von N e u m a n n

algebras.

THEOREM.

Let

G

be the hyperfinite outer

automorphic

The main

be a countable

II 1 factor.

discrete

Any

of d i s c r e t e

groups

on

is the following.

two free

amenable actions

group

of

G

and

let

R

are

on

R

conjugate. An action

Aut M ,

a

of

called o u t e r i.e.

G

on a factor

the group of a u t o m o r p h i s m s

not inner for any

g E G,

conjugate

unitaries

g ~ i.

G,

the above groups

[27, T h e o r e m

=

=

restriction theorem

which

M

a,[:

a unitary

free

if

ag

G --->Aut M

cocycle

u

is

are

for

,

U g a g ( U h)

8 Ad Ug~g8 -I

not hold

,

g E G

for any n o n a m e n a b l e

[26].

in c o n n e c t i o n

sequences)

general

to be i s o m o r p h i c

For such a factor,

are c e n t r a l l y

central

is called

into

Actions

of general

with hyperfinite

factors

3.1].

the factor

predual.

a

is essential:

does

arise n a t u r a l l y

We a c t u a l l y w o r k w i t h more require

M;

G

such that

The a m e n a b i l i t y

amenable

of

of

g E G, w i t h

[g group

is a h o m o m o r p h i s m

if there exists

U g C M,

8 @ Aut M

M

Two actions

Ug h and

actions

result

free

we prove

(i.e. each

factors to

the o u t e r eg,

and a p p r o x i m a t e l y

M® R

and actions. and to have conjugacy

We only separable

for actions

g ~ i, acts n o n - t r i v i a l l y

inner

(i.e.

each

ag

on

is a limit of

inner a u t o m o r p h i s m s ) . For a group

G

duced

in

R,

[21],

II~ factor for outer

(not n e c e s s a r i l y

on

is c o m p l e t e

R0, I,

classification

where

possible

of the group

are c o n j u g a t e

for outer

mod:

a

of a d i s c r e t e

III factors

into

~g if

G

is the m o d u l e

([4]). the

as well.

@ 6 Out M =

intro-

is C o m p l e t e

[6] to obtain

on factors,

Out M = A u t M / I n t M.

exists

A(a),

On the h y p e r f i n i t e

(A(e), mod(~))

lines of

study of G - k e r n e l s G

if there

We show that,

conjugacy.

the

amenable

invariant

Aut R0,1---> ~ +

to go along

for type

We do a p a r a l l e l phisms

action

the s y s t e m of i n v a r i a n t s

conjugacy,

It seems

free)

we show that the c h a r a c t e r i s t i c

0 Bg0 -I

is a d i s c r e t e

which

are h o m o m o r -

Two G - k e r n e l s

with ,

amenable

g

E

group,

G

i

and for a

B,~

G-kernel

~ on

R,

the E i l e n b e r g - M a c L a n e H ~ - o b s t r u c t i o n

a complete c o n j u g a c y invariant, (Ob(8), mod(8))

and for a G-kernel

Ob(8)

8 on

is

R0, 1 ,

is a complete system of invariants to conjugacy.

A result of i n d e p e n d e n t interest o b t a i n e d is the v a n i s h i n g of the 2 - d i m e n s i o n a l unitary v a l u e d c o h o m o l o g y for c e n t r a l l y free actions (the l - c o h o m o l o g y does not vanish for infinite groups:

there are many

examples of outer c o n j u g a t e but not c o n j u g a t e actions). I n v o l u t o r y a u t o m o r p h i s m s of factors have been studied by Davies [8], but the major b r e a k t h r o u g h was done by Connes in c l a s s i f i e d the actions of actions of invariants

Z n on

R,

and in

Z up to outer conjugacy.

[3], w h e r e he

[4], w h e r e he c l a s s i f i e d

A study of the c o h o m o l o g i c a l

for group actions was done by Jones in [21] where he e x t e n d e d

the c h a r a c t e r i s t i c

invariant of

[3] to group actions.

c l a s s i f i e d the actions of finite groups on type actions of

R,

In

[23] Jones

up to conjugacy.

Product

~n of UHF algebras were c l a s s i f i e d by Fack and M a r e c h a l

[ii], and Kishimoto studied by Rieffel

[27], and finite group actions on C * - a l g e b r a s were [39].

C l a s s i f i c a t i o n results for finite group

actions on A F - a l g e b r a s were o b t a i n e d in This paper is an e x t e n s i o n of c o n j u g a c y part of

[17],[18]

by H e r m a n and Jones.

[4], and also g e n e r a l i z e s the outer

[23].

In the first chapter we state the m a i n results in their general setting,

and in the second chapter we use them to obtain,

p r e s e n c e of invariants, II factors.

in the

c l a s s i f i c a t i o n results on the h y p e r f i n i t e type

The proofs of the main results are done in the r e m a i n i n g

part of the paper. The first p r o b l e m is to reduce the study of the group of its finite subsets.

An a p p r o x i m a t e

is an almost invariant finite subset of by means of the F~iner Theorem.

G,

o b t a i n e d from a m e n a b i l i t y

By means of a r e p e a t e d use

of these p r o c e d u r e s we obtain a Paving Structure for p r o j e c t i v e system of finite subsets of

of

G

to one

A link between such subsets is y i e l d e d

by the O r n s t e i n and Weiss Paving Theorem.

G-action.

G

substitute for a finite G-space

G,

G,

w h i c h is a

endowed w i t h an a p p r o x i m a t e

We use this structure to c o n s t r u c t a faithful r e p r e s e n t a t i o n

on the h y p e r f i n i t e

II 1 factor, well p r o v i d e d with a p p r o x i m a t i o n s

on finite d i m e n s i o n a l subfactors. The m a i n ingredients of the c o n s t r u c t i o n are the Mean Ergodic Theorem a p p l i e d on the limit space of the Paving Structure, with a c o m b i n a t o r i a l c o n s t r u c t i o n of m u l t i p l i c i t y

sets.

together

We call the

inner action y i e l d e d by this r e p r e s e n t a t i o n the submodel action.

A

tensor p r o d u c t of c o u n t a b l y many copies of the submodel action is used as the m o d e l of free action of

G.

For

G= Z

this m o d e l is d i f f e r e n t

from the one used in

[4].

An e s s e n t i a l feature of Connes'

a p p r o a c h is the study of a u t o m o r -

phisms in the f r a m e w o r k of the c e n t r a l i z i n g u l t r a p r o d u c t algebra i n t r o d u c e d by Dixmier and McDuff. systematic algebra

study of these techniques and also introduce the n o r m a l i z i n g

Me

as a device for w o r k i n g w i t h both the algebra

c e n t r a l i z i n g algebra

M

and the

Me .

We c o n t i n u e w i t h the main technical r e s u l t of the paper, Rohlin Theorem, groups,

w h i c h yields,

the

for c e n t r a l l y free actions of a m e n a b l e

an e q u i v a r i a n t p a r t i t i o n of the unit into projections.

first part of the proof we obtain some, p o s s i b l y small, system of of projection.

In the

equivariant

The a p p r o a c h is b a s e d on the study of the

g e o m e t r y of the crossed product,

and makes use of a result of S.Popa

on c o n d i t i o n a l e x p e c t a t i o n s in finite factors we put t o g e t h e r such systems of p r o j e c t i o n s unity.

Me ,

In the fifth chapter we make a

[37].

In the second p a r t

to obtain a p a r t i t i o n of

We use a p r o c e d u r e in w h i c h at each step the c o n s t r u c t i o n done

in the p r e v i o u s steps is slightly perturbed.

These methods yield new

proofs of the Rohlin T h e o r e m both for a m e n a b l e group actions on m e a s u r e spaces and for c e n t r a l l y free actions of

~

on von N e u m a n n algebras.

As a c o n s e q u e n c e of the Rohlin Theorem, we o b t a i n in the seventh chapter s t a b i l i t y p r o p e r t i e s groups.

for c e n t r a l l y free actions of a m e n a b l e

We first prove an a p p r o x i m a t e v a n i s h i n g of the one- and two-

d i m e n s i o n a l cohomology. of the 2-cohomology.

The m a i n stability result is the exact v a n i s h i n g

The proof is based on the fact that in any coho-

m o l o g y class there is a cocycle w i t h an a p p r o x i m a t e p e r i o d i c i t y p r o p e r t y w i t h r e s p e c t to the p r e v i o u s l y i n t r o d u c e d Paving Structure.

The

techniques used here y i e l d an a l t e r n a t i v e a p p r o a c h for the study of the 2 - c o h o m o l o g y on m e a s u r e spaces. problem, morphism,

The usual way is to reduce the

by means of the h y p e r f i n i t e n e s s ,

to the case of a single auto-

w h e r e the 2 - c o h o m o l o g y is always trivial.

The final part of the paper deals w i t h the r e c o v e r y of the m o d e l inside g i v e n actions.

We first show that there are m a n y systems of

m a t r i x units a p p r o x i m a t e l y fixed by the action.

F r o m such a system,

t o g e t h e r w i t h an a p p r o x i m a t e l y e q u i v a r i a n t system of p r o j e c t i o n s given by the Rohlin Theorem, we obtain an a p p r o x i m a t e l y e q u i v a r i a n t system of m a t r i x units;

this is p r e c i s e l y how a f i n i t e - d i m e n s i o n a l a p p r o x i m a t i o n

of the submodel looks.

R e p e a t i n g the p r o c e d u r e we o b t a i n an infinite

number of copies of the submodel and thus a copy of the model.

At each

of the steps of this c o n s t r u c t i o n there appear u n i t a r y p e r t u r b a t i o n s . The v a n i s h i n g of the 2 - c o h o m o l o g y permits the r e d u c t i o n of those perturbations a r b i t r a r i l y close to 1 cocycles.

The c o r r e s p o n d i n g results for G - k e r n e l s are o b t a i n e d by r e m o v i n g from the proofs the parts c o n n e c t e d to the 2 - c o h o m o l o g y vanishing. The last chapter c o n t a i n s the proof of the I s o m o r p h i s m Theorem. Under the s u p p l e m e n t a r y a s s u m p t i o n that the action is a p p r o x i m a t e l y inner we infer that on the relative c o m m u t a n t of the copy of the model that we construct, whole action.

the action is trivial;

i.e.

the model contains the

We b e g i n by o b t a i n i n g a global form from the e l e m e n t w i s e

d e f i n i t i o n of a p p r o x i m a t e innerness. are induced by unitaries

A p p r o x i m a t e l y inner a u t o m o r p h i s m s

in the u l t r a p r o d u c t algebra

t e c h n i q u e of V . J o n e s to work,

Me .

by means of an action of

eously w i t h these u n i t a r i e s and with the action itself. ing, in the same way as in the p r e c e d i n g chapter,

We use a

G × G,

simultan-

After construct-

an a p p r o x i m a t e l y equi-

v a r i a n t system of m a t r i x units, we make it contain the u n i t a r i e s that a p p r o x i m a t e the action.

We obtain a copy of the submodel w h i c h contains

a large part of the action, on

M,

in the sense that for m a n y normal states

the r e s t r i c t i o n to the relative c o m m u t a n t of the copy of the

submodel is almost fixed by the action.

This way of dealing w i t h the

states of the algebra,

in view of o b t a i n i n g tensor p r o d u c t s p l i t t i n g of

the copy of the model,

is d i f f e r e n t from the one in

[4], and avoids the

use of spectral techniques. A c h a r a c t e r i s t i c of the f r a m e w o r k of this paper is the superposition at each step of technical d i f f i c u l t i e s coming from the structure of general a m e n a b l e groups, factor.

Nevertheless,

acting on

R,

and from the absence of a trace on the

in a t e c h n i c a l l y simple context like,

e.g. ~2

all the main arguments are still needed.

With t e c h n i q u e s based on the Takesaki duality,

V.Jones

[24]

o b t a i n e d from the above results the c l a s s i f i c a t i o n of a large class of actions of compact abelian groups abelian,

hence amenable,

(the duals of w h i c h are discrete

groups).

A similar a p p r o a c h towards c l a s s i f y i n g actions of c o m p a c t nonabelian groups w o u l d first require a study of the actions of their duals, w h i c h are p r e c i s e l y the discrete

symmetrical Kac algebras.

A n a t u r a l f r a m e w o r k for this e x t e n s i o n is the one of d i s c r e t e amenable Kac algebras, w h i c h includes both the duals of c o m p a c t groups and the d i s c r e t e amenable groups.

It appears

[35] that such an a p p r o a c h can

be done along lines similar to the ones in this paper. is to provide,

A first step

in the g r o u p case, proofs w h i c h are of a global nature,

i.e. deal w i t h subsets rather than w i t h e l e m e n t s of the group; of the Rohlin T h e o r e m given in this paper is such an instance. from that, general,

the proof Apart

the s u b s e q u e n t e x t e n s i o n to the n o n - g r o u p a l case needs,

techniques having no e q u i v a l e n t in the group case.

in

NOTATION Let

M

be a v o n

Neumann algebra.

M h,

M +,

MI,

Z(M),

denote the h e r m i t e a n part, p o s i t i v e part, unit ball, group,

and p r o j e c t i o n lattice of

the predual of If

M

¢ • M,

and

+ ¢ e M,

and

respectively.

M,

and

Proj M

unitary + M, denote

and its p o s i t i v e part.

(~x) (y) = ~(xy); If

M,

U(M),

center,

x,y • M,

then

~x,x¢ • M,

(x¢) (y) = ~(yx). and

x • M,

We let

we let

are defined by

[~,x] = C x - x¢. , ½ llxII¢ = %(x x) ,

IxI¢ =¢(Ixl),

llxII2 = ¢ (½(x*x + xx*) )½.

Chapter l:

M A I N RESULTS

This chapter contains an outline of the results of i n d e p e n d e n t i n t e r e s t o b t a i n e d in the main body of the paper. 1.1

Let

M

be a v o n

Neumann algebra.

called c e n t r a l l y trivial,

8 E

CtM,

An a u t o m o r p h i s m

(xn) • M, i.e. w h i c h is norm b o u n d e d and satisfies for any

¢ E M,,

one has

%(Xn)-X n --> 0 81pM

nonzero 8 - i n v a r i a n t p r o j e c t i o n

in Z(M).

e: G --> Aut M

of

M

is

limll[~,Xn]ll = 0

*-strongly.

p r o p e r l y c e n t r a l l y n o n t r i v i a l if p

@

if for any c e n t r a l i z i n g sequence

@ is called

is not c e n t r a l l y trivial for any A d i s c r e t e group action

is called c e n t r a l l y free if for any

g e G\{I} ,

~g

is

p r o p e r l y c e n t r a l l y nontrivial. The group

G

dealt w i t h in this section will always be a s s u m e d

c o u n t a b l e and discrete. A cocycle crossed action of the group where

e: G --~ Aut M

and

~g~h

=

AdUg,hagh

Ug,h Ugh,k Ul,g

=

v: G --~ U(M)

The cocycle

u

~

on

M

(e,u),

g,h,k E G

'

~g(Uh, k) Ug,hk =

is a pair

satisfy for

'

1

is free w i t h the obvious a d a p t a t i o n

is the c o b o u n d a r y of

v,

u = ~v, if

satisfies Ug,h

In this case (Ad Vgeg).

=

Ug,1

(~,u) is called c e n t r a l l y free if of the definition.

G

u: G × G --~ U(M)

=

~ g ( V ~ ) V g Vg h

(~,u) may be viewed as a p e r t u r b a t i o n of the action

We shall prove in Chapter 7 the following v a n i s h i n g result

NOTATION Let

M

be a v o n

Neumann algebra.

M h,

M +,

MI,

Z(M),

denote the h e r m i t e a n part, p o s i t i v e part, unit ball, group,

and p r o j e c t i o n lattice of

the predual of If

M

¢ • M,

and

+ ¢ e M,

and

respectively.

M,

and

Proj M

unitary + M, denote

and its p o s i t i v e part.

(~x) (y) = ~(xy); If

M,

U(M),

center,

x,y • M,

then

~x,x¢ • M,

(x¢) (y) = ~(yx). and

x • M,

We let

we let

are defined by

[~,x] = C x - x¢. , ½ llxII¢ = %(x x) ,

IxI¢ =¢(Ixl),

llxII2 = ¢ (½(x*x + xx*) )½.

Chapter l:

M A I N RESULTS

This chapter contains an outline of the results of i n d e p e n d e n t i n t e r e s t o b t a i n e d in the main body of the paper. 1.1

Let

M

be a v o n

Neumann algebra.

called c e n t r a l l y trivial,

8 E

CtM,

An a u t o m o r p h i s m

(xn) • M, i.e. w h i c h is norm b o u n d e d and satisfies for any

¢ E M,,

one has

%(Xn)-X n --> 0 81pM

nonzero 8 - i n v a r i a n t p r o j e c t i o n

in Z(M).

e: G --> Aut M

of

M

is

limll[~,Xn]ll = 0

*-strongly.

p r o p e r l y c e n t r a l l y n o n t r i v i a l if p

@

if for any c e n t r a l i z i n g sequence

@ is called

is not c e n t r a l l y trivial for any A d i s c r e t e group action

is called c e n t r a l l y free if for any

g e G\{I} ,

~g

is

p r o p e r l y c e n t r a l l y nontrivial. The group

G

dealt w i t h in this section will always be a s s u m e d

c o u n t a b l e and discrete. A cocycle crossed action of the group where

e: G --~ Aut M

and

~g~h

=

AdUg,hagh

Ug,h Ugh,k Ul,g

=

v: G --~ U(M)

The cocycle

u

~

on

M

(e,u),

g,h,k E G

'

~g(Uh, k) Ug,hk =

is a pair

satisfy for

'

1

is free w i t h the obvious a d a p t a t i o n

is the c o b o u n d a r y of

v,

u = ~v, if

satisfies Ug,h

In this case (Ad Vgeg).

=

Ug,1

(~,u) is called c e n t r a l l y free if of the definition.

G

u: G × G --~ U(M)

=

~ g ( V ~ ) V g Vg h

(~,u) may be viewed as a p e r t u r b a t i o n of the action

We shall prove in Chapter 7 the following v a n i s h i n g result

for the 2-cohomology. THEOREM.

Let

G

be an amenable group,

algebra with separable predual,

let M be a yon Neumann + ~ E M, be faithful. If (a,u)

and let

is a centrally free cocycle crossed action of alZ(M)

preserves

Moreover, and a finite

then

u = ~v

~IZ(M),

given any Kc G

then

u

G

on

M,

such that

is a coboundary.

~ > 0 and any finite

F C G,

there exists

~> 0

such that if

Jlug,h- lll~ < 6

g,h ~ K

Hvg-iIt~

g~

with

< ~

Y

F

A similar result for the l - c o h o m o l o g y holds only if

G

is finite,

in w h i c h case the c l a s s i f i c a t i o n can be carried on up to c o n j u g a c y

1.2

A factor

where

R

M

is called a McDuff

is the h y p e r f i n i t e

factor if it is isomorphic to

II 1 factor.

[23].

R®M,

Several e q u i v a l e n t properties,

due to M c D u f f and Connes are given in 5.2 below. In 8.5 we shall obtain the following result. THEOREM.

Let

G

be an amenable group and let

with separable predual. then

~

If

is outer conjugate

Moreover, exists an

given any

(ag)-cocycle

a: G - - ~ A u t M to

e > 0,

be a M c D u f f factor

is a centrally free action

id R ® a . any finite

(Vg) such that

and

M

K a G,

(Ad Vgag)

and any

,, ~E M +

is conjugate

there

to id R ®

#

Actually,

the central freedom of

obtain cocycles.

~

is b a s i c a l l y used only to

An a l t e r n a t i v e a p p r o a c h b a s e d on Lemma 2.4 w o u l d not

need this assumption.

1.3

In Chapter 4 we c o n s t r u c t a model of free action

for an amenable group

G.

~(0).. G --> Aut R

In 8.6 we show that this model action is

c o n t a i n e d in any c e n t r a l l y free action. THEOREM.

Let

G

be an amenable group and let

with separable predual.

Any centrally free action

M

be a M c D u f f factor

a: G --~ Aut M

i8

outer conjugate Moreover, can be chosen

1.4

Under

inner,

to

a (0) ® a.

as in the p r e c e d i n g arbitrarily

close

the s u p p l e m e n t a r y

the a c t i o n

is shown

theorem,

the cocycle

that appears

to i.

assumption

in 9.3

that each

to be u n i q u e l y

ag

is a p p r o x i m a t e l y

determined

up to

outer conjugacy.

Let

THEOREM.

G

be an amenable

with separable

predual.

a: G --> Aut M

is outer conjugate

Bounds

group and let

Any centrally

on the cocycle

to

be a M c D u f f factor

free approximately

inner action

~(0) ® idM.

may also be obtained.

Any two free actions

COROLLARY.

M

of the amenable

group

G

on

R

are outer conjugate. Proof.

1.5

By results

The study of actions

of G-kernels,

which

defined

THEOREM.

result

Let

separable

predual.

conjugate

to

of g r o u p s

are c e n t r a l l y

for G-kernels.

the a n a l o g o u s

From

is c l o s e l y

and

Int R = A u t R.

connected

to the study

G --~ Out M = Aut M / I n t M.

trivial,

the proof

G

be an amenable

central

of T h e o r e m

freedom

Since

can be

1.2 in 8.8 we o b t a i n

group and

M

free G-kernel

a McDuff factor with B: G --> Out M

is

~.

In the same way we o b t a i n

Theorem

CtR = int R

for G-kernels.

Any centrally

id R ®

[3],

are h o m o m o r p h i s m s

inner a u t o m o r p h i s m s

1.6

of Connes

in 8.9 the f o l l o w i n g

analogue

of

1.3.

THEOREM.

Let

separable

predual.

conjugate

to

Here

G

be an amenable

Any centrally

group and

M

free G-kernel

a McDuff factor with B: G --> Out M

is

~(a (0)) ® B.

a(0):

G --~ Aut R

is the c a n o n i c a l

projection.

is the m o d e l

action

and

~: Aut M --~ Out M

Chapter

We o b t a i n actions

2.1

the o u t e r

on the

the p r e c e d i n g When

invariant

INVARIANTS

conjugacy

II 1 and

II

AND

CLASSIFICATION

classification

hyperfinite

of a m e n a b l e

factors

from

group

the

results

in

chapter.

coming

introduced discrete

type

an a c t i o n

implementing

2:

has

from

it.

groups

part,

the u n i q u e n e s s

This

by C o n n e s

an i n n e r

invariant,

modulo

called

for a c t i o n s

by J o n e s

of

[21].

there

We

appears

a cohomological

a scalar

of the u n i t a r i e s

the c h a r a c t e r i s t i c

~n

in

shall

[3], w a s

briefly

invariant,

defined

describe

for g e n e r a l

it in w h a t

follows. Let first of

~

G.

For

~h = A d v h ~h~k

be an a c t i o n

conjugacy each and

= ~hk '

of a d i s c r e t e

invariant h

N = N(e),

take

and

thus

we For

there

exists

for

g eG

and

The p a i r following

(l,p)

relations

of m a p s for

since

c N,

: =

Ig,hk

Ig,l

where

*

tation group above

denotes

from

the d e f i n i t i o n s

consisting

of all

of

1

Vhk

Izi=l}

some

that implement

such

= ~h ' Ig,h 6 ~

p: N x N - - ~

Pk,Z Ph,k£

ig,h

If,g-lhg

that

we

infer

.

satisfies

C(N)

rid of the d e p e n d e n c e

be the

set of all m a p s

the

~ h , h -I kh ~k,h g,k

Ii, h

=

=

conjugation.

the p a i r s

such

and

and

~.

~h,k

lh, I

=

This We

~g-lhg,g-lkg

=

follows

let

of f u n c t i o n s

Ii, h

Z(G,N)

(I,~)

i

by e a s y

compu-

be the a b e l i a n

satisfying

the

relations. To g e t

let

the c o m p l e x

for

=

g,h

=

= {Z6~I

A

g,j E G:

Ph,kPhk,£

lh,k

vh c N

VhV k

eg~g-lhgeg-1

Ig,hV h

Igf, h

M.

= e-1 (Int M)

~h,k Vhk

I: G × N -->~,

h,k,~

N(~)

both

~h,k 6 ~

h CN,

=

on a f a c t o r

a unitary

h,k E N,

=

ag(Vg_lhg)

G

subgroup

choose

v I = i.

VhVk Similarly

group

is the n o r m a l

of

(I,~)

~: N - - ~

on the c h o i c e with

~i = 1

of

(Vh) , we

and,

for

e C(N),

we let

~

= (I,~)

where

Ig,h

=

~h~g -lhg

!ah, k

=

rlhkrlhr?k

It is easy to see that we denote by the image

A(G,N)

A(~) = [l,~]

of

choice of the unitaries

(I,~)

(eg)-Cocycle and

implements

~h'

N

is a s u b g r o u p of Z(G,N);

Z(G,N)/B(G,N).

in

A(G,N)

h,k E H .

For an action

e,

no longer depends on the

~g = Ad Wg~g , then for

h E N,

If

Vh=WhVh

and it is easy to compute that these unitaries yield (I,~) for

~.

called the c h a r a c t e r i s t i c When

= ~C(N)

,

(Vg) and hence is a c o n j u g a c y invariant.

(Wg) is an

the same pair

B(G,N)

the q u o t i e n t

gEG

is abelian,

Thus

A(e)

is an outer c o n j u g a c y invariant,

i n v a r i a n t of the action. then

[~,~] depends only on

1

and no q u o t i e n t

has to be taken. The c h a r a c t e r i s t i c group extensions.

i n v a r i a n t can also be defined in terms of

Let

~: G --> Aut M

= { (h,u) E N x U(M) I ~ h = A d u }. the maps

~÷N,

t+

(l,t)

Then

and

N ÷N:

with N

N = ~-l(Int M)

is a subgroup of

(h,u) ÷ h

and let N × U(N)

and

yield an exact sequence

1 --~ ~F --~ N --~ N --~ 1 where the induced action of over,

gEG

on

t r i v i a l l y and on

~

acts on

N

N

on

~

by c o n j u g a t i o n is trivial.

by conjugation: N

by

h---> ghg -I,

More-

and if we let it act

(h,u)--~ (ghg -I , ~g(U)),

the above sequence

becomes an exact sequence of G-modules. One can show that the classes of e x t e n s i o n s of action)

in the c a t e g o r y of G - m o d u l e s

Cohomological

invariants

by

~

(trivial

form a group w i t h the Brauer

product and this group is n a t u r a l l y isomorphic to

2.2

N

A(G,N).

for the c o n j u g a c y of G - k e r n e l s w e r e

d e f i n e d in an a l g e b r a i c context by E i l e n b e r g and McLane and a d a p t e d to von N e u m a n n algebras by N a k a m u r a and Takeda Let

B: G --->Out M

e: G --> Aut M

be a section of it, with el = i.

there are u n i t a r i e s

Wg,h E M

[43]°

For each

w h i c h may be a s s u m e d to satisfy

=

Ad w g,h gh Wl,g = W g , l = i.

F r o m the a s s o c i a t i v i t y

(eg~h)~k = ~g(~hek ) one obtains Wg,hWgh,k

g,h E G,

with

~g~h

relation

[32] and S u t h e r l a n d

be a G - k e r n e l on a factor and let

=

~g,h,k ~g(Wh,k) W g , h k

10

for s o m e

~g,h,k E ~.

3-cocycle

relation,

obstruction, Jones is the and

and

has

[~,~]

N(~) = N

and

G-kernel

shown

natural

its

that

N

connecting

to an e i g h t - t e r m

G

R

with

[6] e

a normalized

called

the

B.

discrete

normal

an a c t i o n

for e a c h

group

subgroup

a:

there

if

of

G

N

G -~ Aut

H3(G)

and

R

R

with

exists

a free

Ob(B) = [6].

subgroup

maps

exact

H3(G,~),

for any

exists

and

in

satisifes

for the G - k e r n e l

is a c o u n t a b l e then

there

be a n o r m a l

Ob(B)

invariant

if

A(~) = [~,~],

6: G~--~ ~

class

II 1 factor,

• A(G,N)

B: G --> O u t

Let

function

is a c o n j u g a c y

hyperfinite

any

The

of

to e x t e n d

G

and

the

let

Q = G/H.

One

Hochschild-Serre

exact

can define sequence

sequence

1 - > HI(Q) --> HI(G) --> HI(N) G --~ H2(Q) --> H2(G) --~ A (G,N) -> H3(Q) --> H~(G) For details

2.3

The

see

[19], [22], [38] .

following

lemma

describes

actions

with

trivial

characteristic

invariant. LEMMA.

Let

G

be a f a c t o r with with

a-1(Int

projection. an action

be a countable

separable

M)

= ~-~(Ct M)

If

A(a)

Let

is trivial

then

such

v: N --> U(M),

triviality

v1=l

such

a h = Ad v h Let for

s: Q --> G

q e Q.

If

,

that

=

let

Wq, r = V t ( q , r ) .

ap(g)

for

geG,

V h V k = Vhk of

define

hence

((~q), (Wq,r))

=

We h a v e

vanishing

(Theorem

is a c o c y c l e

of the i.i)

iemma, yields

group

= G/N exist

and

let

M

be an action

be the canonical

an a - c o c y c l e

, p

u

and

g e G

we m a y

choose

e N

a map

we h a v e

ag(Vg-lhg) with

•

s(1) = 1

= vh and

let

~q=

as(q)

by

t (q,r) s (qr)

for

q,r,s =

crossed centrally

a map

,

h,k E N

t(q,r)

t(q,r)t(qr,s)

by the h y p o t h e s i s

there

A(a),

s (q) s (r) and

p: G ~ Q

of

be a s e c t i o n

q , r e Q,

amenable

a: G --> A u t M

that

A d Ugag

By the

Let

= N.

~: Q - + A u t M

Proof.

discrete

predual.

e Q

Ad(s(q)) (t(r,s))t(q,rs) action free.

of The

z: Q -~ U(M),

Q

on

M,

which

2-cohomology z1=l,

with

is,

11

* Zq~q (z r) Wq, rZq, r

Let with

aq = A d Zq~q.

p(g) = p

and

Then

g = hm,

Let

g,f 6 G;

=

m = s(p),

Ad(ZpV~)

to be s h o w n

p=p(g),

r = s(pq) e S(q) c_ G; t(p,q)

=

=

of

we

is that

=

on

For

R.

U g = ZpV h .

gE G

We h a v e

~p

is an a - c o c y c l e .

m=s(p),

k = f n -I,

mnf-lg-1£

=

(Ug)

Q;

Q

let

Ad V h ~ p

q=p(f)

h = g m -I,

m n r -I

q,r E Q

1

is an a c t i o n

h E H,

Ad Ugag and all t h a t r e m a i n s

a

=

n = s(q),

~ = g f r -I ~ N.

mk-lm-lh-IZ

=

We h a v e

Ad(s(P)) (k-1)h -IZ

so that

Wp,q

-

p(V IV V

and we o b t a i n U g a g ( U f ) U g*f The l e m m a

2.4 by

The

Z p V h* V h a-p ( Z q V k ) V h*V £ Z p *q

=

=

Z p ~ p ( Z q ) -W p , q Z p*q

=

1 .

is p r o v e d .

lemma t h a t f o l l o w s

is a d e v i c e

to o b t a i n

cocycles,

inspired

[22]. LEMMA.

Let

~: G --> A u t N and

M,N,P

be f a c t o r s

be a c t i o n s

v: G - + U(M)

be maps

such

there

exists

an

~

(Ad Ugag)

Proof. there exists

Since

Let

to

an i s o m o r p h i s m

u

such

to

to

--~ M

The r i g h t m e m b e r

=

8(8g ® ag) 8 -I

is an action, Zg,h

=

~ ®y

.

B@a

•

to

B®7

hence

~g~g(~h)~h

and to

such that

@(Bg ® Ad V g a g ) @ -I

~g = @(i N ® V g ) V g ; t h e n Ad ~gag

7: G - ~ A u t P

that

is c o n j u g a t e @: N ® M

=

Let

G.

B® ~ ,

is c o n j u g a t e

(Ad Vgag)

Ad Vgag

group

is c o n j u g a t e

cocycle

~: G --> A u t M,

let

that

is c o n j u g a t e

(Ad Vgag) Then

and

of a discrete

~® B®X,

12

is a s c a l a r Once

for

g , h E G.

again,

isomorphism

since

~: N ® M

8

is c o n j u g a t e

--> M

such

A d ~g~g We

let

Ug = @ ( l ® ~ g ) ~ g

=

Ugag(Uh)Ug h

B ® 8,

there

exists

an

that

@ ( B g ® A d ~ g a g ) 8 -I

and

A d Ugag

to

infer

=

e(BgQag)e

O(l®Sg)

-I

,

(Ad Vgag) ( l ® V h ) V g a g
A u t

Proof. group

We k e e p

A(G,N)

actions

be the

conjugacy

plication,

remains

which

that

of any induced

1.4, of

Let R

le F

8 induces

R

then

by T h e o r e m there

The

2.7

that

to B

theorem

The

( a g e 8g)g.

is o u t e r

above

the

Since

conjugate

~,B:

extends

to

M,

Let

be a countable

G

action

We a g a i n

classes

a(G)

with

let

defined

product

Lemma

By L e m m a

on

there

in

is an

conjugate ~.

it

2.3,

Q = G/H.

of

let

multi-

2.5(a)

free

If

action

Since by

Hence

and

R

~: A u t

let

with

the

e G

that

to

and hence

R

then

is the

~(~) ® 8'

such

is c o n j u g a t e

~: G --~

[8] e ~;

R --> Out

8 ~ and

Vg,g

c Int M

N ( e ~) = N

and

are

(Ad V g S g ) g

a ® ~,

Lemma

2.4

[~] is a u n i t

in ~.

map

is w e l l

discrete

predual.

each

~([~

amenable

and

group and let

Two approximately =

a-1(Ct M)

if and only if let

A

=

~ e F,

--~

let

We

let

a~: F

[a t ® ~]

and we h a v e

® a])

=

~([~])

inner

8-1(Int M)

=

A(a) = A(8). be the with

M) = a-l(Ct M) = N.

A(a ~) = ~.

defined

framework.

~: G --~ A u t M,

a-~(Int For

and

following

F = A(G,N)

(~, [a]) This

We

To a p p l y

are o u t e r

~-l(Int M)

[a] of a c t i o n s

[el + A(~).

with

[~] of

morphism;

S: G --> A u t

to the

are outer conjugate

= N

be the m a p

is w e l l

~([a]) = 1

a ® B,

result

G --> A u t M

Proof. conjugacy

classes

class.

of

a

be the

is a s e m i g r o u p

Q-kernels

to

F

N(e) = N.

tensor

to a c o c y c l e

unitaries

be a McDuff factor with separable B-I(Ct M)

Q

N(e) = N(8)

is p r o v e d .

THEOREM.

actions

of

Let

1.6

exist

and

which

with

8': G - ~ Out R.

Thus

R

of the q u o t i e n t

be the m o d e l

conjugate.

shows

R ~

let

conjugacy

~-I(i) = {I}.

lifts

G-action.

fixed;

with

@

of a s i n g l e

projection,

is c o n j u g a t e

and

actions

consists

a Q-kernel

and

action

such

induced

invariant,

~: G --~ A u t

cocycle

G

to

is s u r j e c t i v e .

two

~: Q --~ A u t

be the

of

isomorphic

is u n i t a l

the

if and only if

set of o u t e r

R

classes,

by a free

and

N

is a s e m i g r o u p

~

~

action

is a m e n a b l e , any

preimage

Z

preserves

the c l a s s

Corollary

be the with

of J o n e s

to s h o w

a

Z

R,

subgroup

characteristic

action

Aut

let

classes.

by the r e s u l t s

Q

a normal

~: G --> A u t

~: Z ÷ F outer

and

are outer conjugate

R

set of o u t e r

M

isomorphic

We

let

G --> A u t act on

A

R by

~: A + F be an

M

14

To a p p l y that

@-I(i)

the p r o o f

has

2.5(b)

a single

we h a v e

In the

of

G

the class.

on

same

R

The

from

that

from

the

for

the p r e c e d i n g

A(a ~ ®

The proof

2.8

For

is thus

Let

an infinite is a type

We

such

extends

Then

let

N

that

M

2.4 c o n c l u d e s

nite

II

such

that

1.4

instead

of

G/N

with

on

as in

of its

R

the

preserves

[$] • A,

[a ~n ®

8]

since = A ( ~ ~n)

need

the

following

result.

of a discrete

conjugate

R0,1.

There R0, 1

= mod(@)T

Int R0,1 = ker mod.

[7].

an a c t i o n

= id N

to

group

id F ® a

of

where

F

invariant; conjugacy

(N(a), A ( a ) , m o d ( ~ ) )

shown

a: G - + A u t

is an o u t e r

G

since

since

inner

known

a unitary

8.4,

is i s o m o r p h i c

of a c t i o n s

step A,

mod: trace

[4] t h a t

to

N®N.

on the h y p e r f i Aut

R0,1-->

~+

on R0, I, C t R 0 , 1 = Int R0, 1

the h o m o m o r p h i s m

be a countable RO, 1

of L e m m a

N

by C o n n e s

R0, 1

It is w e l l gE G

mod(a) : G --~ ~ +

automorphisms

have

module

i,

invariant.

are

discrete

outer

amenable

conjugate

group.

if and only

Two

if

F

= (N(B), A ( B ) , m o d ( B ) ) .

We k e e p

a normal

of all h o m o m o r p h i s m s

be the p r o d u c t

T

M.

subfactors).

a homomorphism

a conjugacy

Let

(the p r o o f

dimensional

a semifinite

It w a s

G --~ A u t

of

for e a c h

and

mod(a)

a,8:

exists

exists

yields

THEOREM:

subfactor

the c l a s s i f i c a t i o n

@ 6 Aut

and

I

there

the proof,

for

Proof.

=

is outer

that

factor

To@

the g r o u p

action

be an action

to i n f i n i t e

Let us n o w d e s c r i b e

actions

8]

first

a

Ad Vg~glN

2.9

For

is e s t a b l i s h e d

Theorem

and

a n ) = ~n

be a type and

immediately

Lemma

we

a: G - - > A u t

M = N ® (N'N M)

Vg E M

fact

and

factor.

Proof. that

factors

factor.

I

is a F - m o d u l e

finished.

infinite

LEMMA.

free

theorem,

A

that multiplication

~,~ E F an ®

that

last

using

w a y we o b t a i n

[a t ®

follows

This

theorem,

coming

fact

to s h o w

element.

of the p r e c e d i n g

Corollary. action

Lemma

of the g r o u p s

subgroup

N

~: G --~ ~ + A(G,N)

and

of

G

fixed

with F 0,

and

and

let

N ~ k e r w. let

Z

be

F0 We

be

let

the set of

15

all o u t e r

conjugacy

isomorphic is e a s y tensor of

Z

into

~w:

that

Z

For

R0, 1

G --~ A u t

~:

~ e

By r e s u l t s

--> A u t

R0, 1

Since with

A(G,N)

let

a~

8g = Bw(g) .

satisfies

Then

N(X) = N,

M

R0, 1 ~ R0, 1 ® R0, 1

there

For

with

multiplication

R

with

an a c t i o n

F0 w e d e f i n e

the a c t i o n

A(X) = ~

it

by the

a homomorphism

G --~ A u t

exists

w E

given

yields

be an a c t i o n

[42]

m o d ( ~ t) = t.

by

a: G - ~ A u t M

[a] --~ (A(a), m o d ( a ) )

of T a k e s a k i

with

R ® R0, 1 ~ R0, 1

N(a) = N.

is a s e m i g r o u p

The m a p

F.

A(a ~) = ~.

on

[a] of a c t i o n s

to R0, I, a n d w i t h

to see product.

8: ~ +

classes

an a c t i o n

X = ~

and

® ~w

of

mod(X) = w ,

G

hence

is s u r j e c t i v e . If 2.7, a

~([a]) = i,

then

is u n i q u e l y

a: G --~ A u t

R

a

come

from

to the G / N - k e r n e l

2.6 the

fact 2.8,

factor.

The

thus

acts

2.10

The

that a

and

1.3

using

there

exists

(Bg) = (~(~g)) amenable,

u

is a c o b o u n d a r y

is an action; conjugacy

class

G-kernels

on

as in 2.6,

R

the

THEOREM.

G-kernels

crossed

~: A u t M - - ~ O u t

8,7:

A result THEOREM.

Theorem of

having

Let

G

F

~ Aut

theorem

can be done

instead

of t h e i r

action M

the

of

determined. obstructions

Theorem

the o b s t r u c t i o n ,

(a,u)

of

G

and one

on

M

can

such

suppose that

existence

yields,

1.4

in this

Since

to c o n c l u d e The

1.2

8: G --~ O u t M

is the p r o j e c t i o n . i.i,

by the

analogues

Isomorphism

G-kernels

I

R0, i

is p r o v e d .

factors

inner

1.6 of

hand,

is a type

in the

be a countable discrete amenable group.

analogous

G

where

let

of

that G

is

that the free

same w a y

result.

G --> O u t R

Let

Theorem

on

is t h a t

and

On the o t h e r

the

1.4 can n o w be a p p l i e d

arbitrary

From

1.6

by T h e o r e m

is u n i q u e l y

following

separable predual. G --~ Out M

8

~ E ~

as in the p r o o f

G --> A u t ( R ® F )

By the d e f i n i t i o n

a cocycle

where

by T h e o r e m

with

R.

~®a.

2.5(a),

remark

on

id F ® a

approximately

obstruction.

to

to

1.5 and

The k e y

and h e n c e

we obtain

~®idF:

of G - k e r n e l s

free

of G / N a,

By L e m m a

Theorems

for actions.

for c e n t r a l l y

case

conjugate

Z.

classification

trivial

8,X:

in

by

conjugate

of the a c t i o n

as a u n i t

with

2.11

is o u t e r

inner

e: G --~ A u t R0, 1

action

induced

is o u t e r

class

same methods,

works

a

Let

a free

applied

by L e m m a

is a p p r o x i m a t e l y

determined.

Two free

are conjugate if and only if Ob(B) = O b ( 7 ) .

to 2.7

is the

following.

be as above and let

M

be a McDuff factor with

Two centrally free approximately

are conjugate if and only if

inner G-kernels

Ob(B) = Ob(x).

16

2.12

Since

8: G - ~ Out the

inner

automorphisms

of

R0, 1

the

mod(~):

as

in 2.10

same w a y

Let

THEOREM.

free G-kernels (Ob(S),mod(8))

invariant

G

B,X:

one

can p r o v e

be a countable G --~ Out

RO, 1

We a s s o c i a t e

3.1

of

The

at m o s t

group

invariant m(1) = 1

finite

an

mean,

point

hence

locally

m m

survey with

is a

fln±tely

The

mean

to the

Let

be a group. is

are

Two

write

R

If

the

and

relation

and Day,

see

if

FCC G

R

in

and

e >0

solvable

a

we

intrinsic

is a m e n a b l e , groups group

groups

amenable.

For

as

are a

F/R of

([16]). by

say

IKI

that

if it is f i n i t e

by F~iner.

such

the a m e n a b i l i t y

F

IKI
(l-e) Is I . The gEF of a m e n a b l e g r o u p s w a s g i v e n

for

two g e n e r a t o r s

Kcc L

IS N

to N a m i o k a

but

is a

G-spaces.

m: Z~(G)

is the

finite

Abelian

will

map

£g

which

of left

amenable

linear

amenable.

[15].

"growth

3.2

due

group,

system

sequel

where

additive

for a set

shall

G

is c a l l e d

unique.

with

see and

follows,

of

in the

G

is a g a i n

group

a n d we

G

with

g e G,

are a m e n a b l e ,

ity,

S

a paving

G

the b e h a v i o r

can be c h o s e n

groups

group

G is c o n n e c t e d

subset

In

if and only if

group

An a s c e n d i n g

quotient

free

a free

for

is n e v e r

finite

groups

In w h a t

for a G - k e r n e l

result.

amenable

GROUPS

is a p o s i t i v e

m

of a m e n a b i l i t y F

discrete

i,

can be d e f i n e d .

following

are conjugate

is the H a a r m e a s u r e ,

an a m e n a b l e

amenable.

is d e a l t

which

theorem.

of a m e n a b l e

G --~ ~ + the

approximate

m-£g=

invariant

fixed

module

AMENABLE

and n o n t r i v i a l .

and

groups

that

which

if it exists,

since

with

G

£~(G)

on

mean,

sets

countable

left

3:

to an a m e n a b l e

finite

with tion

have

= (Ob(x),mod(x)).

Chapter

system

R0, 1

its c a r d i n a l -

a nonvoid and

characterization For

a short

proof,

16

2.12

Since

8: G - ~ Out the

inner

automorphisms

of

R0, 1

the

mod(~):

as

in 2.10

same w a y

Let

THEOREM.

free G-kernels (Ob(S),mod(8))

invariant

G

B,X:

one

can p r o v e

be a countable G --~ Out

RO, 1

We a s s o c i a t e

3.1

of

The

at m o s t

group

invariant m(1) = 1

finite

an

mean,

point

hence

locally

m m

survey with

is a

fln±tely

The

mean

to the

Let

be a group. is

are

Two

write

R

If

the

and

relation

and Day,

see

if

FCC G

R

in

and

e >0

solvable

a

we

intrinsic

is a m e n a b l e , groups group

groups

amenable.

For

as

are a

F/R of

([16]). by

say

IKI

that

if it is f i n i t e

by F~iner.

such

the a m e n a b i l i t y

F

IKI
(l-e) Is I . The gEF of a m e n a b l e g r o u p s w a s g i v e n

for

two g e n e r a t o r s

Kcc L

IS N

to N a m i o k a

but

is a

G-spaces.

m: Z~(G)

is the

finite

Abelian

will

map

£g

which

of left

amenable

linear

amenable.

[15].

"growth

3.2

due

group,

system

sequel

where

additive

for a set

shall

G

is c a l l e d

unique.

with

see and

follows,

of

in the

G

is a g a i n

group

a n d we

G

with

g e G,

are a m e n a b l e ,

ity,

S

a paving

G

the b e h a v i o r

can be c h o s e n

groups

group

G is c o n n e c t e d

subset

In

if and only if

group

An a s c e n d i n g

quotient

free

a free

for

is n e v e r

finite

groups

In w h a t

for a G - k e r n e l

result.

amenable

GROUPS

is a p o s i t i v e

m

of a m e n a b i l i t y F

discrete

i,

can be d e f i n e d .

following

are conjugate

is the H a a r m e a s u r e ,

an a m e n a b l e

amenable.

is d e a l t

which

theorem.

of a m e n a b l e

G --~ ~ + the

approximate

m-£g=

invariant

fixed

module

AMENABLE

and n o n t r i v i a l .

and

groups

that

which

if it exists,

since

with

G

£~(G)

on

mean,

sets

countable

left

3:

to an a m e n a b l e

finite

with tion

have

= (Ob(x),mod(x)).

Chapter

system

R0, 1

its c a r d i n a l -

a nonvoid and

characterization For

a short

proof,

17 THEOREM

(left) invariant

one can find an

3.3

An

result

between

in this

towards

more

several

is amenable i.e.

more

S

was

precise

and

F ca G

was

the

absence

constructions

in

which

e> 0

G.

invariant

announced

form,

if and only if it has

if for any of

elaborate

approximately

direction

in a s l i g h t l y

G

subsets,

(e,F)-invariant subset

impediment

of a link

A group

(F~iner).

arbitrarily

subsets

[36].

of

We n e e d

for c o n v e n i e n c e

G.

that

A

result

we p r o v e

in the

sequel. Let us c o n s i d e r , which

the p r o p e r t y gaps

for i n s t a n c e ,

is a p p r o x i m a t e l y

invariant

t h a t one

can

or o v e r l a p p i n g s .

shaped

almost

possible

respect We

invariant

to c o v e r

of a f i n i t e

G,

number

within

N

of

if t h e r e

are

subsets

e.g.

moreover

N

of

iEI,

A large

e,

that

it is translates

large

only

sets

without

an a r b i t r a r i l y

by u s i n g

depends

such

of it,

with

is v e r y

rectangle,

moreover,

Nevertheless

each

finite

has,

translates thing

accuracy

provided

(Si)ic I

S i _c Si,

with same

a "disc".

a given

one;

G= Z 2 .

translations

do the

"discs",

a system

case

the g r o u p

cannot

subset,

to the p r e c e d i n g say t h a t

cover

One

the

to g i v e n

on

with

e.

e-disjoint,

are

IS'.1 I ~> I(l-e) ISi

e > 0,

, and

!

(Si) i are subsets

disjoint.

of

subsets

the g r o u p

L I .... ,L N

(KiLi)i=l ..... N and m o r e o v e r > 0

and

we call

of

are

KCCG

e-pave G,

i,

such

finite

K I, .... K N

subset

S

paving centers,

and

E-cover are

S,

of

of

such

i.e.

any

G

if there

that

are

U K i L i C S,

IS \ ~ K i L i l!< sIS1 ,

e-disjoint.

K l,.. .,K N e - p a v e

finite

If t h e r e

are

(~,K)-invariant

SC G

e-paving system of sets.

(Ornstein

Let

and W e i s s ) .

G

be an amenable group.

such that for any

N > 0,

an e-paving system

system

(KiZ)ZELi

that

an

there is

the

the

called

disjoint

KI,...,K N

e> 0

say t h a t

G

for e a c h

THEOREM

any

We

X> 0

of subsets of

K~,...,K N

G,

and

FaaG,

For

there is

with each K i being

(x,F) -invariant. More let

precisely,

K .... ,K N _ G

Kn = p ~U> n ~

and

invariant

be such

invariance

The

degree

The p r o o f

0 < ~ < ½

that

following

Then

4

N

(61Knl

any

i

> ~ log ~

So_-G

'

e

and

6 = (~)

n )-invariant,

which

is

(6

N

;

where

nUKn )-

by K I, .... K N.

essential (X,F)

that

let

Kn+ 1 is

n = 1 ..... N-I.

is e - p a v e d

Remark.

The

for a n y

imposed

follows lemma

fact

is t h a t on the

is b a s e d

shows

that

N

sets

on the if

S

does

not depend

on the

(Ki) i. ideas

of O r n s t e i n

is i n v a r i a n t

enough

and Weiss. with

18

with respect moreover,

to

K

then

it can s w a l l o w

from the a p p r o x i m a t e

approximate

invariance

enough

invariance

of the r e m a i n i n g

right

of

S

part,

translates

and

K

of

follows

provided

K;

the

this part

is

not too small. LE~MA. LaG

Let

be m a x i m a l e Suppose

invariant

moreover

and

K

IS\KL I > plSl, Proof.

In terms

of

L

Suppose

KL C S

that

that

for

S\KL

is

S' = S

some

is

(½,K)-invariant

(K£)£E L ~ > 0

and

are

and

e-disjoint.

F CCG,

S

If for

p > 0,

~s

let

Then

(6,F)-

(3p-16,F)-invariant.

n k-IS ; we have IS'I ~ ½ S. F r o m the keK it follows that for any £ e S' , IKZ N K L I > elk I . n

of c h a r a c t e r i s t i c

functions XK-I

Integrating

S ca G

and

(~IFI -i, F - l ) - i n v a r i a n t .

~s

then

Let

maximality

0 < ¢ < ½. such

this y i e l d s

* XKL

>

eIKIX S,

we get

IKI II ~IKI Is'I hence

~> ~- Isl

I~ml ~ ~ l s ' l and the f i r s t p a r t Suppose

of the lemma

now that the s u p p l e m e n t a r y

let

S" = S n N k-IS kCF IS"I i> (i-6) IS I and

S'

i

=

S i A

N

kEF

is proved.

k-ISz .

and

assumptions

K' = K n

N kK ; keF IK'I ~> (I-61F-II)IKI.

are fulfilled,

from the h y p o t h e s i s Let

$I = S \ K L

and

Then

!

SI\S 1 _C (S\S") u (F-IKL\KL)

C_ (SkS")

SO

Is~\si{ -< Is',s"l + IK\K'I IF1 ILl < 61sl +~IKI ILl From

the e - d i s j o i n t n e s s

of

(K£)zE L

it follows

that

hence

{~{ l~I -< (I-~)-ItKLI With

and the

-< 21K~I (1-2~)[Kil

G,

Therefore

i(g)=l}i > (i-4 )I iII il

a repeated

that follows

contains

such a structure

is an immediate

the verifications

Let

en > 0

£n-Paving

(i,j)

all the information

done further

consequence

on.

(Paving Structure).

(Kn)i mutually

and for any

it with finite

The proposition

of the Theorem and Corollary

are left to the reader. Let

G

be an amenable

GnCCG be given, for n = 0 , 1 , 2 . . . . . n n systems (Ki) i, i e In, with each K i being

EK n+l j

at the

we need

(fixed once and for all) will be

and

E I n x In+ 1

of a paving

and about the ways of approximating

3.3;

PROPOSITION

use of the Paving Theorem

each level consisting

the basis of all the constructions

and with

ILi[

which pave each of the sets appearing

This structure

about the group subsets.

group

of "levels",

system of subsets higher

The e-disjoint-

(3) we infer

I iI = l{g•K'I

3.4

.

yields

disjoint,

and finite

group.

Then there are

(en,G n)-invariant (L ni,j)i,j,

sets

Such that

f -- 1 nll L ni,jl

3 (i,j) e I n × In+l,

~n+l i,3

=

the sets

{ g 6 Kn+l I there are unique (i,k,Z)~ 6 =ll X[ x LI, j l with g =k£, and for these, i = i} I

satisfy

(2)

:n+l, IKi,jl I> (i- e n)

IKnl ILn,jl-

Kn = i~. Kni ; since (Kl) i are supposed 1 often identify K n with u K n C G. i i Let

For any

n

let

such that for any (i,k,i)

to be disjoint,

~n:

we shall

IIl[.. Knl x L nl,j --> % K n+lj = K n+l be a bijection 3 m 3 k n ( ~ K n × Ln j e In+ 1 , i i,j ) = K n+l j ' and if

e ~ • K n1 × L n1 , ] . with 1

k£ e ~n+l k,j'

then

~n

(i,k,£)

=k£

.

21 For any

g• G g

and

tions"

with

in

i• In

such that if We call

ture

for

K n,

k• Kn l

let us choose

frequently

COROLLARY.

"approximately

left t r a n s l a -

bijections with

in: g Kn--~ K n with gk • K n then £n(k) = gk. i ' g n ~J' (Li,~)i,~

K = (£n' Gn' (Kn)i "_

G ; the n o t a t i o n

tion will

that appears

be used

~n, (Zg)g)n

~n(Kn) = K nl' g

a Paving

in the s t a t e m e n t

Struc-

of the p r o p o s i -

in the rest of the paper.

By the conditions

of the proposition,

for any

g • Gn

(i,j) • I n × In+ 1 ,

and

(3)

I{ (k,Z) • K ni × L n1,3. I ~n (£n(k)'£)g ~ ~n+l g (kn(k'£))}I

that is, on most of the the left

g

centers

K n+l,

for a given

g

and for

Kn l

~< 3en IKnl ILni,j I

n

translation almost coincides with the left

on the plaques

large enough g

translation

product with the identity on the set of paving

Ln • . 1,3

Proof. Let (k, i) • K n1 × Li,j n £gn(k) = gk, (k,£) ~ A.

A

be the set in the left m e m b e r

are such that

kn(gk, Z) = gkl,

gk e K ni'

and

kn(k£) = kl

of

k£,gk£

(3).

If

• ~n+l 1,]

then

zn+l .... = g k g n one infers

Fm

E j e Im

m

grows. measure

group

and so, for any

Theorem

~ gives 1

(i), lim m÷~ n E IN

to the one in

F have arbitrarily

The Mean Ergodic

lim IFml -I ~ Xi'Y m÷ ~ ye Fm from

Im -I im-l,im

1~'m xm 1,3 3

of the amenable

degree when

to the F-ergodic

Hence

xn+l

i m ~ Ira, let

IFml -I E Xi'yn = y eFm

invariance

l

IK~I IL9 .I IKn+iI-1 l,] j

n<m,

(I)

Kn+l

of the conditional

3 =

where

From the equality n f = X i we infer

n m ~ i m 11 n'm - ~il ~j je 1,3

=

0

large

in L1-norm

25

(2)

i ~I n j

m ll~'~ 1,3 - Z~'Z~

for all large enough

< en

m.

The measure ~ being chosen once and for all, we make a last assumption on the paving system K. By refining its levels, i.e. replacing (Kn )n by (Knp )np for some subsequence (np)p of ~ , we may suppose that (2) holds for any n and m > n+l. This can be done without renouncing any of the conditions imposed on the Paving Structure, in view of our assumptions stated in 3.5. n Remark. The above inequality states that the proportion li,j of right translates of K9 in K~ +I almost doesn't depend on j. This l 3 n j is in fact arbitrary. What might be quite surprising since li, actually happens is that the ergodic measure ~ and the level refinement n j is almost indepen"choose" a part of the system (K ni)n,i for which li, dent of j ; on the rest of the diagram, ~ being small, the contribution of the corresponding terms in the sum (2) is negligible. Let us fix bijections

s~ .: Tn l,]

l,j

×

We may also suppose that for each that with M n = j ~ _ Mjn , j e In+ 1 Isn+ll = Isnl IMnl

S~ --> S n+l i

j

"

j e In+ 1 there is a set

and

M~] such

IM~I = ~ + l l M n I

We infer S n -i n IK~I- i Isn+l 1Lni,jl ITS, J I = li,j IK~+II 3 J I I i' =

Ini,j ~j-n+l l~n+l I (~li)-n -i Isnl -I

=

n -n+l,-n,-1 iMnl li, j Hj ~i j

Hence

{ ILni,jl 1Tni,jl - IMnl lj

=

It is possible to choose subsets such that 1,j and a bijection

:

pgl,j c_ L9~,3.× Tgl,j and

I l,jl : min{l

-n R~ P~ . Pi,3: 1,j --> 1,3"

• , 1

1,3

nxnl x R n

S

l, 3

([~i ~+I iln -l' 3 ~ I 'Mnl R~l,j --CM~3

Ln .t IM I} i,jl I 1,3

We have

.

1

Kn x sn x

i

i

=

nxMn

28 and i,j

K~ x p~ x Sn C ~ K n × L~ TO S~ ~-m ~] Kn+l × S~ +I = S n+l 1 l,j i -i l,j × 1,j × 1 J 3 i,j j

where the last map is above.

As

~

~n× ~n1,j ' ~n

Isn+II = IsnI~M~I

7 :n ~n x M n

and

being defined in 2.5 and

IP~ , j I = IRol,jl there is a bijection

= (~i Kn× i S~) × ( ~

M~]) --->- ~n+l

n satisfying for any i • I n , j • In+ 1 , k • Ki, (2)

~n1,j

s • S ni,

n n × sn n ~n × ~n (K~ x p~ S~) ~ (Ki 1 x Ri,j) = 1,j l,j x

=

~j K~+I ×3

sn+13

r • R~l,j " ,

n ( k , s , r ) = (~n × ~ n -n (r) s) i,j ) (k' Pi,j ' The inequality

(i) shows that the cardinality of the elements in

the argument or range of

n

not appearing

in the above equality is

small, i.e. (3)

E i,j

IKnl Isnl (IM31-1Rn

÷

i,j

IKnl

(I Ln,

~
0 let ~n be a factor of dimension IMnl and let ~n+l = g n ® ~n . Let g be the finite factor obtained as weak closure of the UHF-algebra

u ~n

on the GNS representation

associated

n

to its canonical trace. Modulo obvious identifications we may suppose that ~n ~ ~n+l ~ ~ . Since ~n: ~n x M n --> ~n+l , n E ~ , are bijections, we can choose systems (E~

s2) , sl,s 2 • S n,

of matrix units in

~n,

n •IN,

which are

I'

connected via

~n

,

i • e. such that En

=

S l ' S2

with

m • M n,

sl = zn(sl,m),

[ E~ +I m

Sl I S 2

s2 = ~n(s2,m) .

For any g a G and n > i, the "approximate left g-translation" ~n: Kn _>K n defined in 3.4 yields a unitary ung • ~n ' given by g

27 un g where

ie I n ,

image of This

= E E En i (k,s) (k1's)'(k's)

g

(k,s) e K ~ × S~ and k I = Zn(k). One can view l l g in an "approximate left regular representation"

is justified

by the following

all the constructions PROPOSITION. corresponding

proposition,

Let

T be the canonical

L1-norm.

Then the limits Ug = nlim ~ Ugn

~.

For any

(i)

n IUg--UgIT ~ 8e n .

n > 1 and

the following

Gnat

trace on

,

G n ~ G,

it is enough

of

G

to prove

inequalities g 6 Gn ,

(3)

n n n < 2~ IUgU h - Ug hl T n

for

g,h 6 G n with

(4)

IT ( gU )n I < S n

for

g e Gn,

(!) in the proposition

in view of 3.5 we have Let us prove (4).

g @i

gh e G n ,

g ~I.

is easy to obtain

from

(2), since

7en+ 1 + 7Sn+ 2 + ... < e n . For g e G,

T(U~)

is

I'IT the

unitary representation

for

g E Gn,

and

G (see 3.4) we have

(2)

If

g

g6G

n .n+l I T ~ 7e n lug - Ug

Statement

K9 i

g e

In view of the fact that

Proof.

is the goal of

done before.

exist in l'IT-norm and yield a faithful into

which

U n as the g of G.

=

and

Isnl -I E ]S~I l { k e K g l £ n ( k ) i6I I g

k E K~i N g-1 K ni , then

=k}l

In(k) = g k ~ k. g

Since

(Sn,Gn)-invariant, • (U~) ~

Isnl -I ~ IS~I e n IK~I i eI n

=

en

n Let us now prove (3). Let g,h,gh e G n. If k E K i with = £~ h(k) = ghk. hk,ghk e K ni then zn£~(k) g So from the (en,Gn)-invarin ance of K i , it follows that

(5)

I { k E K ni I Zg£~(k)

~ g£nh(k)}l

< en IKnl

We have Ugn u nh - ungh IT

= ~i

E (k, s)

~i

En(k2,s),(kl,s)En(kl,s),(k,s) -E~3,s),(k,s) T

~ En - En (k,s) (k2's)'(k's) (k3's)'(k's)

28

where

i 6 In,

moreover, Hence

(k,s) • K9l × S ni' kl = Z~(k) ' k 2 = Zn(kl) g in the last m e m b e r we sum only for t h o s e k

and

k3 = £ngh (k); for w h i c h k~# k 3.

(5) y i e l d s (6)

n n IUgU h - U g hn l

T

IS n I-i ~. 2SnIK~I Is~I 1


1 a n d pgn • A such that T(p~) < 8g n and Proof.

Let

g • G n and consider the projection n qg

=

En+l s s

~ S•

with

g 6 Gn, t h e r e e x i s t s a p r o j e c t i o n (i -p~)Ug = ( l - p gn) Ugn . in ~:

'

A

A = {s E sn+l I En+is,sUgn # En+is,sun+l~g~ " Then

(I - qg)Ug n n = (i- qgn)[~n+l and a careful inspecuion of the _g proof of the proposition reveals that in view of (2) we have actually shown that IAI Hence

T(qg) < 7en,

~< 7enlsn+ll

and if we let

(i -pg)Ug n

=

pg =

n n (i -pg)Ug

V qk k>~h 9

then

g e Gn

and T(p~)

~

E T(q~) ~ k~n

~. 7E k ~ k~n

8s n

The corollary is proved. Remark. Some words about the ideas that lie behind the proof. mn,i n for i6 In and kl,kze K in . Let Let ~kl,k 2 = E s s ~ E(kl's)'(k2's) (~n,1 )i,kl,k2 be matrix units for an AF-algebra ~ = U ~ n which has ~kllk 2 n n as Bratelli diagram the Paving Structure (Ki)n, i (actually the numbers (IK~l)n,i) , and for which ILnl,jl gives the multiplicity of the arrow

30 Kn i ÷ Kn+l. Let h n be the h o m o m o r p h i s m ~ n ÷ g w h i c h maps ~ n , 1 onto ~n,i 3 I< 1 ,K 2 kl,k 2. Then hn+ II~n is a p p r o x i m a t e l y equal to hn, w i t h even better a p p r o x i m a t i o n as

n

grows.

What we did in 4.3 was an almost

e m b e d d i n g of this A F - a l g e b r a

into the U H F - a l g e b r a

~,

"ergodic"

m o t i v a t e d by the

fact that it is much easier to r e c o n s t r u c t U H F - a l g e b r a s

inside a given

W * - a l g e b r a than AF-algebras. _n+l

The c o r o l l a r y shows that on n+l

Zg

n

~-- £g

×

i i

K.

_n

n

we have Li id , and so we obtamn at the limit a r e p r e s e n t a t i o n of •

.

in the weak closure of

W.

If

-~ /_i K. ×

3

i

IInl = 1

!

for all

n, then

~

G

is an

U H F - a l g e b r a and taking all m u l t i p l i c i t i e s If the p r o p o r t i o n of

Kn lL n 1,3• in

Isnl to be 1 we are done. l • does not depend on j, we can

.n+l ~j

still take the same m u l t i p l i c i t i e s for all

K n and again we are done. 1 the ergodic m e a s u r e ~ on the t o p o l o g i c a l

In the general case in 4.2, dynamical

system

(K*,F) yields a tracial factorial state on

c o n s t r u c t i o n of Krieger,

S t r ~ t i l ~ and V o i c u l e s c u

[41].

~

by the

In this way we

obtain a finite h y p e r f i n i t e factor and the c o m b i n a t o r i c s in 4.3 can be viewed as an e x p l i c i t form of the c l a s s i c a l proof of M u r r a y and yon Neumann

4.5

[31] that such a factor is g e n e r a t e d by an UHF-a!gebra.

Let us recall some n o t a t i o n and results

in this chapter w h i c h are

needed further on in the paper. We have started with a d i s c r e t e c o u n t a b l e amenable group which a Paving Structure was i n t r o d u c e d in 4.3. i E In, the Sn-paving subsets of

G

For

n E N,

G,

with

for (K~),

on the n-th level of the Paving

Structure, we have c o n s t r u c t e d finite sets (S~), i • In , and have set ~n = U n i K ~lx Si" We have c o n s i d e r e d a factor gn w i t h a m a t r i x units basis

En indexed by ~n and have c o n s t r u c t e d unitaries n in gn s,t n. u K~ Ug , a s s o c i a t e d to the a p p r o x i m a t e left g - t r a n s l a t i o n £g. i l ---> U1K 91 in the Paving Structure. s u b a l g e b r a of

~n

We have d e n o t e d by ~ n the m a x i m a l abelian n (Es,s). We call ((E~,t), (U~)) the

g e n e r a t e d by

n-th finite dimensional

submodel.

We have a s s u m e d that n •N,

and have let

~

~n c ~n+l

in such a way that

~n c ~n+l,

be the w e a k closure with respect to the trace

of

u ~n, and ~ be the "diagonal" m a x i m a l a b e l i a n s u b a l g e b r a of n g e n e r a t e d by ~ n " Since ISnl + ~ , ~ is a II 1 h y p e r f i n i t e factor.

For each

g • G,

Ug = n÷~limUgh

* - s t r o n g l y was shown to exist and y i e l d

a faithful r e p r e s e n t a t i o n of and

(~n)' n ~

almost trivially.

submodel

G

in

~.

For each

is a II 1 h y p e r f i n i t e s u b f a c t o r of We call

(~, (Ug)) the submodel

n, g

~ = ~n ® ((~n), N ~)

on w h i c h

and

(Ad Ug)

Ad Ug

acts

the

action.

We let

R

be a c o u n t a b l y infinite tensor p r o d u c t of copies of the

31

submodel

factor

for each

g • G,

~,

taken with respect to the normalized

we let

ag(0) be the c o r r e s p o n d i n g

copies of the submodel action factor and Connes

(a~ °))

Ad Ug.

is an action

[3] is free.

We call

Then

R

G ÷ Aut R the model

R

trace,

and

tensor product of

is the hyperfinite

II 1

by which Lemma 1.3.8 of and

~(0) : G ÷ Aut R

the

model action.

Chapter

5:

U L T R A P R O D U C T ALGEBRAS

We study specific properties machinery 5.1

developed

In what follows

We denote by

of ultraproduct

M

~{(M) its unitary group and by M

once and for all a free ultrafilter Let us consider sequences

~ .

Both

~

Let

~

is in ] ~ in

and

I~ such that for

Ilx~yI,~ + ,lyx~11# < ~, We consider and identify bras of

Mw

M and

on

normalize

to 0; ~,

faithful e> 0

yEM

MAM~

of

I~(IN,M) : ~ ,

the w - c e n t r a l i z i n g for any

sequences

~EM,);

~w' the

]~e, the normalizing

state of

algebra of

ilyll~ 0 and a n e i g h b o r h o o d and

llylI~ < ~

W

of

we have

w EW.

the quotient C*-algebras with

We choose

hence are C*-subalgebras

there is a with

its projections;

~q.

limII[xW,~]II = 0

*-strongly

be a normal

iff for any

~

Proj M

M I its unit ball.

sequences; /~,

(xW) w with ]~

and

the following C * - s u b a l g e b r a s

consisting of the constant sequences~-converging

and use the

type automorphisms.

will be a W*-algebra with separable predual.

M h will be the hermitean part of

(i.e.

algebras

thus far to study ultraproduct

(/#+I~)/I~. = Z(M).

M~ = ~ / ~ 0

This way

Any

% e M,

M

and

and Me

M~ = ] ~ / I ~

are C*-subalge-

gives a form

~

on

Mm

by

~m((xV)w)

= lim ~(x~); its restriction to M e will be denoted by ~ . h)÷~0 For simplicity of notation, we write II" II~ and It" II~ for the norms II'II#~ and LEMMA.

complete

II -I12~0 on

Let

M ~.

~ e M,+

in the topology

Proof. sequential

be faithful and

Then

y e M L°.

given by the seminorm

The above topology being metrizable, completeness.

Let

w IIXn+ 1 -XnlI % +

(Xn) n c (M~) h

(M ~) h

i8

x ÷ llxIIt~+ IlxyIIt~o . it is enough to prove

be a sequence

such that

~ w 2 -n II(Xn+ 1 -Xn)YWll ~
0

yEM

MAM~

of

I~(IN,M) : ~ ,

the w - c e n t r a l i z i n g for any

sequences

~EM,);

~w' the

]~e, the normalizing

state of

algebra of

ilyll~ 0 and a n e i g h b o r h o o d and

llylI~ < ~

W

of

we have

w EW.

the quotient C*-algebras with

We choose

hence are C*-subalgebras

there is a with

its projections;

~q.

limII[xW,~]II = 0

*-strongly

be a normal

iff for any

~

Proj M

M I its unit ball.

sequences; /~,

(xW) w with ]~

and

the following C * - s u b a l g e b r a s

consisting of the constant sequences~-converging

and use the

type automorphisms.

will be a W*-algebra with separable predual.

M h will be the hermitean part of

(i.e.

algebras

thus far to study ultraproduct

(/#+I~)/I~. = Z(M).

M~ = ~ / ~ 0

This way

Any

% e M,

M

and

and Me

M~ = ] ~ / I ~

are C*-subalge-

gives a form

~

on

Mm

by

~m((xV)w)

= lim ~(x~); its restriction to M e will be denoted by ~ . h)÷~0 For simplicity of notation, we write II" II~ and It" II~ for the norms II'II#~ and LEMMA.

complete

II -I12~0 on

Let

M ~.

~ e M,+

in the topology

Proof. sequential

be faithful and

Then

y e M L°.

given by the seminorm

The above topology being metrizable, completeness.

Let

w IIXn+ 1 -XnlI % +

(Xn) n c (M~) h

(M ~) h

i8

x ÷ llxIIt~+ IlxyIIt~o . it is enough to prove

be a sequence

such that

~ w 2 -n II(Xn+ 1 -Xn)YWll ~
2 -n+l

xi

therefore in

As

converges M~

~

for any

y 6 M m,

and so there is separating,

on the dense

is

xy

subset

is a W*-algebra.

M e , and hence

xi

is

Being

M~ ll'iJ%

is a W*-subalgebra.

(M~) h we have

N [x,y]ll~ < 211xJt$ llyll + I1xyli~ + 11xy*II~ The left members they vanish precisely

are thus so-continuous for

seminorms

x 6 ( M ) h we have proved

that

for on M

yEM

(M~) h.

. Since

is a W*-

33

subalgebra

of

M

Problem. For

Is it a l w a y s

x e Me

w e can

representing

sequence

normal

trace

with

~

Me

to

%~(x)

5.2

define for

Further

on w e

constructed sequence

from

into

a

Te

~ ~ M,

restriction

to

the

of

~

Me

(xV)~

to

Z(M),

is a

is a f a i t h f u l

restriction

certain

automorphisms

M.

Suppose

of a u t o m o r p h i s m s

of

M

deal with

This

( ~ (x9) )v.

Me .

restriction For

of

automorphism

of

= e - lim x V • M , w h e r e

~

of

since

.

yields

such

that

an a u t o m o r p h i s m

of

w e are

Me

and

given

~ = iim a m

of

I~(~,M)

Me

a exists

sending

Since

II~(x~)II~2 this

M' n M e = M e ?

automorphisms

in the u - t o p o l o g y . (x~)9

on the

shall

the

(~w)9 • ~

Its

in Z(M).

only

= ~(T~(x)) , x • M

that

Te(x)

x.

values

depends

true


@n+l(X) > 0

l,yll~
llx~yll~ + HyxW,l~# < i/n

6 n(x)

x6M e

for

p(n) > n

of

e

w e Wn(X)

For

n>l

choose

(5)

p(n)

e Wn(X),

(6)

llxP(n)y p(n) - (xy)p(n) II~ < I/n ,

(7)

ll[xp(n), an]II~ < I/n

(8)

I~(anx p(n)) - ~ ( a n T e ( x ) ) I < I/n ,

x ~ N n,

a e F n,

(9)

IiB(xp(n)) - (8e(x))P(n)II~ < i/n ,

x eNn,

8 E Aut M

•

and

xeN. x,y E N n-

,

x 6 N n N Me,

aE F n . ~ e M nwith Be 6 B n .

We d e f i n e (xP(n)) n. hence

By

~

N

of

straightforward

N

LEMMA

into

M.

sequences

~(x)

(6) and

statements

be r e p r e s e n t e d

(8),

so it e x t e n d s

of a p a r t of

with respect

and

M

to a n o t h e r

of

Me

~

leaves

and

~

N

~

is a

to a n o r m a l

T

by

and

injective

of the l e m m a a r e n o w

s l o w e n o u g h to m a k e t h e m p a r t of

M

and to a f a m i l y

N

a countable (a~) w ¢ ~

injective

~ is the identity

(2) (3)

#(N ~ M e) C M W . ~(N) C ( F N M e ) ' n M e

(4)

Ye(a~(x))

M

be a W * - a l g e b r a

and

F

be countably

family

and

with generated

of semiliftable

B = ~+~lim a~,

then

auto-

B E

*-homomorphism

~: N ÷ M

satisfying

on N A M .

= Te(a) Te(x)

= ~(~(x))

Let

Let

invariant.

(I)

~(¢(X))

Trick).

e e B~\~.

of M, euch that if

There i8 a normal

(5)

The

(Slow R e i n d e x a t i o n

predual

and such that

letting

automorphisms.

sub W*-algebras morphisms

(xn)n, and from

homomorphism,

like c o n s t a n t s

of s e m i l i f t a b l e

separable

x=

to o b t a i n .

We can r e i n d e x

behave

for

#(x) 6 Me,

ll'II~ p r e s e r v i n g

*-homomorphism

5.4

on

(5),

,

= ~(S(X))

,

xEN,

aEF

xeN,

e = (~)ve

. W ,

B=

lim ~ .

36 Proof.

We m a y a g a i n

and the r e p r e s e n t i n g previous union

lemma.

suppose

sequences

Moreover,

~,

and r e p r e s e n t i n g

e~ = B

if

a = B~

6n(X)

and

Wn(X)

for some for

that

M c N A F.

take f i n i t e

subsets

sequences

(a~)w

B • A u t M.

Take

x • N,

Choose

for the e l e m e n t s

and c h o o s e

of

N

N n, F n, M n,

as in the

An ~ An+ 1 ~ ~

for any

a •~

with

w i t h all

in the same way as b e f o r e

for any n a t u r a l

n,

p(n) •

such t h a t p(n)

• W n(x)

,

x • Nn

iixP(n) yp(n) _ (xy)p(n)II~ ~s

let

and suppose

S = (Sn) n that

ae S'N M e ,

be a n o r m a l M,.

Let

~ ¼ T(q'). of

q

But

is c o n t r a -

is thus proved,

and from

T((8(q) _q)2) = 2T(q)-2T(qB(q))>

of the p r e c e d i n g

be a c o u n t a b l e

~IS'N M e

a # 0,

and

state on

(s ~n)v

outer,

T((~(q) _ q ) 2 )

and let

be r e p r e s e n t i n g

yields

and thus by the C l a i m >i i/2 .

M

Let us k e e p

l e m m a the h y p o t h e s i s

is p r o p e r l y

is not p r o p e r l y

outer,

that

x ~ S' A M e

n = 1,2, . . . .

qeM e

~-invariant

such that

for

faithful

(a~)~

respectively;

with

1.2.1]

of L e m m a

a ( x ) a = ax

and

T = Te

q e Proj M e

q V B(q) V 8-i(q)

and t h u s the m a x i m a l i t y

= T(B(q))

there exists

~

let

= '4.

To p r o v e * - s u b s e t of

s c a l a r values;

o u t e r and let

Then

q' < 1 - (q V ~(q) V B-I(q))

it we i n f e r

Let

takes

< ¼ T(q).

(q' V 8(q'))(qV 8(q)) = 0

dicted

is,

by

T

x E M e.

~(q~(q))

if not,

same reasoning

for

8 e Aut M

such t h a t

Indeed,

then

is a factor,

[x[T = ~(Ix[)

maximal

q' ~ 0

M

(~n)n be a t o t a l sequences

~E~

that

B = (~)e

there exists

We r e m a r k

fixed.

for

a

By m e a n s e Aut M e

a projection

t h a t in the a l g e b r a

Me

we

have T~(18(q)aV-a~qI2 ) = Te(] (8(q)-q)aV] 2) = Te(laVlZ)T((8(q)-q)) 2

> ~/~ Te(laVl2) H e n c e we can p i c k o u t of a r e p r e s e n t i n g element

q~e M

such that

llqV I[ i 1/21ja~ll~

,

1 ll[q ,Sk]ll# ~< U T h e n the s e q u e n c e

for

and

II~(q~)a~ -a~q~)I1# 1 li[qv,~k]N

thus o b t a i n e d

k=l '

.....

k,~ = i,

an e l e m e n t I/2J]al] z T

shows

~

. .,~

q 6 S' N M e

satisfying

0

that

a

is s t r o n g l y

outer.

40 5.8

The following

result appears,

with a slightly

different

proof,

in

[13, Lemma B.5]. LEMMA.

Let

M

1 E E.

subfactor,

be a factor and

Let

~ E ~\~.

induces an i s o m o r p h i s m

E CM

let

be a finite dimensional

E'n M + M

Then the inclusion

(E'n M)~ -~ M w.

COROLLARY. (i)

If

M

(2)

If

8 e Aut M \ C t M

is M c D u f f then

E'o M

is McDuff.

8(E) = E,

and

(@IE'A M) E Aut(E'N M)\Ct(E'N Proof. E.

Let

For any

(ei,j),

y E M,

i,j EI,

Yi,j = k[ ek,iYej,k If

} eM,

and

be a system of matrix units generating

=

~ ei, j Yi i,j 'J

e E ' n M;

x 6 E ' ~ ] M, [~,x] (y)

:

II[~,x]N ~ and thus the inclusion P: M A E '

÷ M X

If

x E M,

~ ei,j[~,x] (Yi,j) i,j

Hence, ~

if

III2H [(~IE' n M), x]l[

E'm M ÷ M

induces

be the conditional ---> P(X)

IiI-li,j

=

an inclusion

(E'~ M)~ ÷ M~.

expectation .xe.

ei, 3

3,i

•

xEM

.

then P(x) - x

(P(xW))9

IIYi,jl;~< i.

then

hence

Let

M).

lJyll~ i, we have Y

with

then

=

(x~)v E Me, then

(xV)~.

Thus

P

III -~

lim

induces

to the one induced by the inclusion. The lemma is proved.

~ ei,j[x, ej, i] i,j (P(x V) - x v) = 0

a map

M~ ÷

*-strongly

and so

(E'A M)~ that is inverse

41 C h a p t e r 6:

THE ROHLIN T H E O R E M

In this chapter we prove a Rohlin type t h e o r e m for a d i s c r e t e a m e n a b l e group

G

As a consequence,

a c t i n g c e n t r a l l y freely on a yon Neumann algebra. we show that if

H

is a normal subgroup of

Rohlin t h e o r e m holds for the action of the q u o t i e n t fixed points for

6.1

G/H

G,

the

on the almost

H.

Some of the basic tools in the m o d e r n d e v e l o p m e n t s of the ergodic

theory in b o t h m e a s u r e spaces and von N e u m a n n algebras are the various e x t e n s i o n s of the Rohlin Tower Theorem.

The one p r o v e d in the sequel

e s s e n t i a l l y states that for a free enough action of a discrete a m e n a b l e group

G

on a v o n

N e u m a n n algebra

unity in p r o j e c t i o n s G

M,

one can find a p a r t i t i o n of the

indexed by finite subsets

acts on it a p p r o x i m a t e l y

(Ki) i

of

G,

such that

the same way in w h i c h it acts on

by means of the left regular action.

£ (~ K i)

The e q u i v a r i a n t p a r t i t i o n of

unity thus o b t a i n e d is the starting p o i n t of most of the c o n s t r u c t i v e proofs that follow. This t h e o r e m extends,

on the one hand,

O r n s t e i n and W e i s s ' s Rohlin

T h e o r e m for d i s c r e t e amenable groups acting freely on a m e a s u r e space ([36]) and, on the other hand,

the Rohlin T h e o r e m of Connes for single

a u t o m o r p h i s m s of von N e u m a n n algebras centrally-)

x6M.

~

is a trace on the yon N e u m a n n algebra we let

For the sake of simplicity, -I

and

~gah~g h • Int M,

THEOREM

group,

and let

let

M

let

a partition

M

on

M

Let

aIZ(M)

be an e-paving

of unity

IxI~ = %(Ixl),

Ixl~

if

x e M~

~: G ÷ Aut M

with

Let

leaves

Let

tIZ(M)

t

count-

be a crossed action be a faithful normal

invarianto

of subsets

,Nj; k E K i

be a discrete

algebra with separable

a: G ÷ Aut M~

family

(Ei,k)i= 1

G

in

of M~

G.

Let

~> 0

and

Then there is

such that

'''"

X l~i I-I i=l

~

l~k~-1(Ei,~l - Ei,kl, ~ 5 ~ ;

k,£• K i

(2)

[Ei, k, eg(Ej,i)]

(3)

ag~h(Ei,k)

Moreover,

IxI~ for

is a map

be a yon Neumann

N

(ii

we w r i t e

and stron.gly free.

such that

KI,...,K N

[33] to

g,h • G.

~ • ~/~.

which is semiliftable state on

G

(Nonabelian Rohlin Theorem).

able amenable predual,

(not n e c e s s a r i l y

but for a m e n a b l e groups this p r o b l e m is still open.

Recall that a crossed action of ~I = 1

For

free actions the t h e o r e m of Connes was e x t e n d e d in

a b e ! i a n groups, If

([4]).

= 0

f~r all

= egh(El, ) .}[ " f o r

g,i,j,k,£ ;

all

(Ei,k)i, k can be chosen

g,h,i,k .

in the relative

commutant

in M~

42

o f any

given

countable

The estimate

subset

Me .

(i) above is an average estimate.

other types of estimates COROLLARY.

of

In

Below we give

that can be derived from it.

the

conditions

o f the

theorem

we

have

f o r any

g~G (4) For we

any

! k~ lag(Ei,k) - Ei,gkl , ~ 10g ~ , i=l ..... N; ~ > 0

a n d any

AkC K i with

subset8

k E Ki N g

Ki .

IAil ~< 61Kil, i=l ..... N,

have

(5)

~" k[ IEi' kl* ~< 6 + 5 s ½ ,

Proof.

i = I,...,N;

For any i=l,...,N,

leg(Ei,k ) -Ei,gkl # Summing for all

k,~


0

and let

K

of a c o u n t a b l y

Then there exists a p a r t i t i o n of unity

N ' A M~

~IZ(M))

be a finite nonempty

We may

is an action. is

is s-invariant.

subset of

(ei)i=0,..., q

G,

in N 'n M e

~hat (i)

le01%


Ifq+iI% < (I-(I+IKI)-I) q ~< 6 Since

y

in

e 0 .... ,eq

lemma.

IKI)-I) Ifkl# and letting

for all

obtained

For any natural of projections

in

above

n~> i,

Let

(Um)m6 N

k =0, .... q , which

invariant Let Trick.

,

(n)~

tek

j

,

k = 0,...,q

I~ ~ 6 k=l

generating

be a separable

all the projections

by the automorphisms ~: e ÷ M W

If

C

(9) and thus prove the

1

be unitaries

and let

does not

with

O {Ad U m l m E I N } C Aut M e.

contains

q

Trick 5.5 to the projec-

a family

(n) . (n) < 1 e k ~g ~e k ) I~ n '

= {~gigEG}

and

in detail.

let us choose

(n) ek =

le(k0)

Step C is proved.

small,

to make y = 0 in

this procedure

N ' N M~ k

such that

k, thus

e0 = fq+ I,

can be taken arbitrarily

Let us describe

N'A M e

.

depend on it, we may apply the Index Selection tions

k = l,...,q,

gE K

(i + IKI)-I Ifkl~

Ifk+ll#~< ( i - ( i +

Step D.

for

and

lek~g(ek) I~ < Ylekl ~

We have

gE K .

in

(acting

k =0,...,

q

geK

and let ek = (e(kn))n e

and which

be the homomorphism

ek = ~(~k) E Me,

..,q ;

sub C*-algebra

ek A

N,

Let

'"

of

~ ( I~, M~) Z~(IN,M w)

is kept globally

term by term on

yielded then

ek

~(~,M~)).

by the Index Selection are projections

of

sum i, and satisfy le01¢ =

~ (e0)

=

lim ~ (e(0n)) ~< n-~oo

and similarly Iek~g(e k) I~

=

lim n ~

Iek(n)

((n)) = 0 ~g ek ~ '

k=l

'

...,q,

geE.

48 We also have

for all

Ad Um(e k)

and thus

e k E N' N M~.

In the f o l l o w i n g times in

6.4

or

M~

Ad Um(~(ek))

=

~(ek

Mm,

we shall

part

by

E=

G

apply

bE

=

1

~

k=0

the Index

in o r d e r

of the p r o o f

.....

Selection

q

Trick

to get g e n u i n e

of the R o h l i n

of m u t u a l l y and

in the s t a t e m e n t

= ~ IKi l-1 i,k

,

))n )

several

equalities

ones.

(Ei, k)

aE

~ ( ( A d Umte k

is proved.

i E I = {i .... ,N}

of

, (n)

=

ek

as above,

to a family

subsets

and for

=

The lemma

the second

indexed

e-paving

)

out of a p p r o x i m a t e

We b e g i n

associating in

=

in the same m a n n e r

M~

me]q

E

k,~ CK.l

theorem

orthogonal

k E Ki of 6.1)

by

projections

(KI,...,K N b e i n g the f o l l o w i n g

the

numbers

lak£-1 (Ei,z) - E i , k l ~

I~i,kI~

g E G =

C g'E Recall

that

a-invariant

0 < e
b E + (l-e)p- (l-e½)p I> b E + (¢½-s)P >i bE +2¢P and thus

(i0) yields

bE ,-b E > ~ ~

i,k

We have proved the statement

IE:l,k

E i ,k l

(i) in the conclusion

of the lemma.

52 Step B. Let us now prove the second part of the lemma, concerning the equivariance of the Rohlin towers. If i E I and k,m 6 K i we infer (12)

I(~km-I (Ei, m)

_E I i,kI@

~< l(~km-1(Ei,m) -Ei, k) (l-~km-1(f'))I¢ + IEi,k(f'- C~km-l(f'))I¢ + IC~km-1(fi,m) -fi,kI@ ~< ]~km-1(Ei,m) -Ei,kl ¢ + IEi,kfAl¢ + I(km-I Si,m)ASi,kl If[¢ For each

iE I we have (I (m-ISi,m)ALi I + I{k-ISi,m)ALi I) E I(km-1Si,m) A Si,kl < E k,m 6 K i k,m E K i

= 21Kil ~ kEK.1

l(k-lSi,k)ALiI

=

I{i C LiIKi, % { k}I

!

21Ki I ~ kEK,

1

= 21Kil Z l{k e Kilk M KI,Z}I £ e Li ~< 2~ILiIIKi 12 < 2g(i-g)-11Ki I Z IKi,£1 £e Li = 2g(i- g)-1 iKil iKiLil If we take this into (ii) and sum up, we obtain %,

--

' E'i,kl¢ Ii IKil-Ik,m [ l~-1(Ei,m)-

~< ~ IKiI-I I l~km-l(Ei,m)-Ei,kl~ + IfAll + 2£(I-e)-IIK'I Ifl~ l k,m -1 =

aE +

IfAl I + 2 e ( l -

e)

0

In view of (9), (ii), and our assumption

(5) on e,

aE, ~< a E + 2slp + 2e(l- ~)-i ~< aE + (2e1+2e(l-e) -I) (~½- ~)-1(bE, -b E ) .
0

(Ei,k)i, k

in

the I n d e x S e l e c t i o n ~ 0

and

Theorem

6.6

Ei, k

of

E÷ b E

n e t in a t o t a l l y

converge g;

hence

and so,

some a r b i t r a r y

in the g

is i n d u e -

[E I

letting

and

E[,~ = E ~ i,k + E ~ "

and

g CA

in

Cg,E

E

and

Trick

will

respectively

,

j,L

I Cl-L

ACC G

N'n M e

A f G, in o r d e r

the m a p

has a m a x i m a l e l e m e n t E ° . h bE0 ~ i - s 2, w h e r e (i) and (2)

are e s t i m a t e d

i,k

j,ZE I [~g(E~,z), E~] I¢
0 and the e-paving

6 > 0 and

same way as in the construction a system

the estimate

and then come back, by means of the Paving

(K~) towers of subsets

2IT + I ~ 1 T < 34~ ½

T

too.

We want to obtain

small constant. (K~)j


~ K j be a bijection with i,j , i 1, 3 ] Li'j) = Kj for all j, and if (k,i) e K i ×Li, j with

k£ e Ki,j

k (k,Z) =k~. We now apply Step A with

to get a partition

of unity

6 and

i

j

3

(Kj)j standing

(E i ,k)j ~ i ,, k e K j , in ,

~

for

e and

(Ki) i

M e such that

166½

k,Z

(I0) X I IBg(E~,k)- E'j,kl~ "< 346½, g~A j k and, moreover,

analogues

of the commutativity

relations

(6) and

(7)

hold. !

From the

(Kj) indexed

!

partition

of unity

(Ej, m) we obtain

a (K i)

58 indexed one

(Ei,k) by letting for

Ei,k where

:

i • I and

k• K i

Ij I£ E'],m

jeJ, 9~ • Li, j and m = k ( k , £ ) . For g • A we have from (i0)

(ii)

[ [Bg(Ei, k) -Ei,kl T i,k Let

i • I and

k 1,k z • K i.

~< 34@ ½

We infer -i

~klk[ I (Ei,k2) -Ei,kl

:

'

E'~

,)

[j IKjl ~,k X ,~k I£k'-1(~k'k'-1(Ej,k ')- 3,k I i

' E ' . ,) - ~ IKjl- i I , c~k £k'-1(ak'k '-1(Ej,k')3,K2 j i,k 1 2 2

3

xj I%1-Ii,k' I where j • J, up we get

~ • Li, j,

k , • K[3,

~klk21(~k2Zk2-1

(%,k~) ~', ) 3'k2

k'l = k(k1' ~) ' k2, = k(k2,~ ) "

Summing

[i IKi -Ikl,k2[ la klk21- (Ei,k2) - E i,k lIT < 2Ei + 2 E where

ZI =

With

j • J,

I I K ' I - I ,X la, , k , j kl,k, nI

(E~, k ) -E~ 'I ' 3,k, T

k',k'1 • Kj, and !

z2 : where

i e I,

kCKi,

I X

X l~k~k-1(E~,k)-Ej,k'IT

i j k,Z

j e J,

£ E L i , j and

k' = k(k,Z).

We have from

Z 1 < 16@ ½ • On the other hand, from the definition (8) of Ki, j, we remark that if in Z 2 we have k'e Ki,j, then k£ = k' and the corresponding term in

Z 2 vanishes.

Hence i k'

where

j6 j

and

k' 6 Kj\(~ K'i,j) "

T

(9)

L,J I < 4sIK[13

corresponding

~)

8s + 32@ ½


0, and normal state

~

on

< 6enIK~+iI

If for some

M,

g,h,gh ~ Gn+ 1

N U g , h - ill2 < then

ll~g - iII~
(i- en) iKnl .

We infer from

70 Ug,h- i

=

Vg~g(Vh)Ug,hVgh - 1

=

[ ~. (Ug keg(Uh,£)Ug,hUgh, m - l)Ei,peg(Ej, q) i,j k,~ '

=

E I +

E2 +

E 3

n n n n where i,j • In, k • Ki, i• Kj, p = In(k)g , q = i h(1) , m = (Igh)-1(p) • n Ki; in El we sum for i=j and i = m • ~n. in E2 we sum for i' m • K~\~nl Ki, In yields

El

and in

E3

for the remaining indices.

we have i=j,

El= 0.

Z=m,

k = q = hm,

Is~l, < 21 I

i,m

where

i • In , m E Kni \K91 and

the estimates

and the cocycle identity

We have Ei, p

I~

p = inh(m).g

Since

I K1g \ ~ 1 I < SnJK~l

6.1(5) yield I~21, ~

2(5e½n +en)

~

12c~

For the third sum we infer

Iz t, < 2 [

[

IEi,peg(Ej,q) l,

±,3 P,q i n q • Kn] n g- Kj, P • K~l

(i,p) ~ (j,q). We have In the same way we get already estimated an analogous sum in 7.2(2).

where

i,j 6 In,

We have thus obtained for

I~g,h- 114 and thus we have proved

and

g,h,gh • G n

ls,l, + Is~l, + Is~l,


n,

= Ad vP-l(Adg v(P-2)~)g

• Ad vP-l(~p)g

_C #p

and so ~ ½ IIv(P) < 22(9e2p- 2 + 9Sp - 2 ) < 26e~ _ 2 g - v(P-l)ll~ g Hence for

m>p>n,

Since for each

~ e ~

g 6 Gp_ 2 we have m LlVg (m) - v~)ll~ ~ ~ 26~ 2 ~ k=p+l

ep~ 0 and

and

Gpf G,

g E G and satisfies llvg - iII~ ~

and since and from u = Sv.

the *-strong limit for

27en_ 2 ,

(2,p) above, lim u p = 1 p +~ g,h The theorem is proved.

8:

g • Gn_ 2 ,

g , h • G,

(v~P-l)),

we infer

MODEL ACTION SPLITTING

1.2 and 1.3, which assert that

free action of an amenable group "contains",

by an arbitrarily close to 1 cocycle, model action.

~ • ~n

((~g), (Ug,h)) by

*-strongly,

In this chapter we prove Theorems a centrally

Vg = limp Vg" (P) exists

g • Gn_ 2

( ( ~ ) , (U~,h)) is the perturbed of

Chapter

27E~_I

if perturbed

both the trivial action and the

The proofs also yield the analogous results,

Theorems

1.5 and 1.6, for G-kernels. 8.1

We begin with some technical

lemmas.

The first result is due to

Connes ([4, Lemma 1.1.4]). The statement here is slightly stronger but follows from the same proof.

78

LEMMA

1.

be a finite then

Let

M

be a countably

set of normal

there

exists

states

a partial

decomposable

of

M.

isometry

Ilv-fll~

If

vE M

W*-algebra

e,f E Proj with v ' v =

M

e,

and

and

let

e~ f

vv* = f

< 61Ie-fll~

II v*-fll ~ ~ 7;1e-fll # for any

~ e ~.

A similar

LEMMA If

2.

e,f E Proj

with

v'v=

result

holds

Let

be a finite

M

e,

M

with

for the

e~ f

vv* = f

then

Let

Proof. fe

and

let

~ = Ie-fir. e I = w * w < e,

lw-fIT

0 2=


l .

M

el,e 2 .... ,en,...

be a factor and let

subfactors

Suppose

of

M,

M = en®

such that

that for each

t

be mutually

((en) 'O M)

in a total subset

for

M,

~ of

we

have

n >i Then if

e

subfactor

8.3

denotes of

M

(en)'n M

the weak closure

u en n

of

in

M,

e is a finite

M = e ® (e'e M).

and

In all that follows,

the group

assumed

discrete

assumed

to have a separable

G

that is dealt with will be

and at most countable, predual;

and the factor

~

will denote

M will be

a free ultrafilter

on ~ . LEMMA.

Let

G

a: G ÷ Aut M~

Let

be an amenable

Let

5°2.

be matrix partial let

-01 V

Since

M

is a McDuff

I be a finite units

in

isometry

set,

M~m.

in

factor,

let

M

be a M c D u f f factor.

free action.

M e with

M

Then the

is of type II 1 by Theorem

0 E I and let

Then_0,_e0,0

(ei,j),

i,j 6 I,

so let

~ ~g(e0,0);

--0--0.

VgVg

Vg Vg = eg(e0,0),

Let us define

= e0r 0 •

strongly

(M~) a is of the type II I.

fixed point algebra

Proof.

group and let

be a semiliftable

~$

be a

= e0, ° ;

for g=l

the unitary -0

=

Vg

.~ ei,0

V

geg

( e ^u, i )

g

e

G

l

and let obtained

((~g), (~g,h))

be the cocycle

by perturbing

~g(ei,j)

=

the action

-* Vg~g(ei,j)Vg

crossed

(~g) with

=

action of (Vg).

G

on

We infer

M~ for i,j 6 I

ei,0 v°a g g (e^u,1. e.l, 3• e.3,0 ) ~ * e0, j =

ei, j

hence Ad ~g,h(ei,j) and

~g,h e e' N M ,

(ei,j) .

(Vg) c e' n M~ action

where

e

We apply Proposition to an action

(ag) to the action &g(ei, j)

=

We apply Proposition

=

~g~h~gh1(ei, j )

is the subfactor 7.4 to perturb

(ag). (~g),

Since

7.2 to the

M~

e i ,j generated

((~g), (Ug,h)) (eg) cocycle.

=

Ad Vg(ei, j)

(~g) cocycle

by

with

(Vg) = (Vg~g) perturbs

(Vg) is an

Ad Vg(~g(ei,j))

of

=

the

Moreover, =

ei, j

(Vg) and obtain a

80 unitary

units

w • M~

such that

Let us take in M~ and

fi,j

ag(fi,j)

This

ends

8.4

By means

from

M~

the proof

to

able group

= Ad w(ei,j),

=

ag(Ad w(ei,j))

=

Ad w(ei, j)

of the

of the

g • G.

i,j • I.

=

=

lemma

that

fi,j

we can lift c o n s t r u c t i o n s

G

on the factor

i,j • I,

be a centrally free action of the amen-

M.

(Vg) c M ~

Let

IiI < ~,

=

E. ±,j

are matrix units in M,

and

(Vg) w

(e~,j)~

geG

'

Ei,j,

for

• which for

Vg, which for each

w

(ag)-Cocycle in M, such that

(Ad Vgag) (e~,j) Proof.

Step A. By Lemma

Ei, j y i e l d i n g

guished

element

sequence

for

We have

for each

of

Vg

I . 9

~

matrix

of u n i t a r i e s

and

~

E0, 0

By Lemma

Wg Wg =

(Ad Vgag) (e0, 0) , WgWg unitaries

M,

w~g • M -~ Wg

then the sequence

=

(Wg)~

in

and

in

M.

(Vg)9 M,

Let

(eW,j)~

0 be a distin-

be a r e p r e s e n t i n g

E0,0

both

exists

v~1 = i,

9 • IN.

= e0, 0 ; we take

a sequence

w I = e0,0 •

e~ g -w 1,0 w (Ad Vgag) (e0, i)

represents Vg~g)(E 0 i )

represent (Wg) of

and s a t i s f y i n g

by i[

w • IN .

sequences

with

(e0,0) ~

7.1 there

representing

Ei, 0 E O , o ( A d i

let

,

e0, 0

(Ad Vgag) (E0, 0) = isometries

representing

g •G

g

((Ad Vgag)(e0,0)) ~

partial

i,j • I,

units

g C G,

and the sequences

in

ei,j

7.1 choose

For each

consisting

for all

=

(Ad Vgag) -~ ~ o) (e0,

define

M O~ such that

i,j • I

for

(ag) ~°

be a cocycle for

be m a t r i x units in

Then there exist r e p r e s e n t i n g sequences

for

are m a t r i x

Ad(WVg) (ag(ei,j))

follows

(Ad V a ~) ( E i , ) g g J

• N

(fi,j)

lemma.

a: G ÷ Aut M

Let

(Ei,j),

form an

Then

M.

LEMMA. and let

Vg = W*ag(W),

=

1 • Mw

If we

81

and, moreover,

as in the previous lemma, we infer (Ad(Wg -~vg)eg)(e~,3') = e~i,j

Hence

(Vg) = (Wg ~g)

represents

Vg

and

(Ad Vgag) (e~,j) Step B.

= ei, j

Let

~ E ~N and let e ~ be the subfactor of M generated ~ (Ug,h )) be the cocycle crossed action obtained by (ei,j)i,j ° Let ((g), by perturbing the action (ag) by (Vg). Since ~gle = id, we infer U~g,h E (eW)'N M; g,h E G, w E IN, and hence ((~gI(e~) 'N M) , (Ug,h)) is a cocycle crossed action of G on (eV)'n M, which by 5.8 is centrally free. By Theorem 7.4, we can perturb ( ( ~ ) , ( ~ h)) with (Q~) c (e~)'N M to obtain an action (8~). " ~ ~'~ ~w ~~~ ~~*h) ,~ Since the sequence (Ug,h)= (Vgag(Vh)Vg represents Vgeg(Vh)Vg h = I E M ~, we have for each g , h e G lim w÷~

u~ g,h

=

1

*-strongly

and by the estimates in the theorem, we may assume that (Wg) also satisfies lim w÷~

w~ = g

1

*-strongly.

We let v ~g = Wg~~Vg~~. Since for each v, (Vg) perturbs the action (ag) to an action (Sg), (Vg) is an (ag)-Cocycle. For each g a G , (Vg) represents Vg, and for each i,j e I (Ad Vgag) 9 e9 (i,j)

~'~ Vgeg) (e~,j )) = Ad Wg((Ad

~~) ~ j) = Ad Wg(ei,

= e~1,j

and the lemma is proved.

8.5

The following result implies Theorem 1.2. THEOREM.

amenable group

Let

a: G ÷ Aut M

be a centrally free action of the

finite subset of

on the M c D u f f factor M. Let c > O, let + M, and let F be a finite subset of G~

exists a cocycle

(Vg) for (ag) and a II 1 hyperfinite s u b f a c t o r

such that

G

M = R ® (R'N M),

(Ad Vgag) IR = id R

tlVg -IH~

< e

II~ O P R , n M - ~ H In Theorem 1.2 we asserted that

be a

Then there

R C M,

and

~ e ~ , < ~

~

g E F ;

~ e

(Ad VgaglR'N M) is conjugate to

82 (~g),

and this can e a s i l y

id R is c o n j u g a t e Proof. point of

G,

from

(Fn) n,

above,

since

the p r e v i o u s

Me

to

F I = F,

lemma

inductively

to lift f i x e d

M. be an a s c e n d i n g

sequence

of finite

subsets

with

u F n = G, and let (~n)n , ~i = ~, be an a s c e n d i n g s e q u e n c e n subsets of M +,, w i t h Un ~{n total in M, . We c o n s t r u c t

of finite mutually

commuting

and c o c y c l e s for

from the t h e o r e m

to id R ® id R.

We a p p l y

factors Let

be o b t a i n e d

subfactors

(~)for

~i,~2 .... ,~n ....

(@g) = (@g),

(eg) = (Ad Vg -n @gn-i ),...

(~)

such that

for

of

(~)=

if we let

M,

of type

12,

(Ad Vg@g)-1 0 ,..., tVg'-n+l)

en

be the s u b f a c t o r

of M g e n e r a t e d by ~i u ... u ~ n , e ° = C.l, we have for e a c h n>~ 1 n en-i n - n - i ...Vg, @gl = iden_l and vg-n (an-l) ' A M, and letting v n = -VgVg V g0-_ i, we have (i)

IIv g n -Vg n-i

(2)

fl~ o p

# - 1 we h a v e Ien (Ad Vgag) n

thus at the l i m i t w h e n g E G.

8.6

The t h e o r e m

m÷~

mien ag

=

and t h e n

id e n n÷~

g e G , we infer

A d V g a g {R 1 =

idR,

is p r o v e d .

Let us r e c a l l THEOREM.

=

Theorem

Let

1.3 u n d e r a s l i g h t l y

a: G ÷ A u t M

different

form.

be a centrally free action of the

amenable group G on the M c D u f f factor M. Let s > O, let ~ be a finite + subset of M, and let F be a finite subset of G. There exists a cocycle M = R®

(vg) for (R'N M),

model action

ag

and a II 1 hyperfinite

(Ad Vg~g) (R) = R,

F r o m the a b o v e

model

1.3 that

action

(Ad Vg~giR)

R C M,

such that

is conjugate

to the

(4.5) and llVg -iii#

of T h e o r e m

subfactor

{a(0)) g


l

that

be an ascending

~

consists of faithful

family of finite

states of

M.

Let

sets of normal states of

M,

with

~i = ~ and u ~n total in M,, and let ( F n ) n > l be an ascending n family of finite subsets of G, with F I = F and U F n = G. n We inductively construct mutually commuting hyperfinite II 1 subfactors

~i,~2,...

of

M,

with

M = ~n ® ((~n), n M)

and cocycles

(~)for (a~) = (~g), ( ~ ) for ( e ~ ) = -n n-l, for ( ~ ) = (Ad Vgag ;,... such that if e n is the generated by eIu.., u ~n, e ° = C.I, and if Vgn = then

hold.

( l , n ) ~ ( e n) = ~n

and

(aS1 ~n)

is conjugate

(2,n)

( ~ I (en)'n M) is outer conjugate

(3,n)

~n c g

(4,n)

llv~-Vgn - l # ~ < 2-ng

(5,n)

II~-~ o p

Let

(an-l) ' N M

n , (e)AM

n > 1 and suppose,

5$,...,vg-n-i

satisfying

constructed.

Let

[ ×i®~ill

(ag)

if n > l ,

~E~ that

e~ .... ,~n-i

and

for k = l,...,n-i have been

M.

~ 6 ~n

some

X 1,...,Xp e e n-l, and

such that under the i d e n t i f i c a t i o n

II~ -

action

g e Fk , ~e~

II < 2-ne

N = (en-l) ' n

subfactor of M -n-n-i ... ~g, v g° = i, VgVg

to the submodel to

n,

g ~ G

(l,k)-(5,k)

Let us choose for each ~i,...,~ p 6 N,

,

for each

(Ad Vgag)-1 0 ,..-, 0 be such that

85 [ IIXilI ll~ill < 2 -n-2 i for all

~ • ~n"

The action (ag-llN) is by the induction hypothesis outer conjugate to (~g). We apply to it the lemma in this section to obtain a II 1 hyperfinite subfactor sn of N with N = a n ® ((an) 'n N) and a cocycle (v~) for (~g-l) such that with e n = en-i ® ~n c M and (~g) = (Ad Vg~g-n n-l) we have ~g($n)= sn; ( e g e n) is conjugate to the submodel action and (~I (en)'n N) is outer conjugate to (~g-lIN). ling- i,]~ ~< 2-n-ls

for

g • Fn,

~ • ~, where

g Og = Ad Vgn-i , and also

II~ -~ o P(en) ' n NIl ~< ~{}~ II

~ •

Via the inequality 7.7(1), we infer -n # + ffVg-!{f~g) -n # iivgn_vgn-l~#~ = ii(vg-n_!)Vgn-i II~ ~ 2½(IIVg-iIi~ 2 ½ . 2-n-ls For

~ e ~n' with

chosen before, if let


~l

II(4 - ~ o p 2-n~

=

n ' ) OP(en_l) ' M11 (~) AM N e

n>~l The theorem is proved.

8.7

The proof of Lemma 8.6, given in the sequel,

is the crucial point

of this chapter. A c c o r d i n g to 4.4,

the submodel can be a p p r o x i m a t e d by a system of

almost e q u i v a r i a n t m a t r i x units, w h i c h form a finite d i m e n s i o n a l submodel p r o d u c t w i t h a h y p e r f i n i t e

II 1 factor almost fixed by the action.

In Step A below, we c o n s t r u c t an almost e q u i v a r i a n t system of m a t r i x units in

M.

In Steps B and C, we p e r t u r b the action in order to make

the almost e q u i v a r i a n t s.m.u, become equivariant. the w h o l e c o n s t r u c t i o n from

Me

to

M,

In Step D we lift

and in Step E we c o n s t r u c t the

remaining almost i n v a r i a n t part of the submodel. T h r o u g h o u t the proof we shall use the n o t a t i o n s c o n n e c t e d to the Paving Structure for action

G

(3.4)

(4.4(5)) was based.

are the Sn-paving, Paving Structure,

on w h i c h the c o n s t r u c t i o n of the model

Recall that

en > 0,

G n CCG,

(K~), i E I n

(en,G n) invariant sets on the n-th level of the and

n. ui K~l ---~ y~ K~± ~g"

are b i j e c t i o n s a p p r o x i m a t i n g

the left g-translations.

The a s s u m p t i o n s on

based upon the fact that

£n+l

(en)n done in 3.5 and

could be chosen very small w i t h respect

to

e~,...,s n, are used w i t h o u t further mention. Also recall that the n set S i is the m u l t i p l i c i t y w i t h w h i c h K~ enters in the c o n s t r u c t i o n 1 of the submodel (see 4.4) and ~n = ui K~± × Si. n Let us choose n > 4 such that

½

30en_ 4 < ~

and

Gn_ 4 ~ F.

87 Step A.

The Rohlin Theorem provides an almost equivariant parti-

tion of unity in M~

we obtain,

M

; from this together with a fixed point s.m.u,

by diagonal summation,

an almost equivariant

in

s.m.u,

in

Me• Lemma 5.6 shows that the action strongly free.

(ag)~ induced by

For simplicity of notation,

(ag) as well.

Since

(M~) a is of type II.

M

is McDuff,

(ag) on

we shall denote

M~

is

(~g)m by

by Lemma 8.3 the fixed point algebra

We choose a s.m.u.

(Fsl,sz),

sl,s 2

An

in (M~)a.

We apply the Rohlin Theorem 6.1 and get a partition of unity (Fi,k), i e~In_ I,

k e K~-ll

[ i[ k,~

in

M~

I~k ~- ~(Fi,~)-

[~g(gi,k), Fj, m]

i r j E in_l,

0

k,i 6 K~1 -I

'

m 6 K n-l, j

(Esl,s2),

E(kl,sl),(k2,s2 ) = for

0

=

We define a s.m.u.

(kl,s ~ ) , (k2,s2) @ A n

=

< s~½~ n-±

Fi,kIT

Fsl,s2]

[Fi, k,

for

=

such that

s~,s 2 e ~n

sl,s 2 E A n

in

"

M e by

[ F( Zl'sl)'(Zz's2) Fi'h i,h

u3. Kjn × Sjn

i

i6 In_l,

£I = in-I in-i h-1 (kl), Z 2 = h-1 (k2). Since Fsl,s z and Fi,k commute and

h @ K ~1 -I

and

Z ng are bijections,

it is easy

to see that (E(kl,sl),(k2,s2) Fi,h) form a s.m.u, under

Fi,h'

for each fixed

i,h;

hence

(Esl,s2)

are

a s.m.u. Let us take ~n Since

Knl is (i)

=

n s E Sn } {(k,s) E sn I i 6 In , k 6 K n ~ ~ g -i Ki, l g6G n l

(en,G n) invariant,

ISnl >

Let us keep

we have

(i - e n) [snl geGn_l;

(kl,sl),(k2,s 2 ) 6 ~n

eg(E(kl,sl),(k2,s2 )) = =

fixed.

We have

[ F(h-lk i, sl ),(h- Ik2, sz) eg (Fi,h) i,h Z I +

Z 2

88 where

i • In_ 1 , h E K~1 -I ; in

h • K~1 -I N g-IK~-i l

and in

~i we sum for

~2

(i,h) with

for the rest of (i,h).

On the other

hand, we infer E(gk I ,sz),(gk 2, S 2 ) =

X F(k-l gkl ,Sl )'(k-lgk2's2) Fi'k i,k

=

~'

+ ~'

i

2

where

i • In_ 1 , k • K~1 -I ; in E I' we sum for (i,k) with k q gK n-i n K~i -I i and in ~ for the other (i,k). Since K~-ll is (E n_l,Gn_l ) invariant, we have for each i • I n - l ' {K~z -I A g - 1Kn-I i and so, by the estimates

lz~l~


' Ik,sl) -

Ad Ug(E(k,s),(k,s)) )

ZI+Z z

(k,s) E ~n; in Z 1 we sum for

(k,s) e ~ n \ A n .

(k,s) e ~n

In view of the estimate

Iz~l For

(k ° s0),(k,s))

and in

(I) on

A n,

Z 2 for we have

< 21 ~n \ Snl ISnl -~ < 2en- 1

(k,s) 6 ~n, the norm of the corresponding

term in

E(gk's)'(gko'So)W°e g g (E'k ~ o"s o)'(k's) -

E(gk,s), (gko,So)E(gko,so), (gk0,So)E(gko,So) • (gk,s) T
n and for ~ E ~n-i let ~k e (en-l), , ~k E N, ; k = l,...,rn_ 1 be chosen in (6,n-l). With (9) we infer for each k,

II~k o P(~n), n N - ~kI[
u K~ are part of the Paving Structure 3.4. ~g: The n-th finite i l submodel 4.5 had a s.m.u, indexed by ~n = u K~ × $9 . In i i l view of the assumptions 3.5 we make use w i t h o u t further m e n t i o n of the

dimensional fact that

Sk+ 1 is very small with respect to

Step A.

We construct

n-th finite dimensional variant

for

a s.m.u.

submodel

(eg) and is fixed by

ek,

for any k > 0 .

(Es,t) , s , t e ~n, replique

in

M

of the

, which is approximately

(Ad V $ ~ g ) , w h e r e

equi-

V g E M ~ are unitaries

implementing

g Let us begin by choosing,

V g E M ~,

g 6 G,

VI=I,

according

which implement VgV h

The action

(Ad Vg@g):

on

Ad V × Ad V*~.

to

Me

M,

= Vghg-l

G + Aut M

and such that

g,h • G

will be denoted by

(Ad Vgh-1~ h) = (Ad Vg Ad V~eh):

by

algebra (Fs,t),

eg

= Vg h

~g(Vh)

the action

to Lemma 9.2, unitaries

G × G ÷ Aut M e

By Lemma 9.1, the restriction

Ad V*~

and

will be denoted

of this last action

is strongly free, and Lemma 8.3 shows that the fixed point Ad V × A d V*~ (M) is of the type II I. We choose a s.m.u. s , t • ~n in (M~)Ad V × A d V * ~

We now apply the Relative of unity

(Fi,k)

i e In_l,

'

mately equivariant

for

Rohlin Theorem 6.6 to obtain a partition V*~ w h i c h is approxik • K 9l -I in (M~) Ad t

(@g I(M~) Ad V*~) = (AdVg I (M~)AHV*e) : the estimates

in 6.6 being better for small so we may suppose that homonimous

satisfies

the same requirements

as its

in Step A of 8.7.

We proceed out of

(Fi,k)

s than those in the Rohlin T h e o r e m 7.1,

to define the almost e q u i v a r i a n t

(Fs, t) and

The s.m.u.

s.m.u.

(Fi,k) by the same formulae as in 8.7,

(Es,t)

(Es,t), Step A.

thus defined will satisfy

lag(E(kl,sl),(k2,s2) ) - E(gkl,sl),(gk2,s2)IT ~ 22~_iIsnI -I

stt

~n,

104

for

g • Gn_l,

(kl,sl),(k2,s2)

e ~n,

IsnI > (i - £n) IsnI as defined we have

where

in 8.7,

{n ~ ~n,

Step A.

with

Moreover,

in this case

(Es,t) c (Mw) Ad V*~

Step B. This step parallels Step B in 8.7. We construct a unitary perturbation (Wg) c (Mw) Ad V*e for (~g) such that if (Ug) are the approximate

left g-translation US

with

i • In ,

generated

by

(k,s)

=

associated

to

(Es,t),

~ k,s~ E(kg 's)'(k's)

• K~l × S ni'

(Es,t) .

unitaries

kg = £n(k),g and

E CM e

is the subfactor