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 Amenab...

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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 <

h)Vgh

( l ® S g h)

= Zg,h ~(l®
The

lemma

2.5

The p r e c e d i n g

tensoring

chapter

case.

with model

is the

LEMMA. a)

Let

Let

~

Let

F

1

actions

having

be a u n i t a l

A

be a

results

can r e d u c e

in the

to this

invariants.

The

case

by

formal

x,y

A.

group.

4-I(1)

(left)

=

E ~

For

=

x E Z

~:

Then

~ ÷ F ~

=

and

so

xzy

=

xz= l.y

be the

we h a v e

we

be a h o m o m o r ~

is b i j e c t i v e .

find

z y = i, =

be a s u r j e c t i v e

is b i j e c t i v e .

and let ~: A ÷ F -I ~ (i) = i. Then

~(x) = ~(y) = g,

(g,x) --->gx

any

let

card

with

x.l

and {i}.

F-space with

9(xz) = ~(zy) = 1

F x A ÷ A, e

we

opposite

semigroup

with

of F - s p a c e s

Then

Let

classification

situations,

x

e = 4-I(1)

contained

be a d i s c r e t e

a) F o r

Proof. -i = g

b)

=

g,h

following.

phism

~(z)

Z

In m a n y

homomorphism b)

g,h

is proved.

invariantless

setting

z

z 6 E with

hence

y

action

of

%(~(x)-ix)

F = I,

on so

A,

and

~(x)-Ix

let = e

and x Hence

2.6

g + ge: F + A

We b e g i n THEOREM.

=

is the

by c l a s s i f y i n g Let

G

l.x

:

%(x)%(x)-Ix

inverse

actions

be a c o u n t a b l e

map

of

=

~(x)e

~.

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

amenable

II 1 f a c t o r group.

Two

R.

13

actions

a,8:

and

= A(8).

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 <

[12]

For

and q u o t i e n t

is w r i t t e n

denote

and

following

[15].

G

KC L

invariant

groups,

groups

is not

in

shall

transla-

of an a m e n a b l e

hence

subgroup,

of

we

---~ C g

are a m e n a b l e ,

Subgroups

group

left

a

of the M a r k o v - K a k u t a n i

an e x t e n s i o n

If the

K

infinite

of a m e n a b l e

amenable,

be d i s c r e t e , if it has

Haar measure".

groups

by m e a n s

union

ratio"

(e,F)-(left)

N gS I > (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 <

[12]

For

and q u o t i e n t

is w r i t t e n

denote

and

following

[15].

G

KC L

invariant

groups,

groups

is not

in

shall

transla-

of an a m e n a b l e

hence

subgroup,

of

we

---~ C g

are a m e n a b l e ,

Subgroups

group

left

a

of the M a r k o v - K a k u t a n i

an e x t e n s i o n

If the

K

infinite

of a m e n a b l e

amenable,

be d i s c r e t e , if it has

Haar measure".

groups

by m e a n s

union

ratio"

(e,F)-(left)

N gS I > (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 <- 2{s}

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

tsi\s'~l -< 361sL lemma

is proved.

!

U F -z ( K k K ) L

<- 3~-lls~t

and

19

4 1 e )N of the Theorem. Let N > ~-log[, w h i c h implies (i- ~ < e. e )N = e )n @ = (~ and ~n (~ , n = I,...,N. Let S N = S and for Proof

Let

n = N,N-I,...,I, that

supposing

KnL n C_ S n

and

Sn

(Kn%)%@ L

If for some n>~l,

is defined,

are e-disjoint;

I S n _ l ~ < elSnl,

IS \ ~ KpLpl = IS01 ~< eISNI = elS I . hypothesis

ISn_ll > elSnl,

n = N,N-I,...,I i)

Sn

2)

ant. (6n+iI is

is

n+ll

n = 1 ..... N

therefore

Sn_ 1 = S n \ K n L n . be done

since

continue

under

and we show i n d u c t i v e l y

the

for

;

lemma,

(2) r e s u l t s

from

(I) is a h y p o t h e s i s .

,

n+l)-invariant

and

(i) since

Sn

is

(½,Kn)-invari-

For n < N, since Kn+ 1 is ISnl> £1Sn+ll,

= (@n,Kn)-invariant.

Is01

we infer

An i t e r a t i o n

of

that

(2) shows

Sn that

(i- ~e )N IsNI ~< e Isl

<

is proved.

COROLLARY. an e-paving

Suppose

that a group

system of sets,

e > O.

G

L~,...,LnCCG

is infinite

and K I, .... K N are

There is a finite

K' ~ ~, which may be chosen arbitrarily subsets

define

such

e ~< ( I - ~)ISnl.

(3e-16n+iKn)

The t h e o r e m

be m a x i m a l

then we w o u l d

We may

(~n,Kn)-invariant

of the

For n=N,

Ln

that

ISn_ll

By m e a n s

let

~nvariant

subset

K'

of

G,

such that there exist

with

N

l)

~' =

[ I~iI ILil;

2)

for any

j=l

(~,k,%) satisfy Proof.

g < %.

centers

IKI a r b i t r a r i l y From

the sets

Suppose

LI,...,LN.

K'C G

with

ILiA Lil < 2cILil

large of K'.

as desired,

that

of e - p a v i n g

and

~CCG

is e - p a v e d K

i=i}

by K I , . . . , K N

arbitrarily

invariant

~

infer

I~I < (1-e-1

for

~iKilJ~il J

i E I,----N, L i C G

K' ~ IKillLi I = I I" S i n c e IKI was as 3 are no r e s t r i c t i o n s on the i n v a r i a n c e degree

such that

there

(Kil)

we easily

IK'A KI < ~ IKil and for e a c h

One can still k e e p

disjoint,

are u n i q u e

and for t h e s e

We m a y take

(l-e) [l~i11Lil j

with

g=k~,

large.

the d e f i n i t i o n

We can find

K[1 = { g C K ' Ithere

l~i (I- 4e)IEil ILil .

Let

with paving and

i E I,N

E i ~ I K~I .x- . i1,] w i t h

J

the a s s u m p t i o n

£ @ Li

that

are e - d i s j o i n t ;

(KiLi) i are m u t u a l l y we o b t a i n

20

3)

IKi 9 'I < 2 IKil I il

For

ge G

let

ness condition

~i(g) =

l{(k,Z)

• K i × Lilg =k£}I

{g • KiLii$i(g)< With

For an amenable

yields

a sequence level.

G,

of

G

l} > (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
and

and so

We infer

A

_C (K n g-iKn) × L nl,j w ... u

The fact that

(~n)-1 ('KnLni i,j ~n+l)i,j u ...

( -I -l g × ~) (~n)

Kn l

is

.KnL n ~n+l) [ i i,j i,j

(en,Gn)-invariant

~< enlKnl IL n1,31 + 2 e n I K n l

IAI

and the c o r o l l a r y

together

ILn,jl

=

with

(2) y i e l d s

3enlKnl ILnl, 3"I

is proved.

Example. Let us c o n s i d e r for i n s t a n c e the case G = ~N. ([-3n-i)/2, (3n-i)/2]) N C Z N and let L n = {-3 n, 0, 3n} N •

Kn=

KnL n = K n+l, Ln

n• N i.e.

and

so

~n is simply

K n increases

as m u c h

t r a n s l a t l •o n s lations

Kn

modulo

£gn

on

are r e c t a n g l e s the product. as d e s i r e d Kn

(3n~) N.

analogue

of r e c t a n g l e s

take

and let

~n

in g

paving

In the g e n e r a l

n.

paving

degree

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

for instance, case,

since

to use several

be b i j e c t i o n s

with

The i n v a r i a n c e

with

can be taken, we have

K n+l

behaving

Let Then

centers

of e a c h left

G

to be the transG

m a y have no

K91 at e a c h level and w e l l on m o s t of the points.

22 3.5

For later use we are now going to make some assumptions

elements

of the Paving Structure,

on the

made possible by the freedom of choice

that we had at each step of its construction. For a finite group

G

for each

n we let

In = {i}

and take

n 1 = {i} . In what follows we assume that G is infinite K nI = G n = G, LI, For each n we choose Gn+ 1 c G after (K~) i were chosen. We may thus assume that (

U

U

p
) Lp 1,j

U

(

U

U

p~
P P-l KP (K)-I P )U(GnU{l})u Ki(Ki) 3

(u uKP) p
C Gn+l

and also that UGn n Since for each by taking all

IKnl

j E In+l, large,

that

= G n+l I~j I = [ 1

i,j Since we may also suppose

Knl ILn,j I

we may assume,

kn+l j 1

'J that

~< ½ ¢nl --j vn+l I,

IGn+l

we may

assume that IGn+l u i~j After

the choice of

interfering

• n+l

(Kj

with the previous

)j

1,jl ~

and of

Sn

(_n+l,

Li,j)i, j we may, without assumptions, replace them with "K t n+l j gj )j ,

, for some arbitrary ((gj)-1 Li,j) n

respectively

elements

gj E G.

G being assumed infinite, we may use this device to assume that for each n, (Kn+lj )j are mutually disjoint and, moreover, that

j For each simplify

n,

p
i,j

l,]

s n > 0 could be chosen arbitrarily

the constants

and unhelpful

appearing

list of assumptions

small.

in the computations.

We use this to To avoid a long

of the form

an+ 1 < f(n, ¢0,gl,...,Sn) with

f > 0,

we leave it to the reader to check the fact that for each n,

a finite number of such assumptions A last problem:

are done in the rest of the paper.

The final choice of the Paving Structure will be np by possibly taking only a subsequence (K i )p

done in the next chapter, of the levels. refinements

It is easy to see, due to the finite number of possible

of a finite number of levels,

in such a manner such refinement.

that the assumptions

that assumptions

appearing

can be made

above are true for any

23

Chapter

For

a given

structure the w e a k

closure

provided

with

a model

action G,

of

used

G

follows. of the

by C o n n e s

in

as s h o w n

We d e f i n e

[4] and

the

and

maps

-->

~

one

first map

is the n a t u r a l

is the

K*

~

limit

it

it is d i f f e r e n t

from

be a p a v i n g and

structure

fixed

in all

/$ tO be the given

of

G,

that

inductive

limit

by Knl =

Kn

i

inverse

projection.

well from

as a n o n c o m m u t a t i v e

Knl × Lnl,j --> ~

of

Thus

,

Pn:

of

being

into

theory.

i

the b i j e c t i o n the

(kn) n e ~ K n w h i c h s a t i s f y for e a c h n kn(kn+l) = (k n In) E K n × L n in i n, in+ 1 Let

K

K n+l + K n

j the

let

G

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

be v i e w e d

chapter,

of

II 1 factor.

G= Z

in e r g o d i c

limit space with

for

could

used

group

on the p a v i n g

We o b t a i n

essentially

while

in the p r e v i o u s

(Kn)n@]N

based

representation

the r e p r e s e n t a t i o n

reduces

model

Kn+l

where

construct,

approximations.

([23])

be an a m e n a b l e

system

we

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

of J o n e s

of the o d o m e t e r

Let

G

the m o d e l

model

the one

constructed

G

ACTION

a faithful

dimensional

version

4.1

group

of an U H F - a l g e b r a ,

finite

units

THE M O D E L

displayed,

finite

of free

For matrix

amenable

previously

4:

elements

of

~n,

and

the

K*

are

fibers

the

inductive

n E IN: k n e K n in

second

and

•

K*--> K n

be the c a n o n i c a l

of the d i s c r e t e

topologies

projection.

on e a c h

K n,

K*

With becomes

a compact

space

a n d the B o r e l a l g e b r a of B o r e l m e a s u r a b l e s u b s e t s is g e n e r a t e d n U92n, w i t h 99n = {pn1(S) IS _c Kn} . F o r h E N , let F 0 c o n s i s t of n those permutations X n of K n w h i c h are d i r e c t sums of p e r m u t a t i o n s of

by

each

K ni,

i E I n.

Any

7 n E F0n

uniquely

k n ( y n + l ( k n + l ) ) = (in, X n ( k n), in) we o b t a i n

a homomorphism

we d e n o t e

by

ascending

union

4.2

We

Fn c A u t

choose

of

F n into the a u t o m o r p h i s m 0 r a n g e and let F = u F n . n groups, F is a m e n a b l e .

of an

an e x t r e m a l

ity of

Being

extremal,

measure

~

on K*;

this

~

in the

set of all

set is n o n e m p t y

by

In this w a y

Being

measures

n f X9 d~. ~i = j 1

= (in,kn,i n) .

group

finite

xin be the

X n+l E _n+l i0

its

Borel

F.

~n(kn+l)

a

from

K*

probability

Let

if

determines

K*;

F-invariant by the a m e n a b i l -

is F - e r g o d i c .

characteristic

function

of

pn1(K~)

_c K*,

and

let

24

on

For ~n-measurable functions f: K*-+ C, -i Pn ({k}), k • K n, the operators

which take the value

f

~->

IFnl -I

f

~

[ IKOI-ll E fk " xni n i k E Ki

fk

~ f yeF n

are both conditional expectations on the Borel algebra generated by n (Xi)ie in, as one can easily check; hence they are equal. Since U is n F-invariant, this shows that ~ is well determined by (~i)n,i • is the characteristic function For i • I n and j • In+ 1 , X ni .n+l xj of

p~1(K~)

N Pn+l-1(~n(K ~ × L~I,3)).

expectations

for

Irn+ll-i

~n+l

applied

to

E

xn'7

X

=

Y • Fn+ 1

I~ = l,j For

i n E I n,

I n'm in, im

=

so that,

for instance,

the case

m = n+l,

The subsets applied

,

X In n+l i,jDj 3

~ q 19 ~ . ln'in+l in+l,-",lm+ 1

l~ +I ... In+l,in+ 2

19'9 +1 = 19 . . In a way similar 1,3 1,3 for any m > 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,

~<

r IKnl Isnl (~n)-1-n+l ~n _ -n i,j ~J Ai'J ~il IMnl

=

-n+l lln,j-~i -n [ IMnl ~ Isnlpj i,j

~

J I ITn, J I-I Pni'JT Isnl

£ i~n+l I n

4.4 We use the sets constructed in the previous chapter to index the matrix units of an UHF-algebra. Let ~0 be a finite dimensional factor of dimension IS°l= 1 and for n > 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

<

We shall n o w use the r e s u l t s

in 4.3

=

2en

•

to p r o v e

(2).

Let

g6 Gn .

F r o m the d e f i n i t i o n s , un

E En = E l + ~2 g = Ei k,s (k~ ' s),(k,s) = i,j ~ k,s,m ~ E~+l s1's

where

i • In,

(k ' s) E

Kn i

x

S ni '

s = ~n(k,s,m),

sl = z n ( k 1 ' s ' m ) ;

m

E 2 those

R n . and in 1,3

F r o m the a s s u m p t i o n

where

(k,i,t,s) •

= kn(k,£),

~

=

in

for w h i c h

i,j

Z~

= Zn(k) g

kl

m • M n3

In+l'

Zz a p p e a r

those

m • M n \ R n .. 3 i,]

on the b i j e c t i o n

]~ k,~,t,s

'

terms

for w h i c h

We i n f e r

~ (k), i),

n

we i n f e r

_n+l

~(~,~),(~_,~)

Kn × L~ Tn . × S~ 1 l,j × l, 3 l

= kn(Z

j •

1

4.3(2)

~ i,j

"

satisfy

(£,t) •

pn i,j'

and

n ](t,s). s = si,

On the o t h e r hand,

un+l

=

E

g where k2

=

(k,i,t,s)

E 2' for

,

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

s =

In

and

,

~l + E2

E'1 we s u m for

(Z,t) • L91,3 × T~1,3. \ P O1,3.. ~<

k = kn(k,i) (Z,t) 6

E IKnl (ILn,jl ITn . I - IPn, l,] i,j

j I)Isnl

we i n f e r

3.4 we o b t a i n

p~1,j

Therefore

I Z 2 1 T + Iz'21T <~ enlSn+iI l~n+ll -I

With Corollary

,

=

i,j k,~,t,s

Iz~I T

and f r o m 4.3(3)

_n+l

t;(~2,{),(k,~)

E K~ × L~ • × Tn . × sn l l,] 1,] l'

Z.gn + l .L.K n ( k , £ ) )

and in

~--

=

en

Isn+ll -l

29 IZ~-Z~I T < 2 .E l{(k,Z) • Kn×Knll,ji~n(£g(k),i) # %g+l(~n(k,i)) } IT~,jlnisnll~n+ll-1 1,3 ~< 2"3sn ~ l~il ILni,jl l~i,jl ISnl Isn+ll-1 l, 3

=

6s

n

Finally, . n

. n+l,

IU -U

,

,

IT ~ IZI-ZIIT + ]Z21T + I~21T ~ 6~n+E n = 7Cn

and (2) is proved. Let A n be the maximal abelian subalgebra of ~n generated by n ~ • ~n. Then An C An+l and if we let A denote the weak (E~,~), closure of u A n in ~, then A is a maximal abelian subalgebra in ~. n The following result is a consequence of the proof of the proposition. COROLLARY. F o r any n > 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~<1

M.

A sequence

of ] ~ .

(x~)~ e 1~(]N,M)

~ > 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 ~ <

(Xn)~" (YW)w be r e p r e s e n t i n g sequences for x n and y , with all h For each n we can modify x n"~ for ~ outside a n e i g h b o r h o o d xn @ M . Let

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~<1

M.

A sequence

of ] ~ .

(x~)~ e 1~(]N,M)

~ > 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 ~ <

(Xn)~" (YW)w be r e p r e s e n t i n g sequences for x n and y , with all h For each n we can modify x n"~ for ~ outside a n e i g h b o r h o o d xn @ M . Let

32

of

e

such that IlX~n+l- x~nlI~ +

holds

for all

n

and

v.

Then,

(Xn) n is s*-fundamenta! s*-converges

to

hence

x~yW;

ll(Xn+l~ - x~)YWIIn ~ $

since

for

to show that

(t~)~ ~ ~

(x~)w

with all

x ~ e M~,

~,

and

(x~y~) n

n, <

2 -n+l

is e-normalizing.

t ~ e M~,

IixWtWH$ ~

for each fixed

to some

liXWn - x~ll~ + It(x~- x~)yli~ and it remains

2-n

being faithful,

s*-converges

so for all

<

when

But this is true,

W + ~ we infer

lltWll~ ---> 0

IItVxVIl~ < IItWx~II~+ IIx~-x~II~ IItPx~II~ + 2 -n+l for any

n.

We are now in a position Theorem

M~

Let

x = 0,

so

associated algebra

in

~

by

of some

M,,

the following

B(H)

~,

having

and

$~

M

and

is in

normal

(x~)w c M +.

is faithful.

to the faithful

extension

of

[i,

If

M~

(M),

state on

are W*-subalgebra8 and is faithful.

M.

Let

@~(x*x) = 0 then

x>0

in

(xW)w 6 ~

By means of the GNS construction we may suppose

a separating

that

M~

is a C*-sub-

cyclic vector

6.

We show

(Mm) h is so-closed. 1 Let

(xi) i C (M~)~

fundamental xy 6 H

(Me)~

with

to some

complete,

be a so-fundamental

in the topology

M

For any

y,

is so-closed xE

xiY ~ ÷ xYy~.

and so

x e(MW)~;

net;

of the lemma before,

xi~ ÷ xY~,

does not depend on of

$

% be a faithful

(M~) + be represented

that

is a W*-algebra

For any faithful

Proof. and

to prove

2.9].

PROPOSITION. of it.

__> 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

<

leaves

II%

2e

• ~v -~-~II llx~II2 + I(~'~)( x~*x~)I

invariant,

and h e n c e

gives

,

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

As

II[~, a~(xV)]II

=

II[~.~ v, x~]II

< II[~.B, x~]ll + 211~'~ ~- ~'BII llx~ll leaves

Me

invariant.

8emiliftable;

if

M e , respectively For e M,

eg= Me,

x=

(x)w

~(Te(~(x))) therefore

the c e n t e r

if n o n e of

M M,

for all

~=

e M e, =

of

of

([4])

i.e.

We

call

=

8 e,

Me

with

For

are

Te

central

CtM

projection group

=

semi!iftable

8

of

a nonzero

denote with

the

G,

of

M

M

centrally free) if all pro~gerly

centrally

then

if

9(8(Te(x)))

is c a l l e d

invariant

of M e

projection

and call

for

8 • Aut M

under

an

trivial.

e: G --> A u t M @g

properly outer

central

centrally trivial a u t o m o r p h i s m s

is c e n t r a l l y

a map

(respectively

U • ~(Me),

8m.

automorphisms

8 e = id e A u t M e ,

(respectively

If

(e~)w of

preserving.

under

8 • Aut M

a discrete

Me

~=

B = l i m eg,

properly centrally nontrivial if none of its r e s t r i c t i o n s invariant

or

respectively

lira ~(8(xV))

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

let

of

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

(~v)w E A u t M ~,

in p a r t i c u l a r ,

M

that

those

we

automorphisms

then

its r e s t r i c t i o n s

is inner.

~

lira ~ ( a W ( x m ) )

Te.e = 8 - T e ;

Recall

of

B

such

liftable and d e n o t e it by

a semiliftable and

fixing

We call

g~ 1

is c a l l e d

free

are p r o p e r l y

outer

nontrivial).

Ad U 6 Aut M ~

is s e m i l i f t a b l e .

A broader

34

class

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

inner

automorphisms

of u n i t a r i e s

of

automorphisms of

M

with

represents

a unitary

Moreover,

8 = Ad UIM,

by

Let

is o b t a i n e d

8 e

Int M

lim Ad U ~ = 8.

U

in

M ~,

but,

and

from

and

let

It is e a s y

be a s e q u e n c e

to see t h a t

Ad U 6 A u t M ~

of course,

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

(UW)~

(U~)w

is s e m i l i f t a b l e .

A d U is not

uniquely

determined

8. In h i s p a p e r

automorphism and

M.

[4], A. C o n n e s

group

its p r o p e r t y

THEOREM

of a factor,

of b e i n g

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

establishes

that

connections

the r i c h n e s s

McDuff.

These

of

between

the

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

properties

are

algebra

essential

for

follow.

Let

(A. C o n n e s ) .

M

be a factor with separable predual.

The f o l l o w i n g are equivalent: (i)

is McDuff,

M

i.e. M ~ M ® R ,

with

R

tricks

in

the hyperfinite II 1

factor.

5.3

is not abelian.

(2)

Int M / I n t

(3)

Int M ~ at M .

(4)

M~

i8 not abelian.

(5)

M~

is type II I.

We

niques

formalize

M

below

of v o n N e u m a n n ,

The a part

idea

of

constant

Me

of the fast

sequences

LEMMA

some

first

enough with

McDuff,

one

respect

sub W * - a l g e b r a s

of M e , and ~ N

Let

(3)

Te(a~(x))

(4)

8~(~(x))

C_ F'

n

= Te(a)Te(x)

We m a y

suppose

N n C_ N n + 1

Q +i•,

s-dense

in

M

and

tech-

of

Me

sequences

behave

of

like

N

M

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

and

F

be countably g e n e r a t e d

N

of

,

xeN

,

xeN

that N

with

in

aeF

, 8e ~ ~.

N = un N n

fixed

is w - d e n s e

w-dense

,

M c_ N n F.

with

and g l o b a l l y

N N Me

F = Un F n

~: N + M ~

M~

= ~(8~(x))

subsets

in

representing

part

*-homomorphism

~(NnM~)

with

from

a countable family of liftable automor-

(2)

finite

F

come

invariant.

~ is the identity on N n M

w-dense

Let

Trick).

(I)

Proof.

that

to it.

There i8 a normal injective

of

another

e e 8~\~.

phism8,

M~

and Connes.

is to r e i n d e x

to m a k e

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

separable predual and

leaving

useful

Dixmier,

F,

in

by

~,

For

natural

a unital such

that

N A M~ ; f i n i t e

F AM

w-dense

in

n,

N A M

subsets M

we take

*-algebra

and

over

is F n C_ F n + 1

F AM

35 w-dense dense

in

in

FnMe;

M,;

x V =x

x C N

we c h o o s e

~EN, for

Let

subsets

subsets

For each for a n y

finite

finite

IIx~ll~ llxIl,

M n ~ Mn+ 1

B n ! Bn+ 1

of

of ~

a representing

(x*) w = (x~) * ,

M,

with union norm

with union sequence

~.

(x~)w s u c h t h a t

(Ix) v = Ix w

for

I 6 C,

and

xEM. ~

be a f a i t h f u l

n6N

find

in ~

such that

normal

6 n > @n+l(X) > 0

l,yll~ <

s t a t e on

M.

For each

and a n e i g h b o r h o o d

Wn(X)

p(n)

E ~

such t h a t

and

~ Wn+l(X)

----->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~ <~ i/n Nx p(n), aIl~ ~< i/n

,

x,y • N n

a•FnAM

l @ ( T ~ ( a ) x p(n) ) - @ ( T ~ ( a ) T ~ ( X )

,

I ~< 1/n ,

IIB(xp(n)) - (BW(x))P(n)N~ ~< i/n ,

T h e r e are n e i g h b o r h o o d s Vn = @

V n ~ Vn+ 1

and s u c h t h a t for any ll[xp(n)

x e N n,

of

~

w

x • N n,

a • F n,

~•M n

B • Aut M

with

6~•A n •

in ~

with

V l = ~,

w • Vn ,

a~]ll~ < i/n ,

r

XeNnAM

xeN n , a • F n ~ M e

l~(a~x p(n)) - ~ ( T W ( a ) x P ( n ) ) I <

i/n

li~9 (xp(n)) - ~(x p(n))II~ ~< i/n ,

, ~=

xeN n , aEFn (~m)mEA n

and

,

~ e Mn

~ = lim~

TM

m-~0]

We d e f i n e

k: ~

x 6 ~ Nn, w e let of the p r o o f

5.5

In

Me

to o b t a i n LEMMA

separable of

->~

¢(x)

by

k(v) = p ( n )

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

is s i m i l a r

a new representing (Index S e l e c t i o n

and

I°~(3N,M LO) and

A

p a r t s of s e v e r a l

The r e m a i n i n g

Let

Trick).

w e ~/I~.

Let

a countable

acting

Then

is a C*-homomorphism

representing

M

G

be a W*-algebra

be a separable

set of semiliftable

term by term on

C,

Ye(~(x))

(2)

~(x) = x

(3)

~(~) e M m

(4)

~(y)

M~

if

XneM ~ for

n

for all ~e~

and

n y=

with

sub C*-algebra invariant.

such that for any

05

= W - lim T (x n) n÷t0 if X n = X for all

= ~(~(x))

sequences

automorphisms

leave it globally

~: e ÷

k = (Xn) n E d (i)

part

lemma.

sequence.

M ~, which there

w E V n \ V n + 1 , and for

(xk(V)) w.

to the one of the p r e c e d i n g

w e can p u t t o g e t h e r

predual

if

by

(a(Xn)) n

of

.

37

Remark. then

From

Proof. = u Cn n invariant

Let

a unital by

~,

and

M n • M n + l • M,

sequence

any

(5)

~ • M, ,

Let Vn, be

n~ 1

such

be

finite

sets w i t h on

M.

x• M ~

over

limIlXnll n+~

take

union

of

~

norm

~

be n e i g h b o r h o o d s

of

~

= 0,

~ _ (Xy)p(n) H$~< Ilxp !n) Yp(n)

(9)

lla~(Xp(n))-

for

all

The

lemma

and

now

In w h a t

follows,

i/n

as in the such

,

x=

union

M,.

Let

a representing (xW)~,

lemmas

real

above.

that

(Xm) m • C n,

V n D _ V n + I,

be d e f i n e d ~(x) • M e

in

Me

by

,

~ • F n.

V I = IN,

extend

(Xm)m • C n ,

9 = (Ym)m • C n

• C n n 1 ~ (IN,M e ) ,

k(~) = p(n)

by ~

~ •M n

x = (Xm) m • C n

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

is s h o w n

x=

x = (Xm) m ,

,

x = (Xm) m

so we m a y

follows

of some

following

let

in

globally

in

with

~ •~

union

(Xm) m • C n

w II$ ~< l/n, (~(X))p(n)

IN ÷ ~

is i n d e e d

x •e,

(6).

for

"# • M n

9 • V n \ V n + I.

by the

sequence

(Xk(w)~ )~ .

We h a v e

II~(x)ll <

llxll

to all of

C

by c o n t i n u i t y .

easily.

we o f t e n

constructions

have

already

to w o r k done.

in the We

relative

therefore

commutant

need

the

property.

Definition. to the

We

relative

is p r o p e r l y

outer.

strongly free if all Problem. strongly

~=

II [Xp(n), ~ 9]II ~< i/n ,

~

kept

sequences

in IN,

(8)

k:

with

for n > i,

I~(T~(Xp(n))-lim ~ ( T m ( X m ) ) I ~< i/n m+L0

let

~

dense

p(n) ~ n,

- lira ~ ( T m ( X m ) ) I ~< i/n m-~60

that

of

for e a c h

in

C

~ +iQ,

representing

p(n) • ~,

of

is n o r m - d e n s e

subsets

Choose

Wn(X)

choose

C

(7)

Y(x)

0

of

subsets

V n c Wn(Xp(n)),

We

M

all

we

n •IN

x = (Xn) n •

5.6

For

finite

(6)

(i0)

That

be f i n i t e state

be

e n I~(IN,Mm)

A n C An+ 1

I~(T~(Xp(n)))

n V = @ n n

Mm

faithful

*-algebra

that

and n e i g h b o r h o o d s

For

For

n •IN

sub

such

normal

(~w) v.

~n(X) > 0

dense

and Let

be a f a i t h f u l

of

if for some

C n c Cn+l,

e n I=(~,M~).

in

(i),

~(x) = 0.

outer?

call

@ E Aut M

commutant

A discrete eg,

group

g # i, are

Is any p r o p e r l y

strongly outer if the r e s t r i c t i o n

of any

outer

countable action

strongly

8-invariant e

of

G

on

subset Me

is

automorphism

of

of

outer.

semiliftable

M

38 Partial results

affirmative

LE~4A. Let If

Let

~ = (aw)v

Proof.

M

centrally nontrivial,

outer

Let

p

for some c o u n t a b l e a E S' n M e

extending

B

ze M e

is p r o p e r l y

with

is

~E B~/~.

B = lim ~ .

and

a

is strongly outer.

a

to

~-invariant

= ay

,

support

of

centrally

q z = z and

ke Z with

of

Me

S'A M e

Sc Me,

is not

and thus

there

with

be the c e n t r a l

Since

then

that the r e s t r i c t i o n

a(y)a

so there

in the sequel,

be a W * - a l g e b r a with separable predual and

Suppose

is a n o n - z e r o

some

are given

be a s e m i l i f t a b l e a u t o m o r p h i s m of

B i8 properly

properly

answers

of A. Connes.

y E S' n M~ Te(la[ 2)

in

nontrivial

B~(z) - z ~ 0.

M,

and

But

Bk(p) IBe(z) -zl 2 # 0.

and

q = k ~ zBk(P)"

B(q) = q,

there

is

qlB~(z) - zl 2 ~ 0 , Let

x=

(B~k(z);

then

plBe(x) - x I ~ 0. We n o w use the Slow R e i n d e x a t i o n N the s m a l l e s t contains

x,

W*-subalgebra

and let

and the c o u n t a b l e y e S' A Me,

F

Me

S.

a(y)

We send = B~(Y)

Te(la*1218w(y)-yl F r o m our choice

that

A

Let

of

x

~ = {~,B w} 6 A u t ( M e ) ,

leaves

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

subset

ya = a y ,

of

Trick.

of

into

invariant Me

and w h i c h

generated

y = ~(x)

e Me

by

a,

p,

such that

and

2)

=

T

(la*12)Te(IB~(x)-xl

2)

x, 2

pT e([B w(x) - x 12) As

p

is the c e n t r a l

=

support

of

Te(l (Be{Y) - y ) a l 2)

Hence

(B~(y) - Y ) a

Another

strongly

case

outer

LEMMAo

Suppose

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

~.

Then

in w h i c h

is t r e a t e d

a

M e,

M

=

-x

we o b t a i n

=

Te(la*I2)Te(18e(x)-xl

a semiliftable

=

with

automorphism

is a factor and let

a=

is properly

2)

~

0

.

the fact that

a(y)a - a y

lemma.

aw

0

T e(la*I 2 IBe(y) -yl 2)

in the f o l l o w i n g

such that

~

=

a(y)a - y a

i8 strongly outer.

)

T (lal 2) = T e la*I2),

~ 0 , in c o n t r a d i c t i o n

(Be(y) - y ) a

5.7

~e(PlB~(x)

(a~)~

=

of

0

M~

is

be a semiliftable

centrally nontrivial for

39

Proof. a n d let

Since

Claim.

Let

with

be p r o p e r l y

[4, T h e o r e m

as in the p r o o f

by r e p l a c i n g T(q)

2- i~. ( I - 1 4 )

s u b s e t of sn

it w i t h q + q '

the Me

lemma

6.3 below)

and

by the

we g e t a p r o j e c t i o n

T(q'B(q'))

and so

be

= i.

(or, a l t e r n a t i v e l y ,

The c l a i m

> ~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[ <~ 1

(q~)~ r e p r e s e n t s Ho~(q)a-aq[l

and the c o n t r a d i c t i o n

q

an

>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 <~ ~

sequence

2 i>

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,~

<

k e Ki N g

-I

K i and

Z 6 K i,

leg(~k£-1 (Ei,£) -Ei,k) 1% + lagkZ-1 (Ei,£) -Ei,gkl %

as above, we infer

IKil ~ leg(Ei,k) -Ei,gkl % < where

k e A i.

2

[ I~m-1 (Ei,m) -Ei,zI ¢ £,m

g-iK.l and £,m e K i. Hence (4) follows from (i). k 6 Kl Let us now prove (5). For any i=l,...,N, m ~ A i and k e K i, IEi,ml ¢

Summing for all such

<

IEk,kl ¢ + I~mk-1 (Ei, k) -Ei,ml ¢ •

m,k

IKil ~m IEi'ml# <

we get IAil k[ IEi'kl% +

[ IskA-~(Ei'z)-Ei'kl~ k,Z

and thus

; IEi,ml~ <

m where

m e A i,

; 1~i,kl~÷ IKiI-~k,£ [ i~k~-1(~i,~)-Ei,ki~

k

k,i E K i.

Thus

(5) is obtained from

Here are some circumstances theorem is fulfilled: If the algebra M needed,

since

#~

under which the hypothesis

is a factor,

is the canonical

semiliftable automorphisms. In the case when ~: G ÷ Aut M crossed action action;

G ÷ Aut M,

for instance if

M

(1). of the

no assumption on the state trace on

M e,

~ is

and is preserved by

is induced by a centrally free

then by Lemma 5.7, is the hyperfinite

~

is a strongly free

II 1 of I I

factor,

then

•

43

any free action

G ÷ Aut M

is c e n t r a l l y free.

For an abelian algebra measure,

if

X, then

a~

6.2

~

M = L

(X, ~, ~), with

~

a probability

is induced by a measure p r e s e r v i n g free action of

G

on

is strongly free and one gets the O r n s t e i n and Weiss theorem.

The proof of the theorem consists of two parts.

In the first part

we use a global g e o m e t r i c a p p r o a c h based on a lemma of Sorin Popa to obtain a basis for some

(possibly small)

Rohlin tower in

Mm.

In

the second part of the proof we put together such towers in order to get a Rohlin tower filling almost all the space. this part of the proof and the ones in

[4] and

A difference between

[33] is that each time

a new tower is added, one d e s t r o y s a part of the old one, taking care to make the p r o c e d u r e convergent. Let us first state the following result

([37, Lemma 1.3]) of

Sorin Popa. Let

A

ful trace

be a finite von N e u m a n n algebra w i t h a finite normal faithT,

and let

B

be a v o n

N e u m a n n s u b a l g e b r a of

is a unique T - p r e s e r v i n g c o n d i t i o n a l e x p e c t a t i o n One calls

xE A

T(xy) = 0

for any

LEMMA normal

that

Let

trace

A

on it,

the r e l a t i v e

e > 0 and x 1,...,x m c A partition

B

if

PB(X) = 0

PB

of

A

Then there onto

B.

(or e q u i v a l e n t l y if

y E B).

(S. Popa).

faithful

Suppose

o r t h o g o n a l on

A.

be a f i n i t e and

B

a yon N e u m a n n

commutant

are

orthogonal B

to

B,

such

algebra,

subalgebra

B' n

condition

(ej)j=l, .... n in

of unity

yon N e u m a n n

then

A _C B there

T

of

a A.

holds.

exists

If

a

that

n

(i)

II[

ejxie j II

j=l

~< elixiII

~

for

i = 1 ..... m

T

Let us b r i e f l y sketch his proof,

since in our context it will

yield a g e o m e t r i c a l insight into the structure of d i s c r e t e crossed products. One begins by proving an e l e m e n t a r y H i l b e r t space lemma,

asserting

that if

F

(Ug) is a u n i t a r y r e p r e s e n t a t i o n of a d i s c r e t e group

Hilbert space

H,

any

~> 0

to

~E H ~,

and

i.e.

w h i c h has no n o n t r i v i a l fixed points in there exists

such that

gE F

IIUg~- ~II >

for ~ ~ 0 the minimal norm point in fixed by Let

(Ug). p: A-->~(L2 (A,T))

such that

(/5-8)II~II •

Ug~

H,

on the

then for

is 6-orthogonal

if not, one shows that

c-~W{Ug~Ige F} is nonzero and is

be the GNS r e p r e s e n t a t i o n and let

the r e p r e s e n t a t i o n of the unitary group of

B

induced by

p

U

on the

be

44

space for

H = L2 (A,T)0L2 (B,T). U

follows

Hilbert vector

space in

projections with

lemma yields

H)

n=l

a unitary

of

and

procedure

u

yields

in

a c t i o n of a d i s c r e t e A

with

B11

X g E B,

and e x t e n d

expectation Let

of

A

onto

on

This y i e l d s Let

be a finite

partition

Then

is a free r - p r e s e r v i n g

and let

translation A

lg

B

lg E A

for

g~ 1

free,

denote

in L(G).

letting,

for any

of

a: G - - > A u t G,

with

(ej)j=O,l, .... n

in

In v i e w of the p r e c e d i n g

in

of the

is a f i n i t e yon N e u m a n n

the

We i d e n t i f y

for x E A ,

are o r t h o g o n a l

xE B

ag = 0

and for

B

gE G

on

B.

we h a v e

g ~ i, a n d h e n c e

family

be as above.

1 ~ K. B

Then

such

there

that

discussion

{%glgGK}

Let

6>0 and

exists

leol < G

j = 1 ..... n;

gEK

a

and

.

we m a y a p p l y P o p a ' s

to g e t a p a r t i t i o n

of u n i t y

with iifi%gfill T <

g21KI -I

,

g EK

i Thus

for

ge K

w e have

I {Ifi~g(fi)IIT i

and by the C a u c h y - S c h w a r t z

I° =

I T ( f i a g ( f i )f'l-)½ = i

=

~ llfiXgfiHT l

<

~, T ( f i % g f i % ~ f i )½

e21KI -I

inequality

; Ifiag(fi)IT i Let

=

<~

[ II111 IIfiag(fi)N T ~ i T

{ i E I Ifi~g(fi) IT > sT(f i)

for some

B

x = ~ Xglg, g conditional

x + x I is a T - p r e s e r v i n g

and all

B, T and

lemma to the B - o r t h o g o n a l (fi)iei

e ~ , . . . , e n G B,

B

lejeg(ej ) IT < ~lejIT Proof.

for

refinement

the f o l l o w i n g

subset

of unity

(viewed as a

Spectral

(ag)

B ×~ G g

to a trace on

B,

B

The

B.

a = Zaglg E B' N A. Then g Since a was assumed

COROLLARY. K

and

= rag.

B'n A ~ B.

let

G

T,

to the left T

of the l o o k e d

the case w h e n

T(x) = T(Xl).

on

Iluxu*-x!l T ~ llxLIT.

be the c r o s s e d p r o d u c t

with

orthogonal

fixed points B' N A ~ B.

in the lemma.

trace

group

corresponding

agag(X)

xE A

with

condition

(i), and an i n d u c t i v e

the r e s u l t s

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

Let

for any

u@ B

of n o n t r i v i a l

commutant

y i e l d a first v e r s i o n

8 = ~

algebra with normalized

unitary

The a b s e n c e

f r o m the r e l a t i v e

s21KI -I

g E K}.

We i n f e r

45

E

T(f i)

<

E

i6 10 e ° = i~i0__fi'

and so, if For any

E

geK

Ifiag(fi) IT < IKIeZIK1-1

= e2

iEl 0

then

T(e 0) < e .

i E I\l ° we have <

IfiSg(fi)I T

,

eifilT

and all that remains to be done is to relabel

geK

(fi)ie I\i0

as

(ej) j=l ..... n "

6.3

This section contains the first part of the proof of the Rohlin

theorem.

We show that almost all the space can be almost filled up

w i t h m u t u a l l y o r t h o g o n a l projections,

each of them suitable to become

a tower basis for a Rohlin tower. Let us come back to the n o t a t i o n used in the statement of T h e o r e m 6.1.

We shall w o r k in the relative c o m m u t a n t in

g e n e r a t e d s - i n v a r i a n t sub W * - a l g e b r a there exists a u n i t a r y assume that Since

a

N

is strongly free,

finite and the trace

LEMMA. 1 ~ K. such

Ug,h E M e

contains all

Let

~

N

of

such that

Ug,h

M~.

is free;

sIN' n M~ moreover

(which depends only on

~> 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% <

(2)

eiag(e i) = 0

Proof.

Step A.

that there exists

for

1
Let ¥ > 0

and

g E K. f E Proj (N'A M~),

f' E Proj (N'A Mw),

0 # f' ~< f,

Let N

N and

f.

Then

s is free on

Ifol~< ½1fl~

(4)

~ geK

Let

fi = fi f 6 Proj (N'A Me)

M~,

containing

N Mm, and by C o r o l l a r y 6.2 we

(fi)i=0 .... ,m

(3)

We show

gc~

be the s m a l l e s t s - i n v a r i a n t s u b a l g e b r a of

may choose a p a r t i t i o n of unity

f ~ 0.

such that

If'sg(f')L~ < 2~If'l~ , both

g,h 6 G

agS h = Ad Ug,hSg h.

and thus that

sIN, N M~

M

For each

]fi~g(fi)I+< yifl~ IfiI~,

in

N ' A M~

such that

i = l ..... m .

and suppose that for each i = l,...,m

46

Ifi~g(~i)I ¢ > 2Ylfil ~ g~K Then the assumed commutativity relations together with (3) yield m i=l gE K I> ~Lo( ~ ~ Ifieg(fi)I If~g(f) I) i=l g E K m

m

i=l gc K =

i=!

2y~o~((l- fo)f)

I> 2Y(ifI~ - IfoI }) i> yifl~

On the other hand, from (4), m

m

~ Ifi~g(fi) l~ < YIfl# i=~l ,= Ifil~ < YIfI~ i=l gE K The contradiction thus obtained shows that for some

gEK

Ifi~g(fi) I~ <

and thus we may take

2yifil ~

f' = fi"

Step B. We show that for any f E P r o j ( N ' n M there exists e E Proj (N'n Mw) with (5) e ~< f (6) Ieag (e) I~ < YIel~ ' g e K (7)

i E {i ..... m} ,

) and any y > 0

lel~ <~ (l+ IKI)-I Ifl~.

The family of projections e e N'A M~ satisfying (5) and (6) is nonvoid and well ordered, so let e be maximal with these properties. We show that (8)

e

e V(

g

also satisfies eV-K~g(e))V

(l-f>

= 1

If not, let e' be a nonzero projection in N'A M e orthogonal to the left member of (8). By Step A there exists a nonzero projection e" in N'e S~, e"<~e', with Ie"~g(e") I~ < 811e"I~ , g e K. We have e"~< f and e " ~ g ( e ) = 0 for g E K , hence e+e" satisfies (5) and (6). The assumed maximality of e is contradicted, and thus (8) is proved. From (8) we get 1 = e V ( Vg6K ~g(e))V (l-f)I, ~< Is', +gEKE ,~, g(e) , + ,l-f,, = i - IfI~ + (i + IKI) Iel~

47 and

(7) is proved. qe~

be such that

prove a weaker version

Step C.

of the lemma,

exists

Let

a partition

of unity

(i - (i + IKI)-1)q showing

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

< 6.

We now

that for any y >0

in

N'n M~

there

such that

le01~ < 6 (9)

Ieieg(ei) I~ < yieiI~

Let us take according e k < fk'

,

i = 1 ..... q;

fl = 1 and construct

to Step B, projections fk+l = f k - ek'

ek

successively and

fk+l

lekI~

>

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 <

N

Let

Let

generated

eIN' N M e

sub W * - a l g e b r a

~> 0

and

of

M e,

is an action.

be a family of mutually

E = (Ei, k)

N' n M~.

I [ag(Ei 'k ) ' Ej,Z]1%

is a c o u n t a b l y

and such that

LEMMA.

tions in

i,~j k ~

AgOG

be given

orthogonal

and suppose

projecthat

Y~. !

If

then there is a family

b E < 1 - e½

orthogonal

projections

o < e ~

(2)

a E, - a E <

(3)

Cg,E, - Cg,E

i,k

Proof. If all

Ei, k

in

N' o M~

are

If not,

a tower

base

of mutually

such that

IEi, k - E i , k l ~ ~ b E ,- b E 3e½(b E, - b E ) < 36e-i (bE , - b E ) for

The idea of the p r o o f

lemma.

E' = (EL, k )

zero,

then a t o w e r

we c h o o s e

among

and then c o n s t r u c t

of

g e A.

(i) and

(2) is the following.

(E'i,k ) is s u p p l i e d

the p r o j e c t i o n s a tower

yielded

by the p r e v i o u s by that lemma

(fi,k) , such that all

fi,k

49

commute w i t h all Ei,k,• then w i t h E'i,k = Ei,k(l -f) + fi,k" cantly larger than

E,

In

f = i~,nfi,k

we take

(i) it is r e q u i r e d that

i.e. that

Ei,kf

E'

be signifi-

be small w i t h respect to

this is a c h i e v e d by an adequate choice of the tower basis. (2) we should care that be equivariant,

aE,

which measures

the failure of

does not increase too much.

f;

In v i e w of (Ei, k) to

The only p r o b l e m is the

fact that we alter the old tower. If

f was e - i n v a r i a n t then cutting with

l-f

w o u l d not affect a E.

We approximate this by taking a tower indexed by a very large subset K' of

G;

such a tower has a very good global invariance,

we regroup its p r o j e c t i o n s to get the tower For

G=~,

I = {i}

and

(fi,k)

and s u b s e q u e n t l y

indexed by K I .... ,KN-

K l = {1,2 ..... p} c ~, a typical picture

w o u l d be the following:

X

Z

mp

......

E2

m M

In the figure, the old a

' El,

the new tower basis,

E I by taking out

K I tower,

Elf

f.

large invariance degree with respect to

there exists

e:,

0 < e I < e,

is shaded and is o b t a i n e d from

and adding the basis of

i.e. the dark parts of

Let us begin the proof.

f

The p r o j e c t i o n

r e a r r a n g e d as f

has a very

~.

Since we have assumed

0 < e <

IA6,

such that

(4)

bE <

(5)

2(s I + e ( i - 8 ) - I ) (E½ - e) -~ < 3~ ½

(i - 8½) (i - ~i)

We may suppose that

6< (

IKiKill

)-i e I

and that

i Step A.

W

Let

K'

E-paved by K I , . . . , K N.

be a

A D U KiKi I i

(6,A)-invariant subset of

A c c o r d i n g to Lemma 6.3

(with

G,

w h i c h is

(K')-IK'\{I}

50 standing

for

K),

choose

a partition

of unity

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

in

N'n M~

with

leol ~ < ~ ~g(ej)~h(e j) = 0 , [ej, ~g(Ei,k)]

j=l ..... q,

= 0

for all

g,h • K' ,

j,i,k,g

g #h

.

Letting X

=

IK'I -l E e g 1 ( E Ei,k) g • K' i,k

we have

Ixt~

=

~(x)

=

~(~

El, k)

= bE

,

i,k and moreover

x

commutes

with all e 1 . j • {l,...,q} such that

There exists lejxl¢ If not,

adding

~

(i- ~½) lejl¢

the opposite

b E = Ixl~ > and thus contradict . We let f = e3'

•

inequalities

I (i- e0)xl~

the hypothesis f' = geK' ~ ~g(f)

>

for

j=l,...,q

we infer

(i- ~½) (l-le01 ~) >

(4). and

p = If' I#.

(l-e½) (l-•z)

Then

If~l,~(z-~½) Ifl, and so (6)

If'E Ei,kl, = i,k

=

E

l~g(f) E Ei,kl,

g•K'

i,k

E

If~g (EEi,k)l,

geK'

i,k

-i

K' We assumed (L i), i e I

of

that G

(Ki),

and

i E I, e-pave

K[l,~ --C Ki, i E I,

= K'.

=

l~'llfxl,

(i - S ~) P •

Hence there are subsets

£ • Li,

such that if

K = ui KiLi'

then

(7)

Let us now define Si,k

=

for

i E I,

{k£1Z E L i,

k E Ki K'~ ~,n B k}

,

Si

=u

k Si'k

51 Accordingly,

let us take for

fi,k

=

E

i 6 I,

k e K

1

~g(f)

g 6 Si, k

fl• = k[ fi,k = Then

f =

~E~g(f)

~ fi k ~< f' =

g

E ~g(f), gEK'

>

and by

(i-E)IK'I

(7) we have

Ifl¢

=

( 1 - ~;) I f ' I ~ )

,

that is, (8)

IfI¢ > ( l - ~ ) p

Let

u u (K'Ak£-IK'). Since for each i, K' is i6 1 k,£ e K i (el IKiKi~ I-I, KiKil)-invariant, we infer IKAI < 2sIIK' I and thus if we let

KA =

fA =

(9)

~ e (f) geK A g

then

IfAI ¢ ~ 2slp

We are now in a position E'i,k

=

(i - f')Ei, k + fi,k

The amount of modifications (I0)

to define the family

from

E

to

E'

i E I "

I

!

= (Ei, k) by taking k E K.1

is estimated by

E I E i , k - E i kI¢ < If'I¢ + ~ Ifi kl¢ i,k ' i,k ' : If'l¢ +

In view of (8) and (Ii)

'

E

l?I¢ < 20

(6) this gives

bE , = I ~

i,k E'i'k1¢

=

~ Ei,kl li,k

~ + Ifl% -li~,kEi,kf' I~

> 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 ) .<

%+3c

½

( b w - b E)

and the proof is finished.

this yields

53 Step C. We now prove the third statement of the lemma, concerning the mutual approximate commutation of projections of the form eg(Ei,k). Since the tower (fi,k) commutes with all ~g(Ej,£), the only problem that remains is (fi,k) itself. The projections fi,k are sums of mutually orthogonal projections of the tower (~m(f))mE K'" Since K' is almost invariant to gE A, ~g(fi,k ) will be approximately equal to a part of this tower too; but the projections (~m(f))m6K, mutually commute. For g6 A, i • I and k • K l we have eg(fi'k) f' where

h e (gSi,k) N K'.

E

i,k

since

= ~h ~h (f)

Hence

I~g(fi,k) (i- f') I~ ~< ]g(i,kU Si,k) \ K' I If]

K' was assumed (6,A) invariant. i,kE ]Sg(Ei, k ( l - f ' ) ) -

Since

ag (Ei,k) (I- f') I~

E'i,k = Ei,k(l- f') + fi,k E'

i,k j,Z

We also infer

we obtain

E'. 3,

- i,kE J,zE l[~g(Ei,k ) (I- f') +eg(fi,k )f', Ej,£(l-f') +fj,z] I~ 2(6p + 2~p)

= 66p !

Since eg(fi,k) fZ and fj,£ are sums of mutually orthogonal projections from the tower (eh(f))he K'' they commute with each other and with the tower E. We thus have Cg,E'

v,k

j ,Z

6dp + Cg,E ~< Cg,E + 3de -I (bE , - b E ) and the proof of (3) is also finished.

The lemma is proved.

54

6.5

The R o h l i n

maximality

Theorem

argument.

the set of f a m i l i e s tions

in

N'N M e

E=

(Ei,k)iei

a E < 3e ½ b E

(2)

Cg,E < 3@e-lbE is n o n v o i d

E

E<E' and

ordered

E' .

E = E'

and,

above come

b y the

° Ei, k = E i,k

Ei, k is a p a r t i t i o n

since

of u n i t y

a E < 5s ½

(4)

C g , E ~ 3Cg,E0 ~

for

E

g E A,

the n e w t e r m s

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¢ <

F o r any g i v e n uDity

-i

-

and s i m i l a r l y ,

by

k • KI

and

This w a y

and it s a t i s f i e s

96e

I

j,£

~,

~

6.4 h o l d s

Zorn lemma,

- (i,k) # (i,k)

(3)

s u b s e t of

of an e l e m e n t

and 6.4(3)

if

be

projec-

We o r d e r

of L e m m a

for any i n c r e a s i n g

E ° satisfies

f r o m 6.4(2)

~

orthogonal

the null family.

the p r o j e c t i o n s

that

Let

w i t h a s u b s e t of the i n t e r v a l

0 = 1 - Z E~ we have IE0[ ~ E° i,k 1,k ' 0 , we c h o o s e To get rid of E 0 define

of m u t u a l l y

ordered

ismorphism

to the c o m p o n e n t s

6.4 shows

lemma by a

fixed.

or the c o n c l u s i o n

For any t o t a l l y

~,

6 > 0

s a tii s f y i n g

it c o n t a i n s

an o r d e r

s u b s e t of

tively ordered

keK

and

g E A.

and a g a i n by 6.4(1)

s*-topology

Lemma

since

if e i t h e r

is, by 6.4(1), [0,i] c ~ ,

,

n o w f r o m the p r e c e d i n g Acc G

(see 6.3 and 6.4{

(i)

letting for

is o b t a i n e d

Let us k e e p

by

g(E[k ) E ° ,

,

3,£] I~

g,E

Cg,E.

we m a y thus f i n d a p a r t i t i o n

satisfying

(3) and

(4).

(4) w i t h

6= 0

and

of

We may apply

the same w a y as we d i d in 6.3, to o b t a i n

-< c

S t e p D, for

A = G.

Thus

6.1 is p r o v e d .

Suppose

statement

that a discrete

of T h e o r e m

6.1,

amenable

and let

the R o h l i n T h e o r e m

holds

on the f i x e d p o i n t

algebra

H

group

for the a c t i o n (M~) H.

G

acts on

be a n o r m a l

M

subgroup

that the q u o t i e n t

To a v o i d

technical

like in the of G/H

G.

Then

induces

complication

we

55 prove

the result only in the case when

which is what we need in the sequel, same lines to the general algebra

M

is a factor,

case.

the subgroup

but the proof

For simplicity

and we denote

by

~

is a direct extends

along the

we also assume

the canonical

summand, that the

trace

T e on

Me • THEOREM

(Relative

Rohlin

countable amenable groups, dual.

Theorem).

and let

e: G × G ÷ Aut M e

Let

able a n d strongly free. family of subsets of

M

ag =

G

and

Let

e> 0

G

be discrete

be a factor with separable pre-

be a crossed action,

which is semilift-

(Ki)iEI

and let

be an s-paving

G.

Then there exists a p a r t i t i o n such that if

Let

e(g,! ) and

(I)

~ IKi I-I k!

(2)

8g(Ei,k)

(3)

[Ei,k, ag(Ej,£)]

(4)

8(g,h)8 (Z,m)(Ei,k)=

of unity

(Ei,k),

i E I; k E K i

I~k£_1 ( E i , ~ ) - E i , k l T ~ 16e

= Ei,k,

geG, = 0

given countable subset of

Remark. trivial),

ie I,

keK i

g,i,k,j,Z

for all

8(g£,hm)(Ei, k)

g,h,Z,m,i,k.

for all

Proof. products

linear

(i) above

in

(very large)

improves

(!) of 6.1

(if we take

e.

The idea of the proof

of

of any

Me .

The estimate

being

Me

Bg = e(l,g ) then

(Ei, k) can be chosen in the relative commutant in M e

Moreover

in

sets in

is to take Rohlin G ×G,

towers

indexed

by

and then sum after the

coordinate. Step A.

We assume

0 < e < ~,6 and choose

that the theorem holds with (i')

~ iKil-i

(2'

[ i,k

Let

S cc G

18g(Ei,k) - Ei,klT

(Ki × Kj)i,j

34e ½, family

any subset of

S c c G are invariant is a 2e-paving

enough.

family

for

We prove

first

by

16e½

g6A. of subsets

of

It is easy to see that the family

of G × G 2E-paves and

A c e G.

(2) replaced

lak _i (El,z) _Ei,kl T

(Ki)ie ~ be an s-paving

(e,A)-invariant. subsets

[

(i) and

G× G

of the form

This doesn't G ×G,

G,

all of them

(K i x Kj)i, j S × S

of if

imply that

but in the proof of the

Roh!in Theorem we need only the fact that for any invariance

degree,

the given

(and not

family of subsets

of the group s-paved

some subset

56 necessarily all subsets) of the group having that invariance degree. We may thus apply the Rohlin Theorem to obtain a partition of unity (F(i,[),(k,~)) in M~, (5)

with (i,[)e

I × ~ and

(k,k) E K i x Ki' such that

~ IKiI-~I iI ~ _ I~-lS~-1(F(i,i),(~,[))-F(i,l),(k,~)iT i,i k,k,£,£ ~< 5 ×(2e)½ < 8s½

(6)

~ (Z,[)] = 0 [~gSg(F(i,i),(k,k))' F(.3,3),

(7)

~gS~hS~(F(i,l),(k,~)) = ~ghS~[(F(i,[),(k,~)) ~ F(i,l),(k,~) , i E I , Ei, k = [,k

Let us take For any

for all g,g,i,l,k,k,j,[ Z,[

iE I and

keKi,

for all g,g,h,h,i,i,k,k. iEI,

k E K =1.

k,Z e K i we infer

~ki-1 (Ei,z) -Ei, k -- 1_ ( i,£

kr I

-

= -~kZ1 (~ IK[ I-~ ~ (~9/n-i8[/~-I(F(i,i),(m,m)-) - F(i,[),(£,~))) i,m + [ [%1 -I _~_ (@~m-18~r~-I (F(i,l),(m,m))-F(i,[),(k,K)) i

£,m

Hence from (5) [ IKil-1 ~ !~k~-~(Ei,~)-Ei,kl T <~ 2 × 8e ½ = 16 ½ i k,£ and (i') is proved. For g e A we have i,k

E

(Bg(Ei, k) -Ei, k)

=

Z 1 + Bg(Z 2) + l

ZI

=

i,~i" " k,kE(Sg(F(i,i),(k,k)) - F(i,[),(k,gk))

Z2

:

~ ~ F(i,l),{k,~ ) i,i k,[

Z3

=

~ E F( i,i k,m i,[),(k,m)

where

and the sums were done for i ~ A i = ~ i \ g - 1 ~ -i

iE I, i e I,

kE K i,

k e Ki n g - lK i,

and m E gAi.

From the assumed (£,A)-invariance of Ki' we infer IAII < sIKI 1 for all [, hence with the global estimates in Corollary 6.1 corresponding to (4) above, we infer

57 <

and thus for any

2 x 8e ½

16e ½

]E2{ T ~

e + 8e ½ <

9e ½

IEsly

e + 8C ½

9e ½

g e A,

and (2') is proved

of

G

We do this by starting

to

Recall

subsets

(2') with an arbitrarily

with better paving

subsets

Theorem,

from

(K i) towers.

that we are given of

G.

Let

c > 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

<

lBg(ei, k) -Ei,k] Y

i,k

Step B.

<

(K~)jcj

of finite

(Li, j ) i e i , j e j m [K jl

of =

A c c G.

of the Paving

subsets

of

family

(Ki) i 6 I

Let us use Corollary Structure

G,

~-paving

,

j6 J ,

G,

3.3 the

3.4, to obtain and finite

G with [ IKiJ ILi,jl 1

and such that the subsets

(8) Ki,j

= { h E K[ I there are unique (i,k,£) E ~ Z Kv × Lv , with 3 l 1,3 h=kZ and for these ~=i} i e I

satisfy IKi,jl Let k(~Ki×i then

>

(i- 4~) IKil ILi,jl

k:

~ K. × L. • ---> ~ 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

<

tl T

<

2Z l + 2Z 2

A C G, =hat

there (ii),

Step

D,

obtain

(i) and

in the

relative

theorem

so as to m a k e (2).

The w h o l e

commutant

In w h a t

M.

We

dimensional

show

and

in terms

centrally

free,

and

(12)

construction

of any g i v e n

was

for

proved M~

16s + 96@ ½

exists

the

a partition

and

also

same w a y we d i d

6 = 0 and

above

countable

(3) and

could subset

A= G, have of

and

been M~.

thus done

The

(i)

(2)

and

COHOMOLOGY

study

~

the

if

e

the

with

for the

technical

[4, Prop. 1.1.3]

on a v o n

obtain

result

vanishes

on

M

M

and

two-

on the c e n t r a l bounds

on the

if

itself

The

, but

coho-

Neumann

is that

preliminaries. for

valued

the one-

induced

case

The m a i n

2-cohomology

some

G

unitary

free

action

two-dimensional

Let

(A.Connes).

a

is

(Theorem

result

the p r o o f s

i.i).

that

remain

M

be a W*-algebra

with separable

~ 6 ~\~.

Any projection

in

M~

of projections

in

M.

Any partition

has a representing

of unity in projections

sented by a sequence (3)

group

is c e n t r a l l y

vanishes

in the

in

low d i m e n s i o n a l

of an a m e n a b l e

that

VANISHING

too.

PROPOSITION

predual

7:

of the cocycle.

then

L e t us b e g i n

follows valid

we

cohomology

algebra,

solution

7.1

follows

for an a c t i o n

algebra

izing

(ii)

the (9) a b o v e

is proved.

Chapter

mology

in

to

<

(12)

=ion T r i c k in 6.3,

and h e n c e

=ants

in M. Let v be a partial

of partitions isometry in M ~

in

sequence M~

consisting

can be repre-

of unity in projections with

v'v=

e,

vv* = f,

L,J I < 4sIK[13

corresponding

~)

8s + 32@ ½

<

tl T

<

2Z l + 2Z 2

A C G, =hat

there (ii),

Step

D,

obtain

(i) and

in the

relative

theorem

so as to m a k e (2).

The w h o l e

commutant

In w h a t

M.

We

dimensional

show

and

in terms

centrally

free,

and

(12)

construction

of any g i v e n

was

for

proved M~

16s + 96@ ½

exists

the

a partition

and

also

same w a y we d i d

6 = 0 and

above

countable

(3) and

could subset

A= G, have of

and

been M~.

thus done

The

(i)

(2)

and

COHOMOLOGY

study

~

the

if

e

the

with

for the

technical

[4, Prop. 1.1.3]

on a v o n

obtain

result

vanishes

on

M

M

and

two-

on the c e n t r a l bounds

on the

if

itself

The

, but

coho-

Neumann

is that

preliminaries. for

valued

the one-

induced

case

The m a i n

2-cohomology

some

G

unitary

free

action

two-dimensional

Let

(A.Connes).

a

is

(Theorem

result

the p r o o f s

i.i).

that

remain

M

be a W*-algebra

with separable

~ 6 ~\~.

Any projection

in

M~

of projections

in

M.

Any partition

has a representing

of unity in projections

sented by a sequence (3)

group

is c e n t r a l l y

vanishes

in the

in

low d i m e n s i o n a l

of an a m e n a b l e

that

VANISHING

too.

PROPOSITION

predual

7:

of the cocycle.

then

L e t us b e g i n

follows valid

we

cohomology

algebra,

solution

7.1

follows

for an a c t i o n

algebra

izing

(ii)

the (9) a b o v e

is proved.

Chapter

mology

in

to

<

(12)

=ion T r i c k in 6.3,

and h e n c e

=ants

in M. Let v be a partial

of partitions isometry in M ~

in

sequence M~

consisting

can be repre-

of unity in projections with

v'v=

e,

vv* = f,

60 and

let

(4)

of p r o j e c t i o n s

then

there

exists

such

that

v~*v ~ = e w

Any

unitary

in

of u n i t a r i e s

(5)

(fw)w be r e p r e s e n t i n g

(eW)w,

consisting

Any

system

sequence

to infinite

Mw

has

of m a t r i x

of m a t r i x

IxI~ = ~ ( I x I ) .

LEMMA.

result For

any

for

e w ~ fw

(vW) w

sequence

a representing

units

units

sequence

in

in

M~

can

e

and

for

for

f,

all

~,

v

consisting

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

inequalities

of the trace norms. M,

and

~ • B~\~.

This is not necessarily to

by a

M.

deals with several

but its restriction

the following

that

in M.

factor properties

subadditive,

sequences

such

vWv w* = fw.

and

state on the W*-algebra

x • M e,

M

a representing

The rest of this section normal

in

M~

Let

extending

$ be a faithful

We define a norm,

is a trace norm.

for

not being More generally

holds. xl,...,x n • M ~

n

yz,...,y n • Me,

and

We

have

n

I X xiYi] , ~< [ IIxilJ l y i l ,

(6)

i=l

Proof.

i=l

For any

ai,b i • M,

i = 1 .... ,n, consider

the polar decom-

positions bi =

viebiI

,

[ aib i = u HI aibi[ i i

We infer ~(I [ aibiI) i

=

~ ~(u*aivilbil) l

~<

~ l#(Ibi 1½u*aivivbii½) I + .[ [laitIIIbiH½ II[~,Ibi1½]11 1 1

~<

[ IlaiH~(ebil) + [ IIaieIIfbilI½ II[~,Ibii½]iI 1 i

If we apply this to representing This result of unity HxIl~ =

is very useful

Yl,''-,Yn

(½ ~ ( x * x + x x * ) )

in

M e.

sequences

for estimates

Further

means of the inequalities

where

(7)

IIxll ~#

(8)

txi~~< (21eI~)½ iexn~ ~< 2½ IaxJI~

e

+ Ix*i~)[txli)½

is the left support

of

x.

xi,Y i we obtain concerning

(6).

partitions

on we work with the norms

½ , x 6 M e, connected

~< (½11xi

for

to the preceding

ones by

61 Although following

l]'It#

is not unitarily

invariant,

it satisfies

the

inequality:

(9) lluv-lll~~ 2½(IIu-lil~+ IIv-iII#~) for any unitaries

u,v e M m.

This is immediate

H u v - i11#2 + IIvu - IH# 2

=

211u-v*11 #2

from the identity =

4 -uv-vu-u*v*

-v'u*

together with the inequality

llu-v*ll#~ < llu-llI# t + Ilv-llI~ This yields inductively

estimates

as well; we shall use for instance (i0)

7.2

llulu2u3u 4 -llI~

In what follows

able,

and

<~ 2

Recall that a l-cocycle

of

a

au

(Ug) by

v 6 ~(M)

Let

G

of

G

on

Me,

(see 5.6)

there exists a f a i t h f u l normal state

aIZ(M).

coboundary.

(Vg), then

commutes with N

in

N

u1= ! and

,

g,h e G . (Ug) with

g 6 G if

(Ug) ~ i.

and let

let

M

(ag) be an action Assume

that

t

~IZ(M)

is

on

M

such that

(Vg) c M e

for

(~g) is a

is any given countable subset of

v = 9w

with

be a

and semiliftable.

Then any cocycle

M o r e o v e r if

1

be a discrete amenable group,

strongly free

p r e s e r v e d by

=

is the cocycle

) , v

with

i.e.

yon Neumann algebra with separable predual, of

always assumed count-

u: G ÷ ~(M)

Ugag(Uh)Ug h

VUgag(V

(Ug) the coboundary

PROPOSITION.

is a map

is trivial, =

=

group,

algebra with separable predual.

(~U)g,h

Ug and we call

Neumann

for

such that its coboundary

of unitaries

4 [ llui- III# i=l

G will be a discrete

M will be a v o n

The perturbation

for longer products

the fact that for ul,uz,us,u4 e ~(M~),

w

in the relative

M e,

which

commutant of

M~.

Proof.

To give the idea of the proof

contain a copy of the left regular action commuting with (Eg)geG

in

could define

suppose

first that

e would

Ad I: G + Aut(£~(G)),

(Vg), i.e. there w o u l d exist a p a r t i t i o n of unity

{vglg EG}' ~ M e such that ~g(E h) = Eg h, g,h e G. Then we * and thus we would get a unitary satisfying w = ~ VgEg g

82

W* g (w) : k,h[Vk k g (v ) gh This is a form of the Shapiro

Vgh g
lemma in eohomologieal

In our actual framework,

algebra.

the Rohlin theorem is an approximate

of the left regular action containment,

and analogous

approximate

in

vanishing

on the cohomology

Selection Trick we eventually of

G

be given.

theorem Me

Let

Let

6.1 and so we can find a partition

We define

[~g(Ei,k),

v h]

Let

=

Me .

k , I E Ki,

of unity

m EKj,

F G

of the Rohlin

(Ei,k)iEi, k e K i

in

g , h C G we have 5~ ½

agh(Ei,k)

Ej, m]

w

fixed.

=

[~g(Ei,k),

the unitary

in

family of subsets of

[ IKi I-~ [ l ~ k ~ - ~ ( E i , ~ ) - E i , k l , ~ i k,i ~g~h(Ei,k)

By means of the Index

We are under the hypothesis

i,j E I,

form

formulae give an

0 < e < 1 and let a finite subset

(Ki)ie I be an e-paving

(e,F) invariant.

such that for any

Me .

obtain exact vanishing

Let us begin the proof. which are

: Vg

= =

0

0

w E M w by i e I'

! [k Vk* Ei,k

Vg = WVg~g(W*)

k e Kl

be the perturbed cocycle.

Let us keep

gE F

We infer

~g-1 = =

[

[ (V{Vg~g(V~)-l)~i,k~g(E ,9~)

i,j k,~ ZI +

3

Z2 +

Z~

where

i,j e I, k 6 Ki, £ E Kj and in Z I we sum for i = j, Z e K i n g IKi, k = g£; in Z2 we sum for i = j, £ e Kj n g -IK.3' and in

Z~

Z e Kj\ g-iK.. 3 From the cocycle identity we get

k ~ g

for

ZI=0.

Trace norm inequalities

yield IZ3I~

<

Since we have assumed 6.1(5),

2 E IEj 1 j,~ ,~ ~

IKi \ g -i KiI<

j E I,

elKil

we infer

(1)

Iz31~

2(e + 5e ½)

~

12e ½

Z E Kj\ g-IK. 3

from the global estimates

63 On the other hand,

tsl 3 where

i,j E I;

2 ~

k E Ki;

Z

E

X tEi,k~g(Ej

,q)lop

i,j

k,£

K. N

g-iK. and either i # j ]

3

or

k~g~.

We obtain (2)

IZzld? ~< 2

;.

(i-Ej,g£)C~g(Ej

~) I~b

j,~

=

2

~

( i - E j , g Z ) (Ej,g Z - e g ( E j , z ) ) I ¢

j,£ ~< 2 for

j 6 I and

[

[Ej,g Z -e~g(Ej,£)t¢

j,~

£ E K. m g 3

Iz~t~

--1 K..

The estimates

6.1(4)

yield

]

~< 2 . 1 o ~ ½

Summing up, we infer for

=

20~ ½

g C F, 32S ½

g = nI and F = F n C C G, where F n ~ G; n e IN. We n a p e r t u r b a t i o n w (n) such that the c o r r e s p o n d i n g

Let us now take obtain for each perturbed

,~(n) ) satisfy [Vg

cocycles

nl~i m

I Vg ~(n) - if

The Index Selection Trick, of Lemma 6.3, cocycle

for any =

applied

countable

Let

subset of

w

such that the perturbed

G

M

u: G × G + ~(M)

of

G

on

M

such that

is normalized

by

assertion of

Recall

group,

and

M

avon

Neumann

that a cocycle crossed action ~: G ÷ Aut M

and

~i= !, =

Ad Ug,heg h

Ul,g = U g , l = i,

Ug,hUgh, k of

countable

is a pair of maps

dg~h

A perturbation

commutant of a

and thus obtain the supplementary

again be a discrete

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

in the relative

The proof is finished.

algebra with separable predual.

u

the same was as in the proof

Step D, yields a unitary

the proposition.

and

0

is trivial.

We could do the whole proof above

7.3

gE G

=

ge G

eg(Uh,k)Ug,hk

((eg), (Ug,h))

is a family

g,h E G and satisfies g,h,k E G (Vg) of unitaries

in

M,

64

g E G, with

v1=l;

the corresponding

perturbed cocycle crossed action

((&g), (Ug,h)) is given by ~g

= Ad Vgag

Ug,h

= Vgeg(Vh)Ug,hVgh

We omit the simple verification indeed.

We say that

that this is a cocycle crossed action

(Ug,h) is the coboundary of

(Vg) if

Ug,h ~ i.

A simple but very useful remark is that the effect of two consecutive perturbations

of

(~,u), first with

same as the one of perturbation with to (~,u) and

u ~ u ~ i,

then

v

v and then with

vv.

Also, if

([g,h) such that

is the (a,u)

is an a-cocycle.

We next show that we can perturb any cocycle to

v,

v perturbs

(Ug,h) is approximately

(Ug,h) with some

periodic in

(~g)

h with respect

to the plaques of the Paving Structure, i.e. for any p E ~, according K p+I ~ Oi U£ K ~ , to the approximate decomposition of the plaques, j -i e Ip • £ c L 1,3 P . , we have -Ug,h £ = Ug,h l for most h e K~ and £ E L~l,j . Moreover, ~g-i is kept under control. This way if U g , h - i is small We use the for h e V K ni, then u g,h - 1 is small for most h eG. l notation in 3.4 for the Paving Structure. LEMMA

(Almost Periodization

crossed action of the amenable Assume

that a choice

notations (~g) of

Lemma).

group

in 3.4 for its elements.

is made for

G

M.

and use the

Then there exists a p e r t u r b a t i o n

((ag), (Ug,h)) such that the p e r t u r b e d cocycle crossed action

(i) I{h e K93 +I I ~ g , h # U-g , k

nEN,

((~g), (Ug,h)) be a cocycle

on the yon Neumann algebra

of a Paving Structure

((ag), (~g,h)) satisfies for any

Moreover,

G

Let

(~g) and

n k i, for

j E In+ 1 and

kn(h)=(k,Z),

g E Gn

n £ e ui Li'j}l

(~g,h) have the following property.

~ > 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~ <

8n6

IJUg,h - llI~ <

8n~

g @ Gn+ 1 g @ Gn;

h,gh E Gn+ 1

= (U =n+l.. Ln n e IN and let Hn+ 1 1,3~i,j)%(Gn+l U i,jU 1,j)" • This set is contained in u -n+l K. ., the subset of U K.n+l whlch behaves i,j 1,] j 3 well with respect to the approximate decomposition in plaques

Proof.

Let

85 K n+l ~_ u KnL n i i i,j Let definition unique

(see 3.4)

g • Hn+ 1 and let i • I n , j • In+ 1 with g • ~n+l. i,]. . From the of ~n+l -'i,j, i and j are uniquely determined and there exist

(k,Z) • Knx L n l 1,j

and define inductively

(2)

vn

=

with

1

Let

((eg), (Ug,h)) = ((~g), (Ug,h))

the perturbations l un k,~

g

if

1

g E Hn+ 1 and

for the other

g = k£

as above

g e G\Hn+ 1

, n+l, . n+l. (%eg ), %Ug,hJ) be the cocycle crossed action obtained by

and let

n ( ( e ) , (Ug,h)) with

perturbing

n = 1,2,3, . . . . the level bations, for each

n,

n Ug,h

We shall, show that that this property

for

is approximately

periodic

at

is not destroyed by the next perturvg n

is stationary

g E G.

Step A.

We show that if

g = ki

with

g E Hn+l,

and if

(k,£) e Kgl × Li,n J , then

k, ~ e K~l u L~1,j ~ GiHn+I

and so

v~=v~=l;

i 6 In,

u kn+l, £= 1 .

j e In+ 1 are Indeed,

we have

n i v ~ = Uk,

since we have

n n. n. n n* u n v n* i. u~+~, = V k e k ~ V z ) U k , l V k = k,Z g =

we infer

Step B. h,gh 6 Hn+ 1 then since u

n (Vg) ; do this successively

and that the product of the perturbations

such that

Since

g =kZ.

We now prove the approximate and gh=

h =k£

with

(gk)£,

k e K9l'

periodicity.

Z E L ni,j

If

g E Gn,

andn+lif moreoVern+l gk • K~I,

from the Step A we have

Uk, ~ = Ugk, Z = i.

is a cocycle, n+l Ug,h

hence the approximate

=

n+l Ug,k Z

=

n + l , n+l,, n+l n+l g tUk,z) Ug,kUgk,~

periodicity

relation holds

for

=

n+l Ug,k

(g,h).

For given

j E I +i and g E G n we evaluate the cardinality of the subset ~n+l .n+l n 3 ~j , consisting of those h for which (g,h) does not satisfy the conditions

above.

We have

• A n+l C {l,g-1}(K n+l \Hn+ I) 3 --

u

<, K nj+ l

+l, \ g - 1 . n~j

i We have shown in 3.4 that nn+l INn +l \ u Ki,jl 3 i

-n+l Ki, j

>

) 1,3

(I -an) IK~I ILgl,3'l' hence

< e n [ IK~I i

In 3.5 we have assumed that for each

IL~,jl

=

j E In+ 1

n+l s n Kj

of

88 IGn+ I

u

Ln n+l i,Jl < sn Kj

u

i

Kn+l\ J Hn+ll

,

Kn+l 1 2Sn! 3

<

From the left invariance properties

of

K~, K9 +I 3

to

g • G n -c G n+l

we have Kn+l \ -1.n+l 3 g ~9 i~ ~K n

Kn+l ~n+ll 3 KO

so that i

iKni\ g- I Knl IL ni'Jl < sn i[ IKnl ILnl'j I = En Kjn+l

and finally A~] +I

<

(2.2e n + Sn+l + £n) iKn+l ] I <

Since one

n

K93 +I ~ Gn+2 ~ Gn+l "'"' for any for which Vgn ~ i; hence the product nn-i ... VgVg

Vg

g• G

4enIK n+l j 1

there is at most

. Vg

( U K ~J +I) -C Gn+ 2 , is well defined. Again by the assumptions of 2.5, Gna--nd and so if g • G n and h • u K~ +I then g,h,gh • Gn+ 2 ] 3 ' un+P n+l for any p > i. g,h = Ug,h Since (([g), (~g,h)), which is the perturbed of ((~g), (Ug,h)) by (~g), is also equal to the pointwise the conclusion Step C.

We prove the estimates.

assumed in 3.5 that

L ~ Gn+ I.

((e~), (US,h)) when

n ÷ ~,

Let

L =

U u L~ p
=

{ (g,h) • G n × Gn+ 1 Igh E Gn+l}

B

=

{ (g,h) • Gn+l × L [ g h e G n + l }

for

p = 1,2,...,n+i

(3,p)

flup -ill# g,h

(4,p)

IIVg p- li,~ ~< 8P-16

From the definition of (3,1) is true, since are true for some p,

Vgp,

We have

Let us define

A

We prove inductively

(g,h) • A u B.

limit of

(i) is proved.

<

8P-I~

that (g,h) • A u B

g E Gn+ I

(4,p) follows from (3,p).

By the hypothesis,

A o B c_ Gn+l × Gn+l . Suppose that (3,p) and l ~ < p < n , and let us prove (3,p+l). Let

(4,p)

67 Suppose

first that

~h = i. p+l Ug,h

and since

g,gh • Gn+ I,

the inequality

vp u p p* g g,h Vgh

=

we may use

(3,p) and

(4,p)

to conclude

with

h • Gn+ 1.

From

7.1(10)

p+l

~< 6.8P-I~

Now let

Then

v~ ~ 1.

the definition

of

Since

v~

From the assumptions

+ Hu p

<

8P~

(g,h) • A o B, we have

we infer

h • ( U3• Kjp+l )\ (Gp+ 1 U

I1~

p < n.

Since

U LPl , j) , we have

i,j

h ~ q

3.5, (U K~+I)3 n ( U Lq j) j q=p+l l,

=

Hence h ~ L and so (g,h) • A. There exist i • I n , j • In+l, • L~ such that h = kZ. Then the cocycle identity yields m,j u p+l g,h

=

v p ~P(v_)uP P _vP*~ g g n g,n gn

=

vPu p up P* g g,k g k , ~ V g h

We again use assumptions g,gh e Gn+ 1 since we infer we

have

(g,h) 6 A.

(g,k) • A. (gk,i) • B.

=

vP~P(uP ~ P* g g' k,£ )u ,k£ Vgh

3.5 on the Paving Since

Structure.

k e KPl --C Gn+ 1 and

As ~ • LP C L, gk • Gn+ 1 and l, 3 -The induction hypothesis yields

liugp+l,h- iII~ <

k • K~,

We have

gk e GnKPl --C Gn+ 1 gk£ = g h • G n +

1

2(HvP-IIi~ + H v pgh-lil$# + IfuP,k-lll$# + Hu pgk,£ -llJ~)

<

2 × 4 × 8P-16

=

8P6

and thus we have finished

the proof of

hold for all

We have shown in Step B that for

i < p < n+l.

(3,p+l).

Hence

(3,p)

and

(4,p)

g E Gn+ 1

we have ~ _ = v~ for some p < n+l, and for g,h,gh e G n we have g,h-- un+l g,h"~ The estimates in the conclusion of the lemma are thus proved. Remark.

If li'iI~

@

is a trace on

M,

we may work

and use a trace norm equality

instead of prove the following T

assertion.

in Step C with

instead

of 7.1(10)

I'I~ to

68 If for some

~ > 0 and

n > i,

lUg,h- ii¢ ~<

g,h,gh E Gn+ 1

IVg-ll~ < 4n~

g E Gn+ 1

lUg,h-lI~ ~< 4n6

g 6 Gn ,

then

7.4

We now prove a vanishing

by means of the Almost

result

Periodization

for M~-valued

h,gh E Gn+ 1

2-cohomology;

Lemma we are able to obtain bounds

for the solution. PROPOSITION.

Let

G

be a discrete

countable group,

yon Neumann algebra with separable predual and let cocycle crossed action of Let by

~

G

(Ug,h)

Then

u=

v

~n > 0

and

such that Given

~ En_ 2

for

g E Gn ,

~ 18~n_ 2

for

g E Gn_ 2

GnCC G

(Ug,h)

If m o r e o v e r

(Vg) C N' n M e

c N' A M w

the identity

for some countable

its product

lemma,

is fixed

n>2,

if

h,gh C Gn+ 1

lemma - Shapiro

and provides

LEMMA.

3.4.

N C Me, we may take

perturbing

converges

(Ug,h) with a

such that at the limit

cocycle.

The lemma that follows the Rohlin

be a

as well.

of perturbations;

we obtain

tlZ(M)

n E N,

were defined in the Paving Structure

The proof will be done by successively sequence

be a

with

IVg-II~ where

M,

is a coboundary.

lUg,h-iI~

M

semiliftable and strongly free.

be a f a i t h f u l normal state on

el Z(M).

then

M~,

on

let

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

displays

lemma,

the inductive

the result of an application

followed

by the Approximate

step in the proof of the proposition.

By the conditions of the proposition,

((ag), (Ug,h))

that the cocycle crossed action

of

Periodicity

let n~>2 and suppose

satisfies

the f o l l o w i n g

condition: For any such that

g E Gn_ 2 and

IAn(g) I ~< 7en_21K3I

h,gh E W (K~ \ A~(g)) 3

(i)

Ajn (g) c K n3 '

there exists a set

and f o r

any

g E

Gn_ 2

and

we have

lUg,h-ll¢ < ~n-2"

Then for each with

j c In

g E Gn_ 1

and

j E In+ 1

•

n + l

there is a set ~j

_ n+l (g) c ~j ,

IA3+l(g) I ~< 7en_iIK~+l I and there exists a p e r t u r b a t i o n

(Vg) of

89 ((~g), (Ug,h)) such that the perturbed cocycle crossed action ((~g), (Ug,h)) satisfies (2)

I~g,h-iI~

~ en_ 1

for g • Gn_l; h,gh • G n and also for Moreover the perturbation satisfies (3)

IVg - llt < 18£n_ 2,

Proof. Shapiro's

Step A.

lemma,

g e Gn_l,

h,gh • u (Kn+l\An+l(g)). J

g • Gn_ 2 .

We again use the Rohlin theorem and a form of

the same way as for the l-cohomology,

to obtain the

approximate vanishing of the cohomology. Recall from the Paving Structure that (K ni)i • I n was an en-paving family of sets and i ng was the approximate left translation with g E G on

u K~ •

1

1

°

We perturb

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

(Vg) to ((~g), (Ug,h)) such that

(4)

IUg,h - iI~ ( 32E ½n

g,g,gh • Gn ;

(5)

Ivg- iI~ < 16en_ 2

g 6 Gn_ 2 .

According to the Rohlin theorem 6.1 let us choose a partition of unity

(El, k), i • In , k • Kgi such that

[

i

)

k,£

~g (eh (Ei,k)

=

egh (Ei, k)

[~g(Ei,k) , Ej,Z]

=

0

[eg(Ei,k) , Ug,h]

=

0

Let us define the unitary for Vg where

iE In,

gh • G n

=

for all

i,j,k,l,h,g

.

g • G,

~ Ug,k Ei,h i,k

k e Knl and

h = £ g (n ) .k

Let us keep

g,h E G n with

fixed, and for i • I n let ~n = l

Since

kl, <

{k•Knlhk•K

K n is (en,G n) invariant, 1 the definition of Vg and Ug,h

n ghk • K n} i' IKnI i> (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,

<

~

32s~

(4).

Let us evaluate now the perturbation.

Let

g 6 Gn_2.

We

decompose Vg- 1

=

~ * - l)Ei ,h v,k(Ug,k

=

~i + ~ 2

and where j E In, k E K?, h = ljn (k) . In El we sum for k e Kn\An(g) 3 ] in ~2 for k 6 A~(g). We infer from the hypothesis (i) of the lemma

IZll,

~<

en_ 2

On the other hand, IE21 , <

2

~ IEj,hl, j,h

71 where j • In, by hypothesis,

h E in(~(g)). Since for each j, lz~(A~(g))I < 7en_21K~I h 3 J the estimates 6.1(5) of the Rohlin theorem yield [Z21 ~ ~

2(7£n_ 2 + 5s~)

and thus using assumptions

3.5 on

<

15En_ 2

(Sn) n we obtain

IVg- !1% < 15Sn -- 2 + lOs½n

<

16~n - 2

g • Gn_ 2 -

'

Step B. A problem is that we have obtained in (4) l U g , h - l l % small for h E Gn, but in the statement of the lemma we need it small for h in a larger set, for induction reasons. The gap is filled by the Almost Periodization Lemma 7.3. Let us apply it to ((~g), (Ug,h)), to obtain (Vg) perturbing it to ((~g), (Ug,h)). Using the estimates in Remark 7.3, we infer from (4) and

(5) in Step A lUg,h-i! ~ ~< 4n-l.32E~ ~< 4 n-l.32s~

l~g - i[~

<

en_l ,

~< en_l

,

geGn_l, g e

h,gh e G n ;

Gn

We now use the almost periodicity. Let g e Gn_ 1 and for kn(j,h) = (jz,hl,kl) and kn-l(jl,hl) = (j2,h2,k2), where ~n

~

Kin x L n is the approximate 1,j Let in the ip,a3ving Structure.

Kn+l __> ~

h • K n+l 3

let

decomposition

3

,

] defined

Aj! (g)

=

A'~(g)3 = Since the cocycle

{h • K n+l I u ~ u } ] g,h g,h z

{h • Kn+lj I -Ug,hI 7~ Ug,h2}

(Ug,h)

satisfies

the almost periodicity

property

7.3(1), we infer !

IAj(g) l

l~':(g) l 3 So if we take

6enlKO +I 3 ~ (iLn n 1,j I 6en-llKil) •

=

!

.n+l (g) = A'. ~j 3 (g) + Aj" (g) , then

6~n -- 1

_ 3 Ug,h z, and

h 2 6 K3-1

C G n.

lUg,h - iI~ ~< en_l and statement

3

1

.n+l (g) < ~j

. n+l Kn+l I . (6Sn-i + 6an) i~J ] ~< 7Sn-i 3 For h e Kn+l\ A3+l(g) we have, with the notation _ Ug,h

Kn+l

above

Hence ,

g E Gn, 1 ,

(2) in the lemma is proved.

h E K n+13 \ 4J'n+l(g)

72

Let us now prove the perturbed of infer

(3).

Let

Vg = ~gVg,

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

IVg-ll~ <

(Vg).

l~g-ll# + [Vg-ll%

such that

(([g), (Ug,h))

is

From previous estimates we

<

16an_2 + a n _ 1 <

17an_ 2 , g • Gn_ 2

and the proof of the lemma is finished.

7.5

Let us now prove Proposition

7.4, by applying successively

lemma for n,n+l,n+2, . . . . n Let ( ( ~ ) , (Ug,h)) = ((eg), (Ug,h)) and for p > n , ( ( ~ ) , (U~,h)) which satisfies

the

preceding

(1,p) for

suppose given

lu~,h-II~ < ~p-2

g e Gp_2,

h,gh • U (KP\AP(g)), j

with

AP(g) c K p and 3 3

IA (g) I <- 6sp_2JK (g)j. g•Gp_2, J Since ~ Kni C_ Gn+ I, (l,n) is true. We use 7.5 to perturb ((~P) , (uP,h)) with (vp) to ((~P+I) , (Ug,h)), p+l. satisfying (l,p+l) and + l - 11 % ~< ep_ 2 1u pg,h

(2,p)

IvP-II~ Let

~

for

17ep_ 2

" (p) = v p'p-I Vg gVg ... vPg for

g • Gp_I,

for

p~>n.

If

h,gh • Gp

g 6 Gp_ 2 g e Gp_2 , then

Iv(p)g-v(P-1)gI¢ = I(vPg_l)v(gp-1)]¢ = Ivpg-ll¢ < Hence for

m > p i> n-i

and

17ep_ 2 .

g 6 Gp_ 2 ,

m E 17Sk- 2 ~< 18ep- 1 IVg(m) _v(p) g I¢ < k=p+l where

v g(n-l) = i and the assumptions on (an) n have been used. Thus the sequence v (p) converges *-strongly to a unitary Vg e M e for any

g e D Gp_ 2 = G; P

moreover IVg

Since in view of 7.4.

((eP)

•

-

1 I¢ <

18an_2

,

g e Gn_ 2 .

(uP,h)) is the perturbed of

(2,p) we infer

u = ~v.

((eg)

,

(Ug,h)) by (v _ g(p-l))

This ends the proof of Proposition

'

73 7.6

The same techniques

vanishing

with bounds

ing of the 2-cohomology is due to the absence Let us recall Paving

Structure

THEOREM.

aIZ(M).

of

state

G

on

the

Some additional

complication

i.i,

in a form in which the

amenable

on M,

M

group,

which

such

and

let ((ag),(Ug,h))

is centrally

~IZ(M)

that

free.

Let

is kept fixed

by

is a coboundary.

given

nk 2

~ = {Ad W ~ I w E W } ,

HVg- ii[#~

and a finite

for estimates

W c ~(M),

set

g @ Gn_2,

if we have

h,gh e Gn+l,

~ •

u = 9 v with

then

~< 2e ½n-2

for

In the proof of this theorem, 7.1(6)

yielded

also give the vanish-

in the estimates.

ilUg,h- IN~ < En_ 2 , where

M.

sections

M~,

M.

explicitly

action

on

Theorem

be a discrete

normal

Moreover,

on

of a trace on

G

(Ug,h)

Then

with bounds

appears

crossed

t be a f a i t h f u l

in the preceding

for convenience

Let

be a cocycle

which

of the 2-cohomology

in connection

g C Gn_ 2

we use

I'I#

~ •

and the inequality

with the partitions

of unity in M~ The only II'II~ for the rest. H

yielded

by the Rohlin

problem

appears

lemma,

in connection

and the norm

with the estimates

of infinite

products

invariant.

We use the inequality

(I) which

of perturbations,

llxviJ~ ~< 2½(HxH~

is immediate

what allows theorem

7.7

let

llxll#2 v%

=

½~ (x*x + xx* + v*x*xv + v*xx*v) family of norms

lemma depend

only on

at each step,

~IZ(M)

step of the proof of Theorem is an analogue

of Lemma

Let

G,M,

((O,g),(Ug,h), %

be as in the

be finite

sets

of normal

=

and

in the Rohlin (Ad v~) IZ(M).

7.6 is provided

which

n

v e ~{(M)

llxl]~2 +

lemma,

C ~n+l

,

=

to use an ever larger

The inductive

LEMMA. and

x e M

us to do so is the fact that the estimates

and the Shapiro

following

II'H~ is not unitarily

,

+ llxii#dv ~)

the convergence

from the identity

#2 llxvIi~2 + llx*vlJ@

We thus have

since

giving

by the

7.4.

states

theorem. on

M,

Let n>/2

which

on

74 Z(M) coincide with $1Z(M). llUg,h-iII~ < gn-2

Suppose that for

g C Gn_2;

g,gh E U (K~.\An(g)) ; ~$E ~n j

J

n

where the sets An(g) C Kj, g 6 Gn_2; j e In satisfy IA~(g)I ~< 7en_21K3I. Then there exists a perturbation (Vg) of ((ag), (Ug,h)) such that

HVg

iII~ < 9 En_ ½ 2

-

and the perturbed cocycle

g E Gn_2,

((ag), (~g,h)) satisfies

llUg-llI~ ~< En_ l for

g E Gn_l;

t ~ $n+l

h,gh e G n and also for }n+l

~ E ~n

=

g E Gn_l;

{Ad Vg~ig ~ Gn_l,

j e In+l, g 6 Gn_l, the sets IAn+l I ] (g) ~< 7Sn- 1 Kn+lj I-

and for

h,gh e u. (K~n+l)\An+l'. j tg)'), 3 9 e Yn+l}

An+l' j tg)" c K n+l j

satisfy

Proof. The proof will parallel the one of Lemma 7.4. Step A. Let ((Bg), (Ug,h)) , where Bg = (ag)~ e Aut M e be the cocycle crossed action induced by ((ag), (Ug,h)) on M e. Since Ad Ug,hlM ~ = i d , (BgIM~) is an action, which by Lemma 5.6 is strongly free. The Rohlin theorem yields a partition of the unity (Ei,k), n i EIn, k ~ K i, in M e such that i

IK n

i]-

i

~ IBk~-I (Ei ~) -Ei'kI$ k,~

[~g(Ei,k), Ej, Z] = We define the perturbation

vg

=

0

for all

< 59½ i,j,k,Z,g

(Vg) c M e by

X u* Ei i,k g,k ,h

where

ie In, k e K ~1 and h = In(k). g Let ((~g), (Ug,h)) be the cocycle crossed action obtained by perturbing ((~g),(Ug,h)) with (Vg). For further use we need estimates of A d V k ( U g , h - I ) . The estimates of Ug,h- 1 in 7.4(2) were based merely on estimates of the Rohlin partition (Ei, k) and did not involve any estimates on the cocycle (Ug,h) which was perturbed. Since in our present context any Vk commutes with any Ei, h, the same estimates work, letting the inequality 7.1(6) replace the trace norm inequality. In this way we infer IAd Vk(Ug,h- I)I~

<

32e~

75 and similarly ~*(US ,h - I) I~ < 32~½n IAd Vk for

k E G,

g,h,gh E Gn,

~ E ~n+l'

where we have also used the fact

that for ~ E ~n+l' 9 ~ = ~ , since }iZ(M) = @IZ(m). Via the inequality 7.1(7) this easily yields (I)

~* ~ -i),,~ ~ (½(32E½n + 3 2 ~ ) . 2 ) ½ IIAd Vk(Ug,h

for k E G, g,h,gh e Gn, as in 7.4(5), we have

~ E ~n+l"

[Vg - llb ~ 16gn_ 2

=

8E~

On the other hand,

,

in the same way

Iv$ - 1 6 4 < 16an_ 2

and hence (2) for

tlVg - llJ~ < (½(16Sn_ 2 + 16an_ 2).2) ½

g @ Gn_2, Step B.

6e~_ 2

~ E ~nWe apply the Almost Periodization

and perturb it with 7.3 yield from

~

(Vg) to get

Lemma to ((~g),(Ug,h)) The estimates in Lemma

((~g), (Ug,h))-

(i) above

(3)

# ~ 8 n-I " 8e n¼ = 8n Sn¼ [IAd V~*(Vg-l)li k

(4)

IIAd Vk(Ug,h-l)II~

kEG,

,

gEGn,

~ e~n+ 1

and ~< 8n-I "8e¼n = 8n e¼n '

kEG,

g E G n_l,

h,gh e Gn,

~ E ~n+l "

The sets An+l(g) are defined the same way as in 7.4 and, as there, 3 because of the almost periodicity of Ug,h' inequality (4) above holds for g e Gn_l; g,gh e u3 (K(n+I)\A n+l j j (g)) as well. Let

Vg = VgVg,

g E G.

We infer from

(i) and

(2) above, by means

of 7.1(9),

jjVg lli~ ~ 2~(H~g 111~+ H~g-IN~#) ~< 2%'6 ½ + 8n e~) an_ 2 for

g e Gn_ 2 and

~ 6 #n _C ~n+l'

HAd

v*k ( -U g , h - ! ) H ~

9 ½ an-2

where again the assumptions

have been used. On the other hand, the estimates (5)

<

(i) and

=

11(V~kV k)

-<

2(2

on

(sn)

(2) yield, with 7.1(10),

(Ad V*(~g k ,h ) )(V*~ k k V k ) -llI~ + HAd

~< 2(2 • 8 n en + 8 n e n¼) <

en-i

78 for

k c Gn_2, 9 @ ~n+l and either h, gh E ~ .(Kn+l. j \ £n+l j (g)). Let M,

by

(Vg)m be representing

g EGn_I,

h,gh ~ G n or

sequences for Vg, with

v~

g E Gn_l,

unitaries

in

v~ : I, v E ~ . Let (([g), (Ug,h)) be the perturbed of ((ag), (Ug,h)) (Vg). Then (U~,h)~ represents Ug,h' and so we may choose ~ e

such that if

Vg =v~,

[ g = ~g

= u~g,h , then Ug,h

and

IIVg -ill

< 9~i_ 2

g E Gn_ 1 ,

~ E ~n

and also HAd Vk(Ug,h - l)II where either If ~ e ~n+l'

g c Gn_ I, then

h,gh c G n

~ = Ad Vk~ II[g, h

as before.

or

for some

- iII~#

< On_ 1 ,

k E Gn_ 1 , ~ E Tn+l

,Kn+l \ .n+l h,gh E 3~ ~ j Aj (g)). k E Gn_ 1 and ~ E %*n+l' and so

g c Gn_l,

# iIAd v~(u K* - g,h - I ) H ~

:

~<

~n-i

The lemma is proved.

for

g,h

7.8

Let us now prove Theorem 7.6.

cocycle with perturbations

We successively perturb the given

given by Lenuma 7.8 for

n, n+l,n+2,...,

as in

n

the proof of 7.5. Let ((e) , (Ug,h)) = ((~g) , (Ug,h)) and ~n = ~" Suppose for p~> n that we are given for k = n,...,p a centrally free k k cocycle crossed action ((~g), (Ug,h)), a finite set #k of faithful normal states on M and a perturbation (vk) of ( ( k ) , (uk h )) taking it into

• k+l,

(~g

(l,p)

, k+l,

;,tUg,h)),

k=n,...,p-l,

11ug,h p -111~ ~< Sp-2

for

such that ~ E ~p,

g e Gp_2; h,gh E y (KP\AP(g)) 3

AP

6 p_21K l

c_ K pJ and where for g e Gp_2, j @ I~ , we have £P(g) J Kn I j (g)I ~< For p =n, (l,n) holds by hypothesis since u • (n-l) J ] _C Gn+ I. We let (Vg ) =- 1 and for n ~ < k < p we take v (k) = V gk V gk-i . . . V gn g We apply the previous lemma to ((~gP), (uP,h)) with n replaced by p,

.

~p defined inductively above, and ~-±i = {Ad v(P-l)~I g EG~_2, ~ 6 }n}" iJ T ~ We obtain a perturbation (vp) such that if ((~P i), (uP+l)) denotes the P~ (uP,h)) perturbed by (vP), if cocycle crossed action ((~g), then Vg( P ) = v Pgv g(p-I) and if % + 1 = {Ad vP~i~ @ ~p+l' g E G p - l } ' satisfies the condition (2,p)

(i, p+l)

p + l - iIl~ <~ Sp-I IIUg,h

p+l,

(Ug,h)

above, and also for

g e G p-l'

h,gh E Gp and

~ 6 %p+l

77

and ,,vp -i,,~ ~< 9~p_ 2 Using the inequality

,

g • Gp_ 2 ,

7.6(1), we infer for

g • Gp_ 2 and

g where

~ • ~p ~ • %n'

g

~g = Ad v(P-l)9 . g ~g

~g

But if p >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.

77

and ,,vp -i,,~ ~< 9~p_ 2 Using the inequality

,

g • Gp_ 2 ,

7.6(1), we infer for

g • Gp_ 2 and

g where

~ • ~p ~ • %n'

g

~g = Ad v(P-l)9 . g ~g

~g

But if p >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=

<

31e-f ;

Let

f e = wp

fl = ww*

lw-felT

+ +

trace

isometry

T. v 6 M

be the p o l a r

=

Ife-fl~ le-fIT

decomposition

We h a v e

< f.

[w(e-o)l~ le-plT

=

+ If(e-f)

l~

+ E

le(e-f)elT

=

~< le-f[x =

I w-f IT ~< 2g . Since

M

u*u = e-e I As

a normal

a partial

efe < e,

le-pl~ < le-o2f~ hence

with

exists

<

le-pl~ since

W*-algebra there

and

Iv-flt

of

L1-norm.

is finite,

and

g2~< el ~< e,

f-fl ~

uu* = f-fl ' and

e-el"

Let us c h o o s e

let v = u+w.

Then

uEM

v'v=

with

e

and

vv* = f.

we have lul T

lu(e-el) ]T

<

le-ell T

~< l e - p21T

hence

Lv-fi~ <-lw-fl~+lul~ The

8.2 of

lemma

Let M,

is p r o v e d .

M

with

be a v o n

Neumann

normalized

x®y following

of A. Connes,

and

algebra

trace

P e ' ~ M the f a i t h f u l n o r m a l w h i c h e x t e n d s the m a p

The

-< 3~

T.

conditional

--> T ( x ) y result

If

,

and

let

be a f i n i t e

expectation

x E e

is an i m m e d i a t e

is y i e l d e d

e

M = e ® (e' ~ M),

by e s s e n t i a l l y

,

M

onto

y e e' A M

extension the

of

same

subfactor

we d e n o t e

e'N M,

.

of L e m m a proof.

by

2.3.6

[4]

79

LEMMA.

Let

commuting finite each n > 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

# -<< 2-ns II~

-~H

<

g E Fn, 2-ns

with

n>~ 1 and suppose, N = (an-l) ' ~ M

free a c t i o n

of

i,j c {0,i}

in

G

units

in

N,

(3)

on

if n > i, that

have

already

is M c D u f f N.

(N~) 8.

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

and

n

the a b o v e p r o p e r t i e s

factor

and

By Lemma

By Lemma

sequences

and for each

(Ad Vg ~~ egn-i ) (~~ e i , j)

8.3

= M,.

Let

eI

been

an =

~v C M

..,~n-i

(in w h i c h

we take (Vg)

By 5.8 units

(Ei,j),

in

of m a t r i x N

such that

e~l, j

= 0.

This

be the s u b f a c t o r

also holds generated

for each

by

ei, j •

We have P(~))' n S (x) hence

for

=

l,j x e~jV, i ½ i,j ~ ~v.

@oP(@~),@M

=

lim

v+~ v÷e

We may

thus

'

x6M

~ E M ,r lim ~+~

choose

~ ~ ~

the

(Vg)-: i), we may

consisting

cocycle

~g,...,Vg

is a c e n t r a l l y

are m a t r i x

(eg-l)

-n-i

and

constructed.

(Bg) = (~g-llN)

(e~,j)_ for Ei,j' v

,.

8.4 there

lim v~ = 1 *-strongly. 9~ g For each ~ e N,, lim II [ i,j~]ll

~ e e n-I × N ,

;

~ ~

(en) ' N M Let

~ e ~n

such that

½

~'. e~. e~. i,j 3 ' i ~ i'3 i,j

3'lei'jP

=

@

83

IlVgv ng - l -

If w e take is s a t i s f i e d .

vn-lll~ < 2 -n g e

m

Hence

Vg = lim v n + ~ g moreover,

2-n

<

c

*-strongly

R

and y i e l d s

~ e ~ : ~i

be the s u b f a c t o r

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

II 1 f a c t o r

II~ ° PR'NM--~II <

of

M

generated

and s p l i t s

2-ns

=

an

hypothesis

g e Fn+ I (ag) c o c y c l e ;

g e F = FI

M.

by

~ ~n ; by L e m m a

8.2

We have

n~l~ II(~ ° P(en) ' n M - ~ ) ~ n~l

For

g E Fn

~ e ~n+l ' exists

llVg -lil~ ~ ~ We let

'

~n = Qv and let s n = ~w, t h e n the i n d u c t i o n g g F r o m (i) we i n f e r for m k n k 0

llvm n # g - VgII~

R

~ C ~n

o P(en_l),~ M11

s

m k n > 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

<

e

~e

~ ,

geF

s t a t e m e n t we m a y o b t a i n

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

(Ad V g a g I R ' n M)

iSlnCenjugate to

is c o n j u g a t e

(a~u) ® a(0)) g

to

assertion

(ag), since the c onstruction. by

84 The model action submodel action. application

is an infinite

tensor product of copies of the

The proof of the theorem will consist of an inductive

of the lemma that follows,

w h i c h yields a copy of the sub-

model action. LEMMA. (Vg) for

~g

By the conditions of the theorem, there exists a cocycle and a II 1 hyperfinite subfactor e C M , such that (Ad ~g~g) (e) = e, (Ad ~g~gle) is conjugate to the

M = e ® ( e ' n M),

--

submodel action,

!

(Ad Vg~gle N M) is outez conjugate to (~g), and

IIVg -iIl~ < s I19 °PR, N M -

t e ~ , 911 < s

g e F

~ e

The proof of the lemma will occupy the next section.

We first

give the proof of the theorem. We may suppose (~n)n>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 ,..-,
M = en - l ® N

we have

~ 2-n-2

i Let

~ e N

be the set of all those

the above d e c o m p o s i t i o n

for some

~ e ~n'

~i,...,% p which appear in and let

6 > 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

<

2-ns

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

%~,...,%p • ~ C N, as

~ = ~ X i ® ~i • M, , then i

tlO- ~Jl ~< 2-n-2E II~-~ OP(en),nMll <

~

llXillll~i-¢i

°P(en),nNII

~< 6 [ IIXilJ i1~ill < l

2-n-2e

and hence {I~- 9 o P(en)' N M II ~< ll~ -~ll + II(,~-~)OP(en) , n M II+ II~ -~°P(en),nMII ~< 3 • 2-n-2~ <

2-us

The induction hypothesis is thus fulfilled. From the conditions (4,n), for m > n > 0 we infer llv~-vnl g ~# ~ 2-n e therefore the limit

Vg = l ~ v

ng *-strongly

g 6 Fn+ 1 ,

~6

exists and yields a

86

u n i t a r y cocycle for

~g

w h i c h satisfies

IIVg -iII~ < s We let tions m>n

R be the subfactor of

(5,n) show,

>i

g EF M

= FI ,

9 e

g e n e r a t e d by

in view of Lemma 8.2,

that

U ~n. n splits M.

R

The condiFor each

the action m ~n Ad Vg~g 1

n ~n Ad Vg~g I

=

is c o n j u g a t e to the submodel action, to the submodel action,

and thus

n ~n ~gl

=

hence

(Ad Vgagle n) is c o n j u g a t e

(Ad VgeglR)

is c o n j u g a t e to the m o d e l

action. We have,

for

~ E ~ : ~i

114 ~ PR'N M - ~II

n>~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~

<

for the Rohlin Theorem,

I

we infer

lFi,hl T

i,h I~n -1(5en_l + gn_l ) <

6en_llSnI-i

h E K n-I A g- iKn-i , and similarly IE~l~ If we let

n-i

k = gh

El- Zl where

6.1(5)

lsnl -'

<

for

(i - en_l> Kn-i { i

>

i E In_ 1 and

the Rohlin partition Iz1- 711

in

=

I F(h-l kz'sl)'(h-lk2's2)(~g(Fi'h) i,h

h • K~-ll N g- iKn-i l Fi,h

(2)

6.1(6)

for

[ l~g(Fi, h) -Fi,ghlT i,h

Isni-i (5en_l + 5gn_ I)

g E Gn_ 1 and

<

lOen_lISni-i

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

we have

I~g(E(kl,sl),(k2,s2)) - E(gkl,sl),(gk2,s2) IT '

~ . on (Esl,s2)

The estimates

-Fi'gh)

yield

~< l~nl -l

< summing up, for

E l we obtain

~

22en_llSnI-1

We perturb the action (eg) with (Wg) to make it coincide with a copy (Ad Ug) of the n-th finite dimensional submodel.

89 For

g E G,

let

Ug

(Es~,s2) , sl,s 2 e ~n submodel

4.4,

be the unitary

(k,s)

to the s.m.u.

i.e. Ug

where

associated

in the same way as in the n-th finite dimensional

• ~n

element of ~n, -0 Wg such that

=

and kg = £n(k). g and let us choose

~0"

--0

g

Wg

Let

(k0,s 0) be some distinguished

for every

g • G,

a partial

isometry

=

eg (E(ko,So),(ko , so))

o o* Wg Wg

=

Ad Ug(E

~0i = ~(ko According

[ E (kg,S),(k,s) k,s

to Lemma

(k0,s0),(k0,s0))

=

(gk0,s0),(gko,s0)

,S0),(k0,s 0 )

8.1.2,

from

(2) above we may assume

that for

g C G n _ 1 we have -

~

Let us define

~ni-1

the unitary -0

=

Wg where

for any

(Ad Wgeg) (Esl,sz) g E G,

sl,s 2 E ~n.

k,s = where

From the definition

(k,s) E ~n. (4)

i!s Ad U g (E(k ,s) , (k 0 , s O ))Wgeg( we infer

= Ad Ug(Esl,s 2) We estimate

for

g e Gn_ I

(Ad OgIEik,s>,ik°'so ll °g g k0s0>' 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 <

Z I is

90

~< IW°g - E(gk0,s0), (gk0,s0)IT + I~g(E(k0,s0), (k,s)) -E(gk0,s0), (gk,s) IT <

66Sn_ 1 snl

22Sn_ 1 [snl-1

: 88gn_ 1 I

where for the last inequality we have used IzII T < 88~n_ 1 (5)

and thus for any

IWg -lit

<

(2) and (3).

Hence

g E Gn_ 1 ,

IZIIT + IZ21T <

½

88En - 1 + 2Sn - 1 < 90e½n-± "

Step C. We use stability results to further perturb (Ad Wgeg) with (Wg), such that it continues to coincide on (Esl,s2) with Ad Ug, but, moreover, (UgWgWg) is an (~g)-Cocycle. Let E C M~ be the subfactor generated by (Esl,s2). Let us consider the cocycle crossed action ((~g), (Zg,h)) of G on M , obtained by perturbing the action (~g) with (U~Wg). We have from (4) Ad (UgWg) ~gIE and since

id~

U g E E, we infer --*--

g,h

--*--

--*

=

UgWg~g (UhWh) Wgh gh

=

Ad(Ug g) (~g(Uh)) UgWgag(W~)W*hU •~ ~ gn. hug gag

where

=

Zg,h = Wgdg(Wh)WSh.

n

For

gn gn

g g

Ugh( g,h )

g,h,gh E Gn_ 1 we have from (5)

IZg,h-ll~ ~ l~g-11~ + I~h-lr~ + I~gh-lIT ~ 270~_i and since

((Es~,s2), (~g)) is isomorphic to the n-th finite dimensional

submodel,

the inequality 4.4(3) yields for

2s n

IUgh -Ug0hlT Hence for

g,h,gh ~ Gn_ 1 we obtain

IZg,h - ll~ < IUgUhUg h-lI~ _

- . - . -

+

< 2s n + 270e~_ 1 < Since

~glE = ida,

Iid U*gh(Zg,h-l)

IT

en_ 4

we have

Ad ~g,hI~ hence

g,h,gh c Gn_ 1

=

(~gah~gh) . . . . ~ IE

=

id~

(Zg,h) c E' ~ M~ . We apply Proposition 7.4 to obtain

91 (Wg) c E' A M e to an action (6)

perturbing

the cocycle crossed action

(~g) such that

IWg - l I T < 1Sen_ 4

Since

(([g), (Zg,h))

for

Uge E commutes with

g • Gn_ 4 . Wg,

we define

and infer (Ad Wgeg) IE : Since

(Wg) perturbs

(eg)-Cocycle.

(Ad Wg~g) IZ : Ad WglE

the action

We have from

=

id~

(eg) to the action

(5) and

(6) for

(~g), it is an

g • Gn_ 4 ,

IUgWg-iI~ = IWgWg-iIT ~ < !Wg-iI~ + IWg-lly

<

18Sn_ 4 + 90c ½n-±_ ~ 19£n_ 4

and so I]UgWg- iII# = T Step D. us

Let

IIUgWg- III ~< (IUgWg-IIT llUgWg-lll)½ ~< (38En_4)½ < 7s~_ 4 • T

We lift the construction

apply Lemma 8.4 to the action (es

) be s.m.u,

in

M

itS2

cocycles

in

M

representing

done before from

(Ad Wg~g), which keeps

e~

is the subfactor of Ug

where

iE I n ,

with the definition of For any

~ EM,

and

(Ug) represents

~+e

s l,s 2

Ug.

I~s~I,s2~sVl ,s2 - e~sl,s2 %~s2, s ¢)

lira 1~nI-~ ~

II~, e~S l t S

2

]H = 0

Sl is2

~) • IN in a suitable way we may thus obtain a s.m.u.

(~sl,sz) ' sl,s2 • ~n (Wg) for

We define

kg : £g(k) , which, compared

shows that

~) ÷ e

cocycle

--V

(esl,s2).

(kg, s) ,(k, s)

k s

-~11 = lira 1~ni~E (~)' n M

By choosing

(Wg)v be (~g)

we have

limll~oP V÷ e

~g,

Let

id ~

M generated by

(k,s) e K n × S n C_ ~n

M.

(Esl ' sz) fixed.

(Es~ s2) and let I (Wg) such that for each v • N representing

(Ad Wg~g) v le = where

M e to

in

M,

generating a subfactor

(~g) and unitaries

copy of the n-th finite dimensional

(Ug) in

e of

e such that

submodel and

M,

a unitary

(e,~g) is a

92

Ad W g e g l e

=

id~

JlU g W g - i{I~ # ~< 7 En _ 4

S t e p E. subfactor

We c o m p l e t e

f of

M

which

g • Gn-4

the f i n i t e d i m e n s i o n a l

is a l m o s t

f i x e d by

(eg),

submodel

e

to o b t a i n

a c o p y of

with

a

the s u b m o d e l . Let 5.8

N = ~ ' n M.

is c e n t r a l l y

finite

The r e s t r i c t i o n

free,

II 1 s u b f a c t o r

(Zg)C N

for

of the a c t i o n

and thus we m a y obtain, f

(Ad Wg~g)

of

N,

with

N = f ®(f'n

A d (ZgWg) ~gl f

=

idf

(Ad(ZgWg) eg

I f'n

N)

ilzg-lll~

is c o n j u g a t e

~< IAs

[l%oPf,n N-%11

The s u b f a c t o r

choose

e

of

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

the one c h o s e n

tion

(Ug) of

G

M

into

N

and a c o c y c l e

~< I/8e

action

between

e

WgegIN) = (Ad Wg~g)

% • (% O Y ) IN

by

acts,

e u f

is i s o m o r p h i c

and

M = e ® (e'n M).

and the s u b m o d e l

in the p r e v i o u s e,

to

step,

coinciding

we get a u n i t a r y

copy of the m o d e l

to the

representation,

If we on

representasuch that

f r o m 4.4.(1) , lUg-~g[~ for

~ e ~,

yields

For a n y

for

~ = ~ o Pe'n M '

~< 8e n since

g E Gn ~I e

is the n o r m a l i z e d

iU g U g - i I~

~< 8~ n

jUgU;

-< ItU g U*gl

-

and any n o r m a l

states

~,~

I~

* UgUg

• M

#2 ~< llxl[~2 + II~-~II llxll2 llxll~ henc e

trace.

g 6 G n,

xe M

by

a hyper-

g • Gn_ 4 , 9 • YI N

generated

the s u b m o d e l

with

N)

to

1.2,

such t h a t

id~ ® ( A d

f a c t o r on w h i c h

(Ad Wg~g)

by T h e o r e m

ll)

-< 4 e n

This

93

id R

Since

is conjugate

to id R ® i d R, one can actually

assume

that

moreover

(Ad V g a g I R ' ~ M) Towards

is conjugate

(ag)

to

this result one first proves the following analogue of

Lemma 8.4. LEMMA.

group

v1= i,

with

~: G ÷ Aut M

Let

the amenable

and

G

let

be a centrally

on the factor

(Ei,j),

M.

i,j ~I,

free

crossed

(Vg) C M ~

Let

action

of

be unitaries,

II < ~, be a s.m.u,

in

M~

such

that

(Ad Vgag)(Ei, j) = Ei, 3 Then

there

exist

representing

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

sequences

i,j C I ,

sequences

of s.m.u.

(Vg) w w

of unitaries

(Ad Vgag) W (e~

,j

) = e~ 1,j

for

gEG

(eW,j)~

for

Vg, v~ = i ,

i,j E I ,

g 6G ,

(Ei,j), such

that

~ E N .

The proof of this lemma consists merely of Step A of the proof of Lemma 8.4

(actually the group property

of the theorem is obtained action on the c e n t r a l i z i n g

8.9

trivial

, Lemma

G

is not needed).

8.4.

Since a crossed action induces

algebra

Me

THEOREM.

Let

a finite

subset

exists

a family

finite

subfactor

(Ad VgaglR)

~: G ÷ Aut M

IIVg -i11 #

such

that

in

<

~

the supplementary

conjugate

to

(ag) may be obtained

conjugate

to

(~g(0)® e(0)) .

M,

action

~ > O,

let

G.

Then

of

~

be

there

v I = i, and a II 1 hyperM),

(Ad Vgag) (R) = R,

and

geF

~e ~ . assertion

that

(Ad VgagiR' N M) be

since the model action

The proof is similar to the one of Theorem 8.6. an action on

crossed

Let

subset

~E~, ~

free

M.

M = R~(R'n

to the mod~,l action

II~ o pR, A M - ~ I I < Once again,

result.

be a centrally

(Vg), g E G, of u n i t a r i e s

is conjugate

are

8.3 can still be used.

group G on the M c D u f f f a c t o r + of M, and let F be a finite

RCM,

an

(because inner automorphisms

T h e o r e m 1.6 is implied by the following

of the a m e n a b l e

The proof

from the one of Theorem 8.5 by using the

above lemma instead of Lemma centrally

of

M w , Steps A, B and C remain unchanged.

((gO)) is

Since

(ag) induces

In Step D the

94

(7)

[Ixll

Letting

#

tlxtl~ + It@-~11

<

x = U g U q - i,

II~ -~II <

%

Ilxt[

~ e~

and

If@ - ~ o P e ' o 1A£

<

IA ~

+

~ = ~ o Pe'n M ' we have

M II + II (~ - ~ o P f ' m M ) o P e ' n =

M

~4S

we i n f e r w

Since

(Ug) is a r e p r e s e n t a t i o n

(Ad(ZgWg)ag), ~g = UgZgWg, e

globally

(Ug) is an then

the s u b m o d e l

~g

and hence,

and

action. =

for a n y

coincides

on

-*UgZgUgUgWg

=

via the i n e q u a l i t y

Therefore,

The a c t i o n with

and

(Ad Ug),

Ug E e,

if w e let

(Ad 5gag)

leaves

the c o p y of

we i n f e r

-* UgUgZgUgWg

7.1(10)

we obtain

-*

<

e

Zg • ~' o M

2(HUgUg-IH~

+ IiZg-IH~

#)

+ I'~gWg-IH~

g E F ~ Gn_ 4 .

The p r o o f of the l e m m a

8.8

(&g) c o c y c l e .

Since

HSg-IH~

f i x e d by t h e a c t i o n

(Ad(ZgWg)ag)-Cocycle°

(~g) is an

invariant

kept

It w i l l be c o n v e n i e n t

is f i n i s h e d .

for the p r o o f s ,

G-kernels,

which

with their

s e c t i o n s , which we c a l l e d

~: G + A u t M, ing t h e o r e m

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

al = id,

implies

THEOREM.

Let

o f the a m e n a b l e a finite

subset

of

unitaries

V g E G,

RCM

that

such

on

and

g EG,

for

to w o r k

and w h i c h

g , h • G.

be a c e n t r a l l y

the M c D u f f let

with

M = R®(R'O

with are m a p s

The f o l l o w -

1.5.

a: G ÷ A u t M + M,

actions,

a g a h a g h E Int M

Theorem

G

crossed

-I

with

group

i n s t e a d of d e a l i n g

G ~ Out M = A u t M / I n t M,

F

be a f i n i t e

v I : l,

M),

llVg - iii#

factor

and

free

M.

subset

~

H~ OPR, N M-~II

of

<

G.

e

There

~ E ~: .

~

be exist

subfactor

and

~ e ~ ,

action

e > O, let

a II 1 h y p e r f i n i t e

(Ad Vgag) IR = id R <

crossed

Let

g E F

95 the l e m m a in the p r e c e d i n g S t e p E, T h e o r e m

section

8.5 is r e p l a c e d

Chapter

9:

is u s e d i n s t e a d of L e m m a

by T h e o r e m

MODEL ACTION

In t h i s c h a p t e r w e g i v e the p r o o f Theorem

1.4, w h i c h c h a r a c t e r i z e s

approximately

9.1

In this

of a g r o u p

at m o s t c o u n t a b l e , e

Let

(Vg)~

be a sequence

group

for

Proof. forward.

=

by

is a cocycle

and strongly

(8,U)

U(g,h),(k,i ) E M e .

=

inner

For each g • G , let = lim A d W ~g ~÷e Vg ,

•

Aut M e

V*

gkZ-lh-1

crossed action of G x G on (see 5.2, 8(g,h )

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

it is a c o c y c l e

For e a c h

•

Me,

Me

which is

5.6).

is the p e r t u r b a t i o n

~ E M,

is s t r a i g h t -

of the a c t i o n

crossed

action.

~ o ~g

,

g EG

L e t us s h o w

.

, then

~ o ( ~ g h - 1 ~ h ~ k i - l a h - ~ a g-I k £ - 1 h_1)

=

U(g,h),(k,£ ) e M e . Let us s h o w t h a t

for

(g,h) ~ (I,i),

@(g,h) IMe

e

(g,h) ÷ e h

we have

u 9(g,h),(k,~) = Vg9 h- ia h ( V k~£ - 1 ) V g~* ki-lh-1 lim ~ o A d +e u (g, h),(k, 9~) =

hence

M.

Vgh-leh(Vkz-l)

~÷elim ~ o Ad v ~g =

If

free approximately

such that

Ad V g h - l e ~

free

(Vgh-1)(g,h) , a n d h e n c e

that

has a s e p a r a b l e

• G,

The f a c t t h a t e a c h

W e see t h a t

M

is

~.

on the factor in

M

G

V I = i.

U(g,h),(k,Z ) (OIMe, U)

on

action.

t h a t the g r o u p

be a centrally G

g,h,k,£

8 (g,h)

semiliftable

are

idea of d e a l i n g w i t h a p p r o x -

by m e a n s of a G × G

of unitaries

(v~) w • M e ;

Let us take,

Then

free a c t i o n s w h i c h

and t h a t the f a c t o r

~: G ÷ A u t M

of the amenable Vg =

G

be a free u l t r a f i l t e r

action and let

of the m a i n r e s u l t of this paper,

the c e n t r a l l y

this c h a p t e r w e a g a i n a s s u m e

we let

LEMMA.

ISOMORPHISM

s e c t i o n w e i m p l e m e n t V. Jones'

Throughout

predual;

8°8.

inner.

imate inner actions

discrete,

8.4 a n d in

is s t r o n g l y

outer.

95 the l e m m a in the p r e c e d i n g S t e p E, T h e o r e m

section

8.5 is r e p l a c e d

Chapter

9:

is u s e d i n s t e a d of L e m m a

by T h e o r e m

MODEL ACTION

In t h i s c h a p t e r w e g i v e the p r o o f Theorem

1.4, w h i c h c h a r a c t e r i z e s

approximately

9.1

In this

of a g r o u p

we let

e

Let

of the amenable be a sequence

group

for

G

=

Proof. forward. by

is a cocycle

and strongly

=

(8,U)

U(g,h),(k,i ) E M e .

•

(see 5.2, 8(g,h )

inner

For each g • G , let ~g = ~lim ÷ e A d VgW ,

Aut M e

V*

gkZ-lh-1

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

is the p e r t u r b a t i o n

For e a c h

•

Me,

Me

which is

5.6).

it is a c o c y c l e ~ E M,

is s t r a i g h t -

of the a c t i o n

crossed

action.

~ o ~g

,

g EG

(g,h) ÷ e h

.

, then

~ o ( ~ g h - 1 ~ h ~ k i - l a h - ~ a g-I k £ - 1 h_1)

=

U(g,h),(k,£ ) e M e . Let us s h o w t h a t

for

(g,h) ~ (I,i),

@(g,h) IMe

e

L e t us s h o w

we have

u 9(g,h),(k,~) = Vg9 h- ia h ( V k~£ - 1 ) V g~* ki-lh-1 lim ~ o A d +e u (g, h),(k, 9~) =

hence

M.

such that

Vgh-leh(Vkz-l)

~÷elim ~ o Ad v ~g =

If

free approximately

crossed action of G x G on

free

(Vgh-1)(g,h) , a n d h e n c e

that

M

is

• G,

The f a c t t h a t e a c h

W e see t h a t

in

G

has a s e p a r a b l e

~.

on the factor

Ad V g h - l e ~

U(g,h),(k,Z ) (OIMe, U)

on

M

V I = i.

g,h,k,£

8 (g,h)

semiliftable

are

action.

t h a t the g r o u p

be a centrally

of unitaries

(v~) w • M e ;

Let us take,

Then

by m e a n s of a G × G

and t h a t the f a c t o r

~: G ÷ A u t M

(Vg)~

Vg =

G

be a free u l t r a f i l t e r

action and let

free a c t i o n s w h i c h

idea of d e a l i n g w i t h a p p r o x -

this c h a p t e r w e a g a i n a s s u m e

at m o s t c o u n t a b l e ,

LEMMA.

of the m a i n r e s u l t of this paper,

the c e n t r a l l y

s e c t i o n w e i m p l e m e n t V. Jones'

Throughout

predual;

ISOMORPHISM

inner.

imate inner actions

discrete,

8.4 a n d in

8°8.

is s t r o n g l y

outer.

96 If

h ~ i,

then for each

thus by Lemma

5.7,

~,

Ad Vgh-la h

@(g,h) IM~

is centrally

is strongly

outer.

nontrivial

If

h = 1 and

and g # i,

then

~ = ag which is centrally nontrivial and Lemma ~lim ÷ e Ad v gh-l~h shows that @(g,l) is strongly outer. The lemma is proved.

9.2

We show that the approximate

pointwise,

innerness

can be given a global

LEMMA.

a: G ÷ Aut M

Let

be a centrally free a p p r o x i m a t e l y

(Vg)v

r e p r e s e n t e d by sequences

of unitaries

M,

in

inner

Vg

There exist unitarie8

Me

limAd v~ = ~g

with

and such that

VgV h

= Vg h

~ g (Vh) This can be restated for

defined

form.

action of an amenable group on a factor.

V I =i

of a group action,

5.6

g,h E G

= Vghg- 1

as the fact that

(g,h)~->(a~) and implies

the fact that

(Vgh-1)(g,h) e(g,h)

is a cocycle

= Ad Vgh-1~ ~

is an

action. Proof. in

Let

the previous

crossed G ×G

Vg,

action of

is amenable,

a coboundary,

G× G

on

W

Me .

yields

e (~h)(g,h)

the action

for it.

Thus we have for

Since

h= g

and

to an action,

g,h,k,£

hence

(W(g,g))

is strongly

c Me

U

is

such that

(W(g,h) Vgh-1)(g,h)

and hence is a cocycle

e G, = W (gk,hl) Vgk~- lh -I

~ = k we obtain

is a cocycle

free.

free and

that

(W(g,h)) c M e

This way

(~g) as

cocycle

is strongly

7.4 to conclude

a perturbation

W(g,g) ~e(W g (k,k) )

which

M e , implementing

(@IMp, U)

an action.

W (g, h) Vgh- i~h (W (k, £) Vkz- i )

If we let

in

(0,U) be the corresponding

there exists

with

perturbs

(i)

be unitaries

and let

we can apply Proposition

i.e.

(8,U) perturbed

geG,

lemma,

=

7.2 yields

Z* ag(Z) e

We define Vg

=

W (gk,gk)

for the action

Proposition W(g,g)

=

ZW(g,k ) VgZ*

(~g)e

of

a unitary gEG

.

G

on

M~

Z e M e with

97

eg

Since on M.

Vg

differs

from

In (i), if we let

Vg

by unitaries

in

M e,

Vg

also implements

h = £ = 1 we infer

W(g,I)VgW(k,I)V k

= W(gk,l)Vg h

which easily yields VgV k

= Vg k

In (i) we now substitute

hg -I, i, g, g

for

g,h,k,Z

and obtain

g h-iW( g,g ) = W(h,g )Vhg -I

W(hg-1,1) and thus Vgh-I

=

Z* ZW(hg-1,1)Vhg-1

=

* ZW(h,g)Vhg-iW(g,g ) Z*

=

~(Z*) ZW(h,g)Vhg-1ag

In particular, Vg-1

=

ZW(I,g ) Vg-1 ~(Z*)g

hence Vg-1 e~g (Vh)

If, in (i), we let

l,g,h,l

=

ZW(I,g)Vg-I e~g (Z*ZW (h, l)~hZ* )

=

~(W (h,l) Vh z* ) ZW(l,g)Vg-1~g

stand for

g,h,k,Z

W(l,g)Vg_1~g(W(h,l)Vh) which yields in the preceding

= W(h,g)Vhg-1

equality --

Vg-le~(V~) ~ Since

we get

=

~

*

ZW (h,g)Vhg -I ~g(Z ) = Vhg- I

(Vg) was shown to be a representation, eg( V h ) = V*-iVhg-1 g

= Vghg-l

and the lemma is proved.

9.3

Let us recall Theorem

1.4 in a convenient

form.

98 THEOREM.

~: G ÷ Aut M

Let

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

inner action of an amenable group let

F

be a finite subset of

(Vg) for

a cocycle

on a M c D u f f factor M. Let E > 0, + and let ~o E M, . Then there exists

G

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

(Ad Vgag) (R) = R

G

and

=

R ® (R' n

(Ad VgagIR)

llVg -lII~

~

such that

M)

is conjugate

(Ad Vg~g I R' O M)

RaM

=

e ,

to the model action

idR'N

M

gC F

0

From the above by Theorem

statement

1.2 the model

we can easily

action

obtain

Theorem

(~$0))- is outer conjugate

from the above theorem we infer that

1.4. to

Indeed,

(e~))--® i d R ;

is outer conjugate to (a 0)) ® idR, O M and hence to (a~0)) ® id R ®idR, N M = (a~ 0)) ® i d M ; moreover, we have control over all cocycles that appear. We obtain

the copy of the model

successively

the following

Recall

En > 0 and

that

The sets

~n

LEMMA.

(ag)

lemma,

GnCC G

action

in the theorem by applying

which yields

a copy of the submodel. 0

are part of the Paving

Structure

index the n-th finite dimensional

submodel

In the conditions

let

and let ~, ~ and

~

of the theorem,

M = e ® (e'O M), and a cocycle (~g) = (Ad Vg~g)

n k 5, let

p = Isnl

M, ,

~

consisting of states

subfactor

e

of

be finite subsets of

There exists a II 1 hyperfinite

3.4.

4.5.

(Vg) for

M,

such that

(ag), such that letting

we have

~g(e)

= e

(~gl e) is conjugate to the submodel

and

(~gle' n M) is outer conjugate to (ag).

action and

(i)

ling- iII~ <

En_5

(2)

II~ 0 Pe'n M - ~ll

(3)

For each

~E #

~ i e (e'O M),,

9 E ~ , <

2p

sup gE Gn+ 1

there exists

i = 1,2 .... ,p2

g E Gn_ 4 .

~ o ag - ~

~ 6

H i e e,,

{I~ill ~< i,

such that

II~ - [ H i ~ till ~< 6 i ll~io (ag le' AM) - ~ill < 6 The proof

of the lemma,

in this paper to work,

,

i = i, . • - ,P ,

which puts the whole machinery

will occupy the next section.

g E Gn+ 2 . developed

9g We r e m a r k ~ ~ o 8g,

that c o n d i t i o n s

where

concentrated

on

hand,

then,

~ g ~ ~g,

e;

its form allows if for some ~gle

i.e. m o s t of the a c t i o n

us to w o r k

% • M,

further

we w o u l d

and

have

in

[g

is

e ' N M.

% ~ ~ o Pe' N M '

we c o u l d

infer

~ o pe, n M o ~g

~oPe,nM

o ~g

~OBg

the form of c o n d i t i o n

(2) above.

~

is inner

the fact that

Bg ~ i d ,

~ o ~g and h e n c e f o r t h

express

Bg = id e ® ([gI e ' n M),

On the o t h e r since

(3) above

~

~

P r o o f of the T h e o r e m Let We let finite

n~> 5 be large e n o u g h pn = IsnI,

nEN.

sets of states

We c o n s t r u c t hyperfinite

Let in

inductively subfactors

(~gn-i ) = (~g),

such

(~n) ,

n~>n-i

=

for

n = n, ~+i, ~+2, ... m u t u a l l y of

n -n ) ~g(e

=

and

if

e

n

nu ~n

M, and c o c y c l e s

n = (Ad Vgn n-l) (~g) g ,. ..,

that

G[_ 4 D F.

be an a s c e n d i n g

~n-i = @'

~n, ~ + I , . . .

~n®

and

with

is the s u b f a c t o r

e-n , e-n+l , .... ~n, w i t h e n-i = C.I, and if v ~-i = i, then for e a c h n ~> n we have g M

3~_ 5 < s

M,,

, n+l,) for tVg

n n-l. (Ad Vg~g ),...

to p r o v i d e

family

total

commuting (Vg)

.-n+l. ~Vg } for of

M

in

of M, . II 1

for ( ~ng ) =

generated

Vg = -n-n-i VgVg ... Vg

by

with

((~n), N M) -n e

and

n n) is c o n j u g a t e (~gle

submodel

n n , ( ~ g ( e ) n M)

action;

conjugate

to

to the is o u t e r

(ag)

v n • (en-l) ' n M g (4,n)

llvn-vn-lll # g g 90

(5,n)

II~ o P(en), n M - ~II ~< 4e n

7# • ~n-i

(6,n)

There

for any

are

exists

<

2en_ 5

rne N

~k e (en), ,

g • Gn_ 4

such that

ll~kll-<-I,

~ke

~ e ~n

((en)' n M),,

there k = i, .... r n

such that I19 - Z ~k ®6kll k

~< en+ 1

--I --I II~k o (~gl (en)' n M) - ~k 11 ~< Pn+l rn en+l ,

g e Gn+ 2

100

-n -n-i Let n>~n and suppose, if n > n, that e ,...,e and -n (~n-l) (Vg) .... , " g satisfying the above conditions have been constructed. "

Let

N=

condition

(en-l) 'n M.

Let

q e IN be chosen

such that the following

holds:

(7)

For any and

~e~ n

there are

~i e N,;

i = l , .... q

k i e (en-l),, with

llXill ~< 1

II~-~. Xi®~ill ~< ½en.±_~ 1

We assume

that in

(6,n-l)

kept fixed in all that follows The action gate to N

(a~-iIN)

(ag), and hence

is a McDuff

and

(7) above,

for each

~

is made and

involved.

is, by the induction is centrally

a choice

hypothesis,

outer conju-

free and approximately

inner,

and

factor.

Let us apply the lemma in this section in order to obtain a n-i ) and a subfactor ~n of N such that (~n g ) C N for (ag

cocycle

letting (8)

agn = Ad -n Vg@gn-i N = ~n® n(~n) g

the following

assertions

hold:

((~n), N N) = en

and

(~gle n) is conjugate

(~gl n (~n), N

action and

N)

to the submodel

is outer conjugate

to

(~

~-llN)

-n - 1 II#° ~< en_ 5 IlVg Sn- 5

g for

n-i ~g = ~0 o Ad Vg

g e Gn_ 4, with (9)

For any

~6~n_l,

if

~k E (en-l),

k = l,...,rn_ 1 were chosen

in

and

~k e N,,

(6,n-l) , then for

k = l,...,rn_ 1 we have ll~k o (~g-iIN) -~kll . II~k o P(en) 'N N - ~k II ~< 2Pn g esup Gn+ 1

(i0)

For any

~ E ~n'

with

i = l , .... q chosen llqi,jll = i

and

j = 1 .... 'Pn2

in

Xi e (en-l),

(7), there exist

$i,j e ((en) ' A N ) , ,

with

and

#i E N,, ~i,j e (~n), ,

i = 1 ..... q,

101

II%i - ~ ~i,j × ~i,j II ~ ]

½q-1 en+2 --I

--i

II~i,j ° (eg (en) ' n N) - 6i,jll ~< Pn+irn en+ 1 ,

and

g E Gn+ 2 .

From (8) we infer, in view of the inequality 7.7(1), if ~g = 90 o Ad v$ -I "v~-vn-III#g~0

g e Gn_ 4

= II(vg-n-l)v~-l.I #~0 2½(,,~-i',~ + ,,~n-l,,~ ) < 2en_ 5 0 g

and hence (4,n) is proved. We have assumed that ~n-i =~@' hence the statement (5,n) is void for n = n. Suppose n > 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[ <

sup

2Pn ge Gn+ 1

II~k ° (~-IIN) -
2PnPn Irn_Isn-1 = 2r~11en_ hence if

~ = [ [k ® [k E M, , then k II~ o P(~n)' N M - ~li <

211~ -~II + II~ o P(@n),n M

~ H

~< 2e n + k[ II{kli la~k o P(~n), A M - ~ktl

2g n + rn_12rnilgn

=

4~ n

and this way (5,n) is proved. Let ~ e ~n and let X i e (en-l),, ~i E N,, i = l,...,q be chosen as in (7). Further on let ~i,j e (~n), , ~i,j E (en)'n N, i = 1 .... ,q, j = l,...,p~ be chosen as in (i0). We let

{i,j and infer for any

=

Xi ® ~ i , j

6 (e n-I ® e n ) .

=

IIDi,j}l ~<

=

(en).

i,j

H~i,jl i

IIXiil

1

II~- [ %i,j ® ~i,jll ~< II~- [ Xi®¢ill + i,j i

,

[ lIXiH If%i- [ ~i,j ® ~ i , j II i j

--i

~< ½Sn+ 1 + q - ½q

en_ 1 = En+ 1

,

102

II ~i,j o (~g (~n), A N) - ~i,j II < and thus,

if w e r e i n d e x

P n-i + i r n-i en+l

(Ei,j),(~i,j),

k = l , . . . , r n = qp~, we o b t a i n

(6,n)

,

g E Gn+ 2

j = 1 ..... Pn2 w i t h

i = i ..... q;

and end the p r o o f of the i n d u c t i o n

step. From

(4,n) we infer,

since

~ _ 2~n_ 5 < 3e~_ 5 < c a n d n~n

that Vg

exists

for any

gEG

=

lim n~

and y i e l d s

an

IIVg- iIl~

vn g

*-strongly

(~g) c o c y c l e w h i c h

< e

We let show,

and

R

be the w e a k

closure

in v i e w of L e m m a

8.2,

of

u e n in M. The c o n d i t i o n s n R is a II 1 h y p e r f i n i t e f a c t o r

that

M = R ® (R'A M). Let

(8~) be the a c t i o n

of

Bgn For in

satisfies

g E F C Gn- 4

0

(5,n)

~ •~n

and

(6,n), w e let

G

=

on

M

g i v e n by

iden ® ( a n l ( e n ) ' N M) y

~k • (en), '

~ = [ ~k ® ~ k k

g •G

k= l,...,r n

~k • ((en)' a M),,

n _ ~ II

~< 2en+ 1 + [ ]]~kI] ll~k o (agl (en) ' A M ) k ~< 2 £ n + 1 + r n P n + i r n g e Gn+ 2 .

Since

u ~n n

is total

lira II~ o 8g -~I] n+oo Let

chosen

and i n f e r

n

for

U G n = G, n

x E R ' ~ M.

F o r any

=

en+ 1

in

M,

0

3Sn÷l and

u G n = G, n

gEG,

n E IN,

~kiI

we obtain

~EM,

g6G

(Ad Vg~g) n (x)

=

n ~g(X)

(Ad Vg~g) (x)

=

w - lim n+

=

w-lim 8n(x) n-~oo

=

n ~g(X)

hence

and thus

Ad V g e g l R ' A

M = idR, A M .

n (Ad Vg~g) (x) =

x

This ends the p r o o f of the t h e o r e m .

103

9.4

In this last section we give the proof of the !emma stated in 9.3.

The first part of the proof will be similar to the one of the main lemma of the p r e c e d i n g

chapter.

In the second part we make use of the

fact that the action is a p p r o x i m a t e l y unitaries

in

M ~.

inner,

and hence implemented

We let the copy of the submodel

almost contain these unitaries,

by

that we construct

and in this way concentrate

on this

copy of most of the action• To simplify ~

of

~

to

the notation,

in what follows we denote

the extension

M e by

~ . We recall that ~n > 0 G C C G , the en-paving g ' n of G and the approximate left g translations

g g subsets ( K ~ ) i e i n

n ui K ni ---> 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

<

g • Gn_ 1

Then

and -

IWg - lit Step C. Theorem,

we

vanishing

90£~_ 1 ,

With the Relative can repeat

Rohlin

the proof

of the 2-cohomology

of

Theorem

instead

of Proposition (~g) in

(M)AdV*~,

We may proceed as in Step C of 8.7 and construct (~g) Ad V*~ C E' n (M) , such that if Wg then

(Wg) C ( M ) A d V * G

g g g is an

of the Rohlin

7.4 to obtain instead

a unitary

of

the M e-

perturbation

g g g

(eg) cocycle,

Ad Wg~g]E = id~

and

½ (1 )

I)UgWg

Step D. ((Es,t),(Ug))

i IIT

-

-4 '

g

•

G

n-4 the copy

of the n-th finite dimensional

submodel

with a

C Me, such that the unitaries Ug are very close to implementing ~g, i.e. such that E "contains" the

action eg. The unitaries (WgVg)

7£n

The aim of this step is to replace c Me

copy ((Es,t),(Ug)) the unitaries Vg

fixed by

<

(Ad V ~ g ) ,

(Wg) E M e

defined

in Step C formed an

hence they form an

is a representation

of

G

into

(~g) cocycle

(Ad Vg) cocycle as well. Thus Ad v*~ (M) and Ad(WgVg) IE = ida,

g e G. we let

Vg = UgWgVg e

We replace isometrics

Es,t

the partial built

from

(M~) Ad V*

, and obtain

isometrics Vg

Es,t

as follows.

in

Ad VgIE = Ad UgIE. E

by some partial

We begin by making

choice of an element i • K~l for each i • I n . For each i • I n , (k,s) • K ni × S~l and h = ki -~ we have, in view of the fact that

a ~ • K~1

105 and

n h~ = k • K i,

n ¢ Z h(1) = k

and so

Ad Vh(E(~,s),(~,s )) Therefore

= Ad Uh(E(~,s),(~,s ))

=

~(k,s),(k,s)

the formulae (k, s),(m, t) =

E (k, s),(k, s)VhV£

(m, t),(m, t)

n n n n ~.-l where (k,s) • K. ×Si, (m,t) • Kj × Sj, h = k l -I, Z = m 3 , define in (M~) A d V * with the same d i a g o n a l m.a.s.a, as E, i.e. Es,s Let

g be the left g - t r a n s l a t i o n Ug

with

i c In, Since

-

= ES,S

=

•

M~

unitary

s •

a s.m.u.

~n

associated

to

(Es,t)

gEG

~ k { s E ( k g ,s),(k,s

(k,s) 6 Kni × sni' k g = In(k).g (U~Vg) = (WgVg)

is a r e p r e s e n t a t i o n ,

UghVgh

=

g,h E G

UgVgUh h

and in view of the fact that

= Ad(WgVg) IE = id~

Ad(O~Qg)

and

Ug E

we infer --W

(2)

VghVhVg

Let us keep

= Ugh(UgVg)Uh(VgUg g E Gn

~g~-i where

i e I n,

fixed.

--

--*--*

= UghUgU h

We have

= iI k,s X ~(k~,s),(k,s)(~hl hg~*~*-l) -- Z~÷Z 2 1

(k,s) e Knx s n l l'

we sum those terms in w h i c h k e K ni \ g - iK ni • Let ~ 6 M, be a state.

IZ21,

Ug

< {

We have

[ I~'(kl,s),(kl,s)l, llQhiQhVg-lll k,s

< 2{ =

n k ) , h = k i ^-I , hl = kli^-i ; in k I =_£g( gk E K~. ~ and in Z 2 those in w h i c h 1

2 I.

[ l~(k~,sl,(k~,s )I~

k,s

1Kn\g -1Kn~ ISnI Isnl -~

1

2e n ~ IKnl 1

Is~

l~nl -I

=

2£ n

Z

106

where Knl is

k ~ K n \g- IK n , s C S n and we have u s e d the f a c t t h a t l 1 ' (Cn,G n) i n v a r i a n t for the e s t i m a t e s of L e m m a 7.1. i e In,

On the o t h e r hand, h = kl -I

and

for a t e r m in

h I = gk[ -I

so t h a t w i t h ~,

~*

E(kl,s),(kl,s)Vh I h

and thus

~i

Analogue

-

g

=

hence with

- -

--,

E(gk,s),(~,s)

Uk~-IUg

=

E(gk,s),(k,s)

g

=

E(gk,s),(gk,s)

lUgVg-lJ~

Is, I, + ts~l~

estimates

--.

the i n e q u a l i t y

<

7.1(7)

2e n we infer

<

2e n½

(i), we o b t a i n

~< 2½(2e n + 6e n-~ ½ A)

W e lift the w h o l e

g,h 6 G

is an

(~g) c o c y c l e ;

8.4 and o b t a i n

s.m.u.'s

in

M

for

cocycles

generated

by

g e Gn_ 4

<

10E n-4 ½

construction

done before

from

Mm

to

M.

we h a v e V g•~ g ( V *h )

(V;)

for

~ 6 M,.

S t e p E. For

2e n

~

2 ½ (IIU~ g V~* # + JlWgVg - llJ#) g - III~ T

=

for any s t a t e

gE Gn,

~ "~* - llJ~ # + J]V~g V g* - lJJ~) ~< 2½(]]UgVg

lJU g V g - 1 Jl#

(3 )

and

yield

F r o m this t o g e t h e r w i t h

(~g)

= gk,

U g k ~ - I U k ~ -I~*g

=

~ E M,

~, _ iii~# ])UgVg

of

in(k) g

_,

E(gk,s),(gk,s)

for a n y state

jVgUg ~ ~, _ 1 I

Lemma

kI =

E I = 0.

In c o n c l u s i o n ,

hence

we h a v e

(2) w e i n f e r

in

=

moreover,

representing

(Es,t) M

for

(e~s,t) t h e n

V *gA d V g ( V *h ) =

(V;) s u c h t h a t

gh

(Es,t) c (M°~)A d V * ~ (e~s,t)~

sequences

and representing

V*

sequences if

e~

.

consisting

We a p p l y of

(Vg*) c o n s i s t i n g

is the s u b f a c t o r

of

M

107

Ad Vg ~g I For

g e G,

dimensional

submodel

Aut M.

(6)

llUgV~

~~ lim since

E

Then

(~V Ug)

From

(3) we have,

# -lll~ <

½ 9en_ 4 ,

•

~g = Ad Vg a g e

*

S~S

= E

~v

lim ÷ ~ ( s's ) =

(7)

We n o w study

s,s

~

will

~ •~N

for any state

the d e c o m p o s i t i o n

s,tE

finite

We let

~ 6 M,

_ (Es, s) = T (Es, s ) = of

X ns,t

M,

with

respect

be the basis

]snI-I

s • ~n •

to

dual

to

es,t'

Cs,t

s,t

We have and

_{~g• M ~.

• M

~ s) = ~ (Es,

=

represent

g ~ Gn_ 4

~~ M = em ® ((eg) ' A M). Let ~s,t e (e~) * s,t e ~n. For ~ • M, let

with %s,t x • (eV)'NM

g •G,

id

we d e f i n e w i t h the u s u a l f o r m u l a e the u n i t a r y u~ e M g ~%) to (es, t) such that (~e~ ~ ~u~g) ) is a copy of the n-th ' s,t'"

associated

and also,

=

--~gleV = id , hence g

for any

Sn ,

l~s,t(eg(X)

~ ((~g(X) - x)e~,t) I

-x) I =

~(~g(Xe~, t) -reds,t) I <

Ad Vg ÷ ~g

Since

(8) for any

when

~ ÷ w,

O~g-~ll llxll

we have

lim ll~s,t o ~g - %s,t II = ~ E M.,

s,t E ~n

and

Let i = mj-i

and since,

by a s s u m p t i o n s

and so

0

is a l m o s t

it a l m o s t c o m m u t e s w i t h i,j E In, s = (k,s) E K ni × w h e r e i @ Kni and ~ E Kjn E ~ , { V h* _ I

÷id

g EG.

We now show that if a state (~g) then

~

with

( s,t ) . n [ = (m,t) e K~] × S~3' Si, were d e f i n e d in Step D.

=

~: , ~

=

~-s,s- ~ wn h Ad Vh~-1 ( w ~ z) e M~ 3.5,

9

invariant

* h v~ * Vhz_ I

=

E- ~ , ~ U- h* W h V

* W *U

respect

to

h = k ~ -I ' We have ~ V *h~-1

108 h£ -I E Kn(Kn) -~ n n -~ i Kj (Kj) C Gn+ 1 we conclude

that

for any

V* • M ~s,t g Let ~ • M,.

s,t • ~n

there

exists

We have

[snl -~ ( ~ ~ tG

lim InG o p - GI[ = lira ~÷~ (~)' e M ~÷ w

<

s,t

l~nl -~ ;

s,t • ~n , let

the p r e v i o u s

g • Gn+ 1 be such that

discussion

~-centralizing

and let

sequence.

(9)

if

e

then

(es,t),

(i0) (ii)

where and

choice

s,t E ~n

finite

~ * - liB# HUgVg ~

by

~X~Vg el

<

v÷~lim (llx~v~G g( - ~ o A d

~<

lim ~)-~ ~0

=

HG -

of

of

M

v

M

(lIG

generated g EG

(13)

li~s,t O~g-$s,tll

in view of by

~< I/4~

(6)-(9)

(v~) for

(~s,t)

such that

s c ~n ,

and if

(~s,t) c e,

a

such that ~g = Ad Vg~g *

~gl~ = id,

is a g •G.

~e~ ~E~

sup fIG o a g - GII , g eGn+ 1 ~c¢,

above,

(~g)

((es,t),(Ug))

4.5

g • Gn_ 4 ,

3/2 Isn ]

7. ~ s , t ® ~ s , t , with s,t ~s,t = (es,t ~) I%'A M.

sup BiG-~ o ~g[[ g•Gn+ 1

submodel

~(es, s) ~< (i +en) iSni -I Grl <~

+li [xm,~]ll)

~ o eg 11

yields,

I 0 ~ n-4 ½

JiG o P t , A M -

v~)llg +II [x ~, ~]Vgll)

we obtain

dimensional

~

Vgll

-G oAd

and a cocycle

U g E e,

(12)

~ =

lim i i X V V g ~ -

G • M,

in

are u n i t a r i e s

copy of the n-th

, as y i e l d e d

v-~60

for any

is the s u b f a c t o r there

s,t t,s

~÷60

Es, t v *g • M

lim IIG o p - ~ II ~< Isni ~÷~ (e~) ' n M

An a p p r o p r i a t e s.m.u.

-

We infer

,

In conclusion,

,s

(x~)~ = (e~s ,tVg~* ) be the c o r r e s p o n d i n g

lira HI [e~s t' G] II = l)÷t0

,

lim II[es, "~ t, G] II

s,t For

g E Gn+ 1 w i t h

~ E H

s , t E S n, the basis

gEGn+

dual

to

2 (es, t )

109 Step F.

We complete the copy

model with a subfactor

f,

e of the finite dimensional

almost fixed by

sub-

(~g) and thus obtain a copy

of the submodel. Let free. f CN

N = e ' N M.

Then by 5.8,

N

is McDuff and

with

N = f ® (f'N N) and a cocycle (Ad Zg&gif)

(Zg) for

= idf

(14)

llZg- iIi~ <

(15)

II (~IN) o P f , N N

and with

en

g E Gn_ 4 , - {INI!

~s,t = (es,t ~) IN'

<

li~S,~So P f , N N -

(17)

fI~s,t o p f , A N - ~s,tfl

We extend the isomorphism between

submodel representation.

e Let

copy of the diagonal m.a.s.a,

Let

~ E ~,

and the submodel.

Let T

be the copy of the of

e

which is the

Then

a

is generated

f.

It is now

of the subfactor

in Corollary 4.4 instead of Lemma 4.4, the of

e comes from

M~

M,.

g E G n there exists a projection

p g e a,

with

T(pg) ~ Sen,

(i - p g ) U g = (I -pg)Ug. pg = ! es,sPg,s , s e ~n with T(pg)

For

s,t E ~n

(ug) c e

reason being that only the diagonal m.a.s.a,

such that

~E~,

of the submodel.

and thus behaves well with respect to For each

s ES n

be the m.a.s.a,

and by a m.a.s.a,

that we use the estimates

~E~,

e and the finite dimensional

e = e ® f

and let a

~cE

s,t 6 ~n,

~< ¼1snl -~

submodel to an isomorphism between

s e ~n

~E

~S,Sll ~< en!Snl -l

be the normalized trace on

id~ × (~gI f' DN) = (~g)

sup II~ o ~g-~It, g E Gn+ 1

$ 6 M,,

(16)

(es,s),

subfactor

(~g) such that

(Ad Zg~glf'A N) is outer conjugate to

by

(~gIN) is centrally

We apply Theorem 8.5 to construct a II 1 hyperfinite

sES n

and

=

Pg,s

[ Y(es,s)T(Pg,s) s

projections =

g E G n we have, in view of

in

f.

IsnI -I I T(Pg,s) s (i!) and

(17),

Then

110

~ (es, sPg, s)

=

~s,s (Pg,s) ~s,s(Pf,AN

(Pg,s))

~S,s(T(pg,s)) =

+ ll~s, s °Pg, A N - ~S,sll

+ CnlSnl -I

~(es,s)T(Pg,s)

+ enIsnl -I

(i +en) IsnI-Ir(Pg,s ) + enlSnl -I Hence (Pg)

:

[ ~(es,sPg,s ) < s

(i + e n) Isnl -I ~ r(Pg,s) + e n s

=

(i + e n)T (pg) + e n

<

(i + e n)8s n + e n

<

10e n

<

7e n=

We infer (19)

IIUgUg - ill

=

IIpg (UgUg ~ # ~* - l)p_It IlUgUg-III

hence

We have Ad Vg~g * I~ = id, A d ( Z g V g ) ~ g l e = id. Since

for

(~g), and

the submodel As

thus if we let Ad Vg~gle

,½

Zg E e'n M

(Ug) is a r e p r e s e n t a t i o n

for Ad(ZgVg)eg,

(

~ pg)

of

G

~< 2. (10e n) and

i.e.

Ad(ZgVg)egif

in

Vg = UgZgVg

= Ad Ugle ,

e,

i

z g e e ' N M,

(Ug) is a c o c y c l e

then

(Qg) is a cocycle

(Ad Vg~gle)

is conjugate

to

we have,

via 7.1(10),

~, ~ , = +UgUgZgUgvg 11, < ~<

~ * 2 (IIUgUg - 1114# + llZg - ii[~# + llUgVg

2(7e~ + Sn + i0£n-4)

for g E Gn_4, ~ e ~, where we have used (19), The estimate (i) in Lemma 9.3 is now proved. (12) and

(15) for

Lit OPe, N M

statement

<

- iii

#)

en-5 (14) and

(i0) before.

~ e _= we have

-~Li ~< tlt OPe, n M - ~[[ + H (~IN) ~< (Isnl +i)

which proves

= id,

action.

u g e e and

Vg

From

½

o pf,r]N-

sup lit o ~g-~]] g 6 Gn+ 1

(2) in the lemma.

(~ IN) It

111

Let

~ be the normalized

=

[

trace of

f.

For

~s,t ® ~s,t e (SEN).

~ E ~ with

=

M.

=

e,

s,t as in (13) and

We have

(17), we let qs,t

= ~s,t ® o e

%s,t

=

(e Of),

~s,tle'n M .

~s,t o Pf'N N = ~ ® %s,t ' hence

II~ [ ~s,t®~s,tll s,t

=

II I ~'s,t ® ( ~ s , t - ~ S t ° P f ' n N ) II s,t ' ~ ll~s,t-~s,t °Pf'NNII s,t hlsnl 2 Isnl-26

=

¼6

Since ~gle'n M

= Ad(UgZgVg)~gle'n

M

= Ad(ZgVg)~gle' N M

and Ad(z g v*)a g g le = id we infer II~s,t o (~gle' n M) - #s,tll =

II(o ® ~s,t) o(Ad(ZgV$)dgle'n

M) - 0 ® ~s,tl I

~<

211~s,t Pf'N N - ~s,t II+ ll~s,t ° A d Z g - $ s , t

II

+ ll~s,t o (Ad Vg~gle' n M) -~s,tl I <

2 • ~Isnl-26

+ ha + ~@

~<

where we have used (17), (18) and (19). The last estimate in the lemma is thus obtained by reindexing ~s,t,#s,t,

s,t 6 Isnl

with a single index

Which brings us to the END.

k = 1,2 ..... l~n12 = p2.

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A.CONNES: "Outer c o n j u g a c y classes of a u t o m o r p h i s m s of factors," Ann. Sci. Ec. Norm. Sup. 4me serie, t. 8 (1975),, 383-420.

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[8]

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[9]

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[11]

T.FACK and O.MARECHAL: "Sur la c l a s s i f i c a t i o n des a u t o m o r p h i s m e s p ~ r i o d i q u e s de C * - a l g e b r e s U.H.F.," Canadian Jour. of Math. 31, no. 3 (1979), 469-523.

[12]

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[13]

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[14]

T . G I O R D A N O and V.JONES: " A n t i a u t o m o r p h i s m e s involutifs du facteur hyperfini de type IIl," C. R. Acad. Sci. Paris, s.A 290 (1980), 29-31.

[15]

F.P.GREENLEAF: I n v a r i a n t Means on T o p o l o g i c a l Groups, Van N o s t r a n d Math. Studies No. 16.

[16]

R.I.GRIGORCHUK: "Symmetrical r a n d o m walks on discrete groups," Uspehi Math. Nauk. 32 (1977), 217-218.

[17]

R.HERMAN and V.F.R.JONES: "Period two a u t o m o r p h i s m s of UHF C * - a l g e b r a s , " tO appear J. Funct. Analysis.

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R . H E R M A N and V.F.R.JONES: J. Funct. Anal. 45 (1982).

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J.HUEBSCHMANN:

[20]

V.F.R.JONES: "Sur la c o n j u g a i s o n de s o u s - f a c t e u r s de type IIl," C. R. Acad. Sci. Paris, 284 (1977), 597-598.

factor of

factors," Ann. of Math.

"Cohomology theory in a b s t r a c t 48 (1947), 326-341.

Scand.

"Models of finite group actions,"

Dissertation,

E.T.H.,

ZUrich.

113

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V.F.R.JONES: d'Op~rateurs,

[22]

V.F.R.JONES: "Actions of finite abelian groups on the hyperfinite II 1 factor," preprint.

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V.F.R.JONES: "Actions of finite groups on the h y p e r f i n i t e type II factor," M e m o i r s A.M.S. No. 237 (1980).

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V.F.R.JONES:

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V.F.R.JONES: "Prime actions of compact abelian groups on the h y p e r f i n i t e II 1 factor," J. Operator Theory 9 (1983), 181-186.

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V.F.R.JONES: "A converse to Ocneanu's theorem," J. O p e r a t o r Theory 10 (1983), 61-63. V . F . R . J O N E S and S.POPA: "Some p r o p e r t i e s of MASA's in factors," O p e r a t o r Theory: Adv. Appl. 6, (Birkhiuser, 1982).

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W.KRIEGER: "On ergodic flows and the i s o m o r p h i s m of factors," Math. Ann. 223 (1976), 19-70.

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D.McDUFF: "Central sequences and the h y p e r f i n i t e Proc. London Math. Soc. XXI (1970), 443-461.

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F . J . M U R R A Y and J. von NEUMANN: of Math. 44 (1943), 716-808.

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[33]

A.OCNEANU: "A Rohlin t h e o r e m for groups acting on von N e u m a n n algebras," in Topics in M o d e r n O p e r a t o r Algebra, (Birkhauser Verlag, 1981), 247-258.

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A.OCNEANU: "Actions des groupes m o y e n n a b l e s sur les algebres de von Neumann," C. R. Acad. Sci. Paris, 291 (1980), 399-401.

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A.OCNEANU: "Actions in preparation.

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D . O R N S T E I N and B.WEISS: "Ergodic theory of amenable group actions, I," Bull. A.M.S°, vol. 2, No. 1 (1980), 161-163.

[37]

S.POPA: "On a p r o b l e m of R.V. Kadison on m a x i m a l abelian *-subalgebras," Inventiones Math. 65 (1982), 269.

[38]

J.G.RATCLIFFE:

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M.RIEFFEL: "Actions of finite groups on C * - a l g e b r a s , " Math. Scand. 47 (1980), 157-176.

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S.STRATILA: M o d u l a r T h e o r y in O p e r a t o r Algebras, A c a d e m i e i and Abacus Press, 1981).

[41]

S . S T R A T I L A and D.VOICULESCU: "Representations of A F - a l g e b r a s and of the group U(~), '' Lecture Notes in Math., No. 486 (1975).

factor,"

"Rings of operators,

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"On the e x t e n s i o n s of finite factors, (1959), 215-220.

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S.STRATILA and L.ZSIDO: Lectures on (Editura Academiei and Abacus Press,

[43]

C.SUTHERLAND: II," Preprint.

[44]

M.TAKESAKI: von Neumann 249-310.

"Cohomology

and

extensions

"Duality in c r o s s p r o d u c t s algebras of t y p e I I I , " A c t a

NOTATION (Not

von Neumann 1979).

denotes

the

Ct M en

notation

section

in i.i

. . . . . . . . .

3.4

Gn

. . . . . . . . .

3.4

~n

. . . . . . . . .

3.4

K ni

. . . . . . . . .

3.4

~gn

. . . . . . . . .

3.4

L.n . . . . . . . . . 1,3

3.4

A(~)

. . . . . . . .

2.1

. . . . . . .

2.1

+

Mh , M+ , M 1, M , , M,

Not

Me, Me

. . . . . . .

5. i

mod

. . . . . . .

2.9

N(~)

. . . . . . . .

2.1

O b (e)

. . . . . . . .

2.2

Out

. . . . . . . .

I. 5

. . . . . . .

Not

U (M)

. . . . . . . .

Not

Z (M)

. . . . . . . .

Not

Proj

(e)

M M

operator

algebras,

and the structure Math. 131 (1973),

INDEX

. . . . . . . .

A(G,N)

of

Algebras,

I " 1(~, II'II}, II'I12

Not

the

Introduction.)

of

SUBJECT

action:

centrally cocycle

i.i,

. . . . . .

i.i

free

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

automorphism:

factor:

trivial

liftable

. . . . . .

left

free

....

5.6

strongly

outer

5.6

. . . . .

. . . . . . . . . .

subset(s):

. . . . . . . . .

invariant

. . . . . . .

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

structure

sequence:

5.5

strongly

amenable

obstruction

I.i,

5.5

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

group:

4.5 4.5

....

invariant

. . . . . . . . .

2

1

1 2 1

5

3 1 3 1 2

2

3.4

centralizing

. . . . . .

5.1

normalizing

. . . . . .

5.1

E-disjoint

. . . . . .

3.3

(e,F)-invariant c-paving

....

. . . . . . .

5.2

5.2

semiliftable

McDuff

G-kernel

. . . . . . . . .

centrally

characteristic

paving

. . . . . .

model

submodel

mean:

free crossed

INDEX

3.2 3.3

5.2

Restricted orbit equivalence for actions of discrete amenable groups

Read more

Restricted Orbit Equivalence for Actions of Discrete Amenable Groups

Read more

Von Neumann Algebras

Read more

Von Neumann Algebras

Read more

von Neumann algebras

Read more

Von Neumann algebras

Read more

Operator Algebras. Theory of Casterisk-Algebras and von Neumann Algebras

Read more

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

Read more

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

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Operator Algebras: Theory of C*-Algebras and Von Neumann Algebras

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Hochschild cohomology of Von Neumann algebras

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Amenable locally compact groups

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An Invitation to von Neumann Algebras (Universitext)

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Restricted Orbit Equivalence for Actions of Discrete Amenable Groups (Cambridge Tracts in Mathematics, 146)

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Restricted Orbit Equivalence for Actions of Discrete Amenable Groups (Cambridge Tracts in Mathematics (No. 146))

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Amenable, locally compact groups

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Finite von Neumann algebras and masas

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An Invitation to von Neumann Algebras (Universitext)

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Operator Algebras: Theory of C*-Algebras and von Neumann Algebras (Encyclopaedia of Mathematical Sciences)

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Hopf algebras and their actions on rings

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Duality for crossed products of von Neumann algebras

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Duality for Crossed Products of Von Neumann Algebras

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Duality for crossed products of von Neumann algebras

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Operator algebras. Theory of C-star-algebras and von Neumann algebras (bad p480)

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Discrete groups

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Operator Algebras. Theory of C^asterisk-Algebras and von Neumann Algebras

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Operator algebras: theory of C*-algebras and von Neumann algebras, Volume 122

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Continuous crossed products and type III Von Neumann algebras

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von Neumann density theorem

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Lectures on Lie groups and Lie algebras

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