ERGODIC THEORY AND VON NEUMANN ALGEBRAS AN INTRODUCTION

Von Neumann

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ERGODIC THEORY AND VON NEUMANN ALGEBRAS : AN INTRODUCTION CLAIRE ANANTHARAMAN-DELAROCHE Contents 1. Finite von Neumann algebras : examples and some basics 2 1.1. Notation and preliminaries 2 1.2. Measure space von Neumann algebras 5 1.3. Group von Neumann algebras 5 1.4. Standard form 9 1.5. Group measure space von Neumann algebras 11 1.6. Von Neumann algebras from equivalence relations 16 1.7. Two non-isomorphic II1 factors 20 Exercises 23 2. About factors arising from equivalence relations 25 2.1. Isomorphisms of equivalence relations vs isomorphisms of their von Neumann algebras 25 2.2. Cartan subalgebras 25 2.3. An application: computation of fundamental groups 26 Exercise 27 3. Study of the inclusion L∞(T2) ? L∞(T2)o F2 29 3.1. The Haagerup property 30 3.2. Relative property (T) 31 References 35 1

• group


• closed unital


• topology coincides


• dual banach


• hilbert space


• measure space von


• neumann algebras


• topology

Sujets

• Neumann
• Hilbert space
• Topology

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• Publié par: pefav
• Nombre de lectures: 19
• Langue: English

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ERGODIC THEORY AND VON NEUMANN ALGEBRAS :
AN INTRODUCTION
CLAIRE ANANTHARAMAN-DELAROCHE
Contents
1. Finite von Neumann algebras : examples
and some basics 2
1.1. Notation and preliminaries 2
1.2. Measure space von Neumann algebras 5
1.3. Group von Neumann algebras 5
1.4. Standard form 9
1.5. Group measure space von Neumann algebras 11
1.6. Von Neumann algebras from equivalence relations 16
1.7. Two non-isomorphic II factors 201
Exercises 23
2. About factors arising from equivalence
relations 25
2.1. Isomorphisms of equivalence relations vs isomorphisms
of their von Neumann algebras 25
2.2. Cartan subalgebras 25
2.3. An application: computation of fundamental groups 26
Exercise 27
1 2 1 23. Study of the inclusion L (T )L (T )oF 292
3.1. The Haagerup property 30
3.2. Relative property (T) 31
References 35
12 CLAIRE ANANTHARAMAN-DELAROCHE
1. Finite von Neumann algebras : examples and
some basics
This section presents the basic constructions of von Neumann algebras
coming from measure theory, group theory, group actions and equivalence
relations. All this examples are naturally equipped with a faithful trace
and are naturally represented on a Hilbert space. This provides a plentiful
source of tracial von Neumann algebras to play with.
11.1. Notation and preliminaries. LetH be a complex Hilbert space
with inner-producth ; i (always assumed to be antilinear in the rst
variable), and letB(H) be the algebra of all bounded linear operators fromH
toH. Equipped with the involution x7! x (adjoint of x) and with the
operator norm,B(H) is a Banach-algebra with unit Id . We shall denoteH
bykxk, or sometimeskxk , the operator norm of x2B(H). Throughout1
this text, we shall consider the two following weaker topologies onB(H):
the strong operator topology (s.o. topology), that is, the locally
convex topology onB(H) generated by the seminorms
p (x) =kx k; 2H;
the weak operator topology (w.o. topology), that is, the locally
convex topology onB(H) generated by the seminorms
p (x) =j! (x)j; ;2H;; ;
where ! is the linear functional x7!h;x i onB(H).;
This latter topology is weaker than the s.o. topology. One of its important
properties is that the unit ball ofB(H) is w.o. compact. This is an immediate
consequence of Tychono ’s theorem.
This unit ball, endowed with the uniform structure associated with the
s.o. topology is a complete space. In caseH is separable, both topologies
on the unit ball are metrizable and second-countable. On the other hand,
whenH is in nite dimensional, this unit ball is not separable with respect
to the operator norm (Exercise 1.1).
A von Neumann algebraM onH is a-subalgebra ofB(H) which is closed
2in the s.o. topology and contains Id . We shall sometimes write (M;H) toH
specify the Hilbert space on which M acts. The unit Id of M will also beH
denoted 1 or simply 1.M
0Given a subset S ofB(H), we denote by S its commutant inB(H) :
0S =fx2B(H) :xy =yx for all y2Sg:
1In this text,H is always assumed to be separable.
2We shall see in Theorem 1.3 that we may require, equivalently, that M is closed in
the w.o. topology.ERGODIC THEORY AND VON NEUMANN ALGEBRAS 3
00 0Then S is the commutant of S , that is, the bicommutant of S. Note that
0 0 0 S is a s.o. closed unital subalgebra ofB(H); ifS =S , thenS = (S ) and
0therefore S is a von Neumann algebra onH.
The rst example of von Neumann algebra coming to mind is of course
0 nM =B(H). Then,M =C Id . WhenH =C , we get the algebraM (C) ofH n
nn matrices with complex entries, the simplest example of von Neumann
algebra.
We recall that a C -algebra onH is a-subalgebra ofB(H) which is
closed in the norm topology. Hence a von Neumann algebra is aC -algebra,
but the converse is not true. For instance the C -algebraK(H) of compact
operators on an in nite dimensional Hilbert space H is not a von Neumann
algebra onH : its s.o. closure isB(H).
For us, a homomorphism between twoC -algebras preserves the algebraic
operations and the involution. We recall that it is automatically a
contraction and a positive map, i.e. it preserves the positive cones. For a concise
introduction to the theory of C -algebras, we refer to [16].
1.1.1. Von Neumann’s bicommutant theorem. [25]
We begin by showing that, although di erent (for in nite dimensional
Hilbert spaces), the s.o. and w.o. topologies introduced in the rst chapter
have the same continuous linear functionals. For; in a Hilbert spaceH we
denote by! the linear functionalx7!h;x i onB(H). We set! =! .; ;
Proposition 1.1. Let ! be a linear functional onB(H). The following
conditions are equivalent :
Pn(i) there exist ;:::; ; ;::: 2H such that !(x) = ! (x)1 n 1 n ; i=1 i i
for all x2B(H) ;
(ii) ! is w.o. continuous ;
(iii) ! is s.o. continuous.
Proof. (i)) (ii)) (iii) is obvious. It remains to show that (iii)) (i). Let
! be a s.o. continuous linear functional. There exist vectors ;:::; 2H1 n
and c> 0 such that, for all x2B(H),
nX 1=22j!(x)jc kx k :i
i=1
n times
z }| {
nLetH =HH be the Hilbert direct sum of n copies ofH. We
nset = ( ;:::; )2H and for x2B(H),1 n
(x) = (x ;:::;x ):1 n
The linear functional : (x)7! !(x) is continuous on the vector
subnspace (B(H)) ofH . Therefore it extends to a linear continuous
functional on the norm closure K of (B(H)). It follows that there exists