The hyperfinite II1 factor and Connes Embedding conjecture

João Pedro Guerreiro do Carmo Paulos

Thesis to obtain the Master of Science Degree in

Mathematics and Applications

Supervisor: Professor Doutor Paulo Jorge da Rocha Pinto

Examination Committee

Chairperson: Professor Doutor Miguel Tribolet de AbreuSupervisor: Professor Doutor Paulo Jorge da Rocha PintoMember(s) of the Committee: Professor Doutor Pedro Manuel Agostinho Resende

Professor Doutor Nuno Miguel Matos Ramos Martins

June 2015

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Acknowledgments

This research was partially funded by FCT/Portugal through the project EXCL/MAT-GEO/0222/2012. In-

deed, the finantial support and motivation provided by this project was unequivocally an unsurpassable

cornerstone.

I dedicate this thesis to my Mother and to my Father. Furthermore, I would like to acknowledge a nec-

essary condition for completing this thesis and more generally, for completing my studies: the existence

of Malina and of a very rich neurotic heritage constitued by a tight circle of friends. Their names shall

remain obscured to avoid the blasphemy of amnesia.

Last, and certainly not least, I would like to declare myself thankful for having the opportunity to learn

and get inspired by some teachers I luckily had the chance to have classes with. Their competence and

their virtues as humans, definitively shaped my understanding of reality. Among them, I would like to

emphatize the role of my adviser. Since Functional Analysis class, he has been tireless in his efforts

and availability to enrich my mathematical knowledge and his guidance is priceless. Again, other names

shall remain obscured to avoid the blasphemy of amnesia.

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Resumo

Estabelecemos a existencia e unicidade (a menos de ∗-isomorfismo) do factor hiperfinito do tipo II1, R.

Estudamos a relacao entre grupos hiperlineares e soficos e a Conjectura do mergulho de Connes para

grupos, concluindo que LG pode ser mergulhada numa ultrapotencia RU , com U um ultrafiltro livre em

N, sempre queG seja um grupo livre ou uma soma ou produto directo de grupos mediaveis. Exploramos

a relacao entre a Conjectura do mergulho de Connes e a Conjectura WEP (Weak Expectation Property )

em factores do tipo II1 separaveis. Alem disso, esbocamos uma prova de que todas as ultrapotencias

RU sao isomorfas, independentemente da escolha do ultrafiltro livre U em N, numa cooperacao profıcua

com Teoria de Modelos. Por fim, terminamos com uma abordagem conceptualmente diferente ao es-

tudo dos subfactores de R e para atingir tal desiderato, estudamos algumas ferramentas da Teoria de

Galois nao comutativa. Neste contexto, calculamos alguns grupos de Galois, estabelecemos algumas

correspondencias de Galois e introduzimos o ındice de Jones.

Palavras-chave: Factores do tipo II1, Conjectura do mergulho de Connes, Grupos hiperlin-

eares, Conjectura WEP, Teoria de Galois nao comutativa

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Abstract

We study the existence and uniqueness (up to ∗-isomorphism) of the hyperfinite type II1 factor R. We

study the relation between hyperlinear and sofic groups and Connes’ Embedding Conjecture for groups,

sketching why LG is embeddable into a suitable ultrapower RU , for a free ultrafilter U on N, whenever G

is a free group or a direct product of amenable groups. Furthermore, we explore the relation between the

Connes’ Embedding Conjecture and the Weak Expectation Property on separable type II1 factors and

we present the ideas that establish, under the Continuum Hypothesis, the uniqueness of ultrapowers

RU , independetly of the choice of the free ultrafilter U . We finish this thesis within an approach to the

study of subfactorsM ofR with a different flavour, focusing our attention on the inclusionsM⊂ R. This

lead us to the so called Non-Commutative Galois Theory. We compute some Galois Groups and Galois

Correspondences and we briefly present Jones’ index.

Keywords: Type II1 factors, Connes embedding conjecture, Hyperlinear groups, WEP Conjec-

ture, Non commutative Galois theory

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viii

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Introduction 1

1 Projections in von Neumann Algebras 6

1.1 Order in Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Finite, Infinite and Abelian Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 The Theorem of Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.1 Measure-theoretic construction of factors of any type . . . . . . . . . . . . . . . . . 14

1.4.2 Type II1 factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Uniqueness and Existence of the hyperfinite type II1 factor 23

2.1 The Dimension Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Uniqueness of the hyperfinite type II1 factor . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 A model for the hyperfinite type II1 factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Connes Embedding Conjecture 31

3.1 Formulation of the conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Connes Embedding Conjecture for Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 About the definition of hyperlinear groups . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.2 Hyperlinear Groups and Connes Embedding Conjecture . . . . . . . . . . . . . . . 39

3.2.3 Sofic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 CEC and WEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.1 Group C∗-algebras and Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.2 CEC and WEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Other equivalent formulations of Connes Embedding Conjecture . . . . . . . . . . . . . . 55

3.5 Model Theory and Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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4 Non-Commutative Galois Theory 62

4.1 Motivation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.1 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.2 Conjugations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Galois Theory and the hyperfinite type II1-factor . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.1 Galois Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.2 Computing some Galois Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.3 Galois Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.4 Computing some correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A Preliminaries 77

A.1 Facts about von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A.2 AF Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

A.2.1 UHF Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

A.2.2 Inductive Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.3 States, Representations and Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

A.3.1 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

A.3.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.3.3 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.4 Ultrafilters and Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.4.1 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.4.2 Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A.5 Crossed Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

A.6 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

A.6.1 Algebraic Tensor Product of R-modules . . . . . . . . . . . . . . . . . . . . . . . . 88

A.6.2 Tensor Product of Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

A.6.3 Tensor Product of C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

A.6.4 Maximal and Minimal Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . 91

A.6.5 Tensor Product of von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . 91

Bibliography 91

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Introduction

The main purpose of this thesis is to study a certain kind of von Neumann algebras. More specifically,

those that are weak closures of UHF algebras, commonly known as hyperfinite von Neumann algebras.

In particular, we will focus our attention to a very special case of hyperfinite von Neumann algebras, the

hyperfinite type II1 factors. The chapter of greater relevance in this thesis is the third one. Here, we

will explore what is arguably one of the most important open problems in modern Operator Algebras,

the Connes’ Embedding Conjecture. In 1976, the prominent French mathematician Alain Connes in

his seminal work [13], suggested that it ought to be the case that all type II1 factors with separable

dual should be embeddable into a suitable ultrapower of the hyperfinite type II1 factor.1 Since then,

this remark rapidily evolve into an open problem with an overwhelming impact in modern Mathematics.

At the pace of multiple failed attempts to prove (or disprove) the result, it became clear that this con-

jecture has profuse and deep implications with a wide and heterogeneous landscape, often gathering

multidisciplinary efforts and allowing new perspectives on a large spectrum of different areas of mod-

ern Mathematics. Consequently, this is an actively researched problem, situated in the fringe of human

knowledge.

In the first chapter, we deal with some standard results. We introduce important terminology regarding

families of projections. This will lead us to different types of von Neumann algebras. Even though the

content of the first chapter is classic, we establish some fundamental results that implicitly will follow us

during the rest of this work. We could summarize the first chapter as follows:

Given any von Neumann algebra M, one define an equivalence relation in the class of projections P

of M setting two projections E and F equivalent if and only if there is a partial isometry V such that

E = V ∗V and F = V V ∗. Promptly we notice that if we declare E F whenever E is equivalent to

a subprojection of F , we have defined a partial ordering in P. More, if we are dealing with a factor

M, actually is a total ordering in P. Still in this spirit we introduce some typology of projections,

namely finite, infinite and abelian projections. This immediately lead us to the definition of the type of a

von Neumann algebra M. After this, we briskly reach the most fundamental result in this chapter, the

Theorem of Decomposition of von Neumann algebras. This result establishes that any von Neumann

algebraM can be decomposed as a direct sum of von Neumann algebras of different types. In a way,

this means that the characterization we previously defined (in types In with n ∈ N ∪ ∞, II1, II∞ and

1”Apparently such an embedding ought to exist for all II1 factors because it does for the regular representation of free groups”([13], pg. 105)

1

III) constitute the atoms of any von Neumann algebra. More importantly, ifM is a factor, we prove that

M is either of type In, II1, II∞ or III, i.e. it is an atom in our analogy.

We finish the first chapter giving examples of factors of each type. In order to accomplish this, one de-

velops a very interesting measure-theoretic construction that allow us, in a simple way, the construction

of examples. Furthermore, we introduce a very rich class of von Neumann algebras : Given a discrete

group G, we define the von Neumann algebra LG. These, are always finite von Neumann algebras.

Moreover, if G is an infinite conjugacy class (i.c.c.) group, we show that LG is in fact a type II1 factor.

For instance, any Fn is an i.c.c. group and thus, LFn is a type II1 factor. However, it is still and open

problem whether LFn and LFm are isomorphic or not if n 6= m. This motivates the search for families

of non-isomorphic factors of a certain type. We give a brief historical overview of this issue and we

culminate in a sketch of the construction of the first countably infinite family of non-isomorphic type II1

factors, following the work of Dusa McDuff.

In the second chapter, we establish the unicity of the hyperfinite type II1 factor. This is an amazing

result. In the appendix, the reader can check a section dedicated to AF algebras in which we establish

that any two UHF with the same type nj are isomorphic. Moreover, in the same section, we define an

invariant δ for any UHF algebra, regardless the type and we conclude that there are uncountably many

classes of isomorphism for UHF algebras. This makes our result of unicity even more remarkable, since

what we are saying is that there is only one weak closure (up to isomorphism) of a UHF algebra such

that the result is a type II1 factor. We finish this chapter providing a model for the hyperfinite type II1

factor. We could summarize the second chapter as follows :

We develop some techniques involving the so called dimension function. These techniques are actually

related to continuous geometry, introduced by von Neumann, where instead of a dimension of a sube-

space being in a discrete set, it can be any element of an interval. This will lead us to a central result that

allow us a very useful and working criteria to identify factors regarding their type. Particulary, if one has

a finite factor M, we can expeditiously identify its type just by studying the range of its trace restricted

to the set of projections P ofM. In the case of our interest, a factorM is of type II1 if and only if the

range of the restriction of τ to P is the interval [0, 1], where τ denotes the trace of M. We proceed to

prove the unicity of the hyperfinite type II1 factor R and we finish the chapter presenting a model for its

construction, in which we end up working with some results considering the characterization of a type

II1 factor regarding its trace.

In the third chapter, we present the Connes’ Embedding Conjecture (CEC). This is arguably one of

the most important problems in Operator Algebras which makes this chapter of great relevance. The

importance of this conjecture is fully justified by its profuse and deep connections with a wide range of

seemingly not related open problems in different areas of Mathematics, which usually lead us to a mul-

tidisciplinary landscape. The conjecture states that every type II1 separable factor can be embedded

into an ultrapower of the hyperfinite type II1 factor, say RU , for a choice of a free ultrafilter U on N.

Here, by embedding we mean a ∗-homomorphism that is also an isometry. Since under the Continuum

2

Hypothesis one can show that there is only one ultrapower RU (up to isomorphism), if the CEC is true

then it means that there is a type II1 factor (RU ) that contains every type II1 separable factor. On the

other hand, it is known that every type II1, not necessarily separable, contains R. Hence, if CEC is

true and considering only separable type II1 factors, we can see R and RU as, respectively, minimal

and maximal objects. During this chapter, we will approach with some detail the Connes’ Embedding

Conjecture for groups. The latter asserts that for every i.c.c. countable group G, LG can be embedded

into some ultrapower RU . Furthermore, we will establish the equivalence between CEC and the Weak

Expectation Property (WEP) Conjecture, that states that every separable type II1 factor has the WEP,

i.e. every separable type II1 factorM ⊂ L(H) is such that contains a weakly dense C∗-subalgebra A

and an unital completely positive map Φ : L(H)→M such that Φ(a) = a for every element a ∈ A. The

chapter is completed with two more sections. The first one is a small collection of equivalent formula-

tions of the CEC that appear naturally in different frameworks. This should emphatize the relevance of

the conjecture in modern Mathematics. The second one is a short stroll through applications of Model

Theory in Operator Algebras. We could summarize the third chapter as follows :

First, we formulate the Connes’ Embedding Conjecture. In order to do so, we go through some con-

siderations regarding separability in von Neumann algebras and the existence of free ultrafilters on N.

In this context, the Frechet filter arises and by a standard Zorn’s Lemma argument, we establish the

existence of free ultrafilters. Another detail that is worth exploring while formulating the conjecture is

related with the use of free, instead of principal ultrafilters. Here, we establish that an ultrapower of

the hyperfinite type II1 factor, RU , is never hyperfinite if U is chosen to be free. After the formulation

of the CEC, we introduce the concept of a hyperlinear group. Our purpose is to establish the relation

between hyperlinear groups and the CEC for groups. Indeed, the main result from this section states

that a group G is hyperlinear if and only if LG can be embedded into some suitable ultrapowerRU . From

this result, we conclude that if G is a countable i.c.c. group, then the CEC for groups holds if and only

if G is hyperlinear. To establish this alternative formulation of the CEC for groups, we heavily rely on

the ultrafilters machinery. One very important step to show this equivalent formulation of the CEC for

groups is a result that characterizes any hyperlinear group G in terms of the unitary group U(RU ) for a

free ultrafilter U that can be taken on N if G is countable. Indeed, G is hyperlinear if and only if G can

be embedded into some U(RU ). Then, we introduce the concept of a sofic group. We prove that if G

is sofic, then G is hyperlinear. This immediately let us conclude that if every countable i.c.c. group G

is sofic, then the CEC for groups is true. We will see that given a family Gii∈I of sofic groups, then

both⊕

i∈I Gi and∏i∈I Gi are sofic. Joining this results with the fact that any amenable group is sofic

(which includes, for instance all abelian, solvable or finite groups), we easily conclude that if Gii∈I is

a family of amenable groups and G denotes its direct sum or its direct product, then LG is embeddable

into some suitable ultrapower RU . Following this flavour, we conclude this section sketching why LFn

is embeddable into some ultrapower RU , combining our previously established results on sofic and hy-

perlinear groups with the concept of LEF groups (LEF stands for locally embeddable into finite groups).

This result was actually Connes motivation for his conjecture. Furthermore, we provide an alternative

argument that guarantee that RU is never hyperfinite for a free ultrafilter U , but this time assuming the

3

Continuum Hypothesis. After this, we come back to the CEC in its stronger formulation. The goal will be

showing the equivalence between the WEP Conjecture and the CEC. First, we recall some necessary

facts about free products and group C∗-algebras. To establish the desired equivalence, we will prove

some technical results about invariant means. A very important step to obtain our equivalence, is to

prove that given a separable type II1 factorM, there is a ∗-monomorphism ϕ : C∗(F∞)→M such that

ϕ(C∗(F∞))WOT

= M. This will rely in two wonderful results. One that asserts that given a sequence

knn∈N there is always a ∗-monomorphism such that∏n∈NMkn(C) → M (hereM is not necessarily

separable). In particular, this means that any type II1 factor contains the hyperfinite type II1 factor. The

other, that asserts that C∗(F∞) is residually finite dimensional. We continue this chapter with some ex-

amples of equivalent formulations of the CEC. Remarkably, one can show that the CEC is equivalent to

the Kirchberg Conjecture, that asserts that C∗(F∞)⊗minC∗(F∞) is isomorphic to C∗(F∞)⊗maxC∗(F∞).

This is indeed rather unexpected. While Kirchberg Conjecture regards a property of a particular object

(an universal C∗-algebra), the CEC regards a class of objects (type II1 factors). We finish this chapter

with a very short stroll trough applications of Model Theory to Operator Algebras. Our purpose is just to

empathize the usefulness of this approach and we sketch an argument that establishes that under the

Continuum Hypothesis, all ultrapowers RU , where U is a free ultrafilter on N, are isomorphic.

In the fourth and last chapter, we redirect our attention to Non-Commutative Galois Theory. Meanwhile

we digress on the Jones’ index and towers of algebras. We could summarize the fourth chapter as

follows:

We introduce some terminology and prove some results about actions of groups on von Neumann

algebras. These results of technical nature will be very useful throughout the rest of the chapter. One of

the most important result of this preliminary stage, is the existence of an outer action of any finite group

G on the hyperfinite type II1 factor R. Then, we introduce the concept of conjugation of inclusions

of von Neumann algebras. This is in fact an equivalence relation in the class of pairs N ⊂ M of von

Neumann algebras. After establishing some technical results about conjugations of inclusions, we reach

the fundamental definition of a Galois group : given a pair of von Neumann algebras N ⊂M we define

the Galois group to be Gal(M,N ) = α ∈ Aut(M) : α|N = 1N . One of the reasons why it is so

interesting to study the Galois group comes from the fact that any infinite dimensional subfactor N of

R, is necessarily isomorphic to R itself. This is a well known major result from the groundbreaking

works of Murray and von Neumann ([37]) and Connes ([13]). On one hand, Connes established that

any subfactor N ⊂ R is hyperfinite. On the other hand, Murray and von Neummann established the

uniqueness of the type II1 hyperfinite factor. However, we can recover many information just by studying

the inclusion N ⊂ R and thus, with the help of invariants regarding these inclusions, one can distinguish

subfactors of R, even if they are all isomorphic. One of the ways of doing this, is using the Galois group

as an invariant in the sense that two conjugated inclusions, will have the same Galois group. Other

important tool to extract information from an inclusion N ⊂ M of von Neumann algebras is the index

[M : N ]. At this point, we start an intermezzo on Jones’ index and towers of algebras and we sketch

the ideas behind the proof of an amazing result: Let N be a subfactor of a type II1 factor M. Then,

4

either [M : N ] = 4 cos2(πq ) for some integer q ≥ 3 or [M : N ] ≥ 4. More, if M = R, then there is

a subfactor of R for any admissible value of the index. After this intermezzo, we proceed to compute

some Galois groups. After this, we finally approach the Galois correspondence in a non-commutative

framework. Recall that from classical Algebra, given a field extension K ⊂ L, one has a biunivocal

correspondence between the subgroups of the associated Galois group (in the classical sense) and the

subfields M such that K ⊂ M ⊂ L. In our case, given a finite group G and an outer action α : G → R,

we will establish biunivocal correspondences between subgroups H ⊂ G and subfactors between RG

(the algebra fixed by α) and R. We will also establish biunivocal correspondence between subgroups

H ⊂ G and subfactors between R and Roα G and between subgroups of G/H, for a normal subgroup

H, and subfactores between RG and RH . Since we know from the beginning of this chapter that indeed

such actions exist, then all this correspondences exist and provide us useful information.

Throughout the text, I tried to make an effort to expose the contents in a clear and coherent way. The

abundance of remarks along the chapters, materialize this effort, trying to provide a guide for a proof or

even a comment about some detail in a proof. Other remarks, provide examples or serve simultaneously

as a map and as a prologue regarding the sequence of ideas worked on this thesis. To give some few

examples per each section, for instance Remarks 1.3.7, 2.1.10, 3.1.13, 3.2.8, 4.2.23 or 4.2.24. Many

other examples could be given, especially along the third and fourth chapter.

Although the skeleton of all chapters is sustained in the bibliographical references properly indicated,

many details are often ommited in the original proofs. For the sake of clearness, I tried to make an

effort to either complete them or write them in a different manner. Almost every important proposition

or theorem in this thesis has a personal touch, in a way I honestly hope it will not disrupt the readibility.

Moreover, I also tried to give a conceptual review on the topics, often leading to add some results and

to make small detours. As an example, in chapter 3.1 we formulate Connes Embedding Conjecture and

we make some considerations about its meaning or an alternative proof that RU is not hyperfinite when

U is free, given in Theorem 3.2.53. Many other examples could be given.

On the other hand, I tried to keep things as interesting as I could. To do so, some links between concepts

used along these chapters and other areas of Mathematics, are explored. On other occasions, I tried to

provide supplementary information or to sketch the ideas behind some constructions that complement

the central results of this thesis. Some examples include a sketch of McDuff’s construction in chapter

1.4, a digression on Model Theory in chapter 3.5 and on towers of algebras in chapter 4.2, a two-line

proof of Tychonoff theorem using ultrafilters toolbox or a stroll along an invariant for UHF algebras in the

appendix. Many other examples could be given.

Finally, I tried to include a comprehensive and concise appendix, hoping to make this text more self-

contained and to enlarge the range of potential readers.

5

Chapter 1

Projections in von Neumann Algebras

The aim of this chapter is to establish the Decomposition Theorem for von Neumann Algebras and the

existence of factors of all types. Meanwhile, we introduce a partial ordering in equivalence classes of

projections and we develop some useful techniques. The basics on von Neumann algebras can be

found in the first chapter of the Appendix.

1.1 Order in Projections

Definition 1.1.1. Let H be a Hilbert space. We say that P ∈ L(H) is a projection, if P 2 = P (P is

idempotent) and if P ∗ = P (P is self-adjoint).

Given a C∗-algebra A, we say that p ∈ A+ ⊂ A is a positive element if p is self-adjoint and if

σ(p) ⊂ R+0 , where σ(p) denotes the spectrum of p. Thus, any projection P is a positive element.1 Given

this definition, recall that we can consider a partial ordering in S ⊂ L(H), where S is the set of self-

adjoint operators, defining H ≤ K if K − H ∈ L(H)+ for H,K ∈ S. Having this partial ordering in

mind, we can define a lattice structure in the set P ⊂ S of projections, just by restriction. In fact, recall

that there is a natural correspondence between closed subspaces Y ⊂ H and elements of P, since

for every closed subspace Y there is an unique projection E ∈ P such that it range is Y and, on the

other hand, the range of any element of P is a closed subspace of H. One can show that E,F ∈ P are

such that E ≤ F if and only if Y ⊂ Z, where Y, Z are respectively the ranges of E and F (check [26],

Prop.2.5.2.). Thus, one can easily see that the restriction of the partial ordering of S to P can be viewed

as the partial ordering by inclusion of the closed subspaces of H. Now note that given a family Yα of

closed subspaces there exists the biggest closed subspace contained in every Yα - which is just⋂Yα

and will be denoted by ∧Yα - and there exists the smallest closed subspace that contains every Yα -

which is⋃Yα and will be denoted by ∨Yα. From this it follows that given a family Eα in P there exists

the greatest lower bound ∧Eα and the least upper bound ∨Eα for the family Eα, when one considers

the formerly mentioned partial ordering in P. Indeed, ∧Eα is the projection which has ∧Eα(H) as range

1Indeed, let P be a projection. Then, since P 2 = P and since, by Spectral Mapping Theorem, σ(P 2) = (σ(P ))2, we have thatσ(P ) ⊂ R+

0

6

and ∨Eα is the projection which has ∨Eα(H) as range. Since the map E 7→ 1 − E reverts the ordering

of projections, it is immediate to conclude that ∧(1− Eα) = 1− ∨Eα and that ∨(1− Eα) = 1− ∧Eα. As

a consequence, ∨E⊥α = (∧Eα)⊥ and also ∧E⊥α = (∨Eα)⊥.

Recall that we say that V ∈ L(H) is a partial isometry if for all x ∈ (ker(V ))⊥ we have that ||V x|| = ||x||.

In this case, we say that (ker(V ))⊥ is the initial space and that =(V ) is the final space. It should be

remarked that the composition of partial isometries is not necessarily a partial isometry. However, we

have the following useful criteria:

Lemma 1.1.2. Let U and V be partial isometries on a Hilbert spaceH. LetW = UV . Then, the following

are equivalent:

1. W is a partial isometry.

2. The initial space of U is invariant under the projection on the final space of V .

3. The final space of V is invariant under the projection on the final space of U∗.

Proof: The interest reader can check [18].

Definition 1.1.3. Let E and F be projections in a von Neumann algebra R. We say that E and F are

Murray-von Neumann equivalent if there exists a partial isometry V such that E = V ∗V and F = V V ∗.

In this case, we write E ∼ F .

One can verify that ∼ is indeed an equivalence relation in the set of projections P in R. In fact, let

E,F,G ∈ P such that E ∼ F and F ∼ G. Then, we have that E ∼ E, F ∼ E and E ∼ G : Let V and

W be partial isometries such that V ∗V = E, V V ∗ = F , W ∗W = F and WW ∗ = G. Thus, F = (V ∗)∗V

and E = V ∗(V ∗)∗ and so we have that F ∼ E. On the other hand, since E = E2 = E∗E = EE∗, we

conclude that E is a partial isometry with initial and final space =(E). Thus 2,E ∼ E. Finally, note that

(WV )∗WV = V ∗FV = V ∗V = E and that WV (WV )∗ = WEW ∗ = WW ∗ = G and thus, E ∼ G.

Definition 1.1.4. Given E,F ∈ P such that E ≤ F , we say that E is a subprojection of F . If E is

equivalent to a subprojection of F , we say that E is weaker than F and we denote it by E F . If we

write E ≺ F , we mean that E F and that E is not equivalent to F .

Proposition 1.1.5. Let Pαα∈I and Qαα∈I be orthogonal families of projections in R such that for

each α ∈ I we have that Pα Qα. Then,∑Pα

∑Qα. If for each α ∈ I we have that Pα ∼ Qα, then∑

Pα ∼∑Qα.3

Proposition 1.1.6. Let E and F be equivalent projections in R. Then, for each central projection P of

R, we have that PE ∼ PF . If instead E F , then we have that PE PF .

2Recall that V is a partial isometry if and only if V ∗V is a projection. Moreover, if E = V ∗V and F = V V ∗, with E,F ∈ P ,notice that V is a partial isometry with initial space =(E) and final space =(F ).

3Given an index set I and a family Eii∈I of operators, when we write∑Ei we actually refer to limi

∑i∈I Ei, taken in the

SOT topology.

7

Proof: Let V be a partial isometry such that V ∗V = E and V V ∗ = F . Since P ∈ C, where C denotes

the center of R, we have that (PV )∗PV = PE and PV (PV )∗ = PF . Thus, PE ∼ PF . Moreover, if

E ∼ G ≤ F , we conclude that PE ∼ PG ≤ PF .

Proposition 1.1.7. Let E and F be projections in R such that E F and F E. Then, E ∼ F .

Proof: Suppose that E ∼ F1 ≤ F and that F ∼ E1 ≤ E. Let U and V be partial isometries such that

E = U∗U , F1 = UU∗, F = V ∗V and E1 = V V ∗. We shall show that P ∼ Q. Let En and Fn be

families of projections defined by induction: Let E0 = E and F0 = F and define En = V Fn−1V∗ and

Fn = UEn−1U∗. Notice that E1 = V FV ∗, F1 = UEU∗ and since E1 ≤ E and F1 ≤ F , it follows easily

by induction that En and Fn are decreasing families of projections. Now, define E∞ = ∧n∈NEnand F∞ = ∧n∈NFn. Since En → E∞ (SOT) and Fn → F∞ (SOT), we conclude that UE∞U∗ = F∞

and V F∞V∗ = E∞. Therefore, E∞ ∼ F∞ since (UE∞)(UE∞)∗ = F∞ and (UE∞)∗(UE∞) = E∞.4

One can also check that U(En − En+1)U∗ = Fn+1 − Fn+2 and V (Fn − Fn+1)V ∗ = En+1 − En+2 and

analagously conclude that (En −En+1) ∼ (Fn+1 −Fn+2) and (Fn −Fn+1) ∼ (En+1 −En+2). Therefore,

for each n ∈ N we have that (E2n − E2n+1) ∼ (F2n+1 − F2n+2) and (F2n − F2n+1) ∼ (E2n+1 − E2n+2).

On the other side, note that E − E∞ =∑∞n=0(En − En+1) and that F − F∞ =

∑∞n=0(Fn − Fn+1).

Finally, joining these observations, we can conclude that P ∼ Q. Since E∞ ∼ F∞, one has that∑(F2n+1 − F2n+2) ∼

∑(E2n − E2n+1) and that

∑(F2n − F2n+1) ∼

∑(E2n+1 − E2n+2). Thus,

E =

∞∑n=0

(E2n − E2n+1) +

∞∑n=0

(E2n+1 − E2n+2) + E∞

F =

∞∑n=0

(F2n+1 − F2n+2) +

∞∑n=0

(F2n − F2n+1) + F∞.

Proposition 1.1.8. Let E,F,G ∈ P such that E F and F G. Then, E G.

Proof: Let V and W be partial isometries such that V V ∗ = E, V ∗V = E1 ≤ F and such that W ∗W = F

and WW ∗ = F1 ≤ G. Let T = WE1W∗ ≤ F1. Now notice that according to Lemma 1.1.2, WV is a

partial isometry. Indeed, we have that W ∗W (=(V )) ⊂ =(V ). Furthemore, WV has with initial space

=(E) and range =(T ). Therefore, E ∼ T ≤ G and thus E G.

Remark 1.1.9. We just proved that is a partial ordering in the equivalence classes of projections. In

Cor.1.1.14, we will see that if R is a factor, is in fact a total order.

Definition 1.1.10. Let R be a von Neumann algebra and E a projection in R. Let P be the union of

central projections Pα ∈ R such that PαE = 0. We say that the projection 1− P is the central carrier of

E and we denote it by CE . Alternatively we can define CE as the intersection of all central projections

Q such that QE = E. Therefore, if Pα is the family of the central projections that majorate E, we have

that CE = ∧αPα.

4(UE∞)(UE∞)∗ = UE2∞U∗ = F∞ and (UE∞)∗(UE∞) = E∞EE∞ = E∞.

8

Remark 1.1.11. Now we make an important remark : If E and F are equivalent projections in R, then

CE = CF . To show this it is useful to check that the range of a central projection CE is the set RE(H)

(considering that R acts on H). Now let V be a partial isometry such that V ∗V = E and V V ∗ = F

and notice that E = V ∗FV . Thus, RE(H) = RV ∗FV (H) ⊂ RF (H). Similarly we also have that

RF (H) ⊂ RE(H) and we can conclude that RE(H) = RF (H) and therefore, CE = CF .

Proposition 1.1.12. Let E and F be projections in R. Then, E and F have equivalent non-zero subpro-

jections if and only if CECF 6= 0.

Theorem 1.1.13. Let R be a von Neumann algebra. Let E and F be projections in R. Then, there is a

central projection Z such that ZE ZF and (1− Z)F (1− Z)E.

Proof: Let F be the family of all families (Eα, Fα)α∈I of ordered pairs where Eα and Fα are projections

in R such that EαEβ = 0, FαFβ = 0 - with α, β ∈ I - and Eα ≤ E, Fα ≤ F and Eα ∼ Fα. Obviously

F 6= ∅, since (0, 0) ∈ F . Let’s consider F with a partial order such that (Eα, Fα)α∈I ≤ (Eβ , Fβ)β∈Jif and only if (Eα, Fα) : α ∈ I ⊂ (Eβ , Fβ) : β ∈ J. It should be clear that every chain Cαα∈Λ ∈

F has a upper bound in F given by⋃α∈Λ Cα. Thus, by Zorn’s Lemma there is a maximal element

(E0α, F

0α)α∈I in F . Recall that by definition E0

αE0β = 0, F 0

αF0β = 0, E0

α ≤ E, F 0α ≤ F and E0

α ∼ F 0α.

Define now E0 = ∨α∈IE0α and F 0 = ∨α∈IF 0

α. Note that since Eαα∈I and Fαα∈I are orthogonal

families, we have that E0 =∑E0α (SOT) and that F 0 =

∑F 0α (SOT). Therefore, by Prop.1.1.5 we

conclude that E0 ∼ F 0. Since (E0α, F

0α)α∈I is a maximal family in F , there are not non-zero projections

E′, F′

inR such that E′ ≤ (E−E0), F

′ ≤ (F −F 0) and E′ ∼ F ′ , otherwise (E0

α, F0α)α∈I ∪(E

′, F′) ∈

F , which is a contradiction. Thus, by Prop.1.1.12 we can conclude that CE−E0CF−F 0 = CF−F 0CE−E0 =

0. Now, define Z = CF−F 0 . Since CF−F 0CE−E0 = 0, we have that =(CE−E0) ⊂ =(CF−F 0)⊥. Since

=(E−E0) ⊂ =(CE−E0) and =(CF−F 0)⊥ = =(1−CF−F 0), we have =(E−E0) ⊂ =(CF−F 0)⊥. From this

it follows that (1−Z)(E−E0) = E−E0. Thus, ZE = ZE0. Since (F−F 0) ≤ Z we have that Z(F−F 0) =

(F − F 0) ≥ 0 and so, ZF 0 ≤ ZF . On the other hand, E0 ∼ F 0 and, by Prop.1.1.6, we conclude that

ZE = ZE0 ∼ ZF 0 and that ZF 0 ≤ ZF imply that ZE ZF . Finally, from (F − F 0)Z = F − F 0 we

have that F (1 − Z) = F 0(1 − Z) and, analogously, F (1 − Z) ∼ E0(1 − Z) and E0(1 − Z) ≤ E(1 − Z).

Therefore, F (1− Z) E(1− Z).

Corollary 1.1.14. Let R be a factor and E and F be projections in R. Then, either E ≺ F or F ≺ E.

Proof: By Theorem 1.1.13, there is a central projection Z ∈ R such that ZE ≺ ZF and that (1−Z)F ≺

(1− Z)E. Since C = C1, either Z = 0 - and in this case F ≺ E - or Z = 1 - and in this case E ≺ F .

1.2 Finite, Infinite and Abelian Projections

Definition 1.2.1. Let R be a von Neumann algebra and let E be a projection in R. We say that E is an

infinite projection if there is a projection E0 < E such that E ∼ E0. Otherwise, we say that E is a finite

projection5. If E is infinite and for each central projection P either PE = 0 or PE is infinite, we say that5Notice that this definition is totally analagous to the usual definition of finite and infinite set. In this sense, some of the following

results are pretty familiar, being analogous with the ones from Set Theory.

9

E is properly infinite.

Proposition 1.2.2. Let E be a finite projection in R. Then, all subprojections of E are finite. Moreover,

all minimal projections in R are finite and if E ∼ F , then F is finite.

Proof: Suppose E0 is a subprojection of E and V is a partial isometry such that V ∗V = E0 and

V V ∗ = E1, with E1 ≤ E0. We will prove that E0 = E1. For this purpose, is enough to check that

E−E0 +V is a partial isometry with initial space =(E) and final space =(E−E0 +E1). Since E is finite,

we have that E = E −E0 +E1 which shows that E1 = E0 and so, every subprojection of E is also finite.

Now, suppose that E ∼ F and let F0 ≤ F such that F ∼ F0. We will show that F0 = F . Let then V and

W be partial isometries such that V ∗V = E, V V ∗ = F , W ∗W = F and WW ∗ = F0. In this case, one

can check that E ∼ V ∗F0V ≤ E, using the partial isometry V ∗WV and so, since E is finite, we conclude

that V ∗F0V = E. Therefore, F0 = F , since V V ∗F0V V∗ = FF0F = F0 = V EV ∗ = F . We can conclude

now that F is finite. Finally, let G be a minimal projection. If this is the case, the only subprojection of G

is the zero projection, which is clearly finite.

Proposition 1.2.3. Let Pα be a family of central projections and E a projection inR such that for each

α we have that PαE is finite. Let P = ∨αPα. Then, PE is finite.

Proof: Suppose by contradiction that there is F ∼ PE, with F < PE. Then, 0 6= PE − F ≤ P .

Notice that there is some α such that (PE − F )Pα 6= 0, otherwise suppose that for each α we have that

(PE − F )Pα = 0. In this case, 0 = (PE − F )P = PE − F which contradicts our hypothesis. Therefore,

there is some α such PαF < PαPE = PαE, where the last equality comes from Prop.1.1.6. But now we

reached to a contradiction, since PαE is finite. Thus, F = PE and we conclude that PE is finite.

Proposition 1.2.4. Let E be an infinite projection in R. Then, there is a central projection P in R such

that P ≤ CE , PE is properly infinite and (1− P )E is finite. Moreover, if E is properly infinite and E ∼ F ,

then F is also properly infinite.

Proof: Let Qα be a maximal orthogonal family of central projections in R such that QαE is finite

for each α. It is straightforward, using Zorn’s Lemma, to check that such family exists. By Prop.1.2.3

we have that QE is finite, where Q = ∨αQα =∑αQα - since the family is orthogonal. Moreover, by

maximality of Qα it is easy to check that P = 1−Q is such that PE is properly infinite and (1−P )E is

finite. Now, let E be properly infinite, F ∼ E and P a central projection such that PE 6= 0. By Prop.1.1.6

we have that PF ∼ PE 6= 0 and since E is properly infinite, PE is also properly infinite. Now, just

by definition of infinite projection and by Prop.1.2.2 we can conclude that PF is infinite and thus, F is

properly infinite.

Proposition 1.2.5. LetG be a finite projection inR and let Eαα∈A and Fββ∈B be orthogonal families

of subprojections of a given projection E, such that they are maximal to the property that Eα ∼ Fβ ∼ G

for all α ∈ A and β ∈ B. Then, |A| = |B|.

Definition 1.2.6. Let E be a projection in R such that ERE is an abelian von Neumann algebra. In this

case, we say that E is an abelian projection.

10

Let C be the center of R. Let E ∈ R′ be a projection. Then, RE is a von Neumann algebra acting

in E(H) with center CE and commutant ER′E. Before sketching the proof of the former fact, we notice

that if R is a von Neumann algebra acting in H and E is a projection in R, then ERE is a von Neumann

algebra acting in E(H). To see this is enough to check that ERE is the commutant of R′E and that

R′E is a von Neumann algebra, since E ∈ R = R′′ by the Bicommutant Theorem. Now we proceed

to sketch the proof of the first result : It is not hard to check that the map ϕ : A 7→ AE is weakly

continuous. Since R is a von Neumann algebra, we have that B1R is weakly compact - the reader can

recall this and other facts in the appendix. Thus, B1RE is weakly compact due to the fact that ϕ is weakly

continuous. One can check that B1RE = B1

RE and notice that B1RE = B1

REWOT

since being compact

implies being closed (recall that L(H) is Hausdorff with WOT topology ). Thus, B1RE = B1

RE and we can

finally conclude that RE is a von Neumann algebra. It is merely computational to check that the center

of RE is CE and that (RE)′

= ER′E.

Proposition 1.2.7. 1. Every subprojection of an abelian projection in R is the product of that projec-

tion with a central projection.

2. A projection is abelian if and only if it is minimal in the class of projections in R with the same

central carrier.

3. All abelian projections in R are finite.

Proof:

1. Suppose E is an abelian projection in R and F ≤ E. Let R = ER in the latter comment and we

conclude that ERE = CE, since E is abelian. Moreover, since F = C0CEE, it remains to show

that C = C0CE is a central projection.

2. Let E be an abelian projection and let F be such that CF = CE . Suppose F ≤ E. From (1)

we have that F = PE, with P a central projection. Now, notice that CE = CF ≤ P and thus,

PE = E = F and so we conclude that E is minimal in the sense of the hypothesis. Reciprocally,

suppose that E is minimal in the class of projection that have central carrier CE . We will show that

ERE ⊂ CE and thus, E is abelian. Let G ≤ E and so, we have that G ≤ CGE.6 If G < CGE, then

G+(1−CG)E < E and since G+(1−CG)E has central carrier CE , we contradict the minimality of

E. Thus, G = CGE and each projection in ERE is in CE. Since ERE is a von Neumann algebra,

it is generated by its projections and thus we conclude that ERE ⊂ CE.

3. Let E be an abelian projection and let F ∼ E such that F < E. Since the projections are equiv-

alent, CE = CF . But this is impossible, since by (2) we know that E is minimal in the class of

projections with central carrier CE .

Proposition 1.2.8. Let E and F be abelian projections in R. Then, if CE = CF , we have that E and F

are equivalent.6Recall that CG = ∧Pα, where Pα is the family of projections such that GPα = G.

11

Proof: Note that we just need to prove that if CE ≤ CF , we have that E F and the result follows

from Prop.1.1.7. Let’s then assume that CE ≤ CF and suppose that E F . Then, by Theorem 1.1.13

there is a central subprojection P of CE (one just has to check the definition of the central projection

exhibited in the proof of the latter theorem) such that PF ≺ PE. Thus, PF ∼ E1 < PE. But this is

impossible, since PE is abelian and therefore minimal in the class of projections with central carrier CE

and CPF = CPE .7

Proposition 1.2.9. Let Eαα∈A be a family of abelian projections in R, such that CEα is orthogonal.

Then,∑Eα is an abelian projection in R.

Proof: The proof is merely computational and one can check in [27]

Proposition 1.2.10. Let E be a minimal projection in R. Then, E is abelian.

Proof: We will establish that if E is a minimal projection, then ERE = CE and thus, an abelian von

Neumann algebra. Notice that since E is minimal, any projection in ERE is either E or 0. Since ERE

is a von Neumann algebra, it is generated by its projections and thus, we conclude that ERE = CE.

Remark 1.2.11. Let R be a factor and let E be an abelian non-zero projection. In this case, since R is

a factor, we have that CE = 1. By Prop.1.2.7 we know that E is minimal in the class of projections with

central carrier 1 and since the only central projections are 0 and 1, it is immediate that E is a minimal

projection. Joining this with Prop.1.2.10, we conclude that if R is a factor, the abelian projections are

exactly the minimal projections.

1.3 The Theorem of Decomposition

Definition 1.3.1. Let R be a von Neumann algebra. We say that R is of type I if there is an abelian

projection in R with central carrier 1. We say that R is of type In (n ∈ N) if it is of type I and there

are n equivalent abelian projections with sum 1. We say that R is of type I∞ if it is of type I and there

is an infinite family of equivalent abelian projections with sum 1. In general, when we write type In,

we consider that n ∈ N ∪ ∞. If there are no non-zero abelian projections in R but there is a finite

projection with central carrier 1, we say that R is of type II. If R is of type II and 1 is finite, we say that R

is of type II1 and if 1 is properly infinite, we say that R is of type II∞. Finally, if there are no non-zero

finite projections in R, we say that R is of type III.

Remark 1.3.2. Let R be a factor. Taking advantage of Remark 1.2.11 we can translate the latter defini-

tion into an equivalent one. Indeed, R is of type I if there is a minimal projection. Moreover, R is of type

II if there is no non-zero minimal projection but there is a finite projection and R is of type III is there is

no non-zero finite projection.

Theorem 1.3.3. Let R be a von Neumman algebra acting in H. Then, there are central and mutually

orthogonal projections Pn (with n ≤ dim(H)), Pc1 , Pc∞ and P∞ with sum 1 and maximal to the property7In fact, if P is a central projection in R and E ∈ P, we have that CPE = PCE and in this case, CE1

= CPF = P = CPE ,since CPE = PCE = P .

12

that RPn is of type In or RPn = 0, RPc1 is of type II1 or RPc1 = 0, RPc∞ is of type II∞ or RPc∞ = 0

and RP∞ is of type III or RP∞ = 0.

Proof: Let Ea be a family of abelian projections, maximal to the property that CEa is orthogonal.

Such object exists by Zorn’s Lemma. By Prop.1.2.9 we have that∑Ea is an abelian projection with

central carrier∑CEa , that we will denote by Pd.8 Thus, either Pd = 0 or RPd - acting in Pd(H) - is a von

Neumann algebra of type I, since the identity in RPd is Pd. Notice that by maximality of Ea, 1 − Pddoes not have non-zero abelian subprojections. By Prop.1.2.4 there is a central subprojection Pc1 of

1 − Pd, such that Pc1 is finite and 1 − Pd − Pc1 is either 0 or properly infinite. As RPc1 does not have

non-zero abelian subprojections and Pc1 is finite, either RPc1 = 0 or RPc1 is a von Neumann algebra of

type II1 - since Pc1 is the identity of RPc1 . Now, let’s consider a family Gc of finite subprojections of

1 − Pd − Pc1 , maximal to the property that Gc is orthogonal. By Prop.1.2.3, we know that∑Gc is a

finite projection with central carrier∑CGc , that we will denote by Pc∞ . By maximality of Gc, we can

check that 1−Pd−Pc1 −Pc∞ - that we will denote by P∞ - does not have non-zero finite subprojections.

Therefore, if P∞ 6= 0 we have that RP∞ is a von Neumann algebra of type III. Moreover, and since

1 − Pd − Pc1 is either zero or properly infinite, if Pc∞ 6= 0, then since Pc∞ does not have non-zero

abelian subprojections, we can conclude that RPc∞ is a von Neumann algebra of type II∞. It remains

to show that Pd is a sum of a family Pn of central projections, such that each Pn is the sum of n

equivalent abelian projections : Let Qa be an orthogonal family of central subprojections of Pd, where

each Qa is the sum of n - with n ≤ dim(H) - equivalent abelian projections Eaj , with j ∈ 1, ..., n.

As usual, using Zorn’s Lemma, let Qa be maximal to the property that is an orthogonal family. Since

Eai ∼ Eaj and∑j Eaj = Qa, we have that CEa1 = ... = CEan = Qa. Now, define Ej =

∑aEaj

which is abelian by Prop.1.2.9 and has central carrier∑aQa, that we will denote by Pn. Notice that it

follows from Prop.1.2.8 that E1 ∼ ... ∼ En, since CEi = CEj =∑aQa = Pn and each Ei is abelian.

As Pn =∑nj=1Ej =

∑nj=1

∑aEaj =

∑aQa = Pn, either Pn = 0 or RPn is a von Neumann algebra

of type In. Finally, we prove that Pd =∑n Pn : In what follows, we denote P = Pd − ∨nPn. We will

see that if P 6= 0, then P has a non-zero central subprojection that is a sum of n equivalent abelian

projections, which contradicts the maximality of Qa. In this case, we have that Pd = ∨nPn. Then, we

prove that PnPm = 0, for n 6= m and we can conclude that Pd =∑n Pn. Recall, from the beginning of

this proof, that Pd = CE for some abelian projection E. Moreover, CPE = PPd = P . Let’s denote PE by

F and consider an orthogonal family Fb of subprojections of P that are equivalent to F . Let’s assume

that this family is maximal. Thus, by maximality of Fb we have that F P −∑Fb and by Theorem

1.1.13 there is a central projection P0 ≤ P such that P0(P −∑Fb) = P0 −

∑P0Fb ∼ F0 < P0F .

By Prop.1.2.7 - since P0F is abelian - we have that F0 is also abelian and that F0 = CF0P0F . Now,

there are two possible cases. Either F0 = 0 or F0 6= 0. If F0 = 0, then P0 =∑P0Fb and P0 is a

non-zero central subprojection of P which is a sum of equivalent abelian subprojections, since Fb ∼ F

and P0 is central (so we use Prop.1.1.6). On the other hand, if P0 6= 0, we will check that CF0 is a

sum of equivalent abelian projections, which lead us to the latter mentioned contradiction. To do so,8Let Ea be a family of projections and let E = ∨aEa. Since Ea ≤ CE , we have that CEa ≤ CE and thus, ∨aCEa ≤ CE .

On the other hand, let P = CE − ∨aCEa . In this case, for each a we have that PEa = 0 and so, PE = 0. Therefore, P = 0 andthus, if CEa is orthogonal, we conclude that CE =

∑a CEa .

13

just notice that P0 −∑P0Fb = CF0

(P0 −∑P0Fb) = CF0

−∑CF0

Fb ∼ F0, since CF0= CP0−

∑P0Fb .

Therefore, we have that CF0 = (∑CF0Fb) + (CF0 −

∑CF0Fb), which are equivalent abelian projections.

As mentioned before, to finish this proof, we will show that if n 6= m, then PnPm = 0. Indeed, suppose

that PnPm = P 6= 0. Recall that Pk =∑kj=1Ek and so, P is simultaneously the sum of n and m

equivalent abelian projections, as PnPm =∑nj=1EjPm =

∑mi=1EiPn. However, each abelian projection

is finite and so, by Prop.1.2.5 we conclude that n = m, since P is clearly finite, En and Em are

orthogonal and also since EjPm ∼ FiPn.9

Corollary 1.3.4. Let R be a von Neumann algebra. Then, R = RI ⊕R1II ⊕R∞II ⊕RIII , where RI , R1

II ,

R∞II and RIII are either 0 or respectively von Neumann algebras of types I, II1, II∞ and III.

Corollary 1.3.5. Let R be a von Neumann algebra of types In and Im. Then, n = m.

Proof: We just need to repeat the argument used to prove that PnPm = 0 if n 6= m, in Theorem 1.3.3.

In fact, we have that 1 =∑ni Ei =

∑mj=1 Fj , where RPn and RPm are respectively of types n and m.

But we know that Ei and Fj are orthogonal equivalent abelian projections and so, by Prop.1.2.5 we

conclude that n = m.

Corollary 1.3.6. Let R be a factor. Either R is of type In, of type II1, of type II∞ or of type III.

Proof: If R is a factor, then the only central projections in R are 0 and 1. Therefore, it should be clear

that among the projections Pd, Pc1 , Pc∞ and P∞ (that occur in the proof of Theorem 1.3.3), only one

can be non-zero (and thus, the identity). In particular, if Pd = 1, by Cor.1.3.5 we already know that n is

determined.

Remark 1.3.7. As we defined type In factor (see Definition 1.3.1), n ∈ N ∪ ∞. One can check in [27]

(Theorem 6.6.1) that ifM is a type In factor, thenM is ∗-isomorphic to L(H), where H has dimension n.

Since there is a Hilbert space with dimension d for each cardinal number d, in fact isomorphism classes

of type I factors correspond exactly to the cardinal numbers. In particular, there is an uncountable family

of non isomorphic type I∞ factors.

1.4 Examples

1.4.1 Measure-theoretic construction of factors of any type

In this section, we will establish that there are factors of each type. To do so, we will provide a very

concrete way of building them, using measure-theoretic constructions. For the sake of completeness,

we briefly digress on the particular case of type II1 factors, sketching the ideas used to show that there

is an infinity of non-isomorphic type II1 (non-hyperfinite) factors. We may assume results studied in

Chapter 2, mainly concerning the uniqueness (and existence) of the hyperfinite type II1 factor.

9Ej(F1 + ...+ Fm) ∼ Fi(E1 + ...+ En), since EjFk ∼ EkFi, as Ej ∼ Ek and Fi ∼ Fk.

14

Until the end of this section, µ will denote a positive measure defined on a σ-algebra F of subsets

of a set S 6= ∅. G will denote a countable group of one-to-one mappings from S to S, endowed with

composition as the group operation. Moreover, one will assume that:

1. There is a sequence Eii∈I ⊂ F such that µ(Ei) < ∞ and such that for any s 6= t in S, there is

an integer j ∈ N such that s ∈ Ej and t 6= Ej .

2. If g ∈ G and X ∈ P(S), then X ∈ F if and only if g(X) ∈ F . Moreover, given X ∈ F , we have that

µ(X) = 0 if and only if µ(g(X)) = 0.

3. Let g ∈ G \ e. Then, s ∈ G : g(s) = s is a null set.

Remark 1.4.1. The first condition leads to S =⋃∞j=1Ej and to s =

⋂Ej : s ∈ Ej for any s ∈ S.

Thus, any point of S is measurable and since s ∈ S : g(s) 6= s =⋃nj=1Ej \ g−1(Ej), the set that

appears in the third condition is indeed measurable.

Definition 1.4.2. We say that G acts ergodically on S if given X ∈ F such that µ(g(X)\X) = 0 for each

g ∈ G, either µ(X) = 0 or µ(S \X) = 0.

Now, let’s consider a different landscape, from which one will return to the measure-theoretic frame-

work. Consider a Hilbert space H and a maximal abelian ∗-subalgebra of L(H), denoted by A. Let G′

be a discrete group and U : G′ → U(L(H)) be an unitary representation. Moreover, let’s assume that

the following conditions are verified :

(A) U(g)AU(g)∗ = A, for each g ∈ G′.

(B) A ∩ U(g)A = 0, for all g ∈ G′ \ e.

In this context, we say that G′ acts ergodically on A if given some A ∈ A such that U(g)AU(g)∗ = A for

any g ∈ G′, then A = λ1 for some scalar λ.

Starting from this context, let K be the Hilbert space consisting of all mappings x : G′ → H for which∑g∈G′ ||x(g)||2 < ∞. For each T ∈ L(K), we associate in the usual way a matrix [Tp,q]p,q∈G′ .10 Now,

one can define a ∗-isomorphism Φ : L(H) → L(K) such that Φ(S) has matrix [δp,qS]. More, when

g ∈ G′, we denote by V (g) the unitary operator on K that has matrix [δp,gqU(g)]. Finally, to fix notation,

let RA denote the von Neumann algebra generated by the operators Φ(A) and V (g), with A ∈ A and

g ∈ G′. We have the following result :

Theorem 1.4.3. RA is a factor if and only if G′ acts ergodically on A.

Proof: The interested reader can check [27] (Prop.8.6.1.)

Let’s come back to our initial framework. Let’s consider the Hilbert spaceH = L2(S,F , µ) and for each

element u ∈ L∞(S,F , µ), we associate an operator Mu ∈ L(H) of multiplication by u. Here, L2(S,F , µ)

is the set of all measurable complex-valued functions on S for which∫S|f(s)|2dµ(s) < ∞. We set A to

10Let yb : b ∈ B be an orthonormal basis of K. Let S ∈ L(K). Each Syb =∑a∈B sabya, where sab = 〈Syb, ya〉. In this way,

we associate with each S ∈ L(K) a complex-matrix [sab]a,b∈B.

15

be Mu : u ∈ L∞ and it is a well known result that A is a maximal abelian von Neumann algebra of

L(H).

Lemma 1.4.4. For each g ∈ G, there is a non-negative and real-valued measurable function ϕg on

S such that∫Sx(g(s))dµ(s) =

∫Sx(s)ϕg(s)dµ(s), for every non-negative measurable function x on S.

Moreover, ϕg(s) > 0, ϕgh(s) = ϕgϕh(g−1(s)) and ϕe(s) = 1, almost everywhere on S.

With this, we define an unitary representation of G on H, U(g) = Ug. For any x ∈ H and g ∈ G,

let (Ugx)(s) := (ϕg(s))12x(g−1(s)). One can show that A and U defined as above, satisfy the conditions

(A) and (B). The interested reader can check a more detailed approach in [27]. Moreover, we have the

following very important result:

Theorem 1.4.5. G acts ergodically on A if and only if G acts ergodically on S.

Proof: The interested reader can check [27] (Prop. 8.6.9.)

Corollary 1.4.6. If G acts ergodically on S, then RA is a factor.

We are now in condition to present some criteria that will allow us to build examples of factors of type

I, II1, II∞ and III. For a proof of the following theorem, one can check [27] (Prop. 8.6.10.). After the

result, we will readily produce the examples !

Theorem 1.4.7. Let G acts ergodically on S. Then :

1. RA is of type I if and only if µ(s) > 0, for some s ∈ S, being of type In when |G| = n. If G is

infinite, then RA is of type I∞.

2. RA is of type II if and only if µ(s) > 0 for each s ∈ S and there is a non-zero σ-finite measure11

µ0 defined on F such that µ0(g(X)) = µ0(X) for each X ∈ F and g ∈ G and such that µ0(X) = 0

whenever X ∈ F and µ(X) = 0. Moreover, RA is of type II1 if µ0(S) < ∞ and is of type II∞ if

µ0(S) =∞.

3. RA is of type III if and only if there is no measure µ0 satisfying the conditions set out in (ii).

Finally, one can apply the above result to provide the existence of factors of each type :

• Type I factors : Let S be a finite group, with F = P(S). Let µ be the counting measure. Finally, let G

be the group of all left translations Lt : S → S such that Lt(s) = ts. Let’s check conditions (1)−(3) : Since

S is finite, let |S| = n with a bijection F : 1, ...n → S. Let Eii∈N with Ej = Fj for each j ∈ 1, ...n

and Ej = ∅ for j ≥ n + 1 and it verifies (1). To verify (2), let Lt ∈ G and X ∈ P(S). Since F = P(S)

and Lt(X) ∈ P(S), it is clear. Moreover, µ(X) = 0 if and only if X = ∅ if and only if µ(Lt(X)) = 0.

Finally, to check the third condition, let Lt ∈ G, with t 6= e. Then, Ls ∈ G : Lt(Ls) = Ls = ∅, since

Lt(Ls)(y) = tsy = sy = Ls(y) implies that t = e. Thus, µ(Ls ∈ G : Lt(Ls) = Ls) = 0. On the other

hand, G acts ergodically on S, since if µ(Lt(X) \X) = 0, then Lt(X) = X for any t ∈ S and thus, using

additivity of µ, one concludes that µ(S \X) = 0. Therefore, by Theorem 1.4.7, RA is a type In factor.11Recall that µ is a σ-finite measure on X if X is a countable union of measurable sets with finite measure. For instance, the

Lebesgue measure on R is σ-finite, choosing the sets [k, k + 1], k ∈ Z.

16

• Type II1 factors : Let S = [0, 1[⊂ R and let µ be the Lebesgue measure on F , the σ-algebra

of Boreal subsets of S. Let G be the group of all rational translations (modulo 1) of S. Let’s check

conditions (1) − (3) : For the family Eii∈N, choose an enumeration of all sets of the form [a, b[∈ P(S)

with a, b ∈ Q. Indeed, µ(Ei) < ∞ and given s 6= t in S, let d(s, t) = d. It is clear that choosing some

q ∈ Q such that d(q, s) = d10 , we have that s ∈ [q − d

2 , q + d10 [= Eα and t /∈ Eα. To check the other

conditions, we just need to use properties of Lebesgue measure. On top of that, let µ0 be the measure

µ. Then, µ0(X) = µ(X) = µ(g(X)), since Lebesgue measure is invariant under translations. It remains

to check the ergocidity of G : Let Y ∈ F be such that µ(Y ) > 0 and µ(G(Y ) \ Y ) = 0 for each g ∈ G.

One can define another measure µ1 on F setting µ1(X) := µ(X ∩ Y ). Let’s assume for a moment that

we know that there is some constant c such that µ1(X) = cµ(X). Then,

µ(S \ Y ) = c−1µ1(S \ Y ) = c−1µ((S \ Y ) ∩ Y ) = 0.

Therefore, we show thatG acts ergodically on S. To check that indeed µ1(X) = cµ(X) for some constant

c, let’s first suppose that µ1(X) = µ1(g(X)). Note that if Ii, i ∈ 1, 2, are intervals [ai, bi[ with ai, bi ∈ Q,

there is certainly some g ∈ G such that I2 = g(I1) and thus, µ1(I1) = µ1(I2). Let’s choose each Ii in a

way such that µ(Ii) = 1n for some n ∈ N. Clearly, one can express S as a disjoint union of such intervals

and if c = µ1(S) (in fact, c = µ(Y )), then the measure µ1 of each interval is equal to cn . Since F if

generated by such intervals I of lengths 1, 12 ,

13 , ..., one has that µ1(X) = cµ(X) for any element X ∈ F

as we wanted. Since µ0(S) = µ(S) = 1 < ∞, to appeal to Theorem 1.4.7 and conclude that RA is a

type II1 factor, it remains to check that µ1(X) = µ1(g(X)) : Using invariance of Lebesgue measure, we

have that µ1(g(X)) = µ(g(X) ∩ Y ) = µ(X ∩ g−1(Y )). But

µ(X ∩ g−1(X)) ≤ µ(X ∩ (g−1(Y ) \ Y )) + µ(X ∩ Y ) = µ(X ∩ Y ),

since µ(g(Y ) \ Y ) = 0. Thus, µ1(g(X)) ≤ µ(X ∩ Y ) = µ1(X). One the other hand, µ1(X) ≤ µ1(g(X))

and it follows, as we wanted, that µ1(X) = µ1(g(X)).

• Type II∞ factors : Here, we choose S = R and µ = µ0 to be the Lebesgue measure on the

σ-algebra F of Borel subsets of R. Moreover, G is chosen to be the group of all rational translations of

R. All the necessary ( and sufficient) conditions to appeal to Theorem 1.4.7 are verified in a very similar

way as in the latter example. Since µ(S) =∞, we have that RA is a type II∞ factor.

• Type III factors : Choose S,F and µ as in the previous example. Let G be the group of all mappings

of the form g : S → S such that g(s) = as + b, with a, b ∈ Q and a 6= 0. Note that G cointains the group

of all rational translations of S. From this remark, using the same arguments as in the type II∞ case, it

follows that G acts ergodically. Also, in an absolutely analogous ways as in the previous case, conditions

(1) to (3) are verified. Thus, to show thatRA is a type III factor, it remains to prove that there is no non-

zero σ-finite measure µ0 on F such that µ0(g(X)) = µ0(X) and µ0(X) = 0 if µ(X) = 0 for any X ∈ F

17

and g ∈ G. Indeed, if such measure exists, the Radon-Nykodym Theorem 12gives the existence of a

non-negative real-valued measurable function ϕ on S such that µ0(X) =∫Xϕ(s)dµ(s) for any X ∈ F .

Since G acts ergodically, one can show that there is a constant c such that ϕ(s) = c almost everywhere.

On the other hand, for each g ∈ G, one can show that ϕ(g(s)) = ϕ(s), also almost everywhere on S.

Thus, it follows that µ0 = cµ and since we assumed that µ0(g(X)) = µ0(X), one concludes that µ is

invariant under G, which is false, since µ is not invariant under the mapping s 7→ as+ b when |a| 6= 1.

1.4.2 Type II1 factors

Now, we give a very important example of a von Neumann algebra. The group von Neumann algebra.

Recall that given a countable set A, we define l2(A) as the linear space of complex functions on A such

that∑a∈A |x(a)|2 is finite. To this linear space we associate a norm given by ||x|| = (

∑a∈A |x(a)|2)

12 ,

induced by the inner product 〈x, y〉 =∑a∈A x(a)y(a), where x, y ∈ l2(A). With this inner product, l2(A)

is a Hilbert space. It is also useful to recall that given a orthonormal basis hbb∈B for a Hilbert space

H, we have that if x ∈ H, then x =∑b∈B〈x, hb〉hb, where this sum is always countable, even if B is

not. Thus, during this subsection when we mention orthonormal basis, we do not refer to a Hamel basis,

where the linear combinations are always finite by definition. Moreover, during this subsection, A will be

a discrete group G with identity e and H will be l2(G). Recall that given x ∈ H, then x∗(g) = x(g−1). In

this setting, we can define a convolution product:

(x ∗ y)(g0) =∑g∈G

x(g0g−1)y(g).

We note that (x ∗ y) ∈ l∞(G), though is not necessarily an element of l2(G). We now introduce two

important linear maps - given a fixed x ∈ H - Lx and Rx, defined as Lx(y) = x ∗ y and Rx(y) = y ∗x and

such that Lx, Rx : H → l∞(G). Finally, given g ∈ G we define xg ∈ H as the function that is 1 at g and

zero otherwise. It is easy to check that xg is an orthogonal basis for H.

Proposition 1.4.8. Let T ∈ L(H), x ∈ H and 〈Txg, xh〉 = 〈x ∗ xg, xh〉, for every g, h ∈ G. Then, T = Lx.

Proof: Notice that (x ∗ xg)(h) = 〈x ∗ xg, xh〉 = 〈Txg, xh〉, for all g, h ∈ G. On the other hand, since the

continuous linear mappings y 7→ (x ∗ y)(h) and y 7→ 〈Ty, xh〉 take the same values at each element of

the basis xg, it follows that Lxy = Ty.

Proposition 1.4.9. Let LG = Lx : x ∈ H,Lx ∈ L(H) and RG = Rx : x ∈ H,Rx ∈ L(H). Then,

LG and RG are von Neumann algebras with (LG)′

= RG. Moreover, Lxg : g ∈ G and Rxg : g ∈ G,

where xg is an orthonormal basis for H, generate respectively LG and RG.

Proof: One can easily check that Lx + Ly = Lx+y, aLx = Lax, LxLy = Lxy, Lx∗ = L∗x, Le = 1 and that

if Lx = Ly, then x = y. Just to provide an example, we will verify that Lx∗ = L∗x: Note that

〈L∗xxg, xh〉 = 〈xg, Lxxh〉 = (x ∗ xh)(g) = x(gh−1).

12Let (X,µ) be a measure space. If ν is a σ-finite measure absolutely continuous relative to µ, then there is some measurablefunction f : X → [0,∞[ such that for any A ∈ F one has that ν(A) =

∫A fdµ.

18

On the other hand, notice that

〈Lx∗xg, xh〉 = (x∗ ∗ xg)(h) = x∗(hg−1) = x(gh−1).

Thus, by Prop.1.4.8 we can conclude that L∗x = Lx∗ . Therefore we can conclude that LG and RGare involutive algebras. Moreover, one can check that if Lx ∈ LG and Ry ∈ RG, then we have that

LxRyxg = RyLxxg for each g ∈ G and thus, LxRy = RyLx and we conclude that L′G ⊂ RG and

R′G ⊂ LG. On the other hand, suppose that T ∈ R′G and let x = Txe. We have that

〈x ∗ xg, xh〉 = 〈(Txe) ∗ xg, xh〉 = 〈Rxg(Txe), xh〉 = 〈T (Rxgxe), xh〉 = 〈Txg, xh〉.

Thus, by Prop.1.4.8 we can conclude that T = Lx ∈ LG. Therefore, LG = R′G and in an analagous

way, RG = L′G. Finally, since T ∈ LG if T commutes with every Rxg, we conclude that R′ ⊂ LG, where

R is the von Neumann algebra generated by Rxg : g ∈ G. However, R ⊂ RG and so we have that

LG ⊂ R′

and thus, R′ = LG. Therefore, R = RG. The case is completely analagous for LG.

Definition 1.4.10. Given a discrete group G, we say that LG and RG are the group von Neumann

algebras (of G).

Proposition 1.4.11. LG and RG are finite von Neumann algebras. 13

Proof: Let E be a projection in LG such that E ∼ Id and let V be a partial isometry such that V V ∗ = Id

and V ∗V = E. In this case, we have that 〈Exe, xe〉 = 〈V ∗V xe, xe〉 = 〈V V ∗xe, xe〉 = 〈xe, xe〉, since

(V ∗V )∗ = V V ∗ = V ∗V . Thus, 〈(1− E)xe, xe〉 = 0 and (1− E)xe = 0. Since xe is a generator vector for

Rg, it is a separator vector for LG and so, we conclude that E = 1. In an analagous way we conclude

that RG is also finite.

Theorem 1.4.12. Let G be a group with identity e, such that G 6= e and such that for every g ∈ G\e,

the class of conjugacy of g is infinite - we say that G is an i.c.c. group or an infinite conjugacy class

group. Then, LG and RG are type II1 factors.

Proof: We just need to show that LG and RG are factors with infinite dimension. Indeed, by Prop.1.4.11

we already know that LG and RG are finite von Neumann algebras. Thus, LG and RG can only be

either type In factors - if they are finite dimensional, since every type In factor is isomorphic to L(H)

with dim(H) = n ≤ ∞ - or type II1 factors - if they are infinite dimensional, since if there are no

minimal projections (check Remark 1.3.2), one can choose an infinite orthogonal family of non-zero

projections. Let’s check that both algebras are infinite dimensional : Consider g1, ..., gn ⊂ G and

recall that∑ni=1 aiLxgi = L∑n

i=1 aixgi= 0, only if

∑ni=1 aixgi = 0, as one can check during the proof

of Prop.1.4.11. Since (∑ni=1 aixgi)(gj) = aj , we can easily conclude that Lxg : g ∈ G is linearly

independent in LG. Since G is i.c.c., evidently that the latter set is infinite and so, LG (and analagously

RG) is infinite dimensional. It remains to show that RG and LG are indeed factors : If Lx ∈ LG and

commutes with Lxg, then x∗xg = xg ∗x and moreover, x(gg0g−1) = (x∗xg)(gg0) = (xg ∗x)(gg0) = x(g0).

13We say that a von Neumann algebra R is finite/infinite if 1 is a finite/infinite projection.

19

Thus, if Lx ∈ CLG , we have that x is constant on the conjugacy class of g0, for every g0 ∈ G. Since x ∈ H,

we know that x = axe14and thus, Lx = a1. Therefore, LG and analagously RG are factors.

Remark 1.4.13. One can check that the free group generated by n symbols, Fn and S∞ (the group of

permutations of N that only permutes a finite subset of integers - actually S∞ is the inductive limit of the

family of groups Snn∈N) are i.c.c. groups. Thus, by the latter result, we have that LFn and LS∞ are

type II1 factors. More, one can show that they are not isomorphic (c.f.[27], Theorem 6.7.2). In fact, LS∞is hyperfinite but LFn is not (check Cor.3.2.52). However, it is still an open problem whether LFn and

LFm are isomorphic or not, whenever n 6= m.

In consequence of the last remark, one does not know if the family LFnn∈N contains an infinity

of non-isomorphic type II1 factors or not. In fact, the problem of knowing if there is an infinity of non-

isomorphic type II1 factors took some time and a serious effort to solve. After the proof of uniqueness

(and existence) of the hyperfinite type II1 factor (check the next chapter), in [37] Murray and von Neu-

mann introduced a sufficient condition on the i.c.c. group G to ensure that LG is not hyperfinite :

Theorem 1.4.14. Let G be a countable i.c.c. group and suppose that there exist a set F ⊂ G such that :

1. ∃g1 ∈ G : F ∪ g1Fg−11 = G \ e.

2. ∃g2 ∈ G : F, g2Fg−12 and g−1

2 Fg2 are disjoint.

Then, LG is not hyperfinite.

Corollary 1.4.15. LF2is not hyperfinite.

Proof: Just choose F ⊂ F2 to be the set of those g ∈ F2 which when written as a power product of a

and b (the generators of F2) of minimum length, end up with an for n ∈ Z \ 0. Then, apply Theorem

1.4.14.

The latter result gives what was the first example of a type II1 non hyperfinite factor, studied in [37],

in 1943. Surprisingly enough, until 1969 only nine non isomorphic type II1 factors were known (c.f.

[15],[50]). Many ways of building the type II1 hyperfinite factor were known, however it took a strong

and highly technical work to present the first examples of an infinite family of non-isomorphic type II1

factors. That only came up in [33], in an article written by Dusa McDuff, where some techniques used by

Dixmier and Lance (c.f. [15]), inspired by the seminal works of Murray and von Neumann, were brilliantly

explored. Later on, in [34], Dusa McDuff extended the construction and presented for the first time, a

continuum of non-isomorphic type II1 factors.

For the sake of completeness in this historical overview, it is relevant to say that in [51] and [52], S.Sakai

extended some notions developed in [15] and worked out in [33] and [34] to prove the existence of a

continuum of type III and type II∞ non-isomorphic and non-hyperfinite factors. On the other hand,

in [47] Powers developed a construction of a continuum of non-isomorphic hyperfinite type III factors,

14We have that∑g∈G |x(g)|2 <∞ and so, since x is constant on conjugacy classes of g0 that are infinite sets - since G is i.c.c.

- the only possibility is that x(g) = 0, if g 6= [id]. Therefore, x = ae.

20

using infinite tensor products of type I2 factors. Thus, and not like in the type II1 case, there is no

uniqueness of hyperfinite type III factors.

We dedicate the last pages of this section to sketch what are the ideas behind Dusa McDuff construction

of a countable infinity of non-isomorphic type II1 factors. We will merely give the main definitions and

draw the fundamental lines in the arguments presented in [33], ommiting what would be an imense

technical work.

Definition 1.4.16. Let G be an i.c.c. group and H ⊂ G a subgroup. H is said to be strongly residual in

G if there is a subset S ⊂ G\H and elements g1, g2 ∈ G such that g1Hg−11 = H and S∪g1Sg

−11 = G\H.

Moreover, gn2Sg−n2 n∈N forms a family of disjoint subsets of G \H. If A is a type II1 factor, we say that

a subalgebra B ⊂ A is strongly residual (in A) if there is a group G and a strongly residual subgroup

H ⊂ G such that there is an isomorphism Ψ : A → LG such that Ψ|B = LH .

Definition 1.4.17. Let A be a type II1 factor and let B1A denote the unit ball in the tracial norm

||T ||2 = (tr(T ∗T ))12 . A sequence Tnn∈N ⊂ B1

A is said to be ε-approximate central sequence if

lim sup ||[Tn, X]||2 < ε for all X ∈ B1A.

Definition 1.4.18. Let A be a type II1 factor. Then, Ωnn∈N is said to be a strongly residual sequence

in A if :15

1. For each n ∈ N, Ωn is a strongly residual subfactor of A.

2. For each n ∈ N, Ωn 6= C1.

3. 〈Ω′n+1 ∩ Ωn,Ωn+1〉 = Ωn for each n ∈ N and 〈A ∩ Ω′n : n ∈ N〉 = A.

Remark 1.4.19. From condition (3), it follows that 〈Ω′m ∩ Ωn,Ωm〉 = Ωn, for any m > n. Now, using

the fact that given subalgebras A1 and A2 of a type II1 factor A such that A1 ⊂ A′2 ∩ A are such that

〈A1,A2〉 ≈ A1 ⊗A2,16 we have that Ωn = Ωm ⊗ (Ω′m ∩Ωn), since Ω′m ∩Ωn ⊂ Ω′m ∩A. If one writes that

Ωn,m = Ω′m ∩ Ωn, then for any p > m > n, we have that Ωn,p = Ωn,m ⊗ Ωm,p.

Definition 1.4.20. A type II1 factor A is said to be a null factor if C1 is strongly residual in A.

Remark 1.4.21. We denote the set of strongly residual sequences in A by SRS(A). Let Ωnn∈N ∈

SRS(A). We write Ωn for Ωn,n+1 and we write Ωk,n for (Ωkn+1)′ ∩ Ωkn = Ωkn,n+1.

Definition 1.4.22. Let A be a type II1 factor. Let Ωnn∈N ∈ SRS(A) and suppose that for each k,

Ωknn∈N ∈ SRS(Ωk). Then, for any p ∈ N, we say that A,Ωn, ...,Ωn1,...,np is a residual array of order

p, where Ωn1,...,nq,n = Ωn1,...,nqn,n+1 and Ωn1,...,nq

n n∈N ∈ SRS(Ωn1,...,nq )

Remark 1.4.23. Note that if A,Ωn, ...,Ωn1,...,np is a residual array of order p, we have that Ωn1,...,np ⊂

... ⊂ Ωn ⊂ A. We denote this in a more compact way, writing A,Ω.

15Here, 〈X〉 denotes the von Neumann algebra generated by X. Moreover, recall that given a factor M and a ∗-subalgebraN ⊂M with the same identity and such that is still a factor, is called a subfactor ofM.

16The interested reader can check [38]

21

Definition 1.4.24. Let A,Ω be a residual array of order p. The set of subalgebras of order q (for q ≤ p)

is the family Ωn1,...,nq : nk ∈ N. We say that A,Ω has length p if every pth-order subalgebra Ωn1,...np

is a null factor.

Theorem 1.4.25. Let A,Ω and A,Φ be residual arrays of length p and q respectively. Then, p = q.

Proof: The interested reader should check [33].

Remark 1.4.26. The length of any residual array A,Ω is an invariant of A.

We have introduced enough terminology to sketch the idea between the construction of an infinite

family of non-isomorphic type II1 factors, explored by Dusa McDuff in [33] : Let Gnn∈N and Hmm∈Nbe two sequences of groups. We consider 〈G1, ...,H1, ...〉 to be the group generated byGi’s andHi’s with

relations such that Hi and Hj commute element-wise for i 6= j and Gi and Hj commute element-wise for

i ≤ j. LetMn be the subgroup generated by Hm : m ≥ n. To check the details of this construction, one

may see [15]. Each Mn is strongly residual in 〈G1, ...,H1, ...〉, that is i.c.c. More, LMnn∈N is a strongly

residual sequence in LG. Let K0 = F2 and Kk = 〈G1, ...,H1, ...〉 with Gi = Z and Hi = Kk−i, for k ≥ 1.

Following this scheme, we write Mn1,...,npk for the subgroup M

npk−p+1, where Mn

k = (Mk,n+1)′ ∩Mk,n.

Then, let Ak = LKk and Ωnk = LMnk

, Ωn1,n2

k = LMn1,n2k

and so on up to Ωn1,...,nkk = LMn1,...,nk

k. One

has that for each k, Ak,Ωnk , ...,Ωn1,...,nkk is a residual array of order k and length k. Thus, Akk∈N is

a family of non-isomorphic type II1 factors, appealing to Theorem 1.4.25. Then, we just established the

following:

Theorem 1.4.27. There is a countably infinite family of non-isomorphic type II1 factors.

22

Chapter 2

Uniqueness and Existence of the

hyperfinite type II1 factor

The aim of this chapter is to establish the existence and uniqueness (up to ∗-isomorphism) of the hy-

perfinite type II1 factor. Meanwhile, we explore some useful techniques regarding the trace, namely

the dimension function, which is related to continuous geometry. In continuous geometry, the dimen-

sion of a subespace can be any element of an interval, instead of assuming just values in a discrete

set. Historically, the first example of a continuous geometry other than projective space was considering

projections on the hyperfinite type II1 factor, as we will see.

2.1 The Dimension Function

During this subsection, R is a von Neumann algebra with center C and set of projections P.

Definition 2.1.1. Let ∆ : P → C. We say that ∆ is a dimension function if it is a map such that if

E,F ∈ P and Q ∈ C, we have the following properties:

1. ∆(E) > 0, if E 6= 0.

2. ∆(E + F ) = ∆(E) + ∆(F ), if EF = 0.

3. ∆(E) = ∆(F ), if E ∼ F

4. ∆(Q) = Q and ∆(QE) = Q∆(E).

It is easy to check that if such map exists on R, then R is finite : Indeed, let E,F ∈ P such that

E ∼ F ≤ E. Then, ∆(E) = ∆(F ). Moreover, ∆(F ) = ∆(F ) + ∆(E − F ), since F (E − F ) = 0. Thus,

by (1) we conclude that E = F . Reciprocally, if R is finite we can appeal to Theorem A.3.6 and we can

check that the center-valued trace τ restricted to P is in fact a dimension function on R. Thus, R is finite

if and only if there is a dimension function ∆. The main goals of this subsection is to establish the unicity

of ∆ and some properties of this map that will allow us classify factors through the dimension function.

23

Proposition 2.1.2. LetR be a finite von Neumann algebra. Then, there is an unique dimension function

∆ : P → C such that whenever E,F ∈ P and Q ∈ P ∩ C, we have that

1. ∆(E) ≤ ∆(F ) if and only if E F .

2. If Eaa∈A is an orthogonal family of projections in R with sum E, then ∆(E) =∑a∈A ∆(Ea).

Proof: Defining ∆ as the restriction of τ to P, one can check that ∆ is a dimension function that also

verifies property (2). We will see in detail that ∆ verifies property (1) : Let E ∼ E1 < F . In this case,

∆(E) = ∆(E1) < ∆(E1) + ∆(F − E1) = ∆(F ), by properties (1),(3) and (2) of Definition 2.1.1. Thus,

if E F we have that ∆(E) ≤ ∆(F ). Reciprocally, suppose that E F . Then, there is Q ∈ C ∩ P

such that QF ≺ QE (by Theorem 1.1.13) and ∆(QF ) < ∆(QE). Thus, ∆(E) ∆(F ). Now, let

us establish the unicity of ∆, checking that any map ∆′

: P → C that verifies properties (1) to (5) of

Definition 2.1.1, has to coincide with ∆. Let F be a monic projection in R with Fi ∈ R and Q ∈ P ∩ C

such that F ∼ F1 ∼ ... ∼ Fk and that∑ki=1 Fi = Q. Using properties (3),(2) and (4) of Definition

2.1.1 we can obtain that k∆′(F ) =

∑ki=1 ∆

′(Fi) = ∆

′(Q) = Q. Therefore, ∆

′(F ) = k−1Q. Proceeding

in the same way with ∆, we conclude that for every monic projection F we have that ∆(F ) = ∆′(F ).

Now, since R is finite, we know that any E ∈ P is the sum of an orthogonal family of monic projections

in R (see [27]), say Ebb∈B . If A ⊂ B is finite,∑b∈A ∆(Eb) = ∆

′(∑b∈AEb) ≤ ∆

′(E) and thus,

∆(E) =∑b∈B ∆(Eb) ≤ ∆

′(E), since ∆ verifies property (2) of the hypothesis. Repeating the argument

with (1 − E), we have that ∆(1 − E) ≤ ∆′(1 − E) = 1 − ∆

′(E) and thus, ∆

′(E) ≤ ∆(E), since

1−∆(E) = ∆(1− E). Therefore, for each E ∈ P we have that ∆(E) = ∆′(E).

Proposition 2.1.3. Let R be a finite von Neumann algebra and ∆ the dimension function. If R is of

type In, the range of ∆ is the set of operators of the form∑nj=1

jnQj , where Qj are central projections

orthogonal pairwise. If R is of type II1, the range of ∆ are the positive operators in the unitary ball of C,

i.e. B1R ∩ C.

Proof: First, suppose that R is of type In and let F be an abelian projection in R with CF = Q. In

this case, Q is the sum of n abelian projections that are equivalent to F 1 and thus, ∆(F ) = n−1Q.

Now, choose abelian projections F1, ..., Fn with sum 1 and such that CFi = 1.2 If Qini=1 are pairwise

orthogonal projections and E =∑nj Qj(F1 + ...+Fj), we have that ∆(E) =

∑nj=1

jnQj . Therefore, every

operator of this form is in the range of ∆. Reciprocally, given E ∈ P we can choose a family Qini=1

of orthogonal central projections with sum 1, such that QjE is a sum of j abelian projections, each one

with central carrier Qj . Therefore, ∆(E) =∑nj=1 ∆(QjE), with ∆(QjE) = Qj

jn . Now, let’s suppose that

R is of type II1. For each E ∈ P we have that 0 ≤ ∆(E)∆(1) = 1 and so, =(∆) ⊂ B1C+ , since that we

have that ||∆|| = 1. On the other side, consider C ∈ B1C+ and let Eλ be the resolution of identity for

C.3 One can choose an increasing sequence Cn in B1C+ converging, in the norm topology, to C (using

the resolution of identity). Thus, one can write C =∑∞j=1 2−n(j)Pj , where Pj is a central projection

and n(j) is non-negative integer. We will show that there is a sequence of orthogonal projections Gj1Since R is of type In, let Ai be abelian projections such that CF = Q = QA1 + ... +QAn. Therefore, CQAi = QCAi =

Q = CF and thus, QAi ∼ F .21 =

∑ni=1, with Ai ∼ Aj and C1 = 1 = C∑n

i=1 Ai= CAi , imply that CAi = 1.

3Check [26] , Thrm.5.2.2

24

such that ∆(Gj) = 2−n(j)Pj . Notice that if this is the case, defining G =∑∞j=1Gj , we have that

∆(G) =∑∞j=1 ∆(Gj) =

∑∞j=1 2−n(j)Pj = C and we can conclude that the range of ∆ is B1

C+ . Since

R is of type II1 we know that each Pj is a sum of 2n(j) equivalent projections in R, say Fj , such that

∆(Fj) = 2−n(j)Pj .4 Using this fact, we define by induction the sequence Gj, letting G1 = F1. Now,

assuming that we already have G1, ..., Gk−1, one can verify that ∆(Fk) ≤ ∆(1 −∑k−1i=1 Gi) and thus,

there is Gk ∈ P such that Fk ∼ Gk ≤ (1 −∑k−1i=1 Gi). Moreover, note that ∆(Gk) = ∆(Fk) = 2−n(k)Pk.

Remark 2.1.4. Now we focus in the case where R is a finite factor. As C = C1, we easily conclude that

τ(A) = ρ(A)1, where ρ is the tracial state. Therefore, by Prop.2.1.3 we know what is the range of ρ if R

is a finite factor. Indeed, and since ∆ is τ |P (we established the unicity in Prop.2.1.2), if R is of type In

the range of ρ is the set 0, 1n ,

2n , ..., 1, since there are only two cases where the hypothesis are verified:

either Qj = 0 for each j or there is j such that Qj = 1. On the other hand, if R is of type II1, the range

of ρ is the interval [0, 1] - to see this, check the proof of Prop.2.1.3 and notice that C =∑∞j=1 2−n(j)Pj .

And what happens when R is an infinite factor ? Then, we have a different approach since R admits a

center-valued trace if and only if is finite. In this case, we generalize the concept of a tracial state.

Definition 2.1.5. Let R be a von Neumann algebra. A map ρ : R+ → [0,∞] such that for each H,K ∈

R+ and a ≥ 0 verifies ρ(H + K) = ρ(H) + ρ(K) and ρ(aH) = aρ(H), is called a weight in R. If

ρ(A∗A) = ρ(AA∗) for each A ∈ R, we say that ρ is a tracial weight.

One can check that if E,F ∈ P such that E F , then ρ(E) ≤ ρ(F ). Moreover, one can define the

sets Nρ = A ∈ R : ρ(A∗A) < ∞ and Nρ = A ∈ R : ρ(A∗A) = 0 and show that they are ideals in

R, given a certain tracial weight ρ (check [27], Prop. 8.5.1). Moreover, if we denote Mρ = spanFρ,

where Fρ = H ∈ R+ : ρ(H) < ∞, one can check that Mρ coincides with N ∗ρNρ = spanΓ, where

Γ = A∗B : A,B ∈ Nρ.5

Proposition 2.1.6. 1. If ρ is a semi-finite6 tracial weight in a von Neumann algebra R and A ∈ R+ \

0, there exists G ∈ P ∩ Fρ and a > 0 such that A ≥ aG.

2. If ρ is a normal semi-finite tracial weight in a factor R and ρ 6= 0, then ρ is faithful. Moreover, if

E ∈ P and ρ(E) <∞, then E is finite and if ρ(E) =∞, then E is infinite.

Corollary 2.1.7. Let R be a factor of type III. Then, there is no semi-finite normal weight ρ 6= 0 in R.

Proof: If such weight ρ exists, since R do not have any finite projections, it follows from Prop.2.1.6 that

for each E ∈ P, we have that ρ(E) =∞. But this is impossible by Prop.2.1.6.

Proposition 2.1.8. LetR be a factor of type I∞ or II∞. Then, there is a normal semi-finite tracial weight

ρ in R. If R is of type I∞, there is c > 0 such that the range of ρ|P is the set 0, c, 2c, ...,∞. If R is

of type II∞, the range is the interval [0,∞]. Moreover, every normal semi-finite tracial weight in R is a

multiple of ρ by a non-negative escalar.4Let R be a von Neumann algebra without central portion of type I and let E ∈ P. Then, for each integer n, there exist n

orthogonal and equivalent projections with sum E.5Let S be a subset of a vectorial space. We define spanS as being the smallest subspace that contains S, i.e.

∑ki=1 λivi, vi ∈ S, λi ∈ K.

6We say that ρ is semi-finite ifMρWOT

= R.

25

Corollary 2.1.9. Let R be a factor and P the set of projections in R. We have that :

1. If R is of type III, there is no non-zero normal tracial semi-finite weight.

2. If R is not of type III, there is a normal tracial semi-finite weight ρ that is faithful and unique up to

multiplication with positive escalar.

(a) R is of type In if and only if ρ|P has 0, c, 2c, ..., nc as range, for some c > 0.

(b) R is of type I∞ if and only if ρ|P has 0, c, 2c, ...,∞ as range, for some c > 0.

(c) R is of type II1 if and only if ρ|P has the interval [0, c] as range, for some c > 0.

(d) R is of type II∞ if and only if ρ|P has the interval [0,∞] as range.

Remark 2.1.10. Let R be a finite factor. In section A.3.3 of the Appendix, we established the existence

of an unique center-valued trace τ . Since C = C1, in Remark 2.1.4 we noticed how the tracial state ρ

induces the dimension function ∆ in a way that after considering normalization one has that ρ(P) = [0, 1]

ifR is a type II1 factor. Then, we introduced the concept of tracial weight. It is clear that ρ|R+ is a normal

semi-finite tracial weight, whenever R is a finite factor. Indeed, ρ|R+ is normal by a similar argument of

Prop.2.1.2 and since P ⊂ R+, everything is well-defined. This basically constitutes a proof of the latter

result, in the case where R is a finite factor. The importance of this remark is precisely to emphatize

that one can classifies a finite factor R just by studying the range of ρ (or any map which is semi-finite,

normal and a tracial weight, since there is unicity). For the sake of simplicity, in the context of finite

factors, we will refer to ρ just as the trace.

2.2 Uniqueness of the hyperfinite type II1 factor

Let’s introduce a useful notation (Kronecker product) : Given A ∈ Mm×n and B ∈ Mp×q, we denote

A ⊗ B ∈ Mmp×nq to be the matrix [aijB]mp×nq. Moreover, let’s establish that unless something is said

about that, for us an isomorphism between von Neumann algebras, will be in fact a ∗-isomorphism.

Definition 2.2.1. We say that a von Neumann algebra R is hyperfinite if it is the weak closure of an

UHF algebra, i.e. there is a sequence nj of positive integers such that Mnj ⊂Mnj+1 - where Mnj are

factors of type Inj - such that R =⋃jMnj

WOT.

In [37], Murray and von Neumann proved that all type II1 hyperfinite factors are ∗-isomorphic. As we

mentioned before, also in their article [37], they established that not all type II1 factors are hyperfinite,

namely LFn are never hyperfinite. Even though it is still not known if LFn are all isomorphic or not, as

stated in the first chapter we already know that there are uncountably many non isomorphic type II1

factors. This brings diversity to the theory, despite the outstanding isomorphism result that we are about

to prove. Indeed, this is a very strong and deep result. The proof that we will follow relies in two very

technical lemmas that we will only state, without approaching any details. From that angle, what follows

is a large sketch of the argument that all hyperfinite type II1 are ∗-isomorphic.

26

Lemma 2.2.2. Let R be a type II1 factor, let ε, ε′ > 0 and let A = aimi=1 ⊂ R. Suppose that for any

such finite subset A there is a finite dimensional ∗-subalgebra C ⊂ R and elements cimi=1 ⊂ C such

that ||ai − ci|| ≤ ε′. Then, there is a type I2n subfactor N ⊂ R and elements bimi=1 ⊂ N such that

||ai − bi||2 ≤ ε. Moreover, if A1 ⊂ A2 are finite subsets of R, then we can associate subfactors N1 and

N2 such that N1 ⊂ N2.

Proof:The interested reader can check [2], Lemma 7.2.5.

Lemma 2.2.3. Let R be a type II1 factor and let τ be its unique faithful normal tracial state. Let

aimi=1 ⊂ R and p be a projection of R such that τ(p) = 2−n and pai = ai = aip. Then, for any ε > 0

there is a type I2r subfactor N ⊂ R, for some r ≥ n and elements bini=1 ⊂ N such that p ∈ N ,

pbi = bi = bip and ||ai − bi||2 ≤ ε.

Proof:The interested reader can check [2], Lemma 7.2.7.

Lemma 2.2.4. Let R be a type II1 factor and let L ⊂ R be a type I2n subfactor, ε > 0 and aimi=1 ⊂ R.

Then, there is a type I2r (for some r ≤ n) subfactor N ⊂ R and elements bimi=1 ⊂ N such that L ⊂ N

and ||ai − bi||2 ≤ ε.

Proof: Since L is a type I2n subfactor, there is a set pi2n

i=1 of orthogonal and minimal equivalent

projections which add up to 1 and thus, each with τ(pi) = 2−n. Now, choose wj ⊂ L such that

w1 = p1, w∗jwj = p1 and wjw∗j = pj for all 1 ≤ j ≤ 2n. Let p = p1 and let aijk = w∗i akwj . Then,

since paaik = aijkp = aijk for all i, j ∈ 1, ..., 2n and k ∈ 1, ...,m, appealing to Lemma 2.2.3 one

gets that there is a type II2r subfactor N ⊂ R, with r ≥ n and elements bijk ∈ N such that L ⊂ N ,

pbijk = bijkp = bijk and with ||aijk − bijk||2 ≤ δ, for some δ > 0 such that 22nδ ≤ ε. Finally, for the

sequence bkmk=1 ⊂ N we choose bk =∑

1≤i,j≤2n wibijkw∗j for each 1 ≤ k ≤ m. Note that since∑2n

i=1 pi = 1, we have that

ak =∑i,j

piakpj =∑i,j

wiw∗i akwjw

∗j =

∑i,j

wiaijkw∗j .

Thus,

||ak − bk||2 ≤∑

1≤i,j≤2n

||aijk − bijk||2 ≤ 22nδ ≤ ε.

Proposition 2.2.5. Let R be a type II1 factor that admits a countable family of generators.7 If, for each

a1, ..., am ⊂ R and ε > 0 there is a finite dimensional ∗-subalgebra B ⊂ R and b1, ..., bm ⊂ B such

that ||ai − bi||2 ≤ ε for each 1 ≤ i ≤ m, then there is a sequence Mn of subfactors of R such that Mn

is of type I2n and R =⋃n≥1Mn

WOT.

Proof: Let A = ann∈N be a generating set. By Lemma 2.2.2 given a set Ak = a1, ..., ak ⊂ A we can

associate a factor of type I2n , say N0 and then, using Lemma 2.2.3 one can associate a factor of type

I2k , say Nk, in a way such that N0 ⊂ Nk and with elements b(k)i ki=1 ⊂ Nk such that ||ai − b(k)

i || ≤ 1k .

7In the sense that the smallest closed algebra that contains the family is R.

27

This yields a chain Mr1 ⊂ ... ⊂ Mrk ⊂ ...R of type I2rk subfactors of R and each with elements

b(k)i ki=1 ⊂ Mrk such that ||b(k)

i − ai||2 ≤ 1k for any 1 ≤ i ≤ k. Clearly, R =

⋃kMrk

WOTand if

necessary one can refine the chain Mrk.

Proposition 2.2.6. Let R be a factor of type II1. The following assertions are equivalent :

1. R is hyperfinite

2. There is an increasing sequence An of finite dimensional ∗-subalgebras of R such that their

union is weakly dense.

3. R is generated by a countable set and for each a1, ..., am ⊂ R and ε > 0, there is a ∗-subalgebra

B ⊂ R and b1, ..., bm such that ||ai − bi||2 ≤ ε, for each 1 ≤ i ≤ m.

4. R is generated by a countable set and for each a1, ..., am ⊂ R and ε > 0, there is a subfactor

N ⊂ R and b1, ..., bm ⊂ N such that ||ai − bi||2 ≤ ε, for each 1 ≤ i ≤ m.

Proof: (1) ⇒ (2) and (4) ⇒ (3) follows by definition. Moreover, (3) ⇒ (4) is a consequence of Lemma

2.2.2 and furthermore, (3) ⇒ (1) follows immediately from Prop.2.2.5. It remains to establish (2) ⇒ (3)

and the only non trivial detail is to check that R is countably generated. To do so, we appeal to Remark

3.1.11 and we recall that any finite dimensional C∗-algebra A is of the form A =⊕

k∈IMk(C) for some

finite index I.

Proposition 2.2.7. Let U be an UHF algebra. Then, there is an unique tracial state in U .

Proof: First, notice that if U is an UHF algebra of type pn, then U is isomorphic to⊗∞

n=1Mmn , with

m1 = p1 and mn = p−1n−1pn for n ≥ 2. This is indeed the case since (mn)! = pn and thus,

⊗ni=1 = Mpn

and therefore,⊗∞

n=1Mmn is an UHF algebra of type pn. Therefore, by unicity (check Theorem A.2.3)

we conclude that U ≈⊗∞

n=1Mmn . Since each Mmn is a finite factor, there is an unique tracial state ϕn

and thus,⊗

n ϕn is the unique tracial state in U .8

Theorem 2.2.8. All hyperfinite factors of type II1 are isomorphic.

Proof: Let R1 and R2 be two hyperfinite factors of type II1. For i = 1, 2 let ϕi be the normal tracial state

in Ri and let πi, Hi be the faithful cyclic representation of Ri generated by ϕi (in the sense of GNS

representation) with cyclic vector ηi. Then, πi(Ri) is also a hyperfinite factor of type II1. This means

that πi(Ri) =⋃k≥1Nk

WOT, with Nk ≈ M2k(C), using Prop.2.2.5 and Prop.2.2.6. Now, let’s consider

an UHF algebra U of type 2k, i.e. U =⋃k≥1 Uk, with Φki : Uk → Nk an isomorphism. Then, one can

define Φi : U → Φi(U) ⊂ πi(Ri), extending⋃k Φki by continuity in the norm topology. Now let’s define

si : U → C such that si(a) = 〈Φi(a)ηi, ηi〉. One can show that it is a tracial state in U and, by the latter

result, we actually have that 〈Φ1(a)η1, η1〉 = 〈Φ2(a)η2, η2〉, for all a ∈ U . Then, by Prop.A.3.2, we know

that there is an unitary operator u : H1 → H2 such that uπ1(R1)u∗ = π2(R2). Hence, π1(R1) ≈ π2(R2)

and thus, R1 ≈ R2.

Theorem 2.2.9. Let R be a finite von Neumann algebra. If R is hyperfinite, then R is a factor.8Recall that

⊗n ϕn is a tracial state in

⊗n An if and only if ϕn is a tracial state for each An.

28

Proof: Suppose, by absurd, that there is a central projection P in R such that P 6= 0 and P 6= 1. Let

η, ζ ∈ H be such that ||η|| = ||ζ|| = 1 and that P (ζ) = 0 and P (η) = η. Since R is hyperfinite, there

is an UHF algebra U ⊂ R such that UWOT= R. Since R is finite, there is a center-valued trace, say

τ : R → C. Thus, 〈τ(.)η, η〉 and 〈τ(.)ζ, ζ〉 are tracial states in U and, since U is UHF, we know by

Prop.2.2.7 that 〈τ(a)η, η〉 = 〈τ(a)ζ, ζ〉, for each a ∈ U and since UWOT= R, we can extend this equality

to R. In particular, 1 = 〈Pη, η〉 = 〈τ(P )η, η〉 = 〈τ(P )ζ, ζ〉 = 〈Pζ, ζ〉 = 0, since τ(A) = A whenever

A ∈ C, becauseR is a factor. But this is obviously a contradiction and thus, we conclude that every finite

von Neumann algebra R that is also hyperfinite, in indeed a factor.

Corollary 2.2.10. All hyperfinite von Neumann algebras of type II1 are isomorphic.

2.3 A model for the hyperfinite type II1 factor

Let Mn : n ∈ N0 be a family of C∗-algebras with M0 = C and Mn = M2n(C). Consider in :

Mn−1 → Mn to be the inclusion given by A 7→ 12 ⊗ A. For n ≤ k, define Φkn : Mn → Mk to be

Φkn = ik ... in+1, a ∗-isomorphism of Mn into a C∗-subalgebra of Mk. Moreover, if n ≤ k ≤ m,

then we have that Φmn = Φmk Φkn and thus, Mn : n ∈ N0 is a directed system of C∗-algebras. By

Theorem A.2.4 there is a C∗-algebra, sayM2∞ , such that⋃n∈NMn

||.||≈M2∞ . Recall that we say that

M2∞ is the inductive limit of that directed system.

Now let Trn denote the usual trace inMn. For any n ∈ N let z ∈Mn and let k be the smallest index for

which z ∈ Mk. Note that we are deliberately confusing, for the sake of notation, z ∈ Mk with Φnk(z) ∈

Mn. Then, define τn(z) to be 12n−k

Trn(z). Following this reasoning, one defines a positive linear

functional τ on⋃n∈NMn that can be extended toM2∞ in the natural way, by continuity. Therefore, since

M2∞ is a C∗-algebra with an approximation of unity, one can apply the GNS construction associated to

τ . Let (Hτ , πτ ) denote the GNS representation with cyclic and unit vector ξτ (the reader can check A.3).

Finally, define R to be [πτ (M2∞)]′′ ⊂ L(Hτ ). By construction, R is a hyperfinite and infinite-dimensional

von Neumann algebra. Note that we extend τ to a normal tracial state on R by setting τ(x) = 〈xξτ , ξτ 〉

Following the next results, one will conclude that in fact, R is also a type II1 factor. First, we establish

that R is factor and then from τ we define a center-valued trace Γ on R, which allow us to conclude that

R is a finite von Neumann algebra and thus, being infinite-dimensional, we have that it is of type II1.

Lemma 2.3.1. LetM be a von Neumann algebra on a Hibert space H with a cyclic vector ξ ∈ H such

that ζ(x) = 〈xξ, ξ〉 is a tracial state. Then, ζ is faithful. In particular, τ is faithful on R

Proof: Let x ∈M be such that τ(x∗x) = 0. Let a ∈M and since ζ is tracial we have that :

||xaξ||2 = ζ(a∗x∗xa) = ζ(xaa∗x∗)

≤ ||a∗||2ζ(xx∗)2 = ||a||2ζ(x∗x) = 0

Thus, xaξ = 0 and since ξ is cyclic, aξ : a ∈M is dense in H and we prove that x = 0.

29

Remark 2.3.2. Recall that for any n ∈ N, 1nTr is the only tracial state on Mn(C). Indeed, let eijni,j=1

be matrix units in Mn(C). The, each x ∈ Mn(C) can be written as x =∑ni,j=1 xijeij and thus, if τ

is any tracial state on Mn(C), one has that τ(x) =∑ni,j=1 xijτ(eij). Then, it is enough to check that

τ(eij) = 1nδij since if that is the case, we have that τ(x) = 1

n

∑ni=1 xii = 1

nTr(x). Well, on one hand,∑nk=1 ekk = 1 and thus, 1 =

∑nk=1 τ(ekk) = nτ(eii), since τ(eii) = τ(eijeji) = τ(ejieij) = τ(ejj), for any

1 ≤ i, j ≤ n. On the other hand, if i 6= j, then τ(eij) = τ(eijejj) = τ(ejjeij) = τ(δjiejj) = 0.

Lemma 2.3.3. R is a factor.

Proof: First, let’s suppose that R has an unique normal faithful tracial state τ . If that is the case, we will

show that C+ = R+1 and since C is generated by positive elements,9 we establish that R is a factor. Let

h ∈ C+\0 and, since τ is faithful, let’s rescale h such that τ(h) = 1. Now, let’s define τ ′(x) = 〈xξτ , hξh〉.

Notice that for any x, y ∈ R we have that :

τ ′(xy) = τ(hxy) = τ(xhy) = τ(hyx) = τ ′(yx)

Thus, τ ′ is tracial and since it is normal and faithful (because τ is also normal and faithful), by our

assumption we conclude that τ ′ = τ . This means that for every x ∈ R we have that τ(hx) = τ(x) and

we can conclude with some additional computational work that this implies that h = 1. Thus, C+ = R+1

as we wanted to prove.

It remains to show that τ is in fact the unique normal tracial state on R, since by Lemma 2.3.1 we know

that τ is already faithful. Let τ ′ be a normal tracial state on R. Since τ and τ ′ are normal, it suffices to

show that τ and τ ′ agree on the strongly dense ∗-subalgebra R0 =⋃n∈N0

M2n(C). But, appealing to

Remark 2.3.2 we know that τ |M2n(C) = τ ′|M2n(C) for any n ∈ N.

Theorem 2.3.4. R is a hyperfinite type II1 factor.

Proof: By construction, we already noticed that R is hyperfinite and infinite-dimensional. By Lemma

2.3.3 we know that R is a factor. If we establish that R is a finite von Neumann algebra, we can

conclude what we want. To do so, we appeal to A.3 and we recall that if R has a center-valued trace,

then R is necesarily finite. Thus, let Γ : R → C be given by Γ(z) = τ(z)1. It is immediate that Γ is a

center-valued trace on R, since if z ∈ C then z = λ1 and τ(x) = λ〈ξτ , ξτ 〉. But ξτ is a unit vector and

consequently, Γ(z) = z for any z ∈ C.

9The center of any C∗-algebra is still a C∗-algebra and any C∗-algebra is generated by its self-adjoint elements. Furthermore,each self-adjoint element can be obtained as a difference of two positive elements.

30

Chapter 3

Connes Embedding Conjecture

This is the central chapter of this thesis. We state the original formulation of the Connes’ Embedding

Conjecture (CEC), which lead us to some considerations on ultrafilters and ultrapowers. For some

preliminary results on these topics, the reader can check A.4. The main goal of this chapter is, ultimately,

to establish the relation between Connes’ Embedding Conjecture for groups and hyperlinear and sofic

groups. We also explore the relation between CEC and Weak Expectation Property (WEP) on separable

type II1 factors. Finally, we briefly mention other open problems in Operator Algebras that are known

to be equivalent to CEC and we sketch the argument that guarantee us that under the Continuum

Hypothesis, all ultrapowers RU (with free ultrafilter U) are isomorphic, in a very appealing cooperation

with Model Theory ideas.

3.1 Formulation of the conjecture

During this chapter, unless we mention the opposite, R will always denote the hyperfinite factor of type

II1. The aim of this preliminary section is just the formulation of the Connes Embedding Conjecture

(CEC), currently an open problem. It is not our purpose to prove or disprove this conjecture. In fact,

CEC is an incredibly important problem and several assertions from seemingly different origins have

been proved to be equivalent to CEC in the past years. We will approach two of them. After the more

classical topics of last chapters, it feels that approaching - even if it is at the surface - the fringe of mod-

ern mathematical investigation is something that fits the nature of this thesis.

We begin by noticing that it is very restrictive to talk about separability in von Neumann algebras con-

sidering the norm topology. In fact, one can prove that if a given von Neumann algebra is infinite dimen-

sional, there is an infinite family of non-zero mutually orthogonal projections, say pn. For each A ⊂ N

let’s consider qA =∑k∈A pk and notice that for distinct subsets A,B ⊂ N we have that ||pA−pB || ∈ 1, 2

- the distance is 1 if A ⊂ B or B ⊂ A, and 2 otherwise. Since P(N) is uncountable, we have an uncount-

able family of elements such that their distance is greater or equal to 1. Thus, it is immediate that this

algebra is not separable in the norm topology.1 We just concluded that if a von Neumann algebra is sep-

1Recall that a metric space X is separable if and only if it has countable basis. In this case, it is obvious that the algebra doesnot have countable basis.

31

arable in the norm topology, then it is finite dimensional, which shows us that using the norm topology in

separability, is too restrictive. Because of this fact, we will use the weak topology. Now, let’s recall that

given a von Neumann algebraM acting in H, the pre-dual ofM - denoted byM∗ - is the linear space of

linear functionals onM that are weakly continuous in the unit ball B1M (one can check that (M∗)∗ =M,

showing that Ψ :M→ (M∗)∗ given by Ψ(x)(f) = f(x), is a surjective linear isometry.)2.

Proposition 3.1.1. LetM be a von Neumann algebra. The following are equivalent :

1. M∗ is separable in the norm topology.

2. M is weakly separable.

3. There exists a faithful representation ofM in L(H) with H separable.

Definition 3.1.2. LetM be a von Neumann algebra. IfM satisfies one of the above conditions, we say

thatM is separable.

Before formulating the CEC, for the sake of completeness one should establish the existence of a

free ultrafilter on N.

Definition 3.1.3. Let F be a filter on I. We say that F is free if⋂X∈F X = ∅. If F is not free, it is said

to be principal.

Remark 3.1.4. Let F be a principal ultrafilter on I and let A ∈⋂X∈F X. Then, F = U ∈ P(I) : A ⊂ U

and A is necessarily a singleton. Indeed, on one hand, it is clear that F ⊂ U ∈ P(I) : A ∈ U. On

the other hand, let U ∈ P(I) be such that A ⊂ U . Let I \ U ∈ F and note that A ∩ I \ U = ∅. Thus,

A /∈⋂X∈F X and we conclude that, since F is an ultrafilter, U ∈ F . Furthermore, since A 6= ∅, A is

a basis for F . If there are elements a1 6= a2 in A, assume without loss of generality that a1 /∈ F .

Consider X0 = I \ a1. Since F is an ultrafilter, then X0 ∈ F but we reach to the absurd that A 6⊂ X0.

Thus, A is necessarily a singleton.

Definition 3.1.5. Let F = S ∈ P(N) : N \ S is finite. We say that F is the Frechet filter.

Remark 3.1.6. More generally, given any infinite set I, one can define the Frechet filter on I as the set

of all cofinite subsets.

Lemma 3.1.7. The Frechet filter is a free filter on N.

Proof: Let F denote the Frechet filter and let Ac denote N\A. Before proving that F is a free ultrafilter on

N, we make the following useful remark: given A ⊂ N, then A ∈ F if and only if there is somem ∈ N such

that m,m + 1, ... ⊂ A. Indeed, if Ac is finite then Ac = x1, ..., xk and consequently, xk+1, ... ⊂ A

and we choose m = xk+1. Conversely, Ac ⊂ 1, ...,m− 1 and thus, A is cofinite. Now, we check that F

is indeed a free ultrafilter: It is clear that /∈ F , since N is infinite. Moreover, let A,B ∈ F, which means

that A and B are cofinite. Then, (A ∩ B)c = Ac ∪ Bc and thus, A ∩ B is still cofinite. Furthermore, let

A ∈ F and B ⊂ N such that A ⊂ B. Let m be such that m,m+ 1, ... ⊂ A and thus, Bc ⊂ 1, ...,m− 12More generally, the pre-dual of an object A is an object B such that B∗ = A.

32

and so B is also cofinite. Therefore, we proved that F is a filter. It remains to check that it is free: By

contradiction, suppose that a ∈⋂X∈F X. Then, in particular, this means that a ∈ a+ 1, a+ 2, ... which

is obviously false.

Theorem 3.1.8. Every free filter on N can be extended to a free ultrafilter on N.

Proof: Let F0 be a free filter on N and let S = F : F0 ⊂ F and F is a free filter. Obviously that S 6= ∅

since F0 ∈ S. Let’s partially order S by inclusion and let C be a chain in S. Define M to be⋃F∈C F and

we prove that M ∈ S and thus, appealing to Zorn’s Lemma we establish the result. To do so, we see that

M is a free ultrafilter, since by construction it is clear that F0 ⊂ M . On one hand, ∅ /∈ M since if ∅ ∈ M

then ∅ ∈ F for some F ∈ M which is an absurd. Moreover, let A,B ∈ M with A ∈ Fα and B ∈ Fβ .

Since C is a chain, and assuming without loss of generality that Fα ⊂ Fβ , one has that A ∩B ∈ Fβ and

therefore, A∩B ∈M . Furthermore, let A ∈M and let B ∈ P(N) be such that A ⊂ B. Let F ∈ C be such

that A ∈ F and since F is a filter, B ∈ F and thus, B ∈ M . Therefore, we just proved that M is also a

filter. It remains to check that M is free: Assume, by contradiction, that there is some m ∈ N such that

m ∈ ∩X∈MX. If for every F ∈ C there is some XF ∈ F such that m /∈ XF , take X0 =⋃F∈C XF . Since

C is a chain, X0 ∈ M , altough m /∈ X0. Thus, there is some F ∈ C such that m ∈⋂X∈F X, which is an

absurd since each F is free.

Corollary 3.1.9. There exists a free ultrafilter on N.

Proof: This follows from Lemma 3.1.7 and Theorem 3.1.8.

We are now in conditions to formulate the original statement of CEC :3

Connes’ Embedding Conjecture : Every separable type II1 factor can be embedded intoRU , where

U is a free ultrafilter on N and RU is the ultrapower of the hyperfinite factor of type II1 with respect to U .

Here, with embedding we mean an isometry (relative to ||a||2 = τ(a∗a)12 induced by the trace τ )

that is also a ∗-homomorphism. Recall that the topology generated by ||.||2 on a type II1 factor M

is equivalent to the strong operator topology in the unit ball B1M ([2],Lemma 1.11.2) and thus, given a

∗-homomorphism ϕ : M → N between two type II1 factors that is also an isometry, by Kaplansky’s

Density Theorem (check A.1) we have that ϕ(M) is still a von Neumann algebra.

So what does it mean if this conjecture is true? One can show that under the assumption of

the Continuum Hypothesis, all the ultrapowers of type II1 factors with separable pre-dual along a free

ultrafilter on N are isomorphic. Particularly, R is a separable type II1 (check Remark 3.1.11). In fact,

if U is a principal ultrafilter the same happens: Let U be a principal filter, say with basis α, then it is

immediate that RU ≈ R. Indeed, Φ : R → RU defined by x 7→ [(x, ...)] is an isomorphism. The only not

completely trivial thing to check is the surjectivity of Φ. To see this, let x = [(xi)i∈I ] ∈ RU . One has that

3As quoted in Introduction, Connes in his article [13] (pg.105) said : ”Apparently such an embedding ought to exist for all typeII1 factors because it does for the regular representation of groups”. We warn that if the notation is unfamiliar, the reader shouldcheck the section of the Appendix about Filters and Ultraproducts.

33

(xα, ...) is equivalent to x and thus, x = Φ(xα). This is so, because U has basis α and hence, for any

ε > 0, the sets of the form i ∈ I : ||y − xi||τi < ε, belong to U .

However, if U is free, the proof that all ultrapowersRU are isomorphic is way harder and indeed assumes

the Continuum Hypothesis. The interested reader can check it at [20] or follow a sketch of the ideas

behind a possible proof, in Chapter 3.5.

On the other hand, it is known that the ultrapower RU is still a type II1 factor. This is again a difficult

result to prove. The interested reader can check a proof in [35] or an alternative approach in Chapter

3.5. Hence, under the Continuum Hypothesis, if Connes Embbeding Conjecture is true, this means that

there is a type II1 factor (the ultrapower RU ) that contains every separable type II1 factor. N.Ozawa,

proved in [42] that such universal object cannot have separable dual. Thus, if RU is separable, then

we know that under Continuum Hypothesis, this is not our desired object. However, considering the

following result, we can be relieved.

Proposition 3.1.10. Let U be a free ultrafilter on N. Then, RU is not separable.

Proof: We will sketch the proof of this result. It is fundamental to assume the following fact: there

is an uncountable S ⊂ Pfin(N) such that any two elements of S have a finite subset of N as their

intersection - the proof of this fact only takes in consideration elementary properties of the real numbers

and elementary facts about Cardinal Arithmetics. The goal of our proof is to show that if U is an ultrafilter

on N, there is an uncountable set of elements ofRU such that the distance - in the trace norm - between

any two of this elements is√

2. In fact, if this is the case, the result is proven since that one can show

that ifM is a separable von Neumann algebra, the space Hϕ associated to the GNS representation of a

linear, positive and normal functional ϕ is separable - in the sense of the norm topology. In this setting,

it just remains for we to consider ϕ as τU . Coming back to our goal, let τ be the trace in R and choose

a sequence of unitary elements unn∈N ⊂ R such that τ(u∗nuk) = 0, whenever n 6= k. One can show

that it is always possible to choose such sequence. Now, let S ∈ Pfin(N) be the previously mentioned

uncountable family. It is convenient, for moments, to think about the elements of S as sequences and in

this sense, given a, b ∈ S we have that an = bn only on a finite subset of N, according to the definition of

S. Therefore, τ((ubn−uan)∗(ubn−uan)) 6= 2 only on a finite subset of N. Thus, since U is a free ultrafilter,

it cannot contain finite subsets and we conclude that n ∈ N : τn((ubn − uan)∗(ubn − uan)) = 2 ∈ U .

Therefore, τU (ub − ua) = 2 and so, ||ub − ua||τU =√

2.

One should make it very clear the reason why in Conne’s Embedding Conjecture the ultrafilter is

free. Indeed, as we saw, if U is principal, then RU ≈ R. Now, recall that every separable von Neumann

algebra of type II1 contains a copy ofR (check Lemma 3.3.12). Thus, if Conne’s Embedding Conjecture

is true for principal ultrafilters, we would conclude that any separable type II1 factor would be isomorphic

to R, which is not true (for instance, we already mentioned that while LS∞ is hyperfinite, LFn is not -

check also Cor. 3.2.52). Another way to see this is using the fact that any subfactor ofR is still hyperfinite

(c.f.[13]). On the other hand, if U is free, then RU is not hyperfinite, as we see in what follows.

Remark 3.1.11. Recall that R being hyperfinite is of the form R =⋃n∈NMn

WOTwhere each Mn is

34

isomorphic to a full matrix algebra. Thus, for eachMn there is a countable and dense subset Cn. Let

C =⋃n∈N Cn. It is clear that C is still countable. We will check that in fact, CWOT

= R and thus, R is

weakly separable. Then, according to Proposition 3.1.1 and Proposition 3.1.10, one concludes that if

U is a free ultrafilter on N, then RU is never hyperfinite. An alternative proof of this fact, but this time

assuming the Continuum Hypothesis, is given in Theorem 3.2.53. To prove that R = CWOT, on one

hand we have that ⋃n∈NMn =

⋃n∈N

Cn ⊂⋃n∈N

CnWOT ⊂

⋃n∈N

CnWOT

.

Thus,⋃n∈NMn

WOT= R ⊂

⋃n∈N Cn

WOT= CWOT

. Conversely, let (xα) ⊂⋃n∈N Cn be a converging

net to some x ∈ CWOT. Since each Cn ⊂Mn, it follows that x ∈

⋃n∈NMn

WOT, i.e. x ∈ R.

Henceforth, we have that:

Corollary 3.1.12. Given a free ultrafilter U , RU is not hyperfinite. In particular, R and RU are not

isomorphic and thus, they are two distinct type II1 factors.

Remark 3.1.13. LetM be a separable type II1 factor. Assume that the Connes Embedding Conjecture

is true. Let U be a free ultrafilter. Then, joining Lemma 3.3.12 and previous comments, we have that :

R ⊂M ⊂ RU .

Furthermore, these inclusions are as subfactors, since all ∗-homomorphisms involved are monomor-

phisms. Moreover, as remarked before, the image of a type II1 factor under a ∗-isomorphism that is

also an isometry, is still a von Neumann algebra and therefore, M ⊂ RU is an inclusion of factors. To

see that R ⊂M is also an inclusion of factors, one should check Lemma 3.3.12.

3.2 Connes Embedding Conjecture for Groups

Connes Embedding Conjecture for Groups: For every countable i.c.c. group G, the group von Neu-

mann algebra LG can be embedded into a certain ultrapower RU .

In this section we will introduce hyperlinear groups and we will see how strongly they are related with

this conjecture. Indeed, we will see that :

Theorem 3.2.1. Let G be a group. Then, G is hyperlinear if and only if LG can be embedded into a

suitable ultrapower RU . Moreover, U can be chosen to be free.

And from the latter result one can immediately conclude that :

Corollary 3.2.2. Let G be a countable i.c.c. group. Then, the following are equivalent :

1. Connes Embedding Conjecture for Groups.

2. Every countable i.c.c. group is hyperlinear.

35

With this alternative equivalence established, we gain a more algebraic-flavoured perspective to

approach CEC for groups. Notice that was the case when G = Fn (with n > 1) that originally motivated

Connes. Along this chapter, we will be able to give more examples on which the conjecture is verified.

3.2.1 About the definition of hyperlinear groups

Definition 3.2.3. Let G be a group. G is said to be hyperlinear if for every F ⊂ Pfin(G) and ε > 0, there

is some n ∈ N and a map ϕ : G→ U(n) such that 4 :

1. ||ϕ(gh)− ϕ(g)ϕ(h)||2 ≤ ε,∀g, h ∈ F .

2. ||ϕ(g)− ϕ(h)||2 ≥√

2− ε,∀g, h ∈ F such that g 6= h.

3. ||ϕ(1)− 1||2 ≤ ε.

Remark 3.2.4. The original definition of hyperlinear group (c.f. [49]) took only in consideration countable

i.c.c. groups G : such G is said to be hyperlinear if it embbeds into U(RU ) for some suitable ultrapower.

In this subsection we will prove that Definition 3.2.3 is in fact equivalent to a similar, but slightly more

general definition. Indeed, we will prove that any group G (countable or not) is hyperlinear if and only if

it embbeds into some U(RU ) and thus, we will have two seemingly different alternative definitions for a

hyperlinear group.

Lemma 3.2.5. Let G be a group and suppose that for each F ∈ Pfin(G) there is some δF > 0 such that

for any ε > 0 there is some n ∈ N and some ϕ : G→ U(n) such that :

1. ||ϕ(gh)− ϕ(g)ϕ(h)||2 ≤ ε, ∀g, h ∈ F .

2. ||ϕ(g)− ϕ(h)||2 ≥ δF ,∀g, h ∈ F such that g 6= h.

Then, G is a hyperlinear group.

Lemma 3.2.6. For any n,m ∈ N with m ≥ n, there is an injective homomorphism ρn,m : U(n) → U(m)

such that ||u−v||2√2≤ ||ρn,m(u)− ρn,m(v)||2 ≤ ||u− v||2.

Proof: Let k, r ∈ N be such that m = kn + r. Define ρn,m : U(n) → U(m) to be given by ρn,m(u) =

u⊕ ...⊕ u⊕ Idr (with k copies of u). It is immediate that ρn,m is an injective homomorphism. Moreover,

note that trm(ρn,m(u)) = m−rm trn(u) + r

m . Thus, choosing ξ := m−rm ∈] 1

2 , 1], it is clear that ||ρm,n(u) −

ρm,n(v)||2 =√ξ||u− v||2

Proposition 3.2.7. G is hyperlinear if and only if it exists an embedding ϕ : G →∏Ui∈I U(ni) for some

index set I, (ni)i∈I ⊂ N and some ultrafilter U in I.

Proof: Consider I = Pfin(G)× N and suppose G is hyperlinear. Then, by Def.3.2.3 for each (F, n) ∈ I

there is some k(F, n) ∈ N and a map ϕF,n : G→ U(k(F, n)) such that ||ϕF,n(gh)−ϕF,n(g)ϕF,n(h)||2 < ε,

||ϕF,n(g) − ϕF,n(h)||2 >√

2 − ε and such that ||ϕF,n(1) − 1U(k(F,n))||2 < ε. Let’s choose ε := 1n . Now,

4U(n) denotes the group of unitary n × n matrices with complex entries. Here, we consider the so called normalized Hilbert-

Schmidt distance, given by ||u− v||2 :=

√tr((u−v)(u−v)∗)√

n

36

consider the family F = FF,nF∈Pfin(G),n∈N where FF,n = (F ′ , n′) ∈ I : F ⊂ F′, n < n

′ which

clearly has the finite intersection property. Let G be the filter generated by F and let U be an ultra-

filter on I that contains G. Recall that such object exists by Ultrafilter Lemma. Now we can define a

suitable embedding ϕ : G →∏Ui∈I U(k(i)) setting ϕ(g) := (ϕi(g))i∈I . Note that this is well-defined

since limi→U ||ϕi(1) − 1U(k(i))||2 = 0 because 1n → 0 (check Remark 3.2.8). Moreover, ϕ is in fact

an injective homomorphism : Let g ∈ G. Note that limi→U ||ϕi(gh) − ϕi(g)ϕi(h)||2 = 0 and, if g 6= h,

limi→U ||ϕi(g)− ϕi(h)||2 =√

2. Thus, ϕ is multiplicative and injective.

Conversely, let’s suppose that there is an embedding ϕ : G →∏Ui∈I U(ni) for a certain index set

I (and ultrafilter U on I) and a set ni : i ∈ I ⊂ N. We will prove that G verifies the proper-

ties stated in Lemma 3.2.5 and hence G is hyperlinear. Well, let F ∈ Pfin(G) and ε > 0. Choose

δF = 12 mindU (ϕ(g), ϕ(h)), g 6= h, g, h ∈ F, where dU denotes the metric in the ultrapower. Note

that this minimum exists since F is finite and, since ϕ is an embedding, in particular it is injective and

so, δF > 0. Let Ψ = (Ψi)i∈I : G → l∞(I, U(n(i))) be a lifting of ϕ, i.e. π Ψ = ϕ. Now this is a

crucial step in this proof : Let’s define the sets Ag,h = i ∈ I : ||Ψi(gh) − Ψi(g)Ψi(h)||2 ≤ ε and

Bg,h = i ∈ I : ||Ψi(g)−Ψi(h)||2 ≥ δF , g 6= h, for each g, h ∈ F . Since Ψ is a homomorphism, one has

that Ag,h ∈ U . Indeed, dU (Ψ(gh),Ψ(g)Ψ(h)) = 0 and thus, limi→U ||Ψi(gh) − Ψi(g)Ψi(h)||2 = 0 which

means that i ∈ I : ||Ψi(gh)−Ψi(g)Ψi(h)||2 ≤ ε ∈ U . On the other hand, Bg,h ∈ U . Indeed,

dU (Ψ(g),Ψ(h)) = limi→U||Ψi(g)−Ψi(h)||2 ≥ min lim

i→U||Ψi(g

′)−Ψi(h

′)||2 : g

′, h′∈ F.

Thus, dU (Ψ(g),Ψ(h)) ≥ 2δ and henceforth, i ∈ I : ||Ψi(g)−Ψ(h)||2 ≥ 2δ ≥ δ ∈ U .

Since Ag,h, Bg,h ∈ U , there is some j ∈ (⋂g,h∈F Ag,h) ∩ (

⋂g,h∈F Bg,h), as this is a finite intersection.

Now, it just remains to notice that Ψj : G→ U(nj) is exactly the map we’re looking for and therefore, we

can conclude that G is hyperlinear.

Remark 3.2.8. This remark should make very clear how important was our definition of F . Indeed, as

an example, let’s sketch the reason why we have that limi→U ||ϕi(g)− ϕi(h)||2 =√

2. To do so, we have

to check that ξn = i ∈ I : |||ϕi(g)−ϕi(h)||2−√

2| < 1n ∈ U for every n ∈ N. Since U is an ultrafilter, for

a fixed m ∈ N, we just need to show that η = I \ ξm /∈ U . Indeed, if for any α ∈ η we have that α = (L, k)

with g, h /∈ L, notice that η ∩ Fg,h,k = ∅. On the other hand, if g, h ∈ L note that by Def.3.2.3, there is

some r ∈ N such that k < r, since α ∈ η. In this case, just notice that η ∩Fg,h,r+1 = ∅. Both cases are

impossible if η ∈ U . Thus, η /∈ U and we conclude that ξm ∈ U .

Proposition 3.2.9. Let H be a metric group with an invariant metric d. Let’s suppose that H contains

a chain of subgroups (Hk)k∈N such that each Hk is isometrically isomorphic to some U(mk) (for a

certain unbounded sequence (nk)k∈N ⊂ N) and such that the union⋃∞k=1Hk is dense in H. Under this

conditions, a group G is hyperlinear if and only if G can be embedded into a suitable ultrapower of H.

Proof: First, let’s assume that G is hyperlinear. We know from Prop.3.2.7 that there is an index set

I, an ultrafilter U on I and a sequence (mi)i∈I ∈ N for which G can be embedded into∏Ui∈I U(mi).

Let ϕ : G →∏Ui∈I U(mi) be such an embedding and let ik : U(mk) → Hk ⊂ H denote the isometric

isomorphisms stated in the hypothesis. Now, for each i ∈ I, let k(i) ∈ N be such that mi ≤ nk(i) and let

37

ρmi,k(i) : U(mi) → U(k(i)) be the injective homomophisms given by Lemma 3.2.6. Finally, let’s define

Ψ :∏Ui∈I U(mi) →

∏Ui∈I H setting that Ψ([(ui)i∈I ]) = [(ik(i) ρmi,k(i)(ui))i∈I ]. One can check that Ψ

is an injective homomorphism (check Remark 3.2.10). Thus, we have that Ψ ϕ : G →∏Ui∈I H is the

embedding we’re looking for.

Conversely, let Ψ : G →∏Ui∈I H be an embedding. We will show that G is hyperlinear by verifying the

conditions of Lemma 3.2.5. Let F ∈ Pfin(G) and let (Ψi)i∈I be a lifting of Ψ, i.e. Ψ(g) = [(Ψi(g))i∈I ].

Now, define δF = 14 mindU (Ψ(g),Ψ(h)), g, h ∈ F, g 6= h. Since F is finite and since Ψ is injective, δF

exists and is positive. Moreover, for a given ε > 0 and g, h ∈ F 2 ∪ F , let’s define the set Ag,h = i ∈

I : ||Ψi(gh) − Ψi(g)Ψi(h)||2 ≤ ε3 and for g 6= h let’s define Bg,h = i ∈ I : ||Ψi(g) − Ψi(h)||2 ≥ 3δF .

Since Ψ is a homomorphism, we know that Ag,h ∈ U and, by the definition of δF , we have that Bg,h ∈ U .

Hence, there is some i ∈ (⋂g,h∈F 2∪F Ag,h)∩(

⋂g,h∈F,g 6=hBg,h). Now, we make a strategical observation:

since⋃∞k=1Hk is dense in H, for each g ∈ F 2 ∪ F one can choose some ng ∈ N and some ug ∈ Hng

such that dU (Ψi(g), ug) ≤ minδF , ε3. Moreover, for each g ∈ G \ (F 2 ∪ F ), let ug = 1H and let

m = maxng : g ∈ F 2 ∪ F - such m exists since F 2 ∪ F is finite. Thus, for any g ∈ G one has that

ug ∈ Hm, since Hkk∈N is a chain. Finally, let’s define ϕ : G → U(nm) by setting ϕ(g) = i−1m (ug). It

remains to prove that in fact, ϕ satisfies all conditions of Lemma 3.2.5 : Let g, h ∈ F . From one hand,

one has that

||i−1m (ugh)− i−1

m (uguh)||2 = d(ugh, uguh)

≤ d(ugh,Ψi(gh)) + d(Ψi(gh),Ψi(g)Ψi(h)) + d(Ψi(g)Ψi(h), uguh)

<3ε

3= ε.

From the other hand, let g, h ∈ G such that g 6= h. Then,

||i−1m (ug)− i−1

m (uh)||2 = d(ug, uh)

≥ d(Ψi(g),Ψi(h))− d(ug,Ψi(g))− d(Ψi(h), uh)

≥ 3δF − δF − δF = δF .

Remark 3.2.10. More generally, let I be an index set and U an ultrafilter on I. Suppose that for each

i ∈ I, (Gi, dGi) and (Hi, dHi) are metric groups and πi : Gi → Hi are group homomorphisms. If there is

a bounded set λii∈I ⊂ R+ such that dHi(πi(gi), πi(g′i)) ≥ λidGi(gi, g′i), then the map F :

∏Ui∈I Gi →∏U

i∈I Hi given by F ([(gi)i∈I ]) = [(πi(gi))i∈I ] is a well defined homomorphism. Moreover, if in addition

there is a set αii∈I ⊂ R+ such that dGi(gi, g′i) ≤ αidHi(πi(gi), πi(g′i)), then F is injective.

Remark 3.2.11. Notice that ifG is countable, Pfin(G) is still countable. Thus, and since I = Pfin(G)×N,

we have that I is countable and so, one can assume that the index set is N. Therefore, if G is countable,

one has that G is hyperlinear if and only if it can be embbeded into some HU , where U is an ultrafilter in

N.

38

Lemma 3.2.12. Let Mii∈I be a family of finite factors and let U be an ultrafilter in I. Then, the natural

inclusion i :∏Ui∈I U(Mi) → U(

∏Ui∈IMi) is a group isomorphism. In particular, if u ∈

∏Ui∈IMi is unitary,

there are unitaries ui ∈Mi such that u = [(ui)i∈I ].

Lemma 3.2.13. LetM be a von Neumann algebra with a normal and faithful trace τ and let A be some∗-subalgebra. Then, if U(A)

SOT= U(M), we have that U(A)

τ= U(M).

Proof: Let u ∈ U(M) and let (uα)α∈Λ ⊂ U(A) be a net such that uα → u in the SOT. Hence, (u−uα)∗(u−

uα)→ 0 in the WOT. Since τ is weakly continuous, we have that ||u−uα|| = τ((u−uα)∗(u−uα))→ 0.

Proposition 3.2.14. Let G be a group. Then, G is hyperlinear if and only if G can be embedded into

U(RU ) for some ultrafilter U in an index set I.

Proof: It is enough to prove that U(R) verifies the conditions of Prop.3.2.9. Indeed, if that is the case,

we know that G is hyperlinear if and only if it can be embedded into some (U(R))U . However, by Lemma

3.2.12, we have that (U(R))U = U(RU ) and we can conclude what we want.

Recall that R =⋃k∈NMk

WOT, withMk ≈ Mnk(C) for some increasing sequence (nk)k∈N ⊂ N. Thus,

we have that U(Mk) = U(nk). Now, notice that the trace in R restricted toMk, coincides with the trace

in Mk. Hence, the Hilbert-Schmidt distance in U(R) restricted to U(Mk) coincides with the Hilbert-

Schmidt distance in U(Mk), for any k ≥ 1. Therefore, it remains to show that⋃∞k=1 U(Mk) is dense

in U(R) with respect to Hilbert-Schmidt distance. If so, we have just verified that indeed U(R) verifies

the conditions of Prop.3.2.9. First, notice thatM :=⋃∞k=1Mk is a ∗-subalgebra of R and that U(M) =⋃∞

k=1 U(Mk). We will prove that U(R) = U(M)SOT

. Then, using Lemma 3.2.13, we conclude that⋃∞k=1 U(Mk) is dense in U(R) with respect to Hilbert-Schmidt distance, as desired. Well, from one

hand, let v ∈ U(M)SOT

and let (vα) ⊂ U(M) be a net that strongly converges to v. Then, (vα) weakly

converges to v and, since weak converge preserves involution and product, one can conclude that

v ∈ U(R). On the other hand, notice that R =MWOT. Then, as a consequence of Kaplanky’s Density

Theorem, we can conclude that u ∈ U(M)SOT

.5

Remark 3.2.15. In the latter proposition, we proved what we have stated in Remark 3.2.4, concerning

alternative definitions for hyperlinear groups. Thus, if G is a countable group, we have that G is hyper-

linear if and only if it can be embedded into some U(RU ) for some ultrafilter U that can be taken on

N.

3.2.2 Hyperlinear Groups and Connes Embedding Conjecture

In this subsection, we will prove Theorem 3.2.1. To do so, we start with some definitions and technical

results. It is important to be aware that great part of the work was already done in the first subsection,

where we presented a more treatable definition for hyperlinear groups that also extends the original one,

given by Radulescu. Moreover, the relationship between Theorem 3.2.1 and the CEC (for groups) was

already made very clear in Corollary 3.2.2.

5LetA be a C∗-algebra and let u ∈ AWOT be an unitary element. Then, u ∈ U(A)SOT

. This is a consequence of Kaplanksy’sDensity Theorem and the interested reader can check a proof in [26], cor.5.3.7.

39

Definition 3.2.16. Let U and V be filters in I and J respectively. We define the so called filter tensorial

product of U and V to be the set of subsets U ⊂ I × J such that i ∈ I : j ∈ J : (i, j) ∈ U ∈ V ∈ U

and we will denote it by U ⊗ V.

Proposition 3.2.17. Let U and V be filters in I and J respectively. Then, U ⊗V is a filter. Moreover, if U

and V are ultrafilters, then U ⊗ V is still an ultrafilter. Finally, U ⊗ V is free if and only if U or V are free.

Proof: First, we prove that U ⊗V is a filter : Let A,B ∈ U ⊗V. For each i ∈ I, j ∈ J : (i, j) ∈ A∩B ∈ V

if and only if j ∈ J : (i, j) ∈ A ∈ V and j ∈ J : (i, j) ∈ B ∈ V. Thus,

i ∈ I : j ∈ J : (i, j) ∈ A∩B ∈ V = i ∈ I : j ∈ J : (i, j) ∈ A ∈ V∩i ∈ I : j ∈ J : (i, j) ∈ B ∈ V

and we have that i ∈ I : j ∈ J : (i, j) ∈ A ∩ B ∈ V ∈ U and we can conclude that A ∩ B ∈ U ⊗ V.

Now, suppose that A ∈ U ⊗ V and that B ⊂ I × J is such that A ⊂ B. Since

i ∈ I : j ∈ J : (i, j) ∈ A ∈ V ⊂ i ∈ I : j ∈ J : (i, j) ∈ B ∈ V

and U is a filter, we have that B ∈ U ⊗ V. Finally, it should be clear that ∅ /∈ U ⊗ V and therefore, we

conclude that U ⊗ V is a filter.

Now let’s suppose that U and V are both ultrafilters. We will check that U ⊗ V is also an ultrafilter : Let

A ⊂ I × J be such that A /∈ U ⊗ V. We have to show that B = (I × J) \ A ∈ U ⊗ V. Since U is an

ultrafilter, if A /∈ U ⊗ V we have that C = i ∈ I : j ∈ J : (i, j) ∈ A /∈ V ∈ U . On the other hand, since

V is an ultrafilter, if i ∈ C then j ∈ J : (i, j) ∈ B ∈ V. Thus, C ⊂ i ∈ I : j ∈ J : (i, j) ∈ B ∈ V and

since C ∈ U , we conclude that B ∈ U ⊗ V.

Now, without loss of generality, let’s assume that U is free. For each A ∈ U we have that A× J ∈ U ⊗ V

and so,⋂F∈U⊗V F ⊂

⋂A∈U (A× J) = (

⋂A∈U A)× J = ∅ × J = ∅. Thus, U ⊗ V is free. Conversely, let’s

suppose that U and V are principal, say with basis u0 ∈ I and v0 ∈ J respectively, and we check

that U ⊗ V is also principal, with basis (u0, v0). Indeed, let A ∈ U ⊗ V. Since u0 is the basis for U ,

we have that u0 ∈ i ∈ I : j ∈ J : (i, j) ∈ A ∈ V. Thus, j ∈ J : (i, j) ∈ A ∈ V. Since A is arbritrary,

we can conclude that (u0, v0) ∈⋂A∈U⊗V A.

Lemma 3.2.18. Let X be Hausdorff and U and V be filters respectively in I and in J . Let (xi,j) ⊂

X be a sequence such that limi→U (limj→V xi,j) exists. Then, lim(i,j)→U⊗V xi,j exists and, moreover,

lim(i,j)→U⊗V xi,j = limi→U (limj→V xi,j).

Proof: Let xi = limj→V xi,j and x = limi→U xi. Given any U ∈ Nx we have that Γ = i ∈ I :

xi ∈ U ∈ U . On the other hand, for each i ∈ Γ, since limj→V xi,j = xi and xi ∈ U , we have that

∆i := j ∈ J : xi,j ∈ U ∈ V. Now, let F :=⋃i∈Γi × ∆i. Note that if we prove that F ∈ U ⊗ V,

we have that lim(i,j)→U⊗V xi,j = x. Indeed, let X = (i, j) ∈ I × J : xi,j ∈ U. It is clear that

F ⊂ X, thus if F ∈ U ⊗ V, we have that X ∈ U ⊗ V. It remains to prove that F ∈ U ⊗ V. Note

that if i ∈ I \ Γ, then j ∈ J : (i, j) ∈ F = ∅ /∈ V. On the other hand, if i ∈ Γ, we have that

j ∈ J : (i, j) ∈ F = ∆i ∈ V. Thus, i ∈ Γ if and only if j ∈ J : (i, j) ∈ F ∈ V, which means that

i ∈ I : j ∈ J : (i, j) ∈ F ∈ V = Γ ∈ U . Therefore, F ∈ U ⊗ V.

40

Proposition 3.2.19. Let U and V be ultrafilters repectively in I and J . Then, (RU )V ≈ RU⊗V .

Proof: Define ϕ : (RU )V → RU⊗V naturally by setting ϕ([([(ai,j)i∈I ])j∈J ]) := [(ai,j)(i,j)∈I×J ]. Since

operations are defined component-wise, it is enough to verify that ϕ preserves the trace, since the SOT

is equivalent to the topology induced by ||.||2-norm.6 But by Lemma 3.2.18 this is immediate, since we

have that limi→U (limj→V xi,j) = lim(i,j)→U⊗V xi,j . Moreover, it is clear that ϕ is bijective and thus, (RU )V

and RU⊗V are isometrically isomorphic.

Lemma 3.2.20. R ≈ R⊗R and R ≈M2(R).

Proof: Recall that R is the weak closure of an AF algebra, i.e. R =⋃k∈NAk

WOTwith Ak ≈ Mn(k)(C).

Then, it is not hard to see that R ⊗ R and M2(R) are also the weak closure of AF algebras. Thus, by

uniqueness of the type II1 hyperfinite factor, it is enough to prove that R⊗R and M2(C) are type II1-

factors. IfR is a factor, then it is clear thatR⊗R is still a factor. SinceR is a finite von Neumann algebra,

it admits a center-valued trace τ . Now, just note that τ ⊗ τ is a center-valued trace in R⊗R and thus,

R⊗R is a finite von Neumann algebra. Moreover, since dim(R) = ∞, we have that dim(R⊗R) = ∞

and hence, R ⊗ R is a type II1-factor. Now, using similar arguments, given a type II1-factor M and

a finite factor N with center-valued traces τ and σ, it is clear thatM⊗N is a still type II1-factor (with

center-valued trace τ ⊗ σ). Then, we just need to notice that choosing M = R and N = M2(C), we

have thatM⊗N ≈M2(R).

Remark 3.2.21. Obviously, proceeding by a simple induction argument, we have that R ≈⊗k

j=1R and

that R ≈Mk(R) for any k ∈ N.

Lemma 3.2.22. Let G be an i.c.c. group and let ϕ : G→ RU be an injective group homomorphism such

that τ(ϕ(g)) = 0 for every g ∈ G \ 1G. Then, there is an extension Ψ : LG → RU that is an embedding.

Proof: Let’s identify each element g of G with an element Lxg of the basis of LG. Then, define

Ψ to be the extension by linearity of ϕ. Using Prop.1.4.9 it is clear that Ψ is a ∗-homomorphism.

Moreover, and since LG is a type II1-factor (G is i.c.c.), we just need to show that the trace is pre-

served by Ψ and we can conclude that Ψ is indeed an embedding. Just notice that ||Ψ(∑k akgk)||22 =

||∑k akΨ(gk)||22 =

∑k,j ajakτ(g−1

j gk) =∑k |ak|2, since τ(ϕ(g)) = 0 for any g 6= 1G. Therefore, we have

that ||Ψ(∑k akgk)||22 = ||

∑k akgk||22 and we have that Ψ is an isometry.

Remark 3.2.23. LetM be a von Neumann algebra with a faithful trace τ . Let u ∈ U(M) be such that

τ(u) = 1. Then, u = 1. Indeed, ||u− 1||2 = τ(u)− τ(1)− τ(u∗u)− τ(u∗). However, since u is unitary we

have that uastu = 1. Moreover, τ(u∗) = τ(uu∗u∗) = τ(u∗uu) = τ(u). Thus, ||u− 1||2 = 0 and since τ is

faithful, this means that u = 1.

Proof of Theorem 3.2.1: First, let’s suppose that ϕ : LG → RU is an embedding. We will prove that

G is hyperlinear using Prop.3.2.14, defining an embedding Ψ : G → U(RU ). To do so, just set Ψ(g) :=

ϕ(Lxg ). Since ϕ is an embedding and appealing to Prop.1.4.9, it is clear that Ψ is also an embedding

into RU . It remains to show that in fact, Ψ(g) is unitary. Indeed, Ψ(Lxg )∗Ψ(Lxg ) = Ψ(L∗xg )Ψ(Lxg ) =

6[2], 1.11.2

41

Ψ(Lxg−1Lxg ) = Ψ(Lx1) = 1, where we used that L∗xg = Lxg−1 .

Conversely, suppose that G is hyperlinear. By Prop.3.2.14 we know that there is an embedding ϕ : G →

U(RU ) ⊂ RU , for a suitable ultrapower. Moreover, by Remark 3.2.21, let πk :⊗k

i=1M2(R⊗R)→ R be

trace-preserving isomorphisms for each k ∈ N. We can now define a map ρk : R →⊗k

i=1M2(R⊗R)

setting ρk(x) =

x⊗ 1

x⊗ x

⊗ ... ⊗x⊗ 1

x⊗ x

(k times). It is not hard to check that πk ρk :

R → R is a ∗-homomorphism such that tr(πk ρk(x)) = ( tr(x)+tr(x)2

2 )k. Now, let (θi)i∈I : G→ l∞(I,R)

be a lifting of ϕ, i.e. ϕ(g) = π (θi)i∈I(g) and lets consider a free ultrafilter in N, say V.

Our goal is to define an injective homomorphism Ψ : G → Rω, for some ultrafilter ω, and such that

τ(Ψ(g)) = 0 for any g ∈ G \ 1. Then, by Lemma 3.2.22, we know that there is an extension ξ : LG → Rω

that is the embedding we are looking for. For this purpose, we define Ψ : G → U((RU )V) given by

Ψ(g) = [([(πk ρk((θi)(g))i∈I ])k∈N]. Note that Ψ : G → (RU )V and that by Prop.3.2.19, we have that

(RU )V ≈ RU⊗V and thus, we can take ω to be U ⊗ V. It remains to check the other properties of

Ψ. First, let’s check that Ψ is indeed a homomorphism: since ϕ is a homomorphism, we have that

limi→U tr(θi(gh)∗θi(g)θi(h)) = 1. Thus, notice that for

d(Ψ(gh),Ψ(g)Ψ(h))2=2−2 limk→V

limi→U< 1

2k(tr(θi(gh)∗θi(g)θi(h))+tr(θi(gh)∗θi(g)θi(h))2)k = 2−2 lim

k→V

2k

2k=0

and we have that Ψ is a homomorphism. Now, let g 6= h and we have that

d(Ψ(g),Ψ(h))2 = limk→V

limi→U||πk(ρk(θi(g)))− πk(ρk(θi(g)))||22

= 2− 2 limk→V

limi→U< 1

2k(tr(θi(g)∗θi(h)) + tr(θi(g)∗θi(h))2)k. (3.1)

Since g 6= h, we have that [(θi(g)∗θi(h))i∈I ] 6= 1 and thus, by Remark 3.2.23 we can conclude that

γ = limi→U tr(θi(g)∗θi(h)) 6= 1. Hence, 12k

(γ + γ2)k → 0 and we have that d(Ψ(g),Ψ(h)) =√

2 and so,

Ψ in injective and such that τ(Ψ(g)) = 0 if g ∈ G \ 1G, as we wanted to prove.

Corollary 3.2.24. Let G be a countable group. Then, G is hyperlinear if and only if LG can be embedded

into some ultrapower RU , with U a free ultrafilter in N.

Proof: Note that in the above proof, we chose V to be free. Then, by Prop.3.2.17 we have that U ⊗ V is

free. Moreover, U ⊗ V is a filter in I × N and we remarked before (for instance, check Remark 3.2.15)

that whenever G is countable, I can be taken countable. Therefore, I × N can be replaced by N.

3.2.3 Sofic Groups

Definition 3.2.25. Let F be a finite set and let Sym(F ) be the set of its permutations. Let α ∈ Sym(F ).

We define in Sym(F ) the so called Hamming metric,7 setting dF (α, β) = |x∈F :α(x)6=β(x)||F | .

Remark 3.2.26. Actually, dF is a bi-invariant metric, since α(x) 6= β(x) if and only if δα(x) 6= δβ(x) and

since α(x) 6= β(x) if and only if αδ(x) 6= βδ(x). Moreover, it is clear that the Hamming metric is always

7Here, Fix(α) denotes the set of points fixed by α. We have that dF (α, β) =(|F |−|Fix(α−1β)|)

|F |

42

less than or equal to 1.

Definition 3.2.27. A group G is said to be sofic if for each K ∈ Pfin(G) and ε > 0, there is a finite and

non-empty set F and a map ϕ : G→ Sym(F ) such that :

1. dF (ϕ(gh), ϕ(g)ϕ(h)) ≤ ε, ∀g, h ∈ K.

2. dF (ϕ(g), ϕ(h)) ≥ 1− ε,∀g, h ∈ K, g 6= h.

Remark 3.2.28. In the conditions of the above definition, we say that ϕ is a (K, ε)-almost homomor-

phism.

Theorem 3.2.29. If G is a sofic group, then G is hyperlinear.

Proof: Let K ∈ Pfin(G) and ε > 0. Appealing to Lemma 3.2.5, it is enough to define a map ϕ :

G → U(n) for a certain n ∈ N, such that ||ϕ(gh) − ϕ(g)ϕ(h)||2 ≤ ε for all g, h ∈ K and such that

||ϕ(g)−ϕ(h)||2 ≥ 1− ε for any g 6= h. First, note that without loss of generality, we may choose ε < 2√

2.

Take ε′ = min ε2

2 , 1 −12 (√

2 − ε)2 and notice that√

2ε′ ≤ ε and that√

2(1− ε′) ≥√

2 − ε. Since G

is sofic, let ϕ′ be a (K, ε′)-almost homomorphism, with ϕ : G → Sym(F ). Since F is finite, without

loss of generality take F = 1, ..., n, where |F | = n. Finally, consider π : Sym(n) → U(n) to be such

that π(σ)ei = eσ(i), with σ ∈ Sym(n) and eini=1 an orthonormal basis for Cn. Note that we have that

||π(σ)− π(τ)||22 = 2dF (σ, τ). Thus, choose ϕ = π ϕ′ : G→ U(n). We have that

||ϕ(gh)− ϕ(g)ϕ(h)||2 = (2dF (ϕ′(gh), ϕ′(g)ϕ′(h)))12 ≤ (2ε′)

12 ≤ ε.

On the other hand, if g 6= h, one has that

||ϕ(g)− ϕ(h)||2 = (2dF (ϕ′(g), ϕ′(h)))12 ≥ (2(1− ε′)) 1

2 ≥√

2− ε ≥ 1− ε.

Corollary 3.2.30. Let G be a sofic group. Then, LG embedds into RU for some choice of an ultrafilter U

Proof: It is an immediate consequence of Theorem 3.2.29 and Theorem 3.2.1.

Why are sofic groups relevant? To start, as we just stablished, if one can prove that any countable

i.c.c. group is sofic, then Connes’ Embedding Conjecture for groups is true. Moreover, as we will see in

the further results, the class of sofic groups is rather robust. If G is sofic then, for instance, any subgroup

of G is sofic. On the other hand, it is known that any amenable group is sofic!8

Definition 3.2.31. A class of groups C is a collection of groups such that given G ∈ C and any group H

that is isomorphic to G, we have that H ∈ C.

Definition 3.2.32. Let G and H be groups, with K ∈ (P )fin(G). A map ϕ : G → H is said to be a

K-almost homomorphism if ϕ|K is injective and ϕ(gh) = ϕ(g)ϕ(h) for all g, h ∈ K.8Recall that we say that a group G is amenable if there is a left translation invariant state s on L∞(G), i.e s(gt) = s(g), where

gt(a) := g(t−1a).

43

Definition 3.2.33. Given a class C of groups, a group G is locally embeddable in C if for each K ∈

Pfin(G) there is a group H ∈ C and a K- almost homomorphism from G to H.

Proposition 3.2.34. Let C be a class of groups closed under finite direct product. Then, if Gii∈I is a

family of groups which is locally embeddable into C,∏i∈I Gi is still locally embeddable into C.

Proof: Let G =∏i∈I Gi and K ∈ Pfin(G). Let L = k1k

−12 : k1, k2 ∈ K : k1 6= k2. For each l ∈ L,

let j ∈ I be such that πj(l) 6= 1G. Note that there exists such j, since 1G 6= L. Let J ⊂ I be the set of

all such j. Now, define GJ to be∏j∈J Gj and let πJ : G → GJ be given by πJ((gi)i∈I) = (gj)j∈J . By

hypothesis, since J is finite and C is closed under finite direct products, GJ ∈ C. Thus, it suffices to show

that πJ |K is injective, since it is clear that is already a homomorphism. Indeed, πJ(l) 6= 1GJ for all l ∈ L,

by definition. Thus, πJ(k1k−12 ) 6= 1GJ , which means πJ |K is injective.

Corollary 3.2.35. Let Gii∈I be a family of groups which is locally embeddable into C. Then,⊕

i∈I Gi

is still embeddable into C.

Proof: It is an immediate consequence of the latter result, since⊕

i∈I Gi is a subgroup of∏i∈I Gi.

Proposition 3.2.36. The class of sofic groups is closed under finite direct products and under passing

to subgroups. Moreover, if a group is locally embeddable into the class of sofic groups, then it is sofic.

Proof: First, we prove that if a group is embeddable into the class of sofic groups, then the group itself is

sofic. Consider anyK ∈ Pfin(G) and ε > 0, for some locally embeddable groupG. More, letH be a sofic

group and ϕ : G→ H be a K-almost homomorphism. On the other hand, ϕ(K) ∈ Pfin(H) and since H

is sofic, there is some finite set F 6= ∅ and a map ζ : H → Sym(F ) such that dF (ζ(k1k2), ζ(k1)ζ(k2)) ≤ ε

and such that dF (ζ(k1), ζ(k2)) ≥ 1 − ε for any k1, k2 ∈ ϕ(K). Thus, defining Ψ = ζ ϕ, we have that

Ψ : G→ Sym(F ) verifies the conditions of Definition 3.2.27.

Now, we prove that the class of sofic groups is closed under finite direct products. First, we stablish a

very useful remark : Let Fini=1, with n ∈ N and each |Fi| <∞. Let Φ :∏ni=1 Sym(Fi)→ Sym(

∏ni=1 Fi)

be the natural map defined as Φ(α1, ..., αn)(x1, ..., xn) = (αi(xi))ni=1 for each αi ∈ Sym(Fi) and xi ∈ Fi.

Then, d∏ni=1 Fi

(Φ(αi)ni=1,Φ(βi)

ni=1) = 1−

∏ni=1(1− dFi(αi, βi)).9

Now, consider sofic groups G1 and G2 with K ∈ Pfin(G1 × G2) and ε > 0. We will look up for a non-

empty finite set F and some (K, ε)-almost homomorphism ϕ : G1×G2 → Sym(F ). To do so, we choose

some ε′ ∈]0, ε] such that 2ε′−(ε′)2 < ε and let Ki ∈ Pfin(Gi) be such that K ⊂ K1×K2. Moreover, since

each Gi is sofic, let ϕi : G → Sym(Fi) be (Ki, ε′)-almost homomorphisms. Define F = F1 × F2 and

ϕ : G1 ×G2 → Sym(F ) by ϕ(g1, g2)(x1, x2) = (ϕ1(g1)(x1), ϕ2(g2)(x2)), for (g1, g2) ∈ G1 ×G2. It remains

to show that ϕ is a (K, ε)-almost homomorphism. To do so, we prove that ϕ is a (K1 × K2, ε)-almost

homomorphism and since K ⊂ K1 ×K2, we are done.

Notice that by the initial remark, letting ki, li ∈ Ki, with i ∈ 1, 2, we have that

dF (ϕ(l1k1, l2k2), ϕ(l1, l2)ϕ(k1, k2)) = 1−(1−dF1(ϕ1(l1k1), ϕ1(l1)ϕ1(k1)))(1−dF2(ϕ2(l2k2), ϕ2(l2)ϕ2(k2)))

9Let F =∏ni=1 Fi. Since |F | =

∏ni=1 |Fi|, this is equivalent to check that |F |(1− dF (Φ(αi)

ni=1,Φ(βi)

ni=1)) =

∏ni=1 |Fi|(1−

dFi (αi, βi)). Thus, one just needs to check that∏ni=1 |Fix(α−1

i βi)| = |Fix((αi)ni=1)−1(βi)

ni=1)|, which follows from the fact

that Fix(((αi)ni=1)−1(βi)

ni=1) = Fix(α−1

1 β1)...F ix(α−1n βn).

44

Since each ϕi i a (Ki, ε′)-almost homomorphism, we have that (1 − dFi(ϕi(liki), ϕi(li)ϕi(ki))) ≥ 1 − ε′

and therefore,

dF (ϕ(l1k1, l2k2), ϕ(l1, l2)ϕ(k1, k2)) ≤ 1− (1− ε′)2 = 2ε′ − (ε′)2 ≤ ε,

verifying the first condition of Definition 3.2.27. Finally, let (l1, l2) 6= (k1, k2) and again, by the initial

remark we have that

dF (ϕ(l1, l2), ϕ(k1, k2) = 1− (1− dF1(ϕ1(l1), ϕ1(k1))(1− dF2(ϕ2(l2), ϕ2(k2))).

By Remark 3.2.26, and since ϕi is a (Ki, ε′)-almost homomorphism, there is some j ∈ 1, 2 such that

(1− dFj (ϕj(lj), ϕj(kj))) ≤ 1− (1− ε′) = ε′.

From this, it follows - as we wanted - that dF (ϕ(l1, l2), ϕ(k1, k2)) ≥ 1− ε′ ≥ 1− ε.

To show that this class is closed under passing to subgroups, one just follows the definitions.

Corollary 3.2.37. Given a family Gii∈I of sofic groups,∏i∈I Gi and

⊕i∈I Gi are sofic.

Proof: If follows from Prop.3.2.34 and Prop.3.2.36.

Let G be a group. We say that G satisfies the Folner condition if for every K ∈ Pfin(G) and every

ε > 0, there is some F ∈ Pfin(G) \ ∅ such that maxk∈KkF∆F|F | < ε. In [8] (Thrm.2.6.8) it is shown that

G is amenable if and only if it satisfies the Folner condition. This fact is used to define a (K, ε)-almost

homomorphism and thus, stablishing the following :

Theorem 3.2.38. Every amenable group is sofic.

Proof: The interested reader should check [62].

Remark 3.2.39. Recall that any finite, abelian or solvable group, is amenable.

Corollary 3.2.40. Let Gii∈I be a family of amenable groups. Then,⊕

i∈I Gi and∏i∈I Gi are sofic.

Corollary 3.2.41. Let Gii∈I be a family of amenable groups. Let G be⊕

i∈I Gi or∏ni∈I Gi. Then, LG

embeds into RU for some ultrafilter U .

Henceforth, Corollary 3.2.41 guarantees that Connes Embedding Conjecture is verified when-

ever G is an amenable i.c.c. group. Moreover, the result per se is quite more general since the

hypothesis does not require G to have infinite conjugacy classes! One should emphatize the

relevance of this result recalling that abelian, finite and solvable groups are amenable.

Now we briefly address the case LFn . Notice that when n = 1, Fn ≈ Z and because Z is amenable

(since it is abelian) Cor.3.2.41 gives the existence of an embedding - however the reader should notice

45

that Z is not an i.c.c. group. In what follows, we will sketch an argument that will establish that indeed

LFn embedds into some RU and meanwhile, as mentioned in Remark 3.1.11, we will see why RU , with

U a free ultrafilter on N, is not hyperfinite.

Definition 3.2.42. A group G is residually finite if it satisfies one of the following equivalent conditions:

1. For every g ∈ G \ 1 there is a normal subgroup N of finite index such that g /∈ N .

2. For every F ∈ Pfin(G) there is a homomorphism h from G to a finite group such that the restriction

h|F is injective.

3. G is a subgroup of a direct product of a family of finite groups.

In fact, all of these conditions are equivalent: Let’s assume (1) and thus, for each g ∈ G\1 letNg be

a normal subgroup of G of finite index and let πg : G → G/Ng be the quotient map. Note that πg(g) 6= 1

since g /∈ Ng. Consequently, define ξ(x) := (πg(x))g∈G\1 which is a monomorphism between G and∏g∈G\1G/Ng, where each G/Ng is finite since the index of Ng is finite. Therefore, we have (3). On the

other hand, assume (3) and use h chosen as a projection on the product of a suitable finite subfamily

of groups and we have (2). Finally, assuming (2) one can chose F = 1, g and let h : G → L be a

homomorphism such that restricted to F is injective and where L is a finite group. Then, h(g) 6= 1 and

we can establish (1), chosing the kernel of h.

Lemma 3.2.43. Free groups are residually finite.

Proof: Let F be a free group with generator X. We will use condition (1) of the previous definition to

establish that a free group is residually finite. Let a = an...a1 be a reduced word with each ai ∈ X and

let’s define f : X → Sn+1 in the following way: f(x) is the identity if x is not equal to any ai or a−1i . On

the other hand, let A be the set of indexes i such that ai = x and let B be the set of indexes j such

that a−1j = x. Then, f(x) is defined to be a permutation σ that sends i ∈ A to i + 1 and j + 1 to j for

any j ∈ B. Notice that since an element and its inverse cannot occur adjacently in the reduced form,

this is well-defined. Now, since F is free, we extend f to a homomorphism from F to Sn+1. Moreover,

under this homomorphism f(a) sends 1 to n+ 1 and thus, is not the identity. Therefore, the kernel of this

homomorphism is a normal subgroup of F with finite index and such that does not contain a.

Definition 3.2.44. A group G is said to be locally embeddable into finite groups (LEF, for short) if for

every finite subset F ⊂ G there is a partially defined monomorphism i of F into a finite group.

Lemma 3.2.45. Every residually finite group is LEF. In particular, every free group is LEF.

Proof: It is a direct consequence of Definition 3.2.42, namely condition (2).

Theorem 3.2.46. Every LEF group is a sofic group.

Proof: The interested reader can check [46], Theorem 5.3.

Corollary 3.2.47. LFn can be embeddable into a suitable ultrapower RU .10

10Recall, once again, that it was this result that motivated Connes’ Embdedding Conjecture.

46

Proof: It is a direct consequence of Theorem 3.2.46 and Corollary 3.2.30.

Definition 3.2.48. A finite von Neumann algebra M acting on H with faithful trace τ is said to be

amenable if there is a central state Φ on L(H) such that Φ|M = τ .

Proposition 3.2.49. Let G be a discret group. Then, G is amenable if and only if LG is amenable.

Proof: The interested reader can check a proof in [4], Prop.2.12.

Corollary 3.2.50. For every n 6= 1, LFn is not amenable.

Proof: Using Prop.3.2.49 we just need to check that Fn is not amenable whenever n 6= 1 (since F1 ≈ Z

which is abelian and thus amenable). We will check the n = 2 case. Indeed, let F2 = 〈a, b〉 and consider

the partition F2 = 1 ∪ Xa ∪ Xa−1 ∪ Xb ∪ Xb−1 , where Xl is the set of all reduced words starting

with l ∈ a, b, a−1, b−1. Now, suppose by contradiction that F2 is amenable. Then, there is a state µ on

l∞(F2) that is invariant by left translation. Applying µ, we get that 1 = µ(1)+µ(1Xa)+µ(Xa−1)+µ(1Xb−1 ).

Decomposing F2 as Xa ∪ aXa−1 = Xb ∪ bXb−1 , one gets that µ(1Xa) + µ(1Xa−1 ) = 1µ(1Xb) + µ(1Xb−1 ),

by invariance of µ. Then, µ(1) = −1 which is an absurd.

Theorem 3.2.51. Any finite and hyperfinite von Neumann algebra is amenable.

Proof: The interested reader can check [4], Prop.2.13

Corollary 3.2.52. For every n 6= 1, LFn is not hyperfinite.

Proof: It is a direct consequence of Theorem 3.2.51 and Corolary 3.2.50.

Theorem 3.2.53. Under the Continuum Hypothesis, RU is never hyperfinite for a free ultrafilter U on N.

Proof: By Corolary 3.2.52 we know that LFn is not hyperfinite. Moreover, by Corolary 3.2.47 we have

that LFn can be embedded into some ultrapower RU . However, it is a well known result due to Connes

([13]) that every infinite dimensional subfactor of R is hyperfinite and therefore, by uniqueness of R, one

can conclude that RU is not hyperfinite. But, under the Continuum Hypothesis, all ultrapowers of R are

isomorphic (check Corolary 3.5.8).

3.3 CEC and WEP

Definition 3.3.1. Let A and B be C∗-algebras and let ϕ : A → B be a linear map. For each n ∈ N one

can define ϕn : Mn(A) → Mn(B) by setting ϕ([aij ]) = [ϕ(aij)]. We say that ϕ is completely positive if

ϕn is positive for every n.

Definition 3.3.2. 11 Let M ⊂ L(H) be a von Neumann algebra. Let A ⊂ M be a weakly dense C∗-

subalgebra. One says that M has a weak expectation property (WEP) relative to A if there is a unital

completely positive map Φ : L(H)→M such that for each a ∈ A one has that Φ(a) = a.

11This definition is inspired by the notion of injectivity :M⊂ L(H) is said to be injective if there exists a u.c.p. map Φ : L(H)→M such that Φ(a) = a for any a ∈M.

47

In this section, we will follow the proof of this :

Theorem 3.3.3. LetM be a separable type II1 factor. The following are equivalent :

1. Connes Embedding Conjecture holds.

2. M has the weak expectation property (WEP).

In order to do so, we will recall some theory about group C∗-algebras and free products.

3.3.1 Group C∗-algebras and Free Groups

Group C∗-Algebra

Let A be an involutive Banach algebra with approximation of identity. Let’s define ||.||u : A → R such

that ||a||u := supπ∈R ||π(a)||, where R is the class of representations of A. Recall that if A is a normed∗-algebra and B is a C∗-algebra, then any continuous ∗-homomorphism π : A → B is such that ||π|| ≤ 1.

Thus, the least upper bound in the definition of ||.||u exists, by simply appealing to Supremum Axiom,

since R 6= ∅ (by GNS construction). It is easy to check that in fact ||.||u defines a semi-norm that verifies

||ab||u ≤ ||a||u||b||u and ||a∗a||u = ||a||2u. Now, let’s consider the ideal I = a ∈ A : ||a||u = 0. It is

not hard to see that A/I||.||u is a C∗-algebra (the so called enveloping C∗-algebra of A). Let’s denote

the enveloping C∗-algebra of A by E(A). Then, E(A) is characterized (up to isomorphism) by being the

unique C∗-algebra together with a ∗-homomorphism Ψ : A → E(A) such that given any ∗-homomorphism

ϕ : A → B there is a ∗-homomorphism Φ : B → E(A) such that Ψ = Φ ϕ.

Now, let G be a locally compact group. Recall that L1(G), equipped with the Haar measure, is an

involutive Banach algebra with the ||.||1 norm and with involution given by f∗(t) := ∆(t)−1f(t−1), where

∆ is the modular function, i.e. ∆ : G → R given by µ(g−1S) = µ(S)∆(g), with µ denoting the Haar

measure. Moreover, G is unital if and only if G is discrete.12 Nevertheless, L1(G) has always an

aproximation of identity ([14],pg.183) and consequentely, one can consider its enveloping C∗-algebra,

that we denote by C∗(G). This is the so called group C∗-algebra.

Let G continue to be a locally compact group. We can define a C∗-norm in L1(G) by ||f ||r := ||λ(f)||,

where λ ∈ L2(G) is the left regular representation of L1(G). To the closure L1(G)||.||r , denoted by

C∗r (G), we call the reduced C∗-algebra of G. One can show that ||f ||r ≤ ||f ||u for all f ∈ L1(G) and

so, the identity map induces a ∗-homomorphism from C∗(G) onto C∗r (G). Hence, C∗r (G) is isomorphic

to a quotient C∗-algebra of C∗(G). However, if G is abelian, C∗r (G) = C∗(G) (c.f. [14], pg.184). More

generally, C∗(G) = C∗r (G) if and only if G is amenable.

Now, let’s consider any group G and a field K. Let⊕

GK be the free vector space on G over K and

consider Γ ⊂⊕

GK such that Γ = f : G→ K : supp(f) <∞. If one consider the convolution product

in Γ, we have an algebra.13 Then, Γ is usually denoted by K(G) and called the group algebra. If G is

12A good reference is [32]13Just like in L1(G), (f ∗ g)(t) :=

∫G f(s)g(s−1t)ds

48

discrete, indeed K(G) coincides (as a set) with L1(G). Thus, if K = C and if G is discrete, one can

reformulate the definition of group C∗-algebra as being the enveloping C∗-algebra of C(G).

Free Products

Let S be a set and consider S−1 to be the set of symbols such that s ∈ S if and only if s−1 ∈ S−1. Let

T := S∪S−1 and let W be the monoid generated by T . An element of W is called a word. We say that a

reduced word is a word where the sequences ss−1 or s−1s do not occur. The set of reduced words will be

denoted by F (S). Clearly, under concatenation of words and defining the empty word to be the identity

element, F (S) has a group structure. Moreover, F (S) is characterized - up to isomorphism - by the

following universal property : Given any map f : S → G (G as a set) there is an unique homomorphism

ϕ : F (S)→ G such that ϕ iS = f , where iS : S → F (S) is simply the inclusion of symbols.

Remark 3.3.4. It should be clear that F (S) ≈ F (T ) if and only if |T | = |S|. If |S| = n ∈ N, it is usual to

denote F (S) by Fn.

Definition 3.3.5. Let Gii∈I be a family of groups. The free product ∗i∈IGi is the unique group G (up

to isomorphism) with homomorphisms Ψi : Gi → G characterized by the following univeral property :

Given a group H and homomorphisms ϕi : Gi → H, there is an unique homomorphism Φ : G → H

such that ϕi = Φ Ψi.

Notice that one can construct G taking G = g1...gn : gj ∈ Gij \ e, i1 6= ... 6= in, n ∈ N under

multiplication given by concatenation (followed by possible reduction). Then, one defines Ψi : Gi → G

to be the inclusion of symbols and Φ(gi1 ...gik) :=∏kj=1 ϕij (gik).

Definition 3.3.6. Let Aii∈I be a family of unital algebras. Their unital algebra free product, denoted

as ∗i∈IAi is the unique (up to isomorphism) unital algebra A together with unital homomorphisms Ψi :

Ai → A such that given any unital unital algebra B and unital homomorphisms Φi : Ai → B, there exists

an unique unital homomorphism Φ : A → B such that Φ = Φ Ψi.

Remark 3.3.7. 14 Let Gi and H be groups and Ai and B be ∗-algebras. Thanks to the universal

property of ∗i∈IGi, one has that Hom(∗i∈IGi, H) ≈∏i∈I Hom(Gi, H). Again, by the universal prop-

erty of ∗i∈IAi, one has that Hom(∗i∈IAi,B) ≈ πi∈IHom(Ai,B). Furthermore, it is not hard to verify that

Hom(C(Gi),B) ≈ Hom(Gi, U(B)). Thus, we can conclude that

Hom(∗i∈IC(Gi),B) ≈∏i∈I

Hom(C(Gi),B) ≈∏i∈I

Hom(Gi, U(B))

≈ Hom(∗i∈IGi, U(B)) ≈ Hom(C(∗i∈IGi),B).

Definition 3.3.8. Let Ai∈I be a family of unital C∗-algebras. Their unital C∗-algebra free product ∗i∈IAiis the unique (up to isomorphism) unital C∗-algebraA together with unital ∗-homomorphism Ψi : Ai → A

such that given any unital C∗-algebra B and unital ∗-homomorphisms Φi : Ai → B there exists an unique∗-homomorphism Φ : A → B such that Φi = Φ Ψi.

14It should be obvious from the context if we are referring to group homomorphisms or to unital ∗-homomorphisms.

49

Lemma 3.3.9. Let G be a discrete group. Then, C∗(∗i∈IGi) ≈∗i∈I C∗(Gi) (as C∗-algebras).

Proof: We will prove that given any C∗-algebra B one has thatHom(∗i∈IC∗(Gi),B) ≈ Hom(C∗(∗i∈IGi),B).

To do so, notice that since G is discrete, C∗(G) = E(C(G)). Then, using the universal property of en-

veloping algebras and Remark 3.3.7, one has that

Hom(∗i∈IC∗(Gi),B) ≈ Hom(∗i∈IE(C(Gi)),B) ≈

∏i∈I

Hom(E(C(Gi)),B) ≈∏i∈I

Hom(Gi, U(B))

≈ Hom(∗i∈IGi, U(B)) ≈ Hom(C(∗i∈IGi),B) ≈ Hom(E(C(∗i∈IGi)),B)

≈ Hom(C∗(∗i∈IGi),B).

3.3.2 CEC and WEP

Definition 3.3.10. A C∗-algebra A is said to be residually finite dimensional (RFD) if there exists ∗-

homomorphisms πn : A →Mk(n)(C) such that⊕πn : A →

∏Mk(n)(C) is faithful.

Remark 3.3.11. Choi proved in [12] that C∗(F2) is RFD. Thus, one can conclude that C∗(F∞) is also

RFD. To see this, we notice that we can embbed F∞ → F2 = 〈a, b〉. Indeed, recall that a subgroup of

a free group is still free. Now, let’s consider the subgroup of F2 generated by the elements anbnn∈N.

Since the word akbk does not belong to the subgroup generated by albll∈N\k, one can conclude that

there is copy of F∞ inside F2. Since being RFD is a property preserved by subalgebras, we are done.

Lemma 3.3.12. LetM be a type II1 factor. Then, given a sequence knn∈N, there is a ∗-monomorphism

such that∏n∈NMk(n)(C) →M. In particular, the hyperfinite type II1 factorR is contained in every type

II1 factor.

Proof: Since M is a type II1 factor, tr(P) = [0, 1]. Hence, let pn ⊂ P be such that tr(pn) → 1

and define q1 := p1 and qn+1 := pn+1 − pn. We have that qn is a sequence of pairwise orthogonal

projections that add up to 1. Now, we will embbed Mn(C) into qnMqn for any n ∈ N : By ([27], Lemma

6.5.6), since M is a type II1 factor, qj =∑ni=1 ri with rini=1 pairwise equivalent projections. Thus,

there are partial isometries e1r such that e1ke∗1k = r1 and e∗1ke1k = rk. Now, define ekj := e∗1ke1j , for

any k, j ∈ 1, ..., n. It is not hard to check that these elements behave like matrix units. Therefore, let’s

consider the ∗-monomorphism πn : Mn(C) → qnMqn given by a 7→∑nk,j=1 akje

∗1ke1j . Finally, one can

define a ∗-monomorphism π :∏n∈NMn(C)→M by setting π(an) =

∑n πn(an). Notice that the sum

converges since∑qn = 1.

In particular, using again Lemma 6.5.6 from [27] just with the projection 1 (instead of using a sequence

qn), one concludes that we can embbed any Mn(C) into M and thus, using the model exposed in

Chapter 2.3 and Bicommutant Theorem, we conclude that every type II1 contains a copy of R as a

subfactor.

Proposition 3.3.13. Let M be a separable type II1 factor. Then, there is a ∗-monomorphism ϕ :

C∗(F∞) →M such that ϕ(C∗(F∞))WOT

=M.

50

Proof: Let A1 := C∗(F∞) and let An := An−1 ∗ C∗(F∞),∀n > 1. Now, let A be the inductive limit

of An : n ∈ N. By Lemma 3.3.9 and Remark 3.3.4 one concludes that A ≈ C∗(F∞). Moreover,

by Remark 3.3.11, we know that A is RFD and so, let kn ⊂ N and σ : A →∏nMk(n)(C) be an

unital ∗-monomorphism, such that the restriction σ|Ai is an injective map of Ai into∏nMk(n)(C). Our

goal is to define a sequence of unital ∗-homomorphisms ϕi : Ai → M such that each ϕi is injective,

ϕi+1|Ai = ϕi and such that⋃i∈N ϕi(Ai)

WOT= M. Notice that indeed, this is enough since we have

that⋃i∈N ϕi(Ai) = ϕ(C∗(F∞)), where ϕ comes from the universal property of the inductive limit (recall

Theorem A.2.4). It remains to construct such a sequence ϕii∈N.

First, note that one can chose an increasing sequence of projections pnn∈N such that tr(pn)→ 1 (just

recall that tr(P) = [0, 1]). Now, define a sequence of pairwise orthogonal projections qn by setting

q1 := p1 and qn+1 := pn+1 − pn and consider the type II1 factors Qj := qjMqj . Appealing to Lemma

3.3.12 and commposing with σ, let σj be the embeddings of A into Qj ⊂ M. On the other hand, for

each i ∈ N, let πi : C∗(F∞) → piMpi be a ∗-homomorphism such that πi(C∗(F∞))WOT

= piMpi.

Note that such πi exist : Indeed, since M is separable, each piMpi is still separable. Thus, one can

construct such πi by mapping the generators of F∞ to a countable set of unitaries that is weakly dense

in piMpi. Finally, we define ϕ1 = π1 ⊕ (⊕

j≥2 σj |A1) : A1 → p1Mp1 ⊕ (∏j≥2Qj) ⊂M. It is not hard to

check that ϕ1 is a ∗-monomorphism. Now, let θ2 : A2 → p2Mp2 be the product15 of ∗-homomorphisms

x 7→ p2ϕ1(x)p2 and π2. Then, just define ϕ2 : A2 → p2Mp2

⊕(∏j≥3Qj) ⊂M to be θ2 ⊕ (

⊕j≥3 σj |A2).

Note that ϕ2|A1 = ϕ1.

In an analogous manner, we proceed to define the others ϕi verifying the desired conditions. Just remark

that we take⊕

j≥n+1 σj |An since it is not granted that σn is faithful.

Definition 3.3.14. Let A ⊂ L(H) be a C∗-algebra. A tracial state τ on A is called an invariant mean

if there is a state Ψ on L(H) such that is invariant under the action of U(A) on L(H)16 and such that

Ψ|A = τ . Usually, the set of all invariant means on A is denoted by T(A).

Recall that we can always define a (unbounded) trace on L(H) given by Tr(T ) :=∑∞n=1〈Ten, en〉,

where enn∈N is (any) orthogonal basis for H. If H is finite dimensional, say dim(H) = n, we have that

tr(T ) =∑ni=1〈Tei,ei〉

n is the unique tracial state on L(H). Recall that T ∈ L(H) is said to be trace class

if for some (and hence, all) orthonormal basis enn∈N of H, the expression∑n〈Ten, en〉 is absolutely

convergent. Evidently, whenever H is finite dimensional, every operator is trace class. One can show

that the predual of L(H) is the space of trace class operators T . We will denote the L1-norm and

L2-norm respectively by ||.||Tr,1 and ||.||Tr,2. In this setting, we have the following result, known as

Powers-Stormer Inequality :

Proposition 3.3.15. Let h, k ∈ T be positive elements. Then, ||h− k||2Tr,2 ≤ ||h2− k2||Tr,1. In particular,

if u ∈ L(H) and h is positive and with finite rank, then

||uh 12 − h 1

2u||Tr,2 = ||uh 12u∗ − h 1

2 ||Tr,2 ≤ ||uhu∗ − h||12

Tr,1

15Let f : A → B and g : C → D be homomorphisms. Then, there exists a homomorphism h : A ∗ C → B ∗ D, such thath iA = iB f and h ic = iD g, where i• are the natural inclusions. We say that h is the free product of f and g.

16We mean that Ψ(uTu∗) = Ψ(T ), ∀T ∈ L(H), u ∈ U(A)

51

Proof: This result is usually known as the Powers-Stormer Inequality. The reader can check a proof in

[48].

Lemma 3.3.16. Let h ∈ L(H) be a positive, finite rank operator with rational eigenvalues and Tr(h) = 1.

Then, there exists some q ∈ N and a u.c.p. map Φ : L(H)→ Mq(C) such that : tr(Φ(T )) = Tr(hT ) for

all T ∈ L(H) and such that |tr(Φ(uu∗)− Φ(u)Φ(u∗))| < 2||uhu∗ − h|| 12 for all u ∈ U(L(H)).

Proof: We will briefly sketch the proof: Notice that the trace of a matrix defined above, is the sum of its

eigenvalues. Let v1, ..., vk ⊂ H be the eigenvectors associated with the eigenvalues p1q , ...pkq ⊂ Q.

Since all eigenvalues λi are rational, there is a q ∈ Z such that λi = piq for each i. Since by hypothesis

Tr(h) = 1, one can immediately conclude that∑pi = q. Now, take any orthogonal basis wm for

H and we consider the orthogonal subset of H ⊗ H defined by B :=⋃ki=1vi ⊗ w1, ..., vi ⊗ wpi. Let

V = span(B) ⊂ H ⊗ H. Consider the orthogonal projection P : H ⊗ H → V . One can check that

Tr(P (T ⊗ 1)P ) =∑ki=1 pi〈Tvi, vi〉 holds. This formula is useful to stablish our result since we now

define Φ : L(H) → Mq(C) by setting Φ(T ) = P (T ⊗ 1)P . We will now check that Φ verifies the two

desired conditions :

Notice that tr(Φ(T )) = Tr(P (T⊗1)P )q =

∑ki=1〈T

piq vi, vi〉 =

∑ki=1〈Thvi, vi〉 = Tr(Th). Here, we just used

the fact that piq vi = hvi (eigenvalues) and the fact that P (T ⊗ 1)P is represented in basis B by a q × q

block diagonal matrix whose blocks have dimension pi. Moreover, one can show that Φ is u.c.p. and

thus, we stablish the first property. Now, to prove the second property, one can work a little bit and, using

Holder Inequality and some considerations about B and how we represent h12T , Th

12 and h

12Th

12T ∗,

one can conclude that |Tr(h 12Th

12T ∗)−tr(Φ(T )Φ(T ∗))| ≤ ||Th 1

2 ||Tr,2||h12T −Th 1

2 ||Tr,2. Then, assuming

that T ∈ U(L(H)), so that ||Th 12 ||Tr,2 = ||h 1

2 ||Tr,2 = 1, we have that:

|Tr(h 12Th

12T ∗)− tr(Φ(T )Φ(T ∗))| ≤ ||h 1

2T − Th 12 ||Tr,2 = ||Th 1

2T ∗ − h 12 ||Tr,2 ≤ ||ThT ∗ − h||

12

Tr,1,

where the last step is due to Powers-Stormer Inequality. Hence,

|Tr(h 12Th

12T ∗)− tr(Φ(T )Φ(T ∗))| ≤ ||ThT ∗ − h||

12

Tr,1(†)

Then,

|tr(Φ(TT ∗)− Φ(T )Φ(T ∗))| ≤ |tr(Φ(TT ∗)− Tr(h 12Th

12T ∗))|+ |Tr(h 1

2Th12T ∗)− tr(Φ(T )Φ(T ∗))|

≤ |1− Tr(h 12Th

12T ∗)|+ ||ThT ∗ − h||

12

Tr,1

= |Tr((Th 12 − h 1

2T )h12T ∗)|+ ||ThT ∗ − h||

12

Tr,1

≤ ||h 12T ∗||Tr,2||Th

12 − h 1

2T ||Tr,2 + ||ThT ∗ − h||12

Tr,1

≤ 2||ThT ∗ − h||12

Tr,1

Therefore, we have the result. Notice that on the first inequality we just used triangular inequality. We

used † on the second inequality, Cauchy-Schwartz Inequality on forth and Powers-Stormer Inequality on

fifth, since T ∈ U(L(H)) and so, ||h 12T ∗||Tr,2 = 1.

52

Remark 3.3.17. Recall that one can identify the normal functionals of L(H) with T associating ω to

A 7→ Tr(XA), where X ∈ T for a normal functional ω. Since normal states are dense in the set of all

states on L(H), given a state Ψ there is a net hα of normal states such that hα → Ψ. Using the latter

identification, one has that for any T ∈ L(H) we have that Tr(hαT )→ Ψ(T ).

Remark 3.3.18. Let A ⊂ L(H) be a separable C∗-algebra. Recall that any a ∈ A is of the form

a = b + ic, where b and c are self-adjoint. Moreover, any self-adjoint element is a linear combination of

unitary elements. Thus, and sinceA is separable, one can choose an increasing family Unn∈N ⊂ U(A)

such that span(⋃n∈N Un)

WOT= A. Let U denote

⋃n Un.

Proposition 3.3.19. Let τ be a tracial state on A ⊂ L(H), with A a separable C∗-algebra. The following

are equivalent :

1. τ is an invariant mean.

2. There is a sequence of u.c.p. maps Φn : A →Mk(n)(C) such that :

(a) ||Φn(ab)− Φn(a)Φn(b)||Tr,2 → 0

(b) τ(a) = limn→∞ τ(Φn(a))

3. Given any faithful representation ρ : A → L(H), there exists a u.c.p. map Φ : L(H) → (πτ (A))′′

such that Φ(ρ(a)) = πτ (a), where πτ denotes the GNS representation associated to τ

Proof: (1) ⇒ (2) : Let Ψ be a state on L(H) that extends τ and such that Ψ(uTu∗) = Ψ(T ) for all u ∈

U(A). By Remark 3.3.17, let hα be a net such that for any T ∈ L(H) we have that Tr(hαT )→ Ψ(T ),

with each hα with trace 1. Since Ψ(uTu∗) = Ψ(u), it follows that Tr(hαu∗Tu) → Ψ(uTu∗) = Ψ(T ). On

the other hand, Tr(uhαu∗T ) = Tr(hαu∗Tu). Thus, we have that Tr(hαT )− Tr((uhαu∗)T )→ 0. Hence,

we can conclude that∑〈hαTen, en〉−

∑〈(uhαu∗T )en, en〉 → 0, which means that hα−uhαu∗ → 0WOT .

Now, let’s consider sets Un = u1, ..., un ⊂ U(A) like in Remark 3.3.18. Fixed some n, let us consider

the convex hull cn of the set u1hαu∗1, ..., unhαu

∗n.17 Note that since hα−uhαu∗ → 0WOT , we know that

0 ∈ cnWOT . Thus, appealing to Hahn-Banach Separation Theorem, we conclude that cnWOT = cnL1

.18

Therefore, there is a convex combination of hα - say h - such that Tr(h) = 1 (since in any convex

combination the coefficients add up to 1 and Tr(hα) = 1), ||uhu∗ − h||1 < 1n and |Tr(uh)− τ(u)| < 1

n for

all u ∈ Un (since hα − uhαu∗ → 0WOT for all u ∈ U(A) and cnWOT = cnL1

).

One can assume that h is a finite rank operator with rational eigenvalues (check Remark 3.3.20) and

with Tr(h) = 1. Then, by Lemma 3.3.16, let Φn : L(H)→Mk(n)(C) be a sequence of u.c.p. maps such

that Tr(Φn(u)) → τ(u) and |Tr(Φn(uu∗) − Φn(u)Φn(u∗))| → 0 for all u ∈ U (since |Tr(uh) − τ(u)| → 0

and ||uhu∗ − h||1 → 0). We just reached (2(a)) for unitaries. Now, let’s define Φ :=⊕

n Φn : A →∏nMk(n)(C) ⊂ l∞(R). Let π : l∞(R) → RU for any free ultrafilter U on N, be the usual quotient

projection. One can show that for any u ∈ U we have that π Φ is multiplicative, using the fact that

span(U)WOT

= A. Then, by the definition of ultraproduct, it follows that ||Φn(ab) − Φn(a)Φn(b)||Tr,2 →17Recall that given a set X, the convex hull c(X) is defined to be the minimal convex set containing X,i.e. the set of all convex

combinations of elements of X.18Here, ||A||1 := Tr(A)

53

0 for all a, b ∈ A and we established (2(a)). To establish (2(b)), we need to use the property that

Tr(Φn(u))→ τ(u) and extend to any a ∈ A by linear combination.

(2) ⇒ (3) : We will just sketch the idea of the proof. Let’s identify each Mk(n)(C) with a subfactor of

R.19 Thus, we can define ξ : A → l∞(R) by x 7→ (Φn(x))n. By hypothesis, each Φn is assymptotically

multiplicative in the 2-norm and thus, π ξ : A → RU is a τ -preserving ∗-homomorphism (recall that

τU ((xn)) = limn→U τ(xn)). Moreover, one can check that π ξ(A)WOT

≈ (πτ (A))′′. On the other hand,

let K be the representing Hilbert space for l∞(R) and, since l∞(R) is injective, there is a surjective map

F : L(K)→ l∞(R). Let i : L(H)→ L(K) be the natural embedding (i.e such that ξ = ρ i F ). Finally,

let G : RU → (πτ (A))′′

be a conditional expectation.(c.f.[56],Prop2.36) We can now define the desired

map Φ : L(H)→ (πτ (A))′′

by setting Φ := G π F i.

(3) ⇒ (1) : Again, we just sketch the proof. By hypothesis, Φ is multiplicative on A and one has that

Φ(aTb) = πτ (a)Φ(T )πτ (b) for any a, b ∈ A and T ∈ L(H). Now, let τ′′

denote the trace on (πτ (A))′′. It

is not hard to verify that τ′′ Φ is a state on L(H) that extends τ and which is invariant under the action

of the unitary group of A and so, it is an invariant mean.

Remark 3.3.20. One can show that the set of finite rank operators is ||.||1 dense in T . Moreover, fixing

the rank to some d < ∞, one has that the set T ∈ L(H) : rank(T ) = d,with rational eigenvalues is

dense in the set of all rank d operators. Joining these two observations with the fact that tr in the finite

rank case is the sum of the eigenvalues, for the sake of inequalities we can assume that h is a finite rank

operator with rational eigenvalues.

Proof of Theorem 3.3.3: (1)⇒ (2): According to Prop.3.3.13 we can identify C∗(F∞) with a weakly

dense subalgebra ofM, say A. We will prove thatM has a weak expectation relative to A. To do so, we

prove that πτ (M) has a weak expectation relative to πτ (A). Since τ is faithful and weakly-continuous,

then πτ (A) is still weakly dense in πτ (M) and indeed, πτ (A) and πτ (M) work as copies of A and M

respectively. The main step is to prove that τ |A is an invariant mean. In fact, suppose that we already

know that τ |A is an invariant mean and consider πτ (M) ⊂ L(H) such that πτ (A) = πτ |A(A) ⊂ L(H).

By Prop.3.3.19, there is a u.c.p. map Φ : L(H) → (πτ (A))′′

such that Φ(πτ (a)) = πτ (a) for all a ∈ A,

since πτ is a faithful representation (πτ has the role of ρ). Moreover, since πτ (A)WOT

= πτ (A), by the

Bicommutant Theorem we have that (πτ (A))′′

= πτ (M). This means that M has a weak expectation

relative toA, as it is desired to establish. Therefore, it just remains to prove that π|A is in fact an invariant

mean. To do so, we will prove the existence of a sequence of u.c.p. maps Φn : A →Mk(n)(C) such that

||Φn(ab) − Φn(a)Φn(b)||2 → 0 and τ(a) = limn→∞ tr(Φn(a)), for all a, b ∈ A. Then, appealing again to

Prop.3.3.19 we conclude that τ |A is an invariant mean.

As it was mentioned in the beginning, and using our hypothesis, one can view the generators unn∈N of

F∞ as elements of A and thus, we can consider each un ∈ RU . Then, as one can see in Remark 3.3.21,

each un is the ||.||2-limit of a sequence of unitary elements inR and thus, to each n ∈ N one can choose

a sequence of unitary matrices that converge to un in the ||.||2-norm. Having this in mind, let’s define

a map σ that associates to each un ∈ F∞ such a sequence σ(un) ∈∏n∈NMk(n)(C). Since unn∈N

19Let N be a subfactor ofR. Either N ≈ R or N ≈Ml(C) for some l ∈ N. Moreover, for any l ∈ N there is a subfactor N suchthat N ≈Ml(C).(c.f. [13])

54

generate a free group, there are no algebraic relations between each un and so we can easily extend σ

to be a ∗-homomorphism between C∗(F∞) and∏n∈NMk(n)(C) ⊂ l∞(R). It is clear that (π σ)(x) = x

for all x ∈ C∗(F∞) (where π : l∞(R) → RU is the canonical quotient map). This is trust by construction

and convergence in the ||.||2-norm. Finally, let’s consider pr :∏n∈NMk(n)(C) → Mk(r) the projection

map. Note that τ(x) = limn→∞ trn(pn(σ(x))) and thus, using Prop.3.3.19 (setting Φn = pn σ, since

each Φn is already u.c.p. ∗-homomorphism) we conclude that τ |A is an invariant mean.

(2) ⇒ (1) : Let’s suppose M has a weak expectation relative to some weakly dense subalgebra A. Let

then Φ′

: M ⊂ L(H) → L(H) be a u.c.p. map which fixes A. We proceed to identify A with πτ (A)

so we can apply Prop.3.3.19. Indeed, πτ (M) = (πτ (A))′′

= πτ (A)WOT

, πτ is a faithful representation

and our Φ is πτ Φ′. Hence, by Prop.3.3.19, there is a sequence Φn : A → Mk(n)(C) such that

||Φn(ab) − Φn(a)Φn(b)||2 → 0 and τ(a) = limn→∞ trn(Φn(a)) for all a, b ∈ A. Now let’s consider the

canonical quotient map π : l∞(R) → RU . Note that Φn(x)n∈N, for each x ∈ A, is an element of

l∞(R) since τ(x) < ∞. Now, if one consider Ψ : A → RU defined by Ψ(x) := π(Φn(x)), the fact that

||Φn(ab) − Φn(a)Φn(b)||2 → 0, tells us that Ψ is a ∗-homomorphism. Moreover, it preserves τ |A and it

is injective : Indeed, if Ψ(x) = 0 then Ψ(x∗x) = 0 (since Ψ is a ∗-homomorphism) and thus, τ(x∗x) = 0

(since it preserves τ |A). Then, one can conclude that x∗x = 0 and from this, it follows that x = 0.

Therefore, Ψ(A) ⊂ RU is such that Ψ(AWOT) ≈ M ≈ RU and we have proved that M embbeds into

RU .

Remark 3.3.21. Let un ∈ RU . Then, π−1(un) ∈ l∞(R), say π−1(un) = (xn) with each xn ∈ R. On the

other hand, R =⋃n∈NMn

WOTand thus there is a sequence (An) ∈

∏nMn such that (An) → (xn) in

the ||.||2-norm. Indeed, for convex sets, the weak closure coincide with the ||.||2 closure.20

3.4 Other equivalent formulations of Connes Embedding Conjec-

ture

A good way to emphatize the importance of the role of the Connes’ Embedding Conjecture, is to realize

the fact that there are many other conjectures which were proven to be equivalent to CEC. In this

subsection, we will briefly talk about some of these alternative formulations. Although these alternative

descriptions of CEC allow different points of view and the use of different techniques (that eventually

makes CEC a multi-disciplinary conjecture), the problem remains open.

• (Kirchberg Conjecture) One of the most unexpected equivalent formulations of CEC has to be the

Kirchberg Conjecture. E.Kirchberg proved in his seminal article (c.f. [30]) the equivalence between :

(A) C∗(F∞)⊗max C∗(F∞) ≈ C∗(F∞)⊗min C

∗(F∞).

(B) Any separable type II1 factor can be embedded into a suitable ultrapower RU .

20This follows from the fact that AWOT= ASOT if A is convex and from the fact that ||.||2-norm induced topology is equivalent

to the SOT topology (c.f. [2], Lemma 1.11.2)

55

This is simply mindblowing. Not only the apparent farness between the two assertions but also the

seemingly antagonistic nature of them : While (A) regards a property of a particular object (an universal

C∗-algebra), (B) regards a class of objects (type II1 factors). Unfortunately, it seems that an attempt for

a deep, detailed and rigorous approach to this, is slightly out of the scope of this project and so, we will

roughly sketch the ideas behind it.

Definition 3.4.1. (Effros-Marechal Topology) Let H be a Hilbert space and let vN(H) denote the set

of all von Neumann algebras acting on H. The Effros-Marechal topology on vN(H) is the weakest

topology that makes the map N 7→ ||ϕN || continuous on vN(H), for every normal functional ϕ on L(H).

In the special case of interest whereH is separable, one can simplify the definition of Effros-Marechal

topology. Let’s consider a sequence Mn ⊂ vN(H) and let’s define liminf Mn to be the set x ∈ L(H) :

∃(xn) ∈∏nMn : (xn → x)so

∗, where so∗ denotes the strong∗ operator topology. On the other hand,

let’s define limsupMn to be the set (x ∈ L(H) : ∃(xn) ∈∏nMn : supn ||xn|| < ∞, (xn → x)WOT )′′ .

Then, one can show that given Mn ⊂ vN(H), Mn → M (in the Effros-Marechal topology) if and only

if liminf Mn = limsupMn (c.f.[22]). Finally, let’s denote the set of injective factors acting on H by Finj .

Then, we have the following :

Theorem 3.4.2. The following are equivalent :

1. Finj is dense in vN(H), with respect to Effros-Marechal topology.

2. C∗(F∞)⊗max C∗(F∞) ≈ C∗(F∞)⊗min C

∗(F∞).

Proof: Check proof in [23].

This alternative description of (A), allow us to set the wanted equivalence :

Theorem 3.4.3. The following are equivalent :

1. Finj is dense in vN(H) with respect to Effros-Marechal topology.

2. Conne’s Embedding Conjecture.

Proof: Check proof in [23].

Thus, as a corollary, one has that (A) is equivalent to (B). For more details, the interested reader should

check [22],[23]. For the sake of completeness, one should remark that if FAFD ⊂ vN(H) and FI ⊂

vN(H) denote respectively the set of AFD factors acting on H and the set of type I factors acting on H,

then FAFDE−M

= vN(H) if and only if FIE−M

= vN(H) if and only if FinjE−M

= vN(H).

• (QWEP Conjecture) Let A be a C∗-algebra. We say that A has the weak expectation property

(WEP) if there exists a faithful representation π : A → L(H) such that π(A) is relatively weakly injective

in L(H), i.e. if there exists a contractive completely positive map Φ : L(H)→ A∗∗ such that Φ(π(a)) = a,

for all a ∈ A. This definition does not depend on the choice of the faithful representation of A. If A is

the quotient of a C∗-algebra with the WEP, we say that A has the quotient weak expectation property

(QWEP). What follows is a staggering conjecture regarding the structure of a C∗-algebra:

QWEP Conjecture : Every C∗-algebra has the QWEP.

56

Theorem 3.4.4. The following are equivalent :

1. Conne’s Embedding Conjecture.

2. QWEP Conjecture is true for separable von Neumann algebras.

Proof: Check proof in [23] or [43].

Hence, CEC can be seen as a particular case of the QWEP Conjecture.

• (Free Probability and Voiculescu’s Entropy) Is it true that L(F2) ≈ L(F3) ? This seemingly simple

question, remains to be answered. Historically, this problem motivated the birth of free probability theory.

Free probability theory is a line of research which parallels some aspects of classical probability, in a

non-commutative context.

A non-commutative probability space is a unital algebra A over C with a linear functional Φ : A → C

such that Φ(id) = 1. If A is a C∗-algebra and Φ is a state, the pair (A,Φ) is called a C∗-probability space.

In this context, if (A,Φ) is a non-commutative probability space, X ∈ A is called a random variable and

the distribution of X is the linear functional defined by µX(P ) = Φ(P (X)), where P ∈ C[X].

A certain family Aii∈I ⊂ A is said to be independent if for each i, j ∈ I one has that [Ai, Aj ] = 0 and

if Φ(a1...an) = Φ(a1)...Φ(an), for ak ∈ Ak. A family Aii∈I ⊂ A of unital subalgebras of A is said to be

free if Φ(a1...an) = 0, for any ak ∈ Ak with i1 6= ... 6= in and Φ(aij ) = 0, for any j ∈ 1, ..., n.

Now consider A = M to be a finite factor. Let’s fix some m, k ∈ N and ε, R > 0. Let’s consider X1, ..., Xn

to be random free variables on M . The set

(A1, ..., An) ∈ (Mk(C))nsa : ||Aj || ≤ R, |tr(Ai1 ...Aip)− τ(Xi1 ...Xip)| < ε,∀(i1, ..., ip) ∈ 1, ..., np,

where τ denotes the trace in M , is called a set of microstates. According to a choice of parametres,

is denoted by ΓR(X1, ..., Xn;m, k, ε). There is a very important quantity that can be attributed to a n-

tuple of free random variables (X1, ..., Xn). It called the free entropy and it is denoted by χ(X1, ..., Xn).

Its definition involves the Lebesgue measure of a set of microstates for some parametres R,m, k, ε

(c.f.[57],[58],[59]). Surprisingly enough, it happens that this concept is related to the CEC in the following

way :

Theorem 3.4.5. Let M be a type II1 factor. The following are equivalent :

1. Every finite subset X ⊂Msa has microstates.

2. Conne’s Embedding Conjecture holds.

3.5 Model Theory and Ultraproducts

The main goal of this small digression through a seemingly divergent path is to reveal how Model Theory

can be applied to study Operator Algebras, in a very fruitful case of cooperation between two apparent

disjoint areas of Mathematics. We will merely sketch some ideas behind the proofs of two previously

57

mentioned results : the ultrapower of type II1 factors is still a II1-factor and, assuming the Continuum

Hypothesis, the ultrapower RU is independent on the choice of the free ultrafilter U on N.

Recall that a first order language L is a set of relation symbols, function symbols and constant sym-

bols, as well as variables and quantifiers and connective symbols, such as ∀, ∃, (, ), ∧, ∨, ¬,⇒ and⇐.

We will also consider that there is an equality binary relation denoted by ≡. Then, we wish to give some

meaning to this set of symbols. To do so, we make a correspondence between L and an universe set,

say M . This correspondence, takes n-ary relations in L to n-ary relations R ⊂Mn, n-ary functions in L

to n-ary functions f : Mn → M and constants in L to constants x ∈ M . Let’s call this correspondece,

an interpretation and denote it by I. To the pair M = (M, I) we call a model.

Then, we develop the syntax, defining terms, atomic formulas and formulas in an inductive fashion. For

a rigorous exposition of this defintions, check [53]. Recall also that we say that a variable (in a formula)

that is not quantified is said to be free and that a formula without free variables is called a sentence.

Suppose now that we know how to attribute falsehood to a certain sentence ϕ of the language L, given

a model M.21 We will adopt the usual notation M |= ϕ. Finally, recall that the set of sentences of a

language L that are true in M is called the theory of M and denoted by Th(M). We say that two models

M1 and M2 are elementarily equivalent (M1 ≡ M2) if and only if Th(M1) = Th(M2). We are now able

to give a model-theoretic definition of ultraproduct, which is of our interest. But first, the set-theoretic

flavoured one :

Definition 3.5.1. Given a family of sets A = Aii∈I and an ultrafilter U on the index set I, consider the

following equivalence relation in∏i∈I Ai : f ∼ g if and only if i ∈ I : fi = gi ∈ U , where f = (fi)i∈I

and g = (gi)i∈I are elements of the cartesian product. We define ultraproduct to be∏i∈I Ai/ ∼ and

denoted by∏U Ai.

Definition 3.5.2. Let U be an ultrafilter on the index set I and let Mii∈I be a family of models of

L with universe sets Mi. The ultraproduct (model-theoretic construction) of Mii∈I by U is defined

to be the unique model of L with universe set∏UMi. Moreover, denoting the equivalence class of

f ∈∏i∈IMi by fU , the interpretation of a constant symbol c is given by fU such that f(i) = cMi , where

cMi is the interpretation given by Mi to c. The interpretation of a function symbol of arity n, F , is given

by (fU1 , ..., fUn ) 7→ fU , with f(i) = FMi(f1(i), ..., fn(i)). Finally, considering a n-ary relation symbol R :

R(fU1 , ..., fUn ) if and only if i ∈ I : RMi(f1(i), ..., fn(i)) ∈ U . We denote it by

∏U Mi.

It is routine work, using the properties of an ultrafilter, checking that the latter ultraproduct is well-

defined. For the sake of completeness, we mention the Fundamental Theorem of Ultraproducts :

Theorem 3.5.3.∏U Mi |= ϕ(fU1 , ..., f

Un ) if and only if i ∈ I : Mi |= ϕ(f1(i), ..., fn(i)) ∈ U

For a model for metric spaces, the interested reader should check [19] or [10]. For other examples,

let’s briefly describe the C∗-algebras and the tracial von Neumann algebras case. One can think of a

C∗-algebra A as a one-sorted structure U . Then, we have symbols Dn for each n ∈ N that should be

21If the reader is not familiar with the rigorous meaning of this, c.f. [53]

58

interpreted as the n-norm ball. The functions in the language will be the constant 0, for every λ ∈ C an

unary function interpreted as scalar multiplication, an unary symbol ∗ for involution and binary symbols

for sum and product. We denote this language by LC∗ . For the second case, one can treat tracial von

Neumann algebra as one-sorted structure with domains Dn, again interpreted as n-norm balls but here,

the metric arises from the l2-norm, induced by the trace. Additionally to the first case, we have the

constant 1 and two unary symbols trR and tri for the real and imaginary parts of the trace function. We

denote this language as LTr. Again, for the detailed exposition, one can check [19]. Our only purpose

is to mention that indeed such models exists and to stablish notation.

We now arrive to a very important concept. We say that a category C is axiomatizable if there is a

language L, a theory 22 T in L and a collection of conditions Σ such that C is equivalent to the category

of models of T , with morphisms given by maps that preserve Σ. Here, Σ should be seen as the set of

rules that the class of objects should obey. In particular, we say that we have axiomatized a class of

algebras C if there is a theory T such that :

(i) for any A ∈ C there is a model M(A) of T determined up to isomorphism.

(ii) for any model M of T there is A ∈ C such that M is isomorphic to M(A).

(iii) If A,B ∈ C, there is a bijection between Hom(A,B) and Hom(M(A),M(B)).

From this definition, one can check that if the category can be axiomatized, then it is closed under

ultraproducts. At this point, the reader can check for instance in [19] for an axiomatization of C∗-algebras,

tracial von Neumann algebras and for the class of type II1-factors. In particular, this means that RU is

still a type II1 factor. Though we ommit the details, all the work is computational and thus, building an

axiomatization for the class of type II1-factors, lead us to an important result that could also be proven

using more orthodox methods. The interested reader can check this at [16] and compare the more

traditional approach just to emphatize the power of using Model Theory tools.

We now finish this digression, sketching the ideas behind another application of Model Theory to

prove a result in Operator Algebras : assuming the Continuum Hypothesis, the ultrapower RU does not

depend on the choice of the ultrafilter. As it was explained in the first section of the present chapter, this

fact leads to an intuition about Connes’ Embedding Conjecture meaning.

To do so, we introduce some preliminary concepts. Consider a model M of a language L with universe

set M . Let A ⊂ M and let LA = ca : a ∈ A ∪ L, where ca is a constant for each a ∈ A. Define

ThA(M) to be the set of all sentences of LA that are true in M. Now, consider a set of LA-formulas in

free variables v1, ..., vn, say P. We say that P is an n-type if for every finite subset P0 ⊂ P, there are

elements b1, ..., bn ⊂ M such that M |= p0(b1, ..., bn), where p0 ∈ P0. We say that P is a complete

n-type if either ϕ ∈ P or ¬ϕ ∈ P for every formula ϕ of LA that is free on variables v1, ..., vn. More, if

P is a complete n-type, we say that M realizes P if there exists some a ∈ Mn such that M |= ϕ(a) for

all ϕ ∈ P. Finally, we introduce the very important concept of saturation : For any cardinal κ, a model M22Here, theory just means a set of sentences in L

59

is said to be κ-saturated if for all subsets A ⊂ M with |A| ≤ κ, M realizes all complete types over A. In

the particular case where κ = |M |, we say that M is saturated.

If we are dealing with structures over countable languages, we have already introduced all model-

theoretic concepts needed to prove that, assuming the Continuum Hypotheis, any two countable ul-

trapowers over ultrafilters on N of a structure with the cardinality of the continuum over a countable

language, are isomorphic. Indeed, the argument go as follows :any countable ultrapower of the struc-

ture has cardinality of the continuum, so any two such ultrapowers are equipotent. Then, it is a stan-

dard Model Theory result that a countable ultrapower of any structure over a countable language, is

ℵ1-saturated. Here, the Continuum Hypothesis plays a fundamental role and let us conclude that our

countable ultrapower is saturated, since our structure has the cardinality of the continuum. Finally, it is

another classic result that any two structures which are saturated and equipotent, must be isomorphic.

Here we just intent to sketch the nature of the arguments used, however the interested reader can check

[11]. However, when we defined a structure of interpretation for C∗-algebra (and for tracial von Neu-

mann algebras), our languages were not countable and because of that, one has to expand our toolbox

to stablish our isomorphism result for a non countable case.

To simplify the approach, let’s focus on our case of interest and let’s consider a language L and fix vari-

ables x = x1...xn and domains D = D1...Dn. For formulas ϕ and Ψ defined on D, set dTD

(ϕ(x),Ψ(x)) =

supsupx∈(D)n |ϕ(x) − Ψ(x)| : M |= T, where T is a theory in the language L. One can prove that

dTD

is in fact a psedo-metric.23 Let χ(T,D) be the density character24 of this pseudo-metric and define

χ(T ) =∑D χ(T,D). We say that L is separable if the density character of L is countable in respect to

all L-theories. We now state a fundamental result : LTr and LC∗ are separable languages.(c.f. [19])

Proposition 3.5.4. Let Mii∈N be models of some language L and let U be an ultrafilter on N. Then,∏U Mi is countably saturated.

Proposition 3.5.5. Let L be a separable language. If M1 and M2 are elementarily equivalent models of

L with the same uncountable density character, then they are isomorphic.

Lemma 3.5.6. Let S be a countably saturated structure over L and letA and B be separableL-structures,

with B being an elementary submodel of A. Then, any elementary embedding Ψ : B → S can be

extended to an elementary embedding Φ : A → S.

The interested reader, can check [11] and [19] for a proof and a deeper look on these results. We

are finally ready to sketch a proof of the following :

Theorem 3.5.7. Assuming the Continuum Hypothesis, if M is a model of density character χ ≤ 2ℵ0 ,

then all of its ultrapowers over non principal ultrafilters on N are isomorphic.

Proof: Let U and V be two non principal ultrafilters on N. As a consequence of Theorem 3.5.3, there

is an elementary embedding between M and MU and another elementary embedding between M and23Recall that a pseudo-metric d : X × X → R0 is a function such that d(x, x) = 0, d(x, y) = d(y, x) and d(x, z) ≤ d(x, y) +

d(y, z). Unlike a metric, we can have x 6= y such that d(x, y) = 0.24Recall that the density character of a topological space X is the least cardinal κ for which there is a dense subset A ⊂ X.

60

MV (in fact, just take the diagonal embedding). By Continuum Hypothesis and Prop.3.5.4, both MU and

MV are saturated. Thus, by Lemma 3.5.6, MU and MV are elementarily equivalent. Indeed, let B = M,

A = MU and let Ψ be the embedding granted by Lemma 3.5.6, with S = MV . In this way, we have

an elementary embedding Φ : MU → MV . Repeating the argument changing A with S, we reach the

conclusion. Therefore, by Prop.3.5.5, we conclude what we wanted.

Corollary 3.5.8. Assuming the Continuum Hypothesis, the ultrapower RU does not depend on the

choice of the free ultrafilter.

Proof: It follows from the definitions of axiomatizability that M(RU ) = M(R)U . Thus, R has non

isomorphic ultrapowers if and only if the model M(R) has non isomorphic ultrapowers, which is false by

the previous theorem.

61

Chapter 4

Non-Commutative Galois Theory

Due to Connes, any subfactor N of the hyperfinite type II1 factor R is still hyperfinite. Thanks to Murray

and von Neumann, there is only one (up to ∗-isomorphism) hyperfinite type II1 factor. Hence, any infinite

dimensional subfactor N ⊂ R is isomorphic to R. Thus, to study the subfactors of R one can adopt a

different approach that emphatizes the study of how the subfactors are included in R. This leads to the

development of invariants that allow us to distinguish these subfactors just by studying their inclusions

on R. The aim of this chapter is to introduce some of these ideas that are object of study in the realm of

Non-Commutative Galois Theory.

4.1 Motivation and Preliminaries

Recall that in classical algebra it is usual to denote an inclusion of fields K ⊂ L as L : K and to call it a

field extension. It is clear that L is a K-module and we define the index of the extension to be dimK(L).

In this context, given α ∈ Aut(L) such that α(k) = k for all k ∈ K, we say that α is a K-automorphism.

It is immediate to check that the set of K-automorphisms is in fact a group endowed with composition.

When the index of the extension if finite, this is the so called Galois Group (of a given extension L : K)

and it is usually denoted as Gal(L : K). Recall that there is a biunivocal correspondence between the

subgroups of Gal(L : K) and the subfields M such that K ⊂ M ⊂ L, the so called Galois correspon-

dence. All these constructions are object of study in Classic Galois Theory, an area in Algebra that is

indeed a magnificent manifestation of creativity.

In this part of this thesis, we merely scratch the surface of Non-Commutative Galois Theory, that is

the study of analogous concepts and ideas presented in a non commutative landscape. Following the

essence of this thesis, we aim to be able to compute some Galois groups and some Galois correspon-

dences concerning the hyperfinite type II1-factor R. Meanwhile, we briefly roam within the realm of

more sophisticated invariants, such as the Jones index.

62

4.1.1 Actions

Definition 4.1.1. LetM be a von Neumann algebra. We say that ϕ ∈ Aut(M) is an inner automorphism

if there is some unitary u such that ϕ(x) = uxu∗. Otherwise, we say that ϕ is outer.

Henceforth, α : G → Aut(M), where Aut(M) is the set of ∗-automorphisms of a von Neumann

algebraM and G is a topological group, is called an action if the map αx : G→M such that g 7→ αg(x)

is weakly continuous for every x ∈M and if α is a homomorphism.

Definition 4.1.2. Let ϕ ∈ Aut(M), withM a von Neumann algebra. We say that ϕ is free if whenever

xy = ϕ(y)x for every y ∈ M, then either x = 0 or ϕ = 1. Then, we say that an action α : G → Aut(M)

is free if αgg∈G\1 is a family of free automorphisms.

Definition 4.1.3. Given a group G and a von Neumann algebraM, let α : G → Aut(M) be an action

such that αgg∈G\1 is a family of inner/outer automorphisms. Then, we say that α is an inner/outer

action.

Definition 4.1.4. Given a von Neumann algebra M and an action α : G → Aut(M) we define the

algebra fixed by α to be the set x ∈M : αg(x) = x, ∀g ∈ G and we denote it byMG.

Proposition 4.1.5. LetM be a factor. Then, ϕ ∈ Aut(M) is an outer automorphism if and only if it is a

free automorphism.

Proof: First, let’s suppose that ϕ 6= 1 is a free automorphism and we assume, by absurd, that ϕ is

inner. Then, there is an unitary u such that for all x ∈ M we have that ϕ(x) = uxu∗. However, ϕ is free

and since ϕ(x)u = ux and ϕ 6= 1, we reach a contradiction because we would have that u = 0 and,

simultaneously, unitary. Conversely let ϕ ∈ Aut(M) be outer. Hence, ϕ 6= 1. Let x ∈ M be such that

for all y ∈ M we have that ϕ(y)x = xy. We need to show that x = 0. Notice that if x is unitary, we

have that x∗ϕ(y) = yx∗. Thus, since in a C∗-algebra every element is a linear combination of unitaries,

we still have that yx∗ = x∗ϕ(y) even if x is not unitary. From this, it is immediate to check that we have

that yx = xϕ(y)−1 and that ϕ(y)−1x∗ = x∗y. As a consequence, (x∗x)y = x∗(ϕ(y)x) = y(x∗x) and

(xx∗)y = x(ϕ(y)−1x∗) = y(xx∗) and we conclude that xx∗ and x∗x are elements of the centre ofM. By

hypothesis, M is a factor and thus, xx∗, x∗x ∈ λ1 : λ ∈ C. Therefore we can assume, after possible

normalization, that if xx∗ 6= 0 and x∗x 6= 0, x is unitary. But this is impossible, since ϕ is assumed to be

outer. Thus, x = 0.

Corollary 4.1.6. LetM be a factor and let α : G→ Aut(M) be an outer action. Then, α is a free action.

Proposition 4.1.7. LetM be a type II1-factor and let G be a finite group. Let α : G → Aut(M) be an

outer action and N :=MG. Then, N ′ ∩M = C1.

Corollary 4.1.8. Let M be a type II1-factor and let G be a finite group. Let α : G → Aut(M) be an

outer action. Then,MG is a subfactor ofM.

Proof: On one hand, it is easy to verify thatMG is a von Neumann algebra. Moreover, it follows from

Prop. 4.1.7 that it is a factor and clearly,MG ⊂M and they have the same identity.

63

A very important fact for the following sections is the existence of an outer action on the hyperfinite

type II1-factor, R. The proof of such fact it’s not trivial and it would require some technical results.

Thus, we will just sketch the construction of such action. Before we proceed, just recall the construction

of R in the latter sections and notice that we can view R as being the bicommutant of⊗

n∈NMn(C).

Moreover, recall that we say that a family of matrices eij ⊂ Mn(C) such that e∗ij = eji, eijekl = δkleil

and∑ni=1 eii = 1 are called unity matrices. It is immediate to see that Mn(C) is a vector space of

dimension n2 precisely with a canonical basis provided by unity matrices. To simplify notation in further

proof, we will write eji instead of eij .

Definition 4.1.9. Let G be a finite group with |G| = n. Define α : G→ Aut(Mn(C)) setting αg(eji ) := egjgi .

This map is called a left translation.1

Lemma 4.1.10. Let G be a finite group with |G| = n. Then, the left translation is an action.

Proof: First, we verify that each αg is in fact an automorphism of Mn(C). On one hand, αg(ejielk) =

αg(δliejk) = δlie

gjgk = δgl,gie

gjgk = egjgie

glgk = αg(e

ji )αg(e

lk). On the other hand, notice that αg((e

ji )∗) =

αg(eij) = egigj = (egjgi )

∗ = (αg(eji ))∗. Thus, αg ∈ Aut(Mn(C)). Finally, we check that α is a homomor-

phism. Indeed, α1(eji ) = eji and αg(αh(eji )) = αg(ehjhi ) = eghjghi = αgh(eji ).

Proposition 4.1.11. Let G be a finite group and let R be the hyperfinite type II1-factor. Then, there

exists an outer action of G on R. Moreover, such action α is unique.

Proof: Define Ψ : G → Aut(R) to be Ψ :=⊗

n∈N α. Even though this is a very natural expression, it is

not easy to check that Ψ is an outer action. The interested reader should check [5].

4.1.2 Conjugations

Definition 4.1.12. Let Ni and Mi, with i ∈ 1, 2, be von Neumann algebras. If we have inclusions

Ni ⊂ Mi such that there is an isomorphism ϕ : M1 → M2 with the property that ϕ(N1) = ϕ(N2), we

say that they are conjugated inclusions and we usually denote it by (N1 ⊂M1) ∼ (N2 ⊂M2).

Remark 4.1.13. The conjugation of inclusions is in fact an equivalence relation. Indeed, to verify re-

flexivity just choose ϕ = 1. If one has that (N1 ⊂ M1) ∼ (N2 ⊂ M2) by ϕ, we choose ϕ−1 to check

symmetry. Finally, if (N1 ⊂ M1) ∼ (N2 ⊂ M2) by ϕ1 and (N2 ⊂ M2) ∼ (N3 ⊂ M3) by ϕ2, we choose

ϕ2 ϕ1 to check transitivity.

Remark 4.1.14. Let α : G → Aut(R) be an outer action and let H ⊂ G be a normal subgroup. One

can define an action β : G/H → Aut(RH) setting β(gH)(x) := αg(x). Indeed, β is well-defined : let

gH = g′H which means that g−1g′ = h ∈ H. Now, notice that αg(x) = αghh−1(x) = αgg−1g′h(x) =

αg′(αh−1(x)) = αg′(x), since x ∈ RH . Moreover, αh(αg(x)) = αhg(x) = αgh′(x) = αg(x), since x ∈ RH

and gH = Hg. Thus, αg(x) ∈ RH . Furthermore, since α is an action, it is clear that β is also an action.

1Define the action ϕ : G→ Aut(G) by setting ϕg(h) := gh. Since G is finite, there is a bijection between G and 1, ..., n andwe can see ϕg as a permutation of indexes. That’s precisely the meaning of gi and gj in the definition of αg , since i, j ∈ 1, ..., n.

64

Proposition 4.1.15. Let α : G→ Aut(R) be an outer action, with G a finite group and H ⊂ G a normal

subgroup. Then, we have that (RG ⊂ RH) ∼ ((RH)G/H ⊂ RH) ∼ (RG/H ⊂ R).

Proof: We just prove the first equivalence and one could make an analogous argument for the second

inclusion. Let ξ : RH → RH be such that ξ(x) := βgH(x) for some g ∈ G \ H.2 It is enough to show

that ξ(RG) = (RH)G/H . On one hand, let y ∈ RG and since RG ⊂ RH , we have that ξ(y) ∈ RH . We

will verify that ξ(y) ∈ (R)G/H , i.e. ξ(y) = βkH(ξ(y)) for all k ∈ G \H. By definition of ξ this is equivalent

to stablish that βgH(y) = βkH(βgH(y)) for all k ∈ G \H. But βkH(βgH(y)) = αkg(y) and since y ∈ RG,

αkg(y) = y and thus, the condition we want to verify is equivalent to y = y and therefore, a tautology.

Conversely, let z ∈ (RH)G/H , i.e. z ∈ RH is such that αk(z) = z for every k ∈ G \ H. In particular,

let k = g and we conclude that z ∈ ξ(z). Moreover, since z ∈ RH , we have that αk(z) = z for every

k ∈ G \H and thus, z ∈ RG. Hence, z ∈ ξ(RG).

Remark 4.1.16. Recall that given a group G and a normal subgroup H ⊂ G, we define the normalizer

N(G,H) = x ∈ G : xH = Hx. It is easy to check that H ⊂ N(G,H) is a normal subgroup and hence,

by the previous result, we have that

(RN(G,H) ⊂ RH) ∼ ((RH)N(G,H)/H ⊂ RH) ∼ (RN(G,H)/H ⊂ R),

using some maps ϕ1 and ϕ2 respectively. On the other hand, since N(G,H) ⊂ G we have that

(RN(G,H))G/N(G,H) ⊂ (RH)N(G,H)/H and that RG/N(G,H) ⊂ RN(G,H)/H . Appealing to similar argu-

ments presented in the previous result, one can check that ϕ1(RG) = (RN(G,H))G/N(G,H) and that

ϕ2((RN(G,H))G/N(G,H)) = RG/N(G,H). Thus, we can conclude that

(RG ⊂ RH) ∼ ((RN(G,H))G/N(G,H) ⊂ (RN(G,H))H/N(G,H)) ∼ (RG/N(G,H) ⊂ RH/N(G,H)).

4.2 Galois Theory and the hyperfinite type II1-factor

4.2.1 Galois Groups

Definition 4.2.1. Let N ⊂M be von Neumann algebras. We define the Galois group of such a pair by

Gal(M,N ) = α ∈ Aut(M) : α|N = 1N .

Proposition 4.2.2. Given a pair N ⊂M, Gal(M,N ) is a well-defined group.

Proof: The proof is trivial.

Proposition 4.2.3. LetM be a type II1-factor and N ⊂M a subfactor. Then :

1. Gal(M,N ) is invariant by conjugation by elements of Aut(M).

2. The action of Gal(M,N ) onM is free3 if and only if N ′ ∩M = C1.

2We use the notation of Remark 4.1.14.3We consider the natural action α : Gal(M,N )→ Aut(M) such that αx := x.

65

Proof: To prove (1.), notice that Gal(M, ϕ(N ))ϕ = ϕGal(M,N ). Indeed, if ξ ∈ Gal(M, ϕ(N ))ϕ, we

have that ξ = β ϕ, for some β ∈ Aut(M) such that β|ϕ(N ) = 1N . We will verify that there is some

γ ∈ Aut(M) such that γ|N = 1N and such that ξ = ϕ γ. In fact, chose γ = ϕ−1 βϕ. It is clear that

γ ∈ Aut(M) and that γ|N = 1N , since β(ϕ(N )) = ϕ(N ). The reverse inclusion is absolutely analogous

and we can conclude that Gal(M, ϕ(N ))ϕ = ϕGal(M,N ) and from this it follows, as wanted, that

Gal(M, ϕ(N )) = ϕGal(M,N )ϕ−1.

To prove (2.), recall that from Prop.4.1.5 we have that ϕ ∈ Aut(M) is outer if and only if ϕ is free.

Thus, ϕ is free if and only if Inn(M) ∩ Gal(M,N ) = 1.4 We will prove that Gal(M,N ) ∩ Inn(M) ≈

U(N ′ ∩ M)/T. Indeed, let’s suppose that we already know this. Then, if N ′ ∩ M = C1 we have

that Gal(M,N ) ∩ Inn(N ) = 1 and we conclude that the action is free. Conversely, if Gal(M,N ) ∩

Inn(M) = 1, we have that U(N ′ ∩ M)/T = 1, which means that N ′ ∩ M = C1. Therefore,

to prove (2.), we just need to stablish that Gal(M,N ) ∩ Inn(M) ≈ U(N ′ ∩ M)/T. To do so, let

F : U(N ′ ∩M) → Gal(M,N ) ∩ Inn(M) be such that u 7→ Adu, where Adu ∈ Aut(M) is such that

Adu(x) := uxu∗. By definition, F (u) ∈ Inn(M). Moreover, let x ∈ N and since u ∈ U(N ′), one has that

uxu∗ = xuu∗ = x. Thus, Adu|N = 1N and F is well-defined. Furthermore, F is a homomorphism and

it is a surjective map. Indeed, let ξ ∈ Gal(M,N ) ∩ Inn(M). Hence, there is a unitary u ∈ U(M) such

that ξ(x) = uxu∗ and ξ|N = 1N . If y ∈ N , we have that uyu∗ = y and hence, uy = yu, which means that

u ∈ U(N ′ ∩M). Thus, F is surjective. Finally, it is clear that ker(F ) = T and appealing to Isomorphism

Theorem, we conclude what we wanted.

Intermezzo : On Jones’ Index and Towers of Algebras

We now give a look at a ring-theoretic construction of towers of algebras and we introduce an Algebra

flavoured definiton of index. We briefly sketch some of the ideas involved in computing the Jone’s Index

of the hyperfinite type II1-factor R. The reader can skip most of the content of this intermezzo.

Recall that a semi-simple algebra is an Artinian algebra over a field which has a trivial Jacobson radical.5

If A is a semi-simple and finite-dimensional algebra, there is a finite set Eini=1 of idempotent elements

in the center of A such that A ≈ AE1 ⊕ ... ⊕AEn. Whenever every component AEi is isomorphic to a

matrix algebra (over the same ground field K as A), we say that A is a multi-matrix algebra. It is a very

well known result that multi-matrix algebras are precisely the finite-dimensional C∗-algebras.

Let’s consider 1 ∈ N ⊂M, arbitrary algebras over some field K. The so called fundamental construc-

tion associates to the pair N ⊂ M, another pair M ⊂ L, where L := EndN (M) - viewed as a right

N -module - with every x ∈M being identified with the left multiplication operator. The nested sequence

M0 = N ⊂M1 =M⊂ ... ⊂Mk ⊂Mk+1 ⊂ ...,

where each Mk ⊂ Mk+1 is obtained from Mk−1 ⊂ Mk using the fundamental construction, is said

to be the tower induced by the pair N ⊂ M. As usual, rank(Mk|M0) denotes the smallest possible4Inn(M) denote the group of inner automorphisms ofM.5An algebra is Artinian if it satisfies the descending chain conditions on ideals. The Jacobson radical of an algebra is the ideal

of all elements that annihilate every simple ideal, i.e. every ideal without non-zero proper sub-modules.

66

number of generators ofMk as a rightM0-module.

Definition 4.2.4. Given a pair N ⊂M, we define the index of N inM to be the number given by

[M : N ] := limk→∞

sup[rank(Mk|M0)]1k ,

where eachMk comes from the tower induced by the pairM⊂ N .

Remark 4.2.5. IfM and N are isomorphic to matrix algebras, one has that [M : N ] = dimKMdimKN .

Now we introduce the concept of inclusion matrices. Let M and N be multi-matrix algebras, say

M =⊕m

i=1Mpi and N =⊕n

j=1N qj . For each pair (i, j) - with 1 ≤ i ≤ m and 1 ≤ j ≤ n - define

Mij := piqjMpiqj and Nij := piqjNpiqj . Define ΛMN := (λij) to be the m × n matrix such that

λij = 0 whenever piqj = 0 and that λij = [Mij : Nij ]12 , whenever piqj 6= 0. This matrix codifies useful

information. For instance, if N ⊂M is a pair of multi-matrix algebras with inclusion matrix ΛMN , then one

can show that ||ΛMN ||2 = [M : N ].

Proposition 4.2.6. Let N ⊂ M be a pair of K-algebras, with N finite-dimensional. Let tr :M→ K be

a faithful trace with a faithful restriction to N . Then, there is an unique K-linear map E : M→ N such

that :

1. tr(E(x)) = tr(x),∀x ∈M.

2. E(y) = y,∀y ∈ N .

3. E(xy) = E(x)y,∀x ∈M, y ∈ N .

4. E(xy) = 0 for all y, then x = 0.

Indeed, let’s considerM together with a non-degenerate symmetric K-bilinear form (x, y) 7→ tr(x, y).

Since tr and tr|N are faithful, one has that M = N ⊕ N⊥ and we can define E : M → N to be the

orthogonal projection ofM onto N . All the properties of the previous result are verified and we then say

that E is a faithful conditional expectation.

Now let N ⊂M be a pair of multi-matrix algebras and suppose that tr is a faithful trace onM which

restrition to N is still faithful. We already know that E is well-defined. Now let β ∈ K. We say that tr is a

Markov trace (of modulus β) if there is a trace Tr : EndN (M)→ K such that Tr(λ(x)) = tr(x) and that

βTr(λ(x)E) = tr(x), for all x ∈M, where λ :M → L is the usual left multiplication operator.

Now let’s consider again a pair of multi-matrix algebras M0 ⊂ M1 and let (Mk)k∈N0 be the induced

tower. Let’s also suppose that tr1 is a Markov trace of modulus β on the pair M0 ⊂ M1. Let tr2

be the extension Tr (to M2 = EndN (M)) and denote by E1 : M1 → M0 and by E2 : M2 → M1

the associated conditional expectations. One can extend a Markov trace through all Mi of the tower.

Moreover, we have the following :

Lemma 4.2.7. Let M0 ⊂ M1 be a pair of multi-matrix algebras on which exists a Markov trace tr of

modulus β. For each k ≥ 1 we have that :

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1. Mk is generated byM1 and E1, ..., Ek−1.

2. The idempotents E1, ..., Ek satisfy the following relations :

(a) βEiEjEi = Ei, if |i− j| = 1.

(b) EiEj = EjEi, if |i− j| ≥ 2.

Proof: The interested reader can check the proof in [21], Prop.2.7.5.

Lemma 4.2.8. Let β ∈ R∗ and Pnn∈N be an infinite sequence of orthogonal projections on some

Hilbert space such that βPiPjPi = Pi if |i − j| = 1 and that PiPj = PjPi if |i − j| ≥ 2. Then, if P1 6= 0,

one has that either β ≥ 4 or β = 4 cos2(πq ) for some integer q ≥ 3.

Proof: The interested reader can check the proof in [21], Thrm II.16

After this preamble, we have enough motivation to set our framework on von Neumann algebras. Let’s

consider a pairM ⊂ N of finite factors. SinceM is a finite factor, there is a faithful tracial state tr and

we can define on M an inner product, by the usual way, setting 〈a, b〉 := tr(b∗a). Regarding the norm

||x|| := (tr(x∗x))12 , we define [M : N ] to be dimN (L2(M)).

Definition 4.2.9. Given a finite factorM and N ⊂M a subfactor, we define the Jones’ Index as

[M : N ] := dimN (L2(M)).6

If [M : N ] < ∞, the pair N ⊂ M can generate a tower of factors by the fundamental construction

defined as follows:

M0 = N ⊂M1 =M⊂ ... ⊂Mk ⊂Mk+1 ⊂ ...,

where Mk is the von Neumann algebra acting in L2(Mk), generated by Mk and ek, where ek :

L2(Mk) → L2(Mk−1) is a conditional expectation. If M is a type II1-factor, eN : M → N is the

restriction to M of the orthogonal projection of L2(M) onto the closure of N ⊂ L2(M). Moreover, in

this case, defining iteratively the tower we get that eachMk is a type II1-factor.

Proposition 4.2.10. LetN ⊂M be a pair of finite factors such that [M : N ] <∞. Then, as C∗-algebras,

we have that 〈M, eN 〉 ≈ EndN (M).

Proof: The interested reader can check the proof in [21], cor.3.6.5.

Remark 4.2.11. Note that the previous result tells us that the way we defined our fundamental construc-

tion in this setting, agrees with the more general ring-theoretic definition.

Theorem 4.2.12. Let N be a subfactor of a type II1-factorM. Then:

1. Either [M : N ] = 4 cos2(πq ) for some integer q ≥ 3, or [M,N ] ≥ 4.

6Here, given aM-module H, with dimM(H) we mean the coupling constant.

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2. If M is the hyperfinite type II1-factor R, there exist subfactors which index take any value in the

set 4 cos2(πq ), q ∈ N, q ≥ 3 ∪ [4,∞[.

Proof: We will merely present the idea of the proof. Let’s start with (1.). As we previously described,

the fundamental construction yields a type-II1 factor 〈M, eN 〉 when one consider the unique faithful

conditional expectation fromM onto N , which can be viewed as a projection. Moreover, one can show

that the normalized trace of 〈M, eN 〉 has the Markov property with modulus [M,N ]. Now, in a similar

way with what we did before with multi-matrix algebras, one has a tower construction and an increasing

sequence of type-II1 factors M0 = N ⊂ M1 = M ⊂ ... ⊂ Mk ⊂ Mk+1 ⊂ ... and a sequence of

projections Pnn∈N satisfying [M : N ]PiPjPi = Pi whenever |i− j| = 1 and PiPj = PjPi, if |i− j| ≥ 2.

Hence, by Lemma 4.2.8 we can conclude (1.).

To prove (2.), we define a chain of C∗-algebras, say (Aβ,k)k∈N - for any β ∈ R such that either β ≥ 4

or β = 4 cos2(πq ) for some integer q ≥ 3 - with each Aβ,k being generated by its identity and projections

P1, ..., Pk−1 satisfying [M,N ]PiPjPi = Pi if |i− j| = 1 and PiPj = PjPi if |i− j| ≥ 2. Furthermore, each

Aβ,k has a faithful trace tr that satisfies βtr(wPj) = tr(w) for any w ∈ Aβ,k and 1 ≤ j ≤ k. To see this

in the case β ≥ 4, the reader should check section 2.8 of [21] and to see the case when β = 4 cos2(πq ),

check section 2.9 of [21]. Now, since tr is faithful, one can apply the GNS construction through πtr and

then define R1 : πtr = (⋃k∈NAβ,k)

′′. It is not hard to see that, by definition, R1 is a finite and hyperfinite

von Neumann algebra. On the other hand, one can show that R1 is in fact a factor and thus, by unicity,

R ≈ R1. Finally, using the fact that generators of Aβ,k verify the relations of Lemma 4.2.7(2) and the

properties of tr, setting Rβ := 1, P2, P3, ...′′

one has that [R : Rβ ] = β, as the interested read can

check in [21],Lemma 3.4.5. Hence, we just proved that for any value β in 4 cos2(πq ),∈ N, q ≥ 3∪ [4,∞[,

there is a subfactor Rβ ⊂ R1 ≈ R such that [R : Rβ ] = β.

Proposition 4.2.13. Let M be a type II1-factor, α : G → Aut(M) an outer action and N ⊂ M a

subfactor. Then :

1. If [M : N ] <∞ and N ′ ∩M = C1, then |Gal(M,N )| <∞.

2. If P ⊂ N ⊂M is an inclusion of factors, then [N : P][M : N ] = [M : P].

3. If |G| < ∞, then [M : MG] = |G| = [M oα G : M] and if H is a subgroup of G, then [M oα G :

Moα H] = |G|/|H|.

4. If N ⊂M is an inclusion of type II1 factors and [M : N ] = 1, thenM = N .

∗∗ End of intermezzo ∗∗

Proposition 4.2.14. LetM be a type II1-factor and N ⊂ M a subfactor. If [M : N ] = |Gal(M,N )| <

∞, we have that N =MGal(M,N ).

Proof: First, we prove that MGal(M,N ) is a subfactor of M. As we proved in Prop.4.2.3, one has that

Gal(M,N ) ∩ Inn(M) ≈ U(N ′ ∩M)/T. If, by hypothesis, |Gal(M,N )| < ∞, we have that |U(N ′ ∩

M)/T| <∞ which is only possible if N ′ ∩M = C1. On the other hand, since N ⊂MGal(M,N ), we have

69

that (MGal(M,N ))′ ⊂ N ′ and we can conclude that (MGal(M,N ))′ ∩M ⊂ C1. Thus, (MGal(M,N ))′ ∩

MGal(M,N ) ⊂ C1. Conversely, C1 ⊂ MGal(M,N ) and we can conclude that MGal(M,N ) ⊂ M is a

subfactor.

Now, notice that since N ⊂M is a subfactor, we have that in fact, N ⊂MGal(M,N ) ⊂M are inclusions

of subfactors. By Prop.4.2.13, we have that [M : MGal(M,N )][MGal(M,N ) : N ] = [M : N ]. Again, by

Prop.4.2.13 and by hypothesis, we know that [M : MGal(M,N )] = [M : N ]. Therefore, we have that

[MGal(M,N ) : N ] = 1 and applying Prop.4.2.13, we conclude that N =MGal(M,N ).

• LetM be a type II1-factor and N ⊂M a subfactor such that [M : N ] <∞ and that N ′ ∩M = C1.

Let’s define G1 := Gal(M,N ), L1 := MG1 and, proceeding by induction, Gi+1 := Gal(Li,N ) and

Li+1 := LGi+1

i . From the definition, for any i we have thatN ⊂ Li. Moreover, since [M : N ] <∞ we can

use Prop.4.2.13 and conclude that there is some k such that N = Lk and thus, Gk+1 = Gal(M,N ) =

1. Thus, there is some k0 such thatGk0 = 1 and thatGk0−1 6= 1. Furthermore, sinceN ′∩M = C1,

every Gk is a finite group. To the sequence of finite groups (G1, ..., Gk0) we call Galois series. Again, by

Prop.4.2.13, it should be clear that [M : N ] = [M : L1]...[Lk : N ].

Proposition 4.2.15. LetM be a type II1-factor and let N ⊂M be a subfactor such that N ′ ∩M = C1.

Then, |Gal(M,N )| ≤ [M : N ].

Proof: Note that [M : L1] = [M :MGal(M,N )] = |Gal(M,N )|, using Prop.4.2.13. Since [Li : Li+1] ≥ 1,

we conclude the inequality.

4.2.2 Computing some Galois Groups

Theorem 4.2.16. LetM be a type II1-factor and let α be a free action of G, a finite group, onM. Then,

Gal(M,MG) ≈ G.

Proof: It is easy to see that (MG)′ ∩ M = C1 and thus, by Prop.4.2.3 one has that Gal(M,MG)

acts freely on MG. Hence, using Prop.4.1.5 we know that it is an outer action. On the other hand,

one has that |Gal(M,MG)| < ∞ and since M is a type II1-factor, appealing to Prop.4.1.7, we con-

clude that (MGal(M,MG))′ ∩ M = C1. Thus, MG ⊂ MGal(M,MG) ⊂ M is an inclusion of fac-

tors. Now, using Prop.4.2.13, we have that [M : MG] ≥ [M : MGal(M,MG)] and because of that,

|Gal(M,MG)| ≥ |α(G)|. However, by definition we have that |α(G)| ≥ |Gal(M,MG)| and we con-

clude that Gal(M,MG) ⊂ α(G) are equipotent sets. Since by hypothesis G ≈ α(G) is finite, we have

necesarily that Gal(M,MG) ≈ G and we prove what we wanted.

Crash course in Representation Theory

• Let G be a finite group. A representation of G is a homomorphism ϕ : G→ GL(V ).7 A character of G

is a map χ : G→ C such that χ(g) = tr(ϕ(g)), for some representation ϕ of G.

If W ⊂ V is such that for every w ∈ W we have that ϕ(g)w, g ∈ G ⊂ W , we say that W is invariant. A

7Here, V is a vector field and GL(V ) is the group Aut(V ) under composition. Moreover, if dim(V ) = n, we have thatGL(V ) ≈ GL(n, V ).

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representation is said to be irreducible whenever the only invariant subspaces are V and 0.

• If ϕ : G → GL(V ) is a representation with dim(V ) = n, ϕ is said to be n-dimensional. Notice that

χ(1g) = tr(1) = dim(V ) and thus, if χ(1G) = 1, χ is 1-dimensional. Let’s denote the set of 1-dimensional

charaters of G by χ(G).

Lemma 4.2.17. Let ϕ : G → GL(V ) be an irreducible representation. Let Φ : V → V be such that

ϕ(g)(Φ(v)) = Φ(ϕ(g)(v)) for any g ∈ G and v ∈ V . Then, there is some λ ∈ C such that Φ = λ1

The previous result is known as Schur’s Lemma and has uncountable applications in the rich area

of Representation Theory. For instance, let G be abelian and let ρ : G → GL(V ) be an irreducible

representation. Now, define the action α : G → Aut(V ) setting α(g)(v) := ρ(g)(v). Since G is abelian,

it is clear that ρ(g)ρ(h) = ρ(h)ρ(g) for any g, h ∈ G. Then, by Schur’s Lemma, we have that ρg =

λ(g)1 with λ ∈ C∗. Thus, every subspace is invariant. Since ρ was taken to be irreducible, this mean

that necessarily dim(V )=1. Hence, if G is abelian, any irreducible representation is 1-dimensional.

Conversely, it should be clear that any 1-dimensional representation is irreducible.

Lemma 4.2.18. Let Γ denote the set of irreducible representations of G. Then, |G| =∑ϕ∈Γ dim(Vϕ)2.

Notice that joining the observations after Schur’ Lemma and the latter result, one concludes immedi-

ately that if G is abelian, the number of irreducible representations is equal to |G|.

Now, notice that χ(G) is in fact a group.8 Morevoer, thanks to the properties of traces, χ(G) is an

abelian group. Now recall that if G is a finite abelian group, there are generators g1, ..., gr such that

G = 〈g1〉 ⊕ ... ⊕ 〈gr〉. Then, it is not hard to check that one can define an isomorphism between G and

χ(G) by setting χ 7→ (χ(g1), ..., χ(gr). This remark leads to the following useful result :

Proposition 4.2.19. Let G be a finite group. Then, G/[G,G] ≈ χ(G).

∗∗ End of crash course ∗∗

Theorem 4.2.20. LetM be a type II1-factor and let α : G → Aut(M) be an outer action, with G finite.

Then, Gal(Moα G,M) ≈ G/[G,G].

Proof: We will prove that Gal(M oα G) ≈ χ(G). Then, using Prop.4.2.19, we can conclude the result.

Having this in mind, given any ϕ ∈ Gal(M oα G) we will prove that there is an unique map χ : G → C

such that χ(ghg−1) = χ(h), χ(1G) = 1 and ϕ(∑g∈Gmgug) =

∑g∈G χ(g)mgug. Notice that if χ is a

character, the second condition tells us that it is in fact a 1-dimensional character, as we remarked

before. In this case, the first condition tells us that χ is indeed a character and the third condition

establishes its unicity. Then, it is not hard to check that indeed this biunivocal correspondence is in fact

the desired isomorphism. It remains to prove that such χ exists.

First we prove that ϕ(ug)u∗g ∈ M′ ∩ (M oα G). Notice that if this is the case, using Lemma A.5.5 (and

identifying tα(y) with y ⊗ 1) and since α is outer, we have that M′ ∩ (M oα G) = C1MoαG. Thus,

ϕ(ug)u∗g = λ1 and hence ϕ(ug) = λ(g)ug and one define χ : G → C to be χ(g) := λ(g). To check that

8χ1χ2(g) := χ1(g)χ2(g) and χ−1(g) := χ(g)−1

71

indeed we have the desired equality, notice that (ϕ(ug)u∗g)a = a(ϕ(ug)u

∗g) for all a ∈ M is equivalent

to check that ϕ(ug)α∗g(a)u∗g = aϕ(ug)u

∗g since u∗ga = u∗gtα(a) = tα(α∗g(a))u∗g = α∗g(a)u∗g, where we used

Lemma A.5.4. But the latter condition is equivalent to ϕ(ug)α∗g(a) = aϕ(ug). Furthermore, since ϕ|M =

1M, the condition is equivalent to ϕ(ugα∗g(a)) = ϕ(aug) and multiplying by u∗g we get α∗g(a) = u∗gaug

which mean that for any h ∈ G the condition is equivalent to αh(a) = uhau∗h (taking h := g−1). But since

α is outer, this condition is verified and therefore, one can conclude that ϕ(ug)u∗g ∈M′ ∩ (Moα G) and

we can define χ as previously described. It remains to check the mentioned properties of χ.

By construction, it is immediate that ϕ(∑g∈Gmgug) =

∑g∈G χ(g)mgug. On the other hand, we have

that ϕ(a) = aχ(1G)u1G = χ(1G)ϕ(a) and thus, χ(1G) = 1. Finally, we check that χ(ghg−1) = χ(h) : Let

x =∑h∈Gmhuh. Then, ugxu∗g =

∑h∈G αg(mh)ughg−1 . Now, notice that

ϕ(ugxu∗g) =

∑h∈G

αg(mh)uguhu∗g =

∑h∈G

χ(ghg−1)αg(mh)ughg−1 .

On the other hand,

ϕ(ugxu∗g) = ϕ(ug)ϕ(x)ϕ(u∗g) =

∑h∈G

χ(h)αg(mh)ughg−1 ,

which means that χ(h) = χ(ghg−1) as we wanted.

Remark 4.2.21. Suppose that (A1 ⊂ B1) ∼ (A2 ⊂ B2) by the isomorphism ϕ : B1 → B2 such that

ϕ(A1) = A2. Then, Gal(B1,A1) ≈ Gal(B2,A2). To see this, let’s define Ψ : Gal(B1,A1) → Gal(B2,A2)

such that Ψ(ξ) := ϕξϕ−1. First, notice that if z ∈ B2, since ξ(ϕ−1(z)) ∈ A1, we have that ϕ(ξ(ϕ−1(z))) ∈

B2. Thus, Ψ(ξ) ∈ Gal(B2,A2). Moreover, it is clear that Ψ is a bijective homomorphism and thus, an

isomorphism.

Corollary 4.2.22. Let G be a finite group, R the hyperfinite type II1-factor and let α : G → Aut(R) be

an outer action. Let H ⊂ G be a normal subgroup. Then :

1. Gal(RH ,RG) ≈ G/H.

2. Gal(Roα G,Roα H) ≈ (G/H)/([G/H,G/H]).

Proof: To prove (1.) note that by Prop.4.1.15 we know that (RG ⊂ RH) ∼ (RG/H ⊂ R). Then, appealing

to the previous remark and to Theorem 4.2.16, we can conclude that Gal(RH ,RG) = Gal(R,RG/H) ≈

G/H. To prove (2.) we use Theorem 4.2.20 and we immediately obtain the result.

Remark 4.2.23. Let’s consider the inclusions (RS3 ⊂ R) and (RZ6 ⊂ R) and consider an outer action

α (c.f. Prop. 4.1.11). Notice that Gal(R,RS3) ≈ S3 and that Gal(R,RZ6) ≈ Z6, using Theorem

4.2.16. Since Z6 and S3 are not isomorphic, using Remark 4.2.21 we can conclude that these are not

conjugated inclusions. However, if we used Jones Index we would not be able to conclude this: indeed,

using Prop.4.2.13 one has that [R : RS3 ] = [R : RZ6 ] = 6.

Remark 4.2.24. Let’s check that the assumption of H being normal in G in Corollary 4.2.22 is necessary

and meanwhile, for an outer action α of S3 on R we compute that

Gal(Roα S3,Roα Z2) = 1.

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Let G denote Gal(Roα S3,Roα Z2) and G1 denote Gal(Roα S3,R). Clearly, G ⊂ G1 and by Corollary

4.2.22, since [S3, S3] ≈ Z3, one has that G1 ≈ Z2. Hence, either G = Z2 or G = 1. Appealing to

Corollary 4.1.8 and to Prop.A.5.6 one has that the following are inclusions of factors:

(Roα Z2) ⊂ (Roα S3)G ⊂ (Roα S3).

Furthermore, by Prop.4.2.13 we conclude that [(RoαS3) : (RoαZ2)] = 3 and also, again by Prop.4.2.13,

we have that [(Roα S3) : (Roα S3)G] = |G|. Thus, since |G| ∈ 1, 2, using Prop.4.2.13 again, we have

that [(Roα S3)G : (Roα Z2)] is either 3 or 32 . However, since (Roα S3)G is a type II1 factor (it contains

an infinite dimensional factor (R oα Z2) and it is contained in a type II1 factor, (R oα S3)), Theorem

4.2.12 implies that |G| = 1 and therefore, G = 1. On the other hand, if we use Corollary 4.2.22, we

would conclude that |G| = 2.

4.2.3 Galois Correspondence

In this subsection, we will finally approach Galois correspondences in the non-commutative framework.

If a group G acts onM, a type II1-factor through an outer action, we will be able to establish the relation

between subgrops of G and subfactors ofM.

Remark 4.2.25. Let M be a type II1-factor and G a countable group such that α : G → Aut(M) is

an outer action. Now, consider the map u : G → U(M) defined by αg(x)u(h) = αgh(x), for all x ∈ M

and g, h ∈ G. We have that g 7→ ug is an unitary representation. Moreover, using properties of the

trace τ , one can check that ugg∈G is a linearly independent family overM′. Furthermore, notice that

if N =MG, any element y ∈ N ′ can be written (uniquely) as y = x1 + xgug + ...+ xhuh, with xk ∈ M′.

Indeed, let X be the set of all such expressions. Notice that if x ∈ N , then xuh = x for all h ∈ G, just by

definition of N and uh. On one hand, given any y ∈ N one can verify that (xgug)y = y(xgug) : Observe

that xg(ugy) = xgy and that y(xgug) = y(ugxg) = (yug−1)(ugxg) = yxg and thus, (xgug)y = y(xgug) is

equivalent to xgy = yxg which is verified since xg ∈ M′. Thus, X ⊂ N ′. Conversely, it is not hard to

check that X is in fact a subalgebra and by Bicommutant Theorem it’s enough to prove that X ′ ⊂ N ,

which is immediate from the definitions.

Remark 4.2.26. Let M be a type II1-factor and let N ⊂ M be a von Neumann subalgebra. Then,

let’s consider ξ : M → N to be the projection of L2(M) onto L2(N ) restricted to M. The reader can

check that ξ verifies useful properties such as ξ(axb) = aξ(x)b, tr(aξ(x)) = tr(ax), ξ(x∗) = ξ(x)∗ and

ξ(x∗x) ≥ ξ(x)∗ξ(x), for all a, b ∈ N and x ∈M. In this context we say that any such map is a conditional

expectation onM conditioned by N .

Proposition 4.2.27. LetM be a type II1-factor, G a finite group and α : G→ Aut(M) an outer action.

Let N =MG and let P be a von Neumann subalgebra such thatM′ ⊂ P ⊂ N ′. Then, P is generated

by uhh∈H andM′ for a certain subgroup H ⊂ G.

Proof: Let H = k : uk ∈ P ⊂ G, clearly a subgroup of G. Let ∆ be the von Neumann algebra

generated by uhh∈H overM′. Clearly that ∆ ⊂ P. Conversely, let y ∈ P. Then, according to Remark

73

4.2.25, y = x1 +xgug + ...+xhuh with each xi ∈M′ and g ∈ G. Now, considering ξ to be the conditional

expectation on N ′ conditioned by P. Using the properties stated in Remark 4.2.24, one can easily

conclude that either ξ(ug) = 0 or ξ(ug) = ug, i.e. either ug ∈ P⊥ or ug ∈ P. Obviously, the first case is

not possible and hence the definition of H = k : uk ∈ P.

Lemma 4.2.28. Let N ⊂ M be type II1-factors such that M ∩ N ′ = 1. Then, each von Neumann

subalgebra between N and M is a subfactor of M and each von Neumann algebra between M′ and

N ′ is a subfactor of N ′.

Proof: The proof is fairly simple. Just notice that N ⊂ P ⊂ M implies that M′ ⊂ P ′ ⊂ N ′ and thus,

P ∩ P ⊂ N ′ ∩M = C1. The caseM′ ⊂ P ⊂ N ′ is completely analogous.

Theorem 4.2.29. Let G be a finite group, M be a type II1-factor and α : G → Aut(M) be an outer

action. Let N =MG ⊂M. Then :

1. There is a biunivocal correspondence between subgroups of G and subfactors between M′ and

N ′.

2. There is a biunivocal correspondence between subgroups of G and subfactors between M and

N .

Proof: We just need to prove (1.), since taking the commutant and using that (M′)′ = M and that

(N ′)′ = N , lead us immediately to (2.). By Prop.4.1.7 we have that M∩N ′ = C1. Then, by Lemma

4.2.28 we know that every von Neumann algebra P such that M′ ⊂ P ⊂ N ′ is a subfactor of N ′.

Therefore, by Prop.4.2.27, there is a subgroup H of G such that P is generated by M′ and uhh∈H .

Then, we have a surjective map between subgroups H ⊂ G and subfactors P between M′ and N ′.

Moreover, using the fact that uhh∈H is linearly independent overM′ (check Remark 4.2.25), one can

conclude that in fact this is an injective correspondence and thus, biunivocal.

Remark 4.2.30. In the above conditions, let K ⊂ H be subgroups of G. Then, if P and L are, respec-

tively, the associated subfactors between M′ and N ′, we have that P ⊂ L. If one considers that P

and L are, respectively, the associated subfactores between N and M, then we have that L ⊂ P. To

see this, notice that to each L such that N ⊂ L ⊂ M, we can associate L′ (such thatM′ ⊂ L′ ⊂ N ′)

and thus, a subgroup H ⊂ G by the latter result. This means that if N ⊂ L ⊂ P ⊂ M, we have that

M′ ⊂ P ′ ⊂ L′ ⊂ N ′ and we associate subgroups K ⊂ H respectively to P ′ and L′ (and thus, P and L)

and we conclude that the inclusion order is reversed.

4.2.4 Computing some correspondences

It should be remarked that, as stated in Prop.4.1.11, outer actions α : G→ Aut(R), with G a finite group

and R the hyperfinite type II1-factor, exist and are unique. In this sense, all the following correspon-

dences exist and are unique - for the same group G, obviously.

Theorem 4.2.31. Let G be a finite group and let α : G → Aut(R) be an outer action. Then, there is

a biunivocal correspondence between subgroups H ⊂ G and subfactors between RG and R that are

invariant under α|H.

74

Proof: Let H ⊂ G be a subgroup. Then, as seen in Prop.4.2.27 and Remark 4.2.25, we can associate H

with P such thatRG ⊂ P ⊂ R and such that P ′ is generated byR′ and H. We will check that P is indeed

invariant under α|H. Notice that in this way, we have defined an injective map of the set of subgroups

of G into the set of subfactors between RG and R that are invariant under the action of α restricted to a

certain subgroup. Obviously, this map is also surjective and thus, a biunivocal correspondence. Indeed,

from the fact that H belong to the set of generators of P ′ and from the fact that uhh∈H is a linearly

independent family, it follows that this map is injective. It remains to check that indeed, if y ∈ P, then

αh(y) = y for any h ∈ H. Notice that αh(y) = α1h(y) = α1(y)uh = yuh. Since each uh ∈ P ′ we easily

conclude what we wanted.

Theorem 4.2.32. Let G be a finite group and let α : G → Aut(R) be an outer action. Let P be a

subfactor between R and R oα G. Then, P is of the form R oα|H H, with H ⊂ G a subgroup defined

as g ∈ G : ug ∈ P. Thus, there is a biunivocal correspondence between subgroups H ⊂ G and

subfactors P such that R ⊂ P ⊂ Roα G.

Proof: Recall that by Prop.A.5.6 we have that Roα G is a type II1-factor. Hence, let R ⊂ P ⊂ Roα G

be a factor and consider ξ : R oα G → P to be the conditional expectation (conditioned by P). Note

that it’s enough to prove that ξ(ug) = 0 for each g ∈ G \ H. Indeed, if that is the case, and defining

H = g ∈ G : ug ∈ P, let’s see that P = R oα|H H : On one hand, let x ∈ P. Then, x =∑g∈Gmgug

and ξ(x) = x. If ξ(ug) = 0 whenever g ∈ G \H, we have that x =∑h∈H mhξ(uh). However, by definition

of H, we have that ξ(uh) = uh and thus, x =∑h∈H mhuh. Conversely, let x =

∑hmhuh. Again, by

definition of H, we have that

ξ(x) =∑h∈H

mhξ(uh) =∑h∈H

uhuh.

Thus, ξ(x) = x which means that x ∈ P.

It remains to prove that indeed we have that for any g ∈ G \H, ξ(ug) = 0. Let x ∈ R. Then,

xξ(ug) = ξ(xug) = ξ(ugu∗gxug) = ξ(ug)u

∗gxug.

By Lemma A.5.4, we conclude that ξ(ug)u∗g ∈ R′ ∩ (R oα G). Thus, appealing to Lemma A.5.5, there

is some scalar λ such that ξ(ug) = λug. Hence, if λ 6= 0, one has that ξ(ug) ∈ P and thus, g ∈ H.

Moreover, if g ∈ G \H, we have that ξ(ug) = 0 and we prove what we wanted.

Theorem 4.2.33. Let G be a finite group and let α : G → Aut(R) be an outer action. Let H ⊂ G be a

normal subgroup. Then :

1. There is a biunivocal correspondence between subgroups K ⊂ G/H and subfactors between RG

and RH .

2. There is a biunivocal correspondence between subgroups K ⊂ G/H and subfactors between

Roα H and Roα G.

Proof: First let’s prove (1.). If H ⊂ G is normal, G/H is a group. Thus, by Theorem 4.2.31, there is a

correspondence between subgroups of G/H and subfactors between RG/H and R. Now, to conclude

75

(1.), just recall that according to Prop.4.1.15 we have that (RG ⊂ RH) ∼ (RG/H ⊂ R).

To prove (2.), we notice that using analogous arguments (as we did during the proof of Prop.4.1.15) one

can prove that (R oα H ⊂ R oα G) ∼ (R ⊂ R oα (G/H)). Then, appealing to Theorem 4.2.32, we

conclude the proof.

76

Appendix A

Preliminaries

A.1 Facts about von Neumann algebras

We begin by recalling three different topologies in L(X), where X is a Banach space.

(i) Given T0 ∈ L(X), its neighborhood basis in the norm topology is generated by sets of the form

T ∈ L(X) : ||T − T0|| < ε, with ε > 0. In this sense, Tα → T if ||Tα − T || → 0, where the limit is

taken in the operator norm.

(ii) The SOT topology (meaning strong operator topology ) is generated by the family of semi-norms

||.||xx∈X such that ||T ||x = ||T (x)||, with x ∈ X, T ∈ L(X) and the norm of the space X. In

this sense, Tα → T (SOT) if ||Tαx − Tx|| → 0, for all x ∈ X and we say that the net Tα strongly

converges to T .

(iii) The WOT topology (meaning weak operator topology ) is generated by the family of semi-norms

||.||x,ϕx∈X,ϕ∈X∗ , such that ||T ||x,ϕ = |ϕ(Tx)|. In this sense, Tα → T (WOT) if |ϕ(Tαx− Tx)| → 0,

for all x ∈ X and ϕ ∈ X∗ and we say that the net Tα weakly converges to T .

It is immediate to prove that convergence in the norm topology implies convergence in the SOT topology

and, on the other hand, convergence in the SOT topology implies convergence in the WOT topology.

Thus, if Y is a subset of L(X), we have that Y ⊂ Y SOT ⊂ YWOT.

Remark A.1.1. Note that if H is a Hilbert space, by Riesz’s Representation Theorem we have that

Tα → T in the WOT topology if and only if 〈TSx, y〉 = limα〈TαSx, y〉, for all x, y ∈ H and S ∈ L(H).

Before introducing von Neumann algebras, we just recall a very natural topology in X∗, given a

normed space X. Consider the map ϕx : X∗ → R such that ϕx(f) = f(x). The so called weak topology

in X∗ is generated by the subbasis ϕ−1x (U), where U is an open subset of R. It is easy to see that this

is the topology that X∗ inheritates from RX with the product topology, as a subspace. It should also be

clear that f ∈ X∗ : |(f−g)(xi)| < εi for a finite set of indexes i1, ..., in is a basis for the neighborhood

of g ∈ X∗. Moreover it is straightforward to check that X∗ with the weak topology is a Hausdorff space

and that fα → f if and only if fα(x)→ f(x), for all x ∈ X.

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Remark A.1.2. As an example of how useful this topology can be, we give a proof of the Banach-

Alaoglu Theorem, that guarantees that the unit ball B1X∗ is compact (not surprising, if we regard X∗ as

a subspace of RX with the product topology, which is compact by the Tychonoff Theorem ). Indeed, let’s

define the compact set Kx = λ ∈ R : |λ| ≤ ||x|| and K =∏x∈X Kx, which is compact by Tychonoff

Theorem. Now let’s define Ψ : B1X∗ → K, such that Ψ(f) = (f(x))x∈X . It is clear that this is an injective

(and well-defined) map. Moreover, is a homemorphism. In fact, if fα → f we have that fα(x)→ f(x) for

all x ∈ X and so, Ψ(fα)→ Ψ(f). In a similar way we easily check that Ψ has continuous inverse. Finally,

it remains to note that Ψ(B1X∗) is a closed subset of K and so, is compact.

Definition A.1.3. LetH be a Hilbert space and letR be a ∗-subalgebra of L(H) that contains the identity

1 ∈ L(H). We say that R is a von Neumann algebra if R is closed for the WOT topology.

Remark A.1.4. Note that since R is a convex subset of L(H), in the above definition requiring to be

strongly closed is equivalent to require to be weakly closed.

Definition A.1.5. Let R be a von Neumann algebra. The commutant of R is the set T ∈ L(H) : TS =

ST, S ∈ R and we denote it by R′ . Moreover, we say that the set R′′ = (R′)′ is the bicommutant of R.

Definition A.1.6. Let R be a von Neumann algebra. We say that R′ ∩R is the center of R. IfM has a

trivial center (i.e. the multiples of the identity), then R is said to be a factor. If N ⊂ R is a ∗-subalgebra

such that 1N = 1R and with trivial center, N is said to be a subfactor.

It is fairly obvious that if R is a subset of L(H), not necessarily being a von Neumann algebra,

we have that R ⊂ R′′ . Moreover, if R1 and R2 are subsets of L(H), we have that if R1 ⊂ R2, then

R′2 ⊂ R′

1. With these small observations, it is now immediate to establish that R′ = (R′′)′ . Finally, if R

is a self-adjoint subset of L(H), we can briefly conclude that R′ is a von Neumann algebra - and so, the

commutant of a von Neumman algebra is still a von Neumann algebra. Indeed, since 1 ∈ R′ , it remains

to show that R is weakly closed : let Tα → T , with the net Tα in R′ . Noting that 〈TSx, y〉 = lim〈TαSx, y〉

and that 〈TαSx, y〉 = 〈STαx, y〉, we conclude that T ∈ R′ .

Proposition A.1.7. Let R be a self-adjoint C∗-subalgebra of L(H) that contains the identity 1. Given

x ∈ H and T ∈ R′′ , there exists a sequence T xn ∈ R such that ||T xnx − Tx|| → 0. Moreover, given a

certain finite set x1, ..., xn ⊂ H, there is a sequence Tn in R such that ||Tnxi − Txi|| → 0.

Theorem A.1.8. Let R be a C∗-subalgebra of L(H) that contains the identity operator 1. Then, the

following assertions are equivalent :

1. R = R′′ .

2. R = RWOT

3. R = RSOT .

Proof: To prove that (1) ⇒ (2), note that R′′ is weakly closed since by former observations we can

conclude that R′′ is a von Neumann algebra. Thus, and using the hypothesis, we have that RWOT= R.

To prove that (2) ⇒ (1) we just observe that since R is convex, we have that R = RWOT= RSOT .

78

Finally, to prove that (3) ⇒ (1), since we already noted that R ⊂ R′′ , it remains to show the inverse

inclusion. By hypothesis, RSOT = R and so we just have to establish that R′′ ⊂ RSOT

. But that is

immediate since given a neighborhood V of T ∈ R′′ in the SOT topology, we easily see that V ∩ R 6= ∅

by applying Prop.A.1.7.

Remark A.1.9. The last result is known as the von Neumann Bicommutant Theorem and gives a purely

algebraic alternative definition of a von Neumann algebra - recall that our original definition was purely

topological.

Theorem A.1.10. Let R be a C∗-subalgebra of L(H) that contains the identity 1. Then, B1RSOT

=

B1

RSOT.

Remark A.1.11. The last result is known as the Kaplansky’s Density Theorem and offers a slightly

different ( topological) definition of a von Neumann algebra. In fact, let R a C∗-subalgebra of L(H) that

contains 1. Then, R is a von Neumann algebra if and only if B1R = B1

RSOT

. Indeed, if R is a von

Neumann algebra we have that A = RSOT and by Kaplansky’s Density Theorem we conclude that

B1R = B1

RSOT

. Reciprocally, if B1R = B1

RSOT

, then by Kaplansky’s Density Theorem we have that

B1R = B1

RSOTand thus R = RSOT and we conclude that R is a von Neumann algebra.

Following this observation and noting that L(H) is itself a von Neumann algebra, we can conclude

that B1L(H) is weakly closed. This remark is quite useful since it allow us to show that B1

L(H) is weakly

compact by simply adapting the argument used to prove Banach-Alaoglu Theorem (not surprising, since

both theorems are similar, just on different topologies). Indeed, let’s consider this time the compact set

Px,y = z ∈ C : |z| ≤ ||x||||y|| and P =∏x,y∈H Px,y, which is compact by Tychonoff Theorem. Now

define Ψ : B1L(H) → P such that Ψ(T ) = (〈Tx, y〉)x,y∈H . It should be clear that Ψ is a well-defined

injective map. Moreover, Ψ is a homeomorphism, since Tα → T ⇔ 〈Tαx, y〉 → Ψ(T ). Finally, since

B1L(H) is weakly closed we conclude that so is Ψ(B1

L(H)) and we establish the result.

A.2 AF Algebras

A.2.1 UHF Algebras

Definition A.2.1. We say that a C∗-algebra U is an uniformly matricial algebra (or uniformly hyperfinite

algebra, hence the common designation as UHF algebra) if there is a sequence nj of positive integers

and a sequence Uj of C∗-subalgebras of U , such that Ui ( Uj if ni < nj , each Uj is isomorphic to the

algebra of nj × nj matrixes with complex entrances - we say that Uj is a type Inj factor1 - and such that

U =⋃j Uj . In this case, we also say that U is of type nj and that Uj is a generating net of type nj

for U .1Let U be isomorphic toMk, the algebra of k× k matrixes. Consider a faithful representation π : U → L(H) of U . Recall that

this object always exists through GNS construction. Since U and L(H) are C∗-algebras and π is a ∗-homomorphism, it is alsohomeomorphsim in its image and thus, π(U) is strongly closed, since U being isomorphic toMk, makes it also strongly closed.So it makes sense calling U a factor in the same way we did with von Neumann algebras.

79

Note that given j, k ∈ N, if kj = q then each matrix inMk admits a decomposition by blocs, as a j× j

matrix with entrances in Mq. Thus, we can define in this fashion an isomorphism ϕ of Mj into Mk,

setting ϕ(A) to be the matrix with blocs [arsIq]. It should be now clear that if U1 is a type Ij factor and

U2 is a type Ik factor with j|k, then there is an isomorphism between U1 and a C∗-subalgebra of U2. In

fact, we have the following result :

Proposition A.2.2. Let j, k ∈ N be such that j, k > 0 and U1,U2 are respectively factores of types Ij and

Ik. Then we have that:

1. There exists an isomorphism of U1 into U2 if and only if j|k.

2. If U1 is a C∗-subalgebra of U2, then j|k.

3. If B1 and B2 are factors of types Ij and Ik respectively, with B1 being a C∗-subalgebra of B2

and if U1 is a C∗-subalgebra of U2, then any isomorphism φ : U1 → B1 can be extended to an

isomorphism ψ : U2 → B2.

Theorem A.2.3. There exists a C∗-algebra of type nj if and only if nj is a strictly increasing se-

quence of positive integers such that nj |nj+1. Under this conditions and for each sequence nj, all the

C∗-algebras of type nj are isomorphic.

Proof: Let Uj be a generating net of type nj for U . Since Uj ( Uj+1, we conclude -by Prop.A.2.2-

that nj |nj+1 and thus, nj+1 > nj since the inclusion is strict. Reciprocally, let nj be a sequence under

the hypothesis conditions. Let M be the set of all bounded sequences Ar, where Ar ∈ Mr, clearly a

C∗-algebra with ||Ar|| = sup||Ar||. We can show that K ⊂ M , the subset of sequences such that

||Ar|| → 0, is an ideal of M and so we have the canonical quocient projection π : M → M/K. Defining

ij : Mj → M such that ij(A) = (0, ..., 0, A, ϕj(A), ϕj+1(A), ...), where A is at the jth-position and ϕj is

the isomorphism ofMj intoMj+1 - that exists by Prop.A.2.2 - we can show that πij is an isomorphism

ofMj onto C∗-subalgebra Uj of M/K. Moreover, we have that Uj ( Uj+1. Then, if we take the closure

of⋃j Uj in M/K, we have the desired algebra. It remains to show the unicity (up to isomorphism) of

this algebra: Let U and B two uniformly matricial algebras of type nj, with generating nets Uj and

Bj respectively. Applying Prop.A.2.2(3) successively, we obtain isomorphisms ψj : Uj → Bj such that

ψj+1|Uj = ψj . Thus, we have an isomorphism ψ :⋃j Uj →

⋃j Bj such that ψ|Uj = ψj and that we can

extend by continuity to the closure.

A.2.2 Inductive Limits

Let Uα : α ∈ Λ be a family of C∗-algebras, with Λ a directed set. Suppose that whenever α ≤ β in Λ,

there is an isomorphism Φβα of Uα into a C∗-subalgebra of Uβ . Moreover, assume that if α ≤ β ≤ γ in Λ,

we have that Φαγ = Φγβ Φβα. Under these conditions, we say that Uα : α ∈ Λ is a directed system

of C∗-algebras. As an example of this construction, consider a family Uα : α ∈ Λ of C∗-algebras such

that if α ≤ β, we have that Uα ⊂ Uβ and define the isomorphisms Φβα to be the inclusion maps. In

this particular case, we say that Uα : α ∈ Λ is a net of C∗-algebras. However, the following result

establishes that this is not a so particular example.

80

Theorem A.2.4. Let Uα : α ∈ Λ be a directed system of C∗-algebras, with isomorphisms Φβα when-

ever β ≥ α. Then, we have that :

1. There is a C∗-algebra U such that for each α ∈ Λ there is an isomorphism ϕα of Uα into U , such

that ϕα = ϕβ Φβα- if β ≥ α - and such that⋃α ϕα(Uα) is dense in U .

2. The C∗-algebra defined in the above statement, is unique up to isomorphism, i.e. if B is another

C∗-algebra and for each α ∈ Λ we have isomorphisms Ψα like in the above statement, then there

is an isomorphism Θ : U → B, such that for each α we have that Ψα = Θ ϕα.

We say that the algebra U of the previous theorem, is an inductive limit of the directed system

Uα : α ∈ Λ. Note that ϕα(Uα) : α ∈ Λ is a net of C∗-subalgebras of U , with dense union in U . Thus

we can replace any directed system Uα : α ∈ Λ with the example formerly provided -ϕα(Uα) : α ∈ Λ,

in which the maps Φβα are inclusions. Let us also note that if U is a uniformly matricial algebra of type

nj, then it should be clear that U is also the inductive limit of the net of C∗-subalgebras of U formed by

it generating net. After this small observation and recalling the unicity of the inductive limit established

in the previous theorem, it should not be a surprise that we can show the existence of inductive limits in

a similar fashion as we did with showing the existence of the uniformly matricial algebra, given a certain

generating net.

Definition A.2.5. If U is the inductive limit of finite dimensional C∗-algebras, we say that U is approxi-

mately finite dimensional or that U is an AF algebra. It is clear, by the previous remark, that any uniformly

matricial algebra is also an AF algebra.

An Invariant for UHF Algebras

From the Fundamental Theorem of Arithmetic, we know that any natural number is uniquely determined

by its decomposition in prime factors. Having this idea in mind, it is possible to rigourously define

a generalization of natural numbers, the so called supernatural numbers. A supernatural number is

uniquely determined by a product of the form Πp∈Ppnp , where P is the set of prime numbers and np can

be a natural number, zero or infinity. We can indeed extend the notion of divisibility establishing that

ω1|ω2 if np(ω1) ≤ np(ω2) for every p ∈ P. In this sense, we say that two supernatural numbers are equal

if and only if their sequences npp∈P are the same. Now, recall that given an UHF algebra of type nj,

the sequence nj is increasing and with nj |nj+1. Define, for each p ∈ P, εp ∈ N0 ∪ ∞ as the least

upper bound of potencies of p such that ∀j : pεp |nj . We will see that two UHF algebra are isomorphic

if and only if the sequences εp are the same. By different words, given a UHF algebra U of type nj

we define δ(U) = Πp∈Ppεp , then two UFH algebras U1 and U2 are isomorphic if and only if δ(U1) = δ(U2).

Proposition A.2.6. Let U be an AF algebra such that U =⋃n≥1 Un =

⋃n≥1 Bn. Then, for each ε > 0,

there is an unitary operator W in U such that ||W − 1|| < ε and that⋃n≥1 Un = W (

⋃n≥1 Bn)W ∗. In

particular, there are subsequences mi and ni such that Umi ⊂WBniW ∗ ⊂ Umi+1 , for all i ≥ 1, with

WBniW ∗ ⊂ Bni .

81

Corollary A.2.7. If U =⋃n≥1 Un and B =

⋃n≥1 Bn are isomorphic as AF algebras, then

⋃n=1 Un and⋃

n=1 Bn are also isomorphic.

Proof: Let ϕ : U → B be an isomorphism. One can prove that ϕ(⋃n=1 Un) = B =

⋃n=1 ϕ(Un).

Indeed, if zα ∈ ϕ(uαi), with zα → a, we have that a = ϕ(limα uαi) ∈ ϕ(⋃Un). Reciprocally, if a = ϕ(z)

with z = limα zα, with zα ∈ Uαi , it is clear that a ∈⋃ϕ(Un). Therefore, by Prop.A.2.6 we know that⋃

n=1 ϕ(Un) = W⋃n=1 BnW ∗ and we conclude what we wanted.

Theorem A.2.8. Let U and B be UHF algebras. Then, U and B are isomorphic if and only if δ(U) = δ(B).

Proof: Let U and B be two UHF algebras respectively of type kn and ln. From the latter corollary,

we conclude that⋃m≥1 Um and

⋃n≥1 Bn are isomorphic. From Prop.A.2.6, there exist subsequences

mi and ni such that Umi ≈ Mkmi(C) is isomorphic to a subalgebra of Bni ≈ Mlni

(C) and such that

Bni is isomorphic to a subalgebra of Umi+1≈Mkmi+1

(C). From Prop.A.2.2 we can conclude that kmi |lniand that lni |kmi+1

. Therefore, we have that δ(U) = δ(B) : If p ∈ P and pa|kmi , we have that pa|lni and

that pa|kmi+1. Reciprocally, suppose that δ(U) = δ(B). Then, by the definition of supernatural number

we can conclude that there are subsequences mi and ni such that kmi |lni |kmi+1 . To simplify the

notation, let’s assume that mi = ni = i. Let αn be the embedding of Un into Un+1 and βn the embedding

of Bn into Bn+1. We are going to define an isomorphism between⋃n≥1 Un and

⋃n≥1 Bn that we can

extend by continuity to an isomorphism between U and B. Now, let ϕ1 : U1 → B1 be the embedding

uniquely determined - up to unitary equivalence -by the multiplicity of U1 in B1.2 By the same argument,

let Ψ′

1 : B1 → U2 and again, by uniqueness up to unitary equivalence, there is an unitary element W1

such that W ∗1 Ψ′

1ϕ1 = α1. Define Ψ1 = W ∗Ψ′

1. Now, defining maps Ψn : Bn → Un+1 and ϕn : Un → Bnsuch that Ψn ϕn = αn and that ϕn+1 Ψ2 = βn, we have that ϕ =

⋃n≥1 ϕn is an isomorphism between⋃

n≥1 Un and⋃n≥1 Bn.

Remark A.2.9. Using the latter result, we conclude that two UHF algebras U and B associated to

sequences an and bn such that an = npn(ω1) and bn = npn(ω2) - where pn is the th-prime and

ω1 = δ(U1) and ω2 = δ(U2) - are isomorphic if and only if ∀n : an = bn. Thus, there are uncountable

classes of isomorphism of UHF algebras.

A.3 States, Representations and Trace

A.3.1 States

Some of the following definitions and results still make sense if R is just a C∗-algebra, though we will

assume that R is a von Neumann algebra. Recall that a positive linear functional ρ : R → C is called a

state if ||ρ|| = 1. We say that ρ is a normal state in R if given an increasing monotone net of operators

Hα with least upper bound H, we have that ρ(Hα) → ρ(H). We say that ρ is completely addictive

if given an orthogonal family Eα of projections in R, we have that ρ(∑αEα) =

∑α ρ(Eα). One can

show that ρ is a normal state if and only if is a completely addictive state in R (check [26], Thrm.7.1.12).2Check Theorem A.2.3.

82

Given a bounded linear functional ρ on R, we say that ρ is central if ρ(AB) = ρ(BA) for all A,B ∈ R. If

ρ is a positive, linear, bounded and central functional, we say that ρ is a numerical trace on R and if ρ

is a numerical trace with ||ρ|| = 1, we say that ρ is a tracial state. One can show that if R is a finite von

Neumann algebra, then there exists a normal tracial state on R (check [2], Lemma 6.3.9) and if R is a

finite factor, there is an unique faithful normal state on it (see [2], remark after Prop.7.1.2).

Whenever there is a faithful numerical trace ρ in a von Neumann algebra R (or, more generally, if R is

just a C∗-algebra), one can define a norm by ||x||ρ =√|ρ(A∗A)|. This is the so called trace norm.

A.3.2 Representations

Definition A.3.1. Given a C∗-algebra A we say that the pair (X,π), with X a linear space and π : A →

L(X) a ∗-homomorphism, is a representation of A.

In the above definition, whenever ker(π) = 0 we say that the representation is faithful. If a sub-

space Y ⊂ X is such that π(A)Y ⊂ Y , we say that Y is invariant and if X and 0 are the only

invariant subspaces, we say that π is an irreducible representation. We say that the representation is

non-degenerated if π(A)H = H and we say that the representation is cyclic - with cyclic vector η - if

π(A)η = H. It is immediate to check that if (H,π) is non-degenerated, then⋂a∈Aπ(a) = 0.

Proposition A.3.2. Let A be a C∗-algebra and let H1, π1 and H2, π2 be two cyclic representations

with cyclic vectors η1 and η2. Then, if 〈π1(a)η1, η1〉 = 〈π2(a)η2, η2〉 for every a ∈ A, there exists an unitary

operator U : H1 → H2 such that π2(a) = Uπ1(a)U∗ for all a ∈ A.3

Proof: Define U0 : π1(A)η1 → H2 such that π1(a)η1 7→ π2(a)η2. Since ||π2(a)η2||2 = 〈π2(aa∗)η2, η2〉 =

〈π1(a∗a)η1, η1〉 = ||π1(a)η1||2, we have that U0 is an isometry and so, let U be its extension, with

U : H1 → H2 (since π1(A)η1 and π2(A)η2 are dense). Notice that given a, b ∈ A we have that

(Uπ1(a))π1(b)η1 = U0π1(ab)η1 = π2(ab)η2 = (π2(a)U)π1(b)η1 and thus, by density, Uπ1(a) = π2(a)U

for all a ∈ A.

Gelfand-Naimark-Segal construction

Theorem A.3.3. Any C∗-algebra is isometrically isomorphic to a C∗-subalgebra of a certain L(H).

Proof: To give a proof of this result, we will sketch the Gelfand-Naimark-Segal construction (GNS con-

struction). Let ρ be a positive linear functional and define Lρ = a ∈ A : ρ(a∗a) = 0, which is a closed

left ideal of A. Now we consider the quotient A/Lρ with the map 〈., .〉ρ : A/Lρ × A/Lρ → C such that

(a+Lρ, b+Lρ) 7→ ρ(b∗a). One can prove that (A/Lρ, 〈., .〉) is a pre-Hilbert space and so, we consider Hρ,

the completeness ofA/Lρ, which is a Hilbert space with norm ||.||ρ associated to the inner product 〈., .〉ρ.

Now, for each a ∈ A we define the linear operator π(a) : A/Lρ → A/Lρ such that (b+ Lρ) 7→ (ab+ Lρ).

One can check that π(a) can be extended to a bounded linear operator πρ(a) : Hρ → Hρ. In this setting,

we define the GNS representation ρ : A → L(Hρ), given by a 7→ πρ(a). Thus, to each positive linear

functional ρ we have a representation (Hρ, πρ) that one can show that it is cyclic. Finally, let EA be

3We say that π1 and π2 are equivalent.

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the set of all states of A and let’s consider the pair (Hu, πu) = (⊕

ρ∈EA Hρ,⊕

ρ∈EA πρ), the so called

universal representation of A. It is fairly easy to see that the universal representation is faithful : If

πu(a) = 0, then for every state ρ we have that πρ(a) = 0 and so, ρ(a∗a) = 0. But there is a state such

that ρ(a∗a) = ||a||2 6= 0, if a 6= 0 (Check [1], Prop.4.1.10). Now it is done, since a faithful representation

is necessarily an isometry (Check [1], Cor.3.2.3).

A.3.3 Trace

Definition A.3.4. Let R be a von Neumann algebra with center C. We say that a linear map τ : R → C

is a center-valued trace if it verifies the following properties :

1. τ(AB) = τ(BA), if A,B ∈ R.

2. τ(C) = C, if C ∈ C.

3. τ(A) > 0, if A ∈ R+.

Remark A.3.5. From this definition, it is clear that if such τ exists onR thenR is a finite algebra. In fact,

if E and F are projections in R such that E ∼ F ≤ E, let V be a partial isometry such that V ∗V = E

and V V ∗ = F . Then, we have that τ(E − F ) = τ(E)− τ(F ) = τ(V ∗V )− τ(V V ∗) = 0, since τ is linear

and where we used property (1) of the previous definition. Now, by property (3) of the same definition,

we can conclude that E = F .

Recall that we say that a projection E in R is monic if E 6= 0 and if there is k > 0 and projections

E1, ..., Ek in R and Q ∈ C such that E1 ∼ · · ·Ek ∼ E and E1 + · · · + Ek = Q. Note that if every Ej has

central carrier CE , then Q = CE .

We recall that if A is a commutative C∗-algebra, there is a correspondence between the multiplica-

tive linear functionals on A and it maximal ideals (since these ideals are precisely the kernels of the

mentioned functionals). We then define, for each a ∈ A the Gelfand map φa : MA → K, such that

φa(f) = f(a), where MA denotes the set of multiplicative linear functionals. Now we can define the

Gelfand transformation, given by φ : A → C(MA) such that φ(a) = φa, that is an isometric isomorphism.

Thus, and noting that by Banach-Alaoglu Theorem we have thatMA is a compact and Hausdorff space,

we establish the first Gelfand-Naimark Theorem, guaranteeing that every commutative C∗-algebra is iso-

metrically isomorphic to a C∗-subalgebra of C(X), with X a compact and Hausdorff space. Now, recall

that we say that a state ρ of A is pure if ρ is a state and majorates only the states of the form λρ, with

λ ∈ [0, 1]. We can prove that the set of the multiplicative linear functionals on A coincides with the set of

it pure states. After recalling these facts, we can now show that if η is a positive linear map from R to C,

then η is bounded with ||η|| = ||η(1)||. In fact, note that C is a commutative C∗-algebra an so, appealing

to Gelfand transformation we conclude that for each C ∈ C, ||C|| = supρ(C), where ρ is a pure state

of C. Since ρ is a pure state, given a positive linear map η : R → C we have that ||ρ η|| ≤ ||η(1)||. Thus

we have that |ρ(η(A))| ≤ ||η(1)||||A|| for A ∈ R and now, taking the supreme over all pure states, we can

conclude that ||η(A)|| ≤ ||η(1)||||A||.

84

We checked that if a von Neumann algebra R admits a center-valued trace, then R is finite. The recip-

rocal is true and so a finite von Neumann algebra R is finite if and only if it admits a center-valued trace.

Indeed, we have the following result :

Theorem A.3.6. Let R be a finite von Neumann algebra with center C. Then there exists an unique

center-valued trace τ such that :

1. τ(CA) = Cτ(A), A ∈ R, C ∈ C.

2. ||τ(A)|| ≤ ||A||, A ∈ R.

3. τ is ultraweakly continuous.

A.4 Ultrafilters and Ultraproducts

A.4.1 Ultrafilters

Filters are a very useful tool that play a major role in many parts of Mathematics, namely in Topology.

Definition A.4.1. Given a set X, we say that a filter F on X is a collection of subsets of X such that :

1. X ∈ F and ∅ /∈ F

2. If V ∈ F and V ⊂W , then W ∈ F

3. If U, V ∈ F , then U ∩ V ∈ F

Definition A.4.2. If F is a filter on X such that for every A ⊂ X either A ∈ F or X \A ∈ F , we say that

F is an ultrafilter on X.

Remark A.4.3. One can prove that given a filter F on X, there is always an ultrafilter U on X such that

F ⊂ U , using a standard Zorn’s Lemma argument. This result is known as the Ultrafilter Lemma.4

Remark A.4.4. Notice that given a topological space X, for every x ∈ X we have that Nx is a filter on

X. It is then natural to define that F → x if Nx ⊂ F .

Now, let’s consider J ⊂ X, a subset with the finite intersection property. We say that the smallest

filter F that contains J , is the filter generated by J . It is routine work to check that F = U ∈ X :

∃U1, ..., Un ∈ J : ∩ni=1Ui ⊂ U.

Remark A.4.5. One of the reasons why filters are so useful is the fact that many desired topological

properties are true using this framework (similarly, and not by coincidence, as it is using nets). As an

example of this, we have that X is compact if and only if every ultrafilter converges. To see that this is

the case, it is only needed to prove that X is compact if and only if every filter can be extended to a

convergent filter (and then, use Ultrafilter Lemma): On one hand, if X is compact and F a filter on X,

we consider T = U : U ∈ F, which has the finite intersection property. Thus, by compactness, there

4The reader can check the proof of Theorem 3.1.8 and replicate the reasoning to conclude the existence of ultrafilters.

85

exists x ∈⋂T and it is fairly easy to check that F ∪ Nx forms a sub-base for a filter that extends F

and converges to x. Conversely, let C be a collection of closes subsets of X with the finite intersection

property. Now, consider F as the filter generated by C and let, using hypothesis, G be an extension of F

that converges to x. Then, it is easy to check that x ∈⋂C and therefore, X is compact.

Theorem A.4.6. Let Xii∈I be a family of topological compact spaces. Then, with the product topology,

Πi∈IXi is compact.

Proof: Let Fi be a filter on Xi. We define the product filter F on X =∏iXi as the filter generated by

the set of products∏i Ui, where Ui ∈ Fi and Ui = Xi for all but finitely many i. Then, given an ultrafilter

U on X we check that πi(U) is an ultrafilter on Xi. If each Xi is compact, then πi(U) is convergent for

each i and so is U . Hence, we conclude that X is also compact.

Definition A.4.7. Now we define the concept of F-convergence. Given a topological space X, an index

set I and a filter F on I, we say that a family xii∈I ⊂ X converges along the filter F to some x ∈ X, if

for every U ∈ Nx we have that i ∈ I : xi ∈ U ∈ F . In this case, we write that limi→F xi = x.

Proposition A.4.8. Let X be compact, I an index set and U an ultrafilter on I. Then, any subset of X

indexed by I, say xii∈I , converges along U .

Proof: Suppose that for every x ∈ X there is a Ux ∈ Nx such that i ∈ I : xi ∈ Ux /∈ U . By

compactness of X, let Uini=1 be a finite subcover of Ux : x ∈ X, such that i ∈ I : xi ∈ X \ Uj ∈ U ,

for every j ∈ 1, ..., n, since U is an ultrafilter. But now, note that⋂nj=1(X \ Uj) = ∅, which imples that⋂n

j=1i ∈ I : xi ∈ X \ Uj = ∅ ∈ U , a contradiction.

Proposition A.4.9. Every bounded net of reals, is convergent along a given ultrafilter U .

Proof:Let xαα∈Λ ⊂ R be a bounded net. First, we suppose that U is principal on Λ, with basis α0.

It is immediate to conclude that for any ε > 0, the sets Aε = α ∈ Λ : |xα − xα0| < ε ∈ U , since

α0 ∈ Aε. Now, let’s suppose that U is a free ultrafilter and let xαα∈Λ ⊂ [−M,M ]. Let R1 = [−M, 0]

and R2 =]0,M ], with Fi = α ∈ Λ : xα ∈ Ri. Since U is an ultrafilter, either F1 ∈ U or F2 ∈ U - if it is

F2, in the following argument we consider the closure R2. Repeating this process we have a sequence

of non-empty closed sets Rn with diameter converging to zero. Thus, by Cantor’s Intersection Theorem,

let x ∈⋂nRn. It should now be clear that limα→U xα = x.

A.4.2 Ultraproducts

A metric group is a topological group G with a metric d that generates its topology. Whenever d is such

that d(gh, gk) = d(h, k) = d(hg, kg), we say that d is bi-invariant.

Now, suppose that Gii∈I is a family of metric groups with bi-invariant metrics di and identities 1i. Let

U be an ultrafilter on I and consider NU = (gi)i∈I ∈ l∞(I,Gi) : limi→U di(gi, 1i) = 0 5. One can prove

that NU is a normal group of l∞(I,Gi).

5Here, l∞(I,Gi) is the set x ∈∏i∈I Gi : supi∈I di(xi, 1i) < ∞. Also, note that by the last observation in the filters

subsection, this limit always exists.

86

Definition A.4.10. To the quotient l∞(I,Gi)/NU we call the metric ultraproduct of the groups Gii∈Iand we denote it by

∏Ui∈I Gi. In the particular case of Gi = G for every i, we denote it by GU and we say

that it is the ultrapower of G.

One can show that∏Ui∈I Gi is still a metric group with a bi-invariant metric given by

dU ([(gi)i∈I ], [(hi)i∈I ]) = limi→U

di(gi, hi).

Now, let’s consider a family Aii∈I of C∗-algebras, such that τi is a trace6 in Ai for every i ∈ I.

Let’s denote the trace norm in Ai by ||.||τi and let U be an ultrafilter on I. Similarly with the previous

construction, let’s consider A = x ∈∏i∈I Ai : sup ||xi||τi < ∞ and I = x ∈ A : limi→U ||xi||τi = 0,

a closed ideal of A. Thus, the quotient A/I is still a C∗-algebra. One can define a trace in A by

τ((xi)i∈I) = limi→U τi(xi) and verify that this induces a faithful trace in A/I, given by

τU ([(xi)i∈I ]) = τ((xi)i∈I).

Finally, let π be the GNS representation of A/I associated to τU .

Definition A.4.11. To the image π(A/I) we call the tracial ultraproduct of the family Aii∈I of C∗-

algebras and we denote it by∏Ui∈I Ai. Again, in the particular case where Ai = A for each i ∈ I, we

denote it by AU and we call it the tracial ultrapower of A.

Note that π(A/I) and A/I are isomorphic, since τU is faithful and so, we will use only the notation

A/I.

Remark A.4.12. Let Aii∈I be a family of type II1 factors. We already know - check the section of this

Appendix about the trace - that there is a trace τi on each Ai and so, we can consider the ultraproduct∏Ui∈I Ai, given an ultrafilter U on I. In this case, one can prove that the ultraproduct is still a factor of

type II1.(c.f.[35])

A.5 Crossed Products

This section is rather superficial, containing what is strickly necessary for a better understanding of some

chapters of this thesis.

Firstly, we define the so called continuous crossed product. Let G be a locally compact group and let

dG denote the Haar measure in G. LetM∈ L(H) be a von Neumann algebra and let α : G→ Aut(M)

be an action. Let’s denote, as usual, K(G,H) := f ∈ HG: supp(f ) is compact and f is continuous

endowed with the inner product given by 〈f1, f2〉 =∫G〈f1(g), f2(g)〉dG. Finally, let L2(G,H) be the

6For the sake of easing the text, whenever we mention trace in the context of a trace norm, we really mean - using the previousterminology of the latter chapters - a faithful numerical trace.

87

Hilbert space obtained by completeness from K(G,H) and for each x ∈M, g, h ∈ G and ξ ∈ L2(G,H),

we define the operators tα(x) and ug in L(L2(G,H)) such that

(tα(x)ξ)(h) := α−1h (x)ξ(h) (ugξ)(h) := ξ(g−1h) u∗g := ug−1 .

Definition A.5.1. To the bicommutant (tα(x), ugx∈M,g∈G)′′

we call crossed product ofM and G via α

and we denote it byMoα G.

Our second definition is taken when G is a countable group and there is a faithful normal trace τ on

M. One takes an eventually zero sequence (xg)g∈G ⊂ M and considers the symbols∑g∈G xgg and

the set A = ∑g∈G xgg such that (xg) is eventually zero. Then, we define in A the following operations:

(i) (∑g∈G xgg) + (

∑g∈G ygg) :=

∑g∈G(xg + yg)g

(ii) λ(∑g∈G xgg) :=

∑g∈G(λxg)g for any scalar λ

(iii) (∑g∈G xgg)(

∑h∈G yhh) :=

∑g,h∈G(αh(xg)yh)gh

(iv) (∑g∈G xgg)∗ :=

∑g∈G(αg−1(x∗g))g

−1.

With this operations and identifying g ∈ G with 1 ⊗ g and x ∈ M with x ⊗ 1, we have that 1M ⊗ 1G

is the identity.7 It is clear that A is a ∗-algebra. Moreover, one can extend τ from M to A setting that

τ(xgg) := τ(xg) if g = 1G and τ(xgg) = 0 if g 6= 1G. Then, τ(∑g∈G xgg) :=

∑g∈G τ(xgg).

Definition A.5.2. To the weak closure of A, we call crossed product via semi-direct product.

Remark A.5.3. WheneverG is a countable group andM has a faithful and normal trace, both definitions

ofMoα G coincide. Furthermore, we identify talpha(y) with y ⊗ 1G and g ∈ G with 1⊗ g.

Lemma A.5.4. In the above conditions, we have that ugtα(x)u∗g = tα(αg(x)).

Proof: Merely computational proof following the definitions.

Lemma A.5.5. Let G be a finite group and let M be a type II1-factor and let α : G → Aut(M) be an

action. Then, α is outer if and only if (tα(M′)) ∩ (Moα G) = C1MoαG.

Proposition A.5.6. Let M be a type II1-factor and let α be an outer action of a finite group G on M.

Then,Moα G is still a type II1-factor.

Proof: We use Lemma A.5.5 to prove thatMoα G is a factor. Then, we check that τ(Moα G) = [0, 1]

and we use Cor.2.1.9 to conclude thatMoα G is a type II1-factor.

A.6 Tensor Products

A.6.1 Algebraic Tensor Product of R-modules

During this section, we will describe several tensorial products. We will be mainly interested in the

tensorial product of C∗-algebras and in the maximal and minimal tensorial product. However, it is useful7Here, to make notation more clear we denote the symbolic product as xg ⊗ g, instead of just xgg.

88

to recall other constructions. We start with the algebraic tensor product of R-modules, where R is a

commutative ring. Let M,N be R-modules.

The tensor product of R-modules is characterized by the following universal property : Given two R-

modules M and N , there is a R-module M ⊗N and a bilinear map τ : M ×N → M ⊗N , such that for

any R-module T and bilinear map f : M×N → T , there is an unique R-homomorphism f ′ : M⊗N → T

such that f ′ τ = f .

Definition A.6.1. We say that M⊗N is the algebraic tensor product of M and N and we denote τ(m,n)

by m⊗ n.

Remark A.6.2. In fact, given two R-modules M and N , their tensor product always exist : Let F be

the free R-module over the set M × N and let i : M × N → F be the canonical map. Let Y be the

R-submodule generated by the elements of the form i(m1 +m2, n1)− i(m1, n1)− i(m2, n1), i(rm1, n1)−

ri(m1, n1), i(m1, n1 + n2) − i(m1, n1) − i(m1, n2) and i(m1, rn1) − ri(m1, n1), with r ∈ R, ni ∈ N and

mi ∈M . Now, consider the quotient map π : F → F/Y and just define τ := π i. The R-module M ⊗N

is then defined as the image τ(M ×N).

Before moving forward, we will introduce a bit of notation. To avoid further confusion, whenever we

mention the algebraic tensor product of R-modules M and N , we denote it by M N . However, we still

denote τ(m,n) as m⊗ n.

Remark A.6.3. It is useful to remark the following fact : Let A,B and C be C∗-algebras and πA : A→ C

and πB : B → C be ∗-homomorphisms. We denote by πA×πB the homomorphism πA×πB : AB → C

given by (πA × πB)(a ⊗ b) = πA(a)πB(b). If H is a Hilbert space and π : A B → L(H) is a ∗-

homomorphism, one can show that there are ∗-homomorphisms πA : A → L(H) and πB : B → L(H)

such that π = πA × πB . We say that πA and πB are the restrictions of π.

A.6.2 Tensor Product of Hilbert Spaces

Before defining the tensor product of C∗-algebras, we need to define the tensor product of Hilbert spaces.

Let Hini=1 be a finite family of Hilbert spaces and let ϕ be a bounded functional on∏ni=1Hi. Now,

define the sum S =∑y1∈Y1

...∑yn∈Yn |ϕ(y1, ..., yn)|2. One can show that S is independent of the choice

of orthonormal basis Yi for Hi.

Definition A.6.4. If S < ∞, we say that ϕ is a Hilbert-Schmidt functional. Moreover, we denote the set

of all Hilbert-Schmidt functionals for a given∏ni=1Hi, as H.

The tensor product of a finite family of Hilbert spaces, will be a Hilbert space characterized by a

certain universal property. But, to describe this property, we need the definition of a weak Hilbert-

Schmidt functional.

Definition A.6.5. Let L :∏ni=1Hi → K be a multilinear and bounded map, with K a Hilbert space.

We say that L is a weak Hilbert-Schmidt functional if for every z ∈ C we have that the functional

Lz(x1, ..., xn) := 〈L(x1, ..., xn), z〉 is Hilbert-Schmidt and if there is a d ∈ R such that for every u ∈ K we

have that ||Lu||2 ≤ d||u||.

89

Finally, we are able to characterize the tensor product of Hilbert spaces using the following universal

property: Let Hini=1 be a finite family of Hilbert spaces. Then, there is a Hilbert space ⊗ni=1Hi and a

weak Hilbert-Schimdt map ρ :∏ni=1Hi → ⊗ni=1Hi such that given any Hilbert space K and any weak

Hilbert-Schmidt functional L :∏ni=1Hi → K, there is a bounded linear map T : ⊗ni=1Hi → K such that

T ρ = L. For a construction of such Hilbert space, the reader can check [27].

Definition A.6.6. We say that ⊗ni=1Hi is the tensor product of the family Hini=1 of Hilbert spaces.

Lemma A.6.7. Let Hini=1 be a family of Hilbert spaces. Then, the following properties are verified:

1. x1 ⊗ ...⊗ (axm1+ bxm2

)⊗ ...⊗ xn = a(x1 ⊗ ...⊗ xn) + b(x1 ⊗ ...⊗ xn)

2. 〈x1 ⊗ ...⊗ xn, y1 ⊗ ...⊗ yn〉 = 〈x1, y1〉...〈xn, yn〉

3. ||x1 ⊗ ...⊗ xn|| = ||x1||...||xn||

4. (H1 ⊗ ...⊗Hm)⊗ (Hm+1 ⊗ ...⊗Hn) = H1 ⊗ ...⊗Hn

Remark A.6.8. Let Hini=1 and Kini=1 be families of Hilbert spaces and let Aj ∈ L(Hi,Ki). Then,

there is an unique A ∈ L(⊗ni=1Hi,⊗ni=1Ki) such that A(x1 ⊗ ...⊗ xn) = A1x1 ⊗ ...⊗Anxn-

A.6.3 Tensor Product of C∗-algebras

First we consider finite families of represented C∗-algebras, i.e. algebras of operators acting in a Hilbert

space. Let Aini=1 be a family of represented C∗-algebras acting in Hini=1. Let H = ⊗ni=1Hi and let

A0 ⊂ L(H) be the set whose elements are finite sums of operators of the form A1 ⊗ ...⊗ An, with each

Aj ∈ L(Hj).

Definition A.6.9. We define the tensor product of the family Aini=1 of represented C∗-algebras, as the

closure of A0 and we denote this new C∗-algebra by ⊗ni=1Ai.

Now, let’s consider a finite family Aini=1 of not necessarily represented C∗-algebras. One can

prove that there is a C∗-algebra A and a multilinear map ρ :∏ni=1Ai → A such that given faithful

representations πj of Aj , there is an isomorphism ϕ : A → π1(A1) ⊗ ... ⊗ πn(An) such that ϕ ρ =

π1 × ...× πn. This universal property will characterize the tensor product:

Definition A.6.10. We define the tensor product of a finite family Aini=1 of C∗-algebras as the C∗-

algebra A.

Remark A.6.11. It should be clear that the latter definition coincides with the first one in the case that

every Ai is represented.

Finally, we consider arbitrary families Aα : α ∈ Λ of C∗-algebras, not necessarily finite or repre-

sented. Let F ⊂ Pfin(Λ). It should be clear that this is a directed set, considering a partial ordering

by inclusion. Given F ∈ F , let’s define AF = ⊗α∈FAα. It is immediate to verify that AF : F ∈ F

together with the maps ΦGF - whenever F ⊂ G - such that ΦGF (A) = (A ⊗ I), is a directed system of

C∗-algebras.

90

Definition A.6.12. We define the tensorial product of the family Aα : α ∈ Λ, denoted as ⊗α∈ΛAα, as

the inductive limit of the mentioned directed system of C∗-algebras.

A.6.4 Maximal and Minimal Tensor Product

Let A and B be C∗-algebras and let’s consider A B with the norm ||.||max given by

||x||max = sup||π(x)|| : π : A B → L(H),

where the supremum is taken through all ∗-representations π.

Definition A.6.13. The completion8 of A B under the norm ||.||max, denoted by A ⊗max B, is called

the maximal tensor product of A and B.

Remark A.6.14. A⊗max B is a C∗-algebra.

Let A and B be C∗-algebras and let’s consider A B with the norm ||.||min given by

||n∑k=1

ak ⊗ bk||min = ||n∑k=1

π(ak)⊗ ρ(bk)||L(H⊗K),

where π : A → L(H) and ρ : B → L(K) are faithful representations.

Definition A.6.15. The completion of AB under the norm ||.||min, denoted by A⊗min B, is called the

minimal tensor product of A and B.

Remark A.6.16. A ⊗min B is a C∗-algebra and it can be show that the above definition is independent

of the choice of π and ρ.

There is a good reason why we use the terms maximal and minimal. In fact, one can prove that

||.||max is the biggest possible C∗-norm in A B and that ||.||min is the smallest possible C∗-norm in

A B. Thus, if A⊗min B = A⊗max B, there is only one C∗-norm in A B.

A.6.5 Tensor Product of von Neumann algebras

LetM and N be two von Neumann algebras acting in H and K respectively.

Definition A.6.17. We define the tensor product ofM and N as the strong closure of the set

x⊗ y : x ∈M, y ∈ N

and we denote it byM⊗N .

Remark A.6.18. By Bicommutant Theorem, notice that M⊗N coincides with the bicommutant of the

set x⊗ y, x ∈M, y ∈ N, in L(H ⊗K).

8Consider a metric space (X, d). We say that a pair consisting in a complete metric space (X′, d′) and an isometry ϕ : X →

X′

such that ϕ(X) is dense, is a completion. Moreover, given any metric space (X, d) there is always its completion which isunique up to isometry.

91

92

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