arxiv:1309.2034v6 [math.gr] 5 may 2015 · arxiv:1309.2034v6 [math.gr] 5 may 2015 introduction to...

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Introduction to Sofic and Hyperlinear groups

and Connes’ Embedding Conjecture

VALERIO CAPRAROCentrum voor Wiskunde en Informatica

Amsterdam, The Netherlands

MARTINO LUPINIDepartment of Mathematics

York UniversityToronto, Canada

With an appendix by

VLADIMIR PESTOVDepartamento de Matematica

Universidade Federal de Santa CatarinaFlorianopolis-SC, Brasil

http://arxiv.org/abs/1309.2034v6

Preface

Analogy is one of the most effective techniques of human reasoning: When we face newproblems we compare them with simpler and already known ones, in the attempt to usewhat we know about the latter ones to solve the former ones. This strategy is particu-larly common in Mathematics, which offers several examples of abstract and seeminglyintractable objects: Subsets of the plane can be enormously complicated but, as soon asthey can be approximated by rectangles, then they can be measured; Uniformly finitemetric spaces can be difficult to describe and understand but, as soon as they can be ap-proximated by Hilbert spaces, then they can be proved to satisfy the coarse Novikov’s andBaum-Connes’s conjectures.

These notes deal with two particular instances of such a strategy: Sofic and hyperlineargroups are in fact the countable discrete groups that can be approximated in a suitablesense by finite symmetric groups and groups of unitary matrices. These notions, introducedby Gromov and Radulescu, respectively, at the end of the 1990s, turned out to be very deepand fruitful, and stimulated in the last 15 years an impressive amount of research touchingseveral seemingly distant areas of mathematics including geometric group theory, operatoralgebras, dynamical systems, graph theory, and more recently even quantum informationtheory. Several long-standing conjectures that are still open for arbitrary groups weresettled in the case of sofic or hyperlinear groups. These achievements aroused the interestof an increasing number of researchers into some fundamental questions about the natureof these approximation properties. Many of such problems are to this day still open suchas, outstandingly: Is there any countable discrete group that is not sofic or hyperlinear? Asimilar pattern can be found in the study of II1 factors. In this case the famous conjecturedue to Connes (commonly known as Connes’ embedding conjecture) that any II1 factor canbe approximated in a suitable sense by matrix algebras inspired several breakthroughs inthe understanding of II1 factors, and stands out today as one of the major open problemsin the field.

The aim of this monograph is to present in a uniform and accessible way some cor-nerstone results in the study of sofic and hyperlinear groups and Connes’ embedding con-jecture. These notions, as well as the proofs of many results, are here presented in theframework of model theory for metric structures. We believe that this point of view, eventhough rarely explicitly adopted in the literature, can contribute to a better understanding

iii

iv

of the ideas therein, as well as provide additional tools to attack many remaining openproblems. The presentation is nonetheless self-contained and accessible to any studentor researcher with a graduate-level mathematical background. In particular no specificknowledge of logic or model theory is required.

Chapter 1 presents the conjectures and open problems that will serve as common threadand motivation for the rest of the survey: Connes’ embedding conjecture, Gottschalk’s con-jecture, and Kaplansky’s conjecture. Chapter 2 introduces sofic and hyperlinear groups, aswell as the general notion of metric approximation property; outlines the proofs of Kaplan-sky’s direct finiteness conjecture and the algebraic eigenvalues conjecture for sofic groups;and develops the theory of entropy for sofic group actions, yielding a proof of Gottschalk’ssurjunctivity conjecture in the sofic case. Chapter 3 discusses the relationship betweenhyperlinear groups and the Connes’ embedding conjecture; establishes several equivalentreformulations of the Connes’ embedding conjecture due to Haagerup-Winsløw and Kirch-berg; describes the purely algebraic approach initiated by Radulescu and carried over byKlep-Schweighofer and Juschenko-Popovich; and finally outlines the theory of Brown’s in-variants for II1 factors satisfying the Connes’ embedding conjecture. An appendix by V.Pestov provides a pedagogically new introduction to the concepts of ultrafilters, ultralimits,and ultraproducts for those mathematicians who are not familiar with them, and aimingto make these concepts appear very natural.

The choice of topics is unavoidably not exhaustive. A more detailed introduction to thebasic results about sofic and hyperlinear groups can be found in [127] and [128]. The surveys[120, 123, 122] contain several other equivalent reformulations of the Connes’ embeddingconjecture in purely algebraic or C*-algebraic terms.

This survey originated from a short intensive course that the authors gave at the Uni-versidade Federal de Santa Catarina in 2013 in occasion of the “Workshop on sofic andhyperlinear groups and the Connes’ embedding conjecture” supported by CAPES (Brazil)through the program “Science without borders”, PVE project 085/2012. We would like togratefully thank CAPES for its support, as well as the organizers of the workshop DanielGoncalves and Vladimir Pestov for their kind hospitality, and for their constant and pas-sionate encouragement.

Moreover we are grateful to Hiroshi Ando, Goulnara Arzhantseva, Samuel Coskey, IlijasFarah, Tobias Fritz, Benjamin Hayes, Liviu Paunescu, Vladimir Pestov, David Sherman,Alain Valette, and five anonymous referees for several useful comments and suggestions.

Valerio CapraroDepartment of MathematicsUniversity of Southampton

United Kingdom

Martino LupiniDepartment of Mathematics

York UniversityToronto, Canada

Contents

I Introduction 1Valerio Capraro and Martino Lupini

I.1 Von Neumann algebras and II1 factors . . . . . . . . . . . . . . . . . . . . . 1

I.2 Voiculescu’s free entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

I.3 History of Connes’ embedding conjecture . . . . . . . . . . . . . . . . . . . 6

I.4 Hyperlinear and sofic groups . . . . . . . . . . . . . . . . . . . . . . . . . . 7

I.5 Kaplansky’s direct finiteness conjecture . . . . . . . . . . . . . . . . . . . . 8

I.6 Kaplansky’s group ring conjectures . . . . . . . . . . . . . . . . . . . . . . . 10

II Sofic and hyperlinear groups 13Martino Lupini

II.1 Definition of sofic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

II.2 Definition of hyperlinear groups . . . . . . . . . . . . . . . . . . . . . . . . . 18

II.3 Classes of sofic and hyperlinear groups . . . . . . . . . . . . . . . . . . . . . 23

II.4 Closure properties of the classes of sofic and hyperlinear groups . . . . . . 24

II.5 Further examples of sofic group . . . . . . . . . . . . . . . . . . . . . . . . . 28

II.6 Logic for invariant length groups . . . . . . . . . . . . . . . . . . . . . . . . 33

II.7 Model theoretic characterization of sofic and hyperlinear groups . . . . . . 39

II.8 The Kervaire-Laudenbach conjecture . . . . . . . . . . . . . . . . . . . . . . 41

II.9 Other metric approximations and Higman’s group . . . . . . . . . . . . . . 43

II.10 Rank rings and Kaplansky’s direct finiteness conjecture . . . . . . . . . . . 48

II.11 Logic for tracial von Neumann algebras . . . . . . . . . . . . . . . . . . . . 52

II.12 The algebraic eigenvalues conjecture . . . . . . . . . . . . . . . . . . . . . . 56

II.13 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

II.13.1 Entropy of a single transformation . . . . . . . . . . . . . . . . . . . 59

II.13.2 Entropy of an integer Bernoulli shift . . . . . . . . . . . . . . . . . . 60

II.13.3 Tilings on amenable groups . . . . . . . . . . . . . . . . . . . . . . . 61

II.13.4 Entropy of actions of amenable groups . . . . . . . . . . . . . . . . . 63

II.13.5 Entropy of Bernoulli shifts of an amenable group . . . . . . . . . . . 63

II.13.6 Entropy of actions of sofic groups . . . . . . . . . . . . . . . . . . . . 65

v

vi CONTENTS

II.13.7 Bernoulli actions of sofic groups . . . . . . . . . . . . . . . . . . . . . 65

IIIConnes’ embedding conjecture 71Valerio Capraro

III.1 The hyperfinite II1 factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71III.2 Hyperlinear groups and Radulescu’s Theorem . . . . . . . . . . . . . . . . . 73III.3 Kirchberg’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

III.3.1 Effros-Marechal topology on the space of von Neumann algebras . . 76III.3.2 Proof of Kirchberg’s theorem . . . . . . . . . . . . . . . . . . . . . . 79

III.4 Connes’ embedding conjecture and Lance’s WEP . . . . . . . . . . . . . . . 83III.5 Algebraic reformulation of the conjecture . . . . . . . . . . . . . . . . . . . 94III.6 Brown’s invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

III.6.1 Convex combinations of representations into RU . . . . . . . . . . . 98III.6.2 Extreme points of Hom(M,RU ) and a problem of Popa . . . . . . . 102

Conclusions 107

A Tensor product of C*-algebras 111

B Ultrafilters and ultralimits (by V. Pestov) 115

Bibliography 137

Index 148

Chapter I

Introduction

Valerio Capraro and Martino Lupini

I.1 Von Neumann algebras and II1 factors

Denote by B(H) the algebra of bounded linear operators on the Hilbert space H. Recallthat B(H) is naturally endowed with an involution x 7→ x∗ associating with an operator xits adjoint x∗. The operator norm ‖x‖ of an element of B(H) is defined by

‖x‖ = sup {‖xξ‖ : ξ ∈ H, ‖ξ‖ ≤ 1} .

Endowed with this norm, B(H) is a Banach algebra with involution satisfying the identity

‖x∗x‖ = ‖x‖2 (C*-identity)

i.e. a C*-algebra.The weak operator topology on B(H) is the weakest topology making the map

x 7→ 〈xξ, η〉

continuous for every ξ, η ∈ H, where 〈·, ·〉 denotes the scalar product of H. The strongoperator topology on B(H) is instead the weakest topology making the maps

x 7→ ‖xξ‖

continuous for every ξ ∈ H. As the names suggest the strong operator topology is strongerthan the weak operator topology. It is a consequence of the Hahn-Banach theorem that,conversely, a convex subset of B(H) closed in the strong operator topology is also closedin the weak operator topology (see Theorem 5.1.2 of [91]).

A (concrete) von Neumann algebra is a unital *-subalgebra (i.e. closed with respectto taking adjoints) of B(H) that is closed in the weak (or, equivalently, strong) operator

1

2 CHAPTER I. INTRODUCTION

topology. It is easy to see that if X is a subset of B(H), then the intersection of all vonNeumann algebras M ⊂ B(H) containing X is again a von Neumann algebra, called the vonNeumann algebra generated by X. Theorem I.1.1 is a cornerstone result of von Neumann,known as von Neumann double commutant theorem, asserting that the von Neumannalgebra generated by a subset X of B(H) can be characterized in a purely algebraic way.The commutant X ′ of a subset X of B(H) is the set of y ∈ B(H) commuting with everyelement of X. The double commutant X ′′ of X is just the commutant of X ′.

Theorem I.1.1. The von Neumann algebra generated by a subset X of B(H) containingthe unit and closed with respect to taking adjoints coincides with the double commutant X ′′

of X.

A faithful normal trace on a von Neumann algebra M is a linear functional τ on Msuch that:

• τ(x∗x) ≥ 0 for every x ∈M (τ is positive);

• τ(x∗x) = 0 implies x = 0 (τ is faithful);

• τ(xy) = τ(yx) for every x, y ∈M (τ is tracial);

• τ(1) = 1 (τ is unital)

• τ is continuous on the unit ball of M with respect to the weak operator topology (τis normal).

A (finite) von Neumann algebra endowed with a distinguished trace will be called atracial von Neumann algebra. A tracial von Neumann algebra is always finite as in [92,Definition 6.3.1]. Conversely any finite von Neumann algebra faithfully represented on aseparable Hilbert space has a faithful normal trace by [92, Theorem 8.2.8].

The center Z(M) of a von Neumann algebra M ⊂ B(H) is the subalgebra of Mconsisting of the operators in M commuting with any other element of M . A von Neumannalgebra M is called a factor if its center is as small as possible, i.e. it contains only thescalar multiples of the identity. A finite factor has a unique faithful normal trace (see [92,Theorem 8.2.8]). Moreover any faithful unital tracial positive linear functional on a finitefactor is automatically normal, and hence coincides with its unique faithful normal trace.

Example I.1.2. If Hn is a Hilbert space of finite dimension n, then B(Hn) is a finite factordenoted by Mn(C) isomorphic to the algebra of n× n matrices with complex coefficients.The unique trace on Mn(C) is the usual normalized trace of matrices.

It is a consequence of the type classification of finite factors (see [92, Section 6.5]) thatthe ones described in Example I.1.2 are the unique examples of finite factors that are finitedimensional as vector spaces.

I.1. VON NEUMANN ALGEBRAS AND II1 FACTORS 3

Definition I.1.3. A II1 factor is an infinite-dimensional finite factor.

Theorem I.1.4 is a cornerstone result of Murray and von Neumann (see [116, TheoremXIII]), offering a characterization of II1 factors within the class of finite factors.

Theorem I.1.4. If M is a II1 factor, then M contains, for every natural number n, aunital copy of Mn(C), i.e. there is a trace preserving *-homomorphism from Mn(C) to M .

It follows from weak continuity of the trace and the type classification of finite factorsthat if M is a finite factor, then the following statements are equivalent:

1. M is a II1 factor;

2. the trace τ of M attains on projections all the real values between 0 and 1.

The unique trace τ on a finite factor M allows to define the Hilbert-Schmidt norm ‖·‖2on M , by ||x||2 = τ(x∗x)

12 . The Hilbert-Schmidt norm is continuous with respect to the

operator norm ‖·‖ inherited from B(H). A finite factor is called separable if it is separablewith respect to the topology induced by the Hilbert-Schmidt norm.

Let us now describe one of the most important constructions of II1 factors. If Γ isa countable discrete group then the complex group algebra CΓ is the complex algebra offormal finite linear combinations

λ1γ1 + · · · + λkγk

of elements of γ with coefficients from C. Any element of CΓ can be written as∑

γ

aγγ,

where (a)γ∈Γ is a family of complex numbers all but finitely many of which are zero. Sumand multiplication of elements of CΓ are defined by

(∑

γ

aγγ

)+

(∑

γ

bγγ

)=∑

γ

(aγ + bγ) γ

and (∑

γ

aγγ

)(∑

γ

bγγ

)=∑

γ

∑

ρρ′=γ

aρbρ′

γ.

Consider now the Hilbert space ℓ2(Γ) of square-summable complex-valued functions on Γ.Each γ ∈ Γ defines a unitary operator λγ on ℓ2(Γ) by:

λγ(f)(x) = f(γ−1x).

The function γ → λγ extends by linearity to an embedding of CΓ into the algebra B(ℓ2(Γ)

)

of bounded linear operators on ℓ2(Γ).


Definition I.1.5. The weak closure of CΓ (identified with a subalgebra of B(ℓ2(Γ)

)) is a

von Neumann algebra denoted by LΓ and called the group von Neumann algebra of Γ.

The group von Neumann algebra LΓ is canonically endowed with the trace τ obtainedby extending by continuity the condition:

τ

(∑

γ

aγγ

)= a1Γ ,

for every element∑

γ aγγ of CΓ.

Exercise I.1.6. Assume that Γ is an ICC group, i.e. every nontrivial conjugacy class of Γis infinite. Show that LΓ is a II1 factor.

Exercise I.1.7. Show that the free group Fn on n ≥ 2 generators and the group Sfin∞ of

finitely supported permutations of a countable set are ICC.

It is a milestone result of Murray and von Neumann from [116] (see also Theorem6.7.8 of [92]) that the group factors associated with the free group F2 and, respectively,the group Sfin

∞ are nonisomorphic. It is currently a major open problem in the theoryof II1 factors to determine whether free groups over different number of generators haveisomorphic associated factors.

Connes’ embedding conjecture asserts that any separable II1 factor can be approximatedby finite dimensional factors, i.e. matrix algebras. More precisely:

Conjecture I.1.8 (Connes, 1976). If M is a separable II1 factor with trace τM then forevery ε > 0 and every finite subset F of M there is a function Φ from M to a matrix algebraMn(C) that on F approximately preserves the operations and the trace. This means thatΦ(1) = 1 and for every x, y ∈ F :

• ‖Φ(x+ y) − (Φ(x) + Φ(y))‖2 < ε;

• ‖Φ(xy) − Φ(x)Φ(y)‖2 < ε;

•

∣∣τM(x) − τMn(C) (Φ(x))∣∣ < ε.

I.2 Voiculescu’s free entropy

Connes’ embedding conjecture is related to many other problems in pure and appliedmathematics and can be stated in several different equivalent ways. The simplest andmost practical way is probably through Voiculescu’s free entropy.

Consider a random variable which outcomes the set {1, . . . , n} with probabilities p1, . . . , pn.Observe that its Shannon’s entropy, −∑ pi log(pi), can also be constructed through thefollowing procedure:

I.2. VOICULESCU’S FREE ENTROPY 5

1. Call microstate any function f from {1, 2, . . . , N} to {1, 2, . . . , n}. A microstate fε-approximates the discrete distribution p1, . . . pn if for every i ∈ {1, 2, . . . , n}

∣∣∣∣∣

∣∣f−1(i)∣∣

N− pi

∣∣∣∣∣ < ε.

Denote the number of such microstates by Γ(p1, . . . , pn, N, ε).

2. Take the limit of

N−1 log |Γ(p1, . . . , pn, N, ε)|,as N → ∞

3. Finally take the limit as ε goes to zero.

One can show that the result is just the opposite of the Shannon entropy, that is,∑i∈n pi log pi. Over the early 1990s, Voiculescu, in part motivated by the isomorphism

problem of whether the group von Neumann algebras associated to different free groups areisomorphic or not, realized that this construction could be imitated in the noncommutativeworld of II1 factors.

1. Microstates are self-adjoint matrices, instead of functions between finite sets. For-mally, let ε,R > 0, m,k ∈ N, and X1, . . . ,Xn be free random variables∗ on a II1-factor M . We denote ΓR(X1, . . . ,Xn;m,k, ε) the set of (A1, . . . , An) ∈ (Mk(C)sa)n

such that ‖Am‖ ≤ R and

|tr(Ai1 · · ·Aim) − τ(Xi1 · · ·Xim)| < ε

for every 1 ≤ m ≤ p and (i1, . . . , im) ∈ {1, 2, . . . , n}m.

2. The discrete measure is replaced by the Lebesgue measure λ on (Mk(C)sa)n.

3. The limits are replaced by suitable limsups, sups, and infs.

Specifically, set

χR(X1, . . . ,Xn;m,k, ε) := log λ(ΓR(X1, . . . Xn;m,k, ε)),

χR(X1, . . . Xn;m, ε) := lim supk→∞

(k−2χR(X1, . . . Xn;m,k, ε) + 2−1n log(k)),

χR(X1, . . . Xn) := inf{χR(X1, . . . Xn;m, ε) : m ∈ N, ε > 0},∗Freeness is the analogue of independence in the noncommutative framework of II1-factors. For a formal

definition we refer the reader to [149].


Finally, define entropy of the variables X1, . . . ,Xn the quantity

χ(X1, . . . Xn) := sup{χR(X1, . . . Xn) : R > 0}.

The factor k−2 instead of k−1 comes from the normalization. The addend 2−1n log(k) isnecessary, since otherwise χR(X1, . . . Xn;m, ε) would always be equal to −∞.

It is not clear why this construction should give entropy different from −∞. In-deed Voiculescu himself proved that it is −∞ when X1, . . . ,Xn are linearly dependent(see [150, Proposition 3.6]). A necessary condition to have χ(X1, . . . ,Xn) > −∞ is thatΓR(X1, . . . ,Xn,m, k, ε) is not empty for some k, i.e. the finite subset X = {X1, . . . ,Xn} ofMsa has microstates. This requirement turns to be equivalent to the fact that M satisfiesConnes’ embedding conjecture.

Theorem. Let M be a II1 factor. The following conditions are equivalent

1. Every finite subsets X ⊆Msa has microstates.

2. M verifies Connes’ embedding conjecture.

I.3 History of Connes’ embedding conjecture

Connes’ embedding conjecture had its origin in Connes’ sentence: “We now constructan approximate imbedding of N in R. Apparently such an imbedding ought to exist forall II1 factors because it does for the regular representation of free groups. However theconstruction below relies on condition 6” (see [43, page 105]). This seemingly innocentobservation received in the first fifteen years after having been formulated relatively littleattention. The situation changed drastically in the early 1990s, when two fundamentalpapers appeared on the scene: the aforementioned [150] where the Connes embeddingconjecture is used to define free entropy, and [102] where Eberhard Kirchberg obtainedseveral unexpected reformulations of the Connes embedding conjecture very far from itsoriginal statement, such as the fact that the maximal and minimal tensor products of thegroup C*-algebra of the free group on infinitely many generators coincide:

C∗(F ) ⊗min C∗(F ) = C∗(F ) ⊗max C

∗(F ). (I.1)

What is most striking in this equivalence is that Connes’ embedding conjecture concernsthe class of all separable II1 factors, while ((I.1)) is a statement about a single C*-algebra.Even more surprising is the fact that the equivalence of these statements can be proved atopological way as shown in [84].

Following the aforementioned papers by Voiculescu and Kirchberg, a series of papersfrom different authors proving various equivalent reformulations of Connes’ embeddingconjecture appeared, such as [16],[120], [41], [85], [64] contributing to arouse the inter-est around this conjecture. In particular Florin Radulescu showed in [132] that Connes’

I.4. HYPERLINEAR AND SOFIC GROUPS 7

embedding conjecture is equivalent to some noncommutative analogue of Hilbert’s 17thproblem. This in turn inspired work of Klep and Schweighofer, who proved in [104] apurely algebraic reformulation of Connes’ embedding conjecture (see also the more recentwork [90]). In [25] it is observed that Connes’ embedding conjecture could theoreticallybe checked by an algorithm if a certain problem of embedding Hilbert spaces with someadditional structure into the Hilbert space L2(M, τ) associated to a II1 factor has a posi-tive solution. Another computability-theoretical reformulation of the Connes’ embeddingconjecture has more recently been proved in [74]. Other very recent discoveries include thefact that Connes’ embedding conjecture is related to Tsirelson’s problem, a major openproblem in Quantum Information Theory (see [67], [89], and [123]) and that it is connectedto the “minimal” and “commuting” tensor products [97] of some group operator systems[35], [65], [63].

I.4 Hyperlinear and sofic groups

In [133] Radulescu considered the particular case of Connes’ embedding conjecture for II1factors arising as group factors LΓ of countable discrete ICC groups. A countable discreteICC group Γ is called hyperlinear if LΓ verifies Connes’ embedding conjecture. As we willsee in Section II.2 and Section II.6 the notion of hyperlinear group admits several equivalentcharacterizations, as well as a natural generalization to the class of all countable groups.The notion of hyperlinear group turned out to be tightly connected with the notion of soficgroups. Sofic groups are a class of countable discrete groups introduced by Gromov in [80](even though the name “sofic” was coined by Weiss in [152]). A group is sofic if, looselyspeaking, it can be locally approximated by finite permutation groups Sn up to an errormeasured in terms of the Hamming metric Sn (see Section II.1). Elek and Szabo in [55]showed that every sofic group is hyperlinear. Gromov’s motivation to introduce the notionof sofic groups came from an open problem in symbolic dynamics known as Gottschalk’ssurjunctivity conjecture: Suppose that Γ is a countable group and A is a finite set. Denoteby AΓ the set of Γ-sequences of elements of A. The product topology on AΓ with respectto the discrete topology on A is compact and metrizable. The Bernoulli shift of Γ withalphabet A is the left action of Γ on AΓ defined by

ρ · (aγ)γ∈Γ =(aρ−1γ

)γ∈Γ .

A continuous function f : AΓ → AΓ is equivariant if it preserves the Bernoulli action,i.e. f (ρ · x) = ρ · f(x), for every x ∈ AΓ and ρ ∈ Γ. Conjecture I.4.1 was proposed byGottschalk in [77] and it is usually referred to as Gottschalk’s surjunctivity conjecture.

Conjecture I.4.1. Suppose that Γ is a discrete group and A is a finite set. If f : AΓ → AΓ

is a continuous injective equivariant function, then f is surjective.


It is currently an open problem to determine whether Gottschalk’s surjunctivity conjec-ture holds for all countable discrete groups. Gromov proved in [80] using graph-theoreticalmethods that sofic groups satisfy Gottschalk’s surjunctivity conjecture. Another proof wasobtained Kerr and Li in [100] as an application of the theory of entropy for actions of soficgroups developed by Bowen, Kerr, and Li (see [15], [100], and [101]). The countable discretegroups satisfying Gottschalk’s surjunctivity conjecture are sometimes called surjunctive.An example of a monoid that does not satisfy the natural generalization of surjunctivityfor monoids has been provided in [33]. More information about Gottschalk’s surjunctivityconjecture and surjunctive groups can be found in the monograph [31].

Since Gromov’s proof of Gottschalk’s conjecture for sofic groups, many other openproblems have been settled for sofic or hyperlinear groups such as the Kervaire-Laudenbachconjecture (see Section II.8), Kaplansky’s direct finiteness conjecture (see Section I.5 andSection II.10), and the algebraic eigenvalues conjecture (see Section II.12). This showedhow deep and fruitful the notion of hyperlinear and sofic groups are, and contributed tobring considerable attention to the following question which is strikingly still open:

Open Problem I.4.2. Is there any countable discrete group Γ that is not sofic or hyper-linear?

I.5 Kaplansky’s direct finiteness conjecture

Suppose that Γ is a countable discrete group, and consider the complex group algebra CΓdefined in Section I.1. Kaplanksi showed in [93] that CΓ is a directly finite ring. Thismeans that if a, b ∈ CΓ are such that ab = 1 then ba = 1. In [19] Burger and Valette gave ashort proof of Kaplansky’s result by means of the group von Neumann algebra constructionintroduced in Section I.1. Indeed, one can regard CΓ as a subalgebra of the group vonNeumann algebra LΓ and prove that LΓ is directly finite using analytic methods. This isthe content of Theorem I.5.1.

Theorem I.5.1. If M is a von Neumann algebra endowed with a faithful finite trace τ ,then M is a directly finite algebra.

We now prove this theorem. Let M be a von Neumann algebra and τ be a faithfulnormalized trace on M . If x, y ∈ M are such that xy = 1 then yx ∈ M is an idempotentelement such that

τ(yx) = τ(xy) = τ(1) = 1.

It is thus enough to prove that if e ∈M is an idempotent element such that τ(e) = 1 thene = 1. This is equivalent to the assertion that if e ∈M is an idempotent element such thatτ(e) = 0 then e = 0. The latter statement is proved in Lemma I.5.2 (see [19, Lemma 2.1]).

Lemma I.5.2. If M is a von Neumann algebra endowed with a faithful finite trace τ ande ∈M is an idempotent such that τ(e) = 0, then e = 0.

I.5. KAPLANSKY’S DIRECT FINITENESS CONJECTURE 9

Proof. The conclusion is obvious if e is a self-adjoint idempotent element (i.e. a projection).In fact in this case

τ(e) = τ (e∗e) = 0

implies e = 0 by faithfulness of τ . In order to establish the general case it is enough toshow that if e ∈ M is idempotent, then there is a self-adjoint invertible element z of Msuch that f = ee∗z−1 is a projection and τ(e) = τ(f). Define

z = 1 + (e∗ − e)∗ (e∗ − e) .

Observe that z is an invertible element (see [14, II.3.1.4]) commuting with e. It is notdifficult to check that f = ee∗z−1 has the required properties.

The construction of the complex group algebra of Γ introduced in Section I.1 can begeneralized replacing C with an arbitrary field K. One thus obtains the group K-algebraKΓ of Γ consisting of formal finite linear combinations

k1γ1 + . . . + knγn

of elements of Γ with coefficients in K. As before a typical element a of KΓ can be denotedby ∑

γ

aγγ

where the coefficients aγ ∈ K are zero for all but finitely many γ ∈ Γ. The operations onKΓ are defined by (∑

γ

aγγ

)+

(∑

γ

bγγ

)=∑

γ

(aγ + bγ) γ

and (∑

γ

aγγ

)(∑

γ

bγγ

)=∑

γ

∑

ρρ′=γ

aρbρ′

γ.

Conjecture I.5.3 is due to Kaplansky and usually referred to as direct finiteness conjecture.

Conjecture I.5.3. The algebra KΓ is directly finite, i.e. any one-side invertible elementof KΓ is invertible.

Kaplansky’s direct finiteness conjecture was established for residually amenable groupsin [118] and then for sofic groups in [54]. Section II.10 contains a proof of Conjecture I.5.3for sofic groups involving the notion of rank ring and rank ultraproduct of rank rings. Itis a well know fact (see Observation 2.1 in [128]) that any field embeds as a subfield of anultraproduct of finite fields (see Section 2 of for the definition of ultraproduct of fields).It is not difficult to deduce from this observation that a group Γ satisfies Kaplansky’s


direct finiteness conjecture as soon as KΓ is directly finite for every finite field K (this factwas pointed out to us by Vladimir Pestov). This allows one to deduce that Gottschalk’sconjecture is stronger than Kaplansky’s direct finiteness conjecture. In fact suppose that Γis a group satisfying Gottschalk’s conjecture and K is a field that, without loss of generality,we can assume to be finite. Consider the Bernoulli action of Γ with alphabet K. We willdenote an element (a)γ∈Γ of KΓ by

∑γ aγγ, and regard the group algebra KΓ as a subset

of KΓ. Defining (∑

γ

aγγ

)·(∑

γ

bγγ

)=∑

γ

∑

ρρ′=γ

aρbρ′

γ

for∑

γ aγγ ∈ KΓ and∑

γ bγγ ∈ KΓ one obtains a right action of KΓ on KΓ that extendsthe multiplication operation in KΓ and commutes with the left action of Γ on KΓ. Supposethat a, b ∈ KΓ are such that ab = 1Γ. Define the continuous equivariant map f : KΓ → KΓ

by f(x) = x · a. It follows from the fact that b is a right inverse of a that f is injective.Since Γ is assumed to satisfy Gottschalk’s conjecture, f must be surjective. In particularthere is x0 ∈ KΓ such that

x0 · a = f(x0) = 1Γ.

It follows that a has a left inverse. A standard calculation allows one to conclude that a isinvertible with inverse b.

I.6 Kaplansky’s group ring conjectures

Conjecture I.5.3 is only one of several conjectures concerning the group algebra KΓ for acountable discrete torsion-free group Γ attributed to Kaplansky. Here is the complete list:

• Zero divisors conjecture: KΓ has no zero divisors;

• Nilpotent elements conjecture: KΓ has no nilpotent elements;

• Idempotent elements conjecture: the only idempotent elements of KΓ are 0 and 1;

• Units conjecture: the only units of KΓ are kγ for k ∈ K \{0} and γ ∈ Γ;

• Trace of idempotents conjecture: if b is an idempotent element of KΓ then thecoefficient b1Γ corresponding to the identity 1Γ of Γ belongs to the prime field K0 ofK (which is the minimum subfield of K).

All these conjectures are to this day still open, apart from the last one which hasbeen established by Zalesskii in [153]. We will present now a proof of the particular caseof Zalesskii’s result when K is a finite field of characteristic p. Recall that a trace onKΓ is a K-linear map τ : KΓ → K such that τ(ab) = τ(ba) for every a, b ∈ KΓ. If n

I.6. KAPLANSKY’S GROUP RING CONJECTURES 11

is a nonnegative integer and a ∈ KΓ, define τn(a) to be the sum of the coefficients ofa corresponding to elements of Γ of order pn. In particular τ0(a) is the coefficient of acorresponding t the identity element 1Γ of Γ.

Exercise I.6.1. Verify that τn is a trace on KΓ.

Exercise I.6.2. Show that if K is a finite field of characteristic p and τ is any trace onKΓ, then

τ ((a+ b)p) = τ(ap) + τ(bp), (I.2)

for every a, b ∈ KΓ. Thus by induction

τ(ap) =∑

γ

apγτ(γp).

The identity (I.2) can be referred to as “Frobenius under trace” in analogy with thecorresponding identity for elements of a field of characteristic p. Suppose now that e ∈ KΓis an idempotent element. We want to show that τ(e) belongs to the prime field K0 of K.To this purpose it is enough to show that τ(e)p = τ(e). For n ≥ 1 we have, by ExerciseI.6.1 and Exercise I.6.2, denoting by |γ| the order of an element γ of Γ:

τn(e) = τn(ep)

=∑

γ

epγτn(γp)

=∑

|γ|=pn+1

epγ

=

∑

|γ|=pn+1

eγ

p

= τn+1(e)p.

On the other hand:

τ0(e) = τ0(ep)

=∑

γ

epγτ0(γp)

=∑

|γ|=1

epγ +∑

|γ|=p

epγ

= τ0(e)p + τ1(e)

p.

From these identities it is easy to prove by induction that

τ0(e) = τ0(e)p + τn(e)p

n

,


for every n ∈ N. Since e has finite support, there is n ∈ N such that τn(e) = 0. Thisimplies that τ0(e) = τ0(e)

p and hence τ0(e) ∈ K0.The proof of the general case of Zalesskii’s theorem can be inferred from the particular

case presented here. The details can be found in [19].

Chapter II

Sofic and hyperlinear groups

Martino Lupini

II.1 Definition of sofic groups

A length function ℓ on a group G is a function ℓ : G→ [0, 1] such that for every x, y ∈ G:

• ℓ(xy) ≤ ℓ(x) + ℓ(y);

• ℓ(x−1) = ℓ(x);

• ℓ(x) = 0 if and only if x is the identity 1G of G.

A length function is called invariant if it is moreover invariant by conjugation. Thismeans that for every x, y ∈ G

ℓ(xyx−1) = ℓ(x)

or equivalentlyℓ(xy) = ℓ(yx).

A group endowed with an invariant length function is called an invariant length group. IfG is an invariant length group with invariant length function ℓ, then the function

d : G×G→ [0, 1]

defined by d(x, y) = ℓ(xy−1) is a bi-invariant metric on G. This means that d is a metricon G such that left and right translations in G are isometries with respect to d. Converselyany bi-invariant metric d on G gives rise to an invariant length function ℓ on G by

ℓ(x) = d(x, 1G).

This shows that there is a bijective correspondence between invariant length functions andbi-invariant metrics on a group G. If Γ is any group, then the function

13

14 CHAPTER II. SOFIC AND HYPERLINEAR GROUPS

ℓ0(x) =

{0 if x = 1G,

1 otherwise;

is an invariant length function on Γ, called the trivial invariant length function. In thefollowing any discrete group will be regarded as an invariant length group endowed withthe trivial invariant length function.

Consider for n ∈ N the group Sn of permutations over the set n = {0, 1, . . . , n− 1}.The Hamming invariant length function ℓ on Sn is defined by

ℓSn(σ) =1

n|{i ∈ n : σ(i) 6= i} .

Exercise II.1.1. Verify that ℓSn is in fact an invariant length function on Sn.

The bi-invariant metric on Sn associated with the invariant length function ℓSn will bedenoted by dSn .

Definition II.1.2. A countable discrete group Γ is sofic if for every ε > 0 and every finitesubset F of Γ \{1Γ} there is a natural number n and a function Φ : Γ → Sn such thatΦ (1Γ) = 1Sn and for every g, h ∈ F \{1Γ} :

• dSn (Φ(gh),Φ(g)Φ(h)) < ε;

• ℓSn (Φ(g)) > r(g) where r(g) is a positive constant depending only on g.

This local approximation property can be reformulated in terms of embedding into a(length) ultraproduct of the permutation groups. The product

∏

n∈NSn

is a group with respect to the coordinatewise multiplication. Fix a free ultrafilter U overN (see B for an introduction to ultrafilters). Define the function

ℓU :∏

n∈NSn → [0, 1]

by

ℓU ((σn)n∈N) = limn→U

ℓSn(σn).

It is not hard to check

NU =

{x ∈

∏

n∈NSn : ℓU(x) = 0

}

II.1. DEFINITION OF SOFIC GROUPS 15

is a normal subgroup of∏

n Sn. The quotient of∏

n Sn by NU is denoted by

∏USn

and called the ultraproduct relative to the free ultrafilter U of the sequence (Sn)n∈N ofpermutation groups regarded as invariant length groups.

Exercise II.1.3. Show that if (σn)n∈N, (τn)n∈N ∈ ∏n Sn belong to the same coset of NUthen ℓU ((σn)n∈N) = ℓU

((τn)n∈N

).

Exercise II.1.3 shows that the function ℓU passes to the quotient inducing a canonicalinvariant length function on

∏U Sn still denoted by ℓU . If x ∈∏U Sn belongs to the coset

of NU associated with (σn)n∈N ∈ ∏n Sn, then (σn)n∈N is called a representative sequencefor the element x. It is not difficult to reformulate the notion of sofic group in term ofexistence of an embedding into

∏U Sn.

Exercise II.1.4. Suppose that Γ is a countable discrete group. Show that the followingstatements are equivalent:

1. Γ is sofic;

2. there is an injective group *-homomorphism Φ : Γ →∏U Sn for every free ultrafilter

U over N ;

3. there is an injective group *-homomorphism Φ : Γ → ∏U Sn for some free ultrafilter

U over N .

Hint. For 1. ⇒ 2. observe that the hypothesis implies that there is a sequence (Φn)n∈N ofmaps from Γ to Sn such that Φn (1Γ) = 1Sn and for every g, h ∈ Γ \{1Γ}

limn→+∞

dSn (Φn (gh) ,Φn(h)Φn (g)) = 0

andlim infn→+∞

ℓSn (Φn (g)) ≥ r (g) > 0.

Define Φ : Γ →∏U Sn sending g to the element of

∏U Sn having (Φn (g))n∈N as representa-

tive sequence. For 3. ⇒ 1. observe that if Φ : Γ → ∏U Sn is an injective *-homomorphism

and for every g ∈ G(Φn (g))n∈N

is a representative sequence of Φ (g) then the maps Ψn = Φn (1Γ)−1 Φn (g) satisfy thefollowing properties: Ψn (1Γ) = 1Sn and for every g, h ∈ Γ \{1Γ}

limn→U

dSn (Ψn (gh) ,Ψn (g) Ψn(h)) = 0


andlimn→U

ℓSn (Ψn (g)) ≥ ℓU (Φ (g)) .

If F ⊂ Γ is finite and ε > 0 then the maps Ψn for n large enough witness the condition ofsoficity of Γ relative to F and ε > 0.

In view of the fact that a (countable, discrete) group is sofic if and only embeds in∏

U Snfor some free ultrafilter U , the (length) ultraproducts of the sequence of permutations arecalled universal sofic groups (see [55], [128], [147], [112], [126]).

Universal sofic groups are not separable. This fact follows from a well known argument.Two functions f, g : N → N are eventually distinct if they coincide only on finitely manyn’s.

Exercise II.1.5. Suppose that h : N → N is an unbounded function. Show that there is afamily F of size continuum of pairwise eventually distinct functions from N to N such thatf (n) ≤ h (n) for every f ∈ F .

Hint. For every subset A of N define fA : N → N by

fA (n) =∑

k<n

χA (n) 2k

where χA denotes the characteristic function of A. Observe that

F0 = {fA : A ⊂ N}

is a family of size continuum of pairwise eventually distinct functions such that f (n) ≤ 2n

for every n ∈ N.

Exercise II.1.6. Show that for any free ultrafilter U there is a subset X of∏

U Sn of sizecontinuum such that every two distinct elements of X have distance one. Conclude thatany dense subset of

∏U Sn has the cardinality of the continuum.

Hint. By Exercise II.1.5 there is a family F of size continuum of pairwise eventually distinctfunctions such that f (n) ≤ √

n for every n ∈ N and f ∈ F . For every f ∈ F consider theelement xf of

∏U Sn having (

σf(n)n

)n∈N

as representative sequence, where σn is any cyclic permutation of n. Show that

X := {xf : f ∈ F}

has the required property.An amplification argument of Elek and Szabo (see [55]) shows that the condition of

soficity is equivalent to the an apparently stronger approximation property, which is dis-cussed in the following Exercise II.1.8.

II.1. DEFINITION OF SOFIC GROUPS 17

Definition II.1.7. Suppose that G,H are invariant length groups, F is a subset of G,and ε is a positive real number. A function Φ : G → H is called an (F, ε)-approximatemorphism if Φ (1G) = 1H and for every g, h ∈ F :

•

∣∣ℓH(Φ(gh)Φ(h)−1Φ(g)

)∣∣ < ε;

• |ℓH (Φ(g)) − ℓG(g)| < ε.

Exercise II.1.8. Prove that a countable discrete group Γ is sofic if and only if for everypositive real number ε and every finite subset F of Γ there is a natural number n andan (F, ε)-approximate morphism from Γ to Sn, where Γ is regarded as an invariant lengthgroup with respect to the trivial invariant length function.

Hint. If n, k ∈ N and σ ∈ Sn consider the permutation σ⊗k of the set [n]k of k-sequencesof elements of n defined by

σ⊗k (i1, . . . , ik) = (σ(i1), . . . , σ(ik)) .

Identifying the group of permutations of

k times︷︸︸︷n× · · · × n with Snk , the function

σ 7→ σ⊗k

defines a group *-homomorphism from Sn to Snk such that

1 − ℓSnk

(σ⊗k

)= (1 − ℓSn (σ))k .

Using Exercise II.1.8 one can express the notion of soficity in terms of length-preservingembedding into ultraproducts of permutations groups.

Exercise II.1.9. Suppose that Γ is a countable discrete group regarded as an invariantlength group endowed with the trivial invariant length. Show that the following statementsare equivalent:

• Γ is sofic;

• there is a length-preserving *-homomorphism Φ : Γ →∏U Sn for every free ultrafilter

U over N ;

• there is an length-preserving *-homomorphism Φ : Γ → ∏U Sn for some free ultrafilter

U over N .


Hint. Follow the same steps as in the proof of Exercise II.1.9, replacing the condition givenin the definition of sofic group with the equivalent condition expressed in Exercise II.1.8.

In a number of cases it is useful to consider approximate morphisms satisfying conve-nient additional properties. An example of such morphisms with additional properties isconsidered in the following exercise, originally proved in [56, Lemma 2.1].

Exercise II.1.10. Suppose that Γ is a countable discrete sofic group, F is a finite sym-metric subset of Γ containing the identity element, and ε is a positive real number. Showthat one can find an (F, ε)-approximate morphism g 7→ σg from Γ to Sn for some n ∈ Nsuch that, for every g ∈ F , σg has no fixed points and σg−1 = σ−1

g .

Hint. Fix η > 0 small enough and set F = F · F . By Exercise II.1.8 one can consider an(F , η)-approximate morphism g 7→ τg from Γ to Sn for some n ∈ N. For any nonidentity

element g of F consider the largest subset Ag of n such that τg−1τg is the identity on Ag

and τg has no fixed points on Ag. Verify that τg defines a bijection between Ag and Ag−1 .Define an approximate morphism g 7→ σg from Γ to S2n in the following way. Identify

S2n with the group of premutations of n × 2. For any nonidentity element g of F fixa bijection φg from Ag\Ag−1 × 2 to Ag−1\Ag × 2 and a fixed-point free involution ψg of(n\(Ag ∪Ag−1)

)× 2 such that φg−1 = φ−1

g and ψg−1 = ψ−1g . Define then

τg (i, j) =

(σg (i) , j) if i ∈ Ag,φg (i, j) if i ∈ Ag\Ag−1 ,φg−1 (i, j) if i ∈ Ag−1\Ag,ψg (i, j) otherwise.

Verify that for η small enough the assignment g 7→ τg is an (F, ε)-approximate morphismwith the required extra properties.

II.2 Definition of hyperlinear groups

Recall that Mn(C) denotes the tracial von Neumann algebra of n × n matrices over thecomplex numbers. The normalized trace τ of Mn(C) is defined by

τ ((aij)) =1

n

n∑

i=1

aii.

The Hilbert-Schmidt norm ‖x‖2 on Mn(C) is defined by

‖x‖2 = τ(x∗x)12 .

II.2. DEFINITION OF HYPERLINEAR GROUPS 19

An element x of Mn(C) is unitary if x∗x = xx∗ = 1. The set Un of unitary elementsof Mn(C) is a group with respect to multiplication. The Hilbert-Schmidt invariant lengthfunction on Un is defined by

ℓUn(u) =1

2‖u− 1‖2 .

Exercise II.2.1. Show that ℓUn is an invariant length function on Un such that ℓUn(u)2 =12 (1 − Re (τ(u))).

Hyperlinear groups are defined exactly as sofic groups, where the permutation groupswith the Hamming invariant length function are replaced by the unitary groups endowedwith the Hilbert-Schmidt invariant length function.

Definition II.2.2. A countable discrete group Γ is hyperlinear if for every ε > 0 and everyfinite subset F of Γ \{1Γ} there is a natural number n and a function Φ : Γ → Un suchthat Φ (1Γ) = 1Un and for every g, h ∈ F :

• dUn (Φ(gh),Φ(g)Φ(h)) < ε;

• ℓUn (Φ(g)) > r(g) where r(g) is some positive constant depending only on g.

As before this notion can be equivalently reformulated in terms of embedding intoultraproducts. If U is a free ultrafilter over N the ultraproduct

∏U Un of the unitary

groups regarded as invariant length groups is the quotient of∏

n Un with respect to thenormal subgroup

NU =

{(un)n∈N : lim

n→UℓUn (un) = 0

}

endowed with the invariant length function

ℓU ((un)NU ) = limn→U

ℓ(un).

Exercise II.2.3. Suppose that Γ is a countable discrete group. Show that the followingstatements are equivalent:

• Γ is hyperlinear;

• there is an injective *-homomorphism Φ : Γ → ∏U Un for every free ultrafilter U over

N ;

• there is an injective *-homomorphism Φ : Γ →∏U Un for some free ultrafilter U over

N .


As in the case of sofic groups, (length) ultraproducts of sequences of unitary groupscan be referred to as universal hyperlinear groups in view of Exercise II.2.3. ExerciseIII.2.1 in Section III.2 shows that the same conclusion of Exercise II.2.3 holds for (length)ultrapowers of the unitary group of the hyperfinite II1 factor (see Definition III.1.2).

If σ is a permutation over n denote by Pσ the permutation matrix associated with σ,acting as σ on the canonical basis of Cn. Observe that Pσ is a unitary matrix and thefunction

σ 7→ Pσ

is a *-homomorphism from Sn to Un. Moreover

τ (Pσ) = 1 − ℓSn(σ)

and hence

ℓUn (Pσ)2 =1

2ℓSn(σ). (II.1)

It is not difficult to deduce from this that any sofic group is hyperlinear. This is the contentof Exercise II.2.4.

Exercise II.2.4. Fix a free ultrafilter U over N . Show that the function

(σn)n∈N 7→ (Pσn)n∈N

from∏

n Sn to∏

n Un induces an algebraic embedding of∏

U Sn into∏

U Un.

Proposition is an immediate consequence of Exercise II.2.4, together with II.1.4, andII.2.3.

Proposition II.2.5. Every countable discrete sofic group is hyperlinear.

It is currently and open problem to determine whether the notions of sofic and hyper-linear group are actually distinct, since no example of a nonsofic groups is known.

Exercise II.2.6. Show that for any free ultrafilter U there is a subset X of∏

U Un of sizecontinuum such that every pair of distinct elements of X has distance 1√

2.

Hint. Proceed as in Exercise II.1.6, where for every n ∈ N the cyclic permutation σn isreplaced with the unitary permutation matrix Pσn associated with σn. Recall the relationbetween the Hamming length function and the Hilbert-Schmidt length functions given byEquation II.1.

It follows from II.2.6 that a dense subset of∏

U Un has the cardinality of the continuum.In particular the universal hyperlinear groups are not separable.

An amplification argument from [133] due to Radulescu (predating the analogous ar-gument for permutation groups of Elek and Szabo) shows that hyperlinearity is equivalent

II.2. DEFINITION OF HYPERLINEAR GROUPS 21

to an apparently stronger approximation property. In order to present this argument weneed to recall some facts about tensor product of matrix algebras.

If Mn(C) and Mm(C) are the algebras of, respectively, n × n and m × m matriceswith complex coefficients, then their tensor product (see Appendix A) can be canonicallyidentified with the algebra Mnm(C) of nm×nm matrices. Concretely if A = (aij) ∈Mn(C)and B = (bij) ∈Mm(C) then A⊗B is identified with the block matrix

a11B a12B . . . a1nBa21B a22B . . . a2nB

......

. . ....

an1B . . . . . . annB

.

It is easily verified thatτ (A⊗B) = τ(A)τ (B)

and A⊗B is unitary if both A and B are.

Exercise II.2.7. Suppose that u ∈ Un. If u is different from the identity, then the absolutevalue of the trace of (

u 00 1Un

)∈ U2n

is strictly smaller than 1.

Exercise II.2.8. If u ∈ Un then define recursively u⊗1 = u ∈ Un and u⊗(k+1) = u⊗k ⊗u ∈Unk+1 . Show that the function

u 7→ u⊗k

is a group *-homomorphism from Un to Unk such that

τ(u) = τ(u)k.

We can now state and prove the promised equivalent characterization of hyperlineargroups.

Proposition II.2.9. A countable discrete group Γ is hyperlinear if and only for everyfinite subset F of Γ and every positive real number ε there is a natural number n and an(F, ε)-approximate morphism from Γ to Un, where Γ is regarded as invariant length groupswith respect to the trivial length function.

Proof. Suppose that Γ is hyperlinear. If F is a finite subset of Γ and ε is a positive realnumber, consider the map Φ : Γ → Un obtained from F and ε as in the definition ofhyperlinear group. By Exercise II.2.7 after replacing Φ with the map

Γ → U2n

γ 7→(

Φ(γ) 00 1Un

)


we can assume that|τ (Φ(γ))| < 1

for every γ ∈ F . It is now easy to show using Exercise II.2.8 that the map

Γ → Unk

γ 7→ Φ(γ)⊗k

for k ∈ N large enough satisfies the requirements of the statement. The converse implicationis obvious.

As before we can deduce a characterization of hyperlinear groups in terms of length-preserving embeddings into ultraproducts of unitary groups.

Exercise II.2.10. Suppose that Γ is a countable discrete group regarded as an invariantlength group endowed with the trivial invariant length function. Show that the followingstatements are equivalent:

• Γ is hyperlinear;

• there is a length preserving *-homomorphism Φ : Γ →∏U Un for every free ultrafilter

U over N ;

• there is a length preserving *-homomorphism Φ : Γ → ∏U Un for some free ultrafilter

U over N .

We conclude this section with yet another reformulation of the definition of hyperlineargroup in terms of what are usually called microstates. Suppose that Γ is a finitely generatedgroup with finite generating set S = {g1, . . . , gm}. If n ∈ N and ε > 0 then an (n, ε)-microstate for Γ is a map Φ : Γ →Mk (C) for some k ∈ N such that for every word w in xiand x−1

i for i ≤ m of length at most n one has that

|τ (w (Φ (g1) , . . . ,Φ (gm)))| < ε whenever w (g1, . . . , gm) 6= 1Γ,

and|τ (w (Φ (g1) , . . . ,Φ (gm))) − 1| < ε whenever w (g1, . . . , gn) = 1Γ.

A countable discrete groups is hyperlinear if and only if it admits (n, ε)-microstates forevery n ∈ N and ε > 0. In order to verify that such a definition is equivalent to theprevious ones—and, in particular, does not depend on the choice of the generating setS—let us consider the relation R (x) defined by

max {‖xx∗ − 1‖2 , ‖x∗x− 1‖2} .

It is obvious that an element a in Mn (C) is a unitary if and only if R(a) = 0. Moreoversuch a relation has the following property, usually called stability : for every ε > 0 there is

II.3. CLASSES OF SOFIC AND HYPERLINEAR GROUPS 23

δ > 0 such that for every n ∈ N and every element a of Mn (C) if R(a) < δ then there isa unitary u ∈ Mn (C) such that ‖a− u‖2 < ε. In other words any approximate solutionto the equation R (x) = 0 is close to an exact solution. Such a property, which holds evenif one replaces Mn (C) with an arbitrary tracial von Neumann algebra, can be establishedby means of the polar decomposition of an element inside a von Neumann algebra; see [14,Section III.1.1.2].

Exercise II.2.11. Using stability of the relation defining unitary elements, verify that themicrostates formulation of hyperlinearity is equivalent to the original definition.

II.3 Classes of sofic and hyperlinear groups

A classical theorem of Cayley (see [29]) asserts that any finite group is isomorphic to agroup of permutations (with no fixed points) on a finite set. To see this just let the groupact on itself by left translation. This observation implies in particular that finite groups aresofic. This argument can be generalized to prove that amenable groups are sofic. Recallthat a countable discrete group Γ is amenable if for every finite subset F of Γ and for everyε > 0 there is an (H, ε)-invariant finite subset K of Γ, i.e. such that

|hK△K| < ε |K|

for every h ∈ H. Amenable groups were introduced in 1929 in [151] by von Neumann inrelation with his investigations upon the Banach-Tarski paradox. Since then they have beenthe subject of an intensive study from many different perspectives with recent applicationseven in game theory [23], [26]. More information about this topic can be found in themonographs [78] and [124].

Proposition II.3.1. Amenable groups are sofic.

Proof. Suppose that Γ is an amenable countable discrete group, F is a finite subset of Γ,and ε is a positive real number. Fix a finite

(F ∪ F−1, ε

)-invariant subset K of Γ. If γ ∈ F

then defineσγ(x) = γx

for x ∈ γ−1K ∩K and extended σγ arbitrarily to a permutation of K. Observe that thisdefines an (F, 2ε)-approximate morphism from G to the group SK of permutations of Kendowed with the Hamming length (which is isomorphic as invariant length group to Snwhere n is the cardinality of K). This concludes the proof that Γ is sofic.

If C is a class of (countable, discrete) groups, then a group Γ is locally embeddable intoelements of C if for every finite subset F of Γ there is a function Φ from Γ to a group T ∈ Csuch that Φ is nontrivial and preserves the operation on F , i.e. for every g, h ∈ F :

Φ(gh) = Φ(g)Φ(h),


andΦ(g) 6= 1, for all g 6= 1.

The group Γ is residually in C if moreover Φ is required to be a surjective *-homomorphism.Recall also that Γ is locally in C if every finitely generated subgroup of Γ belongs to C.It is clear that if Γ is either locally in C or residually in C, then in particular Γ is locallyembeddable into elements of C.

It is clear from the very definition that soficity and hyperlinearity are property con-cerning only finite subsets of the group. This is made precise in Exercise II.3.2.

Exercise II.3.2. Show that a group that is locally embeddable into sofic groups is sofic.The same is true replacing sofic with hyperlinear.

In particular Exercise II.3.2 shows that locally sofic and residually sofic groups are sofic.Groups that are locally embeddable into finite groups were introduced and studied in

[76] under the name of LEF groups. Since finite groups are sofic, Exercise II.3.2 impliesthat LEF groups are sofic. In particular residually finite groups are sofic. More gener-ally groups that are locally embeddable into amenable groups (LEA)—also called initiallysubamenable—are sofic. It is a standard result in group theory that free groups are resid-ually finite (see [128, Example 1.3]). Therefore the previous discussion implies that freegroups are sofic . Examples of hyperlinear and sofic groups that are not LEA have beenrecently constructed by Andreas Thom [145] and Yves de Cornulier [45], respectively. Wewill present these examples in Section II.5.

It should be now mentioned that it is to this day not know if there is any group whichis not sofic, nor if there is any group which is not hyperlinear.

Open Problem II.3.3. Are the classes of sofic and hyperlinear groups proper subclassesof the class of all countable discrete groups?

It is not even known if every one-relator group, i.e. a finitely presented group with onlyone relation, is necessarily sofic or hyperlinear. This is an open problem suggested by NateBrown. Similarly it is not known whether all Gromov hyperbolic groups are sofic. In factit is not known whether there is a Gromov hyperbolic groups that is not residually finite.An example of a monoid that does not satisfy the natural generalization of soficity formonoids is provided in [32].

II.4 Closure properties of the classes of sofic and hyperlinear

groups

The classes of sofic and hyperlinear groups have nice closure properties.

Proposition II.4.1. The class of sofic groups is closed with respect to the following oper-ations:

II.4. CLOSURE PROPERTIES OF THECLASSES OF SOFIC AND HYPERLINEARGROUPS 25

1. Subgroups;

2. Direct limits;

3. Direct products;

4. Inverse limits;

5. Extensions by amenable groups;

6. Free products;

7. Free products amalgamated over an amenable group;

8. HNN extension over an amenable group;

9. Graph product.

It is clear from the definition that soficity is a local property. Therefore a group is soficif and only if all its (finitely generated) subgroups are sofic. In particular subgroups of soficgroups are sofic. It immediately follows that the direct limit of sofic groups is sofic.

To see that the direct product of two sofic groups is sofic, suppose that Γ0,Γ1 are soficgroups and F0, F1 are finite subsets of Γ0 and Γ1 respectively. If σ0 and σ1 are elements ofSn and Sm respectively, then define σ0 ⊗ σ1 ∈ Snm by

(σ0 ⊗ σ1) (im+ j) = σ0(i)m + σ1 (j)

for i ∈ n and j ∈ m. If Φ0 : Γ0 → Sn is an (F0, ε)-approximate morphism and Φ1 : Γ1 → Smis an (F1, ε)-approximate morphism, then the map

Φ0 ⊗ Φ1 : Γ0 × Γ1 → Snm

defined by

(γ0, γ1) 7→ Φ0 (γ0) ⊗ Φ1 (γ1)

is an (F0 × F1, 2ε)-approximate morphism. This observation is sufficient to conclude thata direct product of two sofic groups is sofic. The result easily generalizes to arbitrary directproducts in view of the local nature of soficity. Since the inverse limits is a subgroup ofthe direct product, it follows from what we have observed so far that the inverse limit ofsofic groups is sofic.

We now prove the result—due to Elek and Szabo [56]—that the extension of a soficgroup by an amenable group is sofic. Suppose that Γ is a group, and N is a normal soficsubgroup of Γ such that the quotient Γ/N is amenable. We want to show that Γ is sofic.Fix ε > 0 and a finite subset F of Γ. Denote by g 7→ g the canonical quotient map from Γ toΓ/N and let r be a (set-theoretic) inverse for the quotient map. Observe that r (g)−1 g ∈ N


for every g ∈ G. Følner’s reformulation of amenability yields a finite subset A of Γ\N suchthat ∣∣Ag\A

∣∣ ≤ ε∣∣A∣∣

for every g ∈ F . Let A be the image of A under r and F be N ∩ (A · F ·A−1). Since N issofic there is a (F , ε)-approximate morphism g 7→ σg from N to Sn for some n ∈ N. Let kbe n|A| = n

∣∣A∣∣ and identify Sk with the group of permutations of n×A. Define the map

g 7→ τg from Γ to Sk by setting

τg (i, h) =

{ (σr(gh)

−1gh

(i) , r(gh))

if gh ∈ A

(i, h) otherwise..

This is well defined since, as we observed above, r(gh)−1

gh ∈ N .

Exercise II.4.2. Show that the map g 7→ τg is an (F, 3ε)-approximate morphism from Γto Sk.

Hint. Suppose that g ∈ F\ {1Γ}. To show that τg has at most εn |A| fixed points, distin-guish the cases when g /∈ N and g ∈ N . In the first case observe that (i, h) is a fixed pointonly if gh /∈ A, and use the fact that

∣∣Ag\A∣∣ ≤ ε

∣∣A∣∣. In the second case observe that (i, h)

is a fixed point only if i is a fixed point for σh−1gh, and use the fact that h−1gh ∈ F and

γ 7→ σγ is a (F , ε)-approximate morphism. To conclude fix g, g′ ∈ F and observe that forall but ε |A| values of h ∈ A, gh and g′gh both belong to A. For such values of h show thatfor all but εn values of i ∈ n one has that τg′τg (i, h) = τg′g (i, h).

It is a little more involved to observe that the free product of sofic groups is sofic, whichis a result of Elek and Szabo from [56]. Suppose that Γ0,Γ1 are sofic groups, F0, F1 arefinite subsets of Γ0 and Γ1, and ε > 0. We identify canonically Γ0 and Γ1 with subgroupsof the free product Γ = Γ0 ∗Γ1. Every element γ of Γ has a unique shortest decomposition

γ = g1h1 · · · gnhn

with gi ∈ Γ0 and hi ∈ Γ1. Fix N ∈ N and set F to be the set of elements of γ of Γ suchthat the unique shortest decomposition of γ has length at most N and its elements belongto F0 ∪ F1. By Exercise II.1.10 one can find n ∈ N, an (F0, ε)-approximate morphism

g 7→ σ(0)g from Γ0 to Sn and an (F1, ε)-approximate morphism g 7→ σ

(1)g from Γ1 to Sn

such that(σ(i)g

)−1= σ

(i)g−1 and σ

(i)g has no fixed points for g ∈ F (i)\ {1Γi

} and i = 0, 1. It

follows from the already recalled fact that free groups are residually finite that there is afinite group V generated by elements vij for i, j ∈ [n] such that the generators satisfy nonontrivial relation expressed by words of length at most N . Equivalently the Cayley graphof V with respect to the generators vij has no cycle of length smaller than or equal to N .

II.4. CLOSURE PROPERTIES OF THECLASSES OF SOFIC AND HYPERLINEARGROUPS 27

Consider then k = n2 |V | and identify Sk with the permutation group of n×n×V . Definefor g ∈ Γ0 the permutation τg of n× n× V by

τg (i, j, w) =(σ(0)g (i) , j, w

).

Similarly define for h ∈ Γ1 the permutation τh of [n] × [n] × V by

τh (i, j, w) =(i, σ

(1)h (j) , w · v−1

ij · vi,σ

(1)h

(j)

).

Finally is γ is an element of F with shortest decomposition g1h1 · · · gnhn for n ≤ N set

τγ = τg1 ◦ τh1 ◦ · · · ◦ τgn ◦ τhn .

Exercise II.4.3. Show that the map γ 7→ τγ defied above is an (F, ε)-approximate mor-phism from Γ to Sk.

Hint. The fact that the generators vij of V satisfy no nontrivial relations expressed bywords of length at most N implies that τγ has no fixed points whenever γ is a nonidentityelement of F . Suppose now that γ, γ′ ∈ F\ {1} and consider the corresponding uniqueshortest decompositions γ = g1h1 · · · gnhn and γ′ = g′1h

′1 · · · g′nh′n. In order to show that

τγ ◦τγ′ and τγγ′ differ on at most εk points of [n]×[n]×V consider different cases dependingwhether both, none, or exactly one between hn and g′1 is equal to the identity.

The result about the free product of sofic groups has been later generalized by Collinsand Dykema, who showed that the free product of sofic groups amalgamated over amonotileable amenable group is sofic [42]. The monotileability assumption has been laterremoved by Elek and Szabo [57, Theorem 1] and, independently, by Paunescu [125].

Suppose that Γ is a group with presentation 〈S|R〉, H is a subgroup of Γ, and α : H → Γis an injective homomorphism. Let t be a symbol not in Γ. The corresponding HNNextension Γ is the group generated by S ∪ {t} subject to the relations R and t−1ht = α(h)for h ∈ H. We claim that Γ is sofic whenever Γ is sofic an H is amenable. Define Γi = t−iΓti

for i ∈ Z, and let S be the subgroup of Γ generated by Γi for i ∈ Z. Then Γ is an extensionof S by Z. In particular to show that Γ is sofic it suffices to show that S is sofic. To showthat S is sofic it is enough to show that for every n ∈ N the group Sn = 〈Γi : i ∈ [−n, n] ∩ Z〉is sofic. This can be shown by induction on n observing that S0 = Γ, and Sn+1 can beobtained as a free product of Sn and isomorphic copies of Γ amalgamated over amenablesubgroups isomorphic to H.

A different generalization of the result about free products consists in considering thegraph products of sofic groups. Suppose that (V,E) is a simple undirected graph and, forevery v ∈ V , Γv is a group. The corresponding graph product is the quotient of the freeproducts of the Γv’s by the normal subgroup generated by the relators [gv, gw] for gv ∈ Γv

and gw ∈ Γw such that v and w are connected by an edge. Clearly free products correspondto graph products where there are no edges. Theorem 1.1 of [38] refines the argument forfree products to show that in fact an arbitrary graph product of sofic groups is sofic.


II.5 Further examples of sofic group

In this section we want to list some interesting examples of sofic and hyperlinear groups.It follows from the discussion in Section II.3 that all residually finite groups are sofic. Inparticular free groups are sofic. To obtain an example of a sofic group that is not resid-ually finite one can consider, for n,m ≥ 2 distinct, the Baumslag-Solitar group BS(m,n)generated by two elements a, b satisfying the relation a−1bma = bn. These groups wereintroduced in [11] to provide examples of simple finitely presented groups that are notHopfian, i.e. admit a surjective but not injective endomorphism. It is known that onehand such groups are not residually finite [114, Theorem C]. On the other hand they areresidually solvable [105, Corollary 2] and, in particular, sofic.

An example of a sofic (and in fact LEF) finitely generated group that is not residuallyamenable is provided in [56, Theorem 3], adapting a construction from [76]. Recall that afunction φ : Γ → C has positive type if whenever λi ∈ C and gi ∈ Γ are for i ≤ n then

∑

i,j≤n

λiλjφ(g−1i gj

)≥ 0.

A group Γ has Kazhdan’s property (T) if any sequence of positive type functions on Gthat converges pointwise to the function 1 constantly equal to 1 in fact converges to 1uniformly. Such a property, originally introduced by Kazhdan in [99], has played a keyrole in the latest developments of geometric group theory; see [12]. Property (T) canbe regarded as a strong antithesis to amenability, since the only amenable groups withproperty (T) are the finite groups. Since property (T) is preserved by quotients, it followsthat an amenable quotient of a property (T) group must be finite. We also recall here thata group with property (T) is finitely generated [12, Theorem 1.3.1].

Consider an infinite hyperbolic residually finite property (T) group K. (Examples ofsuch groups are provided in [79].) Let P be the group of finitely supported permutationsof K, and Q be the group generated by P and the left translations by elements of K. ThenP is a normal subgroup of Q and Q is a semidirect product of P and K. Consistently weidentify K with a subgroup of Q. Let S be a finite set of generators for K and TS be theset of transpositions of the form (1, s) for s ∈ S.

Exercise II.5.1. Show that S ∪ TS is a set of generators for Q.

Hint. Observe that if s ∈ S and g ∈ K then the transposition (g, gs) can be written asg−1 (1, s) g. Recall that the group of permutations on n symbols x1, . . . , xn is generated bythe transpositions (xi, xi+1) for i ≤ n− 1.

We now observe that Q is not residually amenable. Recall that every hyperbolic groupcontains an nontorsion element. Let t ∈ K such an element, and let a ∈ P be the transposi-tion (t, 1). Observe that tnat−n =

(tn+1, t

)6= a for every n ∈ N. Suppose by contradiction

that there is a homomorphism φ : Q→M into an amenable group M such that φ(a) 6= 1M

II.5. FURTHER EXAMPLES OF SOFIC GROUP 29

and φ (t) 6= 1M . Let A be the (simple) subgroup of P of even permutations. Since A issimple, a ∈ A , and φ(a) 6= 1M , φ must be injective on A. As observed before the fact thatK has property (T) implies that the image of K under φ is finite. Let n be the rank ofφ [K] and observe that φ (tn) = 1. Therefore φ (t−nat−n) = φ(a) while t−nat−n 6= a. Thiscontradicts the injectivity of φ on A.

We now show that Q is sofic and, in fact LEF. Denote by Bn(K) the n-ball around 1Kin the Cayley graph of K associated with the generating set S. Define Fn to be the set ofelements of Q of the form kσ where k ∈ Bn(K) and σ ∈ P is supported on Bn(K). Wewant to define an injective map from F2n to a finite group that preserves the operationon Fn. Since K is residually finite it has a finite-index normal subgroup Nn such thatNn ∩ B2n(K) = {1K}. Define Hn to be the (finite) group of permutations on K/Nn.Denote by πn the quotient map from K onto K/Hn and define ψn : F2n → Hn by setting

ψn (kσ) = ψn (k)ψn (σ) .

Here ψn (k) is the left translation by π (k), while ψn (σ) is defined by ψn (σ) (πn(h)) =π (σ(h)) if h ∈ B2n(K) and acts as the identity otherwise. Clearly ψn is injective on F2n.

Exercise II.5.2. Show that ψn (xy) = ψn (x)ψn (y) for x, y ∈ Fn.

Hint. Want to show that ψn (xy) (πn(h)) = ψn (x)ψn (y) (πn(h)) for every h ∈ K. Distin-guish the cases when h ∈ B2n(K) and h /∈ B2n(K).

Another example of a not residually finite sofic group which is moreover finitely pre-sented (and in fact one-relator) was provided by Jon Bannon in [8]; see also [9] for othersimilar examples. Consider the group Γ generated by two elements a, b subject to the rela-tion a =

[a, ab

]where ab = bab−1 and

[a, ab

]is the commutator of a and ab. Such a group

was introduced by Baumslag in [10] as an example of a non-cyclic one-relator group all ofwhose finite quotients are cyclic. In particular Γ is not residually finite and, in fact, notresidually solvable since a belongs to all the derived subgroups of Γ. We now show that itis sofic.

Set x = a−1 and y = bab−1. Then the relator a−1[a, ab

]becomes x−2y−1xy. Observe

that the group H =⟨x, y|y−1xy = x2

⟩is the Baumslag-Solitar group B (1, 2) and, in

particular, sofic. Moreover Γ is the HNN extension of H with respect to the isomorphismxn 7→ yn between the subgroups 〈x〉 and 〈y〉 of H. It follows that Γ is sofic since HNNextensions of sofic groups over amenable groups are sofic.

We would now like to present examples of hyperlinear and sofic groups that are not LEA.Recall that a group is locally embeddable into amenable groups (LEA) if, briefly, every finiteportion of its multiplication table can be realized as a portion of the multiplication tableof an amenable group. The first example of a hyperlinear not LEA group was constructedby Thom in [145], adapting a construction due to de Cornulier [48] and Abels [1].

If n,m ∈ N denote by Mn,m

(Z[1p

])the space of n ×m matrices over Z

[1p

]and by

SLn

(Z[1p

])the group of n×n matrices of determinant 1. Let Γ be the group of matrices


in SL8

(Z[1p

])of the form

1 a12 a13 a140 a22 a23 a240 0 a33 a340 0 0 1

where the diagonal blocks are square matrices of rank 1, 3, 3, 1 and a22, a33 ∈ SL3

(Z[1p

]).

The center C of Γ consists of the matrices of the form

1 0 0 a0 I 0 00 0 I 00 0 0 1

where I is the 3×3 identity matrix and a ∈ K. Thus C is isomorphic to the additive group

of Z[1p

]. In particular C contains a subgroup Z isomorphic to Z.

It is shown in [48, Proposition 2.7] that Γ is a lattice in a locally compact group withproperty (T). Since property (T) passes to lattices [12, Theorem 1.7.1], it follows that Γ hasproperty (T) and, in particular, is finitely generated. It is moreover shown in [48, Section3] that Γ and (hence) Γ/Z are finitely presented. We present here the argument to showthat Γ/Z is not Hopfian, i.e. it has a surjective endomorphism with nontrivial kernel.

Denote by Z[1p

]×the multiplicative group of invertible elements of Z

[1p

]. Then Z

[1p

]×

identified with the group of matrices of the form

a 0 0 00 1 0 00 0 1 00 0 0 1

naturally acts on Γ by conjugation. Considering p ∈ Z[1p

]×gives an automorphism β of

Γ that maps the center Z of Γ to its property subgroup Zp. Thus β induces a surjectiveendomorphism β of Γ/Z whose kernel is the (nontrivial) subgroup C/Z of Γ/Z; see [48,Lemma 2.3]. As a consequence Γ/Z is not Hopfian.

As observed in [145, 3.1] a finitely presented LEA group that has property (T) isnecessarily Hopfian. In fact a finitely presented LEA group is residually amenable by [31,Proposition 7.3.8]. As recalled before an amenable quotient of a property (T) group isnecessarily finite. Therefore a property (T) residually amenable group is residually finite.Finally a residually finite finitely generated group is easily seen to be Hopfian. It followsfrom this argument that the group Γ/Z, being finitely presented, property (T), and notHopfian, is not LEA.

II.5. FURTHER EXAMPLES OF SOFIC GROUP 31

To conclude one needs to observe that Γ/Z is hyperlinear. Considering the reductionmodulo q where q is an arbitrary positive integer prime with p shows that the group Γis residually finite and, in particular, hyperlinear. The argument is concluded by showingthat hyperlinearity is preserved by taking quotients by central subgroups; see [145, Remark3.4].

It is not known whether the group Γ/Z above is sofic. We conclude this sequence ofexamples with an example due to de Cornulier of a finitely presented sofic group that isnot LEA [46]. Let Γ be the group of matrices

a b u02 u03 u04c d u12 u13 u140 0 pn2 u23 u240 0 0 pn3 u340 0 0 0 1

with

•

[a bc d

]∈ SL2 (Z),

• uij ∈ Z[1p

], and

• n2, n3 ∈ Z.

Let M be the normal subgroup of Γ consisting of matrices of the form

1 0 0 0 m1

0 1 0 0 m2

0 0 1 0 00 0 0 1 00 0 0 0 1

with m1,m2 ∈ Z[1p

]. The normal subgroup MZ is defined similarly with m1,m2 ∈ Z.

Then one can consider the quotient Γ/MZ. It is shown in [46] that such a group is soficand finitely presented but not LEA.

In order to prove soficity, it is enough to show that Γ can be obtained as an extensionof a sofic group by an amenable group. Consider the normal subgroup Υ of elements of Γfor which n2 = n3 = 0. Observe that Γ/Υ is isomorphic to Z2. Therefore it remains toshow that Υ/MZ is sofic.

Fix m ∈ N and consider the subgroup Υm of elements of Υ for which

• u02, u12, u23, u34 ∈ p−mZ,


• u03, u13, u24 ∈ p−2mZ, and

• u04, u14 ∈ p−3mZ.

It is clear that⋃

m Υm = Υ. Therefore by the local nature of soficity we are left withthe problem of showing that Υm/MZ is sofic for every m ∈ N. Such a group is in factresidually finite and hence sofic. Consider the subgroup Λ of elements of Γ for which theblock [

a 00 b

]∈ SL2 (Z)

is the identity matrix. Observe that (Υm ∩ Λ) /MZ is a normal subgroup of Υm/MZ, andmoreover Υm/MZ is isomorphic to the semidirect product (Υm ∩ Λ) /MZ ⋊ SL2 (Z). NowΥm ∩ Λ is solvable and finitely generated, and hence residually finite [81, 2.1]. The proofis concluded by observing that a semidirect product of a finitely generated residually finitegroup by a residually finite group is residually finite.

Observe that the quotient map Γ 7→ Γ/MZ restricted to the subgroup (isomorphic toSL2 (Z)) of elements

a b 0 0 0c d 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

is injective. Thus Γ/MZ contains a copy of SL2 (Z) and, hence, of the free group on 2generators F2. (Recall that the matrices

A =

[1 20 1

]and B =

[1 02 1

]

generate a copy of F2 inside SL2 (Z); see [128, page 3].)

By [31, Corollary 7.1.20] in order to show that Γ/MZ is not LEA it is enough to showthat it is an isolated point in the space of marked groups. Fix n ∈ N. An n-markedgroup is a group Γ endowed with a distinguished generating n-tuple (γ1, . . . , γn). Let Gn

be the space of n-marked groups. Given an n-marked group one can consider the kernelof the epimorphism Fn → Γ mapping the canonical free generators of Fn to the γi’s.Conversely any normal subgroup N of Fn gives rise to the n-marked group Fn/N wherethe distinguished n-tuple of generators is the image of the free generators of Fn. Thisargument shows that one can identify the space Gn of n-marked group with the spaceof the normal subgroups of Fn. Identifying in turn a normal subgroup of Fn with itscharacteristic function yields an inclusion of Gn into 2Fn as a closed subspace. This definesa compact metrizable zero-dimensional topology on Gn. Corollary 7.1.20 in [31] shows thata group is LEA if and only if it is a limit of amenable groups in the space of marked groups.

II.6. LOGIC FOR INVARIANT LENGTH GROUPS 33

Therefore since Γ/MZ is not LEA, it is enough to show that Γ/MZ is isolated, i.e. it is anisolated point in the space of marked groups.

It is shown in [47, Lemma 1] that being isolated is indeed a well defined propertyof a group, independent of the marking. Proposition 2 in [47] provides the followingcharacterization of being isolated: a group is isolated if and only if it is finitely presented andmoreover finitely discriminable, i.e. it contains a finite subsets that meets every nontrivialnormal subgroup in a nonidentity element. The proof that Γ/MZ satisfies these conditionsis presented in [46, Section 3] and [47, 5.4].

Other examples of finitely presented sofic groups that are not LEA are provided in [94].

II.6 Logic for invariant length groups

The logic for metric structures is a generalization of the usual first order logic. It is anatural framework to study algebraic structures endowed with a nontrivial metric and theirelementary properties (i.e. properties preserved by ultrapowers or equivalently expressibleby formulae). In the sequel we introduce a particular instance of logic for metric structuresto describe and study groups endowed with an invariant length functions.

A term t(x1, . . . , xn) in the language of invariant length groups in the variables x1, . . . , xnis a word in the indeterminates x1, . . . , xn, i.e. an expression of the form

xn1i1. . . xnl

il

for l ∈ N and ni ∈ Z for i = 1, 2, . . . , l. For example

xyx−1y−1

is a term in the variables x, y. The empty word will be denoted by 1. If G is an invariantlength group, g1, . . . , gm are elements of G, and t(x1, . . . , xn, y1, . . . , ym) is a term in thevariables x1, . . . , xn, y1, . . . , ym, then one can consider the term t(x1, . . . , xn, g1, . . . , gm)with parameters from G, which is obtained from t (x1, . . . , xn, y1, . . . , ym) replacing formallyyi with gi for i = 1, 2, . . . ,m. The evaluation tG(x1, . . . , xn) in a given invariant lengthgroup G of a term t (x1, . . . , xn) in the variables x1, . . . , xn (possibly with parameters fromG) is the function from Gn to G defined by

(g1, . . . , gn) 7→ t (g1, . . . , gn)

where t (g1, . . . , gn) is the element of G obtained replacing in t(x1, . . . , xn) every occurrenceof xi with gi for i = 1, 2, . . . , n. For example the evaluation in a invariant length groupG of the term xyx−1y−1 is the function from G2 to G that assigns to every pair (g, h)of elements of G their commutator ghg−1h−1. The evaluation of the empty word is thefunction on G constantly equal to 1G.


A basic formula ϕ in the variables x1, . . . , xn is an expression of the form

ℓ (t(x1, . . . , xn))

where t(x1, . . . , xn) is a term in the variables x1, . . . , xn. The evaluation ϕG(x1, . . . , xn) ofϕ(x1, . . . , xn) in an invariant length group G is the function from Gn to [0, 1] defined by

(g1, . . . , gn) 7→ ℓG(tG (g1, . . . , gn)

)

where ℓG is the invariant length of G. For example

ℓ(xyx−1y−1)

is a basic formula whose interpretation in an invariant length group G is the functionassigning to a pair of elements of G the length of their commutator. This basic formulacan be thought as measuring how much x and y commute. The evaluation at (g, h) of itsinterpretation in an invariant length group G will be 0 if and only if g and h commute.

Finally a formula ϕ is any expression that can be obtained starting from basic for-mulae, composing with continuous functions from [0, 1]n to [0, 1], and taking infima andsuprema over some variables. In this framework continuous functions have the role of log-ical connectives, while infima and suprema should be regarded as quantifiers. With theseconventions the terminology from the usual first order logic is used in this setting. A for-mula is quantifier-free if it does not contain any quantifier. A variable x in a formula ϕ isbound if it is within the scope of a quantifier over x, and free otherwise. a formula withoutfree variables is called a sentence. The interpretation of a formula in a length group G isdefined in the obvious way by recursion on its complexity. For example

supx

supyℓ(xyx−1y−1)

is a sentence, with bound variables x and y. Its evaluation in an invariant length group Gis the real number

supx∈G

supy∈G

ℓG(xyx−1y−1).

This sentence can be thought as measuring how much the group G is abelian. Its inter-pretation in G is zero if and only if G is abelian. This example enlightens the fact thatthe possible truth values of a sentence (i.e. values of its evaluations in an invariant lengthgroup) are all real numbers between 0 and 1. Moreover 0 can be thought as “true” whilestrictly positive real numbers of different degrees as “false” . In this spirit we say that asentence ϕ holds in G if and only if its interpretation in G is zero. Using this terminologywe can assert for example that an invariant length group G is abelian if and only if theformula

supx

supyℓ(xyx−1y−1)


holds in G. Observe that if ϕ is a sentence, then 1 − ϕ is a sentence such that ϕ holds inG if and only if the interpretation of 1 − ϕ in G is 1. Thus 1 − ϕ can be though as a sortof negation of the sentence ϕ. Another example of sentence is

supx

min {|ℓ(x) − 1| , |ℓ(x)|} .

Such sentence holds in an invariant length group G if and only if the invariant lengthfunction in G attains values in {0, 1}, i.e. it is the trivial invariant length function on G.It is worth noting at this point that for any sentence ϕ as defined in the logic for invariantlength groups there is a corresponding formula ϕ0 in the usual (discrete) first order logicin the language of groups such that the evaluation of ϕ0 in a discrete group G coincideswith the evaluation of ϕ in G regarded as an invariant metric group endowed with thetrivial invariant length function. For example the (metric) formula expressing that a groupis abelian corresponds to the (discrete) formula

∀x∀y (xy = yx).

Sentences in the language of invariant length groups allow one to determine whichproperties of an invariant length group are elementary. A property concerning lengthgroups is elementary if there is a set Φ of sentences such that an invariant length grouphas the given property if and only if G satisfies all the sentences in Φ. For example theproperty of being abelian is elementary, since an invariant length group is abelian if andonly if it satisfies the sentence supx,y ℓ(xyx

−1y−1). Two invariant length groups G and G′

are elementarily equivalent if they have the same elementary properties. This amounts tosay that any sentence has the same evaluation in G and G′. A class C of invariant lengthgroups will is axiomatizable if the property of belonging to C is elementary. The previousexample of sentence shows that the class of abelian length groups is axiomatizable by asingle sentence. Elementary properties and classes are closely related with the notion ofultraproduct of invariant length groups.

Suppose that (Gn)n∈N is a sequence of invariant length groups and U is a free ultrafilterover N. The ultraproduct

∏U Gn of the sequence (Gn)n∈N with respect to the ultrafilter U

is by definition quotient of the product∏

nGn by the normal subgroup

NU =

{(gn)n∈N : lim

n→UℓGn (gn) = 0

}

endowed with the invariant length function

ℓU ((gn)NU ) = limn→U

ℓGn (gn) .

As before a sequence (gn)n∈N in∏

nGn is called representative sequence for the correspond-ing element in

∏U Gn. Observe that ultraproducts of the sequence (Sn)n∈N of permutation


groups endowed with the Hamming invariant length function as defined in Section II.1 andultraproducts of the sequence (Un)n∈N of unitary groups endowed with the Hilbert-Schmidtinvariant length function as defined in Section II.2 are particular cases of this definition.When the sequence (Gn)n∈N is constantly equal to a fixed invariant length group G theultraproduct

∏U Gn is called ultrapower of G and denoted by GU . Observe that diagonal

embedding of G into GU assigning to g ∈ G the element of GU having the sequence con-stantly equal to g as representative sequence is a length preserving group homomorphism.This allows one to identify G with a subgroup of GU .

The ultraproduct construction behaves well with respect to interpretation of formulae.This is the content of a theorem proven in the setting of the usual first order logic by Losin [109]. Its generalization to the logic for metric structures can be found in [13] (Theorem5.4). We state here the particular instance of Los’ theorem in the context of invariantlength groups.

Theorem II.6.1 ( Los). Suppose that ϕ(x1, . . . , xk) is a formula with free variables x1, . . . , xk,(Gn)n∈N is a sequence of invariant length groups, and U is a free ultrafilter over N. Ifg(1), . . . , g(k) are elements of

∏nGn then

ϕ∏

U Gn

(g(1), . . . , g(k)

)= lim

n→UϕGn

(g(1)n , . . . , g(k)n

)

where g(i) is any representative of the sequence(g(i)n

), for i = 1, 2, . . . , k. In particular if

ϕ is a sentence then

ϕ∏

U Gn = limn→U

ϕGn .

Theorem II.6.1 can be proved by induction on the complexity of the formula ϕ. Theparticular instance of Theorem II.6.1 when the sequence (Gn)n∈N is constantly equal toan invariant metric group G shows that G and any ultrapower GU of G are elementarilyequivalent. Moreover the diagonal embedding of G into GU is an elementary embedding,i.e. it preserves the value of formulae possibly with parameters from G.

A particularly useful property of ultraproducts is usually referred to as countable sat-uration. Roughly speaking countable saturation of a group H asserts that whenever onecan find elements of H approximately satisfying any finite subset of a given countablyinfinite set of conditions, then in fact one can find an element of H exactly satisfying si-multaneously all the conditions. In order to precisely define this property, and prove it forultraproducts, we need to introduce some model-theoretic terminology, in the particularcase of invariant length groups.

Definition II.6.2. Suppose that H is a group endowed with an invariant length function.A countable set of formulae X in the free variables x1, . . . , xn and possibly with parametersfrom H is:


• approximately finitely satisfiable in H or consistent with H if for every ε > 0 andevery finite collection of formulae ϕ1 (x1, . . . , xn) , . . . , ϕm(x1, . . . , xn) from X thereare a1, . . . , an ∈ H such that

ϕHi (a1, . . . , an) < ε

for every i = 1, 2, . . . , n;

• realized in H if there are a1, . . . , an ∈ H such that

ϕH(a1, . . . , an) = 0

for every formula ϕ(x1, . . . , xn) in X . In this case the n-tuple (a) is called a realizationof X in H.

In model-theoretic jargon, a set of formulae X as above is called a type over A, whereA is the (countable) set of parameters of the formulae in X . Clearly any set of formulaewhich is realized in H is also approximately finitely satisfiable H. The converse to thisassertion is in general far from being true. For example suppose that H is the direct sumG⊕N of countably many copy of a countable group G with trivial center endowed with thetrivial length function. Consider the set X of formulae

max{|1 − ℓ(x)| , ℓ

(xax−1a−1

)}

in the free variable x where a ranges over the elements of G⊕N. Being the center of G⊕N

trivial, the set of formulae X is not realized in G⊕N. Nonetheless the fact that one can findin H a nonidentity element commuting with any given finite subset of G⊕N shows that Xis approximately finitely satisfiable in H.

Definition II.6.3. An invariant length group is countably saturated if any countable setof formulae X with parameters from H which is approximately finitely satisfied in H isrealized in H.

Thus in a countably saturated invariant length group a countable set of formulae isapproximately finitely satisfiable if and only if it is realized. Moreover it can be easily shownby recursion on the complexity that in the evaluation of a formula in a countably saturatedstructure infima and suprema can be replaced by minima and maxima respectively. Moregenerally one can define κ-saturation for an arbitrary cardinal κ replacing countable typeswith types with less than κ parameters (it is not difficult to check that countable saturationis the same as ℵ1-saturation). An invariant length group G is saturated if it is κ-saturatedwhere κ is the density character of G, i.e. the minimum cardinality of a dense subset ofG. The notions of saturation and countable saturation here introduced are the particularinstances in the case of invariant length groups of the general model-theoretic notions ofsaturation and countable saturation. Being countably saturated is one of the fundamentalfeatures of ultraproducts. The proof of this fact in this context can be easily deduced from Los’ theorem and is therefore left as an exercise.


Exercise II.6.4. Suppose that (Gn)n∈N is a sequence of invariant length groups, and U isa free ultrafilter over N. Show that the ultraproduct

∏U Gn is countably saturated.

Saturated structures have been intensively studied in model theory and have someremarkable properties. In the case of usual first order logic it is a consequence of the socalled Chang-Makkai’s theorem (see [34, Theorem 5.3.6]) that the automorphism group ofa saturated structure of cardinality κ has 2κ elements. The generalization to this result tothe framework of logic for metric structures is an unpublished result of Ilijas Farah, BraddHart, and David Sherman. Proposition is the particular instance of such result in the caseof invariant length groups of density character ℵ1.

Proposition II.6.5. Suppose that G is an invariant length group. If G is countably satu-rated and has a dense subset of cardinality ℵ1 then the group of automorphisms of G thatpreserve the length has cardinality 2ℵ1 .

Proposition II.6.6 is an immediate consequence of Proposition II.6.5 and Exercise II.6.4.Recall that the Continuum Hypothesis asserts that the cardinality of the power set of Ncoincides with the first uncountable cardinal ℵ1. A cornerstone result in set theory assertsthat the Continuum Hypothesis is independent from the usual axioms of set theory, i.e.can not be neither proved nor disproved (see [73], [39], [40]).

Proposition II.6.6. Suppose that (Gn)n∈N is a sequence of separable invariant lengthgroups. If the Continuum Hypothesis holds then

∏U Gn has 2ℵ1 length preserving automor-

phisms for any free ultrafilter U over N. In particular∏

U Gn has outer length-preservingautomorphisms.

Proof. Assuming the Continuum Hypothesis∏

U Gn has a dense subset of cardinality ℵ1.Moreover by Exercise II.6.4

∏U Gn is countably saturated. It follows from Proposition

II.6.5 that∏

U Gn has 2ℵ1 length-preserving automorphisms.

Corollary II.6.7 is the particular instance of Proposition II.6.6 when the sequence(Gn)n∈N is the sequence of permutation groups or the sequence of unitary groups, onecan obtain

Corollary II.6.7. Assume that the Continuum Hypothesis holds. For any free ultrafilter Uover N the universal sofic group

∏U Sn has 2ℵ1 automorphisms. As a consequence

∏U Sn

has outer automorphisms. The same is true for the universal hyperlinear group∏

U Un.

We do not know if there exist models of set theory where some universal sofic orhyperlinear groups have only inner automorphisms. It is conceivable that this could beproved using ideas and methods from [138]. For example in [110] Lucke and Thomas showusing a result from [138] that there is a model of set theory where some ultraproduct ofthe permutation groups regarded as discrete groups has only inner automorphisms.

II.7. MODEL THEORETICCHARACTERIZATIONOF SOFIC AND HYPERLINEARGROUPS 39

Corollary II.6.7 and the discussion that follows address a question of Paunescu from[126]: Theorem 4.1 and Theorem 4.2 of [126] show that all automorphisms of the universalsofic groups preserve the length function and the conjugacy classes. Paunescu then asks if itis possible that all automorphisms of the universal sofic groups are in fact inner. CorollaryII.6.7 in particular implies that such assertion does not follow from the usual axioms of settheory.

An application of Theorem 2 from [51] (known as Dye’s theorem on automorphisms ofunitary groups of factors) allows one to prove the analogue of Theorem 4.1 and Theorem 4.2from [126] in the case of universal hyperlinear groups. More precisely all automorphisms ofthe universal hyperlinear groups preserve the length function, while the normal subgroupof automorphisms that preserves the conjugacy classes has index 2 inside the group of allautomorphisms. The not difficult details can be found in [111].

Finally let us point out another consequence of countable saturation of ultraproducts.Suppose that Γ is a discrete group of size ℵ1 such that every countable (or equivalentlyfinitely generated) subgroup of Γ is sofic. It is not difficult to infer from Exercise II.1.9 andcountable saturation of

∏U Sn that for any ultrafilter U over N there is a length-preserving

homomorphism from Γ (endowed with the trivial length function) to∏

U Sn. For examplethis implies that

∏U Sn contains a free group on uncountably many generators. Analogue

facts hold for hyperlinear groups and ultraproducts∏

U Un.

II.7 Model theoretic characterization of sofic and hyperlin-

ear groups

In this section we shall show that the classes of sofic and hyperlinear groups are axioma-tizable in the logic for invariant metric groups. Equivalently the properties of being soficor hyperlinear are elementary. Recall that the quantifiers in the logic for invariant metricgroups are inf and sup. More precisely sup can be regarded as the universal quantifier,analogue to ∀ in usual first order logic, while inf can be seen as the existential quantifier,which is denoted by ∃ in the usual first order logic. A formula is therefore called univer-sal if it only contains universal quantifiers, and no existential quantifiers. The notion ofexistential sentence is defined in the same way. A quantifier-free formula is just a formulawithout any quantifier. Say that two formulae ϕ and ϕ′ are equivalent if they have thesame interpretation in any invariant length group. It can be easily proved by induction onthe complexity that any universal sentence is equivalent to a formula of the form

supx1

. . . supxn

ψ(x1, . . . , xn)

where ψ(x1, . . . , xn) is quantifier-free. An analogous fact holds for existential sentences.It is easy to infer from this that if ϕ is a universal sentence, then 1 − ϕ is equivalentto an existential sentence, and vice versa. Exercise II.7.1 together with Los’ theorem on


ultraproducts shows that universal and existential formulae have the same values in anyultraproduct of the symmetric groups regarded as invariant length groups.

Exercise II.7.1. If ϕ is a universal sentence then the sequence(ϕSn

)n∈N of its evaluation

in the symmetric groups converges. Infer that the same is true for existential formulae.Deduce that the same holds for existential formulae.

Proof. Fix n ∈ N and ε > 0. If N > n write

N = kn+ r

for k ∈ N and r ∈ n. Define the map ι : Sn → SN by

ι(σ) (ik + j) =

{in+ σ (j) if i ∈ k and j ∈ n

in+ j otherwise.

Observe that ι is a group homomorphism that preserves the Hamming length function upto 1

k . This means that

|ℓSn(σ) − ℓSN(ι(σ))| ≤ 1

k.

Deduce that if ϕ is a universal sentence and ε > 0 then∣∣ϕSN − ϕSn

∣∣ < ε.

for N ∈ N large enough.

Theorem 5.6 of [112] asserts that the same conclusion of Exercise II.7.1 holds for for-mulae with alternation of at most two quantifiers. It is currently an open problem if thesame holds for all formulae. A positive answer to this question would imply that the uni-versal sofic groups, i.e. the length ultraproducts of the permutation groups, are pairwiseisomorphic as invariant length groups if the Continuum Hypothesis holds. Theorem 1.1 of[147] asserts that if instead the Continuum Hypothesis fails there are 22

ℵ0 ultraproductsof the permutation groups that are pairwise nonisomorphic as discrete groups.

We can now show that the property of being sofic is axiomatizable. Suppose that Γ isa sofic group. By Exercise II.1.9 there is a length preserving embedding of Γ into

∏U Sn

for any free ultrafilter U , where Γ is endowed with the trivial length function. It is easilyinferred from this that

ϕΓ ≤ ϕ∏

U Sn = limn→+∞

ϕSn

for any universal sentence ϕ in the language of invariant length groups. If ϕ is instead anexistential sentence then

ϕΓ ≥ limn→+∞

ϕSn .

In particular if ϕ is an existential sentence that holds in Γ, then limn→+∞ ϕSn . ExerciseII.7.2 shows that this condition is sufficient for a group to be sofic.

II.8. THE KERVAIRE-LAUDENBACH CONJECTURE 41

Exercise II.7.2. Suppose that Γ is a (countable discrete) group with the property thatfor any existential sentence ϕ that holds in Γ the sequence

(ϕSn

)n∈N of the evaluation of

ϕ in the symmetric groups is vanishing. Show that Γ is sofic.

Hint. Fix ε > 0 and a finite subset F = {g1, . . . , gn} of Γ. Write an existential sentenceϕ witnessing the existence of elements g1, . . . , gn with the multiplication rules given by Γ.Infer from the fact that ϕ holds in Γ that ϕ approximately holds in Sn for n large enough.Use this to construct an (F, ε)-approximate morphism Φ : Γ → Sn.

It is easy to deduce from Exercise II.7.2 the following characterization of sofic groups,showing in particular that soficity is an elementary property.

Proposition II.7.3. If Γ is a group, the following statements are equivalent

1. Γ is sofic;

2. if ϕ is an existential sentence that holds in Γ, then limn→+∞ ϕSn = 0;

3. if ϕ is a universal sentence such that limn→+∞ ϕSn = 0, then ϕΓ = 0.

All the results of this section carry over to the case of hyperlinear group, when onereplaces the permutation groups with the unitary groups (see [112]). The analogue ofExercise II.7.1 for the unitary groups can be proved in a similar way, considering foru ∈ Un and N = kn+ r the unitary matrix

(u⊗ Ik 0

0 1Ur

)∈ UN .

II.8 The Kervaire-Laudenbach conjecture

Let Γ be a countable discrete group and γ1, . . . , γl ∈ Γ. Denote w(x) the monomial

w(x) = xn1γ1 . . . xnlγl,

where ni ∈ Z for i = 1, 2, . . . , l. Consider the following problem: Determine if the equation

w(x) = 1

has a solution in some group extending Γ. The answer in general is “no”. Consider forexample the equation

xax−1b−1 = 1

If a and b have different orders then clearly this equation has no solution in any groupextending Γ. Assuming that the sum

∑li=1 ni of the exponents of x in w(x) is nonzero is

a way to rule out this obstruction. A conjecture attributed to Kervaire and Laudenbach


asserts that this is enough to guarantee the existence of a solution of the equation w(x) = 1in some group extending Γ.

We will show in this section that the Kervaire-Laudenbach conjecture holds for hyper-linear groups (see Definition II.2.2). This result, first observed by Pestov in [127], will bea direct consequence of the following theorem by Gerstenhaber and Rothaus (see [70]).

Theorem II.8.1. Let Un denote the group of unitary matrices of rank n. If a1, . . . , , akare elements of Un then the equation

xs1a1 · · · xskak = 1

has a solution in Un as long as∑k

i=1 si 6= 0.

Proof. Let w(x, a1, . . . , ak) := xs1 · · · xskak and consider the map f : Un → Un defined by

b 7→ w(b, a1, . . . , an).

We just need to prove that f is onto. Recall that Un is a compact manifold of dimensionn2. Thus the homology group Hn2 (Un) is an infinite cyclic group. (A standard referencefor homology theory is [68, Chapter 23].) Being continuous (and in fact smooth) f inducesa homomorphism

f∗ : Hn2 (Un) → Hn2 (Un) .

If e is a generator of Hn2 (Un) thenf∗(e) = de

for some d ∈ Z called the degree of f . In order to show that f is onto, it is enough to showthat its degree is nonzero. We claim that d = sn, where s =

∑ni=1 si. Being Un connected,

the map f is homotopy equivalent to the map

fs : Un → Un

defined byb 7→ bs.

By homotopy invariance of the degree of a map, f and fs have the same degree. Thereforewe just have to show that fs has degree sn. This follows from the facts that the genericelement of Un has sn s-roots of unity, and that the degree of a map can be computedlocally.

Theorem II.8.1 is in fact a particular case of [70, Theorem 2], where arbitrary compactLie groups and system of equations with possibly several variables are considered.

Let us now discuss how one can infer from Theorem II.8.1 that the Kervaire-Laudenbachconjecture holds for hyperlinear group. Consider a word

w(x, y1, . . . , yk) ≡ xs1y1xs2y2 · · · xskyk,

II.9. OTHER METRIC APPROXIMATIONS AND HIGMAN’S GROUP 43

where∑k

i=1 si 6= 0. By Theorem II.8.1, formula

supy1,...,yk

infxℓ (w(x, y1, . . . , yk))

evaluates to 0 in any unitary group Un. By Los’ theorem on ultraproducts, the sameformula evaluates to 0 in any ultraproduct

∏U Un of the unitary groups. Thus if a1, . . . , ak

are elements of∏

U Uk

infx∈∏U Un

ℓ (w(x, a1, . . . , ak)) = 0.

By countable saturation of∏

U Un one can find b ∈∏U Un where the infimum is achieved.This means that

ℓ (w(b, a1, . . . , ak)) = 0

and hence b is a solution of the equation

xs1a1 · · · xskak.

This shows that any ultraproduct∏

U Un satisfies the Kervaire-Laudenbach conjecture.Since hyperlinear groups are subgroups of

∏U Un, they will satisfy the Kervaire-Laudenbach

conjecture as well.

II.9 Other metric approximations and Higman’s group

The notions of sofic and hyperlinear groups as defined in Section II.1 and Section II.2respectively admit natural generalizations where one considers approximate morphismsinto different classes of invariant length groups. Suppose that C is a class of groups endowedwith an invariant length function.

Definition II.9.1. A countable discrete group Γ has the C-approximation property if forevery finite subset F of Γ \{1Γ} and every ε > 0 there is an (F, ε)-approximate morphism(as defined in Definition II.1.7) from Γ endowed with the trivial length function to a groupT in C , i.e. a function Φ : Γ → T such that Φ (1Γ) = 1T and for every g, h ∈ F :

• ℓT(Φ(gh)Φ(h)−1Φ(g)−1

)< ε;

• ℓT (Φ(g)) > 1 − ε.

The following characterization of groups locally embeddable into some class of invariantlength groups can be proved with the same arguments as Section II.6, where the notionsof universal and existential sentence are introduced.

Proposition II.9.2. The following statements about a countable discrete group Γ (endowedwith the trivial length function) are equivalent:


1. Γ has the C-approximation property;

2. There is a length-preserving group homomorphism from Γ to an ultraproduct∏

U Gn

of a sequence (Gn)n∈N of invariant metric groups from the class C;

3. For every existential sentence ϕ in the language of invariant length groups

ϕΓ ≥ inf{ϕG : G ∈ C

}

where ϕG denotes the evaluation of ϕ in the invariant length group G;

4. For every universal sentence ϕ in the language of invariant length groups

ϕΓ ≤ sup{ϕG : G ∈ C

}.

This characterization in particular shows that the C-approximation property is elemen-tary in the sense of Section II.6

Observe that when C is a class of groups endowed with the trivial length function, theC-approximation property coincides with the notion of local embeddability into elementsof C as defined in Section II.3. It is immediate from the definition that a group is sofic ifand only if it has the C-approximation property where C is the class of symmetric groupsendowed with the Hamming invariant length function. Analogously hyperlinearity can beseen as the C-approximation property where C is the class of unitary groups endowed withthe Hilbert-Schmidt invariant length functions. Weakly sofic groups as defined in [71] areexactly C-approximable groups where C is the class of all finite groups endowed with aninvariant length function. More recently Arzhantseva and Paunescu introduced in [6] theclass of linear sofic groups, which can be regarded as C-approximable groups where C isthe class of general linear groups endowed with the invariant length function

ℓ(x) = N(x− 1)

where N is the usual normalized rank of matrices.Giving a complete account of the notion of local metric approximation in group theory

(not to mention other areas of mathematics) would be too long and beyond the scope ofthis survey. It should be nonetheless mentioned that it can be found in nuce in the workof Malcev. More recently it has been considered by Gromov in the paper [80] that leadto the introduction of sofic groups. The notion of C-approximation is defined implicitly byPestov in [127, Remark 8.6] and explicitly by Thom in [146, Definition 1.6], as well as byArzhantseva and Cherix in [5].

A more general notion of metric approximation called asymptotic approximation hasbeen more recently defined by Arzhantseva for finitely generated groups in [4, Definition9]: A finitely generated group Γ is asymptotically approximated by a class of invariantlength groups C if there is a sequence (Xn)n∈N of finite generating subsets of Γ such that


for every n ∈ N there is a(BXn(n), 1n

)-approximate morphism as in Definition II.1.7 from

Γ endowed with the trivial invariant length function to an element of C, where BXn(n)denotes the set of elements of Γ that have length at most n according to the word lengthassociated with the generating set Xn. In the particular case when the finite generating setXn does not depend on n one obtains the notion of C-approximation as above. It is showedin [4, Theorem 11] that several important classes of finitely generated groups, includingall hyperbolic groups and one-relator groups, are asymptotically residually finite, i.e. havethe asymptotic C-approximation property where C is the class of residually finite groupsendowed with the trivial length function. In particular these groups are asymptoticallysofic. (It is currently not known if these groups are in fact sofic.)

To this day no countable discrete group is known to not have the C-approximationproperty as in Definition II.9.1 when C is any of the classes of invariant length groupsmentioned above. In an attempt to find an example of a countable discrete group failingto have the C-approximation property with respect to some natural large class of invariantlength groups, Thom introduced in [146] the class Fc of finite commutator-contractiveinvariant length groups, i.e. finite groups endowed with an invariant length function ℓsatisfying

ℓ(xyx−1y−1) ≤ 4ℓ(x)ℓ(y).

Examples of such groups, besides finite groups endowed with the trivial length function, arefinite subgroups of the unitary group of a C*-algebra A endowed with the length function

ℓ(x) =1

2‖1 − x‖ .

Corollary 3.3 of [146] shows that groups with the Fc-approximation property form a propersubclass of the class of all countable discrete groups. More precisely Higman’s group Hintroduced by Higman in [86] does not have the Fc-approximation property. This is one ofthe few currently know examples of a group failing to have the C-approximation propertyfor some broad class C of groups endowed with a (nontrivial) invariant length function.

Higman’s group H is the group with generators hi for i ∈ Z /4Z subject to the cyclicrelations

hi+1hih−1i+1 = h2i ,

for i ∈ Z /4Z where the sum i+1 is calculated modulo 4. It was first considered by Higmanin [86], who showed that H is an infinite group with no nontrivial finite quotients, thusproviding the first example of a finitely presented group with this property. Higman’s proofin fact also shows that H is not locally embeddable into finite groups. Theorem 3.2 andCorollary 3.3 from [146] strengthen Higman’s result, showing that H does not even havethe Fc-approximation property.

We will now prove, following Higman’s original argument from [86], that Higman’sgroup H is nontrivial and in fact infinite. Consider for i ∈ Z /4Z the group Hi generated


by hi and hi+1 subjected to the relation

hi+1hih−1i+1 = h2i ,

where again i + 1 is calculated modulo 4. It is not hard to see that every element of Hi

can be expressed uniquely as hni hmi+1 for n,m ∈ Z. In particular hi and hi+1 respectively

generate disjoint cyclic free groups in Hi. Define H01 to be the free product of H0 and H1

amalgamated over the common free cyclic subgroup generated by h1, and H23 to be the freeproduct of H2 and H3 amalgamated over the common free cyclic subgroup generated byh3. By Corollary 8.11 from [117] {h0, h2} generates a free subgroup of both H01 and H23.Define H to be the free product of H01 and H23 amalgamated over the subgroup generatedby {h0, h2}. Again by Corollary 8.11 from [117] {h1, h3} generates a free subgroup of H.In particular H is an infinite group.

As mentioned before, Higman showed that the group H is not locally embeddable intofinite groups. Equivalently the system RH(x0, x1, x2, x3) consisting of the relations

xi+1xix−1i+1 = x2i ,

for i ∈ Z /4Z has no nontrivial finite models. This means that if F is any finite group andai ∈ F for i ∈ Z /4Z satisfy the system RH , i.e. satisfy

ai+1aia−1i+1 = a2i ,

for i ∈ Z /4Z , thena0 = a1 = a2 = a3 = 1F .

In fact suppose by contradiction that one of the ai’s is nontrivial. Define p to be thesmallest prime number dividing the order of one of the ai’s. Without loss of generality wecan assume that p divides the order of a0. From the equation

a1a0a−11 = a20,

one can obtain by inductionan1a0a

−n1 = a2

n

0 .

In particular if n is the order of a1 we have

a2n−1

0 = 1F ,

which shows that 2n − 1 is a multiple of the order of a0 and hence of p. In particular p isan odd prime and

2n ≡ 1 (modp) .

Thus by Fermat’s little theorem p−1 divides n, contradicting the assumption that p is thesmallest prime dividing the order of one of the ai’s.


A similar argument from [143] shows that the system RH has no nontrivial solution inGLn(C) for any n ∈ N. In fact suppose that ai ∈ GLn(C) for i ∈ Z /4Z satisfy the systemRH . The relation

ai+1aia−1i+1 = a2i

implies that ai and a2i are conjugate and, in particular, have the same eigenvalues. Thisimplies that the eigenvalues of ai are roots of unity. Considering the Jordan canonical formof ai shows that the absolute value of the entries of ani grows at most polynomially in n.Now the relation

ani+1aia−ni+1 = a2

n

i

shows that also the absolute value of the entries of a2n

i grows at most polynomially in n.It follows that ai is diagonalizable and, having roots of unity as eigenvalues, a finite orderelement of GLn(C). The same argument as before now shows that the ai’s are equal to theunity of GLn(C).

Andreas Thom showed in [146, Corollary 3.3] thatH does not have the Fc-approximationproperty where Fc is the class of finite groups endowed with a commutator-contractive in-variant length function. This is a strengthening of Higman’s result that H is not locallyembeddable into finite groups, since the trivial invariant length function on a group iscommutator-contractive. The following proposition is the main techinical result involvedin the proof; see [146, Theorem 3.2].

Proposition II.9.3. Suppose that G is a finite commutator contractive invariant lengthgroup and ε is a positive real number smaller than 1

176 . If ai for i ∈ Z /4Z are elements ofG satisfying the system RH up to ε, then

ℓ(ai) < 4ε

for every i ∈ Z /4Z .

In view of Proposition II.9.2 in order to conclude that Higman’s group H does not havethe Fc-approximation property, it is enough to show that there is a universal sentence ϕsuch that ϕG = 0 for every invariant length group G but ϕΓ = 1. Write

ψ(x0, x1, x2, x3) ≡ maxi∈Z/4Z

ℓ(xi) − 4 maxi∈Z/4Z

ℓ(xi+1xix−1i+1x

−2i )

where i+ 1 is calculated modulo 4, and define ϕ to be the universal sentence

supx0,x1,x2,x3

f(ψ)

where f is the functionx 7→ min {max {x, 0} , 1} .

Observe that by Proposition II.9.3 the interpretation of ϕ in any commutator-contractiveinvariant length group is 0, while the interpretation of ϕ in Higman’s group endowed withthe trivial length function is 1.


II.10 Rank rings and Kaplansky’s direct finiteness conjec-

ture

Recall that Kaplansky’s direct finiteness conjecture for a countable discrete group Γ assertsthat if K is a field, then the group algebra KΓ is directly finite. This means that if a, bare elements of KΓ such that ab = 1 then also ba = 1. This conjecture has been confirmedwhen Γ is a sofic group by Elek and Szabo in [55]. An alternative proof of this result hasbeen obtained in [30, Corollary 1.4] using the theory of cellular automata; see also [31,Section 8.15]. The proof of this result can be naturally presented within the framework ofrank rings.

Definition II.10.1. Suppose that R is a ring. A function N : R→ [0, 1] is a rank functionif:

• N(1) = 1;

• N(x) = 0 iff x = 0;

• N(xy) ≤ min {N(x), N(y)};

• N(x + y) ≤ N(x) +N(y).

If N is a rank function on R then

d(x, y) = N(x− y)

defines a metric that makes the function x 7→ x + a isometric and the functions x 7→ xaand x 7→ ax contractive for every a ∈ R. A ring endowed with a rank function is called arank ring.

In the context of rank rings, a term in the variables x1, . . . , xn is just a polynomial inthe indeterminates x1, . . . , xn. A basic formula is an expression of the form

N (p(x1, . . . , xn))

where p(x1, . . . , xn) is a term in the variables x1, . . . , xn. . formulae, sentences and theirinterpretation in a rank ring can then be defined starting from terms analogously as in thecase of invariant length groups. Also the notion of elementary property, and axiomatizableclass, saturated and countably saturated structure carry over without change.

Suppose that (Rn)n∈N is a sequence of rank rings and U is a free ultrafilter over N. Theultraproduct

∏U Rn is the quotient of the product ring

∏nRn by the ideal

IU =

{(xn) ∈

∏nRn

∣∣∣∣ limn→UNn(xn) = 0

}.

II.10. RANK RINGS AND KAPLANSKY’S DIRECT FINITENESS CONJECTURE 49

The function

NU (xn) = limn→U

Nn(xn)

induces a rank function in the quotient, making∏

U Rn a rank ring. Then all the Rn

coincide with the same rank ring R the corresponding ultraproduct will be called ultrapowerof R. The notion of representative sequence of an element of an ultraproduct of rank rings isdefined analogously as in the case of length groups. Los’ theorem and countable saturationof ultraproducts can be proved in this context in a way analogous to the case of invariantlength groups. In particular:

Theorem II.10.2 ( Los). Suppose that ϕ(x1, . . . , xk) is a formula for rank rings with freevariables x1, . . . , xk, (Rn)n∈N is a sequence of rank rings, and U is a free ultrafilter over N.If a(1), . . . , a(k) are elements of

∏nRn then

ϕ∏

U Rn

(a(1), . . . , a(k)

)= lim

n→UϕRn

(a(1)n , . . . , a(k)n

)

where a(i) is any representative sequence of(a(i)n

)n∈N

for i = 1, 2, . . . , k. In particular if ϕ

is a sentence then

ϕ∏

U Rn = limn→U

ϕRn .

A rank ring R such that for every x, y ∈ R

N(xy − 1) = N (yx− 1)

is called a finite rank ring . Clearly any finite rank ring is a directly finite ring. Moreoverthe class of finite rank rings is axiomatizable by the formula

supx

supy

|N(xy − 1) −N (yx− 1)| .

It follows that an ultraproduct of finite rank rings is a finite rank ring and in particulara directly finite ring. Exercise II.10.3 shows that there is a tight connection between rankfunctions on rings and length functions on groups.

Exercise II.10.3. Suppose that N is a rank function on a ring R. Define

ℓ(x) = N(x− 1),

for x ∈ R. Show that ℓ is a length function on the multiplicative group R× of invertibleelements of R, which is invariant if and only if R is a finite rank ring.

A natural example of finite rank rings is given by rings of matrices over an arbitraryfield K. Denote by Mn(K) the ring of n×n matrices with coefficients in K Suppose that ρ


is the usual matrix rank on Mn(K), i.e. ρ(x) for x ∈Mn(K) is the dimension of the rangeof x regarded as a linear operator on a K-vector space of dimension n. Define

N(x) =1

nρ(x)

for every x ∈Mn(K).

Exercise II.10.4. Prove that N is a rank function on Mn(K) as in Definition II.10.1.Show that moreover Mn(K) endowed with the rank N is a finite rank ring.

Another natural example of finite rank rings comes from von Neumann algebra theory:If τ is a faithful normalized trace on a von Neumann algebra M , then

Nτ (x) = τ (s(x)) ,

where s(x) is the support projection of x, is a rank function on M . Moreover M endowedwith the rank function Nτ is a finite rank ring.

Fix a free ultrafilter U over N. In this section the symbol∏

U Mn(K) will denote theultraproduct with respect to U of the sequence of matrix rings Mn(K) regarded as rankrings. By Exercise II.10.4 and Los’ theorem on ultraproducts

∏U Mn(K) is a finite rank

ring (and in particular a directly finite ring).

The rest of this section is dedicated to the proof from [54] that sofic groups satisfyKaplansky’s direct finiteness conjecture. The idea is that soficity of Γ together with thealgebraic embedding of Sn into Mn(K) obtained by sending σ to the associated permutationmatrix Pσ allows one to construct an injective *-homomorphism from the group algebraKΓ to the ultraproduct

∏U Mn(K). In order to do this one needs some relations between

the rank of a linear combination of permutation matrices and the lengths of the associatedpermutations. Explicit upper and lower bounds of the former in term of the latter onesare established in Exercise II.10.5 and Exercise II.10.6 respectively.

In the following for σ ∈ Sn denote by Pσ ∈ Mn(K) the associated permutation matrixas in Section II.2.

Exercise II.10.5. Show that

N (Pσ − I) ≤ ℓSn(σ)

where ℓSn is the Hamming invariant length function.

Hint. Define c (σ) the number of cycles of σ (including fixed points). Show that

N (Pσ − I) = 1 − c (σ)

n

by induction on the number of cycles.

II.10. RANK RINGS AND KAPLANSKY’S DIRECT FINITENESS CONJECTURE 51

Exercise II.10.6. Suppose that σ1, . . . , σk ∈ Sn and λ1, . . . , λk ∈ K \{0} . Define

ε = min1≤i≤k

(1 − ℓ(σi)) .

Prove that

N

(k∑

i=1

λiPσi

)≥ 1 − εk

k2.

Hint. Denote by {ei | i ∈ n} the canonical basis of Kn. Recall that Sn is assumed to act onthe set n = {0, 1, . . . , n− 1}. Define D to be a maximal subset of n such that for s, t ∈ Xand 1 ≤ i, j ≤ k such that either s 6= t or i 6= j one has

σi (s) 6= σj (t) .

Observe that if x ∈ span {ei | i ∈ D} then

k∑

i=1

λiPσi(x) 6= 0.

Infer that

N

(k∑

i=1

λiPσi

)≥ |D|

n.

By maximality of X for every s ∈ n there are t ∈ D and 1 ≤ i, j ≤ k such that

s = σ−1i σj (t)

When i, , j vary between 1 and k and t varies in D the expression

σ−1i σj (t)

attains at most εnk + k2 |D| values. Infer that

|D|n

≥ 1 − εk

k2.

Exercise II.10.5 and Exercise II.10.6 allow one to define a nontrivial morphism fromthe group algebra K (

∏U Sn) to

∏U Mn(K). This is the content of Exercise II.10.7.

Exercise II.10.7. Define, using Exercise II.10.5, a ring morphism Ψ : K (∏

U Sn) →∏U Mn(K). Prove using Exercise II.10.6 that if x1, . . . , xk ∈∏U Sn are such that ℓU (xi) =

1 for i = 1, 2, . . . , k and λ1, . . . , λk ∈ K \{0} then

N (λ1x1 + · · · + λkxk) ≥ 1

k.


One can now easily prove that a sofic group Γ satisfies Kaplansky’s finiteness conjecture.In fact if Γ is sofic then Γ embeds into

∏U Sn in such a way that the length of any element

in the range of Γ \{1Γ} is 1. This induces a ring morphism from KΓ into K (∏

U Sn).By composing it with the ring morphism described in Exercise II.10.7, we obtain a ringmorphism from KΓ into

∏U Mn(K) that is one to one by the second statement in Exercise

II.10.7. This shows that KΓ is isomorphic to a subring of the directly finite ring∏

U Mn(K).In particular KΓ is itself directly finite.

II.11 Logic for tracial von Neumann algebras

In the context of tracial von Neumann algebras a term p(x1, . . . , xn) in the variablesx1, . . . , xn is a noncommutative *-polynomial in x1, . . . , xn, i.e. a polynomial in the non-commuting variables x1, . . . , xn and x∗1, . . . , x

∗n. A basic formula is an expression of the

formτ (p(x1, . . . , xn)) ,

where p(x1, . . . , xn) is a noncommutative *-polynomial. General formulae can be obtainedfrom basic formulae composing with continuous functions or taking infima and supremaover norm bounded subsets of the von Neumann algebra or of the field of scalars. Moreformally if ϕ1, . . . , ϕm are formulae and f : Cm → C is a continuous function then

f (ϕ1, . . . , ϕm)

is a formula. Analogously if ϕ(x1, . . . , xn, y) is a formula then

inf‖y‖≤1

Re (ϕ (x1, . . . , xn, y))

andinf|λ|≤1

Re (ϕ (x1, . . . , xn, λ))

are formulae. Similarly one can replace inf with sup. The interpretation of a formula in atracial von Neumann algebra is defined in the obvious way by recursion on the complexity.For example

(τ(x∗x))12

is a formula usually abbreviated by ‖x‖2 whose interpretation in a tracial von Neumannalgebra (M) is the 2-norm on M associated with the trace τ . Analogously

sup‖x‖≤1

sup‖y‖≤1

‖x− y‖2

is a sentence (i.e. a formula without free variables) that holds in a tracial von Neumannalgebra M iff M is abelian, while

sup‖x‖≤1

inf|λ|≤1

‖x− λ‖2

II.11. LOGIC FOR TRACIAL VON NEUMANN ALGEBRAS 53

is a sentence which holds in (M) iff M is one-dimensional (i.e. isomorphic to C).The notion of elementary property and axiomatizable class of tracial von Neumann

algebras are defined as in the case of length groups or rank rings. In particular the pre-vious examples shows that the property of being abelian and the property of being one-dimensional are elementary. Exercise II.11.1 and Exercise II.11.2 show that the propertyof being a factor and, respectively, the property of being a II1 factor are elementary.

Recall that the center Z(M) of a von Neumann algebra M is the set of elements thatcommute with any other element of M . This is a weakly closed subalgebra of M and henceit is itself a von Neumann algebra. The von Neumann algebra M is called a factor if itscenter contains only the scalar multiples of the identity.

The unitary group U(M) of M is the multiplicative group of unitary elements of M , i.e.elements u satisfying uu∗ = u∗u = 1. Recall that it can be seen using the Borel functionalcalculus [14, I.4.3] that the set of linear combinations of projections of a von Neumannalgebra is dense in the σ-weak topology.

Exercise II.11.1. Suppose that M is a von Neumann algebra endowed with a faithfultrace τ . Show that M is a factor if and only if for every x ∈M

‖x− τ(x)‖2 ≤ supy∈M1

‖xy − yx‖2 . (II.2)

Conclude that the property of being a factor is elementary.

Hint. If M is not a factor then a nontrivial projection p in Z(M) violates Equation (II.2).If M is a factor and x is an element of the unit ball of M consider the strong closure ofthe convex hull of the orbit

{uxu∗ : u ∈ U(M)}of x under the action of U(M) on M by conjugation. Observe that such set endowedwith the Hilbert-Schmidt norm is isometrically isomorphic to a closed subset of the Hilbertspace L2 (M, τ) obtained from τ via the GNS construction [14, II.6.4]. It therefore has aunique element of minimal Hilbert-Schmidt norm x0, which by uniqueness must commutewith every element of U(M). Since the convex hull of U(M) is dense in the unit ball ofM by the Russo-Dye theorem [14, II.3.2.17], x0 belongs to the center of M . Moreoverby normality of the trace x0 must coincide with τ(x). Thus τ(x) can be approximatedin the Hilbert-Schmidt norm by convex combinations of elements of the form uxu∗, withu ∈ U(M). Equation (II.2) easily follows from this fact.

Exercise II.11.2. Fix an irrational number α ∈ (0, 1). Using the type classification offactors prove that a factor M is II1 if and only there is a projection of trace α. Deducethat the property of being a II1 factor is elementary.

Hint. Recall that by Theorem I.1.4 the trace in a II1 factor attains on projections all thevalues between 0 and 1.


Let us now define the ultraproduct∏

U Mn of a sequence (Mn) of tracial von Neumannalgebras with respect to a free ultrafilter U over N. Define ℓ∞(M) the C*-algebra ofsequences

(an)n∈N ∈∏

nMn

such thatsupn

‖an‖ < +∞

endowed with the norm‖(an)‖ = sup

n‖an‖ .

The ultraproduct∏

U Mn is the quotient of ℓ∞(M) with respect to the norm closed idealIU of sequences (a) such that

limn→U

τ (a∗nan) = 0

endowed with the faithful trace

τ ((an) + IU ) = limn→U

τ(an).

Being the quotient of a C*-algebra by a norm closed ideal,∏

U Mn is a C*-algebra. ExerciseII.11.3 shows that in fact

∏U Mn is always a von Neumann algebra. When the sequence

(Mn) is constantly equal to a tracial von Neumann algebra M , the ultraproduct∏

U Mn iscalled ultrapower of M and denoted by MU . The notion of representative sequence of anelement of an ultraproduct of tracial von Neumann algebras is defined analogously as inthe case of invariant length groups.

The notions of approximately finitely satisfiable (or consistent) and realized set offormulae and countably saturated structure introduced in Definition II.6.2 and II.6.3 forinvariant length groups admit obvious generalization to the setting of tracial von Neumannalgebra. An adaptation of the proof of Exercise II.6.4 shows that an ultraproduct of a se-quence of tracial von Neumann algebras is countably saturated. If moreover the elements ofthe sequence are separable, and the Continuum Hypothesis is assumed, then ultraproductsare in fact saturated (cf. the discussion after Exercise II.6.4).

Exercise II.11.3. Show that the unit ball of∏

U Mn is complete with respect to the2-norm. Infer that

∏U Mn is a von Neumann algebra.

Hint. Recall that an ultraproduct of a sequence of tracial von Neumann algebra is countablysaturated. Suppose that (xn)n∈N is a sequence in the unit ball of

∏U Mn which is Cauchy

with respect to the 2-norm. Define for every n ∈ N

εn = sup{‖xi − xj‖2 : i, j ≥ n

}.

Observe that (εn)n∈N is a vanishing sequence. Consider for every n ∈ N the formula ϕn (y)

inf‖y‖≤1

max {‖xn − y‖2 − εn, 0} .

II.11. LOGIC FOR TRACIAL VON NEUMANN ALGEBRAS 55

Argue that the set X of formulae containing ϕn (y) for every n ∈ N is approximately finitelysatisfiable and hence realized in

∏U Mn. A realization x of X is such that

‖xn − x‖2 ≤ εn

for every n ∈ N and hence the sequence (xn)n∈N converges to x. In order to conclude that∏U Mn is a von Neumann algebra it is enough to observe that, by completeness of its unit

ball with respect to the 2-norm and Kaplansky’s density theorem (see [14, I.9.1.3]),∏

U Mn

coincides with the von Neumann algebra generated by the GNS representation of∏

U Mn

associated with the canonical trace τ of∏

U Mn. Los’ theorem on ultraproducts also holds in this context without change.

Theorem II.11.4 ( Los’ for tracial von Neumann algebras). Suppose that ϕ (x1, . . . , xk)is a formula with free variables x1, . . . , xk, (Mn)n∈N is a sequence of tracial von Neumannalgebras, and U is a free ultrafilter over N. If a(1), . . . , a(k) are elements of

∏nMn then

ϕ∏

U Mn

(a(1), . . . , a(k)

)= lim

n→UϕMn

(a(1)n , . . . , a(k)n

),

where a(i) is any representative sequence of(a(i)n

)n∈N

for i = 1, 2, . . . , k. In particular if ϕ

is a sentence thenϕ∏

U Mn = limn→U

ϕMn .

In particular Los’ theorem implies that a tracial von Neumann algebra M is elementarilyequivalent to any ultrapower MU of M . This means that if ϕ is any sentence, then ϕ hasthe same evaluation in M and in MU . It follows that any two ultrapowers of M areelementarily equivalent. If moreover M is separable and the Continuum Hypothesis isassumed, then any two ultrapowers of M , being saturated and elementarily equivalent,are in fact isomorphic by a standard result in model theory (see Corollary 4.14 in [61]).Conversely if the Continuum Hypothesis fails then, assuming that M is a II1 factor, byTheorem 4.8 in [60] there exist nonisomorphic ultrapowers of M (and in fact 22

ℵ0 many byProposition 8.2 of [62]).

Theorem II.11.4 together with the fact that the property of being a II1 factor is elemen-tary (see Exercise II.11.2) shows that an ultraproduct of II1 factors is again a II1 factor.Analogously an ultraproduct of factors is a factor and an ultraproduct of abelian tracialvon Neumann algebras is abelian.

The trace on the von Neumann algebra naturally induces an invariant length functionon U(M), as shown in the following exercise.

Exercise II.11.5. Suppose that M is a tracial von Neumann algebra. Show that thefunction ℓ : U(M) → [0, 1] defined by

ℓ(u) =1

2‖u− 1‖2

is an invariant length function on U(M).


In particular if M is the algebra Mn(C) of n×n complex matrices, then U(M) coincideswith the group Un of n×n unitary matrices, and the induced length function on Un coincideswith the one considered in Section II.2.

Recall that in Section II.2 we introduced the following relation

max {‖x∗x− 1‖2 , ‖xx∗ − 1‖2} .

This is a formula ϕu in the logic for tracial von Neumann algebras. We have also mentionedthat the polar decomposition shows that such a formula is stable, i.e. every approximatesolution of ϕu (x) = 0 is close to an exact solution (and the estimate is uniform over alltracial von Neumann algebras). In model theoretic jargon this means that the zero-set ofthe interpretation ϕu in a given von Neumann algebra M—i.e. the unitary group U(M)—isa definable set as in [13, Definition 9.16]; see also [13, Proposition 9.19]. In particular it canbe inferred that the unitary group of an ultraproduct of tracial von Neumann algebras isthe ultraproduct of corresponding unitary groups. This is the content of Exercise II.11.6.

Exercise II.11.6. Suppose that (Mn)n∈N is a sequence of tracial von Neumann algebras,and U is a free ultrafilter on N. Show that any element of the unitary group of theultraproduct

∏U Mn admits a representative sequence of unitary elements. Conclude that

U (∏

U Mn) is isomorphic as invariant length group to the ultraproduct∏

U U(Mn) of thesequence of unitary groups of the Mn’s endowed with the invariant length described inExercise II.11.5.

In particular the unitary group of∏

U Mn(C) can be identified with the group∏

U Un

introduced in Section II.2.

Exercise II.11.7. The unitary group of∏

U Mn(C) contains a subset X of size continuumsuch that ‖u− v‖2 =

√2 for every pair of distinct elements u, v of X. Deduce that the

same conclusion holds for the unitary group of any ultraproduct∏

U Mn of a sequence ofII1 factors.

Hint. The first statement follows directly from Exercise II.11.6 and Exercise II.2.6. For thesecond statement observe that by Theorem I.1.4, if (Mn)n∈N is a sequence of II1 factors,and U is a free ultrafilter on N, then

∏U Mn (C) embeds into

∏U Mn.

II.12 The algebraic eigenvalues conjecture

Suppose that Γ is a (countable, discrete) group. Considering the particular case of thegroup algebra construction for the field C of complex numbers as in Section I.1 one obtainsthe complex group algebra CΓ of formal finite linear combinations

λ1γ1 + · · · + λkγk

II.12. THE ALGEBRAIC EIGENVALUES CONJECTURE 57

where λi ∈ C and γi ∈ Γ. The group ring ZΓ is the subring of CΓ of finite linear combina-tions

n1γ1 + · · · + nkγk

where ni ∈ Z and γi ∈ Γ. The natural action of CΓ on the Hilbert space ℓ2Γ defines aninclusion of CΓ into B

(ℓ2Γ). A conjecture due to Dodziuk, Linnell, Mathai, Schick, and

Yates known as algebraic eigenvalues conjectures (see [50]) asserts that elements x of ZΓregarded as linear operators on ℓ2Γ have algebraic integers as eigenvalues. Recall that acomplex number is called an algebraic integer if it is the root of a monic polynomial withinteger coefficients. The algebraic eigenvalues conjecture has been settled for sofic groupsby Andreas Thom in [144]. The proof involves the notion of ultraproduct of tracial vonNeumann algebras and can be naturally presented within the framework of logic for metricstructures.

The complex group algebra CΓ can be endowed with a linear involutive map x 7→ x∗

such that(λγ)∗ = λγ−1.

Recall that the trace τ on CΓ is defined by

τ

(∑

γ

λγγ

)= λ1Γ .

The weak closure LΓ of CΓ in B(ℓ2Γ)

is a von Neumann algebra containing CΓ as a *-subalgebra. The trace of CΓ admits a unique extension to a faithful normalized trace τ onLΓ. Moreover CΓ is dense in the unit ball of LΓ with respect to the 2-norm of LΓ definedby ‖x‖2 = τ(x∗x)

12 .

In the rest of the section the matrix algebra Mn(C) of n×n matrices with complex coef-ficients is regarded as a tracial von Neumann algebra endowed with the (unique) canonicalnormalized trace τn. If U is an ultrafilter over N then

∏U Mn(C) denotes the ultraproduct

of Mn(C) as tracial von Neumann algebras. (Note that this is different from the ultra-product of Mn(C) as rank rings.) Denote by

∏U Mn(Z) the closed self-adjoint subalgebra

of∏

U Mn(C) consisting of elements admitting representative sequences of matrices withinteger coefficients. Recall that Un denotes the group of unitary elements of Mn(C). If σis a permutation over n, then the associated permutation matrix Pσ is a unitary elementof Mn(C) such that τ (Pσ) = 1− ℓ(σ). This fact can be used to solve Exercise II.12.1. Theargument is analogous to the proof that sofic groups are hyperlinear (see Exercise II.2.4).

Exercise II.12.1. Suppose that Γ is a sofic group and U is a nonprincipal ultrafilterover N . Show that there is a trace preserving *-homomorphism of *-algebras from CΓ to∏

U Mn(C) sending ZΓ into∏

U Mn(Z).

Hint. Use, as at the end of Section II.2, the embedding of Sn into Mn (C) sending σ to itsassociated permutation matrix Pσ, recalling that τ(Pσ) = 1 − ℓ(σ).


Since the *-homomorphism from CΓ to∏

U Mn(C) obtained in Exercise II.12.1 is trace-preserving, it extends to an embedding of LΓ into

∏U Mn(C). This can be seen using

the GNS construction associated with τ (see [14, Section II.6.4]), or observing that byKaplansky’s density theorem [14, I.9.1.3] the operator norm unit ball of CΓ is dense inthe norm unit ball of LΓ with respect to the σ-strong topology (which coincides with thetopology induced by the Hilbert-Schmidt norm associated with τ).

Suppose now that M is a von Neumann algebra, and x is an operator in M . Consideringthe spectral projection p ∈ M on Ker (x− λI) shows that a complex number λ is aneigenvalue of x if and only if there is a nonzero projection p ∈ M such that (x− λ) p =0. It follows that if Ψ : M → N is an embedding of von Neumann algebra, then xand Ψ(x) have the same eigenvalues. This observation together with Exercise II.12.1 andits following observation allow one to conclude that in order to establish the algebraiceigenvalues conjecture for sofic groups it is enough to show that the elements of

∏U Mn(Z)

have algebraic eigenvalues. This is proved in the rest of this section using Los’ theorem onultraproducts together with a characterization of algebraic integers due to Thom.

Suppose in the following that λ ∈ C is not an algebraic integer. Theorem II.12.2 is aconsequence of Theorem 3.5 from [144].

Theorem II.12.2. Suppose that ε is a positive real number, and that N is a naturalnumber. There is a positive constant M(λ,N, ε) depending only on λ, N , and ε with thefollowing property: For every monic polynomial p with integer coefficients such that all thezeros of p have absolute value at most N the proportion of zeros of p at distance less than

1M(λ,N,ε) from λ is at most ε.

Corollary II.12.3 is a direct consequence of Theorem II.12.2.

Corollary II.12.3. Suppose that ε is a positive real number, and that N is a naturalnumber. Denote by M(λ,N, ε) the positive constant given by Theorem II.12.2. For everyfinite rank matrix with integer coefficients A of operator norm at most N there is a complexmatrix B of the same size of operator norm at most M (λ,N, ε) such that

‖B (λ−A) − I‖2 ≤ ε.

Proof. If A is a finite rank matrix with integer coefficients of norm at most N , then theminimal polynomial pA of A is a monic polynomial with integer coefficients whose zeroshave all absolute value at mostN . Since λ is not an algebraic integers, λ is not an eigenvalueof pA and hence λ− A is invertible. Moreover by the choice of M(λ,N, ε) the proportionof zeros of p at distance at least 1

M(λ,N,ε) from λ is at least 1 − ε. This means that if p

is the projection on the eigenspace corresponding to these eigenvalues, then τ (p) > 1 − ε.Define B = p (λ−A)−1, and observe that B has operator norm at most M (λ,N, ε) and

‖B (λ−A) − 1‖2 = ‖1 − p‖2 = τ (1 − p) ≤ ε.

II.13. ENTROPY 59

Define for every M > 0 the formula ϕM (x) in the language of tracial von Neumannalgebras by

inf‖y‖≤1

‖My (λ− x) − 1‖2 .

By Corollary II.12.3 if a is any element of Mn(Z) of operator norm at most N , then

ϕM(λ,N,ε)(a) ≤ ε

where M(λ,N, ε) is the constant given by Theorem II.12.2. By Los’ theorem on ultraprod-ucts the same is true for any element a of

∏U Mn(Z) of operator norm at most N . It follows

that for every a ∈∏U Mn(Z) and every positive real number ε there is b ∈∏U Mn(C) suchthat ‖b‖ ≤M(λ, ‖a‖ , ε) and

‖b (a− λ) − 1‖2 ≤ ε.

Therefore if p is a projection such that (a− λ) p = 0, then

‖p‖2 ≤ ‖(b (a− λ) − 1) p‖2 + ‖b (a− λ) p‖≤ ‖b (a− λ) − 1‖2≤ ε.

Being this true for every positive real number ε, p = 0. This shows that λ is not aneigenvalue of a for any element a of

∏U Mn(Z), concluding the proof that the elements of∏

U Mn (Z) have algebraic eigenvalues.

II.13 Entropy

II.13.1 Entropy of a single transformation

Suppose that X is a zero-dimensional compact metrizable space, i.e. a compact space witha countable clopen basis. Denote by T a transformation of X, i.e. a homeomorphismT : X → X. A (necessarily finite) clopen partition P of X is generating for T if for everypair of distinct points x, y of X there are C ∈ P and n ∈ N such that T nx ∈ C andT ny /∈ C. Suppose that T has a generating clopen partition P (observe that in generalthis might not exist). Consider P as a coloring of X and Pn as a coloring of Xn. Denotefor n ≥ 1 by Hn (P, T ) the number of possible colors of partial orbits of the form

(x, Tx, T 2x, . . . , T nx

)∈ Xn.

Equivalently Hn (P, T ) is the cardinality of the clopen partition of X consisting of sets

C0 ∩ T−1C1 ∩ . . . ∩ T−nCn,

where Ci ∈ P for i = 0, 1, 2, . . . , n.A sequence (an)n∈N of real numbers is called submultiplicative if an+m ≤ an · am for

every n,m ∈ N.


Exercise II.13.1. Show that the sequence

(Hn (P, T ))n∈N

is submultiplicative.

Fekete’s lemma from [66] asserts that if (an)n∈N is a submultiplicative sequence thenthe sequence

(1n log(an)

)n∈N converges to infn∈N 1

n log(an).

Exercise II.13.2. Prove Fekete’s lemma.

It follows from Fekete’s lemma and Exercise II.13.1 that the sequence

(1

nlog (Hn+m (P, T ))

)

n∈N

has a limit h (T ) that is by definition the entropy of the transformation T . Despite beingdefined in terms of the partition P, the entropy h (T ) of T does not in fact depend on thechoice of the clopen partition of X generating for T .

Exercise II.13.3. Show that the entropy of T does not depend from the choice of P .

Observe that h (T ) is at most log |P| for any clopen partition P of X generating forT . Recall that transformations T and T ′ of compact Hausdorff spaces X and X ′ aretopologically conjugate if there is a homeomorphism f : X → X ′ such that T ′ ◦ f = f ◦ Tfor every x ∈ X. It is not difficult to verify that entropy is a topological conjugacy invariant,that is, topologically conjugate transformations have the same entropy.

II.13.2 Entropy of an integer Bernoulli shift

Suppose that A is a finite alphabet of cardinality k and X is the space AZ. Consider theBernoulli shift T on X defined by

T (ai)i∈Z = (ai−1)i∈Z .

For a ∈ A consider the clopen set

Xa = {(ai)i∈Z : a0 = a}

and observe that P = {Xa | a ∈ A} is a clopen partition of X generating for T . It is nothard to verify that

Hn (P, T ) = kn,

for every n ∈ N, and hence

h (T ) = log(k).

II.13. ENTROPY 61

Gottschalk’s conjecture for Z asserts that if f : X → X is a continuous injectivefunction such that T ◦ f = f ◦T then f is surjective. In view of the conjugation invarianceof entropy, in order to establish Gottschalk’s conjecture for Z it is enough to show that if Yis any proper closed T -invariant subspace of X, then the entropy h (T |Y ) of the restrictionof T to Y is strictly smaller then the entropy h (T ) of X. Suppose that Y is a proper closedT -invariant subspace of X. Observe that

Q = {Xa ∩ Y : a ∈ A}is a generating partition for T |Y . It is easy to see that

Hn (Q, T |Y )

is the number of A-words of length n that appear in elements of Y . Since Y is a properclosed T -invariant subspace of X there is some n ∈ N such that

Hn (Q, T |Y ) < kn

and hence

h (T |Y , Y ) = infn

1

nlog (Hn (Q, T |Y )) < log(k) = h (T,X) .

This concludes the proof of Gottschalk’s conjecture for Z.

II.13.3 Tilings on amenable groups

Suppose in the following that Γ is a (countable, discrete) group. If E,F are subsets of Γ,then define

• F−E =⋂

γ∈E Fγ−1 = {x ∈ Γ |xE ⊂ F };

• F+E =⋃

γ∈E Fγ = {x ∈ Γ |xE ∩ F 6= ∅};

• ∂EF = F+E∖F−E .

The subset F of Γ is (E, δ)-invariant for some positive real number δ if

|∂EF ||F | < δ.

The group Γ is amenable if and only if for every finite subset E of Γ and every positive realnumber δ there is a finite (E, δ)-invariant subset F of Γ. This is equivalent to the existenceof a Følner sequence, i.e. a sequence (Fn)n∈N of finite subsets of Γ such that

limn

|∂EFn||Fn|

= 0

for every finite subset E of Γ.Suppose in the following that Γ is an amenable group. A function φ from the set [Γ]<ℵ0

of finite subsets of Γ to the set R+ of positive real numbers is called:


• subadditive if φ(a ∪ b) ≤ φ(a) + φ (b) and φ (∅) = 0;

• right invariant if φ(a) = φ(aγ) for every γ ∈ Γ.

Lindenstrauss and Weiss proved in [107] using the theory of quasi-tilings for amenablegroups introduced by Ornstein and Weiss in [119] that if φ : [Γ]<ℵ0 → R+ is a right-

invariant subadditive function, then the function E 7→ |φ(E)||E| has a Følner limit ℓ(φ). This

means that for every ε > 0 there is a finite subset E of Γ and a positive real number δ suchthat for every (E, δ)-invariant finite subset F of Γ

∣∣∣∣φ(F )

|F | − ℓ(φ)

∣∣∣∣ < ε.

If E and E′ are subsets of Γ, then an (E,E′)-tiling is a subset T of Γ such that the family{γE |γ ∈ T } is made of pairwise disjoint sets, while the family {γE′ |γ ∈ T } is a cover ofΓ.

Exercise II.13.4. If 1 ∈ E and E′ = EE−1, then there is an (E,E′)-tiling of Γ. Moreoverif T is any (E,E′)-tiling, then for every finite subset F of Γ

∣∣T ∩ F−E∣∣

|F | ≥ 1

|E′| −|∂E′(F )|

|F | .

Hint. To show existence consider any maximal set T with the property that the family{Tγ |γ ∈ E } contains pairwise disjoint elements. For the second statement, observe that

{γE′

∣∣∣γ ∈ T ∩ F+E′}

covers F , while

(T ∩ F+E′

)∖(T ∩ F−E

)⊂ F+E′

∖F−E′

= ∂E′(F ).

Thus

∣∣T ∩ F−E∣∣

|F | ≥

∣∣∣T ∩ F+E′∣∣∣− |∂E′(F )||F |

≥ 1

|E′| −|∂E′(F )|

|F | .

II.13. ENTROPY 63

II.13.4 Entropy of actions of amenable groups

Suppose that Γ y X is an action of an amenable group Γ on a zero-dimensional compactspace X admitting a generating clopen partition. This means that P is a clopen partitionof X such that for every x, y ∈ X distinct there are γ ∈ Γ and C ∈ P such that γ · x ∈ Cand γ · y /∈ C. Regard P as a coloring of X and, for any finite subset F of Γ, PF as acoloring of XF . Denote by HF (P,Γ,X) the number of possible colors of partial orbits

(γ · x)γ∈F ∈ XF .

Equivalently HF (P,Γ,X) is the cardinality of the clopen partition of X consisting of setsof the form ⋂

γ∈Fγ−1Cγ

for Cγ ∈ P.

Exercise II.13.5. The function F 7→ log (HF (P,Γ,X)) is right-invariant and subadditive.

It follows from Exercise II.13.5 and the Lindenstrauss-Weiss theorem on right-invariantsubadditive functions that there is a positive real number h (Γ,X), called entropy of theaction Γ y X, such that for every ε > 0 there is a finite subset E of Γ and a positive realnumber δ such that if F is an (E, δ)-invariant subset of Γ then

∣∣∣∣h (Γ,X) − 1

|F | log (HF (P,Γ,X))

∣∣∣∣ < ε.

As for the case of integer actions, it can be verified that the entropy h (Γ,X) does notdepend on the chosen generating clopen partition. Recall that two actions Γ y X andΓ y X ′ are topologically conjugate if there is a homeomorphism f : X → Y such that forevery γ ∈ Γ

f(γx) = γf(x).

It is easy to verify that topologically conjugate actions have the same entropy.

II.13.5 Entropy of Bernoulli shifts of an amenable group

Suppose that Γ is an amenable group, A is a finite alphabet of cardinality k, and AΓ is thespace of Γ-sequences of elements of A. The Bernoulli action of Γ on X is defined by

γ (ah)h∈Γ =(aγ−1h

)h∈Γ .

The clopen partition P of X containing for every a ∈ A the set

Xa ={

(ah)h∈Γ |a1 = a}


is generating for the Bernoulli action. It is not hard to verify that for every finite subsetF of Γ

HF

(P,Γ, AΓ

)= k|F |

and hence

h(Γ, AΓ

)= log(k).

Gottschalk’s conjecture for the group Γ asserts that if f : AΓ → AΓ is a continuousinjective function such that f (γx) = γf(x) for every γ ∈ Γ and x ∈ AΓ then f is surjective.As in the case of integers, in order to establish Gottschalk’s conjecture it is enough to showthat if Y is any proper closed invariant subspace of X then the entropy h (Γ, Y ) of theBernoulli action of Γ on Y is strictly smaller than log(k). The theory of tilings is useful toshow that a proper Bernoulli subshift has strictly smaller entropy.

Suppose that Y is a proper Bernoulli subshift. If F ⊂ Γ is finite, define YF to be theset of restrictions of elements of Y to F . Observe that HF (Γ, Y ) = |YF |. Moreover sinceY is a proper subshift of X there is a finite subset E of Γ such that 1 ∈ E and YE is aproper subset of AE . Define E′ = EE−1.

Exercise II.13.6. Show that for any finite subset F of Γ

1

|F | log |YF | ≤ logk −(

1

|E′| −|∂E′(F )|

|F |

)log

(k|E|

k|E| − 1

).

Hint. Pick an (E,E′)-tiling T . If F is a finite subset of Γ, then define

T− = T ∩ F−E

and

F ∗ = F\⋃

γ∈T−

γE.

Observe that

YF ⊂ AF ∗ ×∏

g∈T−

YgE. (II.3)

Deduce from Equation (II.3) and from Exercise II.13.4 that the conclusion holds.

It follows from Exercise II.13.6 that

h (Γ, Y ) ≤ log(k) − 1

|E′| log

(k|E|

k|E| − 1

)< log(k).

II.13. ENTROPY 65

II.13.6 Entropy of actions of sofic groups

Suppose that Γ y X is an action of a sofic group on a zero-dimensional space X. As beforewe will assume that there is a clopen partition P that is generating for the action, and wewill regard P as a coloring of X and Pn as a coloring of Xn. If F is a finite subset of Γand σ is a function from Γ to Sn, define

HF,δ (σ,P,Γ,X)

to be the number of colors (Ci)i∈n in Pn such that there is a sequence (xi)i∈n ∈ Xn suchthat for every γ ∈ F and for a proportion of at least (1 − δ) indexes i ∈ n, γ−1xi has colorCσ−1

γ (i). Suppose that Σ is a sofic approximation sequence of Γ, i.e. a sequence (σn)n∈N of

maps σn : Γ → Sn such that for every γ, γ′ ∈ Γ

limn→+∞

d(σn(γγ′), σn(γ)σn

(γ′))

= 0

andlim

n→+∞d (σn(γ), 1) = 1,

for all γ 6= 1Γ. Define hΣ,F,δ (P,Γ,X) to be

lim supn→+∞

1

nlog (HF,δ (σn,P,Γ,X)) .

The entropy hΣ (Γ,X) of the action Γ y X relative to the sofic approximation sequence Σis the infimum of hΣ,F,δ (P,Γ,X) when F varies among all finite subsets of Γ and δ variesamong all positive real numbers. Observe that, as before, hΣ (Γ,X) is at most log |P|,it does not depend on the generating finite clopen partition chosen, and it is invariantby topological conjugation. It is shown in Section 5 of [101] using the so called Rokhlinlemma for sofic approximations of countable amenable groups (see Section 4 of [101]) thatthe sofic entropy associated with any sofic approximation sequence of an amenable groupcoincide with the classical notion of entropy for actions of amenable groups. Nonethelessthe entropy of an action of a nonamenable sofic group can in general depend on the choiceof the sofic approximation sequence.

II.13.7 Bernoulli actions of sofic groups

Suppose that Γ is a sofic group, A is a finite alphabet, and Γ y AΓ is the Bernoulli actionof Γ with alphabet A of cardinality k. The entropy hΣ

(Γ, AΓ

)of the Bernoulli action with

respect to any sofic approximation sequence Σ is log(k). In fact consider as before theclopen partition P of AΓ consisting of the sets

Xa ={

(aγ)γ∈Γ |a1Γ = a}


for a ∈ A.In the following we will say that σ : Γ → Sn is a good enough sofic approximation if

d (σγη, σγση) < ε

for every η,γ ∈ F , where F is a large enough finite subset of Γ and ε is a small enoughpositive real number. It is not difficult to see that if σ is a good enough sofic approximationthen

HF,δ

(P,Γ, AΓ

)= kn.

It follows thathΣ(Γ, AΓ

)= log(k).

As in the case of amenable groups, Gottschalk’s conjecture for sofic groups can be provedby showing that a proper subshift of the Bernoulli shift has entropy strictly smaller thanlog(k). Suppose thus that Y is a proper closed invariant subspace of AΓ. It is easyto see that if some element of A does not appear as a digit in any element of Y thenhΣ (Γ, Y ) ≤ log (k − 1) < log(k). Thus without loss of generality we can assume that allelements of A appear as digits in some element of Y . Since Y is a proper closed subset ofX there is a finite subset F of Γ such that the set YF of restrictions of elements of Y to Fis a proper subset of AF . We will prove that, if N is the cardinality of F , then

infδ>0

hΣ,F,δ (P,Γ, Y ) ≤ log(k) − 1

N2log

(kN

kN − 1

).

Fix an element (bγ)γ∈F of AF \YF , a function σ : Γ → Sn for some n ∈ N, and η, δ > 0

such that δ |F | < η < 12|F |2+1

Lemma II.13.7. Suppose that (ci)i∈n ∈ An. If there is (xi)i∈n ∈ Y n such that for everyγ ∈ F

γ−1xi ∈ Xcσ−1γ (i)

for a proportion of i ∈ n larger than (1 − δ) then

1

n

∣∣∣∣∣∣⋂

γ∈Fσγ [{i ∈ n : ci = bγ}]

∣∣∣∣∣∣< δ |F | .

Proof. Define

B =⋂

γ∈Fσγ [{j ∈ n : cj = bγ}]

and suppose by contradiction that

|B|n

≥ δ |F | .

II.13. ENTROPY 67

Observe that there is a subset C of n such that

1

n|C| > 1 − δ |F |

and for every i ∈ C and γ ∈ Fγ−1xi ∈ Xc

σ−1γ (i)

.

It follows that there is i ∈ C ∩B. Define y = xi and observe that for every γ ∈ F

y ∈⋂

γ∈FγXσ−1

γ (i)

wherecσ−1

γ (i) = bγ .

This contradicts the fact that (bγ)γ∈F /∈ YF .

Denote by Z the set of i ∈ n such that for every distinct γ, γ′ ∈ F one has

σγ(i) 6= σγ(i).

Assuming that σ is a good enough sofic approximation we have

1

n|Z| > 1 − η

For every i ∈ Z consider the set

Vi ={σ−1γ (i) : γ ∈ F

}

and observe that |Vi| = F . Take a maximal subset Z ′ of Z subject to the condition thatVi and Vj are disjoint for distinct i and j in Z ′. Then by maximality

Z ⊂⋃

γ,γ′∈Fσ−1γ σγ′

[Z ′]

and hence1

n

∣∣Z ′∣∣ ≥ |Z|n |F |2

≥ 1 − η

|F |2

Denote by S the set of choices of colors c = (ci)i∈n ∈ An for which there is some Z ′′ ⊂ Z ′

such that1

n

∣∣Z ′′∣∣ > η

and for every γ ∈ F and i ∈ Z ′′ one has that cσ−1γ (i) = bγ . For any such c one has that

⋂

γ∈Fσ−1γ ({j ∈ n : cj = bγ}) ⊃ Z ′′


and hence

1

n

∣∣∣∣∣∣⋂

γ∈Fσ−1γ ({j ∈ n : cj = bγ})

∣∣∣∣∣∣≥ 1

n

∣∣Z ′′∣∣ > η.

By Lemma II.13.7 this shows that when σ is a good enough sofic approximation

H (Y, F, δ, σ) ≤ |An \S | .

Observe that if c = (ci)i∈n ∈ An \S then for every Z ′′ ⊂ Z ′ such that |Z ′′| > ηn there is

i ∈ Z ′′ such that the sequences (bγ)γ∈F and(cσ−1

γ (i)

)γ∈F

are distinct. This implies that

there is W ⊂ Z ′ such that |W | = |Z ′|−⌊ηn⌋ and for every i ∈W the sequence(σ−1γ (i)

)γ∈F

differs from the sequence (bγ)γ∈F . Therefore the number of elements of An \S is boundedfrom above by

( |Z ′||Z ′| − ⌊ηn⌋

)(|A||F | − 1

)|Z′|−⌊ηn⌋|A|n−(|Z′|−⌊ηn⌋)|F | . (II.4)

Define the function ξ (t) = −tlog (t) for t ∈ [0, 1], and observe that ξ is a concave function.By Stirling’s approximation formula (see [134] for a very short proof) the expression II.4is in turn bounded from above by

Cexp

(∣∣Z ′∣∣ ξ(

1 − ηn

|Z ′|

)+∣∣Z ′∣∣ ξ

(ηn

|Z ′|

))|A|n

(|A||F |

|A||F | − 1

)−(|Z′|−ηn)

(II.5)

for some constant C not depending on |Z ′| or n. From the fact that ξ is a concave functionand

1

n

∣∣Z ′∣∣ ≥ (1 − η)

|F |2> 2η

we obtain the estimate

ξ

(1 − ηn

|Z ′|

)+ ξ

(ηn

|Z ′|

)≤ ξ

(1 − η |F |2 n

1 − η

)+ ξ

(η |F |2 n1 − η

n

).

It follows that the quantity (II.5) is smaller than or equal to

Cexp

(nξ

(1 − η |F |2

1 − η

)+ nξ

(η |F |21 − η

))|A|n

(|A||F |

|A||F | − 1

)−(

1−η

|F |2−η

)

n

.

II.13. ENTROPY 69

Thus

1

nlog (HF,δ (σ,P,Γ, Y ))

≤ ξ

(1 − η |F |2

1 − η

)+ ξ

(η |F |21 − η

)+ log |A| −

(1 − η

|F |2− η

)log

(|A||F | − 1

|A||F |

)+ o(1).

Since this is true for every good enough sofic approximation σ, if Σ is any sofic approxi-mation sequence then

hΣ,δ,F (P,Γ, Y )

= lim supn→+∞

1

nlog (HF,δ (σn,P,Γ, Y ))

≤ ξ

(1 − η |F |2

1 − η

)+ ξ

(η |F |21 − η

)+ log |A| −

(1 − η

|F |2− η

)log

(|A||F | − 1

|A||F |

).

Being this true for every δ, η < 0 such that δ |F | < η < 12|F |2+1

, it follows that

infδ>0

hΣ,δ,F (P,Γ, Y ) ≤ log |A| − 1

|F |2log

(|A||F | − 1

|A||F |

)

as desired.

Chapter III

Connes’ embedding conjecture

Valerio Capraro

III.1 The hyperfinite II1 factor

Recall that a von Neumann algebra, as defined in Section I.1, is a weakly closed *-subalgebra of the algebra B (H) of bounded linear operators on a Hilbert space H. Afactor is just a von Neumann algebra whose center consists only of scalar multiples of theidentity. An infinite-dimensional factor endowed with a (necessarily unique and weaklycontinuous) faithful normalized trace is a II1 factor.

Definition III.1.1. Suppose that M and N are two factors. If F is a subset of M and εis a positive real number, then an (F, ε)-approximate morphism from M to N is a functionΦ : M → N such that Φ (1) = 1 and for every x, y ∈ F :

• ‖Φ(x+ y) − (Φ(x) + Φ(y))‖2 < ε;

• ‖Φ(xy) − Φ(x)Φ(y)‖2 < ε;

• |τM(x) − τN (Φ(x))| < ε.

A II1 factor M satisfies Connes’ embedding conjecture (or CEC for short) if for everyfinite subset F of M and every positive real number ε there is a natural number n and an(F, ε)-approximate morphism Φ : M →Mn(C).

Definition III.1.2. A finite von Neumann algebra is called hyperfinite if it contains anincreasing chain of copies of matrix algebras whose union is weakly dense.

Exercise III.1.3. Show that a separable hyperfinite II1 factor satisfies the CEC.

71

72 CHAPTER III. CONNES’ EMBEDDING CONJECTURE

Hyperfiniteness is a much stronger property than satisfying the CEC. In fact it is acornerstone result of Murray and von Neumann from [116] that there is a unique separablehyperfinite II1 factor up to isomorphism, usually denoted by R. The separable hyperfi-nite II1 factor admits several different characterizations. It can be seen as the group vonNeumann algebra (as defined in Section I.1) of the group Sfin

∞ of finitely supported permu-tations of N. Alternatively it can be described as the von Neumann algebra tensor product⊗

nM2(C) of countably many copies of the algebra of 2 × 2 matrices. This descriptionenlightens the useful property that R is tensorially self-absorbing, i.e. R ≃ R⊗R. A deepresult of Connes from [43] asserts that R is also the unique II1 factor that embeds in anyother separable II1 factor.

In the following all II1 factors are assumed to be separable, apart from ultrapowers ofseparable II1 factors that are never separable by Exercise II.11.7. Moreover R will denotethe (unique up to isomorphism) hyperfinite separable II1 factor.

The CEC can be equivalently reformulated in terms of local representability into R.

Definition III.1.4. A II1 factor M is locally representable in a II1 factor N if for everyfinite subset F of M and for every positive real number ε there is an (F, ε)-approximatemorphism from M to N .

Exercise III.1.5. Show that a separable II1 factor satisfies the CEC if and only if it islocally representable into R.

Local representability can be equivalently reformulated in terms of embedding into anultrapower, or in terms of values of universal sentences. (The notion of formula and sentencefor tracial von Neumann algebras has been defined in Section II.11.) The arguments areanalogous to the ones seen in Section II.6 and are left to the reader as Exercise III.1.6. Asin Section II.6 a formula is called universal if it is of the form

supx1

supx2

. . . supxn

ψ(x1, . . . , xn)

where no sup or inf appear in ψ.

Exercise III.1.6. Suppose that N and M are a separable II1 factors. Show that thefollowing statements are equivalent:

1. M is locally representable in N ;

2. M embeds into some or, equivalently, every ultrapower of N ;

3. ϕM ≤ ϕN for every universal sentence ϕ.

The universal theory of a II1 factor M is the function associating with any universalsentence its value in M . The existential theory of M is defined similarly, where universal

III.2. HYPERLINEAR GROUPS AND RADULESCU’S THEOREM 73

sentences are replaced by existential sentences. Since R embeds in any other II1 factor, theuniversal theory of R is minimal among the universal theory of II1 factors, i.e. ϕR ≤ ϕM forany II1 factor M . In view of this observation and Exercise III.1.5, the particular instance ofExercise III.1.6 when N is the hyperfinite II1 factor implies that the following statementsare equivalent:

1. M satisfies the CEC;

2. M embeds into some or, equivalently, every ultrapower of R;

3. M has the same universal theory as R.

In the terminology of [59], a II1 factor N is called locally universal if every II1 factor isfinitely representable in N . Thus CEC asserts that the separable hyperfinite II1 is locallyuniversal.. The existence of a locally universal II1 factor, which can be regarded as a sortof weak resolution of the CEC, is established in [59, Example 6.4]. It should be noted thaton the other hand by [121, Theorem 2] there is no universal separable II1 factor, i.e. thereis no separable II1 factor containing any other separable II1 factor as a subfactor

III.2 Hyperlinear groups and Radulescu’s Theorem

Hyperlinear groups were defined in Section II.2 in terms of local approximations by unitarygroups of matrix algebras (see Definition II.2.2). They can also be equivalently character-ized in terms of local approximations by the unitary group of the hyperfinite II1 factor.This is discussed in Exercise III.2.1.

Exercise III.2.1. Suppose that Γ is a countable discrete group. Regard Γ as an invariantlength group with respect to the trivial length function, and the unitary group U(R) ofR as an invariant length group with respect to the invariant length described in ExerciseII.11.5. Then the following statements are equivalent:

1. Γ is hyperlinear;

2. for every finite subset F of Γ and every positive real number ε there is an (F, ε)-approximate morphism (as in Definition II.1.7) from Γ to U(R);

3. there is an injective group homomorphism from Γ to U(R)U for some or, equivalently,any free ultrafilter U over N;

4. there is a length preserving group homomorphism from Γ to U(R)U for some or,equivalently, any free ultrafilter U over N.


Hint. The equivalence of 1 and 2 can be can be shown using the characterization ofhyperlinear groups given by Proposition II.2.9. The equivalence of 1, 3, and 4 can beestablished as in Exercise II.2.3 and Exercise II.2.10.

Observe that, by Exercise II.11.6, the ultrapower U(R)U of the unitary group of Rwith respect to the free ultrafilter U can be identified with the unitary group U(R) ofthe corresponding ultrapower of R. Therefore, in view of Exercise III.2.1, U(R) can beregarded as a universal hyperlinear group, analogously as the ultraproduct

∏U Un of the

finite rank unitary groups as we have seen in Section II.2. Proposition II.6.6 allows one toinfer (cf. Corollary II.6.7) that if the Continuum Hypothesis holds then the unitary groupof any ultrapower of R has 2ℵ1 outer automorphisms.

It is a consequence of condition (4) in Exercise III.2.1 that if Γ is a countable discretehyperlinear group, then the group von Neumann algebra LΓ of Γ (se Definition I.1.5)satisfies the CEC (this a result of Radulescu from [133]). In fact a length preservinghomomorphism from Γ to U(R) extends by linearity to a trace preserving embedding ofCΓ to RU . This in turn induces an embedding of LΓ into RU , witnessing that LΓ satisfiesthe CEC.

With a little extra care one can show that any (not necessarily countable) subgroup Γ ofthe unitary group of some ultrapower of R has the property that the group von Neumannalgebra LΓ of Γ embeds into a (possibly different) ultrapower of R. This is the content ofTheorem III.2.2, established by the first-named author and Paunescu in [24].

Theorem III.2.2. For any group Γ, the following conditions are equivalent:

1. Γ admits a group monomorphism into U(R) for some free ultrafilter U on N;

2. The group von Neumann algebra LΓ embeds into RV for some (possibly different)free ultrafilter V on N.

If the Continuum Hypothesis holds then, as discussed in Section II.11, all the ultra-powers of R as isomorphic. In particular one can always pick the same ultrafilter. It is notclear that this is still possible under the failure of the Continuum Hypothesis (see [21]).

In the rest of this section we will present the proof of Theorem III.2.2 that involves thenotion of product of ultrafilters. Let us denote for k ∈ N and B ⊂ N the vertical section{n ∈ N : (k, n) ∈ B} of B corresponding to k by Bk.

Definition III.2.3. Suppose that U , V are free ultrafilters on N. The (Fubini) productU × V is the free ultrafilter on N × N such that B ∈ U × V if and only if the set of k ∈ Nsuch that Bk ∈ V belongs to U .

Exercise III.2.4. Show that the operation × is not commutative.

Exercise III.2.5. Suppose that U , V are free ultrafilters on N. Show that U × V is a freeultrafilter on N× N. Moreover if (an,m)n,m∈N is a double-indexed sequence in R, then

limn→U

limm→V

an,m = lim(n,m)→U×V

an,m.

III.2. HYPERLINEAR GROUPS AND RADULESCU’S THEOREM 75

It follows from Exercise III.2.5 that an iterated ultrapower can be regarded as a singleultrapower. More generally:

Proposition III.2.6. If (Mn,m, τn,m) is a double-indexed sequence of tracial von Neumannalgebras, then ∏

U

(∏VMn,m

)≃∏

U×VMn,m.

In particular (MV)U ≃MU×V .

Exercise III.2.7. This exercise describes an amplification trick to be used in the proof ofTheorem III.2.2.

1. Let z ∈ C such that |z| ≤ 1 and |z + 1| = 2. Prove that z = 1.

2. Let u be a unitary in a II1 factor with trace 1. Show that u = 1.

3. Let u be a unitary in a II1 factor N and let M2(N) be the factor of two-by-twomatrices with entries in N and denote by τ its normalized trace. Consider theelement

u′ =

(u 00 1

).

Show that |τ(u′)| < 1.

Proof of Theorem III.2.2. (2) implies (1) is trivial. Conversely, let Γ be a group and θ :Γ → U(RU ) a group monomorphism. Define the new group homomorphism θ1 : Γ →U(M2(RU )) by

θ′(γ) =

(θ(γ) 0

0 1Γ

).

By Exercise III.2.7, this monomorphism verifies the property that |τ(θ′(γ))| < 1, for allγ 6= 1. Recall that R is up to isomorphism, the unique (separable) hyperfinite II1 factor(see Definition III.1.2). In particular R is isomorphic to R⊗R, as well as to the algebraM2(R) of 2 × 2 matrices over R. As a consequence one obtains an isomorphism betweenRU , (R⊗R)U , and M2(R). This allows one to regard θ1 as a group monomorphism fromΓ to the unitary group of RU itself.

Now define θ2 to be the map from Γ to the unitary group of (R)U defined in thefollowing way: If θ1(γ) has a representative sequence of unitaries (un)n∈N ∈ U(R) (thisexists by Exercise II.11.6) then θ2(γ) has representative sequence

(un ⊗ un)n∈N .

Identifying (R⊗R)U with RU one can regard θ2 as a map from Γ to the unitary groupof RU . Define analogously θn : Γ → U(R) for every natural number n taking the tensor


product of θ1 by itself (coordinatewise). Observe that θn is a monomorphism that moreoverhas the property that

|τ(θn(γ))| = |τ(θ1(γ))|n

for every γ ∈ Γ. Next define θ∞ from Γ the unitary group of the iterated ultrapower(RU )U (which by Exercise III.2.5 is isomorphic to RU×U) assigning γ to the element of(RU )U having

(θn(γ))n∈N

as representative sequence. It is easy to check that θ∞ is a group monomorphism verifyingthe additional property that τ (θ(γ)) = 0 for every γ 6= 1Γ. It is easy to infer from this, asin the discussion after Exercise III.2.1, that there is a trace-preserving embedding of LΓinto (RU )U ≃ RU×U .

III.3 Kirchberg’s theorem

As mentioned in the Introduction, Connes’ embedding conjecture can be reformulatedin several different ways. The purpose of this section is to present probably the mostunexpected of such reformulations, originally proved by Kirchberg in [102].

Theorem III.3.1. The following statements are equivalent:

1. Connes’ embedding conjecture holds true,

2. C∗(F∞) ⊗max C∗(F∞) = C∗(F∞) ⊗min C

∗(F∞).

We refer the reader to Appendix A for all definitions needed to understand the state-ment of Kirchberg’s theorem, as the minimal and maximal tensor product of C*-algebrasand the full C*-algebra associated to a locally compact group.

We are not going to present the original proof, but a more recent and completely differ-ent one provided by Haagerup and Winsløw [83], [84]. This is fundamentally a topologicalproof relying on the use of the Effros-Marechal topology on the space of von Neumannalgebras. To present a completely self-contained proof of this theorem is pretty much im-possible, since it relies on the Tomita-Takesaki modular theory. However, we will presentthe idea in quite many details, leaving out a few technical lemmas, whose proof is highlynon-trivial.

III.3.1 Effros-Marechal topology on the space of von Neumann algebras

Let H be a Hilbert space and vN(H) be the set of von Neumann algebras acting on H,that is the set of von Neumann subalgebras of B(H). The Effros-Marechal topology onvN(H), first introduced in Effros in [53] and Marechal in [113], can be defined is severaldifferent equivalent ways and we will in fact be playing with two different definitions.

III.3. KIRCHBERG’S THEOREM 77

Let us first recall some terminology. Given a net {Ca}a∈A on a direct set A, we say thata property P is eventually satisfied if there is a ∈ A such that, for all b ≥ a, Cb satisfiesproperty P. We say that P is frequently satisfied if for all a ∈ A there is b ≥ a such that Cb

satisfies property P .

First definition of the Effros-Marechal topology. Let X be a compact Hausdorffspace, c(X) be the set of closed subsets of X and N (x) be the filter of neighborhoods of apoint x ∈ X. N (x) is a direct set, ordered by inclusion. Let {Ca} be a net of subsets ofc(X), define

limCa = {x ∈ X : ∀N ∈ N (x), N ∩Ca 6= ∅ eventually} , (III.1)

limCa = {x ∈ X : ∀N ∈ N (x), N ∩ Ca 6= ∅ frequently} . (III.2)

It is clear that limCα ⊆ limCα, but the other inclusion is far from being true in general.

Exercise III.3.2. Find an explicit example of a sequence Cn of closed subsets of the realinterval [0, 1] such that limCn ( limCn.

Effros proved in [52] that there is only one topology on c(X), whose convergence isdescribed by the condition:

Ca → C if and only if limCa = limCa = C. (III.3)

Exercise III.3.3. Let M be a von Neumann subalgebra of B(H). Denote by Ball(M)the unit ball of M . Prove that Ball(M) is weakly compact in Ball(B(H)).

The previous exercise allows to use Effros’ convergence in our setting.

Definition III.3.4. Let {Ma} ⊆ vN(H) be a net. The Effros-Marechal topology is de-scribed by the following notion of convergence:

Ma →M if and only if limBall(Ma) = limBall(Ma) = Ball(M). (III.4)

Second definition of the Effros-Marechal topology. Recall that the strong*topology on B(H) is the weakest locally convex topology making the maps

x 7→ ‖xξ‖ and x 7→ ‖x∗ξ‖

continuous for every ξ ∈ H. Let x ∈ B(H) and let so∗(x) denote the filter of neighborhoodsof x with respect to the strong* topology.


Definition III.3.5. Let {Ma} ⊆ vN(H) be a net. We set

lim inf Ma = {x ∈ B(H) : ∀U ∈ so∗(x), U ∩Ma 6= ∅ eventually} . (III.5)

Observe that lim inf Ma is obviously so*-closed, contains the identity and it is closed un-der involution, scalar product and sum, since these operations are so*-continuous. Haagerupand Winsløw proved in [83], Lemma 2.2 and Theorem 2.3, that lim inf Ma is also closedunder multiplication. Therefore lim inf Ma is a von Neumann algebra. Moreover, by The-orem 2.6 in [83], lim inf Ma can be seen as the largest von Neumann algebra whose unitball is contained in limBall(Ma). This suggests to define lim supMa as the smallest vonNeumann algebra whose unit ball contains limBall(Ma), that is clearly (limBall(Ma))′′.Indeed, the double commutant of a subset of B(H) is always a *-algebra and the doublecommutant theorem of von Neumann states that this is the smallest von Neumann algebracontaining the set. So we are led to the following

Definition III.3.6. Let {Ma} ⊆ vN(H) be a net. We set

lim supMa :=(limBall(Ma)

)′′. (III.6)

Definition III.3.7. The Effros-Marechal topology on vN(H) is described by the followingnotion of convergence:

Ma →M if and only if lim inf Ma = lim supMa = M.

These two definitions of the Effros-Marechal topology are shown to be equivalent in[83], Theorem 2.8.

Connes’ embedding conjecture regards separable II1 factors, that is, factors with afaithful representation into B(H), with H is separable. Assuming separability on H, theEffros-Marechal topology on vN(H) turns out to be separable and induced by a completemetric (i.e. vN(H) is a Polish space). One possible distance is given by the Hausdorffdistance between the unit balls:

d(M,N) = max

{sup

x∈Ball(M)

{inf

y∈Ball(N)d(x, y)

}, supx∈Ball(N)

{inf

y∈Ball(M)d(x, y)

}}, (III.7)

where d is a metric on the unit ball of B(H) which induces the weak topology [113].Since vN(H) is separable, we may express its topology by using sequences instead of nets.This turns out to be particularly useful, since, if {Ma} = {Mn} is a sequence, then thedefinition of the Effros-Marechal topology may be simplified by making use of the seconddefinition. One has


lim inf Mn ={x ∈ B(H) : ∃{xn} ∈

∏Mn such that xn →s∗ x

}. (III.8)

We now state the main theorem of [84]. A part from being of intrinsic interest, it allowsto reformulate Kirchberg’s theorem in the form that we are going to prove.

Let us fix some notation: FIfin is the set of finite factors of type I acting on H, thatis the set of von Neumann factors that are isomorphic to a matrix algebra. FI is the setof type I factors acting on H, that is the set of von Neumann factors that are isomorphicto some B(H), with H separable. FAFD is the set of approximately finite dimensional orAFD factors acting on H, that is the set of factors containing an increasing chain of matrixalgebras whose union is weakly dense. Finally, Finj is the set of injective factors acting onH, that is the set of factors that are the image of a bounded linear projection of norm 1,P : B(H) →M .

Theorem III.3.8. (Haagerup-Winsløw) The following statements are equivalent:

1. FIfin is dense in vN(H),

2. FI is dense in vN(H),

3. FAFD is dense in vN(H),

4. Finj is dense in vN(H),

5. Connes’ embedding conjecture is true.

Moreover, a separable II1 factor M is embeddable into RU if and only if M ∈ Finj.

Since FIfin ⊆ FI ⊆ FAFD, the implications (1) ⇒ (2) ⇒ (3) are trivial. The implication(3) ⇒ (1) follows from the fact that AFD factors contain, by mere definition, an increasingchain of type Ifin factors, whose union is weakly dense and from the first definition of theEffros-Marechal topology (Definition III.3.4). The equivalence between (3) and (4) is atheorem by Alain Connes proved in [43]. What is really new and important in TheoremIII.3.8 is the equivalence between (4) and (5), proved in [84], Corollary 5.9, and the proofof the last sentence, proved in [84], Theorem 5.8.

III.3.2 Proof of Kirchberg’s theorem

By using Theorem III.3.8, it is enough to prove the following statements:

1. If FIfin is dense in vN(H), then C∗(F∞) ⊗min C∗(F∞) = C∗(F∞) ⊗max C

∗(F∞).

2. If C∗(F∞) ⊗min C∗(F∞) = C∗(F∞) ⊗max C

∗(F∞), then Finj is dense in vN(H).


Proof of (1). Let π be a *-representation of the algebraic tensor product C∗(F∞)⊙C∗(F∞)into B(H). Since C∗(F∞) is separable, we may assume that H is separable. In this way

A = π(C∗(F∞) ⊙ C1) and B = π(C1 ⊙ C∗(F∞))

belong to B(H), with H separable. Let {un} be the universal unitaries in C∗(F∞), as inExercises A.4 and A.5. Let

vn = π(un ⊗ 1) ∈ A and wn = π(1 ⊗ un) ∈ B.

Set M = A′′ ∈ vN(H). By hypothesis, there exists a sequence {Fm} ⊆ FIfin such thatFm → M in the Effros-Marechal topology. Therefore, A ⊆ M = lim inf Fm = lim supFm.Thus, we have

A ⊆ lim inf Fm.

It follows that

{vn} ⊆ U(A) ⊆ U(lim inf Fm) = limBall(Fm) ∩ U(B(H)),

where the equality follows from [83], Theorem 2.6. Let w(x) and so∗(x) respectively thefilters of weakly and strong* open neighborhoods of an element x ∈ B(H). We have justproved that for every n ∈ N and W ∈ w(vn), one has W ∩ Ball(Fm) ∩ U(B(H)) 6= ∅eventually in m. Let S ∈ so∗(vn), by [83] Lemma 2.4, there exists W ∈ w(vn) such thatW ∩ Ball(Fm) ∩ U(B(H)) ⊆ S ∩ Ball(Fm) ∩ U(B(H)). Now, since the first set must beeventually non empty, also the second one must be the same. This means that we canapproximate in the strong* topology vn with elements vm,n ∈ U(Fm).

In a similar way we can find unitaries wm,n in F ′m such that wm,n →so∗ wn. Indeed, we

have

B ⊆ A′ = M ′ = (lim supFm)′ = lim inf F ′m,

where the last equality follows by the commutant theorem ([83], Theorem 3.5). Therefore,we may repeat word-by-word the previous argument.

Now let m be fixed, πm,1 be a representation of C∗(F∞) mapping un to vm,n and πm,2

be a representation of C∗(F∞) mapping un to wm,n. We can find these representationsbecause the u′ns are free and because any representation of G extends to a representationof C∗(G). Notice that the ranges of these representations commute, since vn ∈ A andwn ∈ B and A,B commute. More precisely, the image of πm,1 belongs into C∗(Fm) andthe image of πm,2 belongs to C∗(F ′

m). So, by the universal property in Proposition A.10,there are unique representations πm of C∗(F∞) ⊗max C

∗(F∞) such that

πm(un ⊗ 1) = vm,n and πm(1 ⊗ un) = wm,n, m, n ∈ N,

whose image lies into C∗(Fm, F′m).


Exercise III.3.9. Prove that C∗(Fm, F′m) = Fm ⊗min F

′m (Hint: use Theorem 4.1.8(iii) in

[91] and Lemma 11.3.11 in [92]).

By Exercise III.3.9, C∗(Fm, F′m) = Fm ⊗min F

′m and thus πm splits: πm = σm ⊗ ρm, for

some σm, ρm representation of C∗(F∞) in C∗(Fm, F′m). Consequently, by the very definition

of minimal C*-norm, ||πm(x)|| ≤ ||x||min for all m ∈ N and x ∈ C∗(F∞) ⊙ C∗(F∞). Now,by Exercise A.5, the sequence {un} is total and therefore πm converges to π in the strong*point-wise sense. Namely, for all x ∈ C∗(F∞) ⊙ C∗(F∞), one has πm(x) →so∗ π(x).

Exercise III.3.10. Prove that if xn ∈ B(H) converges to x ∈ B(H) in the strong*topology, then ||x|| ≤ lim inf ||xn||.

Therefore

||π(x)|| ≤ lim inf ||πm(x)|| ≤ ||x||min, ∀x ∈ C∗(F∞) ⊙ C∗(F∞).

Since π is arbitrary, it follows that ||x||max ≤ ||x||min and the proof of the first implicationis complete.

In order to prove (2) we need a few more definitions and preliminary results. Given two*-representation π and ρ of the same C*-algebra A in the same B(H), we say that they areunitarily equivalent, and we write π ∼ ρ, if there is u ∈ U(B(H)) such that, for all x ∈ A,one has uπ(x)u∗ = ρ(x). Given a family of representations πa of the same C*-algebra inpossibly different B(Ha), we may define the direct sum

⊕a πa to be a representation of A

in B(⊕Ha) in the obvious way, that is,

(⊕πa

)(x)ξ =

⊕(πa(x)ξa) , for all ξ = (ξa) ∈

⊕Ha.

A family of representation is called separating if their direct sum is faithful (i.e. injective).We state the following deep lemma, proved by Haagerup and Winsløw in [84], Lemma 4.3,making use of Voiculescu’s Weyl-von Neumann theorem.

Lemma III.3.11. Let A be a unital C*-algebra and λ, ρ representations of A in B(H).Assume ρ is faithful and satisfies ρ ∼ ρ ⊕ ρ ⊕ · · · . Then there exists a sequence {un} ⊆U(B(H)) such that

unρ(x)u∗n →s∗ λ(x), ∀x ∈ A.

The other preliminary result is a classical theorem by Choi (see [37], Theorem 7).

Theorem III.3.12. Let F2 be the free group with two generators. Then C∗(F2) has aseparating family of finite dimensional representations.

Exercise III.3.13. Show that F∞ embeds into F2.


Proof of (2). By using Choi’s theorem and Exercise III.3.13 we can find a sequence σnof finite dimensional representations of C∗(F∞) such that σ = σ1 ⊕ σ2 ⊕ · · · is faithful.Replacing σ with the direct sum of countably many copies of itself, we may assume thatσ ∼ σ⊕σ⊕· · · . Now, ρ = σ⊗σ is a faithful (by [142] IV.4.9) representation of C∗(F∞)⊗min

C∗(F∞), because ρ splits. Moreover, ρ still satisfies ρ ∼ ρ ⊕ ρ ⊕ · · · . Now, given M ∈vN(H), let {vn}, {wn} be strong* dense sequences of unitaries in Ball(M) and Ball(M ′),respectively. Let {zn} be the universal unitaries representing free generators of F∞ inC∗(F∞), as in Exercises A.4 and A.5. Since the zn’s are free, we may find *-representationsλ1 and λ2 of C∗(F∞) in B(H) such that

λ1(zn) = vn and λ2(zn) = wn.

Since the ranges of these representations commute, we can apply the universal property inProposition A.10 to find a representation λ of C∗(F∞) ⊗min C

∗(F∞) such that

λ(zn ⊗ 1) = vn and λ(1 ⊗ zn) = wn, ∀n ∈ N,

where we can use the minimal norm instead of the maximal one thank to the hypothesisof the theorem. This means that λ and ρ satisfy the hypotheses of Lemma III.3.11 andtherefore there are unitaries un ∈ U(B(H)) such that

unρ(x)u∗n →so∗ λ(x), ∀x ∈ C∗(F∞) ⊗min C∗(F∞).

DefineMn = unρ(C∗(F∞) ⊗ C1)′′u∗n.

Therefore, using also Equation (III.8), we have

λ(C∗(F∞) ⊗ C1) = lim inf unρ(C∗(F∞) ⊗ C1)u∗n ⊆ lim inf Mn. (III.9)

Now, observe thatunρ(C1 ⊗C∗(F∞))u∗n ⊆M ′

n

and therefore

λ(C1 ⊗ C∗(F∞)) ⊆ lim inf M ′n. (III.10)

Since lim inf Ma is always a von Neumann algebra, the inclusions in (III.9) and (III.10)still hold passing to the weak closure. Therefore, by (III.9), we obtain

M = λ(C∗(F∞) ⊗ C1)′′ ⊆ lim inf Mn, (III.11)

where the equality with M follows from the fact that the vn’s are strong* dense in Ball(M).Analogously, by (III.10), we obtain

M ′ = λ(C1 ⊗ C∗(F∞))′′ ⊆ lim inf M ′n, (III.12)

III.4. CONNES’ EMBEDDING CONJECTURE AND LANCE’S WEP 83

where the equality with M ′ follows from the fact that the wn’s are strong* dense inBall(M ′).

Now, using (III.11) and (III.12) and applying the commutant theorem ([83], Theorem3.5), we get

M = M ′′ ⊇(lim inf M ′

n

)′= lim sup(Mn)′′ = lim supMn.

Therefore

lim supMn ⊆M ⊆ lim inf Mn,

i.e. Mn → M in the Effros-Marechal topology. Therefore, we have proved that everyvon Neumann algebra M can be approximated by von Neumann algebras Mn that areconstructed as strong closure of faithful representations that are direct sum of countablymany finite-dimensional representations. Such von Neumann algebras are injective andtherefore, we have proved that vNinj(H) is dense in vN(H). Now, by [83], Theorem 5.2,vNinj is a Gδ subset of vN(H) and by [83], Theorem 3.11, and [84], Theorem 2.5, the setof all factors F(H) is a dense Gδ-subset of vN(H). Since vN(H) is a Polish space, we canapply Baire’s theorem and conclude that also the intersection vNinj(H) ∩ F(H) = Finj(H)must be dense.

Let K denote the class of groups Γ satisfying Kirchberg’s property

C∗(Γ) ⊗min C∗(Γ) = C∗(Γ) ⊗max C

∗(Γ)

The following seems to be an open and interesting problem [20].

Problem III.3.14. Does K contain a countable discrete non-amenable group?

Another interesting question comes from the observation that the previous proof aswell as Kirchberg’s original proof uses free groups in a very strong way.

Problem III.3.15. For which groups Γ, does the property Γ ∈ K implies Connes’ embed-ding conjecture?

III.4 Connes’ embedding conjecture and Lance’s WEP

Kirchberg’s theorem III.3.1 is an astonishing link between the theory of von Neumannalgebras and the theory of C*-algebras. In the same paper, Kirchberg found anotherprofound link between these two theories: Connes’ embedding conjecture is a particularcase of a conjecture regarding the structure of C*-algebras, the QWEP conjecture, askingwhether any C*-algebra is a quotient of a C*-algebra with Lance’s WEP. It is then naturalto ask whether there is a direct relation between Connes’ embedding conjecture and WEP.Nate Brown answered this question in the affirmative, proving that Connes’ embeddingconjecture is equivalent to the analogue of Lance’s WEP for separable type II1 factors.


Definition III.4.1. Let A,B be C*-algebras. A linear map φ : A → B is called positiveif φ(a∗a) ≥ 0, for all a ∈ A.

Definition III.4.2. Let A,B be C*-algebras and φ : A → B be a linear map. For everyn ∈ N we can define a map φn : Mn(A) →Mn(B) by setting

φn[aij ] = [φ(aij)]

φ is called completely positive if φn is positive for every n.

Exercise III.4.3. Show that any *-homomorphism between C*-algebras is completelypositive.

Definition III.4.4. Let A ⊆ B be two C*-algebras. We say that A is weakly cp com-plemented in B if there exists a unital completely positive map φ : B → A∗∗ such thatφ|A = IdA.

Let A be a C*-algebra. We recall that there is always a faithful representation ofA into a suitable B(H), for instance the GNS representation. In other words, one canalways regard an abstract C*-algebra as a sub-C*-algebra of B(H) through a faithfulrepresentation. Hereafter, we use the notation A ⊆ B(H) to say that we have fixed oneparticular such representation. As we will see, the concepts we are going to introduce areindependent of the representation.

Definition III.4.5. A C*-algebra A has the weak expectation property (WEP, for short)if it is weakly cp complemented in B(H) for a faithful representation A ⊆ B(H).

The weak expectation property was introduced by Lance in[106].

Exercise III.4.6. Show that WEP does not depend on the faithful representation A ⊆B(H).

Representation-free characterizations of WEP have been recently proven in [63], [96],and [64] leading to new formulations of Connes’ Embedding Problem [64].

Definition III.4.7. A C*-algebra A has QWEP if it is a quotient of a C*-algebra withWEP.

QWEP conjecture states that every C*-algebra has QWEP. As mentioned above,in[102], Kirchberg proved also the following theorem.

Theorem III.4.8. Let M be a separable II1 factor. The following statements are equiva-lent

1. M is embeddable into some RU ,


2. M has QWEP.

We refer to the original paper by Kirchberg [102] and to the more recent survey byOzawa [120] for the proof of this theorem. In these notes we focus on a technically easierbut equally interesting topic: the von Neumann algebraic analogue of Lance’s WEP andthe proof of Brown’s theorem.

Definition III.4.9. Let M ⊆ B(H) be a von Neumann algebra and A ⊆ M a weaklydense C*-subalgebra. We say that M has a weak expectation relative to A if there exists aucp map Φ : B(H) →M such that Φ|A = IdA.

Theorem III.4.10. (Brown, [16] Th.1.2) For a separable type II1 factorM the followingconditions are equivalent:

1. M is embeddable into RU .

2. M has a weak expectation relative to some weakly dense subalgebra.

Observe that the notion of injectivity for von Neumann algebras can be stated also thisway: M ⊆ B(H) is injective if there exists a ucp map Φ : B(H) →M such that Φ(x) = x,for all x ∈ M . So relative weak expectation property is something just a bit less thaninjectivity. In fact Brown’s theorem can be read by saying that weak expectation relativeproperty is the “limit property of injectivity”.

Corollary III.4.11. For a separable type II1 factor the following conditions are equivalent:

1. M has a relative weak expectation property,

2. M has QWEP,

3. M is Effros-Marechal limit of injective factors.

Proof. It is an obvious consequence of Theorems III.4.10 and III.3.8.

The equivalence between 2. and 3. has been recently generalized to every von Neumannalgebra (see [2], Theorem 1.1).

We now move towards the proof of Theorem III.4.10. Let A be a separable C*-algebra.

Definition III.4.12. A tracial state on A is map τ : A+ → [0,∞] such that

1. τ(x + y) = τ(x) + τ(y), for all x, y ∈ A+;

2. τ(λx) = λτ(x), for all λ ≥ 0, x ∈ A+;

3. τ(x∗x) = τ(xx∗) for all x ∈ A;


4. τ(1) = 1.

A tracial state extends to a positive functional on the whole A. We will often identify thetracial state with its extension.

Definition III.4.13. A tracial state τ on A ⊆ B(H) is called invariant mean if thereexists a state ψ on B(H) such that

1. ψ(uTu∗) = ψ(T ), for all u ∈ U(A) and T ∈ B(H);

2. ψ|A = τ .

Note III.4.14. Theorem III.4.19 shows that the notion of invariant mean is independentof the choice of the faithful representation A ⊆ B(H).

In order to prove Brown’s theorem we need a characterization of invariant means thatwill be proven in Theorem III.4.19.

In order to prove such result we need two classical theorems and one lemma. Recallthat Lp(B(H)) stands for the ideal of linear and bounded operators a such that ||a||p :=

Tr(

(a∗a)p2

) 1p

is finite, where Tr is the canonical semifinite trace on B(H). The notation

Lp(B(H))+ stands for the cone of operators a in Lp(B(H)) that are positive, that is,(aξ, ξ) ≥ 0, for all ξ ∈ H. The following well known inequality is sometimes called Powers-Størmer inequality (see[130]).

Theorem III.4.15. Let h, k ∈ L1(B(H))+. Then

||h− k||22 ≤ ||h2 − k2||1,

In particular, if u ∈ U(B(H)) and h ≥ 0 has finite rank, then

||uh1/2 − h1/2u||2 = ||uh1/2u∗ − h1/2||2 ≤ ||uhu∗ − h||1/21 .

We now use Powers-Størmer’s inequality to prove a lemma. Recall that Mq (C) standsfor the von Neumann algebra of q-by-q matrices with complex entries.

Lemma III.4.16. Let H be a separable Hilbert space and h ∈ B(H) be a positive, finiterank operator with rational eigenvalues and Tr(h) = 1. Then there exists a ucp mapΦ : B(H) →Mq (C) such that

1. tr(Φ(T )) = Tr(hT ), for all T ∈ B(H);

2. |tr(Φ(uu∗) − Φ(u)Φ(u∗))| < 2||uhu∗ − h||1/21 , for all u ∈ U(B(H));

where tr stands for the normalized trace on Mq (C).


We prove the two statements separately.

Proof of (1). Let v1, . . . , vk ∈ H be the eigenvectors of H and p1q , . . . ,

pkq be the correspond-

ing eigenvalues. Thus

1. hvi = piq ;

2.∑k

i=1piq = tr(h) = 1. It follows that

∑pi = q.

Let {wm} be an orthonormal basis for H and let

B = {v1 ⊗ w1, . . . , v1 ⊗ vp1} ∪ {v2 ⊗ w1, . . . , v2 ⊗ wp2} ∪ · · · ∪ {vk ⊗ w1, . . . vk ⊗ wpk}

Let V be the subspace of H ⊗H spanned by B and let P : H ⊗H → V be the orthogonalprojection. Given T ∈ B(H), observe that the following formula holds

Tr(P (T ⊗ 1)P ) =

k∑

i=1

pi〈Tvi, vi〉.

Indeed P (T ⊗ 1)P is representable (in the basis B) by a q × q block diagonal matrixwhose blocks have dimension pi with entries ETE, where E : H → span{v1, . . . vk} is theorthogonal projection.

Now define Φ : B(H) →Mq (C) by setting Φ(T ) = P (T ⊗ 1)P . We have

tr(Φ(T ))

=1

qTr(P (T ⊗ 1)P )

=

k∑

i=1

〈T piqvi, vi〉

=k∑

i=1

〈Thvi, vi〉 = Tr(Th).

The following exercise concludes the proof of the first statement.

Exercise III.4.17. Prove that Φ is ucp.

Proof of (2). By writing down the matrix of P (T ⊗ 1)P (T ∗ ⊗ 1)P in the basis B we have

Tr(P (T ⊗ 1)P (T ∗ ⊗ 1)P ) =

k∑

i,j=1

|Ti,j |2 min(pi, pj),


where Ti,j = 〈Tvj, vi〉. Analogously, by writing down the matrices of h1/2T, Th1/2 andh1/2Th1/2T ∗ in any orthonormal basis beginning with {v1, . . . vk} we have

Tr(h1/2Th1/2T ∗) =

k∑

i,j=1

1

q(pipj)

1/2|Ti,j |2.

By using these formulae, we can make the following preliminary calculation

|Tr(h1/2Th1/2T ∗) − tr(Φ(T )Φ(T ∗))|

=

∣∣∣∣∣∣

k∑

i,j=1

1

q(pipj)

1/2|Ti,j|2 −1

qTr(P (T ⊗ 1)P (T ∗ ⊗ 1)P )

∣∣∣∣∣∣

=

∣∣∣∣∣∣

k∑

i,j=1

1

q|Ti,j|2((pipj)

1/2 −min(pi, pj)

∣∣∣∣∣∣

≤k∑

i,j=1

1

q|Ti,j|2p1/2i

∣∣∣p1/2j − p1/2i

∣∣∣

≤

k∑

i,j=1

1

q|Ti,j |2pi

1/2

k∑

i,j=1

1

q|Ti,j |2

(p1/2i − p

1/2j

)

1/2

= ||Th1/2||2||h1/2T − Th1/2||2,

where the first inequality follows by the property min(pi, pj) ≤ pi and the second one isobtained by applying the classical Holder inequality. Now suppose that T ∈ U(B(H)), sothat ||Th1/2||2 = ||h1/2||2 = 1. Using again the Powers-Størmer inequality, we can continuethe previous computation as follows:

||h1/2T − Th1/2||2= ||Th1/2T ∗ − h1/2||2≤ ||ThT ∗ − T ||1/21

We can now conclude the proof. Indeed, using the triangle inequality, the previous com-


putation and the Cauchy-Schwarz inequality, we find

|Tr(Φ(TT ∗) − Φ(T )Φ(T ∗))|≤ |1 − Tr(h1/2Th1/2T ∗)| + ||ThT ∗ − h||1/21

= |Tr(ThT ∗) − Tr(h1/2Th1/2T ∗)| + ||ThT ∗ − h||1/21

= |Tr((Th1/2 − h1/2T )h1/2T ∗)| + ||ThT ∗ − h||1/21

≤ ||h1/2T ∗||2||Th1/2 − h1/2T ||2 + ||ThT ∗ − h||1/21

≤ 2||ThT − h||121 ,

where the last inequality follows by using the fact that T is unitary and by applyingPowers-Størmer inequality once again.

The last preliminary result needed for the proof of Theorem III.4.19 is classical theoremby Choi [36].

Theorem III.4.18. Let A,B be two C*-algebras and Φ : A→ B be a ucp map. Then

{a ∈ A : Φ(aa∗) = Φ(a)Φ(a∗),Φ(a∗a) = Φ(a∗)Φ(a)}

= {a ∈ A : Φ(ab) = Φ(a)Φ(b),Φ(ba) = Φ(b)Φ(a),∀b ∈ A}.We are now ready to prove a useful characterization of invariant means, first proved in

[103], Proposition 3.2.

Theorem III.4.19. Let τ be a tracial state on A ⊆ B(H). Then the following are equiv-alent:

1. τ is an invariant mean.

2. There exists a sequence of ucp maps Φn : A→ Mk(n) such that

(a) ||Φn(ab) − Φn(a)Φn(b)||2 → 0 for all a, b ∈ A,

(b) τ(a) = limn→∞tr(Φn(a)), for all a ∈ A.

3. For any faithful representation ρ : A → B(H) there exists a ucp map Φ : B(H) →πτ (A)′′ such that Φ(ρ(a)) = πτ (a), for all a ∈ A, where πτ stands for the GNSrepresentation associated to τ∗.

∗We briefly recall the GNS construction (see [69] and [136]). Setting 〈x, y〉 := τ (x∗y), one obtains apossibly singular inner product on A. Let I be the subspace of elements x such that 〈x, x〉 = 0 and considerthe Hilbert space H obtained by completing A/I with respect to 〈·, ·〉. Using the fact that I is a left ideal,one sees that the operator πτ (x) defined by πτ (x)(y) = xy passes to a linear and bounded operator fromH to itself and then the mapping x → πτ (x) is a representation of A into B(H). It turns out that thisrepresentation is always faithful: if A is unital, then this is immediate. Otherwise, one has to use a moresubtle argument using approximate identities in C*-algebras.


Proof of 1) ⇒ 2). Let τ be an invariant mean with respect to the faithful representationρ : A → B(H). Thus we can find a state ψ on B(H) which extends τ and such thatψ(uTu∗) = ψ(T ), for all u ∈ U(A) and for all T ∈ B(H). Since the normal states areweak* dense in the dual of B(H) and they can be represented in the form Tr(h·), with h ∈L1(B(H)), we can find a net hλ ∈ L1(B(H)) such that Tr(hλT ) → ψ(T ), for all T ∈ B(H).Moreover, the representatives hλ can be chosen to be positive and with trace 1. Now, sinceψ(uTu∗) = ψ(T ), it follows that Tr(uhλu

∗T ) = Tr(hλu∗Tu) → ψ(u∗Tu) = ψ(T ) and thus

Tr(hλT )−Tr((uhλu∗)T ) → 0, for all T ∈ B(H), i.e. hλ−uhλu∗ → 0 in the weak topologyof L1(B(H)). Now let {Un} be an increasing family of finite sets of unitaries whose unionhave dense linear span in A and ε = 1

n . Let Un = {u1, . . . , un}. Fixed n, let us consider theconvex hull of the set {u1hλu∗1−hλ, . . . , unhλu∗n−hλ}. Its weak closure contains 0 (becauseof the previous observation) and coincide with the 1-norm closure, by the Hahn-Banachseparation theorem. Thus there exists a convex combination of the hλ’s, say h, such that

1. Tr(h) = 1,

2. ||uhu∗ − h||1 < ε,∀u ∈ Un,

3. |Tr(uh) − τ(u)| < ε,∀u ∈ Un.

Moreover, since finite rank operators are norm dense in L1(B(H)), we can assume withoutloss of generality that h has finite rank with rational eigenvalues. Now we can apply LemmaIII.4.16 in order to construct a sequence of ucp maps Φn : B(H) → Mk(n) such that

1. Tr(Φn(u)) → τ(u),

2. |Tr(Φn(uu∗)) − Φn(u)Φn(u∗)| → 0,

for every unitary in a countable set whose linear span is dense in A. So the property 2.(b)is true for unitaries and consequently, being a linear property, it holds for all operators.

In order to prove 2.(a), we observe that Φn(uu∗) − Φn(u)Φn(u∗) ≥ 0 and thus thefollowing inequality holds

||1 − Φn(u)Φn(u∗)||22 ≤ ||1 − Φn(u)Φn(u∗)||tr(Φn(uu∗) − Φn(u)Φn(u∗))

Since the right hand side tends to zero, also the left hand side must converge to 0. Nowdefine Φ = ⊕Φn : A → ∏

Mk(n) ⊆ ℓ∞(R) and compose with the quotient map p :

ℓ∞(R) → RU . The previous inequality shows that if u is a unitary such that ||Φn(uu∗) −Φn(u)Φn(u∗)||2 → 0 and ||Φn(u∗u)−Φn(u∗)Φn(u)||2 → 0, then u falls in the multiplicativedomain of p ◦ Φ. But such unitaries have dense linear span in A and hence the whole Afalls in the multiplicative domain of p ◦ Φ (by Choi’s theorem III.4.18). By definition ofultraproduct this just means that ||Φn(ab) − Φn(a)Φn(b)||2 → 0, for all a ∈ A.


Proof of 2) ⇒ 3). Let Φn : A→ Mk(n) be a sequence of ucp maps with the properties statedin the theorem. By identifying each Mk(n) with a unital subfactor of R we can define a ucp

map Φ : A → ℓ∞(R) by x → (Φn(x))n. Since the Φ′ns are asymptotically multiplicative

in the 2-norm one gets a τ -preserving *-homomorphism A → RU by composing with thequotient map p : l∞(R → RU . Note that the weak closure of p◦Φ(A) into RU is isomorphicto πτ (A)′′. Thus we are in the following situation

AΦ //

ρ

��

ℓ∞(R)p

// RU ⊇ p ◦ Φ(A)w ∼= πτ (A)′′

B(H)

i��

B(K)

where K is a representing Hilbert space for ℓ∞(R) and i is a natural embedding inducedby an embedding H → K, whose existence is guaranteed by the fact that H is separable.Since ℓ∞(R) is injective, there is a projection E : B(K) → ℓ∞(R) of norm 1. Let F :RU → πτ (A)′′ be a conditional expectation (see[142], Proposition 2.36), one has

AΦ //

ρ

��

l∞(R)p

// RU F// πτ (A)′′ ∼= p ◦ Φ(A)w

B(H)

i��

B(K)

E

DD✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠

Define Φ : B(H) → πτ (A)′′ by setting Φ = FpEi. One has Φ(ρ(a)) = πτ (a).

Proof of 3) ⇒ 1). The hypothesis Φ(a) = πτ (a) guarantees that Φ is multiplicative onA, since πτ is a representation. By Choi’s theorem III.4.18 it follows that Φ(aTb) =πτ (a)Φ(T )πτ (b), for all a, b ∈ A,T ∈ B(H). Let τ ′′ be the vector trace on πτ (A)′′ andconsider τ ′′ ◦ Φ. Clearly it extends τ . Moreover it is invariant under the action of U(A),indeed

(τ ′′ ◦ Φ)(u∗Tu) = τ ′′(πτ (u)∗Φ(T )πτ (u)) = τ ′′(Φ(T )) = (τ ′′ ◦ Φ)(T )

Hence τ is an invariant mean.

The following proposition was also proved by Nate Brown in[16].

Proposition III.4.20. Let M be a separable II1 factor. There exists a *-monomorphismρ : C∗(F∞) →M such that ρ(C∗(F∞)) is weakly dense in M .


Proof. Observe that C∗(F∞) can be viewed as inductive limit of free products of copies ofitself. This can be proven by partitioning the set of generators in a sequenceXn of countablesets, by defining An = C∗(X1, . . . ,Xn) and by observing that An = An−1 ∗ C∗(Xn) ∼=C∗(F∞), where ∗ stands for the free product with amalgamation over the scalars. Now,by Choi’s theorem III.3.12 we can find a sequence of integers {k(n)}n∈N and a unital *-monomorphism σ : A→ ∏

n∈N Mk(n). Note that we may naturally identify each Ai with asubalgebra of A and hence, restricting σ to this copy, get an injection of Ai into

∏Mk(n).

Assume we can prove the existence of a sequence of unital *-homomorphism ρi : Ai →M such that:

1. Each ρi is injective;

2. ρi+1|Ai= ρi where we identify Ai with the ”left side” of Ai ∗ C∗(F∞) = Ai+1;

3. The union of {ρi(Ai)} is weakly dense in M .

then we would have completed the proof. Indeed, it would be enough to define ρ as theunion of the ρi’s.

The purpose is then to prove existence of such a sequence ρi. To this end we firstchoose an increasing sequence of projections of M such that τM(pi) → 1. Then we definethe orthogonal projections qn = pn − pn−1 and consider the II1 factors Qi = qiMqi. Now,by the division property of II1 factors (see Theorem I.1.4), we can find a unital embedding∏

Mk(n) → Qi ⊆ M . By composing with σ, we get a sequence of embeddings A → M ,which will be denoted by σi. Now piMpi is separable and thus there is a countable familyof unitaries whose finite linear combinations are dense in the weak topology. Hence wecan find a *-homomorphism πi : C∗(F∞) → piMpi with weakly dense range (take thegenerators of F∞ into C∗(F∞) and map them into that total family of unitaries). Now wedefine

ρ1 = π1 ⊕

⊕

j≥2

σj|A1

: A1 → p1Mp1 ⊕ (Πj≥2Qj) ⊆M

ρ1 is a *-monomorphism, since each σi is already faithful on the whole A.Now define a *-homomorphism θ2 : A2 = A1 ∗C∗(F∞) → p2Mp2 as the free product of

the *-homomorphism A1 → p2Mp2 defined by x→ p2ρ1(x)p2 and π2 : C∗(F∞) → p2Mp2.We then set

ρ2 = θ2 ⊕

⊕

j≥3

σj |A2

: A2 → p2Mp2 ⊕ (Πj≥3Qj) ⊆M

Clearly ρ2|A1 = ρ1. In general, we construct a map θn+1 : An ∗ C∗(F∞) → pn+1Mpn+1 asthe free product of the cutdown (by pn+1) of ρn and πn. This map need not be injectiveand hence we take a direct sum with ⊕j≥n+2σj |An+1 to remedy this deficiency. These mapshave all the required properties and hence the proof is complete (note that the last propertyfollows from the fact that the range of each θn is weakly dense in pn+1Mpn+1).


Theorem III.4.21. (N.P. Brown [16]) Let M be a separable II1 factor and U be a freeultrafilter on the natural numbers. The following conditions are equivalent:

1. M is embeddable into RU .

2. M has the weak expectation property relative to some weakly dense subalgebra.

Prood of (1) ⇒ (2). Let M be embeddable into RU . By Prop.III.4.20, we may replace Mwith a weakly dense subalgebra A isomorphic to C∗(F∞). We want to prove that M hasthe weak expectation property relative to A. Let τ the unique normalized trace on M ,more precisely we will prove that πτ (M) has the weak expectation property relative toπτ (A). Indeed τ is faithful and w-continuous and hence πτ (M) and πτ (A) are respectivelycopies of M and A and πτ (A) is still weakly dense in πτ (M). We first prove that τ |A isan invariant mean. Take {un} universal generators of F∞ into A. Let n be fixed, sinceun ∈ RU , then un is || · ||2-ultralimit of unitaries in R (see Exercise II.11.6). On the otherhand, the unitary matrices are weakly dense in U(R) and hence they are || · ||2-dense inU(R) (since weakly closed convex subsets coincide with the || · ||2-closed convex ones (see,e.g.,[88])). Thus we can find a sequence of unitary matrices which converges to un in norm|| · ||2. Let σ be the mapping which sends each un to such a sequence. Since the un’s haveno relations, we can extend σ to a *-homomorphism σ : C∗(F∞) → ∏

Mk (C) ⊆ ℓ∞(R).Let p : ℓ∞(R) → RU be the quotient mapping. By the 2-norm convergence we have(p ◦ σ)(x) = x for all x ∈ C∗(F∞). Let pn :

∏∞k=1Mk (C) → Mn (C) be the projection, by

the definition of the trace in RU , we have

τ(x) = limn→U

trn(pn(σ(x))),

where trn is the normalized trace on Mn (C). Now we can apply Theorem III.4.19,2) bysetting φn = pn ◦ σ (they are ucp since they are *-homomorphisms) and conclude thatτ |A is an invariant mean. Now consider πτ (M) ⊆ B(H) and πτ (A) = πτ |A(A) ⊆ B(H).By Theorem III.4.19 there exists a ucp map Φ : B(H) → πτ (A)′′ = πτ (M) such thatΦ(a) = πτ (a). Thus M has the weak expectation property relative to C∗(F∞).

Proof of (2) ⇒ (1). Let A ⊆M ⊆ B(H), with A weakly dense in M , and Φ : B(H) →M aucp map which restricts to the identity on A. Let τ be the unique normalized trace on M .After identifying A with πτ (A), we are under the hypothesis of Theorem III.4.19.3) andthus τ |A is an invariant mean. By Theorem III.4.19 it follows that there exists a sequenceφn : A→ Mk(n) such that

1. ||φn(ab) − φn(a)φn(b)||2 → 0 for all a, b ∈ A,

2. τ(a) = limn→∞ trn(φn(a)), for all a ∈ A.

Let p : ℓ∞(R) → RU be the quotient mapping. The previous properties guarantee thatthe ucp mapping Φ : A→ RU , defined by setting Φ(x) = p({φn(x)}) is a *-homomorphism


which preserves τ |A. It follows that Φ is injective. Indeed, Φ(x) = 0 ⇒ Φ(x∗x) = 0 ⇒τ(x∗x) = 0 ⇒ x = 0. Observe now that the weak closure of A into RU is isomorphic toM (they are algebraically isomorphic and have the same trace) and hence M embeds intoRU .

III.5 Algebraic reformulation of the conjecture

In this section we present a new line of research that has been initially designed byRadulescu, with his proof that Connes’ embedding conjecture is equivalent to a non-commutative analogue of Hilbert’s 17th problem, and continued by Klep and Schweighoferfirst and Juschenko and Popovich afterwards, who arrived to a purely algebraic reformu-lation of Connes embedding conjecture. This section is merely descriptive and serves tointroduce the reader to a new field of research, which, though motivated by the Connesembedding conjecture, is quite far from geometric group theory and operator theory, whichare the main topics of this monography. The reader interested in technical details of thisapproach, is referred to the original papers [132],[104],[90].

We begin with a short description of the original formulation of Hilbert’s problem andwe show hot to get to Radulescu’s formulation through a series of generalizations.

Let R[x1, . . . , xn] denote the ring of polynomials with n indeterminates and real coeffi-cients and R(x1, . . . , xn) denote its quotient field. A polynomial f ∈ R[x1, . . . , xn] is callednon-negative if, for all (x1, . . . , xn) ∈ Rn, one has f(x1, . . . , xn) ≥ 0.

Problem III.5.1 (Hilbert’s 17th problem). Can every non-negative polynomial be ex-pressed as sum of squares of elements belonging to R(x1, . . . , xn)?

This problem was solved in the affirmative by Emil Artin [3], who provided an abstractproof of existence of such a sum. More recently, Delzell[49] provided an explicit algorithm.

More recently, scholars have been looking for challenging generalizations of this prob-lem, the most intuitive of which is the one concerning matrices. Consider positive semi-definite matrices with entries in R[x1, . . . , xn], that is, matrices that are positive semi-definite for all substitution (x1, . . . , xn). Observe that the matrix analogue of a square, isa positive semi-definite symmetric matrix. Indeed, every matrix B such that B = A∗A, isalso symmetric, that is B∗ = B; conversely, every positive semi-definite symmetric matrixB can be rooted (by functional calculus) and so it is a square: B = (

√B)2.

Problem III.5.2. Can all positive semi-definite matrices with entries in R[x1, . . . , xn] bewritten as sum of squares of symmetric matrices with entries in R(x1, . . . , xn)?

This problem was solved in the affirmative independently by Gondard and Ribenoim[75] and Procesi and Schacher [131]. A constructive solution has been provided much laterby Hillar and Nie [87].

In order to present the version of this problem using operator theory, we need to passthrough a geometric analogue. To this end, observe that a polynomial f ∈ R[x1, . . . , xn]

III.5. ALGEBRAIC REFORMULATION OF THE CONJECTURE 95

is just a function from Rn to R. So one may attempt to extend Hilbert’s 17th problemfrom polynomial on Rn to more general functions defined on manifolds. Recall that ann-manifold M is called irreducible if for any embedding of the n-sphere Sn−1 into M thereexists an embedding of the n-ball Bn into M such that the image of the boundary of Bn

coincides with the image of Sn−1.

Problem III.5.3 (Geometric analogue of Hilbert’s 17th problem). Let M be a paracom-pact irreducible analytic manifold and f : M → R be a non-negative analytic function.Can f be expressed as a sum of squares of meromorphic functions?

Recall that meromorphic functions are those functions which are analytic on the wholedomain expect for a set of isolated points, which are their poles. So, rational functions aremeromorphic and one can thus recognize a generalization of Hilbert’s 17th problem. Inits generality, this problem is still open. A complete solution is known only for dimensionn = 2 (see [28]) and for compact manifolds (see [135]).

The basic idea to get to Radulescu’s analogue in terms of operator theory is to generalizeanalytic functions with formal series. Let Y1, . . . , Y n be n indeterminates. Define

In = {(i1, . . . , ip), p ∈ N, i1, . . . , ip ∈ {1, . . . , n}}.

For each I = (i1, . . . , ip) ∈ In, set YI = Yi1 · . . . · Yip and define the space

V =

∑

I∈InαIYI , αI ∈ C : ∀R > 0,

∥∥∥∥∥∑

I∈InαIYI

∥∥∥∥∥R

:=∑

I

|αI |R|I| <∞

.

Radulescu showed in [132], Proposition 2.1, that V is a Frechet space and so it carries anatural weak topology σ(V, V ∗) induced by its dual space. To generalize the notion of‘square’ and ‘sum of squares’ to this setting, we need to introduce a notion of symmetryin V . Since V is a space of formal series, this is done in the obvious way. We set (Yi1 · . . . ·Yip)∗ := Yip · . . . · Yi1 and α∗ := α. This mapping can clearly be extended to an adjointoperation on V . Also, we observe that formal series are, by definition, possibly infinitesums and so there is no hope, in general, to have them expressed as a finite sum of squares.This motivates the need to use weak limits of sum of squares, instead of just the finitesums of squares.

Definition III.5.4. We say that a formal series q ∈ V is a sum of squares if it is in theweak closure of the set of the elements of the form p∗p, for p ∈ V .

We now observe that the original formulation of Hilbert’s 17th problem concerns ma-trices with real entries and its geometric variant concerns real valued analytic functions.Recalling that the operator analogue of real valued functions are self-adjoint operators, itfollows that the right setting in which Hilbert’s 17th problem can be generalized is that ofself-adjoint operators. We then introduce the space Vsa = {v ∈ V : v∗ = v}. It remainsonly to generalize the notion of positivity.


Definition III.5.5. A self-adjoint operator v ∈ Vsa is called positive semidefinite if forevery N ∈ N and for every N -tuple of self-adjoint matrices X1, . . . ,XN , one has

tr(p(X1, . . . ,XN )) ≥ 0.

The cone of positive semi-definite operators is denoted V +sa .

One last step is needed to get to the operator analogue of Hilbert’s 17th problem.Indeed, in case of polynomials, one has YI − YI = 0, for every permutation I of I. Sincethis is no longer the case in the non-commutative world of formal series, we need to identifyseries which differ by a permutation.

Definition III.5.6. Two elements p, q ∈ V +sa are called cyclic equivalent if p − q is weak

limit of sums of scalars multiples of monomials of the form YI − YI , where I is a cyclicpermutation of I.

Problem III.5.7 (Non-commutative analogue of Hilbert’s 17th problem). Is every elementof V +

sa cyclic equivalent to a weak limit of sums of squares?

As mentioned, the interest in this problem comes from the fact that it is equivalent toConnes’ embedding conjecture.

Theorem III.5.8 (Radulescu). The following statements are equivalent:

1. Connes’ embedding conjecture is true;

2. Problem III.5.7 has a positive answer.

The proof of this theorem is quite involved and we refer the interested reader to theoriginal paper by Radulescu. Here we move on to further developments, which led to apurely algebraic reformulation of Connes’ embedding conjecture. Such a purely algebraicreformulation is unexpected since all formulations we have discussed so far have a strongtopological component. We discuss two different, though similar, purely algebraic refor-mulations of Connes’ embedding conjecture; one due to Klep and Schweighofer [104], theother to Juschenko and Popovich [90].

We start by discussing Klep and Schweighofer’s approach. This differs from Radulescuin many parts, the most important of which is that they do not use formal series but theyget back to polynomials. Using finite objects instead of infinite objects is the ultimatereason why they are able to get rid of every topological condition.

Let K be either the real or the complex field and let V denote the ring of polynomials onn indeterminates with coefficients in K. Instead of using Radulescu’s adjoint operation theydefine the adjoint operation acting identically on monomials and switching each coefficientwith its conjugate. As before, the set of self-adjoint elements is senoted Vsa. Moreover,instead of using the cyclic equivalence, they define two polynomial p and q to be equivalentwhen their difference is a sum of commutators.

III.5. ALGEBRAIC REFORMULATION OF THE CONJECTURE 97

Definition III.5.9. A polynomial f ∈ V is called positive semidefinite if for every N ∈ Nand for every contractions A1, . . . , An ∈ MN (R) one has

tr(f(A1, . . . , An)) ≥ 0.

The set of positive semidefinite elements is denoted by V +.

The introduction of the quadratic module is the major difference with Radulescu’sapproach.

Definition III.5.10. A subset M ⊆ Vsa is called quadratic module if the following hold:

1. 1 ∈M ;

2. M +M ⊆M ;

3. p∗Mp ⊆M , for all p ∈ V .

The quadratic module generated by the elements 1 −X21 , . . . , 1 −X2

n is denoted by Q.

Theorem III.5.11 (Klep-Schweighofer). The following statements are equivalent:


2. For every f ∈ V + and for every ε > 0, there exists q ∈ Q such that f+ε is equivalentto q, in the sense that f + ε− q is a sum of commutators.

One may wish to be able to replace the quadratic module by the more standards squares,namely elements of the form v∗v. We conclude this section by describing the approach byJuschenko and Popovich, which is indeed aimed to this.

One way to reformulate Klep and Schweighofer’s approach is by considering the freeassociative algebra K(X) generated by a countable family of self-adjoint elements X =(X1,X2 . . .}. Thus, a polynomial f ∈ V is just an element of K(X). Klep-Schweighofer’stheorem affirms that Connes’ embedding conjecture is equivalent to the statement thatevery positive semidefinite element of K(X) cab be written as an element of the quadraticmodule, up to a sum of commutators and ε, with ε arbitrarily small.

Instead of considering the free associative algebra K(X), Juschenko and Popovich con-sidered the group *-algebra F of the countably generated free group F∞ = 〈u1, u2, . . .〉.With this choice they were able to simplify Klep and Schweighofer’s theorems in two ways.First, instead of using the quadratic module, they were able to use standard squares; sec-ond, instead of considering every polynomial f , they were able to consider only polynomialsof degree at most two, in the variables ui. Before presenting the theorem, we redefine thenotion of positivity in this new context.


Definition III.5.12. An element f ∈ F with n indeterminates is called positive semidef-inite if for all m ≥ 1 and all n-tuples of unitary matrices U1, . . . , Un of dimension m, onehas

tr(f(U1, . . . , Un)) ≥ 0.

Theorem III.5.13 (Juschenko-Popovich). The following statements are equivalent:


2. For every self-adjoint positive semidefinite f ∈ F and for every ε > 0, one hasf + ε = g + c, where g is a sum of squares (elements of the forv v∗v) and c is a sumof commutators.

III.6 Brown’s invariant

Most of research about Connes’ embedding conjecture has been focusing on impressivereformulations of it, that is, on finding apparently very far statements that turn out to beeventually equivalent to the original conjecture.

Over the last couple of years another point of view has been also taken, mostly due toNate Brown’s paper[17]. He assumes that a fixed separable II1 factor M verifies Connes’embedding conjecture and tries to tell something interesting about M . In particular,he managed to associate an invariant to M , now called Brown’s invariant, that carriesinformation about rigidity properties of M . The purpose of this section is to introduce thereader to this invariant.

III.6.1 Convex combinations of representations into RU

Let M be a separable II1 factor verifying Connes’ embedding conjecture and fix a freeultrafilter U on the natural numbers. The set Hom(M,RU ) of unital morphisms M → RU

modulo unitary equivalence is non-empty. We shall show that this set, that is in factBrown’s invariant, has a surprisingly rich structure.

We can equip Hom(M,RU ) with a metric in a reasonably simple way. Since M isseparable, it is topologically generated by countably many elements a1, a2 . . ., that we mayassume to be contractions, that is ||ai|| ≤ 1, for all i. So we can define a metric onHom(M,RU ) as follows

d([π], [ρ]) = infu∈U(RU )

( ∞∑

n=1

1

22n||π(an) − uρ(an)u∗||22

) 12

,

since the series in the right hand side is convergent. A priori, d is just a pseudo-metric,but we can use Theorem 3.1 in [139] to say that approximately unitary equivalence is thesame as unitary equivalence in separable subalgebras of RU . This means that d is actually

III.6. BROWN’S INVARIANT 99

a metric. Moreover, while this metric may depend on the generating set {a1, a2, . . .}, theinduced topology does not. It is indeed the point-wise convergence topology.

Hom(M,RU ) does not carry any evident vector space structure, but Nate Brown’sintuition was that one can still do convex combinations inside Hom(M,RU ) in a formalway. There is indeed an obvious (and wrong) way to proceed: given *-homomorphismsπ, ρ : M → RU and 0 < t < 1, take a projection pt ∈ (π(M) ∪ ρ(M))′ ∩ RU such thatτ(pt) = t and define the “convex combination” tπ + (1 − t)ρ to be

x 7→ π(x)pt + ρ(x)p⊥t .

Since the projection pt is chosen in (π(M) ∪ ρ(M))′, then tπ + (1 − t)ρ is certainly anew unital morphism of M in RU . Unfortunately this procedure is not well defined onclasses in Hom(M,RU ) and the reason can be explained as follows: if p ∈ RU is a nonzeroprojection, then the corner pRUp is still a hyperfinite II1-factor and so, by uniqueness, it isisomorphic to RU . Thus the cut-down pπ can be seen as a new morphism M → RU . Theproblem is that the isomorphism pRUp→ RU is not canonical and this reflects on the factthat convex combinations as defined above are not well-defined on classes in Hom(M,RU ).The idea is to allow only particular isomorphisms pRUp→ RU that are somehow fixed byconjugation by a unitary. This is done by using the so-called standard isomorphisms, thatrepresent Nate Brown’s main technical innovation.

Definition III.6.1. Let p ∈ RU be a non-zero projection. A standard isomorphism is anymap θp : pRUp→ RU constructed as follows. Lift p to a projection (pn) ∈ ℓ∞(R) such thatτR(pn) = τRU (p), for all n ∈ N, fix isomorphisms θn : pnRpn → R, and define θp to be theisomorphism on the right hand side of the following commutative diagram

ℓ∞(pnRpn) //

⊕θn��

pRUp

∼=��

ℓ∞(R) // RU

Definition III.6.2. Given [π1], . . . , [πn] ∈ Hom(N,RU ) and t1, . . . , tn ∈ [0, 1] such that∑ti = 1, we define

n∑

i=1

ti[πi] :=

[n∑

i=1

(θ−1i ◦ πi

)],

where θi : piRUpi → RU are standard isomorphisms and p1, . . . , pn ∈ RU are orthogonalprojections such that τ(pi) = ti for i ∈ {1, . . . , n}.

We can explain in a few words why this procedure of using standard isomorphismsworks. It has been originally proven by Murray and von Neumann that there is a unique


unital embedding of Mn (C) into R up to unitary equivalence. Since R contains an in-creasing chain of matrix algebras whose union is weakly dense, it follows that all unitalendomorphisms of R are approximately inner. Now, if we take an automorphism Θ of RU

that can be lifted (i.e. it is of the form (θn)n∈N where θn is an automorphism of ℓ∞(R)), itfollows that Θ is just the conjugation by some unitary, when restricted to a separable subal-gebras or RU . Now, Nate Brown’s standard isomorphisms are exactly those isomorphismspRUp→ RU that are liftable and therefore it is intuitively clear that, after passing to thequotient by the relation of unitary equivalence, the choice of the standard isomorphismshould not affect the result. The formalization of this rough idea leads to the followingtheorem.

Theorem III.6.3. (N.P. Brown[17])∑n

i=1 ti[πi] is well defined, i.e. independent of theprojections pi, the standard isomorphisms θi and the representatives πi.

To prove this result we need some preliminary observations.

Lemma III.6.4. Let p, q ∈ R be projections of the same trace and θ : pRp → qRq be aunital *-homomorphism, that is θ(p) = q. Then there is a sequence of partial isometriesvn ∈ R such that:

1. v∗nvn = p,

2. vnv∗n = q,

3. θ(x) = limn→∞ vnxv∗n,

where the limit is taken in the 2-norm.

Proof. Since p, q have the same trace, we can find a partial isometry w such that w∗w = qand ww∗ = p. Consider the unital endomorphism θw : pRp → pRp defined by θw(x) =wθ(x)w∗. Since R is hyperfinite, every endomorphism is approximately inner in the 2-norm, that is, we can find unitaries un ∈ pRp such that wθ(x)w∗ = limn→∞ unxu

∗n.

Defining vn = w∗un completes the proof.

Proposition III.6.5. Assume p, q ∈ RU are projections with the same trace, M ⊆ pRUp isa separable von Neumann subalgebra and Θ : pRUp→ qRUq is a unital *-homomorphism.Let (pi), (qi) ∈ ℓ∞(R) lifts of p and q, respectively, such that τR(pi) = τR(qi) = τRU (p), forall i ∈ N. If there are unital *-homomorphisms θi : piRpi → qiRqi such that (θi(xi)) is alift of Θ(x), whenever (xi) ∈ ΠpiRpi is a lift of x ∈ M , then there are a partial isometryv ∈ RU such that:

1. v∗v = p,

2. vv∗ = q,


3. Θ(x) = vxv∗, for all x ∈M .

Proof. We shall prove the proposition only in the case M = W ∗(X) is singly generated.

Let (xi) ∈ ΠpiRpi be a lift of X. By Lemma III.6.4, there are partial isometriesvi ∈ R such that v∗i vi = pi, viv

∗i = qi and ||θi(xi) − vixiv

∗i ||2 < 1/i. Observe that

(vi) ∈ ℓ∞(R) drops to a partial isometry v ∈ RU with support p and range q. To showthat Θ(X) = vXv∗, fix ε > 0 and consider the set

Sε = {i ∈ N : ||θi(xi) − vixiv∗i ||2 < ε}.

This set contains the cofinite set {i ∈ N : i ≥ 1ε} and therefore Sε ∈ U .

Exercise III.6.6. Prove Proposition III.6.5 in the general case. (Hint: pick the vi’sto obtain inequalities of the shape ||θi(Yi) − viYiv

∗i ||2 < 1/i on a finite set of Yi’s that

corresponds to lifts of a finite subset of a generating set of M).

Proof of Theorem III.6.3. Let σi : qiRUqi → RU be standard isomorphisms, built by pair-wise orthogonal projections qi of trace ti and assume [ρi] = [πi]. By Proposition III.6.5,applied to the standard isomorphism σ−1

i ◦ θi : piRUpi → qiRUqi, there are partial isome-tries vi ∈ RU such that v∗i vi = pi, viv

∗i = qi and

vi(θ−1i ◦ πi

)(x)v∗i =

(σ−1i ◦ πi

)(x), for all x ∈M.

Now since [πi] = [ρi], we can find unitaries ui such that ρi = uiπiu∗i . The proof is then

completed by the following exercise.

Exercise III.6.7. Show that u :=∑σ−1i (ui)vi is a unitary conjugating

∑θ−1i ◦ πi over

to∑σ−1i ◦ ρi.

Having a notion of convex combinations on Hom(N,RU ), it is natural to ask (i) whetherthis set can be embedded into a vector space; and, if so, (ii) what can be done with thisvector space. Nate Brown himself proved in [17] that his notion of convex combinationsverifies the axioms of a so-called convex-like structure. Afterwards, Capraro and Fritzshowed in [22] that every convex-like structure is linearly and isometrically embeddableinto a Banach space, but their proof is strongly based on Stone’s representation theorem[140], which provides an abstract embedding into a vector space. Therefore, taken together,these two results provide only an abstract embedding of Hom(N,RU ) into a Banach space.In the appendix, the interested reader can find a sketch of a concrete embedding, using aconstruction appeared in [18].


III.6.2 Extreme points of Hom(M,RU) and a problem of Popa

In this section we present one application of the Hom space. Given a separable II1 factorM that embeds into RU , Sorin Popa asked whether there is always another representationπ such that π(M)′ ∩RU is a factor. The following theorem by Nate Brown[17] shows thatthis problem is equivalent to a geometric problem on Hom(M,RU ).

Theorem III.6.8. Let π : M → RU be a representation. Then π(M)′ ∩RU is a factor ifand only if [π] is an extreme point of Hom(M,RU ).

In this section we prove only the “only if” part: if [π] is an extreme point, then π(M)′∩RU is a factor.

Definition III.6.9. We define the cutdown of a representation π : M → RU by a pro-jection p ∈ π(M)′ ∩ RU to be the map M → RU defined by x → θp(pπ(x)), whereθp : pRUp→ RU is a standard isomorphism.

Lemma 3.3.3 in [17] shows that this definition is independent by the standard isomor-phism, hence we can denote it by [πp].

Lemma III.6.10. Let π : M → RU be a morphism p, q ∈ π(M)′ → RU be projections ofthe same trace. The following statement are equivalent:

1. [πp] = [πq],

2. p and q are Murray-von Neumann equivalent in π(M)′∩RU , that is, there is a partialisometry v ∈ π(M)′ ∩RU such that v∗v = p and vv∗ = q.

3. there exists v ∈ RU such that v∗v = p, vv∗ = q and vπ(x)v∗ = qπ(x), for all x ∈M .

Exercise III.6.11. Prove the equivalence between (2) and (3) in Lemma III.6.10.

Exercise III.6.12. Given projections p, q and a partial isometry v such that v∗v = p andvv∗ = q, show that there exist lifts (pn), (qn), (vn) ∈ ℓ∞(R) such that pn, qn are projectionsof the same trace as p, and v∗nvn = pn, vnv

∗n = qn, for all n ∈ N.

Proof of (3) ⇒ (1). Let pn, qn, vn as in Exercise III.6.12 and fix isomorphisms θn : pnRpn →R and γn : qnRqn → R and use them to define standard isomorphisms θ : pRUp → RU

and γ : qRUq → RU and use them to define πp and πq. The isomorphism on the right handside of the following diagram† is liftable by construction and so Proposition III.6.5 can beapplied to it, giving unitary equivalence between πp and πq.

†The notation Adu in the diagram stands for the conjugation by the unitary operator u.


ℓ∞(R) //

⊕θ−1n

��

RU

θ−1��

ℓ∞(pnRpn) //

⊕Advn��

pRUp

Adv��

ℓ∞(qnRqn) //

⊕γn��

qRUq

γ

��ℓ∞(R) // RU

Exercise III.6.13. Use a similar idea to prove the implication (1) ⇒ (3).

We recall that a projection p ∈ M is called minimal if pMp = C1. A von Neumannalgebra without minimal projections is called diffuse.

Exercise III.6.14. Let M be a diffuse von Neumann algebra with the following property:every pair of projections with the same trace are Murray-von Neumann equivalent. Showthat M is factor. (Hint: show that every central projection is minimal).

Proof of the “only if” of Theorem III.6.8. Let [π] be an extreme point of Hom(M,RU ) andp ∈ π(M)′ ∩RU . Since

[π] = τ(p)[πp] + τ(p⊥)[πp⊥ ],

it follows that [πp] = [π], for all p ∈ π(M)′ ∩RU , p 6= 0. By Lemma III.6.10, it follows thattwo projections in π(M)′∩RU are Murray-von Neumann equivalent into π(M)′∩RU if andonly if they have the same trace. Since π(M)′ ∩RU is diffuse, Exercise III.6.14 completesthe proof.

From Theorem III.6.8 we obtain the following geometric reformulation of Popa’s ques-tion.

Problem III.6.15. (Geometric reformulation of Popa’s question) Does Hom(M,RU )have extreme points?

This problem is still open. There are two obvious ways to try to attack it, leading totwo related problems, whose positive solution would imply a positive solution of Popa’squestion.

1. Since Hom(M,RU ) is a bounded, closed and convex subset of a Banach space onecannot apply Krein-Milman’s theorem and conclude existence of extreme points. Nev-ertheless, one can ask the question whether the Banach space into which Hom(M,RU )


embeds is actually a dual Banach space. In this case, Hom(M,RU ) would be compactin the weak*-topology and one could apply Krein-Milman’s theorem.

Problem III.6.16. Does Hom(M,RU ) embed into a dual Banach space?

Note that one way to try to attack this problem is by observing that M is the dualBanach space (every von Neumann algebra is a dual Banach space) of a uniqueBanach space, usually denoted by M∗. Consequently there might be some possibilityto express representations M → RU in terms of dual representations of RU

∗ into M∗.

We mention that Chirvasitu [95] has shown that Hom(M,RU ⊗B(H)) is Dedekind-complete with respect to the order induced by the cone Hom+(M,RU ⊗B(H)). SinceDedekind-completeness is a necessary condition for Banach spaces of the form CR(K),with K compact Hausdorff, to have a predual, we can consider Chirvasitu’s result asa small measure of evidence that Hom(M,RU ⊗B(H)) is a dual Banach space.

2. The second approach is through a simple observation about geometry of Banachspaces. Recall that a Banach space B is called strictly convex if b1 6= b2 and ||b1|| =||b2|| = 1 together imply that ||b1 + b2|| < 2.

Exercise III.6.17. Let B be a strictly convex Banach space and C ⊆ B be a convexsubset. Fix c0 ∈ C and assume that there is c ∈ C such that d(c0, c) is maximized inC. Show that c is an extreme point of C.

Exercise III.6.18. Let M = W ∗(X) be a singly generated II1 factor which embedsinto RU . Fix [π0] ∈ Hom(M,RU ). Show that the function Hom(M,RU ) ∋ [π] →d([π0], [π]) attains its maximum.

With a little bit more effort one can extend Exercise III.6.18 to every separable II1factor. Consequently, Exercises III.6.17 and III.6.18 together imply that if the convex-like structure on Hom(M,RU ) were strictly convex, then Popa’s question would haveaffirmative answer.

Problem III.6.19. Does Hom(M,RU ) embed into a strictly convex Banach space?

Observe that Theorem III.6.21 suggests that Hom(M,RU ) itself should not be strictlyconvex.

The study of the extreme points of Hom(M,RU ) is not only interesting in light of Popa’squestion, but also because it provides a method to distinguish II1 factors. For instance,Brown proved in [17], Corollary 5.4, that rigidity of an RU -embeddable II1 factor M withproperty (T) reflects on the rigidity of the set of the extreme points of its Hom(M,RU ),that turns out to be discrete. Property (T) for von Neumann algebras is a form of rigidityintroduced by Connes and Jones in [44] and inspired to Kazhdan’s property (T) for groups[98]. A simple way to define property (T) for von Neumann algebras is through the followingdefinition.

105

Definition III.6.20. A II1 factor M with trace τ has property (T) if for all ε > 0, thereexist δ > 0 and a finite subset F of M such that for all τ -preserving ucp maps φ : M →M ,one has

supx∈F

||φ(x) − x||2 ≤ δ ⇒ supBall(M)

||φ(x) − x||2 ≤ ε.

The interpretation of property (T) as a form of rigidity should be clear: if a trace-preserving ucp map is closed to the identity on a finite set, then it is actually close to theidentity on the whole von Neumann algebra.

Classical examples of factors with property (T) are the ones associated to SL(n,Z),with n ≥ 3. The following result was proved by Nate Brown in[17], Corollary 5.4.

Theorem III.6.21. If M has property (T), then the set of extreme points of Hom(M,RU )is discrete.

Proof. Popa proved in [129], Section 4.5, that for every ε > 0, there is a δ > 0 such thatif [π], [ρ] ∈ Hom(M,RU ) are at distance ≤ δ, then there are projections p ∈ π(M)′ ∩ RU

and q ∈ ρ(M)∩RU and a partial isometry v such that v∗v = p, vv∗ = q, τ(p) > 1− ε, andvπ(x)v∗ = qρ(x), for all x ∈ M . This implies that pπ and qρ are approximately unitarilyequivalent and consequently, by countable saturation, [πp] = [ρq].

Now fix ε > 0 assume that [π] and [ρ] are δ-close extreme points and take projectionsp ∈ π(M)′ ∩ RU and q ∈ ρ(M)′ ∩ RU such that [πp] = [ρq]. Since [π] and [ρ] are extremepoints, we can apply Theorem III.6.8] and conclude that [π] = [πp] and [ρ] = [ρq], that is,[π] = [ρ].

Remark III.6.22. Observe that Theorem III.6.21 tells that the set of extreme points isdiscrete but, as far as we know, it might be empty. Indeed the problem of proving thatextreme points actually exist for RU -embeddable II1 factors is open even for factors withproperty (T).

We conclude mentioning that also examples of factors with a continuous non-empty setof extreme points are also known in (see [17] Corollaries 6.10 and 6.11).

106 CONCLUSIONS

Conclusions

The problem of whether every countable discrete group is sofic or hyperlinear is currentlyof paramount importance. Along this monograph, we have shown that answering theseproblems would automatically settle a number of conjectures from different fields of pureand applied mathematics.

In the sofic case this problem seems to boil down to understanding which relations (or,more generally, existential formulas in the language of invarant length groups; see SectionII.6) are approximately satisfiable in the permutation groups endowed with the Hammingdistance. This connection has been made explicit in [72] and [7]. Similar arguments holdsfor hyperlinear groups.

Suppose that w (x) is a word, where x = (x1, . . . , xn), and G is an invariant lengthgroup. A tuple g = (g1, . . . , gn) in G is a solution of the equation ℓ (w (x)) = 0 ifℓG (w (g)) = 0 or, equivalently, w (g) equals the identity of G. A δ-approximate solution ofℓ (w (x)) = 0 is a tuple (g) such that ℓG (w (g)) < δ.

Suppose now that C is a class of invariant length groups. The formula ℓ (w (x)) is stablewith respect to C if, roughly speaking, every approximated solution of such an equationis close to an exact solution. More precisely for any given ε > 0 there is δ > 0 such thatwhenever G is an element of the class C and g is δ-approximate solution of the equationℓ (w (x)) = 0 in G, there is an exact solution h in G such that maxi ℓG

(gih

−1i

)< ε.

More generally if ϕ (x) is an arbitrary formula in the language of invariant lengthgroups, then one can define as above the notion of solution and δ-approximate solution ofthe equation ϕ (x) = 0. The formula ϕ (x) is stable with respect to a class C if for anyε > 0 there is δ > 0 such that whenever G is an element of C and g is a tuple in G such thatϕ (g) < δ there is a tuple h in G such that ϕ

(h)

= 0 and maxi ℓG(gih

−1i

)< ε. Stability is

a key notion in model theory for metric structures, being tightly connected to the conceptof definability [27, 58]. (This use of the word “stability” should not be confused with theclassical model-theoretic stability theory as in [137].) In the field of operator algebras,stability and the related notion of (weak) semiprojectivity are of fundamental importance;see [108].

Recall from Section II.9 that a countable discrete group has the C-approximation prop-erty if it admits a length-preserving embedding into a length ultraproduct of elements of C,i.e. an embedding such that the image of every element other than the identity has length

107

108 CONCLUSIONS

1. For a discrete group, soficity is equivalent to the C-approximation property, where C isthe class of permutation groups endowed with the Hamming distance. Similarly a discretegroup is LEF (respectively LEA) if and only if it has the C-approximation property, whereC is the class of finite (resp. amenable) groups endowed with the trivial length function.

Stability of a formula with respect to a class C of invariant length groups allows oneto perturb a C-approximation to obtain a Cdisc-approximation, where Cdisc is the class ofgroups from C endowed with the trivial length function. The following proposition is aconsequence of this observation.

Proposition. Suppose that

Γ = 〈g1, . . . , gn : wi (g) = 1 for i ≤ k〉

is a finitely presented group. If Γ has the C-approximation property, and the formula

maxi≤k

ℓ (wi (x)) + (1 − minj≤n

ℓ (xj))

is stable with respect to the class C, then Γ has the Cdisc-approximation property.

In the particular case when C is the class of permutation groups endowed with theHamming length function one obtains the following consequence; see also [72, Proposition3] and [7, Theorem 7.3]. (Recall that a finitely presented LEF group is residually finite.)

Corollary (Glebsky-Rivera, Arzhantseva-Paunsecu). Suppose that

Γ = 〈g1, . . . , gn : wi (g) = 1 for i ≤ n〉

is a finitely-presented group that is not residually finite. If the formula

maxi≤n

ℓ (wi (x)) + 1 − minj≤n

ℓ (xj)

is stable with respect to the class of permutation groups endowed with the Hamming lengthfunction, then Γ is not sofic.

This provides a possible line of attack to the soficity problem of some groups, firstsuggested in [72] and [7]. There are in fact many finitely-presented groups that are knownto be not residually finite. Among these Higman’s group which we have considered inSection II.9. Recall that this is the group H with presentation

⟨h1, h2, h3, h4 : hi+1hih

−1i+1h

−2i = 1 for i ≤ 4

⟩

where the sum i+1 is evaluated modulo 4. Thus the corollary above provides the followingsufficient conditions for H being not sofic: the fomula

maxi≤k

ℓ(xi+1xix

−1i+1x

−2i

)+ 1 − min {ℓ (x1) , ℓ (x2) , ℓ (x3) , ℓ (x4)}

109

is stable with respect to the class of permutation groups endowed with the Hamming lengthfunction. This means that for every ε > 0 there is δ > 0 such that whenever n ∈ N andσ1, σ2, σ3, σ4 ∈ Sn are such that ℓSn (σi) > 1 − δ and

ℓSn

(σi+1σiσ

−1i+1σ

−2i

)< δ

for i ≤ 4, then there are τ1, τ2, τ3, τ4 ∈ Sn such that τi+1τiτ−1i+1 = τ2i and ℓSn

(τiσ

−1i

)< ε

for i ≤ 4.Another candidate for this line of approach is Thompson’s group F . This is a very

famous groups with several equivalent descriptions. For our purposes it can be regardedas the finitely presented groups with presentation

⟨a, b :

[ab−1, a−1ba

]=[ab−1, a−2ba2

]= 1⟩

.

While it is known that F is not elementarily amenable, it is a famous open problemwhether F is amenable. If fact it even seems to be unkown whether F is sofic. In view ofthe corollary above one can conclude that a sufficient condition to refute soficity of F isthe stability of the formula

max{ℓ([xy−1, x−1yx

]), ℓ([xy−1, x−2yx2

])}+ 1 − min {ℓ (x) , ℓ (y)}

with respect to the class of permutation groups endowed with the Hamming length function.Concretely this means that for every ε > 0 there is δ > 0 such that whenever n ∈ N andσ, ρ ∈ Sn are such that ℓSn (σ) > 1 − δ, ℓSn (ρ) > 1 − δ, ℓSn

([σρ−1, σ−1ρσ

])< δ, and

ℓSn

([σρ−1, σ−2ρσ2

])< δ, there are τ, λ ∈ Sn such that τλ−1 commutes with τ−1λτ and

τ−2λτ2, ℓSn

(στ−1

)< ε, and ℓSn

(ρλ−1

)< ε.

These reformulations show the importance of determining stability of formulas in per-mutation groups. This problem seems to be currently not well understood. Among the fewpapers dedicated to this subject we can mention [72, 115, 7]. In particular it is shown in[7] that, remarkably, the commutator relation ℓ

(xyx−1y−1

)= 0 is stable in permutation

groups. This seems to be the first natural step towards determining whether the nonsoficitycriterion above applies to Higman’s and Thompson’s group.

110 APPENDICES

Appendix A

Tensor product of C*-algebras

A normed *-algebra is an algebra A (over C) equipped with:

1. an involution ∗ such that

• (x + y)∗ = x∗ + y∗,

• (xy)∗ = x∗y∗,

• (λx)∗ = λx∗.

2. a norm || · || such that ||xy|| ≤ ||x||||y||.

A Banach *-algebra is a normed *-algebra that is complete.

Definition A.1. A C*-algebra is a Banach *-algebra verifying the following additionalproperty, called C*-identity :

||x∗x|| = ||x||2, ∀a ∈ A.

The C*-identity is a relation between algebraic and topological properties. It indeed im-plies that sometimes algebraic properties imply topological properties. A classical exampleof this interplay is the following fact.

Fact. A *-homomorphism from a normed *-algebra to a C*-algebra is always a con-traction.

Example of C*-algebras certainly include the (commutative) algebra of complex valuedfunctions on a compact space equipped with the sup norm and B(H) itself. But, exactly asin case of von Neumann algebras, one can start from a group and construct a C*-algebrain a natural way.

Example A.2. Let G be a locally compact group. Fix a left-Haar measure µ and constructthe convolution *-algebra L1(G) as follows: the elements of L1(G) are µ-integrable complex-valued functions on G. The convolution is defined by (f ∗ g)(x) =

∫G f(y)g(y−1x)dµ(y)

111

112 APPENDICES

and the involution is defined by f∗(x) = f(x−1)∆(x−1), where ∆ is the modular function,i.e. the (unique) function ∆ : G → [0,∞) such that for all Borel subsets A of G one hasµ(Ax−1) = ∆(x)µ(A). The full C*-algebra of G, denoted by C∗(G) is the enveloping C*-algebra of L1(G), i.e. the completion of L1(G) with respect to the norm ||f || = supπ||π(f)||,where π runs over all non-degenerate *-representations of L1(G) in a Hilbert space∗.

Exercise A.3. Show that in fact || · || is a norm on L1(G).

In particular, we can make this construction for the free group on countably manygenerators, usually denoted by F∞. Observe that this group is countable and so its Haarmeasure is the counting measure which is bi-invariant. Consequently, the modular functionis constantly equal to 1.

Exercise A.4. Let δg : G → R be the characteristic function of the point g. Show thatg → δg is an embedding F∞ → U(C∗(F∞)).

Exercise A.5. Prove that the unitaries in Exercise A.4 form a norm total sequence inC∗(F∞).

Given two C*-algebras A1 and A2, their algebraic tensor product is a *-algebra in anatural way, by setting

(x1 ⊗ x2)(y1 ⊗ y2) = x1x2 ⊗ y1y2,

(x1 ⊗ x2)∗ = x∗1 ⊗ x∗2.

Nevertheless it is not clear how to define a norm to obtain a C*-algebra.

Definition A.6. Let A1, A2 be two C*-algebras and A1⊙A2 their algebraic tensor product.A norm || · ||β on A1 ⊗A2 is called C*-norm if the following properties are satisfied:

1. ||xy||β ≤ ||x||β ||y||β , for all x, y ∈ A1 ⊙A2;

2. ||x∗x||β = ||x||2β , for all x ∈ A1 ⊙A2.

If || · ||β is a C*-norm on A1 ⊙A2, then A1 ⊗β A2 denotes the completion of A1 ⊙A2 withrespect to || · ||β. It is a C*-algebra.

Exercise A.7. Prove that every C*-norm β is multiplicative on elementary tensors, i.e.||x1 ⊗ x2||β = ||x1||A1 ||x2||A2 .

The interesting thing is that one can define at least two different C*-norms, a minimalone and a maximal one, and they are indeed different in general. To define these norms,let us first recall that a representation of a *-algebra A is a *preserving algebra-morphismfrom A to some B(H). We denote Rep(A) the set of representation of A.

∗A representation π : L1(G) → B(H) is said to be non-degenerate if the set {π(f)ξ : f ∈ L1(G), ξ ∈ H}is dense in H .

A. TENSOR PRODUCT OF C*-ALGEBRAS 113

Definition A.8.

||x||max = sup {||π(x)|| : π ∈ Rep(A1 ⊙A2)} (A.1)

Exercise A.9. Prove that || · ||max is indeed a C*-norm. (Hint: to prove that ||x||max <∞,for all x, take inspiration from Lemma 11.3.3(iii) in [92]).

The norm || · ||max is named maximal norm, or projective norm, or Turumaru’s norm,being first introduced in [148]. The completion of A1 ⊙A2 with respect to it is denoted byA1 ⊗max A2.

Let A be a C*-algebra and S ⊆ A. Denote by C∗(S) the C*-subalgebra of A generatedby S. The maximal norm has the following universal property (see [142], IV.4.7).

Proposition A.10. Given C*-algebras A1, A2, B. Assume πi : Ai → B are *-homomorphismswith commuting ranges, that is for all x ∈ π1(A1) and y ∈ π2(A2), one has xy = yx. Thenthere exists a unique *-homomorphism π : A1 ⊗max A2 → B such that

1. π(x1 ⊗ x2) = π1(x1)π2(x2)

2. π(A1 ⊗max A2) = C∗(π1(A1), π2(A2))

Exercise A.11. Prove Proposition A.10.

We now turn to the definition of the minimal C*-norm. The idea behind its definitionis very simple. Instead of considering all representations of the algebraic tensor product,one takes only representations which split into the tensor product of representations of thefactors.

Definition A.12.

||x||min = sup {||(π1 ⊗ π2)(x)|| : πi ∈ Rep(Ai)} (A.2)

Exercise A.13. Prove that || · ||min is a C*-norm on A1 ⊙A2.

This norm is named minimal norm, or injective norm, or Guichardet’s norm, being firstintroduced in [82]. The completion of A1⊙A2 with respect to it is denoted by A1⊗minA2.

Remark A.14. Clearly || · ||min ≤ || · ||max, since representations of the form π1 ⊗ π2are particular *-representation of the algebraic tensor product A1 ⊙ A2. These norms aredifferent, in general, as Takesaki showed in [141]. Notation || · ||max reflects the fact thatthere are no C*-norms greater than the maximal norm, that follows straightforwardly fromthe GNS construction. Notation || · ||min has the same justification, but it is much harderto prove:

Theorem A.15. (Takesaki, [141]) || · ||min is the smallest C*-norm on A1 ⊙A2.

114 APPENDICES

Appendix B

Ultrafilters and ultralimits

by Vladimir G. Pestov ∗

B.1 Let us recall that the symbol ℓ∞ denotes the linear space consisting of all boundedsequences of real numbers. This linear space is sometimes viewed as an algebra, mean-ing that it supports a natural multiplication (the product of two bounded sequences isbounded). Besides, ℓ∞ is equipped with the supremum norm,

‖x‖ =∞

supn=1

|xn|

and the corresponding metric.

B.2 The usual notion of the limit of a convergent sequence of real numbers can beinterpreted as a mapping from a subset of ℓ∞ consisting of all convergent sequences (thissubset is usually denoted c, it is a closed normed subspace of ℓ∞) to R. This map,

c ∋ x = (xn)∞n=1 7→ limn→∞

xn ∈ R,

has the following well-known properties (refer to first-year Calculus for their proofs):

1. limn→∞(xn + yn) = limn→∞ xn + limn→∞ yn.

2. limn→∞(xnyn) = limn→∞ xn · limn→∞ yn.

3. limn→∞(λxn) = λ limn→∞ xn.

4. If xn ≤ yn, then limn→∞ xn ≤ limn→∞ yn.

∗ Departamento de Matematica, Universidade Federal de Santa Catarina, Campus Universitario

Trindade, CEP 88.040-900 Florianopolis-SC, Brasil (Pesquisador Visitante Especial do CAPES, processo085/2012) and Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue,

Ottawa, Ontario, K1N6N5 Canada (Permanent address).

115

116 APPENDICES

5. The limit of a constant sequence of ones is 1:

limn→∞

1 = 1,

where 1 = (1, 1, 1, . . .).

B.3 In a sense, the main shortcoming of the notion of a limit is that it is not definedfor every bounded sequence. For instance, the following sequence has no limit in the usualsense:

0, 1, 0, 1, 0, 1, 0, . . . , 0, 1, 0, . . . (B.1)

As we all remember from the student years, there are very subtle cases where proving ordisproving the convergence of a particular sequence can be tricky!

Can one avoid such difficulties and devise the notion of a “limit” which would haveall the same properties as above, and yet with regard to which every sequence will have alimit, even the one in formula (B.1)?

B.4 The response to the question in (B.3) turns out to be very simple and disappointing.It is enough, for example, to define a “limit” of a sequence (xn) by selecting the first term:

φ(x) = x1.

This clearly satisfies all the properties listed in (B.2). But this is not what we want: sucha “limit” does not reflect the asymptotic behaviour of a sequence at the infinity. Andof course this unsatisfying response is due to a poorly-formulated question. We have,therefore, to reformulate the question as follows.

B.5 Does there exist a notion of a “limit” of a sequence which

• is defined for all bounded sequences,

• has all the same properties listed in (B.2), and

• coincides with the classical limit in the case of convergent sequences?

B.6 Answering this question is the main goal of the present Appendix. Meanwhile,the sequence in Eq. (B.1) will serve as a guiding star, or rather as a guinea pig. Indeed,assuming such a wonderful specimen of a limit does exist, it will in particular assign a limitto this concrete sequence, so perhaps it would be a good idea, to begin by trying to guesswhat it will be? Say, will 1/2 be a reasonable suggestion?

Let us analyse this question in the context of all binary sequences, that is, sequencestaking values 0 and 1 only. The family of all binary sequences is {0, 1}N+ , the Cantor space

B. ULTRAFILTERS AND ULTRALIMITS 117

(though the metric and topology on this space will not play any role for the moment).Every binary sequence is just a map

x : N+ → {0, 1},

and so can be identified with the characteristic function of a suitable subset A ⊆ N+ ofnatural numbers, namely the set of all n where xn takes value one:

χA(n) =

{1, if n ∈ A,

0, otherwise.

For instance, the sequence in Eq. (B.1) is the characteristic (indicator) function of the setof all even natural numbers.

B.7 Suppose the desired limit exists. Let us denote it, for the time being, simply bylim. What are the properties of this limit on the set of binary sequences?

First of all, it must send the constant sequence of ones (the indicator function of N+)to one, this was one of the rules required:

B.7.1 limχN+ = lim(1, 1, 1, 1, . . .) = 1.

This implies that the limit of the zero sequence (the indicator function of the emptyset) is zero: indeed, the sum of the two sequences is the sequence of ones, and the additiveproperty of the limit implies

1 + limχ∅ = limχN+ + limχ∅ = limχN+ = 1.

We summarize:

B.7.2 limχ∅ = lim(0, 0, 0, 0, . . .) = 0.

The next observation is that the only possible values of our limit on binary sequencesare 0 or 1. Indeed, for a binary sequence (εn), where εn ∈ {0, 1}, one has ε2n = εn, and so

lim(εn) = lim(ε2n) = (lim εn)2,

but there are only two real numbers satisfying the property ε2 = ε.

B.7.3 For every subset A ⊆ N+, one has limχA ∈ {0, 1}.

This excludes a “natural” possibility to assign the limit 1/2 to the sequence in Eq.(B.1).

Now let χA be any binary sequence. We can “flip” the sequence and replace all oneswith zeros, and vice versa. This corresponds to the indicator function of the complementAc = N+ \ A. What about the limit of this sequence? It turns out the limit flips as well.

118 APPENDICES

B.7.4 lim(χAc) = 1 − limχA.

Indeed, χA + χAc = χN+ is the identical sequence of ones whose limit is one, and nowone has

limχAc = limχN+ − limχA = 1 − limχA.

Since the limit of every binary sequence must exist by assumption, this property impliesthe next one:

B.7.5 Let A be any subset of the positive natural numbers. Then either limχA = 1, orlimχAc = 1, but not both at the same time.

Let us address the following situation. Suppose we have a sequence χA whose limit weknow to be one. Now we take a subset B of the natural numbers which contains A:

B ⊇ A,

and consider the sequence χB. Thus, some zeros in χA have been (possibly) replaced withones, and all the ones in χA keep their values. What about the limit of the new sequence,χB?

B.7.6 If A ⊆ B and limχA = 1, then limχB = 1.

Indeed, χA ≤ χB, and according to the property (5) in Subs. B.2, we conclude:

limχA ≤ limχB.

However, limχA = 1, and limχB cannot be strictly greater than one. We conclude.

The next situation to consider is as follows. Suppose A and B are two subsets of thenatural numbers. Their intersection corresponds to the product of the indicator functions,as is well known and easy to see:

χA · χB = χA∩B.

It follows that

limχA · limχB = limχA∩B.

In particular:

B.7.7 If limχA = limχB = 1, then limχA∩B = 1.


B.8 Now it is time to put together some of the properties proved. Notice that instead ofbinary functions, one can just as well talk of subsets of the set N+. To some subsets A thereis associated the value one (when limχA = 1), to others, the value zero (if limχA = 0). Allsubsets of N+ are therefore grouped in two classes. Let us denote the class of all subsetsA to which we associate the limit one by U :

U = {A ⊆ N+ : limχA = 1}.

The properties established above immediately translate into the following.

1. ∅ /∈ U .

2. For every subset A ⊆ N+, either A ∈ U , or Ac ∈ U .

3. If A,B ∈ U , then A ∩B ∈ U .

B.8.1 Ultrafilters. A collection of subsets of natural numbers (or, in fact, of any fixednon-empty set) satisfying the above three axioms is called an ultrafilter (on this set).

There is no need to include more axioms in the definition of an ultrafilter.

B.8.2 Exercise. Convert the rest of the properties that we have established in (B.7.1)–(B.7.7) into a set-theoretic form and deduce them from the three axioms above.

B.8.3 Example. The property (B.7.6) becomes: if A ⊆ B and A ∈ U , then B ∈ U .

⊳ This follows from the axioms of an ultrafilter: assuming the contrary, one must haveN \B ∈ U (axiom 2), meaning ∅ = A ∩ (X \B) ∈ U , a contradiction. ⊲

B.9 Thus, each time we have the concept of a limit with the properties that we havespecified in Subs. B.2, we get an ultrafilter on the set of natural numbers.

B.9.1 Principal ultrafilters. In particular, to the disappointing example of a “limit” de-scribed in Subs. B.4, there corresponds the following ultrafilter:

U = {A ⊆ N : 1 ∈ A}.

In other words, a binary sequence has limit one if and only if the first term of this sequenceis one. This happens exactly when A ∋ 1, where our sequence is χA.

More generally, one can repeat the construction given any chosen element n ∈ N+. Theresulting ultrafilter is denoted (n),

(n) = {A ⊆ N+ : n ∈ A},

and called a trivial (or: principal) ultrafilter.

120 APPENDICES

B.9.2 Non-principal ultrafilters. This is, in a sense, a non-interesting situation. What isan interesting one? An ultrafilter U is called free, or non-principal, if its elements have nopoints in common: ⋂

U = ∅.

B.9.3 Every ultrafilter is either trivial or free.

⊳ Suppose U is an ultrafilter which is not trivial. This means: for every n ∈ N+,U 6= (n). This can mean two things: either there is A ∋ n which is not in the ultrafilter(in which case Ac is), or else there is A ∈ U so that A 6∋ n. In both cases, we can find anelement A of U not containing n. Since this holds for all n, it is clear that the intersectionof all members of U is an empty set, and the ultrafilter is non-principal. ⊲

B.10 How to establish the existence of free ultrafilters? We need the following notion. Asystem C of subsets of a certain non-empty set is called a centred system if the intersectionof every finite collection of elements of C is non-empty:

∀n, ∀A1, A2, . . . , An ∈ C, A1 ∩A2 ∩ . . . ∩An 6= ∅.

B.10.1 Every ultrafilter is a centred system.

This follows immediately from one of the axioms of an ultrafilter.

B.10.2 Every ultrafilter is a maximal centred system. In other words, if U is an ultrafilterand A /∈ U , then the system U ∪ {A} is not centred.

⊳ Indeed, since A /∈ U , we must have Ac ∈ U , and since A and Ac are both members ofthe system U ∪ {A} and their intersection is empty, the system is not centred. ⊲

B.10.3 Ultrafilters are exactly maximal centred systems.More precisely, let C be a collection of subsets of a non-empty set (in our case, we are

interested in the set of natural numbers, N+). Then C is an ultrafilter if and only if C isa maximal centred system.

We have already established ⇒ in the previous paragraph, now let us prove the otherimplication.

⊳ Suppose C is a maximal centred system. “Maximal” means that if we add to C anysubset A of N+ which is not in C , then the resulting system of sets

C ∪ {A}

is no longer centred. Let us verify all three properties of an ultrafilter.(1) Clearly, ∅ /∈ C , because otherwise C would not be a centred system to begin with:

for example,∅ ∩ ∅ = ∅.


(2) Let A,B ∈ C . We need to prove A ∩B ∈ C . Consider the system

C ∪ {A ∩B}.This system is centred: indeed, if A1, . . . , An are elements of this system, then, if A ∩ Bis not among them, their intersection is clearly non-empty (as C is centred). If we addA ∩B, then

A1 ∩ . . . ∩An ∩ (A ∩B) = A1 ∩ . . . ∩An ∩A ∩B 6= ∅,because C is centred and all sets A1, . . . , An, A,B belong to C .

We conclude: the system C ∪ {A ∩ B} cannot be strictly larger than C , because ofmaximality of the latter. But this means that

A ∩B ∈ C .

(3) Let A ⊆ N+ be any subset. We want to show that either A or its complement Ac =N+ \A belong to C . Suppose neither holds, towards a contradiction. The assumption thatA /∈ C implies that the system C∩{A} is not centred, and so there exist A1, A2, . . . , Am ∈ C

withA1 ∩A2 ∩ . . . ∩Am ∩A = ∅.

Note that this observation can be rewritten as

A1 ∩A2 ∩ . . . ∩Am ⊆ Ac. (B.2)

Likewise, the assumption Ac /∈ C implies the existence of elements B1, B2, . . . , Bk ∈ C

satisfyingB1 ∩B2 ∩ . . . ∩Bk ∩Ac = ∅,

or, in an equivalent form,B1 ∩B2 ∩ . . . ∩Bk ⊆ A. (B.3)

Together, the equations (B.2) and (B.3) imply

A1 ∩A2 ∩ . . . ∩Am ∩B1 ∩B2 ∩ . . . ∩Bk ⊆ A ∩Ac = ∅,which contradicts the fact that C is a centred system. We are done. ⊲

B.11 Existence of free ultrafilters.

B.11.1 Zorn’s lemma. Let (X,≤) be a partially ordered set. Suppose that X has thefollowing property: every totally ordered subset C of X has an upper bound. (One says thatX is inductive.) Then X has a maximal element.

An element x is maximal if there is no y ∈ X which is strictly larger than x. It doesnot mean that x is necessarily the largest element of X.

The statement of Zorn’s lemma is an equivalent form of the Axiom of Choice, which isa part of the standard system of axioms of set theory ZFC.

122 APPENDICES

B.11.2 Exercise. Let C be a family of centred systems on a set N+ which is totallyordered by inclusion. Prove that the union of this family, ∪C, is again a centred system.

B.11.3 Every centred system is contained in a maximal centred system.

⊳ Let C be a centred system of subsets of a certain non-empty set (let us again assumethat this set is N+, this does not affect the argument in any way.) Denote X the family ofall centred systems of subsets of N+ which contain C . This family is ordered by inclusion.Due to the previous exercise, it is inductive. Since clearly ∪C contains C , it follows that∪C belongs to X and it forms an upper bound for the chain ∪C.

Zorn’s lemma implies the existence of a maximal element, D , in X. This D is a centredsystem which contains C and which is not contained in any strictly larger centred systemcontaining C . Since every centred system containing D will automatically contain C aswell, the latter statement can be cut down to: D is a centred system which contains C

and which is not contained in any strictly larger centred system. In other words, D is amaximal centred system containing C , as required. ⊲

Let us reformulate this result as follows.

B.11.4 Every centred system is contained in an ultrafilter.

B.11.5 Consider the following collection of subsets of N+:

C = {N+ \ {n} : n ∈ N+}.

In other words, C consists of all complements to singletons. This system is clearly centred,and its intersection is empty. According to the previous result, there is an ultrafilter Ucontaining the system C . One has:

∩U ⊆ ∩C = ∅.

Thus, U is a non-principal (free) ultrafilter.

In fact, one can show that N+ supports 2c pairwise different free ultrafilters. They areextremely numerous.

B.11.6 Solution to Exercise B.11.2. Let n ∈ N+ and

A1, A2, . . . , An ∈⋃

C

be arbitrary elements of C. For every i = 1, 2, . . . , n fix a centred system Ci which belongsto the family C and contains Ai:

Ai ∈ Ci ∈ C.


Elements of the collection Ci, i = 1, 2, 3, . . . , n of centred systems are pairwise comparableby inclusion between themselves because C is totally ordered by inclusion. Every finitetotally ordered set has the largest element: for some i0 = 1, 2, . . . , n one has

∀i = 1, 2, . . . , n, Ci ⊆ Ci0 .

This means in particular that

∀i = 1, 2, . . . , n, Ai ∈ Ci0 .

Since Ci0 is a centred system, one concludes

A1 ∩A2 ∩ . . . ∩An 6= ∅,

as required.

B.12 Ultralimits. Now let us re-examine the existence of our conjunctural limits makingsense for every bounded sequence. Notice first that if (εn) is a binary sequence, then,according to the initial way we defined ultrafilters in B.8,

lim εn = 1 ⇐⇒ {n : εn = 1} ∈ U . (B.4)

This definition depends on the choice of an ultrafilter, and for this reason, the correspondinglimit is called the ultralimit along the ultrafilter U . It is denoted

limn→U

εn.

(This notation actually makes good sense, as we will see later.)This definition in the above form will clearly not work in a general case (simply because

it is possible that all the terms of a convergent sequence are different from the limit).However, it extends readily to sequences taking finitely many distinct values (Exercise.).How to extend this definition to an arbitrary bounded sequence? In fact, such an extensionis quite natural and moreover unique.

B.12.1 Exercise. Let U be an ultrafilter on natural numbers. By approximating agiven bounded sequence (xn) with sequences taking finitely many different values in theℓ∞ norm, extend the definition of the ultralimit to all bounded sequences in such a waythat it respects all of our required axioms. Show that this extension is well defined (doesnot depend on an approximation).

B.12.2 Exercise. Conclude that, given an ultrafilter U on the natural numbers, there isa unique way to define the ultralimit along U on all bounded sequences which satisfies theproperties listed in (B.2) as well as the property in Eq. (B.4).

124 APPENDICES

B.12.3 Exercise. Conclude that there is a natural correspondence between the ultrafilterson the set of natural numbers and the limits on bounded sequences satisfying the propertiesstated in (B.2).

At the same time, the definition of an ultralimit based on approximations is not veryconvenient. We are going to reformulate it now.

B.13 A workable definition of an ultralimit. The right definition in a usable formis obtained by adjusting in an obvious way the classical concept of a convergent sequencexn → x:

∀ε > 0, ∃N, ∀n ≥ N, |x− xn| < ε.

This means that, given an ε > 0, for “most” values of n the point xn is within ε from thelimit. In this context, “most” means that the set of such n is cofinite.

If U is an ultrafilter, “most” values of n means that the set of n with this propertybelongs to U . So we say that a sequence (xn) of real numbers converges to a real numberx along the ultrafilter U ,

x = limn→U

xn,

if∀ε > 0, ∃A ∈ U , ∀n ∈ A, |x− xn| < ε.

Since every superset of such an A also belongs to U , an equivalent statement is:

∀ε > 0, {n ∈ N+ : |x− xn| < ε} ∈ U .

B.13.1 Exercise. Show that the definition of the ultralimit given in this subsection isequivalent to the definition in the language of approximations in Exercise B.12.1.

B.13.2 Example. If U = (n0) is a trivial ultrafilter generated by the natural number n0,then the statement

limn→U

xn = x

means that for every ε > 0,

{n ∈ N+ : |x− xn| < ε} ∋ n0,

that is, simply∀ε > 0, |x− xn0 | < ε,

which meansx = xn0 .

This is an “uninteresting” case of an ultralimit. The interesting cases correspond to freeultrafilters.


B.13.3 Exercise. Show that if limn→∞ xn = x (in a usual sense), then for every freeultrafilter U on N+,

limn→U

xn = x.

B.13.4 Exercise. Show that, if one has

limn→U

xn = x

for every free ultrafilter U on N+, then

limn→∞

xn = x.

Thus, the familiar symbol ∞ essentially means all the free ultrafilters lumped together,and the existence of a limit in the classical sense signifies that all the free ultrafilters onN+ agree between themselves on what the value of this limit should be. If there is no suchagreement, then the limit in the classical sense does not exist, and the sequence (xn) isdivergent. However, every free ultrafilter still gives its own interpretation of what the limitof the sequence is.

B.13.5 Example. Recall the alternating sequence

εn = (−1)n+1

of zeros and ones as in Eq. (B.1) Every ultrafilter U on N either contains the set of oddnumbers, or the set of even numbers. In the former case, the ultralimit of our sequence is0, in the latter case, 1. Because of this, the ultrafilters “disagree” between themselves onwhat the limit of the sequence should be. Consequently, limn→∞ εn does not exist and thesequence is divergent in the classical sense.

B.13.6 Theorem. Every bounded sequence of real numbers has an ultralimit along everyultrafilter on N+.

⊳ Let (xn) be a bounded sequence of real numbers, which we, without loss in generality,will assume to belong to the interval [0, 1]. Let U be an ultrafilter on N+. The proofclosely resembles the proof of the Heine–Borel theorem about compactness of the closedunit interval, but is actually simpler.

Subdivide the interval I0 = [0, 1] into two subintervals of length half, [0, 1/2] and [1/2, 1].Set

A = {n ∈ N+ : xn ∈ [0, 1/2]}.Either A belongs to U , or else its complement, Ac, does. In the latter case, since

{n : xn ∈ [1/2, 1]} ⊇ Ac,

126 APPENDICES

the set {n : xn ∈ [1/2, 1]} is in U as well. We conclude: at least one of the two intervals oflength 1/2, denote it I1, has the property

{n ∈ N+ : xn ∈ I1} ∈ U .

Continue dividing the intervals in two and selecting one of them. At the end, we will haveselected a nested sequence of closed intervals Ik of length 2−k, k = 0, 1, 2, 3, . . ., with theproperty that for all k,

{n : xn ∈ Ik} ∈ U .According to the Cantor Intersection Theorem, there is c ∈ [0, 1] so that

∞⋂

k=0

Ik = {c}.

We claim this c is the ultralimit of (xn) along U . Indeed, let ε > 0 be arbitrary. For asufficiently large k, the interval Ik is contained in the interval (c− ε, c+ ε) (indeed, enoughto take k = − log2 ε+ 1). Now one has

{n : c− ε < xn < x+ ε} ⊇ {n : xn ∈ Ik} ∋ U ,

from where one concludes that the former set is also in U . ⊲

B.13.7 Solution to Exercise B.13.3. We will verify the definition of an ultralimit. Letε > 0. For some N and every n ≥ N , one has |x− xn| < ε. Let us write

N = {1} ∪ {2} ∪ . . . ∪ {N − 1} ∪ {N,N + 1, N + 2, . . .}.

One of these sets must belong to U . If it were one of the singletons, {i}, then every otherelement A ∈ U must meet {i}, therefore contain i, so U would be a principal ultrafiltergenerated by i. One concludes:

{N,N + 1, N + 2, . . .} ∈ U ,

and so{n : |x− xn| < ε} ∈ U ,

because the set on the left hand side is a superset of {N,N + 1, N + 2, . . .}.

B.13.8 Solution to Exercise B.13.4. We have just seen that if xn → x in the classical

sense, then we have convergence xnU→ x along every free ultrafilter U . This means that

the only case to eliminate is where xnU→ x along every free ultrafilter U , yet at the same

time xn 6→ x in the classical sense. Assume that xn does not converge to x. Then for some


ε > 0 and every N there is n = n(N) ≥ N with d(x, xn) ≥ ε. The set I = {n(N) : N ∈ N}is infinite. There is a free ultrafilter U on N+ containing I, namely an ultrafilter containingthe centred system {I \ {n} : n = 1, 2, 3, . . .}. Since I ∈ U ,

{n : d(x, xn) < ε} /∈ U ,

because the set above is disjoint from I. We conclude: xn 6 U→ x.

B.14 Ultralimits in metric spaces. So far, we have only considered ultralimits ofsequences of real numbers. Of course the definition extends readily to an arbitrary metricspace, as follows. A sequence (xn) of elements of a metric space (X, d) converges to a pointx along an ultrafilter U on the set of positive natural numbers if for every ε > 0 the set

{n ∈ N+ : d(x, xn) < ε}

belongs to U .

B.14.1 Exercise. Prove that a sequence (xn) of elements of a metric space X can haveat most one ultralimit along a given ultrafilter. That is, let U be an ultrafilter on N+.Prove that the ultralimit limn→U xn, if it exists, is unique.

B.14.2 At the same time, let us note that in an arbitrary metric space, not everybounded sequence of elements needs to have an ultralimit. For instance, if X is a metricspace with a 0-1 metric and (xn) is a sequence of pairwise different elements of X, then forevery free ultrafilter U on N+, the ultralimit limn→U xn does not exist.

Indeed, let x ∈ X be arbitrary. The set

{n ∈ N+ : d(x, xn) < 1}

contains at most one element (in the case where x = xn for some n) and so does not belongto U . We conclude: xn does not converge to x along U . Since this argument applies toevery point x ∈ X, the sequence (xn) does not have an ultralimit.

B.14.3 Since not every bounded sequence in a metric space X needs to have an ultralimit,a natural question to ask is, can be enlarge X in such a way that every sequence has anultralimit, much in the same way as we can form a completion of X so that every Cauchysequence is assigned a limit in it?

The following result provides a negative answer to this question.

128 APPENDICES

B.14.4 Theorem. For a metric space X, the following conditions are equivalent.

1. X is compact.

2. Every sequence of elements in X has an ultralimit along every free ultrafilter U onN+.

3. There exists a free ultrafilter U on N+ with the property that every sequence of ele-ments in X has an ultralimit along U .

⊳ (a) ⇒ (b): very much the same proof as for the interval. For every ε > 0, choose a finitecover of X with open ε-balls. One of these balls, say B, has the property

{n : xn ∈ B} ∈ U .

Proceeding recursively, we obtain a centred sequence of open balls Bi of radius convergingto zero satisfying

Ai = {n : xn ∈ Bi} ∈ U .

The ball centers form a Cauchy sequence and so converge to some limit, x. Every neigh-bourhood V of x contains some Bi for i large enough, and so the set

{n : xn ∈ V }

contains Ai and so belongs to U .

(b)⇒(c): trivial, given that free ultrafilters on N exist. Now just pick any one of them.

(c)⇒(a): by contraposition. If X is not compact, then either it is not totally bounded(in which case there is a sequence of points at pairwise distances ≥ ε0 > 0 from eachother, which has no ultralimit due to the argument in B.14.2), or non-complete. In thelatter case, select a Cauchy sequence (xn) without a limit. We will show that it does notadmit an ultralimit along U either. Assume x is such an ultralimit. For every ε > 0, theset {n : d(xn, x) < ε} is in U , so non-empty, and using this, one can recursively select asubsequence of xn converging to x in the usual sense. But (xn) is a Cauchy sequence, sothis would mean xn → x. ⊲

B.14.5 Exercise. Deduce that a metric space X metrically embeds into a space X inwhich every bounded sequence has an ultralimit if and only if every ball in X is totallybounded. In this case, X is the metric completion of X, and every closed ball in X iscompact.

This is of course a rather restrictive condition, which, for instance, is failed by anyinfinite-dimensional normed space.


B.14.6 Solution to Exercise B.14.1. Suppose x, y ∈ X and

limn→U

xn = x and limn→U

xn = y.

For every ε > 0, both sets

{n ∈ N+ : d(x, xn) < ε} and {n ∈ N+ : d(y, xn) < ε}

belong to the ultrafilter U , and so does their intersection. As a consequence, this intersec-tion is non-empty, and one can find n ∈ N with

d(x, xn) < ε and d(y, xn) < ε.

By the triangle inequality,

d(x, y) < 2ε,

and since this holds for every ε > 0, we conclude: x = y.

B.15 Ultraproducts. In view of (B.14.5), we cannot hope to attach a “virtual” ultralimitto every bounded sequence of points of a metric space, X. If the space is not totallybounded, some sequences are destined to diverge along some ultrafilters in every metricextension of X. Still, we can assign to every such sequence an ideal new point, which willbe the limit (hence, ultralimit) in case of a Cauchy sequence, and otherwise will registerthe asymptotic behaviour of the sequence with regard to the given ultrafilter. Now we willbriefly examine this important construction.

B.15.1 Space of bounded sequences in a metric space. Let X = (X, dX ) be a metricspace. Fix an ultrafilter U on the integers. Since we are going to assign an “ideal” point toevery bounded sequence, the right place to start will be the set ℓ∞(N+;X) of all boundedsequences with elements in X. (For example, the space ℓ∞ is, in this notation, ℓ∞(N+;R).)We would formally identify the space of “ideal” points with ℓ∞(N+;X), or rather with itsquotient space under a suitable equivalence relation — just like we do when we constructthe completion of a metric space beginning with the set of all Cauchy sequences.

Observe that the set ℓ∞(N+;X) admits a structure of a metric space with regard tothe ℓ∞-distance:

d∞(x, y) = supn∈N+

dX(xn, yn).

B.15.2 Exercise. Show that the metric space ℓ∞(N+;X) is complete if and only if Xis complete.

130 APPENDICES

B.15.3 It is quite natural to make two bounded sequences x and y share the same “idealpoint” if the distance between the corresponding terms of those sequences converges tozero along the ultrafilter U :

xU∼ y ⇐⇒ lim

n→UdX(xn, yn) = 0.

The resulting equivalence relationU∼ on ℓ∞(N+;X) agrees with the metric in the sense that

the quotient metric on the quotient set is well defined. For an x ∈ ℓ∞(N+;X) denote

[x]U = {y ∈ X : xU∼ y}

the corresponding equivalence class.

B.15.4 Exercise. Show that the rule

d([x], [y]) = inf{d∞(x′, y′) : x′ ∈ [x]U , y′ ∈ [y]U}

defines a metric on the quotient set X/U∼, and that this quotient metric can be alternatively

described byd([x], [y]) = lim

n→UdX(xn, yn).

B.15.5 Metric ultrapowers. The quotient metric space ℓ∞(N+;X)/U∼ is called the metric

ultrapower of X with regard to the ultrafilter U , and denoted XNU .

B.15.6 Exercise. Show that the original metric space X canonically isometrically embedsinside of its metric ultrapower under the diagonal embedding

X ∋ x 7→ [(x, x, x, . . .)]U ∈ XNU ,

no matter what the ultrafilter U is. In particular, the metric ultrapower of a non-trivialspace is itself non-trivial.

B.15.7 Exercise. Show that, if U = (n) is a principal ultrafilter, the metric ultrapowerXN

U is canonically isometric to X.

B.15.8 Exercise. Let U be a non-principal ultrafilter, and suppose that X is a separablemetric space. Show that the metric ultrapower XN

U is isometric to X (not necessarily in acanonical way!) if and only if X is compact.

Before establishing a couple of other properties of the ultrapower, let us extend thedefinition to the case where terms of sequences come from possibly different metric spaces:for every n,

xn ∈ Xn,

where (Xn), n = 1, 2, . . ., are metric spaces which may or may not differ from each other.


B.15.9 Pointed metric spaces. In this situation, how do we define a bounded sequencex = (xn)? The notion of boundedness becomes purely relative: one can say that a sequencey is bounded with regard to another sequence, x, if the values dXn(xn, yn), n ∈ N, form abounded set. However, there is no absolute notion of boundedness. For this reason, it isnecessary to select a sequence x∗ = (x∗n), x∗n ∈ Xn, thus fixing a class of sequences boundedwith regard to x∗. In other words, we are dealing with a family of pointed metric spaces,(Xn, x

∗n).

For example, in cases of considerable interest, where Xn are normed spaces or metricgroups, the selected points are usually zero and the neutral element, respectively. If allspaces Xn = X are the same, in order to obtain the usual notion of (absolute) boundednessof a sequence, one chooses any constant sequence (x∗, x∗, . . .), all of them giving the sameresult. If the spaces Xn have a uniformly bounded diameter, the choice of a distinguishedsequence does not matter.

B.15.10 Ultraproduct of pointed metric spaces. Now it is clear how to reformulate thedefinitions. The metric space consisting of all relatively bounded sequences with regard tox∗ is the ℓ∞-type sum of pointed spaces (Xn, x

∗n), denoted

⊕ℓ∞

n (Xn, x∗n).

In cases like the above where the choice of distinguished points is clear or does not matter,these points are suppressed.

The equivalence relation and the distance on the quotient metric space are defined inexactly the same fashion as above. The resulting metric space

⊕ℓ∞

n (Xn, x∗n)/

U∼

is called the metric ultraproduct of (pointed) metric spaces (xn, x∗n) modulo the ultrafilter

U , and denoted (∏

n

(Xn, x∗n)

)

U.

B.15.11 Exercise. Assuming that U is a non-principal ultrafilter, show that there is acanonical isometry between the metric ultraproducts

(∏

n

(Xn, x∗n)

)

Uand

(∏

n

(Xn, x∗n)

)

U,

where Xn denotes the completion of the metric space Xn.

⊳ Hint: observe that the ultraproduct on the left admits a canonical isometric embed-ding into the one on the right. ⊲

132 APPENDICES

B.15.12 Exercise. Given a Cauchy sequence in the metric ultraproduct of spaces Xn,show that it is the image under the quotient map of some Cauchy sequence in the ℓ∞-typesum of the spaces Xn.

⊳ Hint: use the following equivalent definition: a sequence (an) is Cauchy if and onlyif for every ε > 0 there is N such that the open ball Bε(xN ) contains all xn with n ≥ N . ⊲

The following stands in contrast to Exercise B.15.2.

B.15.13 Exercise. Deduce from the previous two exercises that the metric ultraproductof a family of arbitrary metric spaces, formed with regard to a non-principal ultrafilter, isalways a complete metric space.

Recall that the covering number N(X, ε), where ε > 0, of a metric space X is thesmallest size of an ε-net for X.

B.15.14 Exercise. (∗) Prove that if the metric ultraproduct X of a family (Xn) of separa-ble metric spaces is separable, then it is compact. Moreover, in this case for every ε > 0 thesequence of covering numbers N(Xn, ε) is essentially bounded, that is, uniformly boundedfrom above for all n belonging to some element A = A(ε) of the ultrafilter.

Otherwise, X has density character continuum.

B.16 The space of ultrafilters. In conclusion, we want to give a clear literal meaning tothe symbol

limn→U

xn,

used for an ultralimit along an ultrafilter U . In fact, an ultralimit of a sequence can beinterpreted as a limit in the usual sense.

B.16.1 Denote by βN the set of all ultrafilters on the set N of natural numbers. IdentifyN with a subset of βN by assigning to each natural number n the corresponding principalultrafilter, (n) = {A ⊆ N : A ∋ n}:

N → βN, N ∋ n 7→ (n) ∈ βN.

This mapping is clearly one-to-one, and in this sense we will often refer to N as a subsetof βN.

B.16.2 For every A ⊆ N, denote by A the family of all ultrafilters ξ ∈ βN containing Aas an element:

A = {U ∈ βN : A ∈ U}.


B.16.3 Exercise. Prove that the collection

{A : A ⊆ N}forms a base for a topology on βN.

B.16.4 Exercise. Prove that the above described topology on βN induces the discretetopology on N as on a topological subspace.

B.16.5 Exercise. Prove that the topology as above on βN is Hausdorff.

B.16.6 Exercise. Prove that βN with the above topology is compact.

B.16.7 Exercise. Prove that N forms an everywhere dense subset in βN, that is, theclosure of N in βN is the entire space βN.

B.16.8 Exercise. Let X be an arbitrary compact space, and let f : N → X be an

arbitrary mapping. Prove that f extends to a continuous mapping f : βX → X.

B.16.9 Exercise. Prove that a continuous mapping f as above is unique for any given f ,assuming the space X is compact Hausdorff. In other words, if g : βN → X is a continuousmapping with g|N = f , then g = f .

B.16.10 The compact space βN as above is called the Stone–Cech compactification (orelse universal compactification) of the discrete space N.

B.16.11 Exercise. Let U ∈ βN be an ultrafilter, and let x = (xn) be a sequence of pointsin a compact metric space X, in other words, a function x : N → X. Prove that theultralimit of (xn) along U is exactly the value of the unique continuous extension x at U ,that is, the usual classical limit of the function x as n→ U :

limn→U

xn = x(U) = limn→U

x.

B.16.12 Exercise. Generalize the above as follows: suppose U is an ultrafilter on N,and let (xn) be a sequence of points in a metric space X. Prove that the following areequivalent:

1. The sequence (xn) converges along the ultrafilter U to some point x ∈ X.

2. The function n 7→ xn has a classical limit x ∈ X as n approaches the point U in thetopological space βN.

Moreover, if the two limits in (1) and (2) exist, they are equal.

134 APPENDICES

B.16.13 Solution to Exercise B.16.3. Since every ultrafilter U on N contains N and thus

U ∈ N, one concludes that N = βN and therefore

∪A⊆NA = βN.

Now let A,B ⊆ N, and let U ∈ A ∩ B. Then U ∋ A and U ∋ B, therefore U ∋ A ∩ B. It

means that U ∈ A ∩B. On the other hand, clearly A ∩B ⊆ A∩B because, more generally,if C ⊆ D ⊆ N, then C ⊆ D (un ultrafilter containing C necessarily contains D as well).

We have proved that A ∩B = A ∩ B, from where the second axiom of a base follows.

B.16.14 Solution to Exercise B.16.4. Here one should notice that N is naturally identifiedwith a subset of βN through identifying each element n ∈ N with the principal (trivial)ultrafilter, (n), generated by n:

N ∋ n 7→ (n) = {A ⊆ N : A ∋ n} ∈ βN.

This mapping is clearly one-to-one, and so we will often think of N as a subset of βN.To establish the claim, it is enough to show that every singleton in N is open in the

induced topology. Let n ∈ N. Then it is easy to see that

{n} = {(n)},

because if an ultrafilter U contains the set {n}, it must coincide with the trivial ultrafilter(n). Since the singleton {(n)} is open in βN, it is also open in N. ⊲

B.16.15 Solution to Exercise B.16.5. Let U , ζ ∈ βN, and let U 6= ζ. What is means, isthis: there is an A ⊆ N such that A belongs to one ultrafilter (say U) and not the other(that is, A /∈ ζ). A major property of ultrafilters implies then that N \ A ∈ ζ. The sets

A and N \ A are both open in βN, contain U and ζ respectively, and are disjoint: indeed,

assuming κ ∈ A∩N \ A would mean that the ultrafilter κ contains both A and N\A, whichis impossible.

B.16.16 Solution to Exercise B.16.6. Let γ be an arbitrary open cover of βN. Then

β := {A : for some V ∈ γ, A ⊆ V }

is also an open cover of βN: indeed, if U ∈ βN, then for some V ∈ γ one has U ∈ V ,and by the very definition of the topology on βN, there is an A ⊆ N with U ∈ A ⊆ V .Notice that it is now enough to select a finite subcover, say β1, of β: this done, we willthen replace each A ∈ β1 with an arbitrary V ∈ γ such that V ⊇ A, and thus we will geta finite subcover of γ. From now on, we can forget of γ and concentrate on β alone, andthe cover β consists of basic open sets.


Consider the system of subsets of N,

δ := {A ⊆ N : A ∈ β}.

This is clearly a cover of N: ∪δ = N. If we assume that δ contains no finite subcover, it isthe same as to assume that the system of complements,

{N \ A : A ∈ δ},

is centred. Being centred, it is contained in an ultrafilter, say U . One has

∀A ∈ δ, N \A ∈ U ,

and consequently,

∀A ∈ δ, A /∈ U ,which can be rewritten as

∀A ∈ δ, U /∈ A,

that is,

U /∈ ∪β,a contradiction. One concludes: there are finitely many elements A1, A2, . . . , An ∈ δ with

A1 ∪A2 ∪ · · · ∪An = N.

By force of a familiar property of ultrafilters, if U is an arbitrary ultrafilter on N, itmust contain at least one of the sets Ai, i = 1, . . . , n, or, equivalently, U ∈ Ai for somei = 1, 2, . . . , n. It follows that

A1 ∪ A2 ∪ · · · ∪ An = βN,

which of course finishes the proof by supplying the desired finite subcover β1 of β.

B.16.17 Solution to Exercise B.16.7. It is enough to find an element of N in an arbitrarynon-empty open subset of βN, say U . Since U is the union of basic open subsets, for someA ⊆ N, A 6= ∅, one has A ⊆ U . Let a ∈ A be arbitrary. Then A ∈ (a), that is, (a) ∈ A ⊆ U ,as required.

B.16.18 Solution to Exercise B.16.8. Let U ∈ βN be arbitrary. Since the space X is

compact, there is a limit, limn→U f(x) ∈ X. Denote this limit f(U). It remains to provethe continuity of f . Let U ⊆ X be open, and let f(U) ∈ U for some U ∈ βN. We want tofind a neighbourhood V of U in βN such that f(V ) ⊆ U .

136 BIBLIOGRAPHY

Every compact space (T1 or not) is regular: this is easily proved using an argumentinvolving finite subcovers of covers of closed sets with open neighbourhoods. Therefore,one can find an open set, W , in X with

f(U) ∈W ⊆ clW ⊆ U.

The condition f(U) ∈ W , that is, limn→U f(x) ∈ W , implies that f−1(W ) ∈ U . Set A :=f−1(W ). It is a non-empty subset of N, and therefore V := A forms an open neighbourhoodof U in βN. If now ζ ∈ V , that is, ζ ∋ A = f−1(W ), then every neighbourhood of the limit,f(ζ), in X meets W , meaning that f(ζ) ∈ clW and consequently f(ζ) ∈ U , as required.We conclude: f(V ) ⊆W . The continuity of the mapping

f : βN → X

is thus established.

B.16.19 Solution to Exercise B.16.9. This follows at once from a much more generalassertion, making no use of compactness whatsoever: if f, g : X → Y are continuous map-pings between two topological spaces, where Y ∈ T2, and if Z ⊆ X is everywhere dense inX, and if f |Z = g|Z , then f = g.

Indeed, assume that for some x ∈ X one has

f(x) 6= g(x).

Find disjoint open neighbourhoods V and U in Y of f(x) and g(x), respectively. Theirpreimages, f−1(V ) and g−1(U), form open neighbourhoods of x in X, and so does theirintersection. Since Z is everywhere dense in X, there is a z ∈ Z such that z ∈ f−1(V ) ∩g−1(U). Now one has

V ∋ f(z) = g(z) ∈ U,

a contradiction since V ∩ U = ∅.Without the assumption that X be Hausdorff, the statement is no longer true. A simple

example is this: let X = {0, 1} be a two-element set with indiscrete topology (that is, only∅ and all of X are open). Clearly, such an X is compact. Define a map f : N → X as theconstant map, sending each n ∈ N to 0. Since an arbitrary map from any topological spaceto an indiscrete space is always continuous, the map f admits more than one extensionto a map to X, and all of them are continuous. (E.g., one can send all elements of theremainder βN \ N to 1, or else to 0.)

Thanks to Tullio G. Ceccherini-Silberstein and Michel Coornaert for a number of re-marks on this Appendix.

Bibliography

[1] H. Abels. An example of a finitely presented solvable group. In Homological grouptheory (Proc. Sympos., Durham, 1977), volume 36 of London Math. Soc. LectureNote Ser., pages 205–211. Cambridge Univ. Press, Cambridge-New York, 1979.

[2] H. Ando, U. Haagerup, and C. Winslow. Ultraproducts, QWEP von Neumann Al-gebras, and the Effros-Marechal Topology. Preprint, available at arXiv:1306.0460v1,2013.

[3] E. Artin. Uber die zerlegung definiter functionem in quadrate. Abh. Math. Sem.Univ. Hamburg, (5), 85-89 1927.

[4] G. Arzhantseva. Asymptotic approximations of finitely generated groups. Preprint,available at http://www.mat.univie.ac.at/~arjantseva/Abs/asymptotic.pdf,2012.

[5] G. Arzhantseva and P.-A. Cherix. Quantifying metric approximations of groups.Unpublished.

[6] G. Arzhantseva and L. Paunescu. Linear sofic groups and algebras. Preprint, availableat arXiv:1212.6780, 2012.

[7] G. Arzhantseva and L. Paunescu. Almost commuting permutations are near com-muting permutations. arXiv:1410.2626, October 2014. arXiv: 1410.2626.

[8] J. Bannon. A non-residually solvable hyperlinear one-relator group. Proceedings ofthe American Mathematical Society, 139(4):1409–1410, 2011.

[9] J. Bannon and N. Noblett. A note on non-residually finite solvable hyperlinear groups.Involve, 3:345–347, 2010.

[10] G. Baumslag. A non-cyclic one-relator group all of whose finite quotients are cyclic.Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statis-tics, 10:497–498, 1969.

137

http://arxiv.org/abs/1306.0460

http://www.mat.univie.ac.at/~arjantseva/Abs/asymptotic.pdf


138 BIBLIOGRAPHY

[11] G. Baumslag and D. Solitar. Some two-generator one-relator non-hopfian groups.Bulletin of the American Mathematical Society, 68(3):199–201, 1962.

[12] B. Bekka, P. de la Harpe, and A. Valette. Kazhdan’s property (T), volume 11 of NewMathematical Monographs. Cambridge University Press, Cambridge, 2008.

[13] I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyatsov. Model theory formetric structures. In Model theory with applications to algebra and analysis. Vol.2, volume 350 of London Math. Soc. Lecture Note Ser., pages 315–427. CambridgeUniv. Press, Cambridge, 2008.

[14] B. Blackadar. Operator algebras, volume 122 of Encyclopaedia of Mathematical Sci-ences. Springer-Verlag, Berlin, 2006. Theory of C∗-algebras and von Neumannalgebras, Operator Algebras and Non-commutative Geometry, III.

[15] L. Bowen. Measure conjugacy invariants for actions of countable sofic groups. J.Amer. Math. Soc., 23(1):217–245, 2010.

[16] N. P. Brown. Connes’ embedding problem and Lance’s WEP. Int. Math. Rev. Notices,10:501–510, 2004.

[17] N. P. Brown. Topological dynamical systems associated to II1-factors. Adv. Math.,227(4):1665–1699, 2011. With an appendix by Narutaka Ozawa.

[18] N. P. Brown and V. Capraro. Groups associated to II1-factors. J. Funct. Anal.,264(2):493–507, 2013.

[19] M. Burger and A. Valette. Idempotents in complex group rings: theorems of Zalesskiiand Bass revisited. J. Lie Theory, 8(2):219–228, 1998.

[20] V. Capraro. Cross norms on tensor product of universal C*-algebras and Kirchberg property.Mathoverflow question, 2012.

[21] V. Capraro. Double ultrapower of the hyperfinite II1-factor. Mathoverflow question,2013.

[22] V. Capraro and T. Fritz. On the axiomatization of convex subsets of a Banach space.Proceedings of the AMS, 141(6), 2127-2135 2013.

[23] V. Capraro and K. Morrison. Optimal strategies for a game on amenable semigroups.International J. Game Theory, In press.

[24] V. Capraro and L. Paunescu. Product between ultrafilters and applications toConnes’ embedding problem. Journal of Operator Theory, 68(1):165–172, 2012.

http://mathoverflow.net/questions/90686/cross-norms-on-tensor-product-of-universal-c-algebras-and-kirchberg-property

http://mathoverflow.net/questions/133437/double-ultrapower-of-the-hyperfinite-ii-1-factor

BIBLIOGRAPHY 139

[25] V. Capraro and F. Radulescu. Cyclic Hilbert spaces and Connes’ embedding problem.Complex Anal. Oper. Theory, 7:863–872, 2013.

[26] V. Capraro and M. Scarsini. Existence of equilibria in countable games: An algebraicapproach. Games and Economic Behavior, 79:163–180, 2013.

[27] K. Carlson, E. Cheung, I. Farah, A. Gerhardt-Bourke, B. Hart, L. Mezuman, N. Se-queira, and A. Sherman. Omitting types and AF algebras. Archive for MathematicalLogic, 53(1-2):157–169, 2014.

[28] A. Castilla. Artin-Lang property for analytic manifolds of dimension two. Math. Z.,217, 5-14 1994.

[29] A. Cayley. On the theory of groups, as depending on the symbolic equation θn = 1,part II. Philosophical Magazine Series 4, 7(47), 408-409 1854.

[30] T. Ceccherini-Silberstein and M. Coornaert. Injective linear cellular automata andsofic groups. Israel Journal of Mathematics, 161(1):1–15, 2007.

[31] T. Ceccherini-Silberstein and M. Coornaert. Cellular automata and groups. SpringerMonographs in Mathematics. Springer-Verlag, Berlin, 2010.

[32] T. Ceccherini-Silberstein and M. Coornaert. On sofic monoids. Semigroup Forum,89(3):546–570, December 2014.

[33] T. Ceccherini-Silberstein and M. Coornaert. On surjunctive monoids.arXiv:1409.1340, September 2014.

[34] C. C. Chang and H. J. Keisler. Model theory. Princeton University Press, 1966.

[35] A. Chirvasitu. Dedekind complete posets from sheaves on von Neumann algebras.Preprint, available at arXiv:1307.8400v1, 2013.

[36] M. D. Choi. A Schwarz inequality for positive linear maps on C∗-algebras. IllinoisJ. Math., 18:565–574, 1974.

[37] M. D. Choi. The full C∗-algebra of the free group on two generators. Pacific J.Math., 87:41–48, 1980.

[38] L. Ciobanu, D.F. Holt, and S. Rees. Sofic groups: graph products and graphs ofgroups. to appear in Pac. J. Math.

[39] P. J. Cohen. The independence of the continuum hypothesis. Proc. Nat. Acad. Sci.U.S.A., 50:1143–1148, 1963.

[40] P. J. Cohen. The independence of the continuum hypothesis. II. Proc. Nat. Acad.Sci. U.S.A., 51:105–110, 1964.

http://arxiv-web3.library.cornell.edu/abs/1307.8400v1

140 BIBLIOGRAPHY

[41] B. Collins and K. Dykema. Linearization of Connes’ embedding problem. New YorkJ. Math., 14:617–641, 2008.

[42] B. Collins and K. Dykema. Free products of sofic groups with amalgamation overmonotileably amenable groups. Munster J. Math., 4:101–118, 2011.

[43] A. Connes. Classification of injective factors. Cases II1, II∞, IIIλ, λ 6= 1. Ann. ofMath. (2), 104(1):73–115, 1976.

[44] A. Connes and V. Jones. Property T for von Neumann algebras. Bull. London Math.Soc., 17(1):57–62, 1985.

[45] Y. Cornulier. A sofic group away from amenable groups. Math. Ann., 350(2):269–275,2011.

[46] Y. Cornulier. A sofic group away from amenable groups. Mathematische Annalen,350(2):269–275, June 2011.

[47] Y. de Cornulier, L. Guyot, and W. Pitsch. On the isolated points in the space ofgroups. Journal of Algebra, 307(1):254–277, 2007.

[48] Yves de Cornulier. Finitely presentable, non-hopfian groups with kazhdans property(t) and infinite outer automorphism group. Proceedings of the American Mathemat-ical Society, 135(4):951–959, 2007.

[49] C.N. Delzell. A continuous, constructive solution to Hilbert’s 17th problem. Inven-tiones Mathematicae, 76(3), 365-384 1984.

[50] J. Dodziuk, P. Linnell, V. Mathai, T. Schick, and S. Yates. Approximating L2-invariants and the Atiyah conjecture. Comm. Pure Appl. Math., 56(7):839–873, 2003.Dedicated to the memory of Jurgen K. Moser.

[51] H. A. Dye. On the geometry of projections in certain operator algebras. Ann. ofMath. (2), 61:73–89, 1955.

[52] E. Effros. Convergence of closed subsets in a topological space. Proceedings of theAMS, 16:929–931, 1965.

[53] E. Effros. Global structure in von Neumann algebras. Trans. Amer. Math. Soc.,121:434–454, 1966.

[54] G. Elek and E. Szabo. Sofic groups and direct finiteness. Journal of Algebra, 280:426–434, 2004.

[55] G. Elek and E. Szabo. Hyperlinearity, essentially free actions and L2-invariants. thesofic property. Math. Ann., 332(2):421–441, 2005.

BIBLIOGRAPHY 141

[56] G. Elek and E. Szabo. On sofic groups. Journal of Group Theory, 9(2):161–171,2006.

[57] G. Elek and E. Szabo. Sofic representations of amenable groups. Proceedings of theAMS, 139:4285–4291, 2011.

[58] I. Farah, B. Hart, M. Lupini, L. Robert, A. P. Tikuisis, A. Vignati, and W. Winter.Model theory of nuclear C*-algebras. In preparation.

[59] I. Farah, B. Hart, and D. Sherman. Model theory for operator algebras III: Elemen-tary equivalence and II1 factors. Preprint, available at arXiv:1111.0998, 2011.

[60] I. Farah, B. Hart, and D. Sherman. Model theory for operator algebras I: Stability.Bull. London Math. Soc., In press.

[61] I. Farah, B. Hart, and D. Sherman. Model theory for operator algebras II: Modeltheory. Israel Journal of Mathematics, In press.

[62] I. Farah and S. Shelah. A dichotomy for the number of ultrapowers. J. Math. Log.,10(1-2):45–81, 2010.

[63] D. Farenick, A.S. Kavruk, and V.I. Paulsen. C*-algebras with the weak expectationproperty and a multivariable analogue of Ando’s theorem on the numerical radius.J. Operator Theory, To Appear.

[64] D. Farenick, A.S. Kavruk, V.I. Paulsen, and I.G. Todorov. Characterisations of theweak expectation property. Available at arXiv:1307.1055, 2013.

[65] D. Farenick and V. Paulsen. Operator system quotients of matrix algebras and theirtensor products. Math. Scand., III:210–243, 2012.

[66] M. Fekete. Uber die verteilung der wurzeln bei gewissen algebraischen gleichungenmit. ganzzahligen koeffizienten. Math. Zeit., 17:228–249, 1923.

[67] T. Fritz. Tsirelson’s problem and Kirchberg’s conjecture. Rev. Math. Phys.,24:1250012, 67pp., 2012.

[68] W. Fulton. Algebraic Topology: A First Course. Springer, 2008.

[69] I. Gelfand and M. Neumark. On the imbedding of normed rings into the ring ofoperators in Hilbert space. In C*-algebras: 1943–1993 (San Antonio, TX, 1993),volume 167 of Contemp. Math., pages 2–19. Amer. Math. Soc., Providence, RI, 1994.Corrected reprint of the 1943 original [MR 5, 147].

[70] M. Gerstenhaber and O. S. Rothaus. The solution of sets of equations in groups.Proc Natl Acad Sci U S A, 48(1531-1533), 1962.

http://arxiv.org/pdf/1111.0998.pdf


142 BIBLIOGRAPHY

[71] L. Glebsky and L. M. Rivera. Sofic groups and profinite topology on free groups. J.Algebra, 320(3512-3518), 2008.

[72] L. Glebsky and L.M. Rivera. Almost solutions of equations in permutations. Tai-wanese Journal of Mathematics, 13(2A):493–500, 2009.

[73] K. Godel. The Consistency of the Continuum Hypothesis. Annals of MathematicsStudies, no. 3. Princeton University Press, Princeton, N. J., 1940.

[74] I. Goldbring and B. Hart. A computability-theoretic reformulation of the ConnesEmbedding Problem. Preprint, available at arXiv:1308.2638, 2013.

[75] D. Gondard and P. Ribenoim. Le 17eme probleme de Hilbert pour les matrices. Bull.Sci. Math., 98, 49-56 1974.

[76] E. I. Gordon and A. M. Vershik. Groups that are locally embeddable in the class offinite groups. St. Petersburg Math. J., 9:49–67, 1998.

[77] W. Gottschalk. Some general dynamical notions. In Recent advances in topologicaldynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972;in honor of Gustav Arnold Hedlund), pages 120–125. Lecture Notes in Math., Vol.318. Springer, Berlin, 1973.

[78] F. P. Greenleaf. Invariant Means on Topological Groups. Number 16 in Van NostrandMathematical Studies. Van Nostrand-Reinhold Co., 1969.

[79] M. Gromov. Hyperbolic groups. In Essays in group theory, volume 8 of Math. Sci.Res. Inst. Publ., pages 75–263. Springer, New York, 1987.

[80] M. Gromov. Endomorphism in symbolic algebraic varieties. J. Eur. Math. Soc.,1(2):109–197, 1999.

[81] K. W. Gruenberg. Residual properties of infinite soluble groups. Proceedings of theLondon Mathematical Society. Third Series, 7:29–62, 1957.

[82] A. Guichardet. Tensor products of C∗-algebras. Dokl. Akad. Nauk SSSR, 160:986–989, 1965.

[83] U. Haagerup and C. Winslow. The Effros-Marechal topology in the space of vonNeumann algebras. American Journal of Mathematics, 120:667–617, 1998.

[84] U. Haagerup and C. Winslow. The Effros-Marechal topology in the space of vonNeumann algebras, II. Journal of Functional Analysis, 171:401–431, 2000.

[85] D. Hadwin. A noncommutative moment problem. Proceedings of the AMS,129(6):1785–1791, 2001.


BIBLIOGRAPHY 143

[86] G. Higman. A finitely generated infinite simple group. J. London Math. Soc., 26:61–64, 1951.

[87] C.J. Hillar and J. Nie. An elementary and constructive solution to hilbert’s 17thproblem for matrices. Proc. Amer. Math. Soc., 136, 73-76 2008.

[88] V. F. R. Jones. Von Neumann algebras. Course lecture notes, 2010.

[89] M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V. B. Scholz, and R. F.Werner. Connes’ embedding problem and Tsirelson’s problem. J. Math. Phys.,52:012102, 12 pp., 2011.

[90] K. Juschenko and S. Popovych. Algebraic reformulation of Connes’ embedding prob-lem and the free group algebra. Israel Journal of Mathematics, 181:305–315, 2011.

[91] R. V. Kadison and J. R. Ringrose. Fundamentals of the theory of operator algebras.Vol. I, volume 15 of Graduate Studies in Mathematics. American MathematicalSociety, Providence, RI, 1997. Elementary theory, Reprint of the 1983 original.

[92] R. V. Kadison and J. R. Ringrose. Fundamentals of the theory of operator algebras.Vol. II, volume 16 of Graduate Studies in Mathematics. American MathematicalSociety, Providence, 1997.

[93] I. Kaplansky. Fields and rings. The University of Chicago Press, Chicago, Ill.-London,second edition, 1972. Chicago Lectures in Mathematics.

[94] A. Kar and N. Nikolov. A non-LEA sofic group. arXiv:1405.1620, 2014.

[95] A. S. Kavruk. Nuclearity related properties in operator systems. Preprint, availableat arXiv:1107.2133, 2011.

[96] A.S. Kavruk. The weak expectation property and riesz interpolation. Available atarXiv:1201.5414, 2012.

[97] A.S. Kavruk, V.I. Paulsen, I.G. Todorov, and M. Tomforde. Tensor products ofoperator systems. J. Funct. Anal., 261(2):267–299, 2011.

[98] D. Kazhdan. Connection of the dual space of a group with the structure of its closedsubgroup. Functional Anal. and its Applic., 1:63–65, 1967.

[99] D. A. Kadan. On the connection of the dual space of a group with the structureof its closed subgroups. Akademija Nauk SSSR. Funkcional\cprime nyi Analiz i egoPriloenija, 1:71–74, 1967.

[100] D. Kerr and H. Li. Entropy and the variational principle for actions of sofic groups.Invent. Math., 186(3):501–558, 2011.

http://math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf



144 BIBLIOGRAPHY

[101] D. Kerr and H. Li. Soficity, amenability, and dynamical entropy. American Journalof Mathematics, 135(3):721–761, 2013.

[102] E. Kirchberg. On semisplit extensions, tensor products and exactness of group C∗-algebras. Invent. Math., 112:449–489, 1993.

[103] E. Kirchberg. Discrete groups with Kazhdan’s property T and factorization propertyare residually finite. Math. Ann., 299, 551-563 1994.

[104] I. Klep and M. Schweighofer. Connes’ embedding problem and sums of hermitiansquares. Adv. Math., 217:1816–1837, 2008.

[105] P. H. Kropholler. Baumslag-solitar groups and some other groups of cohomologicaldimension two. Commentarii Mathematici Helvetici, 65(4):547–558, 1990.

[106] E. C. Lance. On nuclear C∗-algebras. Journal of Functional Analysis, 12:157–176,1973.

[107] E. Lindenstrauss and B. Weiss. Mean topological dimension. Israel Journal of Math-ematics, 115(1):1–24, 2000.

[108] T. A. Loring. Lifting solutions to perturbing problems in C∗-algebras, volume 8 ofFields Institute Monographs. American Mathematical Society, Providence, RI, 1997.

[109] J. Los. Quelques remarques, theoremes et problemes sur les classes definissablesd’algebres. In Mathematical interpretation of formal systems, pages 98–113. North-Holland Publishing Co., Amsterdam, 1955.

[110] P. Lucke and S. Thomas. Automorphism groups of ultraproducts of finite symmetricgroups. Communications in Algebra, 39:3625–3630, 2011.

[111] M. Lupini. Class-preserving automorphisms of universal hyperlinear groups. Avail-able at arXiv:1308.3737, 2013.

[112] M. Lupini. Logic for metric structures and the number of universal sofic and hyper-linear groups. Proceedings of the AMS, In press.

[113] O. Marechal. Topologie et structure borelienne sur l’ensemble des algebres de vonNeumann. Compt. Rend. Acad. Sc. Paris, pages 847–850, 1973.

[114] S. Meskin. Nonresidually finite one-relator groups. Transactions of the AmericanMathematical Society, 164:105–114, 1972.

[115] R. Moreno and L.M. Rivera. Blocks in cycles and k-commuting permutations.arXiv:1306.5708, 2013.

http://arxiv.org/pdf/1308.3737v3.pdf

BIBLIOGRAPHY 145

[116] F. J. Murray and J. von Neumann. On rings of operators. IV. Ann. of Math. (2),44:716–808, 1943.

[117] H. Neumann. Generalized free products with amalgamated subgroups. AmericanJournal of Mathematics, 70:590–625, 1948.

[118] P. Ara K. C. O’Meara and F. Perera. Stable finiteness of group rings in arbitrarycharacteristic. Adv. Math., 170(2):224–238, 2002.

[119] D. Ornstein and B. Weiss. Entropy and isomorphism theorems for actions of amenablegroups. Journal d’Analyse Mathematique, 48:1–141, 1987.

[120] N. Ozawa. About the QWEP conjecture. Internat. J. Math., 15(5):501–530, 2004.

[121] N. Ozawa. There is no separable universal II1-factor. Proc. Amer. Math. Soc.,132(2):487–490 (electronic), 2004.

[122] N. Ozawa. About the Connes embedding conjecture. Jpn. J. Math., 8(1):147–183,2013.

[123] N. Ozawa. Tsirelson’s problem and asymptotically commuting unitary matrices. J.Math. Phys., 54:032202 (8 pages)., 2013.

[124] A. L. T. Paterson. Amenability, volume 29 of Mathematical Surveys and Monographs.American Mathematical Society, Providence, RI, 1988.

[125] L. Paunescu. On sofic actions and equivalence relations. J. Funct. Anal., 261(9):2461–2485, 2011.

[126] L. Paunescu. All automorphisms of the universal sofic group are class preserving.Preprint, available at arXiv:1306.3469, 2013.

[127] V. Pestov. Hyperlinear and sofic groups: a brief guide. The Bulletin of SymbolicLogic, 14:449–480, 2008.

[128] V. Pestov and A. Kwiatkowska. An introduction to hyperlinear and sofic groups, vol-ume 406 of London Mathematical Society Lecture Notes, pages 145–186. CambridgeUniversity Press, 2012.

[129] S. Popa. Correspondences. INCRES, 1986.

[130] R. T. Powers and E. Størmer. Free states of the canonical anticommutation relations.Comm. Math. Phys., 16:1–33, 1970.

[131] C. Procesi and M. Schacher. A non commutative real Nullstellensatz and Hilbert’s17th problem. Ann. of Math., 104, 395-406 1976.


146 BIBLIOGRAPHY

[132] F. Radulescu. A non-commutative, analytic version of Hilbert’s 17th problem in typeII1 von Neumann algebras. In Von Neumann algebras in Sibiu, volume 10 of ThetaSer. Adv. Math., pages 93–101. Theta, Bucharest, 2008.

[133] F. Radulescu. The von Neumann algebra of the non-residually finite Baumslag group⟨a, b∣∣ab3a−1 = b2

⟩embeds into Rω, pages 173–185. Hot Topics in Operator Theory.

Theta Ser. Adv. Math., 2008.

[134] D. Romik. Stirling’s approximation for n!: The ultimate short proof? The AmericanMathematical Monthly, 107(6):556–557, 2000.

[135] J.M. Ruiz. On Hilbert’s 17th problem and real Nullstellensatz for global analyticfunctions. Math. Z., 190, 447-454 1985.

[136] I. E. Segal. Irreducible representations of operator algebras. Bull. Amer. Math. Soc.,53:73–88, 1947.

[137] S. Shelah. Classification theory and the number of nonisomorphic models, volume 92of Studies in Logic and the Foundations of Mathematics. North-Holland PublishingCo., Amsterdam, second edition, 1990.

[138] S. Shelah. Vive la difference, III. Israel Journal of Mathematics, 166:61–96, 2008.

[139] D. Sherman. Notes on automorphisms of ultrapowers of II1 factors. Studia Math.,195(3):201–217, 2009.

[140] M. H. Stone. Postulates of barycentric calculus. Ann. Mat. Pura Appl., 29(1):25–30,1949.

[141] M. Takesaki. On the cross-norm of the direct product of C∗-algebras. Tohoku Math.J., 16:111–122, 1964.

[142] M. Takesaki. Theory of operator algebras I. Springer, 2001.

[143] T. Tao. Finite subsets of groups with no finite models. Post in the blog “What’snew”, 2008.

[144] A. Thom. Sofic groups and Diophantine approximation. Comm. Pure Appl. Math.,61:1155–1171, 2008.

[145] A. Thom. Examples of hyperlinear groups without factorization property. Groups,Geometry and Dynamics, 4(1):195–208, 2010.

[146] A. Thom. About the metric approximation of Higman’s group. Journal of GroupTheory, 15(2):301–310, 2013.

http://terrytao.wordpress.com/2008/10/06/finite-subsets-of-groups-with-no-finite-models/

INDEX 147

[147] S. Thomas. On the number of universal sofic groups. Proceedings of the AMS,138(7):2585–2590, 2010.

[148] T. Turumaru. On the direct-product of operator algebras. II. Tohoku Math. J. (2),5:1–7, 1953.

[149] D. Voiculescu. Symmetries of some reduced free product C*-algebras. In Springer,editor, Operator algebras and their connections with topology and ergodic theory.,volume 1132 of Lecture Notes in Math., pages 556–588, 1985.

[150] D. Voiculescu. The analogues of entropy and of Fischer’s information measure in freeprobability theory, ii. Invent. Math., 118:411–440, 1994.

[151] J. von Neumann. Zur allgemeinen theorie des masses. Fund. Math., 12:73–116, 1929.

[152] B. Weiss. Sofic groups and dynamical systems. Sankhya Ser. A, 62(3):350–359, 2000.Ergodic theory and harmonic analysis (Mumbai, 1999).

[153] A. Zalesskiı. On a problem of Kaplansky. Dokl. Akad. Nauk SSSR, 203:477–495,1972.

Index

Los’ theoremfor rank rings, 49

for tracial von Neumann algebras, 55 Los’ theorem

for invariant length groups, 36

algebraic integer, 57approximate morphism

of factors, 71, 72of invariant length groups, 17, 21, 23,

25, 41, 43, 73

approximation property, 43axiomatizable class, 40, 49

of invariant length groups, 35of rank rings, 48

of tracial von Neumann algebras, 53

Banach space, 103dual, 104

strictly convex, 104basic formula

for invariant length groups, 34for rank rings, 48

for tracial von Neumann algebras, 52Bernoulli shift, 7

amenable, 63integer, 60

sofic, 65

C*-algebra, 1, 111full, 112

weakly cp complemented, 84

C*-identity, 111center, 2, 53commutant, 2

double, 2duble, 2

conjecture

algebraic eigenvalues, 57Connes’ embedding, 4, 6, 7, 71–73, 76,

78, 79, 83, 98direct finiteness, 9, 10, 50Gottschalk’s surjunctivity, 7, 10, 61, 64,

66idempotent elements, 10Kervaire-Laudenbach, 41, 42nilpotent elements, 10trace of idempotents, 10

units, 10zero divisors, 10

countably saturatedinvariant length group, 37rank ring, 48tracial von Neumann algebra, 54

definable set, 56

elementary equivalence, 35elementary property, 35, 53

of rank rings, 48

of tracial von Neumann algebras, 53entropy

amenable group action, 63single transformation, 60

148

INDEX 149

sofic group action, 65

equivariant function, 7

eventually, 77

extension, 25, 31

HNN, 25, 27, 29

factor, 2

AFD, 79

injective, 79, 85

locally representable, 72locally universal, 73

separable, 3

type I, 79

type Ifin, 79

type II1, 3

Fekete’s lemma, 60

formula

existential, 39

for invariant length groups, 34

for rank rings, 48

for tracial von Neumann algebras, 52

interpretation, 34, 48quantifier-free, 34, 39

universal, 39, 72

frequently, 77

Frobenius under trace, 11

function

positive type, 28

generating clopen partition, 59, 63, 65

Gerstenhaber-Rothaus theorem, 42

group

amenable, 23, 61

Baumslag-Solitar, 28, 29finite, 23

finitely discriminable, 33

finitely generated, 28

finitely presented, 33

free, 4, 24

Higman’s, 45

Hopfian, 28, 30

hyperbolic, 24, 28hyperlinear, 7, 8, 19–22, 24, 29, 39, 41–

44, 73, 74ICC, 4initially subamenable, 24invariant length, 13isolated, 33LEA, 24, 29LEF, 24, 28, 29linear sofic, 44locally embeddable, 23marked, 32one-relator, 24, 29property (T), 28, 30residually amenable, 28residually finite, 24, 28, 32sofic, 7–9, 14–17, 20, 23, 24, 28, 31, 39–

41, 43, 44, 48, 52, 57, 65, 66solvable, 32special linear, 29surjunctive, 8universal hyperlinear, 20, 38, 74universal sofic, 16, 38–40weakly sofic, 44

group algebra, 9, 10complex, 3, 8, 56, 57

group von Neumann algebra, 4, 8, 74

invariant mean, 86

Kirchberg’s property, 83

lattice, 30length function, 13

commutator contractive, 45Hamming, 14Hilbert-Schmidt, 19invariant, 13unitary group, 55

limidirect, 25

limit

150 INDEX

direct, 25

inverse, 25linear map

completely positive, 84positive, 84

microstate, 5, 22minimal projection, 103

negation, 35

normC*-, 112

Hilbert-Schmidt, 3maximal, 113

minimal, 113operator, 1

normed *-algebra, 111

polar decomposition, 23product

direct, 25fraph, 25

free, 25, 26free (amalgamated), 25, 27

graph, 27of ultrafilters, 74

property (T), 105

QWEP, 84

rank function, 48

rank ring, 48finite, 49

representationcutdown, 102

direct sum, 81separating family of, 81

unitarily equivalent, 81

sentencefor invariant length groups, 34–36, 40

for rank rings, 48, 49

for tracial von Neumann algebras, 52,53, 55

sequencerepresentative, 15, 35, 49, 54, 57sofic approximation, 65submultiplicative, 59

set of formulaeapproximately finitely satisfiable, 37, 54consistent, 37, 54realized, 37, 54

stability, 22standard isomorphism, 99

termfor invariant length groups, 33for rank rings, 48for tracial von Neumann algebras, 52

theoryexistential, 72universal, 72

tiling, 62topology

Effros-Marechal, 77, 78strong operator, 1weak operator, 1

trace, 2, 3tracial state, 85type, 37

ultrapowerof invariant length groups, 36of rank rings, 49of tracial von Neumann algebras, 54

ultraproductof invariant length groups, 35of permutations groups, 14of rank rings, 48of tracial von Neumann algebras, 54of unitary groups, 19

unitary group, 53

von Neumann algebra, 1

INDEX 151

diffuse, 103finite, 2hyperfinite, 71tracial, 2

weak expectation property, 84relative, 85

arxiv:1309.2034v6 [math.gr] 5 may 2015 · arxiv:1309.2034v6 [math.gr] 5 may 2015 introduction to...

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