Outer automorphism groups and bimodule categories of type II_1 factors

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type II_1 factors

Sébastien Falguières

To cite this version:

Sébastien Falguières. Outer automorphism groups and bimodule categories of type II_1 factors. Operator Algebras [math.OA]. Université Catholique de Louvain, 2009. English. �tel-00685967�

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KATHOLIEKE UNIVERSITEIT LEUVEN

Faculteit Wetenschappen

Departement Wiskunde

Outer automorphism groups and

bimodule categories of type II

₁

factors

S´ebastien Falgui`eres

Promotor :

Prof. dr. Stefaan Vaes

Proefschrift ingediend tot het behalen van de graad van Doctor in de Wetenschappen

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Dankwoord

Ainsi s’achève ma thèse et avec elle mes trois années belges. Je me sou-viens très bien du jour où Stefaan m’annon¸cait que je pouvais partir faire ma thèse en Belgique sous sa direction ; c’était il y a déjà trois ans, juste après la soutenance de mon mémoire de DEA. C’est justement au cours de cette année de DEA que j’ai commencé à m’intéresser aux Algèbres de von Neumann. La possibilité que m’offrait Stefaan de continuer à tra-vailler dans ce domaine était alors une formidable opportunité.

Ainsi, la première personne que je tiens à remercier c’est toi Stefaan. Même si avec le temps ne restent que les bons souvenirs, quitter la France et avec elle, famille et amis n’a pas été chose facile au début. Heureuse-ment tu m’as tout de suite orienté vers des problèmes mathématiques mo-tivants et tu as toujours su être disponible. C’est grâce aux nombreuses idées et conseils avisés que tu m’as souvent donnés que j’ai pu mener à bien mes travaux. En effet, nos discussions, mathématiques ou non, ont été fréquentes et m’ont toujours redonné le courage et la motivation qui à certains moments de la thèse viennent à manquer. Ce manuscrit tel qu’il est aujourd’hui, résumant mon travail de thèse, doit beaucoup à la relec-ture minutieuse que tu en as faite et je t’en remercie. Je suis fier d’avoir été ton élève.

Je tiens maintenant à remercier Claire Anantharaman, Dietmar Bisch et Johan Quaegebeur pour tous leurs commentaires pertinents concernant une version préliminaire de cette thèse. Enfin, je suis reconnaissant à Raf Cluckers, Mark Fannes et Alain Valette d’avoir accepté de faire partie de mon jury.

Je souhaite remercier chaleureusement tous mes amis fran¸cais qui m’ont soutenu depuis mon départ. Je pense notamment à David et Laam, mais aussi à Matthieu mon compagnon cycliste ainsi qu’à Fred et Caro. Je pense aussi à Cyril avec qui je suis resté en contact constant même depuis la Californie et avec qui j’ai toujours eu plaisir à discuter. Son enthou-siasme communicatif pour les mathématiques qu’il pratique m’a souvent redonné confiance. Enfin, je n’oublie pas Frédéric et Omar avec qui j’ai étudié à Paris mais aussi, Aude et Gérald et bien évidemment Carine. Merci à vous tous !

I also want to address a special thank to Anselm, Reiji and Yoshiko for all the fun time we had when they lived in Leuven. I always took benefit

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this was a precious help to start writing Chapter 7 ! Also many thanks to all my friends from the mathematics department at K.U.Leuven and a special word for my officemate Steven, the only person I know speaking fluently so many computer languages !

Je remercie également Claire Anantharaman, Jean Renault et Eric Ricard de m’avoir invité à venir exposer mes travaux dans leurs séminaires. I am also grateful to Yasuyuki Kawahigashi for inviting me for a long stay at Tokyo university. I discovered there a very pleasant place to work and a great city !

Je conclurai en remerciant bien évidemment toute ma famille pour m’avoir toujours accompagné et soutenu durant ces années d’études.

S´ebastien Falgui`eres. Leuven, le 22 juin 2009.

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Introduction

Von Neumann algebras are unital ∗-subalgebras of the algebra of bounded operators on a Hilbert space that are closed under the weak operator topology. They are characterized, by von Neumann’s bicommutant theo-rem, as being the unital ∗-subalgebras of the algebra of bounded operators on a Hilbert space that are equal to their bicommutant. Murray and von Neumann classified these algebras in three types. They proved in [58] that classifying von Neumann algebras amounts to classifying those with trivial center. The von Neumann algebras for which the center only con-sists of scalar multiples of the identity and which are thus the most non-commutative ones, are called factors and are classified in types I, II and III. Factors are the von Neumann algebras that cannot be decomposed into the direct sum of two other von Neumann algebras.

In this thesis, we focus on factors of type II1. These factors are infinite

dimensional factors endowed with a faithful normal trace. We call a trace normal if its restriction to the unit ball is weakly continuous. Consider now a countable group Γ such that the conjugacy classes {hgh−1 | h ∈ Γ} are infinite for all g 6= e. Such groups with infinite conjugacy classes (called ICC groups) appear frequently. Indeed, the free groups Fn, the

infinite symmetric group S∞:=S_n≥1Snand the groups PSL(n, Z), with

n ≥ 2, are all ICC. The von Neumann algebra generated by the left regular representation of the ICC countable group Γ yields a type II1 factor L(Γ),

called the group von Neumann algebra of Γ. Although this construction is very natural, distinguishing between group factors is usually very hard and many problems about group von Neumann algebras are open. For example, it is still unknown whether the group von Neumann algebras L(F2) and L(F3) are isomorphic or not. In order to distinguish between

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Murray and von Neumann defined the fundamental group F (M ) of the II1 factor M as the following subgroup of R∗+

F (M ) := {t > 0 | Mt∼= M } .

In the case where t ≤ 1, the amplification Mt is defined by Mt := pM p, where p is a projection of trace t in M ; see section 6.1.2 for the precise definition. It can be proven that F (M ) is a multiplicative subgroup of R∗+. This group has a fascinating history. Murray and von Neumann

at their time were only able to prove that the fundamental group of L(S∞) is the whole of R∗+. One had to wait forty years, until the work of

Connes to obtain the existence of II1 factors for which the fundamental

group is different from R∗+. In [7] Connes proved that the fundamental

group of the group factor of an ICC property (T) group is countable. However, even if the fundamental group of L(SL(3, Z)) follows countable by Connes’ result, there are still no computations of it, even nowadays. Voiculescu proved in [56] that the fundamental group of L(F∞) contains

Q+ and R˘adulescu proved in [45] that it is the whole of R∗+. Although

it is still unknown if the the factors L(Fn) and L(Fm) are isomorphic

or not, for n 6= m R˘adulescu in [44] and Dykema in [14] proved the following alternative. Either all L(Fn) are isomorphic and in that case

their fundamental group is the whole of R∗+, either they are two by two non

isomorphic and their fundamental group is trivial. The first examples of II1 factors with trivial fundamental group were given by Popa in [36, 37].

He proved that L SL(2, Z) n Z2 has trivial fundamental group, thus answering a long standing open problem. He went even further in the understanding of fundamental groups since he proved in [34] that every countable subgroup of R∗+ can arise as the fundamental group of a II1

factor. Some other constructions of factors with prescribed countable fundamental group were also obtained later by Ioana, Peterson, Popa [21] and Houdayer [20]. Very recently, Popa and Vaes [42, 41] proved that the fundamental group actually ranges over a large family of uncountable subgroups of R∗+, which was also an open problem for many years.

The outer automorphism group of M , defined by

Out(M ) := Aut(M ) Inn(M )

is another important invariant of M . The outer automorphism group of the hyperfinite II1 factor R is a huge group, it contains for example

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every locally compact second countable group and every countable group acts via outer automorphisms on R. The outer automorphism group of a II1 factor is also hard to compute in general. In [7] Connes proved that

the outer automorphism group of the group factor of an ICC property (T) group is countable. Then, one had to wait until the work of Ioana, Peterson and Popa in 2004 to obtain really new results. They answered a long standing open problem, proving the existence of II1 factors for

which every automorphism is inner. They proved, more generally, that every abelian compact group arises as the outer automorphism group of a II1 factor. These results were still only existence results. But later,

Popa and Vaes [43] gave explicit examples of II1 factors with trivial outer

automorphism group. They even proved that every finitely presented group can be explicitly realized as the outer automorphism group of a II1 factor. Popa and Vaes also gave examples of ICC groups Γ for which

Out(L(Γ)) ∼= Char(Γ) o Out(Γ). In his ten problem list [23] Jones had asked whether this isomorphism could hold for arbitrary ICC property (T) groups and this is still unknown. These results were refined later by Vaes [53] where he proved that every countable group can arise as the outer automorphism group of a II1 factor.

One of the richest invariants of a II1 factor M is the bimodule category

Bimod(M ) consisting of all M -M -bimodules MHM of finite Jones index:

dim(MH) < ∞ and dim(HM) < ∞. See Chapter 4 for the definition of

the Jones index. Equipped with the Connes tensor product, Bimod(M ) is a C∗-tensor category; see Chapter 3 and Chapter 6 for these notions. The category Bimod(M ) completely encodes both the fundamental group F (M ) and the outer automorphism group Out(M ) (see Proposition 6.5). Furthermore, Bimod(M ) also encodes, in a certain sense, all subfactors M0⊂ M of finite Jones index [24]: performing Jones’ basic construction,

we get M0 ⊂ M ⊂ M1 and obtain the M -M -bimodule ML2(M1)M; see

section 4.3 concerning the basic construction.

As a result, it seemed until recently quite hopeless to explicitly compute Bimod(M ) for any II1 factor M . Vaes obtained in [54] the existence of

II1 factors with trivial bimodule category and hence also trivial

subfac-tor structure, trivial fundamental group and trivial outer automorphism group. This result was still an existence result, following the work of [21]. The first concrete II1 factors with trivial bimodule category were also

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factors where Bimod(P ) is a Hecke-like category.

Description of the chapters and statement of the

main results

This thesis is divided in three parts. The first part is an introductory one. In the first chapter we give a general and basic introduction to von Neumann algebras. We also mention some key results that were obtained in this field. We do not try to provide an exhaustive list of the recent results, we would rather highlight some important connections with other areas of mathematics such as infinite group theory and ergodic theory. In the second chapter we go deeper into the theory of von Neumann algebras and focus on specific topics that will form the general background material for this thesis such as Popa’s relative property (T) for von Neumann algebras, outer actions of countable groups on II1 factors or the algebra

of operators affiliated with a II1 factor.

The second part of the thesis is entirely devoted to bimodules over II1

factors and consists of five chapters. This part does not contain new results and is included in order to make the text rather self-contained. Chapter 3 is an introduction to bimodules. The material gathered in this chapter is taken from Bisch’s paper [1]. In Chapter 4, we explain how modules (not bimodules) over a II1 factors are completely classified

by a number between 0 and +∞ called dimension or coupling constant, already introduced by Murray and von Neumann in [30]. We give basic properties of the coupling constant and the Jones index of an inclusion of II1 factors. In Chapter 5, we investigate properties of bimodules with

finite left and right coupling constant. These bimodules are called finite index bimodules. It will become clear throughout the text that finite index bimodules are much richer, and also more difficult to analyze, than the left or right modules. This chapter is meant to be a toolbox providing concrete lemmas to work with bimodules. At the end of the chapter, we recall the construction of a dimension function on the class of finite index bimodules, using the language of C∗-tensor categories with conjugates. In Chapter 6, finite index bimodules over II1 factors are analyzed in a more global way.

It will be clear, after the foregoing chapters, that the class of finite index bimodules over II1 factors, modulo unitary equivalence, has a natural

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notions and constructions associated to fusion algebras such as freeness or free product of fusion algebras. We also prove that the fundamental group and the outer automorphism group of M are entirely encoded by finite index M -M -bimodules. In the last chapter of this second part we study minimal actions of compact groups on II1 factors. We recall

the construction of a fully faithful tensor functor between the category Rep(G) of finite dimensional representations of the compact group G and the category Bimod(MG) of finite index MG-MG-bimodules, where MG denotes the fixed-point algebra. This result was obtained in the 80’s (see [47]), and formulated in the context of algebraic quantum field theory. The third part of the thesis contains the results I obtained during my Ph.D research. I provide computations of the outer automorphism group and the bimodule category of certain type II1 factors. All these factors

are amalgamated free product von Neumann algebras. In Chapter 8, I recall the construction and some basic properties of amalgamated free product von Neumann algebras. I also state some important rigidity re-sults for amalgamated free product von Neumann algebras obtained by Ioana, Peterson and Popa in [21]. This chapter also recalls a few essential results from [54] where Vaes proves the existence of II1factors with trivial

bimodule category. For the convenience of the reader, I give a proof of most of the statements. In Chapter 9 and Chapter 10 I give a detailed version of the papers [15, 16]. These two papers were written in collabora-tion with Stefaan Vaes. I prove the existence of type II1 factors for which

the outer automorphism group can be any prescribed second countable compact group. I also prove the existence of type II1 factors with

bimod-ule category isomorphic to the representation category of any prescribed second countable compact group. The starting point of the whole of the work done in this thesis is the paper [21] by Ioana, Peterson and Popa in which they proved, among many other deep results, that the outer auto-morphism group of a type II1 factor could be any abelian compact group.

As a consequence, they obtained the first examples of II1 factors having

only inner automorphisms. Their results are existence results. Popa has developed a theory of deformation and rigidity for group actions and von Neumann algebras. He extended the notion of relative property (T) for groups to the von Neumann setting. An inclusion N ⊂ M having relative property (T) in the sense of Popa is called rigid, see section 2.3. One can prove that L(Λ) ⊂ L(Γ) is rigid if and only if the pair of groups Λ ⊂ Γ has the relative property (T). His deformation/rigidity techniques led to

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sev-eral breakthroughs in the theory of von Neumann algebras during the last few years. His results also found important applications in ergodic theory. In his strong rigidity theorem obtained in [35], he was able to deduce for the first time the conjugacy of group actions out of their von Neumann equivalence; see section 1.7 for a precise statement. The same defor-mation and rigidity techniques are the core of all the above mentioned results [36, 37, 34, 21, 20, 42, 41, 53] on the fundamental group and the outer automorphism group of II1factors. Amalgamated free products von

Neumann algebras allow lots of deformations. Ioana, Peterson and Popa applied the deformation/rigidity techniques in this context and proved striking rigidity results : they proved, for example, that any rigid von Neumann subalgebra Q ⊂ M0∗N M1 must be “weakly contained” in M0

or M1, see Theorem 8.7 and Theorem 8.8. In the terminology of Popa,

saying that Q is weakly contained in M0 inside M := M0 ∗N M1 means

that there exists a Q-M0-subbimodule of L2(M ) which is finitely

gener-ated as right M0-module. In the “good cases”, this weak containment

leads to actual unitary conjugacy. This technique is known as Popa’s intertwining by bimodules technique; see Theorem 5.19. The previous result locating rigid subalgebras of amalgamated free product von Neu-mann algebras was a crucial ingredient to prove the existence of II1factors

with prescribed abelian compact outer automorphism group. The factors they constructed are of the form M0∗N M1, where N is hyperfinite and

Mi = N o Γi, for countable groups Γi acting outerly on N (see section

2.4 for the notion of outer action). The details of the strategy they used can be found in section 8.2 and a precise statement of their theorem is given in Theorem 8.15. Note that the factor M they construct is in fact the crossed product of N by the group Γ0 ∗ Γ1. Their assumptions on

M are of three different types. There are assumptions on the groups, assumptions on the actions and they also make a “freeness assumption” in the following sense. They suppose that Γ1, viewed as a subgroup of

Out(N ) is free with respect to another countable subgroup of Out(N ) so that, roughly, Γ1 and the normalizer of Γ0 in Out(N ) live far away from

each other. They also suppose that N ⊂ N o Γ0 is rigid. Following their

constructions, we could generalize their methods and prove that even non-abelian compact groups can arise as the outer automorphism group of a II1 factor. Consider a minimal action σ of G on the hyperfinite II1 factor

M1 and denote N := M1G. Let Γ0 be a countable group acting by outer

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The leg M0 is identical to Ioana, Peterson and Popa’s work. We assume

that M0 has property (T). We exploit the different behavior of M0 and

M1 (one is property (T), the other is hyperfinite) to prove, using Ioana,

Peterson and Popa’s results, that every automorphism α ∈ Aut(M ) sat-isfies, up to unitary conjugacy in M , that α(N ) = N and α(M0) = M0.

Here again, our assumptions on the actions of G and Γ0 are linked by a

freeness assumption. A careful use of this freeness assumption will force the automorphism α to preserve M1 globally and imply that α fixes N

pointwise, all this up to unitary conjugacy in M . Then, by some classical theory of outer actions and minimal actions, α|M0 is of the form αω, for

some character ω of Γ, where αω(auγ) = ω(γ)uγ and α|M1 = σg, for some

element g ∈ G. The precise theorem is the following. We use the notation FAlg(N ) to denote the collection of all finite index N -N -bimodules of the II1 factor N , modulo unitary conjugacy.

Theorem I.1 (Theorem 9.1). Let M1 be the hyperfinite II1 factor and

G a compact group acting on M1. Denote N = M1G, the von Neumann

algebra of G-fixed points in M1. Let Γ be an ICC group acting on N .

Denote M0 := N o Γ. Assume that

1. the action σ : G y M1 is minimal,

2. the action Γ y N is outer and M0 has the property (T),

3. the natural images of Rep G ,→ FAlg(N ) and Aut(N ⊂ M0) restr

−→ Out(N ) ⊂ FAlg(N ) inside the fusion algebra FAlg(N ), are free in the sense of Definition 6.14.

Then, the homomorphism

Char(Γ) × G → Aut(M0 ∗

NM1) : (ω, g) 7→ αω∗ σg

induces an isomorphism Char(Γ) × G ∼= Out(M0 ∗ NM1).

We also prove the following.

Corollary I.2 (Corollary 9.2). Let G be a compact, second countable group and σ : G y R a minimal action on the hyperfinite II1 factor R.

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point algebra RG, such that for M given as the amalgamated free product M = (RGo Γ) ∗

RGR, the natural homomorphism

G → Aut(M ) : g 7→ id ∗ σg

induces an isomorphism G ∼= Out(M ).

We deduce the previous corollary from Theorem I.1 above and another crucial ingredient that we explain now. We make use of the fact that the fusion algebra of the hyperfinite II1 factor is huge, in the sense that

it contains many free fusion subalgebras. Because of [54, Theorem 5.1] (see Theorem 6.15), given any countable fusion subalgebras F0 and F1

of FAlg(R) there always exists, via a Baire category argument, an auto-morphism α of the hyperfinite II1 factor such that F0 “conjugated” by

α becomes free with respect to F1. This theorem is a key point for our

proofs and is responsible of the fact that we only obtain existence results: we make two fusion subalgebras free, via a Baire category result. Ioana, Peterson and Popa’s theorem also relies on a similar Baire category type result (see [21, Lemma 1.2]) based on the fact that Out(R) contains many free countable subgroups. In both cases, for [54, Theorem 5.1] and [21, Lemma 1.2], the key ingredients come from [38]. This kind of argument is also used in [54] where Vaes proves the existence of factors with trivial bimodule category.

In [16], we managed to prove a much stronger result. We prove the exis-tence of a II1 factor (M, τ ) and minimal action of a given compact group

G on M such that writing P := MG, every finite index P -P -bimodule is isomorphic with PMor(L2(M ), Hπ)P for a uniquely determined finite

dimensional unitary representation π : G → U (Hπ). Here, the space

Mor(L2(M ), Hπ) :={T : L2(M ) → Hπ | T (σg(ξ)) = π(g)T (ξ),

for all ξ ∈ L2(M ), g ∈ G}

endowed with the scalar product hS, T i := τ (S∗T ) is a P -P -bimodule, for the following left and right P -actions:

(a·S·b)(ξ) := S(a∗ξb∗), for all S ∈ Mor(L2(M ), Hπ), a, b ∈ P, ξ ∈ L2(M ).

More precisely, we prove that

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defines an equivalence of C∗-tensor categories.

The proof is considerably more involved than our first results. We want to understand all finite index bimodules, not only the ones for which the left and right dimension is one (i.e the automorphisms). Still, our factors of the form MG, where M is again an amalgamated free product over the hyperfinite II1 factor. We have to impose many conditions to be able to

control all such finite index MG-MG-bimodules. We do not give here all the technical assumptions and refer to section 10.2 for more information. However we make the following remarks. Let G be a second countable compact group acting on the hyperfinite II1 factor M1. Denote N = M1G

and fix an inclusion N ⊂ M0 into the II1 factor M0. We are interested

in the II1 factor M := M0∗N M1 and extend the action G y M1 to an

action G y M by acting trivially on M0. We denote by P := MG the

fixed point II1 factor. We will assume that M0 contains a subfactor N0

with property (T). The first step of our proof relies on a careful analysis of all finite index M0-P -bimodules. As for the analysis of the automorphisms

in Theorem I.1, we start with the “property (T) leg M0” and use here

again the rigidity results of Ioana, Peterson and Popa. We also make the following construction. Consider the group

Γ := (Q3⊕ Q3) o SL(3, Q) ,

defined by the action A · (x, y) = (Ax, (At)−1y) of SL(3, Q) on Q3⊕ Q3_.

Let Λ := Z3 ⊕ Z3_{. Choose an irrational number α ∈ R and define the}

2-cocycle Ω ∈ Z2(Γ, S1) such that

Ω (x, y), (x0, y0) = exp iα(hx, y0i − hy, x0i), for all (x, y), (x0, y0) ∈ Q3⊕ Q3 _,

Ω(g, A) = Ω(A, g) = Ω(A, B) = 1, for all g ∈ Γ , A, B ∈ SL(3, Q) .

Theorem I.3 (Theorem 10.1). Let G be a compact, second countable group. Define N := LΩ(Λ) and M0 := LΩ(Γ) with the groups Λ, Γ and

the cocycle Ω as above. Let M1 be the hyperfinite II1 factor. Denote

M := M0∗NM1. Then, there exists a minimal action G y M such that

Rep(G) → Bimod(MG) is an equivalence of C∗-tensor categories. Once again, this theorem uses [54, Theorem 5.1], leading again to an existence result.

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Some notations and

conventions

Generalities

We use the Kronecker symbol on the set I defined by

δi,j : I → {0, 1} : k 7→

0 if i 6= j 1 if i = j

Let A ⊂ H be a subset of the Hilbert space H. We denote by [A] the closed vector space generated by A.

We will always denote by Tr the non-normalized trace on Mn(C) and by

tr the normalized one.

Hilbert spaces and operators

All our inner products h·, ·i are linear in the second variable. The norm associated is denoted by k · k.

All Hilbert spaces are assumed to be separable. We denote by ei ∈ Cn the natural vectors.

The algebraic tensor product is denoted by . The tensor product of the Hilbert spaces H0 and H1 is denoted by H0⊗ H1.

We denote by B(H, K) the vector space of bounded operators from H to K. We denote by B(H) := B(H, H) for the algebra of bounded operators on H.

If ξ ∈ H, we denote by ξ∗ : H → C : η 7→ hξ, ηi. We denote by H the dual Hilbert space H := {ξ∗| ξ ∈ H}.

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We write Cn(Cm)∗instead of B(Cm, Cn). We implicitly consider Cn(Cm)∗ as a Hilbert space with scalar product hξ, ηi = Tr(ξ∗η). Obviously, Cn(Cm)∗ is an Mn(C)-Mm(C)-bimodule.

We use the leg numbering notation defined as follows. Let H1, H2, H3 be

Hilbert spaces. Denote by Σ the flip map on H2⊗ H3. Then we have the

following bounded operators on H1⊗ H2⊗ H3.

• T ∈ B(H₁⊗H₃) =⇒ T13:= (1⊗Σ∗)(T ⊗1)(1⊗Σ) ∈ B(H1⊗H2⊗H3),

• T ∈ B(H₁⊗ H₂) =⇒ T12:= T ⊗ 1 ∈ B(H1⊗ H2⊗ H3),

• T ∈ B(H₂⊗ H₃) =⇒ T23:= 1 ⊗ T ∈ B(H1⊗ H2⊗ H3).

von Neumann algebras

All von Neumann algebras are assumed to have separable predual. We denote by M∗ the predual of M .

For any von Neumann algebra M we denote Mn:= Mn(C) ⊗ M .

Let M be a von Neumann algebra. We say that elements (eij) ∈ M form

a system of matrix units if they satisfy the following.

• eijekl= δj,keil .

• e∗_ij = eji.

• P

ieii= 1.

All ∗-homomorphisms between von Neumann algebras are implicitly as-sumed to be normal.

We denote by (M, τ ) the von Neumann algebra endowed with the faithful normal trace τ . We also call M a finite von Neumann algebra or a tracial von Neumann algebra.

Let (M, τM) be a tracial von Neumann algebra. Unless explicit mention,

all inclusions of von Neumann algebras (N, τN) ⊂ (M, τM) are assumed

to satisfy

• τ_{M |N} = τN,

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Let M0 and M1 be von Neumann algebras. The algebraic tensor product

of M0 and M1 is denoted by M0 M1. The tensor product of M0 and

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Part I

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Chapter 1

An introduction to

von Neumann algebras

1.1 Bounded operators on a Hilbert space

Let (H, h·, ·i) be Hilbert space. We denote by k · k the norm given by the inner product h·, ·i. The linear maps T : H → H are continuous whenever they are bounded and the (operator) norm of T is defined by

kT k := inf{M > 0 | kT (ξ)k ≤ M kξk, for all ξ ∈ H} .

Such maps are called bounded operators on H. The collection of all bounded operators on H, denoted by B(H), is a unital algebra. The algebra B(H) is complete for the operator norm which satisfies in partic-ular

kST k ≤ kSkkT k, for all S, T ∈ B(H) .

An algebra with such properties is called a Banach algebra . Furthermore, B(H) is endowed with an operation

∗ : B(H) → B(H) : T 7→ T∗ defined by

hT ξ, ηi = hξ, T∗ηi, for all ξ, η ∈ H .

The operator T∗ is called the adjoint of T . The ∗-operation is 2-periodic, anti-linear, anti-multiplicative and norm-preserving. An application with

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such properties is called an involution. An element x ∈ B(H) such that x = x∗ is called selfadjoint. An element x ∈ B(H) such that x∗x = 1 = xx∗is called a unitary. The unitary elements form a group denoted U (H). Note that an element x ∈ B(H) is an isometry (i.e. x preserves the inner product) if and only if x∗x = 1. We also have that

kT∗T k = kT k2, for every T ∈ B(H) .

A Banach algebra endowed with an involution satisfying the foregoing equality is called a C∗-algebra. In fact, any unital C∗-algebra is a norm closed unital ∗-subalgebra of B(H) for some Hilbert space H. The algebra of continuous functions on a compact space is a unital C∗-algebra and actually, every commutative C∗-algebra arises in this way. The algebra B(H) can be endowed with many other topologies that are weaker than the one inherited from the norm.

We define some weak topologies on B(H). The following list is not ex-haustive.

• The strong topology is the topology generated by the family of semi-norms (x 7→ kxξk)ξ∈H. So,

xi → x iff k(xi− x)ξk → 0, for all ξ ∈ H .

• The weak topology is the topology generated by the family of semi-norms (x 7→ |hxξ, ηi|)ξ,η∈H. So,

xi→ x iff |h(xi− x)ξ, ηi| → 0, for all ξ, η ∈ H .

• The ultraweak topology is the topology generated by the family of semi-norms x 7→ P n∈Nhxξn, ηni (ξn),(ηn)∈`2(N)⊗H. So, xi → x iff X n∈N h(xi− x)ξn, ηni → 0, for all (ξn), (ηn) ∈ `2(N) ⊗ H .

• The ultrastrong topology is the topology generated by the family of semi-norms x 7→P n∈Nkxξnk2 (ξn)∈`2(N)⊗H. So, xi → x iff X n∈N k(xi− x)ξnk2 → 0, for all (ξn) ∈ `2(N) ⊗ H .

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1.2 von Neumann algebras 19

Since |hxξ, ηi| ≤ kxξkkηk ≤ kxkkξkkηk, the weak topology is weaker than the strong one, both being weaker than the norm topology. The weak topology is weaker than the ultraweak, itself weaker than the ultra-strong. These topologies in general do not behave well with respect to the algebraic structure. For example, the ∗-operation and the multiplication map (S, T ) 7→ ST are not strongly continuous. Furthermore, the strong and ultrastrong topologies coincide on the unit ball of B(H). Similarly, the weak and ultraweak topologies coincide on the unit ball.

1.2 von Neumann algebras

From now on, we will focus on the study of unital ∗-subalgebras of B(H) that are closed under the weak operator topology defined above. These algebras are called von Neumann algebras. If a = a∗ in the von Neumann algebra M , the Borel functional calculus of a belongs to M as well. This very important fact allows to do lots of computations inside the von Neu-mann algebra. Note that B(H) itself is a von NeuNeu-mann algebra. It can be proven that every abelian von Neumann algebra is of the form L∞(X, µ), where (X, µ) denotes a standard probability space. This first definition of von Neumann algebras is purely analytic but von Neumann algebras can be defined in two other equivalent ways as we explain now.

Von Neumann algebras can be characterized in a purely algebraic manner, by von Neumann’s bicommutant theorem. For every non-empty subset M ⊂ B(H), one defines its commutant M0 by

M0 := {T ∈ B(H) | xT = T x, for all x ∈ M } .

The commutant of a von Neumann algebra M is still a von Neumann algebra.

Theorem 1.1. Let M ⊂ B(H) be a unital ∗-subalgebra, where 1M is the

identity operator on B(H). Then, the following are equivalent.

• M is a von Neumann algebra.

• M = (M0)0.

Von Neumann algebras can also be characterized without referring to any representation on a Hilbert space. Indeed, Sakai proved that the space

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M∗, called the predual of M , and consisting of ultraweakly continuous

linear functionals on M is the unique Banach space such that M = (M∗)∗.

Von Neumann algebras are the C∗-algebras that are the dual of a Banach space. By construction of M∗, the ultraweak topology on M coincides

with the weak∗-topology on M , viewed as the dual of M∗. This

space-free characterization of a von Neumann algebra proves that the ultraweak topology does not depend on the choice of the Hilbert space where the von Neumann algebra is represented.

In fact, the ultraweakly continuous linear functionals on a von Neumann algebra can be characterized by an algebraic property, called normality. Before giving this property, we make a little digression to talk about positive elements in a C∗-algebra. We know that a matrix a is positive if and only if a is selfadjoint with positive eigenvalues. There is an analogous notion of positivity in arbitrary C∗-algebras. An element a of a unital C∗-algebra is said to be positive if a is selfadjoint and

Sp(a) := {λ ∈ C | a − λ1 is non-invertible} ⊂ R+.

The subset Sp(a) is called the spectrum of a. The positive elements of a C∗-algebra A form a convex cone that we denote A+. An element a of a C∗-algebra is positive and and only if it can be written a = b∗b, for some b ∈ A. The set of self-adjoints elements in A is endowed with the partial ordering given by

a ≤ b if b − a ∈ A+.

The following properties are often useful. Let a, b ∈ A be self-adjoint elements.

• kak ≤ k ⇐⇒ −k1 ≤ a ≤ k1,

• a ≤ b ⇒ x∗ax ≤ x∗bx, for all x ∈ A.

If M is a von Neumann algebra, the positive cone M+ has the following extra property. If (xi) is an increasing and bounded net in M+ (in the

sense that sup_ikx_ik is bounded), then, sup x_i ∈ M+_{. We can now define}

the notion of normal linear functional on M .

Definition 1.2. A linear functional ω : M → C on the von Neumann algebra M is said to be

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1.3 Murray and von Neumann’s classification 21

• normal if it is positive and ω(sup xi) = sup ω(xi) for every increasing

and bounded net (xi) ∈ M .

Linear functional are very important for the analysis of von Neumann algebras and among them, the normal ones are of particular interest to us since they define the ultraweak topology. Indeed, it can be proven that a linear functional is normal if and only it is positive and ultraweakly continuous.

1.3 Murray and von Neumann’s classification

Murray and von Neumann classified von Neumann algebras in three types. They proved in [58] that classifying von Neumann algebras amounts to classifying those for which the center Z(M ) := M ∩M0 is trivial. The von Neumann algebras for which the center only consists of scalar multiples of the identity and which are thus the most non-commutative ones, are called factors and are classified in types I, II and III. Factors are the indecomposable von Neumann algebras, indeed, one can prove that M is a factor if and only if M is not the direct sum of two other von Neumann algebras. The classification of factors relies on a careful study of the geometry of the self-adjoint idempotents, called projections. Projections are crucial objects in von Neumann algebra theory, for example, the linear span of all projections in the von Neumann algebra M is norm-dense in M .

Before giving more details concerning the classification of factors, we review some important definitions and properties concerning projections. Projections are positive elements and we have p ≤ q if and only if Im p ⊂ Im q. So we have that p ≤ q if and only if pq = p. The projections p and q are called orthogonal if Im p and Im q are orthogonal vector spaces. Note that p is orthogonal to q if and only if pq = 0. Then, it is easily verified that the sum of orthogonal projections is still a projection. The set of projections in a von Neumann algebra form a lattice because for every family of projections (pi)i∈I ∈ M , the projections

^ i∈I pi, with range \ i∈I Im pi

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and

_

i∈I

pi, with range span

[

i∈I

Im pi

are respectively the largest projection in M smaller than all the pi and

the smallest projection in M majorizing all the pi.

Definition 1.3. Let M be a von Neumann algebra and x ∈ M .

• The smallest projection p ∈ M such that px = x is called the left support projection of x.

• The smallest projection p ∈ M such that xp = x is called the right support projection of x.

• The smallest central projection z such that x = zx is called the central support of x.

Let M ⊂ B(H) be a von Neumann algebra and x ∈ M . Note that the left (resp. right) support of x is the projection on the closure of xH (resp. x∗H).

An element v ∈ B(H) is called a partial isometry if v is an isometry on a closed subspace of H and identically zero on its orthogonal complement. Equivalently, v is a partial isometry if and only if v∗v is a projection, which is called the initial projection of v. In this case vv∗ is also a projection, called the final projection of v. Every bounded operator a on a Hilbert space admits a polar decomposition given by a := u|a|, where |a| := (a∗a)1/2 and u is a partial isometry with initial projection equal to the right support projection of a. An important fact is that the partial isometry coming from the polar decomposition of an element of a von Neumann algebra M still belongs to M .

Definition 1.4. Let M be a von Neumann algebra and p, q projections in M .

• p and q are said to be equivalent, and we write p ∼ q if there exists a partial isometry v ∈ M such that p = v∗v and q = vv∗.

• p is said to be sub-equivalent to q, and we write p . q if there exists p0 ≤ q such that p ∼ p0.

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1.3 Murray and von Neumann’s classification 23

It is easy to verify that ∼ is an equivalence relation. We have the Cantor-Bernstein type result saying that if p . q and q . p then, p ∼ q. Having equivalent projections in a von Neumann algebra is not an exceptional phenomenon. Indeed, the partial isometry arising in the polar decompo-sition the element a of the von Neumann M entails the equivalence of the initial and final support projections of a. In a factor, two arbitrary projections p, q can be compared: either p . q, either q . p.

A projection p in the von Neumann algebra M is called an atom (or a minimal projection ) if pM p = Cp, or if it satisfies, equivalently,

q ≤ p ⇒ q = p or q = 0, for every projection q ∈ M .

A von Neumann algebra is called diffuse if it contains no minimal projec-tions. In fact, a factor is atomic if and only if it is isomorphic to B(H), for some Hilbert space H, which may be finite dimensional. Those factors are called type I factors. If M is an infinite dimensional type I factor, we say that M is a type I∞ factor.

We say that a projection p in a von Neumann algebra M is finite if

p ∼ p0≤ p ⇒ p0= p .

A projection which is not finite is called infinite. So, an infinite projection is equivalent to a strictly smaller sub-projection. This definition should be compared with the definition of an infinite set, which is set in bijective correspondence with a strictly smaller subset. Note that every minimal projection is finite. A factor is said to be of type II if it has finite projec-tions but no minimal projecprojec-tions. A factor is of type III if every non-zero projection is infinite. Among type II factors, we have the following dis-tinction. A type II factor for which the identity is a finite projection is called a type II1 factor. A type II factor for which the identity is an

infinite projection is called a type II∞factor.

Before giving an equivalent formulation of Murray and von Neumann’s classification, we introduce some important terminology, constantly used in the sequel.

Definition 1.5. Let M be a von Neumann algebra. A positive linear functional τ : M → C is called a state if it satisfies τ (1) = 1. A state τ is called trace if it satisfies τ (xy) = τ (yx), for all x, y ∈ M .

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We say that the positive linear functional ϕ on the von Neumann algebra M is faithful if ϕ(x∗x) = 0 implies that x = 0, for every x ∈ M . On a factor, the faithfulness of normal states is automatic. We also define the notion of infinite traces. We already know that the linear functional

Tr : B(H)+→ [0, +∞] : T →X

i

hT e_i, eii ,

where (ei) is an orthonormal basis of H, is a trace. The trace Tr only

take finite values on trace-class operators. More generally, we have the following definition.

Definition 1.6. An ultraweakly lower semi-continuous positive-linear functional Tr : M+ → [0, +∞] on the von Neumann algebra M satis-fying

Tr(x∗x) = Tr(xx∗), for all x ∈ M ,

is called a normal trace. We say that the trace Tr is semi-finite if

{x ∈ M+| Tr(x) < +∞} is ultraweakly dense in M+.

Then, we can state the following theorem, due to Murray and von Neu-mann.

Theorem 1.7. Let M be a factor. Then

• M is of type I if and only if M ∼= B(H) for some Hilbert space H.

• M is of type II1 if and only if M is infinite dimensional and admits

a normal tracial state.

• M is of type II∞ if and only if M admits a semi-finite trace Tr,

such that Tr(1) = +∞ and M is not isomorphic to B(H) for some infinite dimensional Hilbert space H.

• M is of type III if every normal trace is zero.

It can be proven that the trace on a II1 factor is necessarily unique.

Murray and von Neumann started by constructing a dimension function on the set of projections in a factor M , which is a map D : M → [0, +∞] such that

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1.4 Gelfand-Naimark-Segal construction 25

1. p ∼ q ⇐⇒ D(p) = D(q),

2. p . q ⇐⇒ D(p) ≤ D(q),

3. D(p + q) = D(p) + D(q), for orthogonal projections p and q,

4. D takes finite values on finite projections.

The dimension function they constructed is unique up to multiplication by a scalar. Then, they proved in [29], that in the II1 case this dimension

function could be extended to a trace on M . The trace on a II1factor gives

a notion of continuous dimension in the sense that for every t ∈ [0, 1], there exists a projection p ∈ M such that τ (p) = t. From now on, we will focus on type II1 factors. We give some classical constructions and

discuss the problem of distinguishing between type II1 factors, which is

in general a very hard task.

1.4 Gelfand-Naimark-Segal construction

By definition, von Neumann algebras are represented on a a Hilbert space. It is sometimes crucial to be able to construct another representation of the same von Neumann algebra on another Hilbert space. See for example Chapter 8 where we construct amalgamated free product von Neumann algebras: the important point of the construction is to build an appropri-ate representation. We recall here the Gelfand-Naimark-Segal construc-tion, called GNS construcconstruc-tion, for a von Neumann (M, τ ) endowed with the faithful normal tracial state τ . This construction holds in a more gen-eral setting, see for example [52, Theorem 9.14]. We refer to the section 9 of the first chapter of [52] for the proofs of all the results in this section.

Definition-Proposition 1.8. Let (M, τ ) be a von Neumann algebra with a faithful normal finite trace τ . There exists a unique (up to unitary equivalence) representation λτ : M → B(Hτ) on the Hilbert space Hτ

with a vector ξτ satisfying

1. [λτ(M )ξτ] = Hτ and

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The Hilbert space Hτ is denoted L2(M, τ ) or simply L2(M ), when no

confusion is possible and is obtained as the completion of M with respect to the inner product given by

ha, bi := τ (a∗b), for all a, b ∈ M .

We denote by k · k2 the norm inherited from this inner product. It is

easy to see that the left multiplication on M extends to the required ∗-representation λτ.

Note that since τ is faithful, the representation λτ is also faithful so, we

can see M ⊂ B(L2(M )).

A very interesting fact is that M is a subset of the Hilbert space L2(M ). The map x 7→ x∗ extends to an anti-unitary operator JM : L2(M ) →

L2(M ) and we have the anti-representation ρτ : M → B(L2(M )) given by

ρτ(x) := JMx∗JM. It is easy to see that ρτ and λτ commute.

Further-more, we have the following important relation

λτ(M )0∩ B(L2(M )) = ρτ(M ) .

Example 1.9. Let Γ be an countable and ICC group. Since the vector δe ∈ `2(Γ) is cyclic for the left regular representation λ : Γ → B(`2(Γ)),

we have that L2(L(Γ)) = `2(Γ).

1.5 Group von Neumann algebras

Let Γ be a countable group. The left regular representation of λ : Γ → B(`2(Γ)) is defined by λg(δh) = δgh, where (δg) denotes the canonical

orthonormal basis of `2(Γ). The group von Neumann algebra L(Γ) of the group Γ is defined by

L(Γ) := {λg| g ∈ Γ}00.

This von Neumann algebra is endowed with the faithful normal trace given by τ (x) = hδe, xδei. The group von Neumann algebra L(Γ) is a

factor (and thus of type II1) if and only if the conjugacy classes {hgh−1 |

h ∈ Γ} of every element g 6= e are infinite. Groups with infinite conjugacy classes are called ICC groups. Such groups appear frequently, for example the free groups Fn, the infinite symmetric group S∞ := S_n≥1Sn and

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1.5 Group von Neumann algebras 27

the groups PSL(n, Z), with n ≥ 2, are all ICC. One can prove, using the Fourier transform, that for a countable abelian group Γ, the group von Neumann algebra L(Γ) is isomorphic with L∞(Char(Γ), Haar), where Char(Γ) denotes the group of characters of Γ. So, for example, we have that L(Z) ∼= L∞(S1_).

The GNS space associated to L(Γ) is given by `2(Γ) since the vector δe

is cyclic of L(Γ). Every element x ∈ L(Γ) can be written as the L2 -convergent sum x = P

g∈Γxgug where xg ∈ C and the ug are unitaries

satisfying ugh= uguh. Writing elements of L(Γ) in such way is very useful

for concrete computations.

Murray and von Neumann proved in [31] the existence of a unique (up to isomorphism) hyperfinite II1 factor, defined as the bicommutant of

an increasing union of matrix algebras. The hyperfinite finite II1 factor,

always denoted by R, can be realized as the group von Neumann algebra L(S∞). Murray and von Neumann were able to distinguish between the

two II1 factors L(S∞) and L(F2), using an invariant that they called

property (Γ). A II1 factor is said to have property (Γ) if it admits

non-trivial central sequences (un) ∈ U (M ) in the sense that k[x, un]k2 tends

to zero, for every x ∈ M . We say that a central sequence (un) ∈ U (M )

is trivial when kun− τ (un)1k2 tends to zero. Murray and von Neumann

proved that all central sequences of L(F2) are trivial while L(S∞) admits

non-trivial central sequences. Distinguishing between group factors is usually very hard and many problems about group von Neumann algebras remain open. For example, it is not known whether the group factors of F2 and F3 are isomorphic or not. Due to a celebrated theorem of

Connes, rigidity phenomena deducing the isomorphism of the groups Γ and Λ from the isomorphism of their group von Neumann algebras L(Γ) and L(Λ), should only be expected in the non-amenable setting. We explain this fact. A countable group is said to be amenable if the left regular representation admits a sequence of almost invariant unit vectors. For example, the group Z but also all abelian and all solvable groups are amenable. The amenability property can be translated in the von Neumann setting as follows. It can be proven that the group Γ is amenable if and only if the von Neumann algebra L(Γ) is injective:

Definition 1.10. A von Neumann algebra M ⊂ B(H) is injective if there exists a conditional expectation of B(H) onto M i.e a linear map EM : B(H) → M satisfying the following.

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• E_M is positive and EM(1) = 1

• E_M(aT b) = a EM(T )b, for every a, b ∈ M and T ∈ B(H).

The very deep result proven by Connes in [8] is that a II1factor is injective

if and only if it is hyperfinite. So, all the group von Neumann algebras arising from ICC amenable groups are isomorphic.

One way of contradicting amenability is to have Kazhdan property (T) (see section 2.3). All SL(n, Z), with n ≥ 3 are property (T) groups. Connes and Jones extended the notion of property (T) to the von Neu-mann level in [9] (see section 2.3). They also proved that a property (T) II1 factor cannot be embedded into a free group factor. Connes

conjec-tured in [6] that ICC property (T) groups are isomorphic if and only if their group von Neumann algebras are isomorphic and this conjecture is completely open. As a consequence of his strong rigidity theorem (see Theorem 1.13) Popa proves that wreath-products of Z with ICC w-rigid groups are distinguished by their group von Neumann algebras. We call wreath product of Z with Γ the group Z o Γ := (L

ΓZ) o Γ where Γ acts

on L

ΓZ via Bernoulli shift : g · (xh)h := (xg−1_h)_h. A group is called

w-rigid if it contains an infinite normal subgroup with relative property (T); see section 2.3. The group SL(2, Z) n Z2 is w-rigid.

1.6 Murray and von Neumann’s group measure

space construction

Let Γ be a countable group and M ⊂ B(H) a von Neumann algebra. We say that the group Γ acts on M , via σ, and we write Γ y M , if σ is a homomorphism Γ → Aut(M ). Given such action, we construct a von Neumann algebra, called crossed product , encoding at the same time the data of the group Γ and the action Γ y M . We represent M on H ⊗`2(Γ) via the representation π given by

π(a)(ξ ⊗ δg) := σg−1(a)ξ ⊗ δ_g, for all a ∈ M, ξ ∈ H, g ∈ Γ .

We represent L(Γ) on H ⊗`2(Γ) via id⊗λ, where λ denotes the left regular representation of Γ:

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1.6 Murray and von Neumann’s group measure space

construction 29

The crossed product of M by Γ is the von Neumann algebra

M o Γ := {π(M ), (id ⊗ λ)(Γ)}00 ⊂ B(H ⊗ `2(Γ)) .

We will denote by ug := 1 ⊗ λg the unitaries defining the action of Γ on

H ⊗ `2(Γ) and we identify a ∈ M with π(a). Then, one can verify the following commutation relation

ugau∗g = σg(a), for all a ∈ M, g ∈ Γ . (1.1)

Note that in the case where M = C and σ is the trivial action, the crossed product M oΓ is just the group von Neumann algebra L(Γ). Furthermore, if Γ is a countable group acting by automorphisms on the countable group Λ, then one can prove that L(Λ o Γ) ∼= L(Λ) o Γ. For example, we have that L(Z2o SL(2, Z))∼= L(Z2) o SL(2, Z) ∼= L∞(S2) o SL(2, Z).

We are interested in the situations where M o Γ is a factor. We introduce the following terminology concerning actions of groups on von Neumann algebras.

Definition 1.11. Let σ : Γ y M be an action of the countable group Γ on the von Neumann algebra M .

• The action σ is said to be properly outer, if ax = σg(x)a, for all x ∈

M ⇒ a = 0.

• The action σ is said to be ergodic, if {x ∈ M | σg(x) = x} = C1.

It can be proven that Γ acts on M properly outerly if and only if M o Γ ∩ M0 = Z(M ). If the action is also ergodic, then M o Γ is a factor. In section 2.4 we study outer actions of countable groups on II1 factors, in

that case, the resulting crossed product is factorial.

Suppose that M is a tracial von Neumann algebra. The GNS space associated to M oΓ is given by L2(M )⊗`2(Γ). So, in particular, L2(M oΓ) is the orthogonal direct sum of copies of L2(M ) indexed by Γ given by L2(M ug). Every element x ∈ M o Γ can be written as the L2-convergent

sum x =P

g∈Γxgug, where xg ∈ M .

Crossed products can also be built out of the action of a countable group on a standard Borel measured space. Let Γ act on a standard Borel measured space (X, B, µ). We say that the action Γ y (X, µ) is

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• essentially free if {x | g · x = x} is µ-negligible, for all g 6= e,

• ergodic if it is indecomposable in the sense that any Γ invariant subspace of X is either of measure 0 or 1,

• non-singular if µ(B) = 0 ⇒ µ(g · B) = 0, for all g ∈ Γ.

In this setting all statements only hold up to measure zero. Every count-able group Γ admits an essentially free, ergodic and non-singular action on the standard probability space given by the Bernoulli shift action Γ yQ

Γ(X, µ) where g · (xh)h= (xg−1_h)_h.

Let Γ y (X, µ). Then, Γ acts on L∞(X, µ) via (g · F )(x) := F (g−1· x). So, we can build the crossed product L∞(X, µ) o Γ also known as Mur-ray and von Neumann’s group-measure-space construction, which already appeared in [30]. One can check that Γ y (X, µ) essentially freely (resp. ergodically) if and only if Γ y L∞(X, µ) properly outerly (resp. ergod-ically). Then, every essentially free and ergodic action of the countable group Γ on (X, µ) yields a factor L∞(X, µ) o Γ. The following actions are all essentially free, ergodic and will give examples of factors of all types, by Theorem 1.12 due to Murray and von Neumann.

1. Let θ 6∈ Q and let Z y R

Z via k · z := e 2iπkθ_z.

2. Let θ 6∈ Q and let Z + θZ y R by translation. 3. Let Q∗n Q y R via (p, q) · x := px + q.

Theorem 1.12. Let Γ y (X, µ) be an essentially free, ergodic, non-singular action. Denote M := L∞(X, µ) o Γ. Then,

• M is a type I factor if and only if Γ acts transitively.

• M is a type II₁ factor if and only if there exists a non-atomic Γ-invariant finite measure in the class of µ.

• M is a type II∞ factor if and only if there exists a non-atomic

Γ-invariant infinite measure in the class of µ.

• M is a type III factor if and only if there exists no non-trivial Γ-invariant measure in the class of µ.

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1.7 von Neumann algebras and ergodic theory 31

1.7 von Neumann algebras and ergodic theory

In this section (X, µ) and (Y, ν) denote standard non-atomic probability spaces and we only consider essentially free, ergodic and probability mea-sure preserving actions of countable groups. The II1 factor L∞(X) o Γ is

a von Neumann algebra containing a copy of L∞(X) and a copy of L(Γ) satisfying the covariance relation (1.1). So, L∞(X) o Γ captures infor-mation concerning the group Γ and the action Γ y (X, µ). Two natural questions arise:

• What kind of isomorphism of actions gives rise to isomorphic crossed product II1 factors?

• Conversely, what does an isomorphism L∞_{(X, µ)oΓ ∼}= L∞(Y, ν)oΛ tell us concerning the groups and the actions?

It is proven in [49, 17] that a probability space isomorphism ∆ : (X, µ) → (Y, ν) extends to an isomorphism of the crossed products L∞(X, µ) o Γ ∼= L∞(Y, ν) o Λ if and only if ∆(Γ · x) = Λ · ∆(x), for µ-almost every x ∈ X. In that case the the actions Γ y (X, µ) and Λ y (Y, ν) are called orbit equivalent. Note that the groups Γ and Λ may not be isomorphic for orbit-equivalent actions Γ y (X, µ) and Λ y (Y, ν). There is a stronger way of comparing actions which goes as follows.

We say that actions Γ y (X, µ) and Λ y (Y, ν) are conjugate if there exists an isomorphism of probability spaces ∆ : (X, µ) → (Y, ν) and an isomorphism δ : Γ → Λ such that ∆(g · x) = δ(g) · ∆(x), for all g ∈ Γ and µ-almost every x ∈ X. Then, the map

X g∈Γ xgug7→ X g∈Γ ∆∗(xδ(g))uδ(g)

extends to an isomorphism L∞(X, µ)oΓ ∼= L∞(Y, ν)oΛ sending L∞(X, µ) to L∞(Y, ν). In particular, conjugate actions are orbit equivalent. Conjugacy implies orbit equivalence which in turn implies von Neumann equivalence (isomorphism of the crossed products). All converse results are called rigidity results. To deduce orbit equivalence from von Neu-mann equivalence one needs to have an isomorphism of crossed products sending L∞(X) to L∞(Y ). This phenomenon is usually unexpected: for

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example Connes and Jones in [10] gave examples of von Neumann equiv-alent actions that are not orbit equivequiv-alent. Moreover, a lot of information can be lost in the passage to the orbit equivalence; indeed Dye [13] proved that all free ergodic actions of infinite abelian groups are orbit equivalent. The celebrated theorem of Ornstein and Weiss [32] proves that in fact all free ergodic actions of infinite amenable groups are orbit equivalent. We end this section with the following breakthrough rigidity result ob-tained by Popa in 2004 (see [34, 35]). With this theorem, Popa deduces for the first time and for a large class of groups, the conjugacy of actions out of their von Neumann equivalence. To obtain this very powerful result Popa developed a theory of deformation/rigidity which was the starting point for many other rigidity results concerning group actions and von Neumann algebras.

Theorem 1.13 (Popa, Theorem 7.1 in [35]). Let Γ be an ICC group acting on (X, µ) via Bernoulli shift. Let Λ y (Y, ν) be an essentially free, ergodic measure preserving action of the w-rigid group Λ. If

L∞(X, µ) o Γ ∼= L∞(Y, ν) o Λ ,

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Chapter 2

Some topics in von

Neumann algebra theory

2.1 Normalizers and Quasi-normalizers

Let (M, τ ) be a tracial von Neumann algebra and N ⊂ M a von Neumann subalgebra.

• The normalizer of N inside M is defined as:

NormM(N ) := {u ∈ U (M ) | uN u∗ = N } .

• The inclusion N ⊂ M is called regular if NormM(N )00 = M .

• The quasi-normalizer of N inside M is defined by

QN_M(N ) = n

a ∈ M

∃a1, . . . , an, b1, . . . , bm∈ M such that N a ⊂ n X i=1 aiN and aN ⊂ m X i=1 N bi o . (2.1)

• The inclusion N ⊂ M is called quasi-regular if QN_M(N )00= M .

Remark that the quasi-normalizer of N ⊂ M is a unital ∗-subalgebra of M containing N and we have

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Let Γ be a group and Λ ⊂ Γ a subgroup.

• The commensurator of Λ ⊂ Γ is defined as

CommΓ(Λ) := {g ∈ Γ | gΛg−1∩ Λ has finite index

in gΛg−1 and in Λ} .

• The inclusion Λ ⊂ Γ is called almost normal if CommΓ(Λ) = Γ.

A typical example of an almost normal subgroup is SL(n, Z) ⊂ SL(n, Q). Remark that the inclusion L(Λ) ⊂ L(Γ) is

• regular if Λ is a normal subgroup of Γ,

• quasi-regular if the inclusion Λ ⊂ Γ is almost normal.

We conclude this section on quasi-normalizers with the following useful lemma; see [55, Lemma 6.5].

Lemma 2.1. Let Q ⊂ M be an inclusion of finite von Neumann algebras and p ∈ Q, a non-zero projection. Then

QN_{pM p}(pQp)00= p QN_M(Q)00p .

2.2 Group von Neumann algebras twisted by a

2-cocycle

In this section, we recall definitions and basics facts concerning twisted group von Neumann algebras. We start with the definition of the 2-cohomology of a countable group with values in S1, the complex numbers of modulus 1. In this section, the group Γ always denotes a countable group.

Definition 2.2. The set Z2_{(Γ, S}1_{) of S}1_{-valued 2-cocycles on the group}

Γ is defined by

Z2(Γ, S1) := {Ω : Γ × Γ → S1 | Ω(g, h)Ω(gh, k) = Ω(g, hk)Ω(h, k) for all g, h, k ∈ Γ} .

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2.2 Group von Neumann algebras twisted by a 2-cocycle 35

The set ∂2(Γ, S1) of S1-valued coboundaries on the group Γ is defined by

∂2(Γ, S1) := {Ω ∈ Z2(Γ, S1) | ∃a : Γ → S1, such that for all g, h ∈ Γ a(g)a(h) = Ω(g, h)a(gh)} .

For every function a : G → S1, the map defined by (∂a)(g, h) := a(gh)a(g)a(h)

is a 2-cocycle with vales in S1, called the boundary of a. Note that a is a character if and only if ∂a = 1.

The set Z2(Γ, S1) is an abelian group and the set ∂2(Γ, S1) is a normal subgroup. The 2-cohomology of Γ with values in S1 the group defined by

H2(Γ, S1) := Z

2_{(Γ, S}1₎

∂2_{(Γ, S}1₎ .

All examples of 2-cocycles are built from projective representations as we explain now.

• Let π : Γ → U (H) be a projective representation. Then, there exists a 2-cocycle Ωπ : Γ × Γ → S1 such that, for all g, h ∈ Γ, π(g)π(h) =

Ωπ(g, h)π(gh). We sometimes refer to Ωπ as the obstruction cocycle

of π.

• Let π : Γ → U (H) be a unitary representation and a : Γ → S1_.

Then, the formula ρ(g) := a(g)π(g) defines a projective representa-tion of Γ. The obstrucrepresenta-tion cocycle Ωρ is a coboundary for the map

a.

Let Γ be a countable group acting by automorphisms on the countable group Λ. The following lemma gives a technique to extend Γ-invariant 2-cocycles on Λ into 2-cocycles on Γ n Λ.

Lemma 2.3. Let Γ, Λ be countable groups. Let σ : Γ y Λ, by automor-phisms. Let Ω ∈ Z2(Λ, S1) such that

Ω σs(g), σs(h) = Ω(g, h), for all g, h ∈ Λ and s ∈ Γ .

Then, ˜

Ω (s, g), (t, h) := Ω g, σs(h), for all g, h ∈ Λ and s ∈ Γ

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We omit the proof which is a straightforward computation.

Let Ω ∈ Z2(Γ, S1). The left Ω-regular representation λΩ : Γ → U (`2(Γ))

defined by

λΩ(g)δh = Ω(g, h)δgh

is a projective representation of Γ with cocycle Ω. The twisted group von Neumann algebra LΩ(Γ) is defined as the von Neumann algebra generated

by the λΩ(g) . This von Neumann algebra is not always a factor but is

endowed with the normal tracial state

τ (x) := hδe, xδei, x ∈ LΩ(Γ) .

As for the group von Neumann algebra, we easily prove that LΩ(Γ) is

a factor when Γ is ICC. The converse does not hold anymore but the following proposition gives an easy way of building factorial LΩ(Γ).

Notation 2.4. We denote by Char(Γ) the group of characters of the abelian countable group Γ. If G is a compact group, we still denote by Char(G) for the subset of Irr(G) consisting of one-dimensional unitary representations of G. If G is a compact abelian group, then Char(G) is the set of continuous group homomorphisms ω : G → S1. We recall that for an abelian countable group Γ, the group Char(Γ) is compact abelian and we have the Pontryagin duality Char(Char(Γ)) ∼= Γ.

Proposition 2.5. Let Γ be an abelian countable group. Suppose that π : Γ → Char(Γ) is a homomorphism. Then:

ˆ

π : Γ → Char(Γ) : ˆπ(h)(g) = π(g)(h)

is a homomorphism and the following hold.

• Ωπ(g, h) := π(g)(h) defines a 2-cocycle,

• Ω_π 6∈ ∂2_{(Γ, S}1_{) if there exists g ∈ G such that π(g) 6= ˆ}_{π(g) ,}

• L_Ω_π(Γ) is the hyperfinite II1 factor if π(g) 6= ˆπ(g) for all g 6= e.

Proof. One can easily check that Ωπ defines a 2-cocycle. We prove the

second assertion. Suppose that a(g)a(h) = Ωπ(g, h)a(gh) for some map

a : G → S1. Abelianness of Γ implies that Ωπ(g, h) = Ωπ(h, g) and thus

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2.2 Group von Neumann algebras twisted by a 2-cocycle 37

Take x in the center LΩπ(Γ) and write the L

2_{-convergent sum x :=}

P

gagλΩπ(g). An immediate computation yields that

agΩ(h, g) = Ω(hgh−1, h)ahgh−1for all g, h ∈ Γ.

Since Γ is abelian, we have that agΩπ(g, h) = agΩπ(h, g), for every g, h ∈

Γ. Whenever g 6= e, π(g) 6= ˆπ(g) yields the existence of an element h ∈ Γ such that Ωπ(g, h) 6= Ωπ(h, g). Then, ag = 0, for all g 6= e.

If there exists a non-trivial g ∈ Γ such that π(g) = ˆπ(g), we have that Ωπ(g, h) = Ωπ(h, g), for all h ∈ Γ. This equality and the fact that Γ is

abelian easily imply that λΩπ(g) is a non-trivial element in the center of

LΩπ(Γ).

We need to get examples of injective morphisms π : Γ → Char(Γ) to pro-duce factorial twisted group von Neumann algebras. We first intropro-duce some terminology.

Definition 2.6. Let Γ be an abelian countable group. A 2-cocycle Ω ∈ Z2(Γ, S1) is called a bi-character when Ω(·, g) and Ω(g, ·) are elements of the group of characters Char(Γ) of Γ, for all g ∈ Γ.

Let Ω be a bi-character on an abelian group Γ. Then, by definition, the map

π : Γ → Char(Γ) : (h 7→ Ω(g, h)) is a homomorphism.

Definition 2.7. Let Ω be a bi-character on an abelian group Γ. The cocycle Ω is called non-degenerate when the associated homomorphism π : Γ → Char(Γ) is injective with dense range.

Non-degenerate bi-characters are very useful to produce concrete real-izations of the hyperfinite II1 factor as a twisted group von Neumann

algebra.

We end this section with a standard example. Let z ∈ S1 _{. We define the}

following bi-character on Z2 : Ωz (x, y), (x0, y0) := zxy 0_−yx0 . Then, if z 6∈ exp(2iπQ), LΩz(Z 2_{) is a II} 1 factor.

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2.3 Property (T) and relative property (T) for

II

1

factors

The notion of property (T) and relative property (T) for II1 factors uses

the concept of bimodule over II1 factors, also called Connes’

correspon-dences. See [5] and also Popa’s notes [40]. Bimodules are intensively studied in this thesis starting from Part II but we give the definition now.

2.3.1 Bimodules over von Neumann algebras

Definition 2.8. Let N, M ⊂ B(H) be von Neumann algebras. Let π : M → B(H) be a normal representation and π0 : N → B(H) a normal continuous anti-representation .

• The Hilbert space H is a left M -module, for the left action given by

a · ξ := π(a)ξ .

• The Hilbert space H is a right N -module, for the right action given by

ξ · a := π0(a)ξ .

• When π and π0 commute, the Hilbert space H is called an M N -bimodule.

The fact that the representation π and the anti-representation π0commute states the associativity of the left and right actions:

a · (ξ · b) = (a · ξ) · b .

We recall the notion of opposite algebra Mopassociated to a von Neumann algebra M . As vector spaces M and Mop are the same but, if we denote by xo the element x ∈ M viewed in Mop, the von Neumann algebra Mop is equipped with the product given by xo _{· y}o _{:= (yx)}o_{. So, an M N}

-bimodule is also the data of commuting normal representations of M and Nop.

Let (M, τ ) be a tracial von Neumann algebra. The GNS construction L2(M, τ ) provides the easiest example of an M -M -bimodule, with left and right actions given by the representation λτ and the anti-representation

Outer automorphism groups and bimodule categories of type II_1 factors

HAL Id: tel-00685967

https://tel.archives-ouvertes.fr/tel-00685967

type II_1 factors

Sébastien Falguières

To cite this version:

Outer automorphism groups and

bimodule categories of type II

1

factors

Dankwoord

Contents

Introduction

Description of the chapters and statement of the

main results

Some notations and

conventions

Part I

Chapter 1

An introduction to

von Neumann algebras

1.1

Bounded operators on a Hilbert space

1.2

von Neumann algebras

1.3

Murray and von Neumann’s classification

1.4

Gelfand-Naimark-Segal construction

1.5

Group von Neumann algebras

1.6

Murray and von Neumann’s group measure

space construction

1.7

von Neumann algebras and ergodic theory

Chapter 2

Some topics in von

Neumann algebra theory

2.1

Normalizers and Quasi-normalizers

2.2

Group von Neumann algebras twisted by a

2-cocycle

2.3

Property (T) and relative property (T) for

II

factors

₁