Seminar operator algebras 2008-2009

Seminars

Abstracts

Countable equivalence relations and von Neumann algebras Part I

by Sébastien Falguières

In a series of talks I will give an easy introduction to Borel equivalence relations and group actions on probability spaces. In this first lecture we will focus on group actions on standard probability spaces. After defining the notion of standard Borel space and standard measure space, we will prove essential freeness and ergodicity of the irrational rotation and Bernoulli actions. In the second part of the talk, we will discuss about Orbit Equivalence and comparing group actions. The action of a countable group Γ on a standard Borel probability space yields an equivalence relation R_Γ, the equivalence classes being the orbits of the Γ-action. The equivalence relation R_Γ will be our model for studying, in the next lecture, countable standard Borel equivalence relations.

Countable equivalence relations and von Neumann algebras Part II

by Sébastien Falguières

In this second lecture, we define abstract countable standard Borel equivalence relations and focuss in particular on type II₁ equivalence relations. We will explain Feldman and Moore's construction of the von Neumann algebra L(R) associated to such equivalence relation R, generalizing Murray and von Neumann's group- measure-space construction. The von Neumann algebra L(R) of the equivalence relation R ⊂ X × X contains L^∞(X) as a Cartan subalgebra. Feldman and Moore proved that this construction describes (up to twisting L(R) by a 2-cocycle) every inclusion of a Cartan subalgebra in a II₁ factor. Thus there exists a bijective correspondence between pairs (Cartan subalgebra ⊂ II₁ factor) and (X,R) where R is a type II₁ equivalence relation on X. Invariants of R yields invariants of pairs (Cartan ⊂ II₁ factor).

Countable equivalence relations and von Neumann algebras Part III

by Sébastien Falguières

For this last lecture, we study an invariant of (ergodic measure-preserving) equivalence relations R called the cost of R. This invariant was introduced by D.Gaboriau in 2000. We will define the cost C(Γ) of a group Γ and prove that C(F_p) = p, C(SL(2,Z)) = 1 + 1/12. Finally, we will prove the following result, due to Gaboriau: Bernoulli shifts actions of free groups with different numbers of generators are not Orbit Equivalent.

A classification of all subfactors of some crossed product factors

by Steven Deprez

For a class of crossed product factors, we are able to describe all finite index bimodules between any two elements of this class. Based on this result, we will present a classification of all finite index subfactors of any factor in this class, up to unitary conjugation. For some explicit examples M, this classification allows us to calculate ℭ(M). Here ℭ(M) is the set of all indices of irreducible subfactors of M. I will give a proof for a special case of these results.

Rigidity for II₁ factors: fundamental groups, bimodules and subfactors

by Stefaan Vaes

I will give an overview of several recent rigidity results for II₁ factors, with a particular emphasis on computing invariants, like the fundamental group, the outer automorphism group and the bimodule category of concrete families of II₁ factors.

Planar Algebras and Subfactors

by Ved Gupta

We shall discuss the formalism of Jones' Planar Algebras in detail and see some examples of them (one of which will be appear in the next talk). Further, we shall state a theorem of Jones that relates a certain class of Planar Algebras with a certain class of Subfactors.

Planar Algebra of the Subgroup-Subfactor

by Ved Gupta

In this talk, I will mainly present an overview of my thesis, with an outline of the proof of the main theorem, which identifies the planar algebra of a subgroup-subfactor with a planar subalgebra of the planar algebra of the bipartite graph with one even and n odd vertices, where n is the index for the pair H ⊂ G of finite groups appearing in the subgroup-subfactor. We conclude with some questions that evolve from this and still remain unanswered.

Isometric coactions of compact quantum groups on compact quantum metric spaces I

by Marie Sabbe

In this talk I will try to quantize the definition of an isometric action of a group on a metric space. Therefore I need to introduce the concepts of both compact quantum groups (CQG) and compact quantum metric spaces (CQMS). Then we can propose a definition of an 'isometric' coaction of a CQG on a CQMS. We will show why this definition coincides with the well known definition in the classical case. I will also explain Banica's definition of isometric coactions of CQGs on small metric spaces. Proving why the definition we propose also generalizes Banica's definition, will be something for the next talk.

Isometric coactions of compact quantum groups on compact quantum metric spaces II

by Marie Sabbe

In this talk I will try to quantize the definition of an isometric action of a group on a metric space. Therefore I need to introduce the concepts of both compact quantum groups (CQG) and compact quantum metric spaces (CQMS). Then we can propose a definition of an 'isometric' coaction of a CQG on a CQMS. We will show why this definition coincides with the well known definition in the classical case. I will also explain Banica's definition of isometric coactions of CQGs on small metric spaces. Proving why the definition we propose also generalizes Banica's definition, will be something for the next talk.

From Subfactor Planar Algebras to Subfactors I

by Ved Gupta

Jones' theorem associates a subfactor planar algebra to a finite index extremal subfactor. The converse is also true, i.e., every subfactor planar algebra comes via Jones' theorem. Recently, three proofs of the converse came up, involving somewhat different techniques. We shall discuss the proof given by Kodiyalam and Sunder, which primarily involves planar algebra techniques.

From Subfactor Planar Algebras to Subfactors II

by Ved Gupta

Jones' theorem associates a subfactor planar algebra to a finite index extremal subfactor. The converse is also true, i.e., every subfactor planar algebra comes via Jones' theorem. Recently, three proofs of the converse came up, involving somewhat different techniques. We shall discuss the proof given by Kodiyalam and Sunder, which primarily involves planar algebra techniques.

Subfactors, quantum groupoids and tensor categories

by Leonid Vainerman (Université de Caen)

I will recall the relations between three topics mentioned in the title : a characterization of finite index and finite depth subfactors via quantum groupoid actions, reconstruction theorems for quantum groupoids. Then I will explain how recent progress in the study of fusion categories can be used in the study of quantum groupoids and subfactors. Some concrete examples will be treated.

Finite index normal sub-equivalence relations

by Aurélien Alvarez (EPFL, Lausanne)

Given a group G and a finite index subgroup H, it is well known that one can find a finite index normal subgroup of G included in H. I will prove an analogous result in the context of Borel/measured equivalence relations.

Equivalence relations generated by profinite actions

by Todor Tsankov (Institut de Mathématiques de Jussieu, Paris)

Profinite actions of countable groups arise naturally as actions on the space of ends of a finitely splitting rooted tree on which the group acts by automorphisms. There is a natural probability measure preserved by the action and hence, the orbit equivalence relation is of type II_1. I will discuss a recent inherent characterization of II_1 equivalence relations that arise in this way, as well as some older results (joint with I. Epstein) that show that a large class of equivalence relations are not of this type.

Property (T) and affine isometric actions

by An Speelman

We introduce the notion of an affine isometric action of a countable group on an affine Hilbert space and prove the Delorme-Guichardet theorem: a countable group has property (T) if and only if every affine isometric action has a fixed point.

Property (T) and finite presentability (after Shalom and Lafforgue)

by Stefaan Vaes

We present Vincent Lafforgue's proof of Shalom's theorem saying that every countable property (T) group is the quotient of a property (T) group of finite presentation. Recall that a group is said to be of finite presentation if it can be written as a free group with finitely many generators subject to finitely many relations.

Orbit equivalence rigidity up to countable classes (after Popa)

by Jan Keersmaekers

Suppose that a property (T) group acts measure preservingly, ergodically and essentially freely on a probability space. We prove that there are at most countably many group actions that are mutually non-conjugate and orbit equivalent to the originally given action.

On the corepresentation theory of complexified free quantum groups

by Sven Raum (WWU Münster)

In my talk I will discuss a method to determine the corepresentation theory of a certain class of quantum groups, so called free quantum groups. I will first give a short introduction to the terminology of compact matrix quantum groups and their corepresentations. Then I will introduce free fusion rings and free quantum groups and will explain subsequently a method introduced by T. Banica and R. Vergnioux which can be applied to determine the corepresentation theory of free quantum groups with free fusion rings. Finally I will apply this method to a new example and will explain how to deal with new problems arising during the investigation of this example. If there is time left, I will briefly introduce another example of a free quantum group forcing us to refine our methods in another aspect.

Interpolated free group factors

by An Speelman

The interpolated free group factors were discovered by Dykema and Radulescu, as a continuation of work of Voiculescu. In this talk we introduce this family of type II₁ factors and explain its importance with respect to the isomorphism problem of free group factors. We give a brief introduction to the notion of freeness and show how free semicircular elements can be used to model free group factors. In the final part of the talk the interpolated free group factors will be constructed. Instead of imitating the ideas Dykema or Radulescu, we use Shlyakhtenko's approach for the construction.

Fundamental groups of equivalence relations and II₁ factors

by Jan Keersmaekers

In this talk, we give an introduction to the recent results of Sorin Popa and Stefaan Vaes concerning fundamental groups of type II₁ factors and countable equivalence relations. We will recall the notion of property (T) for both groups and von Neumann algebras. We will give an idea of the proof of one of their main results and prove that many uncountable subgroups of R^*₊ arise as fundamental groups of II₁ factors.

Subfactors, groups and a new planar algebra

by Dietmar Bisch (Vanderbilt University)

Symmetries of von Neumann algebras are naturally captured by certain bimodules. They form fusion categories whose intertwiner spaces carry an action of Jones' planar operad. This gives rise to surprising algebraic-combinatorial structures, including a new notion of free product (joint work with Vaughan Jones). We will explain these concepts and present some examples.

Twisting of compact quantum groups

by Kenny De Commer

We present an explicit example of a compact quantum group and a 2-cocycle on it, such that twisting the quantum group with this 2-cocycle produces a non-compact, but still locally compact quantum group. We give some motivation why this result is interesting (it shows that the monoidal category of representations of a quantum group does not necessarily remember the topology, and also that compact quantum groups can have irreducible projective representations on infinite dimensional Hilbert spaces), and why it is surprising (it is a purely quantum phenomenon, in that it can not occur for compact quantum groups of Kac type, and it is a purely analytical phenomenon, in that it can not occur for 'smooth' 2-cocycles).

Strongly solid II₁ factors with an exotic MASA

by Cyril Houdayer

Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid II₁ factor M containing an "exotic" maximal abelian subalgebra A: as an A-A-bimodule, L²(M) is neither coarse nor discrete. Thus we show that there exist II₁ factors with such property but without Cartan subalgebras. It also follows from Voiculescu's free entropy results that M is not an interpolated free group factor, yet it is strongly solid and has both the Haagerup property and the complete metric approximation property. This is joint work with Dimitri Shlyakhtenko.