Seminar operator algebras 2009-2010

Seminars

Abstracts

Proper isometric group actions on Hilbert spaces

by Alain Valette

By a classical result of Bieberbach, a finitely generated group acting properly isometrically on a finite-dimensional Euclidean space, is crystallographic. The situation changes dramatically upon replacement of finite-dimensional Euclidean space by Hilbert space: the class of groups admitting a proper isometric action on a Hilbert space is huge, containing free groups and amenable groups. We will explain why this class of groups is interesting (links with ergodic theory and operator algebras) and how geometry can be used to construct examples of groups in this class.

Group measure space decomposition of II₁ factors and W*-superrigidity

by Stefaan Vaes

I give a series of introductory lectures on my joint paper with Sorin Popa available here. The first one or two lectures will be devoted to background material on II_1 factors, Jones' basic construction and Popa's intertwining by bimodules.

The Effros-Ruan conjecture for bilinear forms on C*-algebras

by Ranjana Jain, Pierre Fima

We will explain the proof by Haagerup and Musat of the Effros-Ruan conjecture for bilinear forms on C*-algebras.

An inner amenable group whose von Neumann algebra does not have property Gamma

by Stefaan Vaes

A II₁ factor M has property Gamma of Murray and von Neumann if M admits a non-trivial central sequence. Effros proved in 1975 that property Gamma of the group von Neumann algebra L(G) of an ICC group G, implies the inner amenability of G (the existence of a mean on G that is invariant under the inner automorphisms Ad g and that is different from the Dirac mass in e). Effros asked whether the converse holds and we provide a concrete counterexample.

Affine representations over a planar algebra

by Ved Gupta and Shamindra Gosh

Lecture 1: We will recall the defintion of planar algebra and describe Jones's prescription of the planar algebra associated to the standard invariant of every finite index extremal subfactor. We will then define the affine category over a planar algebra and their representations.

Lecture 2: Given a finite index extremal subfactor, we will show that the space of affine morphism at the 'zero level' is given by the fusion algebra of the bimodule 2-category appearing in the standard invariant of the subfactor. This is a joint work with Ved Prakash Gupta and Paramita Das.

The quantum automorphism group and the quantum isometry group of finite dimensional compact quantum metric spaces

by Marie Sabbe

In this seminar, we will look for a notion of the quantum isometry group of compact quantum metric spaces. We will prove that this quantum isometry group exists if there is a quantum automorphism group. Also, I will tell something about the existence of the quantum automorphism group of a finite dimensional C*-algebra. This was shown by S. Wang.

Quantum Isometry Groups for Spectral Triplets

by Johan Quaegebeur

We present the approach of Bhowmick and Goswami to study "quantum isometry groups". They describe the metric on a quantum space by means of a spectral triplet. They show how a quantum isometry group can be associated to a spectral triplet. An explicit example (the Chakraborty-Pal spectral triplet build on the Podles sphere) shows that the quantum isometry group (in the sense of Bhowmick and Goswami) does not always act continuously (i.e. the action is not always a C*-action).

Descriptive set theory and von Neumann algebras

by Roman Sasyk

We use the notion of Borel reducibility from descriptive set theory and Popa's deformation-rigidity techniques to show that the set of factors of types II_1, II_\infty and III_\lambda (0 \leq \lambda \leq 1) are not classifiable by countable structures. This is joint work with A. Tornquist.

Turbulence and Araki-Woods factors

by Roman Sasyk

Using the notion of turbulence developed by Hjorth, we show that the isomorphism relation of Araki-Woods factors is not classifiable by countable structures. This is joint work with A. Tornquist

Constructing subfactors from Q-systems

by Pinhas Grossman

The notion of a Q-system, due to Longo, is a characterization of the canonical endomorphism (or basic construction) of a subfactor as a certain type of Frobenius algebra in a C^* tensor category. There is a method, due to Izumi, of constructing a subfactor from an automorphism in the principal graph of a known subfactor using Q-systems. We will discuss work in progress with Asaeda on applying this method to the Asaeda-Haagerup subfactor with index (5 + sqrt(17))/2. This would give an example of a quadrilateral of factors whose planar algebra is generated by two cocommuting biprojections and has small first relative commutant.

Sofic groups, equivalence relations and Connes' embedding problem

by Liviu Paunescu

We shall introduce the notion of sofic group and sofic equivalence relation, starting from Connes' embedding problem. Also we shall present Ozawa's proof to a result of Elek and Lippner that Bernoulli shifts of sofic groups are sofic.

Modular properties of matrix coefficients of corepresentations of a locally compact quantum group

by Martijn Caspers

Locally compact (l.c.) quantum groups have been introduced in a von Neumann algebraic setting by Stefaan Vaes and Johan Kustermans in 2000. One of their main motivations was a generalization of the Pontrjagin duality theorem for abelian l.c. groups. So to every l.c. quantum group one can associate a dual l.c. quantum group, such that the double dual is the l.c. quantum group itself. The definition of a l.c. quantum group includes a von Neumann algebra together with two normal, semi-finite, faithful weights. In this talk we will see how the modular automorphism group of the weight can be expressed in terms of matrix coefficients of corepresentations. Furthermore, there exist (Schur) orthogonality relations between matrix coefficients. As a consequence there are relations between the weights of a quantum group and the dual weights. Also it gives a tool to determine the quantum group version of the Plancherel measure as was proved by Pieter Desmedt in 2003.

Rigidity for von Neumann algebras and their invariants

by Stefaan Vaes

We give a survey of recent classification results for von Neumann algebras arising from measure preserving group actions on probability spaces. This includes II₁ factors with uncountable fundamental groups and the construction of W*-superrigid actions where the von Neumann algebra entirely remembers the initial group action.