Seminar operator algebras 2010-2011

Seminars

Abstracts

A type II_1 factor with many non-unitarily conjugate Cartan subalgebras

by An Speelman

In their paper [OP08] Ozawa and Popa gave many examples of group measure space II_1 factors with at least two non-unitarily conjugate Cartan subalgebras. Inspired by their construction we present an example of a type II_1 factor M such that the equivalence relation "being unitarily conjugate" on the set of Cartan subalgebras of M is not smooth.

The equivariant Hilbert space compression of a free product of groups

by Dennis Dreesen, K.U. Leuven (Kortrijk)

Let G be a finitely generated group equipped with the word length metric relative to some finite symmetric generating subset. The Hilbert space compression of G is a number between 0 and 1, describing how close a uniform embedding f:G \rightarrowl²(Z) can be to being quasi-isometric. The equivariant Hilbert space compression only takes into account the uniform embeddings which are G-equivariant relative to some affine isometric action of G on l²(Z) and the left multiplication action of G on itself. Given the equivariant Hilbert space compression of two finitely generated groups G₁ and G₂, we calculate the equivariant Hilbert space compression of the amalgamated free product G₁*_C G₂ when C is finite.

Actions on median spaces, Kazhdan and Haagerup properties, applications to the mapping class groups

by Cornelia Drutu, Oxford University

Actions on median spaces turn out to be related to both Kazhdan and Haagerup properties, and to their generalized version in terms of affine actions on L^p- spaces. Surprisingly the median structure appears also in topology: the mapping class groups of surfaces bear an asymptotic median structure. This allows to discuss spaces of homomorphisms of groups with property (T) to mapping class groups. The talk is on joint work with I. Chatterji and F. Haglund (first part), and J. Behrstock and M. Sapir (second part).

Approximation properties and structural results for free Araki-Woods factors

by Cyril Houdayer, ENS Lyon

The free Araki-Woods factors were introduced by D. Shlyakhtenko in the mid-90's as type III analogs of the free group factors. They provide a very large family of non-amenable factors of type III_1. In this talk, I will survey some recent results I obtained on the free Araki-Woods factors, in a joint work with E. Ricard. This includes approximation properties (CMAP and Haagerup property) and structural results, that is, the absence of Cartan subalgebras, for all of these factors. I will then discuss on the classification problem for certain factors of type III_1.

Turing machines and measurable graphings

by Lukasz Grabowski, Georg August Universität Göttingen

In the first part I'll define a Turing dynamical system essentially as a measurable graphing together with some extra data. The extra data gives rise to an operator in the associated von Neumann algebra of the relation generated by the graphing. We will see that the language accepted by a (classical) Turing machine can be recovered from the kernel of the associated operator, and von Neumann dimension of the kernel measures how large the accepted language is. In the second part I'll present an application - using Pontryagin duality and properties of classical Turing machines we will see that every non-negative number is an l2-Betti number of some manifold M with some free co-compact action of a discrete group. I will also explain when M can be taken to be a universal cover, and a group to be finitely presented.

Amenable homogeneous spaces

by Claire Anantharaman-Delaroche, Université d'Orléans

We shall begin with a short survey on the notion of amenability for the homogeneous space G/H (where H is a closed subgroup of the locally compact group G). Whether this notion pass to subgroups or not was left open for some 30 years and solved negatively at the beginning of this century by Monod-Popa and Pestov. We shall describe their counterexample and discuss the particular case where H is an almost normal subgroup of G, studied through the notion of Schlichting completion of the pair (G,H).

A notion of geometric complexity and application to topological rigidity

by Romain Tessera, ENS Lyon

We prove that all subgroups of GLn(K), elementary amenable groups, and many others satisfy a new metric property called "finite decomposition complexity". As an application we prove the bounded Borel and the stable Borel conjectures for these groups.

Affine modules over finite depth planar algebras with weight zero

by Shamindra Kumar Ghosh

I will start with recalling the notion of `affine category over a planar algebra P' and modules over it (referred as `affine modules'). If P comes from a subfactor, we will define two special modules with `weight zero' (using the space of `affine morphisms at zero level' which was shown to be isomorphic to the fusion algebra of the bicategory associated to P in an earlier seminar). If P has finite depth, then each irreducible affine module with weight zero turns out to be a submodule of either of these special modules. In the case of irreducible depth two subfactors, we establish an equivalence between the representation category of the affine modules and the center of the bimodule category. This is a joint work with Paramita Das and Ved Gupta.

The equivalence relation of Takesaki

by Arnaud Brothier, Institut Mathematiques de Jussieu

M. Takesaki introduced a measure theoretical invariant for maximal abelian subalgebras (MASA) of von Neumann algebras. In this talk I will give several new caracterizations of this invariant, in particular I will prove that it is equal to the equivalence relation induced by the normalizer, answering a question of M. Takesaki.

Francqui lecture I: Metric embeddings in Hilbert and Banach spaces

by Alain Valette, Université de Neuchâtel

In the last years, there was a remarkable convergence between three seemingly remote fields of mathematics: theoretical computer science, geometry of Banach spaces, K-theory of C*-algebras. The common theme is embeddings of discrete metric spaces into Hilbert or Banach spaces. Learning of techniques from other fields allowed for mutual cross-fertilization, and it is the purpose of this set of lectures to present some recent developments in this fascinating subject.

Towards the dynamical quantum group SU_q(2) on the level of operator algebras

by Thomas Timmermann, University of Münster

Dynamical quantum groups were introduced by Etingov and Varchenko as an algebraic framework for the study of the quantum dynamical Yang-Baxter equation. They fit into the theory of Hopf algebroids developed by Boehm and others, and form a particular class of quantum groupoids. The simplest example of a dynamical quantum group is a variant of the compact quantum group SU_q(2) of Woronowicz. On the algebraic level, its representation theory and relations to special functions were studied by Koelink and Rosengren. In this talk, I start with some introduction to dynamical quantum groups, and then focus on the variant of SU_q(2), associating to it a measured quantum groupoid in the sense of Enock and Lesieur, and a proper reduced C*-quantum groupoid.