Seminar operator algebras 2007-2008

Seminars

An example of a solid von Neumann algebra (after Taka Ozawa)

by Cyril Houdayer (UCLA)
June 25, 2008, 16h00-18h00 in room B.01.07 (abstract)

Prime factors and amalgamated free products (joint with Ionut Chifan)

by Cyril Houdayer (UCLA)
June 24, 2008, 16h00-18h00 in room B.01.07 (abstract)

Representations of quantum permutation algebras

by Teo Banica (Univ. Toulouse)
June 11, 2008, 16h00-18h00 in room B.00.16 (abstract)

Approximation of center-valued Betti numbers

by Anselm Knebusch
May 19, 2008, 10h30-12h30 in room B.01.16 (abstract)

Monoidal equivalence for locally compact quantum groups

by Kenny De Commer
May 15, 2008, 13h30-15h30 in room B.01.16 (abstract)

Relative property (T) actions and trivial outer automorphism groups (after Gaboriau)

by Sébastien Falguières
May 5, 2008, 10h30-12h30 in room B.01.16 (abstract)

Free Araki-Woods factors (after Shlyakhtenko)

by Kenny De Commer
April 14, 2008, 10h30-12h30 in room B.01.16 (abstract)

Free Araki-Woods factors (after Shlyakhtenko)

by Steven Deprez
April 7, 2008, 10h30-12h30 in room B.01.16 (abstract)

Free Araki-Woods factors (after Shlyakhtenko)

by Steven Deprez and Kenny De Commer
March 17, 2008, 10h30-12h30 in room B.01.16 (abstract)

On Poisson boundaries of discrete quantum groups

by Reiji Tomatsu
March 10, 2008, 10h30-12h30 in room B.01.16 (abstract)

On Poisson boundaries of discrete quantum groups

by Reiji Tomatsu
March 3, 2008, 10h30-12h30 in room B.01.16 (abstract)

On Poisson boundaries of discrete quantum groups

by Reiji Tomatsu
February 25, 2008, 10h30-12h30 in room B.01.16 (abstract)

Measured Quantum Groupoids : axiomatics, examples, actions, geometrical construction

by Michel Enock (CNRS Paris, Institut de Mathématiques de Jussieu, France)
February 21, 2008, 10h00-12h00 in room B.00.18 (abstract)

Introduction to type III factors

by Stefaan Vaes
February 12, 2008, 14h00-16h00 in room B.01.14 (abstract)

Introduction to type III factors

by Stefaan Vaes
February 1, 2008, 12h30-14h30 in room B.02.18 (abstract)

Introduction to type III factors

by Stefaan Vaes
January 25, 2008, 9h30-11h30 in room B.02.18 (abstract)

Introduction to Bass-Serre theory

by Sébastien Falguières
December 7, 2007, 9h00-11h00 in room B.02.18 (abstract)

Introduction to Bass-Serre theory

by Sébastien Falguières
November 30, 2007, 9h00-11h00 in room B.02.18 (abstract)

The Hecke algebra of Bost-Connes revisited

by Magnus B. Landstad (NTNU, Trondheim, Norway)
November 19, 2007, 14h00-16h00 in room B.02.18 (abstract)

The property of rapid decay (RD) for discrete groups

by Tom Melotte
November 16, 2007, 9h00-11h00 in room B.02.18 (abstract)

The property of rapid decay (RD) for discrete groups

by Tom Melotte
November 9, 2007, 9h00-11h00 in room B.02.18 (abstract)

Homology theory for von Neumann algebras

by Anselm Knebusch
October 26, 2007, 9h00-11h00 in room B.02.18 (abstract)

Homology theory for von Neumann algebras

by Anselm Knebusch
October 19, 2007, 9h00-11h00 in room B.02.18 (abstract)

Homology theory for von Neumann algebras

by Anselm Knebusch
October 12, 2007, 9h00-11h00 in room B.02.18 (abstract)

Homology theory for von Neumann algebras

by Anselm Knebusch
October 5, 2007, 9h00-11h00 in room B.02.18 (abstract)

Abstracts

Monoidal equivalence for locally compact quantum groups

by Kenny De Commer

Recently, monoidal equivalence has been studied for compact quantum groups by Bichon, De Rijdt and Vaes, to provide examples of compact quantum group actions with large quantum multiplicity. It has also been applied succesfully by Vaes and Vander Vennet to determine Poisson boundaries of some (non-amenable) discrete quantum groups. In this talk, I will discuss a theory of monoidal equivalence for locally compact quantum groups (on the level of von Neumann algebras). As in the purely algebraic setup, the main notion is that of a Galois object which links both quantum groups. I show how the Galois object and one of the locally compact quantum groups can be used to reconstruct the other locally compact quantum group, the main problem being the construction of the invariant weights. On the other hand, the Galois object and both locally compact quantum groups can also be combined in a single object, namely a measured quantum groupoid (in the sense of Lesieur and Enock). This (basic) viewpoint seems the most convenient one to treat the 'monoidality' of the strong Morita equivalence of the dual C*-algebraic quantum groups.

Representations of quantum permutation algebras

by Teo Banica (Univ. Toulouse)

We develop a general theory of random matrix representations of As(n), with a two-fold motivation: (1) we conjecture that any quantum permutation algebra appears as "image" of such a representation, (2) by using Hadamard matrices, this leads to new algebras Aq with |q|=1, which seem to depend on the arithmetic properties of q (a bit like the Drinfeld-Jimbo algebras do). This is joint work with Bichon and Schlenker.

Prime factors and amalgamated free products (joint with Ionut Chifan)

by Cyril Houdayer (UCLA)

I will show that any non-amenable factor arising as an amalgamated free product over an abelian von Neumann algebra is prime, i.e. cannot be written as a tensor product of two diffuse factors. I will moreover discuss some applications in orbit equivalence.

An example of a solid von Neumann algebra (after Taka Ozawa)

by Cyril Houdayer (UCLA)

I will prove the following recent result of Taka Ozawa: the von Neumann algebra of the group Z2 ⋊ SL(2,Z) is solid, i.e. the relative commutant of any diffuse von Neumann subalgebra is amenable. I will moreover discuss some applications of this result in ergodic theory.

Approximation of center-valued Betti numbers

by Anselm Knebusch

To calculate L2-Betti-numbers a useful tool is given by an approximation theorem of W. Lück. L2-Betti-numbers measure the dimension of the homology modules of the universal covering of a manifold, using the standard trace coming with the group von Neumann algebra. Universal or center-valued Betti-numbers are finer invariants, measuring the dimension in terms of the center-valued trace. In the talk we will briefly describe Lück's method, and then show how his main ideas have to be modified to gain a similar approximation theorem for universal Betti-numbers.

Relative property (T) actions and trivial outer automorphism groups (after Gaboriau)

by Sébastien Falguières

To a standard, countable, probability measure preserving equivalence relation R one associates a von Neumann algebra M(R) generalizing Murray and von Neumann's "group-measure-space construction" and a Cartan subalgebra A of M(R). The equivalence relation R is said to have relative property (T) when the inclusion of A in M(R) is rigid in the sense of S.Popa. I will explain the following very recent result, due to Gaboriau: every non-amenable free product of groups admits free, ergodic, probability measure preserving actions with relative property (T).

Actions of F whose II1 factors and orbit equivalence relations have prescribed fundamental group (joint work with Sorin Popa)

by Stefaan Vaes

We exhibit a family S of subgroups of the real line R containing R itself, all of its countable subgroups, as well as uncountable subgroups of any Hausdorff dimension in (0,1), satisfying the following property: given any H in S, there exist continuously many free, ergodic, measure preserving actions σi of the free group on infinitely many generators F on non-atomic probability measure spaces (Xii), such that their associated group measure space II1 factors Mi and orbit equivalence relations Ri have fundamental group equal to exp(H) and with Mi (respectively Ri) stably non-isomorphic. In addition, these actions can be taken so that Mi and Ri have, to a certain extent, prescribed outer automorphism group. Note that this result provides in particular the first separable II1 factors and equivalence relations with uncountable fundamental group different from R+.

Free Araki-Woods factors (after Shlyakhtenko)

by Steven Deprez and Kenny De Commer, Steven Deprez, Kenny De Commer

As a class of examples of type III factors, we study the free Araki-Woods factors, as introduced by Shlyakhtenko. He associates a von Neumann algebra to an orthogonal 1-parameter group on a real Hilbert space. This construction defines a functor from the class of orthogonal representations to the class of von Neumann algebras with a given state, sending direct sums to free products. In a first seminar we study free products of von Neumann algebras and orthogonal 1-parameter groups, before we introduce the free Araki-Woods factors, and give some basic properties. We prove the factoriality of these von Neumann algebras and give their classification according to type, using however the 'absorption property' of these von Neumann algebras as a black box. This important fact (due to Shlyakhtenko) will be proven in the second talk, where we will also give a complete classification of the free Araki-Woods factors associated to quasi-periodic 1-parameter groups. This does not extend easily to a full classification of all free Araki-Woods factors. We give one example where the usual invariants fail to detect non-isomorphism.

On Poisson boundaries of discrete quantum groups

by Reiji Tomatsu

For a random walk on a discrete quantum group, one can introduce Poisson boundary abstractly. The main problem there is how a Poisson boundary is realized as a more concrete non-commutative measurable space, that is, a von Neumann algebra. I will present an overview begining from an introductory part of compact quantum groups and their duals. Then in the 2nd talk, I will introduce right coideals and quantum subgroups, and in the final talk, I will give an answer to the above realization problem for discrete quantum groups with some properties.

Introduction to type III factors

by Stefaan Vaes

This series of three lectures is intended as a preparation for a series of lectures on Shlyakhtenko's free Araki Woods factors. I will introduce the type classification of Murray and von Neumann, the basics of Tomita Takesaki modular theory and Connes' invariants for type III factors. I will present plenty of examples (infinite tensor products, crossed products, free products).

Introduction to Bass-Serre theory

by Sébastien Falguières

Consider an orientation-preserving action of a group G on an oriented tree T. There is then a natural notion of quotient tree G\T. The theory of Bass-Serre shows how to reconstruct the group G when one knows the quotient tree G\T, the stabilizers of the vertices and the stabilizers of the edges. We will show that the group G can be identified with the fundamental group of a graph of groups, carried by G\T. Conversely, every graph of group arises in this way. Finally, we will use Bass-Serre theory to give simple proofs of the classical Schreier's and Kurosh's theorems.

The property of rapid decay (RD) for discrete groups

by Tom Melotte

Groups of rapid decay (or property RD) have been introduced by Jolissaint in 1990, but to trace part of its history we have to go back to the work of Grothendieck in 1955. After refreshing a few prerequisites about the regular representation of discrete groups and lengths, we will give several equivalent formulations for property RD. Of course we will discuss accessible examples and constructions under which property RD is preserved. At the end we will give some applications of groups with property RD in relation with the Baum-Connes conjecture and Compact Quantum Metric Spaces.

Homology theory for von Neumann algebras

by Anselm Knebusch

Distinguishing von Neumann algebras is in general not an easy task, many very different constructions can lead to the same von Neumann algebra. For example all amenable ICC groups generate isomorphic factors and it is not yet known if free groups with a different number of generators lead to different factors. A. Connes and D. Shlyakhtenko developed a homology theory for von Neumann algebras, in order to get a good tool for this purpose. The lectures will start introducing the basic concepts of homological algebra, like functors, limits, projectivety, resolutions, derived functors, etc. Afterwards we will develop a suitable dimension theory, in order to define Betti numbers (dimension of a homology module) and finally we will define a homology theory for von Neumann algebras, see some of the basic properties, and make some first calculations.

Measured Quantum Groupoids : axiomatics, examples, actions, geometrical construction

by Michel Enock (CNRS Paris, Institut de Mathématiques de Jussieu, France)

Highly inspired by the theory of locally compact quantum groups, a theory of measured quantum groupoids is here presented and developped; in particular, the notion of actions, crossed products, are introduced; a biduality theorem is obtained, and it is proved that the inclusion of the initial algebra into its crossed product is depth 2, which leads to another measured quantum groupoid. In particular, we associate to each action of a locally compact quantum group an example of a measured quantum groupoid.

The Hecke algebra of Bost-Connes revisited

by Magnus B. Landstad (NTNU, Trondheim, Norway)

The Hecke algebra introduced by Bost and Connes has been an inspiration to many. We shall look at certain aspects of the Hecke algebras associated to pairs (G, H) where G is a group and H a subgroup. We shall see that such a pair has a Schlichting completion (G*, H*) with H* a compact open subgroup of G* (due to Tzanev). To a Hecke algebra there is also a Banach *-algebra and a C*-algebra. To study the representation theory of these 3 algebras both Fell's and Rieffel's version of Morita equivalence is needed. Old results about Banach *-algebras reappear. We may also look at generalised Hecke algebras and it turns out that the Schlichting completion of (G, H) then may be different. Illustrations will be with various versions of the ax+b-group. This is joint work with S. Kaliszewski, John Quigg and Nadia S. Larsen.