# Workshop Classification and dynamical systems II: Von Neumann Algebras

April 18 - 22, 2016

Program Committee: Sorin Popa, Dimitri Shlyaktenko, Stefaan Vaes

### Monday April 18

09:50-10:00 Welcome

10:00-10:45 Sorin Popa

Title: In search of a good cohomology theory for II_1 factors

Abstract: One of the most important problems in II$_1$ factors is to find a

’’good’’ cohomology theory, i.e., one that’s non-vanishing (and if possible calculable) and that could detect important properties of II$_1$ factors, like absence of regularity, or infinite generation.

I will review the various attempts to construct such a theory.

11:15-12:00 Rémi Boutonnet

Title: II_1 factors with non-isomorphic ultrapowers

Abstract: In this talk I will present recent joint work with Ionut Chifan and Adrian Ioana showing that there exist infinitely many II1 factors with non-isomorphic ultrapowers. More precisely, we construct a concrete family of factors whose ultrapowers can be distinguished by computing a new invariant for von Neumann algebras. This invariant relies on central sequences.

14:00-14:45 Cyril Houdayer

Title: Strong solidity and classification of free Araki-Woods factors

Abstract: I will present new results regarding the structure and the classification of Shlyakhtenko's free Araki-Woods factors. Firstly, I will show that all free Araki-Woods factors are strongly solid, meaning that the normalizer of any diffuse amenable subalgebra with normal expectation remains amenable. This provides the first class of nonamenable strongly solid type III factors (joint work with R. Boutonnet and S. Vaes). Secondly, I will present a complete classification of a large family of non-almost periodic free Araki-Woods factors up to *-isomorphism (joint work with D. Shlyakhtenko and S. Vaes).

15:15-16:00 Stefaan Vaes

Title: Cohomology and L^2-Betti numbers for subfactors and quasi-regular inclusions

Abstract: I present a joint work with S. Popa and D. Shlyakhtenko introducing a cohomology theory for quasi-regular inclusions of von Neumann algebras. In particular, we define L^2-cohomology and L^2-Betti numbers for such inclusions. Applying this to the symmetric enveloping inclusion of a finite index subfactor, we get a cohomology theory and a definition of L^2-Betti numbers for finite index subfactors, as well as for arbitrary rigid C*-tensor categories. For the inclusion of a Cartan subalgebra in a II_1 factor, we recover Gaboriau's L^2-Betti numbers for equivalence relations. At the end, I will also discuss a recent joint work with Yuki Arano around the C*-tensor categories arising from totally disconnected groups and the associated subfactors.

### Tuesday April 19

10:00-10:45 Kate Juschenko

Title: Cycling amenable groups and soficity

Abstract: I will give introduction to sofic groups and discuss a possible strategy towards finding a non-sofic group. I will show that if the Higman group were sofic, there would be a map from Z/pZ to itself, locally like an exponential map, satisfying a rather strong recurrence property. The approach to (non)-soficity is based on the study of sofic representations of amenable subgroups of a sofic group. This is joint work with Harald Helfgott.

11:15-12:00 Sven Raum

Title: Operator algebras of locally compact groups acting on trees

Abstract: I will present my work on C*-simplicity of locally compact groups, focusing on its relevance for studying locally compact groups acting on trees. First, I will summarising results that I could obtain in 2015 on simplicity and non-simplicity of reduced group C*-algebras of locally compact groups. After describing some examples of C*-simple groups acting on trees, I will describe the type I conjecture for closed subgroups of Aut(T) and how different C*-algebraic and von Neumann algebraic results can contribute to its clarification.

14:00-14:30 Anna Krogager

Title: A class of II$_1$ factors with exactly two crossed product decompositions

Abstract: We construct the first II$_1$ factors having exactly two group measure space decompositions up to unitary conjugacy. More generally, for every positive integer $n$, we construct a II$_1$ factor $M$ that has exactly $n$ group measure space decompositions up to conjugacy by an automorphism. This is joint work with Stefaan Vaes.

15:00-15:30 Peter Verraedt

Title: Bernoulli crossed products without almost periodic weights

Abstract: The noncommutative Bernoulli crossed products, introduced by Connes in 1974, were the first examples of full factors of type III. In our previous work, we could classify the Bernoulli crossed products (P, φ)^Λ \rtimes Λ for all almost periodic states φ, and for Λ belonging to a large class of groups. To classify Bernoulli crossed products that are not built with an almost periodic state, a new approach is necessary. In this talk, I will explain how one still can retrieve a strong classification result for states that are far from being almost periodic. It will be shown that the family of factors (P, φ) \rtimes Λ with P an amenable factor, φ a weakly mixing state (i.e. a state for which the modular automorphism group is weakly mixing) and Λ belonging to a large class of groups, is classified by the group Λ and the action of Λ on (P, φ)^Λ , up to state preserving conjugation of the action. I will show how this result yields two nonisomorphic Bernoulli crossed products, that previously were not known to be nonisomorphic.

15:45-16:15 Andreas Aaserud

Title: Approximate equivalence of actions

Abstract: We will discuss some new notions of equivalence of measure preserving group actions on probability spaces, including approximate versions of conjugacy and orbit equivalence, which can most easily be defined in the framework of ultrapowers of von Neumann algebras, but the definitions of which turn out not to depend on a choice of ultrafilter. During the talk, we will for instance see that certain non-amenable groups without Kazhdan's property (T) have many mutually non-conjugate actions all of which are mutually approximately conjugate, whereas approximate conjugacy is the same as conjugacy for actions of groups with Kazhdan's property (T). We will also discuss superrigidity in the context of approximate conjugacy and approximate orbit equivalence. This talk is based on joint work with Sorin Popa.

### Wednesday April 20

9:30-10:15 Dietmar Bisch

Title: Singly generated planar algebras

Abstract: Vaughan Jones and I classified subfactor planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of 3-boxes is at most 13 several years ago. In recent joint work with Jones and Liu, we settled the case of dimension 14. We find one depth 3 subfactor planar algebra coming from quantum SO(3), and a one-parameter family coming from quantum Sp(4). They are all BMW.

10:45-11:30 Xin Li

Title: Continuous orbit equivalence

Abstract: We introduce the notion of continuous orbit equivalence. This notion builds bridges between topological dynamics, C*-algebras, and geometric group theory. We also discuss rigidity phenomena.

11:45-12:30 Hiroshi Ando

Title: Unitarizaibility, Maurey-Nikishin factorization and Polish groups of finite type

Abstract: In the seminal work of cocylcle superrigidity theorem, Sorin Popa introduced the class of finite type Polish groups. A Polish group $G$ is finite type, if it is embeddable into the unitary group of a separable II_1 factor.

Popa proposed a problem of finding abstract characterization of finite type Polish groups. There are two conditions which are clearly necessary for a Polish group $G$ to be of finite type, namely that

(a) $G$ is unitarily representable (i.e., is embeddable into the full unitary group of $\ell^ 2$,

and

(b) $G$ is SIN, i.e., it admits a two-sided invariant metric compatible with the topology.

Popa asked whether these two conditions are actually sufficient.

In 2011 I and Yasumichi Matsuzawa obtained several partial positive answers for some classes of Polish groups.

In this talk, we report our recent progress on this problem based on the Maurey-Nikishin Theorem on bounded maps from a Banach space of Rademacher type 2 to the space of all measurable maps on a probability space.

This is joint work with Yasumichi Matsuzawa, Andreas Thom and Asger Tornquist.

### Thursday April 21

10:00-10:45 Narutaka Ozawa

Title: A remark on fullness of some group measure space von Neumann algebras

Abstract: Recently C. Houdayer and Y. Isono have proved among other things that every biexact group $\Gamma$ has the property that for any non-singular strongly ergodic action on a standard measure space $(X,\mu)$ the group measure space von Neumann algebra $\Gamma \ltimes L^\infty(X)$ is full.

In this talk, I will prove the same property for a wider class of groups, notably including SL(3,Z). I also prove that for any connected simple Lie group $G$ with finite center, any lattice $\Gamma \le G$, and any closed non-amenable subgroup $H \le G$, the translation action of $\Gamma$ on $G/H$ is strongly ergodic and the von Neumann factor $\Gamma \ltimes L^\infty(G/H)$ is full.

arXiv:1602.02654

11:15-12:00 Huaxin Lin

Title: A short visit to the classification of simple C*-algebras of finite rank

Abstract: In this talk, we plan to discuss one part of the proof of the isomorphism theorem of the recent classification theorem for unital separable finite simple C*-algebras with finite nuclear dimension.

We will focus on asymptotic (one parameter) unitary equivalence of homomorphisms from one classifiable C*-algebras to another.

14:00-14:45 Alain Valette

Title: Expanders and box spaces

Abstract: Expanders, especially those coming from box spaces of residually finite groups, have been used to test various forms of the coarse Baum-Connes conjecture. The first construction of a pair of expanders, one not coarsely embedding in the other, was provided by Mendel and Naor in 2012. This was extended by Hume in 2014 who constructed a continuum of expanders with unbounded girth, pairwise not coarsely equivalent. In joint work with A. Khukhro, we construct a continuum of expanders with geometric property (T) of Willett-Yu, as box spaces of $SL(3,\Z)$. We will discuss the following results: if box spaces of groups G, H are coarsely equivalent, then the groups G, H are quasi-isometric (Khukhro and myself), and moreover G and H are uniformly measure equivalent (K. Das).

15:15-16:00 Stuart White

Title: Structure of simple nuclear C*-algebras: a von Neumann prospective

Abstract: I'll discuss recent developments in the structure of simple nuclear C*-algebras and illustrate how these are connected to the results of Connes, Haagerup and Popa on amenable von Neumann algebras.

### Friday April 22

9:30-10:15 Makoto Yamashita

Title: Unitary half-braiding and bimodules over the Longo-Rehren/SE-inclusion

Abstract: Popa's symmetric enveloping algebra and Drinfeld double of quantum groups are essential in the study of approximation properties for subfactors and quantum groups. Based on the notion of Drinfeld center of ind-objects in C*-tensor categories, we give a categorical description of certain bimodules over the generalized Longo-Rehren inclusion. This generalizes Izumi's result for fusion categories, and provides a 'global' route to the correspondence between the above two theories. Based on joint work with S. Neshveyev.

10:45-11:30 Philip Dowerk

Title: Bounded normal generation for projective unitary groups

Abstract: In this talk I will present joint work with Andreas Thom on bounded normal generation (BNG) for projective unitary groups of von Neumann algebras. We say that a group has (BNG) if the conjugacy class of every nontrivial element and of its inverse generate the whole group in finitely many steps. I will focus on (BNG) for the projective unitary group G of a finite factor and present applications such as invariant automatic continuity (that is, every homomorphism from G to any separable SIN group is continuous) and uniqueness of the Polish group topology.

11:45-12:30 Dimitri Shlyakhtenko

Title: Free entropy dimension and the first L^2 Betti number

Abstract: In this talk we discuss some old and new results on the connection between Voiculescu’s free entropy dimension and the first L^2 Betti number of a discrete group. In particular, we discuss some work related to recent results of K. Jung on strongly one-bounded von Neumann algebras.