Beyond Amenability: Groups, actions and Operator Algebras

Abstract: We will describe several characterizations of amenable groups or C* algebras, in connection with similarity problems going back to Dixmier (1950) and Kadison (1955). A group G is called unitarizable if all uniformly bounded representations of G in the linear group of a Hilbert space are unitarizable (i.e. are similar to unitary representations). Dixmier proved that amenable implies unitarizable and asked whether the converse is true. We give two results showing that certain kinds of strengthenings of unitarizability are indeed equivalent to amenability for discrete groups. Analogous results are valid for C* algebras.

Abstract: In this talk, I will first recall some of Ed\\\'s earlier work on operator spaces. I will then talk about some of recent applications, in particular, the application to harmonic analysis and locally compact quantum groups

Abstract: We will present two concepts from operator space theory and show how they can be used in noncommutative analysis. The first concept is the theory of completely 1-summing norms, introduced by Effros and Ruan. We show how to combine this concept with tools from free probability to obtain estimates for potentials with respect to quasi-free states. The second concept is the idea of a maximal function. Starting from the maximal ergodic theorem, we derive estimates for the \'\'carree du champs operator\'\'.

Abstract: I will give an introduction to a theory of complexity of classification problems in mathematics and discuss its connections with set theoretic versions of rigidity phenomena for measure preserving actions of countable groups.

Abstract: I will introduce some classifcation result of the mapping class groups up to measure equivalence and explain geometric objects, the curve complex and the Thurston boundary, on which the mapping class group acts naturally. I will talk about what these objects are and how these can be used for the classification.

Abstract: A survey of amenability of equivalence relations in both the measure theoretic and Borel contexts.

Abstract: We propose a new approach to Superrigidity theorems (old, recent and new), which is based on convenient notion of a boundary for a given group G, and an associated generalized Weyl group. Amenability is one of the defining properties of boundaries. Based on a joint work with Uri Bader and Ali Shaker.

Abstract: Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group $\\\\Gamma$ admits a special linear representation with non-amenable $\\\\Re$-Zariski closure if and only if it acts on an Abelian group $A$ (of finite nonzero $\\\\Q$-rank) so that the corresponding group pair $(\\\\Gamma \\\\ltimes A,A)$ has relative property (T). The proof is constructive. The main ingredients are Furstenberg\\\'s celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.

Abstract: We present evidence for treating highly constrained physical systems using the Connes tensor product of correspondences. The idea is that the constraints should force some observables on one system to be identified with observables on the other system.

Abstract: I will present a construction of continuous families of non-isomorphic, irreducible,finite index subfactors of the hyperfinite II_1 factor with the same standard invariant.This is joint work with Remus Nicoara and Sorin Popa.

Abstract: Power series with non-commuting coefficients and variables are ubiquitous in such areas as free probability, non-commutative geometry, and renormalization theory. Computing with these novel objects can often seem impossibly difficult. The use of combinatorial labels such as trees and more general graphs has proved to be very helpful. The most dramatic instance of this may be found in renormalization theory, where Feynman diagrams are extensively employed for that purpose. More recently it has been discovered that many of the ad hoc calculations of physics could be further rationalized by using Hopf algebraic methods. In a recent paper, Anshelevich, Effros, and Mihai Popa have shown that there is a free analogue F of the Faa di Bruno algebra that can be used to invert non-commutative power series. A challenging problem is to determine the embeddings of F into Foissy\'s free Connes-Kreimer algebra. Some remarkable approaches to this have been discovered by Bergbauer and Kreimer in their studies of the combinatorial, and by Mihai Popa, who employed techniques of Foissy. We will survey some of these intriguing new developments.

Abstract: I will describe an example of a quasi-trace on a Type I algebra that is not a 2-quasi-trace. (The main point is that local traces on commutative algebras with one-dimensional spectrum are additive, ... plus some result on abelian subalgebras of extensions.) An open problem on purely infinite algebras is connected with the problem of the existence of 2-quasi traces that are not additive.

Abstract: We plan to present automatic local and global trivialization results for continuous fields of Kirchberg algebras and a computation of the homotopy groups of the automorphism group of Kirchberg algebras. As a corollary we show that a separable unital C(X)-algebra A over a finite dimensional compact Hausdorff space X all of whose fibers are isomorphic to the same Cuntz algebra O(n) is locally trivial. If n=2 or if n is infinite then A is isomorphic to C(X,O(n)). For other values of n, A is isomorphic to C(X,O(n)) if and only if (n-1)[1_A]=0 in K_0(A).

Abstract: Lifting projections from quotients of C*-algebras is of interest in a number of situations. Classically it is the problem of extending vector bundles from a subspace. This is usually dealt with by considering homological obstructions. But there is also a metric approach which does not seem to have been discussed much. If two metric spaces are metrically close together, even if their topological features are quite different, vector bundles on one which are suitably controlled by the metric have counterparts on the other. This works for non-commutative C*-algebras too. Classically it involves the question of extending vector-valued Lipschitz functions. Most of my talk will concern the interesting features of that classical subject.

Abstract: I will survey the current state of the Classification Program, focusing on inductive limits.

Abstract: We consider the Alugthe transform T~=|T|^{1/2}U|T|^{1/2} of a Hilbert space operator T, where T=U|T| is the polar decomposition of T. Interest in this transform is in part due to considerations related to invariant subspace problems. If T belongs to a finite von Neumann algebra with fixed faithful trace, we show that the Brown measure is unchanged by the Aluthge transform. (This prompts a conjecture concerning iterated Aluthge transforms.) We consider the special case when U implements an automorphism of the von Neumann algebra generated by the positive part |T| of T, and we prove that the iterated Aluthge transform converges to a normal operator whose Brown measure agrees with that of T (and we compute this Brown measure). This proof relies on a theorem that is an analogue of von Neumann\'s mean ergodic theorem, but for sums weighted by binomial coefficients. (Joint work with Hanne Schultz.)

Abstract: TBA

Abstract: Suppose that M is a von Neumann algebra embeddable into the ultraproduct of the hyperfinite $\\mathrm{II}_1$-factor and $X$ is an n-tuple of selfadjoint generators for M. Denote by $\\Gamma(X;m,k,\\gamma)$ the microstate space of X of order $(m,k,\\gamma)$. We say that X is tubular if for $\\epsilon >0$ there exists an $m$ and $\\gamma$ such that if $(x_1,\\ldots, x_n), (y_1,\\ldots, y_n) \\in \\Gamma(X;m,k,\\gamma)$, then there exists a $k\\times k$ unitary u satisfying $|ux_i u^* - y_i|_2 < \\epsilon$ for each $1 \\leq i \\leq n$. We show that the following conditions are equivalent: 1) M is injective; 2) X is tubular; 3) Any two embeddings of M into the ultraproduct of the hyperfinite $\\mathrm{II}_1$-factor are conjugate by a unitary $u$ in the ultraproduct algebra. We introduce seemingly weaker notions of tubularity and show that they all coincide with injectivity as well.

Abstract: We will prove that for any finitely generated commutative ring R (with 1), the group EL(n,R) generated by the elementary nxn matrices over R, has Kazhdan\\\'s property (T), once n > 1 + Krull dim R. In fact a sharper result holds, and the key ring theoretical tool relevant here turns out to be the stable range of R. The proof combines several ingredients previously introduced into this game: the relative property (T) over general rings, bounded generation, and reduced cohomology of unitary representations.

Abstract: We shall consider certain C*-dynamical systems arising from Hecke algebras and classify their KMS-states.

Abstract: I will present recent rigidity results for quotients of generalized Bernoulli actions of property (T) groups. Using Popa\'s cocycle superrigidity theorem, these actions are shown to be orbitally rigid: the orbit structure entirely remembers the group and the action. We completely classify certain families of quotients of generalized Bernoulli actions. We finally study the crossed product II_1 factors associated with generalized Bernoulli actions of property (T) groups. This yields explicit continuous families of II_1 factors without outer automorphisms. Note that their existence has been shown before by Ioana, Peterson and Popa.

Abstract: I will present a new approach for showing primeness in von Neumann algebras. Specifically I will apply Popa\\\'s deformation/rigidity techniques in the context of Sauvageot\\\'s deformations arising from closable derivations to conclude that all free product II$_1$ factors, as well as all group factors arising from groups with positive first $L^2$-Betti number are prime. These techniques also give a new approach to Ozawa\\\'s result that all nonamenable subfactors of a free group factor are prime.

Abstract: We consider II$_1$ factors of the form $M=\\\\overline{\\\\bigotimes}_{G}N\\\\rtimes G$, where either i) $N$ is a non-hyperfinite II$_1$ factor and $G$ is an ICC amenable group or ii) $N$ is a weakly rigid II$_1$ factor and $G$ is an ICC group and where $G$ acts on $\\\\overline{\\\\bigotimes}_{G}N$ by Bernoulli shifts. We prove that isomorphism of two such factors implies cocycle conjugacy of the corresponding Bernoulli shift actions.

Abstract: A group is strongly bounded if every isometric action on any metric space has bounded orbits. Although this notion has been introduced by Bergman only recently, there are many non-trivial examples, including full symmetric groups on infinite sets and infinite powers of finite perfect groups.

Abstract: My lecture will be on multiplicative properties of positive linear maps of C*- and von Neumann algebras. There is always a Jordan subalgebra corresponding to ChoiŽs multiplicatve domain, with respect to which a positive map acts as a Jordan module homomorphism. There is also a smaller Jordan algebra on which a map acts as a Jordan automorphism. IŽll discuss properties of these algebras and also their relation to a result of Arveson.

Abstract: About two decades ago, G-C Rota described the (Reduced) Incidence Hopf Algebras associated to partially ordered sets. One can find remarkable examples that fall in this category, such as the Connes-Kreimer Hopf Algebras of rooted trees. New interesting examples are inspired by the recent work on tree Hopf algebras in Quantum Electrodynamics and by some aspect of Free Probability.

Abstract: Shannon\\\'s problem on monotonicity of entropy was recently solved by Artstein, Ball, Barthe, and Naor. They showed that if (X_j) are independent copies of a random variable X, then the entropy of n^{-1/2}(X_1+...+X_n) increases as n increases. Shortly after, Shlyakhtenko solved Shannon\\\'s problem in the free case. That is, if (x_j) are freely independent copies of a self-adjoint non-commutative random variable x, then the free entropy of n^{-1/2}(x_1+...+x_n) increases as n increases. In both cases, the classical and the free, the entropy is constant when X (x, resp.) is Gaussian (semicircular, resp.), and in fact, as we have shown, if the functions considered above are not strictly increasing, then X (x, resp.) is necessarily Gaussian (semicircular, resp.).

Abstract: I will survey the theory of cocycle and outer conjugacy problems of group (outer) actions on a factor including the new paradigme of classification theory in functional analysis.