Biexact von Neumann algebras
Abstract
The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. We introduce the notion of biexactness for von Neumann algebras, which allows us to place many previous solidity results in a more systematic context, and naturally leads to extensions of these results. We will also discuss examples of solid factors that are not biexact. This is a joint work with Jesse Peterson.
Connes's Bicentralizer Problem
Abstract
In the world of von Neumann algebras, the factors that do not have a trace, the so-called type III factors, are the most difficult to study. Some of their key structural properties are still not well-understood. In this talk, I will give a gentle introduction to Connes's Bicentralizer Problem, which is the most important open problem in the theory of type III factors. I will then present some recent progress on this problem and its applications.
Asymptotic freeness in tracial ultraproducts
Abstract
I will present novel freeness results in ultraproducts of tracial von Neumann algebras. As a particular case, I will show that if a and b are the generators of the free group F_2, then the relative commutants of a and b in the ultraproduct of the free group factor are free with respect to the ultraproduct trace. The proof is based on a surprising application of Lp-boundedness results of Fourier multipliers in free group factors for p > 2. I will describe applications of these results to absorption and model theory of II_1 factors. This is joint work with Adrian Ioana.
11:00
L-open and l-closed C*-algebras
Abstract
This talk concerns some ideas around the question of when a *-homomorphism into a quotient C*-algebra lifts. Lifting of *-homomorphisms arises prominently in the notions of projectivity and semiprojectivity, which in turn are closely related to stability of relations. Blackadar recently defined the notions of l-open and l-closed C*-algebras, making use of the topological space of *-homomorphisms from a C*-algebra A to another C*-algebra B, with the point-norm topology. I will discuss these properties and present new characterizations of them, which lead to solutions of some problems posed by Blackadar. This is joint work with Dolapo Oyetunbi.
Simplicity of crossed products by FC-hypercentral groups
Abstract
Results from a few years ago of Kennedy and Schafhauser attempt to characterize the simplicity of reduced crossed products, under an assumption which they call vanishing obstruction.
However, this is a strong condition that often fails, even in cases of finite groups acting on finite dimensional C*-algebras. In this work, we give complete C*-dynamical characterization, of when the crossed product is simple, in the setting of FC-hypercentral groups.
This is a large class of amenable groups that, in the finitely-generated setting, is known to coincide with the set of groups with polynomial growth.
Quantized differential calculus on quantum tori
Abstract
We discuss Connes’ quantized calculus on quantum tori and Euclidean spaces, as applications of the recent development of noncommutative analysis.
This talk is based on a joint work in progress with Xiao Xiong and Kai Zeng.
Quasidiagonal group actions and C^*-lifting problems
Abstract
I will give an introduction to quasidiagonality of group actions wherein an action on a C^*-algebra is approximated by actions on matrix algebras. This has implications for crossed product C^*-algebras, especially as pertains to finite dimensional approximation. I'll sketch the proof that all isometric actions are quasidiagonal, which we can view as a dynamical Petr-Weyl theorem. Then I will discuss an interplay between quasidiagonal actions and semiprojectivity of C^*-algebras, a property that allows "almost representations" to be perturbed to honest ones.
Classifiability of crossed products
Abstract
To every action of a discrete group on a compact Hausdorff space one can canonically associate a C*-algebra, called the crossed product. The crossed product construction is an extremely popular one, and there are numerous results in the literature that describe the structure of this C* algebra in terms of the dynamical system. In this talk, we will focus on one of the central notions in the realm of the classification of simple, nuclear C*-algebras, namely Jiang-Su stability. We will review the existing results and report on the most recent progress in this direction, going beyond the case of free actions both for amenable and nonamenable groups.
Parts of this talk are joint works with Geffen, Kranz, and Naryshkin, and with Geffen, Gesing, Kopsacheilis, and Naryshkin.