C2

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.

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.

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.

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.
 

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.  

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. 

The (finite) Hilbert matrix is arguably one of the single most well-known matrices in mathematics. The infinite Hilbert matrix H was introduced by David Hilbert around 120 years ago in connection with his double series theorem. It can be interpreted as a linear operator on spaces of analytic functions by its action on their Taylor coefficients. The boundedness of H on the Hardy spaces H^p for 1 < p < ∞ and Bergman spaces A^p for 2 < p < ∞ was established by Diamantopoulos and Siskakis. The exact value of the operator norm of H acting on the Bergman spaces A^p for 4 ≤ p < ∞ was shown to be π /sin(2π/p) by Dostanic, Jevtic and Vukotic in 2008. The case 2 < p < 4 was an open problem until in 2018 it was shown by Bozin and Karapetrovic that the norm has the same value also on the scale2 < p < 4. In this talk, we introduce some background, review some of the old results, and consider the still partly open problem regarding the value of the norm on weighted Bergman spaces. We also consider a generalised Hilbert matrix operator and its (essential) norm. The talk is partly based on a joint work with Mikael Lindström, David Norrbo, and Niklas Wikman (Åbo Akademi University).
 

In ring theory, Morita equivalence is an invariant for many properties, generalising the isomorphism of commutative rings. A strong Morita equivalence for selfadjoint operator algebras was introduced by Rieffel in the 60s, and works as a correspondence between their representations. In the past 30 years, there has been an interest to develop a similar theory for nonselfadjoint operator algebras and operator spaces with much success. Taking motivation from recent work of Connes and van Suijlekom, we will present a Morita theory for operator systems. We will give equivalent characterizations of Morita equivalence via Morita contexts, bihomomoprhisms and stable isomorphisms, while we will highlight properties that are preserved in this context. Time permitted we will provide applications to rigid systems, function systems and non-commutative graphs. This is joint work with George Eleftherakis and Ivan Todorov.

A homogeneous space G/H is called a reductive symmetric space if G is a (real) reductive Lie group, and H is a symmetric subgroup of G, meaning that H is the subgroup fixed by some involution on G. The representation theory on reductive symmetric spaces was studied in depth in the 1990s by Erik van den Ban, Patrick Delorme, and Henrik Schlichtkrull, among many others. In particular, they obtained the Plancherel formula for the L^2 space of G/H. An important aspect is that this generalizes the group case, obtained by Harish-Chandra, which corresponds to the case when G = G' x G' and H is the diagonal subgroup.

In our collaborative efforts with A. Afgoustidis, N. Higson, P. Hochs, Y. Song, we are studying this subject from the perspective of noncommutative geometry. I will describe this exciting new development, with a particular emphasis on describing what is new and how this is different from the traditional group case, i.e. the reduced group C*-algebra of G.

Named after mathematical physicists Kubo, Martin, and Schwinger, KMS states are a special class of states on any C$^*$-algebra admitting a continuous action of the real numbers. Unlike in the case of von Neumann algebras, where each modular flow has a unique KMS state, the collection of KMS states for a given flow on a C$^*$-algebra can be quite intricate. In this talk, I will explain what abstract properties these simplices have and show how one can realise such a simplex on various classes of simple C$^*$-algebras.