BGU Probability and Ergodic Theory (PET) seminar Ical Atom

In this talk, we shall focus our attention on intermediate subalgebras of $C(X)\rtimes_r\Gamma$ (and $L^{\infty}(X,\nu)\ltimes\Gamma$). We begin by describing the construction of the commutative crossed product $C(X)\rtimes_r\Gamma$ and how the group contributes to its structure. We shall talk about various (generalized) averaging properties in this context. As a first application, we will show that every intermediate $C^*$-subalgebra $\mathcal{A}$ of the form $C(Y)\rtimes_r\Gamma\subseteq\mathcal{A}\subseteq C(X)\rtimes_r\Gamma$ is simple for an inclusion $C(Y)\subset C(X)$ of minimal $\Gamma$-spaces whenever $C(Y)\rtimes_r\Gamma$ is simple. We shall also show that, for a large class of actions of $C^*$-simple groups $\Gamma\curvearrowright X$, including non-faithful action of any hyperbolic group with trivial amenable radical, every intermediate $C^*$-algebra $\mathcal{A}$, $C_r^*(\Gamma)\subset \mathcal{A}\subset C(X)\rtimes_r\Gamma$ is a crossed product of the form $C(Y)\rtimes_r\Gamma$, $C(Y)\subset C(X)$ is an inclusion of $\Gamma$-$C^*$-algebras.

A topological dynamical system is said to be Bohr chaotic if for any bounded sequence it possesses a continuous function that correlates with the sequence when evaluated along some orbit. The theme of the lecture will be the relation of this property to an abundance of invariant measures of the system.

In a recent work, I introduced the notion of allosteric actions: a minimal action of a countable group on a compact space, with an ergodic invariant measure, is allosteric if it is topologically free but not essentially free. In the first part of my talk I will explain some properties of allosteric actions, and their links with Invariant Random Subgroups (IRS) and Uniformly Recurrent Subgroups (URS). In the second part, I will explain a recent result of mine: the fundamental group of a closed hyperbolic surface admits allosteric actions.

While there are earlier works on bounded cohomology, the topic was popularized by Gromov in 1982. In this introductory talk, we will give definitions of bounded cohomology of discrete groups with trivial coefficients. We will interpret bounded cohomology in low degrees in terms of quasimorphisms and central extensions. Then we will mention a few examples of how bounded cohomology has proved useful in applications, before concentrating on the classification of circle actions