### Seminar talk by Salomon: Combinatorial and operator algebraic aspects of proximal actions

The Operator Algebras and Operator Theory Seminar is back (sort of). This semester we will have the seminar on Thursdays 15:30 (Israel time) about once in a while. Please send me an email if you want to join the mailing list and get the link for the zoom meetings. Here are the details for our first talk :

Title: Combinatorial and operator algebraic aspects of proximal actions

Speaker: Guy Salomon (Weizmann Institute)

Time: 15:30-16:30,Thursday Nov. 12, 2020

(Zoom room will open about ten minutes earlier, and the talk will begin at 15:30)

Abstract:

An action of a discrete group $G$ on a compact Hausdorff space $X$ is called proximal if for every two points $x$ and $y$ of $X$ there is a net $g_i \in G$ such that $\lim(g_i x)=\lim(g_i y)$, and strongly proximal if the natural action of $G$ on the space $P(X)$ of probability measures on $X$ is proximal. The group $G$ is called strongly amenable if all of its proximal actions have a fixed point and amenable if all of its strongly proximal actions have a fixed point.

In this talk I will present relations between some fundamental operator theoretic concepts to proximal and strongly proximal actions, and hence to amenable and strongly amenable groups. In particular, I will focus on the C*-algebra of continuous functions over the universal minimal proximal $G$-flow and characterize it in the category of $G$-operator-systems.

I will then show that nontrivial proximal actions of $G$ can arise from partitions of $G$ into a certain kind of “large” subsets. If time allows, I will also present some relations to the Poisson boundaries of $G$. The talk is based on a joint work with Matthew Kennedy and Sven Raum.