Guy Salomon, Department of Pure Mathematics, University of Waterloo
"The mysterious connection between proximal actions and Poisson boundaries"
An action of a discrete group $G$ on a compact Hausdorff space $X$ is said to be proximal if for every two points $x,y \in X$ there is a net $g_\alpha \in G$ such that $\lim g_\alpha x=\lim g_\alpha y$, and strongly proximal if the natural action of $G$ on the space $P(X)$ of probability measures on $X$ is proximal. $G$ is said to be 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 strongly amenable and amenable groups. In particular, I will focus on the $C^*$-algebra of continues functions over the universal minimal proximal $G$-action and characterize it in the category of $G$-operator-systems. I will then present some relations to the Poisson boundaries of the groups. The talk is based on a joint work with Matthew Kennedy and Sven Raum.
MC 5417