The Unique stationary boundary property

Abstract: Let $G$ be a locally compact group and $\mu$ an admissible probability measure on $G$. Let $(B,\nu)$ be the universal topological Poisson $\mu$-boundary of $(G,\mu)$ and $\Pi_s(G)$ the universal minimal strongly proximal $G$-flow. This note is inspired by a recent result of Hartman and Kalantar. We show that for a locally compact second countable group $G$ the following conditions are equivalent: (i) the measure $\nu$ is the unique $\mu$-stationary measure on $B$, (ii) $(B,G) \cong (\Pi_s(G),G)$.