Double ergodicity of the Poisson boundary and applications to bounded cohomology

Abstract

We prove that the Poisson boundary of any spread out non-degenerate symmetric randomwalk on an arbitrary locally compact second countable group G is doubly $\mathcal{M}$^sep-ergodic with respect to the class $\mathcal{M}$^sep of separable coefficient Banach G-modules. The proof is direct and based on an analogous property of the bilateral Bernoulli shift in the space of increments of the random walk. As a corollary we obtain that any locally compact s-compact group G admits a measure class preserving action which is both amenable and doubly $\mathcal{M}$^sep-ergodic. This generalizes an earlier result of Burger and Monod obtained under the assumption that G is compactly generated and allows one to dispose of this assumption in numerous applications to the theory of bounded cohomology.