Countable amenable identity excluding groups
Canadian Mathematical Bulletin , Volume 47 - Issue 2 p. 215- 228
A discrete group G is called identity excluding if the only irreducible unitary representation of G which weakly contains the 1-dimensional identity representation is the 1-dimensional identity representation itself. Given a unitary representation π of G and a probability measure μ on G, let P μ denote the μ-average ∫ π(g)μ(dg). The goal of this article is twofold: (1) to study the asymptotic behaviour of the powers P μ n and (2) to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure μ on an identity excluding group and every unitary representation π there exists and orthogonal projection E μ onto a π-invariant subspace such that s-lim n→(P μ n - π(a) nE μ) = 0 for every a ∈ supp μ. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of FC-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic.