20040601
Countable amenable identity excluding groups
Publication
Publication
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 1dimensional identity representation is the 1dimensional identity representation itself. Given a unitary representation π of G and a probability measure μ on G, let P μ denote the μaverage ∫ 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 πinvariant subspace such that slim 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 FChypercentral 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.
Additional Metadata  

Canadian Mathematical Bulletin  
Organisation  School of Mathematics and Statistics 
Jaworski, W. (2004). Countable amenable identity excluding groups. Canadian Mathematical Bulletin, 47(2), 215–228.
