Countable Amenable Identity Excluding Groups

Jaworski, Wojciech

Canadian mathematical bulletin, Tome 47 (2004) no. 2, pp. 215-228

Résumé

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 $\pi $ of $G$ and a probability measure $\mu $ on $G$ , let ${{P}_{\mu }}$ denote the $\mu $ -average $\int{\pi (g)\mu (dg)}$ . The goal of this article is twofold: (1) to study the asymptotic behaviour of the powers $P_{\mu }^{n}$ , and (2) to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure $\mu $ on an identity excluding group and every unitary representation $\pi $ there exists and orthogonal projection ${{E}_{\mu }}$ onto a $\pi $ -invariant subspace such that $s-{{\lim }_{n\to \infty }}\,(P_{\mu }^{n}\,-\,\pi {{(a)}^{n}}\,{{E}_{\mu }})\,\,=\,0$ for every $a\,\in $ supp $\mu $ . 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 $\text{FC}$ -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.

Détail

DOI : 10.4153/CMB-2004-021-1

Mots-clés : 22D10, 22D40, 43A05, 47A35, 60B15, 60J50