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# Countable Amenable Identity Excluding Groups

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http://dx.doi.org/10.4153/CMB-2004-021-1
Canad. Math. Bull. 47(2004), 215-228
Published:2004-06-01
Printed: Jun 2004
• Wojciech Jaworski
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## Abstract

A discrete group $G$ is called \emph{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 $s$-$\lim_{n\to\infty}\bigl(P_\mu^n- \pi(a)^nE_\mu\bigr)=0$ 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 $\FC$-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.

MSC Classifications:  22D10 - Unitary representations of locally compact groups 22D40 - Ergodic theory on groups [See also 28Dxx] 43A05 - Measures on groups and semigroups, etc. 47A35 - Ergodic theory [See also 28Dxx, 37Axx] 60B15 - Probability measures on groups or semigroups, Fourier transforms, factorization 60J50 - Boundary theory
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