Abstract view
Countable Amenable Identity Excluding Groups


Published:20040601
Printed: Jun 2004
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, 22D40, 43A05, 47A35, 60B15, 60J50 show english descriptions
Unitary representations of locally compact groups Ergodic theory on groups [See also 28Dxx] Measures on groups and semigroups, etc. Ergodic theory [See also 28Dxx, 37Axx] Probability measures on groups or semigroups, Fourier transforms, factorization Boundary theory
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
