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Countable Amenable Identity Excluding Groups

  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, 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
 

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