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Countable Amenable Identity Excluding Groups

Published online by Cambridge University Press:  20 November 2018

Wojciech Jaworski
Wojciech Jaworski*
Affiliation:
School of Mathematics and Statistics Carleton University Ottawa, Ontario K1S 5B6, e-mail: wjaworsk@math.carleton.ca
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Abstract

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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.

Keywords

22D1022D4043A0547A3560B1560J50
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 47 , Issue 2 , 01 June 2004 , pp. 215 - 228
Copyright
Copyright © Canadian Mathematical Society 2004

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