On the asymptotic behaviour of iterates of averages of unitary representations
type: Item
instance of: scholarly article
title: On the asymptotic behaviour of iterates of averages of unitary representations
zbMATH Open document ID: 1057.22006
author: Wojciech Jaworski
published in: Illinois Journal of Mathematics
publication date: 2005-03-01T00:00:00Z
review text: Let \(G\) be a locally compact group and \(\mu\) a probability measure on \(G\). Given a unitary representation \(\pi\) of \(G\) in a Hilbert space \({\mathcal H}\), let \(P_{\mu}\) denote the \(\mu\)-average \(\int_G \pi (y)\mu (dy)\). \(\mu\) is called neat if for every unitary representation \(\pi\) and every \(a\) in the support of \(\mu\), \(s\)-\(\lim_{n\rightarrow\infty}(P^n_{\mu}-\pi (a)^n E_{\mu})=0\), where \(E_{\mu}\) is an orthogonal projection. \(G\) is called neat if every almost aperiodic probability measure on \(G\) is neat. In the paper the asymptotic behaviour of the products \(P_{\mu_n }P_{\mu_{n-1}}\dots P_{\mu_1 }\) is investigated. Main result: To every sequence \(\{ a_n \}^{\infty}_{n=1}\) of elements of \(G\) there exists a sequence \(\{ a_n \}^{\infty}_{n=1}, a_n \in G\), such that \(\forall k=0,1,\dots \) the sequence \(\pi (a_n )P_{\mu_n }P_{\mu_{n-1}}\dots P_{\mu_{k+1}}\) converges in the strong operator topology. When \(G\) is second countable and \(\{ Y_n \}^{\infty}_{n=1}\) is a sequence of independent \(G\)-valued random variables such that \(\mu_n\) is the distribution of \(Y_n\), then \(\forall k=0,1,\dots \) the sequence \(\pi (Y_n Y_{n-1} \dots Y_1 )^{-1}P_{\mu_n}P_{\mu_{n-1}}\dots P_{\mu_{k+1}}\) converges almost surely in the strong operator topology. As applications of this result the neatness of solvable Lie groups, connected algebraic groups, Euclidean motion groups, [SIN] groups, and extensions of abelian groups by discrete groups is established. The neatness of ergodic probability measures on compact groups is proven.
reviewed by: Victor Sharapov
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 2140229
zbMATH Keywords: unitary representations
zbMATH Keywords: \(\mu\)-averages
zbMATH Keywords: neatness of measures
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2140229