On the asymptotic behaviour of iterates of averages of unitary representations (Q1766831)

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.