On the asymptotic behaviour of iterates of averages of unitary representations. (English) Zbl 1057.22006

Jaworski, Wojciech

Ill. J. Math. 48, No. 4, 1117-1161 (2004).

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.

Reviewer: Victor Sharapov (Volgograd)

MSC:

22D40	Ergodic theory on groups
22D10	Unitary representations of locally compact groups
60B15	Probability measures on groups or semigroups, Fourier transforms, factorization
43A05	Measures on groups and semigroups, etc.
47A35	Ergodic theory of linear operators

Keywords:

unitary representations; \(\mu\)-averages; neatness of measures