Averages of unitary representations and weak mixing of random walks

Michael Lin, Rainer Wittmann

Research output: Contribution to journalArticlepeer-review

Abstract

Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; Un converges weakly for every continuous unitary representation of G; U is weakly mixing for any ergodic group action in a probability space. (ii) If μ is ergodic on G metrizable, and Un converges strongly for every unitary representation, then the random walk is weakly mixing: n−1∑nk=1|⟨μk∗f,g⟩|→0 for $g ∈ L_∞(G)$ and $f ∈ L_{1}(G)$ with ʃ fdλ = 0. (iii) Let G be metrizable, and assume that it is nilpotent, or that it has equivalent left and right uniform structures. Then μ is ergodic and strictly aperiodic if and only if the random walk is weakly mixing. (iv) Weak mixing is characterized by the asymptotic behaviour of μn on $UCB_{l}(G)$
Original languageEnglish
Pages (from-to)127-145
Number of pages19
JournalStudia Mathematica
Volume2
Issue number114
StatePublished - 1995

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