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Title

Averages of unitary representations and weak mixing of random walks

Authors 1, 2

Affiliations

  1. Ben-Gurion University of the Negev, Beer-Sheva, Israel
  2. Institut für Mathematische Stochastik, Lotzestrasse 13, Göttingen, Germany.

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∑k=1n|⟨μk⋅f,g⟩|→0 for g∈L∞(G) and f∈L1(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 UCBl(G)

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Studia Mathematica
1995/114/2
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Pages:
127-145
Main language of publication
English
Received
1994-03-31
Accepted
1994-12-13
Published
1995
Exact and natural sciences