Convergence of representation averages and of convolution powers

Michael Lin, Rainer Wittmann

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

Let S be a locally compact (σ-compact) group or semi-group, 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=∫T(t)xdμ(t). Our main results for random walks on a group G are: (i) if μ is adapted and strictly aperiodic, and generates a recurrent random walk, then U n (U-I) converges strongly to 0. In particular, the random walk is completely mixing. (ii) If μ×μ is ergodic on G×G, then for every unitary representation T(.) in a Hilbert space, U n converges strongly to the orthogonal projection on the space of common fixed points. These results are proved for semigroup representations, along with some other results (previously known only for groups) which do not assume ergodicity. (iii) If μ is spread-out with support S, then {Mathematical expression} if and only if e {Mathematical expression}.

Original languageEnglish
Pages (from-to)125-157
Number of pages33
JournalIsrael Journal of Mathematics
Volume88
Issue number1-3
DOIs
StatePublished - 1 Oct 1994

ASJC Scopus subject areas

  • Mathematics (all)

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