## 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 language | English |
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Pages (from-to) | 125-157 |

Number of pages | 33 |

Journal | Israel Journal of Mathematics |

Volume | 88 |

Issue number | 1-3 |

DOIs | |

State | Published - 1 Oct 1994 |

## ASJC Scopus subject areas

- Mathematics (all)