# Averages of unitary representations and weak mixing of random walks

Studia Mathematica (1995)

- Volume: 114, Issue: 2, page 127-145
- ISSN: 0039-3223

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topLin, Michael, and Wittmann, Rainer. "Averages of unitary representations and weak mixing of random walks." Studia Mathematica 114.2 (1995): 127-145. <http://eudml.org/doc/216184>.

@article{Lin1995,

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; $U^n$ 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 $U^n$ converges strongly for every unitary representation, then the random walk is weakly mixing: $n^\{-1\} ∑_\{k=1\}^n |⟨μ^\{k\}*f,g⟩| → 0$ 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 $μ^n$ on $UCB_\{l\}(G)$},

author = {Lin, Michael, Wittmann, Rainer},

journal = {Studia Mathematica},

keywords = {locally compact -compact group; almost periodicity; ergodicity; unitary representations; convolution operators; metrizable groups; nilpotent groups},

language = {eng},

number = {2},

pages = {127-145},

title = {Averages of unitary representations and weak mixing of random walks},

url = {http://eudml.org/doc/216184},

volume = {114},

year = {1995},

}

TY - JOUR

AU - Lin, Michael

AU - Wittmann, Rainer

TI - Averages of unitary representations and weak mixing of random walks

JO - Studia Mathematica

PY - 1995

VL - 114

IS - 2

SP - 127

EP - 145

AB - 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; $U^n$ 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 $U^n$ converges strongly for every unitary representation, then the random walk is weakly mixing: $n^{-1} ∑_{k=1}^n |⟨μ^{k}*f,g⟩| → 0$ 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 $μ^n$ on $UCB_{l}(G)$

LA - eng

KW - locally compact -compact group; almost periodicity; ergodicity; unitary representations; convolution operators; metrizable groups; nilpotent groups

UR - http://eudml.org/doc/216184

ER -

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