Uniformly ergodic probability measures

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Jorge Galindo Pastor
Universitat Jaume I. Departament de Matemàtiques
Enrique Jordá
Universitat Politècnica de València
https://orcid.org/0000-0003-2980-1699
Alberto Rodríguez-Arenas
Universitat Jaume I. Departament de Matemàtiques
https://orcid.org/0000-0002-4571-2031

Let G be a locally compact group and µ be a probability measure on G. We consider the convolution operator λ1(µ): L1(G) → L1(G) given by λ1(µ)f = µ∗f and its restriction λ 0 1 (µ) to the augmentation ideal L0 1 (G). Say that µ is uniformly ergodic if the Ces`aro means of the operator λ 0 1 (µ) converge uniformly to 0, that is, if λ 0 1 (µ) is a uniformly mean ergodic operator with limit 0, and that µ is uniformly completely mixing if the powers of the operator λ 0 1 (µ) converge uniformly to 0. We completely characterize the uniform mean ergodicity of the operator λ1(µ) and the uniform convergence of its powers, and see that there is no difference between λ1(µ) and λ 0 1 (µ) in these regards. We prove in particular that µ is uniformly ergodic if and only if G is compact, µ is adapted (its support is not contained in a proper closed subgroup of G), and 1 is an isolated point of the spectrum of µ. The last of these three conditions can actually be replaced by µ being spread out (some convolution power of µ is not singular). The measure µ is uniformly completely mixing if and only if G is compact, µ is spread out, and the only unimodular value in the spectrum of µ is 1.

Paraules clau
Ergodic measure, Uniformly ergodic measure, Random walk, Mean ergodic operator, Uniformly mean ergodic operator, Convolution operator, Locally compact group, Measure algebra

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Galindo Pastor, Jorge et al. “Uniformly ergodic probability measures”. Publicacions Matemàtiques, vol.VOL 68, no. 2, pp. 593-1, https://raco.cat/index.php/PublicacionsMatematiques/article/view/430130.
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