Uniformly ergodic probability measures
[EN] Let G be a locally compact group and mu be a probability measure on G. We consider the augmentation ideal L01(G). Say that mu is uniformly ergodic if the Ces & aacute;ro means of the operator A01(mu) converge uniformly to 0, that is, if A01(mu) is a uniformly mean ergodic operator with...
| Autores: | , , |
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| Formato: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Recursos: | Universitat Politècnica de València (UPV) |
| Repositorio: | RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia |
| Idioma: | inglés |
| OAI Identifier: | oai:riunet.upv.es:10251/221978 |
| Acesso em linha: | https://riunet.upv.es/handle/10251/221978 |
| Access Level: | acceso abierto |
| Palavra-chave: | Ergodic measure Uniformly ergodic measure Random walk Mean ergodic operator Uniformly mean ergodic operator Convolution operator Locally compact group Measure algebra |
| Resumo: | [EN] Let G be a locally compact group and mu be a probability measure on G. We consider the augmentation ideal L01(G). Say that mu is uniformly ergodic if the Ces & aacute;ro means of the operator A01(mu) converge uniformly to 0, that is, if A01(mu) is a uniformly mean ergodic operator with limit 0, and that mu is uniformly completely mixing if the powers of the operator A01(mu) converge uniformly to 0. We completely characterize the uniform mean ergodicity of the operator A1(mu) and the uniform convergence of its powers, and see that there is no difference between A1(mu) and A01(mu) in these regards. We prove in particular that mu is uniformly ergodic if and only if G is compact, mu is adapted (its support is not contained in a proper closed subgroup of G), and 1 is an isolated point of the spectrum of mu. The last of these three conditions can actually be replaced by mu being spread out (some convolution power of mu is not singular). The measure mu is uniformly completely mixing if and only if G is compact, mu is spread out, and the only unimodular value in the spectrum of mu is 1. |
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