Topological Wiener-Wintner theorems for amenable operator semigroups
The paper unifies and extends some known Wiener-Wintner type theorems [\textit{E. A. Robinson jun.}, J. Lond. Math. Soc., II. Ser. 49, No. 3, 493--501 (1994; Zbl 0827.28010); \textit{P. Walters}, Ergodic Theory Dyn. Syst. 16, No. 1, 179--206 (1996; Zbl 0861.28013); \textit{S. I. Santos} and \textit{C. Walkden}, ibid. 27, No. 5, 1633--1650 (2007; Zbl 1124.37002); \textit{D. Lenz}, Contemp. Math. 485, 91--112 (2009; Zbl 1200.78008); \textit{D. Lenz}, Commun. Math. Phys. 287, No. 1, 225--258 (2009; Zbl 1178.37011)] to the case of amenable semigroups of Markov operators on the space \(C(K)\) where \(K\) is a compact topological space.NEWLINENEWLINEOne of the major results of the paper (Theorem 2.7) is a characterization of mean ergodicity for an amenable representation \(\{S_g : g\in G\}\) of a semitopological semigroup \(G\) as Markov operators, i.e., as positive operators satisfying the condition \(S_g\mathbf{1}=\mathbf{1}\), for all \(g\in G\). In proving this result the author builds upon his previous work [Semigroup Forum 86, No. 2, 321--336 (2013; Zbl 1273.43009)] where the mean ergodicity of a more general bounded (not necessarily Markov) amenable representations has been characterized. The latter result already has a very interesting corollary which the author points out in the current paper (Corollary 2.3.): a bounded amenable semigroup \(\mathcal{S} = \{S_g : g\in G\}\) of operators is uniquely ergodic if and only if it is mean ergodic and Fix \(\mathcal{S} = \mathbb{C} \cdot \mathbf{1}\).NEWLINENEWLINEIf \(\chi : G\to \mathbb{T}\) is a character of \(G\), one can consider a semigroup \(\chi \mathcal{S}\) formed out of \(\mathcal{S}\). The mean ergodicity question of this semigroup has been studied in the literature (e.g., in [loc. cit.]). In the second part of the paper, the author studies amenable representations \(\mathcal{S}\) of semitopological semigroups as Koopman operators (i.e., operators of type \(T:f\to f\circ \phi \) on \(C(K)\) where \(K\) is compact and \(\phi : K\to K\) is a continuous transformation). A similar characterization of mean ergodicity is obtained for the modified semigroup \(\gamma \mathcal{S}\) where (instead of a character) \(\gamma : G\times K\to U(N)\) is a continuous cocycle into the group of unitary operators on \(\mathbb{C}^N\).NEWLINENEWLINEIn the last part of the paper, the author characterizes the mean ergodicity of semigroups associated to skew product actions of compact group extensions. In particular, extending the result of Furstenberg (in the case of actions of the semigroup \(\mathbb{N}\)) to the actions of amenable semigroups, the author proves that an ergodic skew product action corresponding to a uniquely ergodic action is uniquely ergodic.
scientific article; zbMATH DE number 6366535
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