A note on a wiener-wintner theorem for mean ergodic markov amenable semigroups

We prove a Wiener-Wintner ergodic type theorem for a Markov representation S = {S_g: g ∈ G} of a right amenable semitopological semigroup G. We assume that S is mean ergodic as a semigroup of linear Markov operators acting on (C(K),·_sup), where K is a fixed Hausdorff, compact space. The main result of the paper is necessary and sufficient conditions for mean ergodicity of a distorted semigroup {χ(g)S_g: g ∈ G}, where χ is a semigroup character. Such conditions were obtained before under the additional assumption that S is uniquely ergodic.