Ergodic theorems for certain Banach algebras associated to locally compact groups

Abstract

In this thesis, we establish some ergodic theorems related to Ap(G), the Figà-Talamanca-Herz algebra of a locally compact group G. This thesis is divided in two main portions. The first part is primarily concerned with the study of ergodic sequences in Ap(G) and with a newly introduced notion of ergodic multipliers. After presenting a full description of the non-degenerate *-representations of Ap(G) and of their extensions to the multiplier algebra MAp(G), it is shown that, for all locally compact groups, the weakly ergodic sequences in MAp(G) coincide with the strongly ergodic ones, and that they are, in a sense, approximating sequences for the topologically invariant means on some spaces of linear functionals on Ap(G). Next, motivated by the study of ergodic sequences of iterates, we introduce a notion of ergodic multipliers, and we provide a solution to the dual version of the complete mixing problem for probability measures, The second part is of a more abstract nature and deals with some ergodic and fixed point properties of ϕ-amenable Banach algebras. Among other things, we prove a mean ergodic theorem, establish the uniqueness of a two-sided ϕ-mean on the weakly almost periodic functionals, and provide a simpler proof of a fixed point theorem which is well known in the context of semigroups. We also study the norm spectrum of some linear functionals on Ap(G) and present a new characterization of discrete groups.