Ergodic theorems for certain Banach algebras associated to locally compact groups

Sébastien M. Guex
In this thesis, we establish some ergodic theorems related to A p (G), the Figà-Talamanca-Herz algebra of a locally compact group G. This thesis is divided in two main portions. The rst part is primarily concerned with the study of ergodic sequences in A p (G) and with a newly introduced notion of ergodic multipliers. After presenting a full description of the non-degenerate * -representations of A p (G) and of their extensions to the multiplier algebra MA p (G), it is shown that, for all
more » ... y compact groups, the weakly ergodic sequences in MA p (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 A p (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 xed 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 xed point theorem which is well known in the context of semigroups. We also study the norm spectrum of some linear functionals on A p (G) and present a new characterization of discrete groups.
doi:10.7939/r35g7w fatcat:3cznlllxmjdw5of4y5nzctu2li