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Ergodic theorems for certain Banach algebras associated to locally compact groups Open Access


Other title
locally compact group
Figà-Talamanca-Herz algebra
norm spectrum
ergodic multiplier
ergodic sequence
Type of item
Degree grantor
University of Alberta
Author or creator
Guex, Sébastien M.
Supervisor and department
Lau, Anthony To-Ming (Mathematical and Statistical Sciences)
Examining committee member and department
Troitsky, Vladimir (Mathematical and Statistical Sciences)
Litvak, Alexander (Mathematical and Statistical Sciences)
Runde, Volker (Mathematical and Statistical Sciences)
Sit, Jeremy (Electrical and Computer Engineering)
Forrest, Brian (Pure Mathematics, Univeristy of Waterloo)
Department of Mathematical and Statistical Sciences
Date accepted
Graduation date
Doctor of Philosophy
Degree level
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.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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