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Ergodic theorems for certain Banach algebras associated to locally compact groups

  • Author / Creator
    Guex, Sébastien M.
  • 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.

  • Subjects / Keywords
  • Graduation date
    2013-06
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/R35G7W
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
  • Language
    English
  • Institution
    University of Alberta
  • Degree level
    Doctoral
  • Department
    • Department of Mathematical and Statistical Sciences
  • Specialization
    • Mathematics
  • Supervisor / co-supervisor and their department(s)
    • Lau, Anthony To-Ming (Mathematical and Statistical Sciences)
  • Examining committee members and their departments
    • Forrest, Brian (Pure Mathematics, Univeristy of Waterloo)
    • Runde, Volker (Mathematical and Statistical Sciences)
    • Troitsky, Vladimir (Mathematical and Statistical Sciences)
    • Litvak, Alexander (Mathematical and Statistical Sciences)
    • Sit, Jeremy (Electrical and Computer Engineering)