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Strongly amenable semigroups and nonlinear fixed point properties

  • Author / Creator
    Bouffard, Nicolas
  • Left amenability, in it's modern form, was introduced by M. M. Day, in the 1940s. Amenability of groups and semigroups turned out to be quite common, and many interesting results are known, which motivated the introduction of extreme left amenability by Granirer in the 1960s. Extreme amenability turn out to be equivalent to a very strong nonlinear fixed point property, but examples of topological groups having this property are rather hard to construct. The purpose of this thesis is to study an intermediate property that we call strong left amenability. If S is a semi-topological semigroup, and A denotes either AP(S), WAP(S) or LUC(S) (the spaces of almost periodic, weakly almost periodic or left uniformly continuous functions on S respectively), then we say that A is strongly left amenable (SLA) if there is a compact left ideal group in the spectrum of A. We then say that S is SLA if LUC(S) is SLA. The first part of the thesis investigates the structure of such semigroups. We give some elementary properties, and characterize those semigroups for AP(S), WAP(S) and LUC(S). We also characterize the strong left amenability of a semigroup when S is discrete, compact or connected. Finally, we show that homomorphic images of an SLA semigroup is SLA and so is the product of an extremely left amenable semigroup by a compact group. We conclude the first part of the thesis by giving some examples. Amenability in general is closely related to non linear fixed point properties, and strong amenability is no exception. In the second part of this thesis, we characterize strong amenability in terms of a fixed compact set. We then obtain various fixed point properties related to jointly continuous actions and non-expansive mappings. We then extend some results on ultimately non-expansive mappings, a concept introduced by Kiang and Edelstein, to right reversible semigroups, and show that one of the conditions is always satisfied when the semigroup is indeed strongly amenable.

  • Subjects / Keywords
  • Graduation date
    2011-11
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/R3Q39T
  • 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
  • Supervisor / co-supervisor and their department(s)
    • Lau, Anthony To-Ming (Mathematical and statistical sciences)
  • Examining committee members and their departments
    • Poliquin, René (Mathematical and statistical sciences)
    • Troitsky, Vladimir (Mathematical and statistical sciences)
    • Sims, Brailey (School of mathematical and physical sciences, University of Newcastle, Australia)
    • Al-Hussein, Mohamed (Civil and Environmental engineering)
    • Dai, Feng (Mathematical and statistical sciences)