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Strongly amenable semigroups and nonlinear fixed point properties Open Access


Other title
fixed point
Type of item
Degree grantor
University of Alberta
Author or creator
Bouffard, Nicolas
Supervisor and department
Lau, Anthony To-Ming (Mathematical and statistical sciences)
Examining committee member and department
Al-Hussein, Mohamed (Civil and Environmental engineering)
Dai, Feng (Mathematical and statistical sciences)
Poliquin, René (Mathematical and statistical sciences)
Troitsky, Vladimir (Mathematical and statistical sciences)
Sims, Brailey (School of mathematical and physical sciences, University of Newcastle, Australia)
Department of Mathematical and Statistical Sciences

Date accepted
Graduation date
Doctor of Philosophy
Degree level
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.
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