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Amenability and fixed point properties of semitopological semigroups of nonexpansive mappings in Banach spaces

 Author / Creator
 Salame, Khadime

In this thesis we are interested in fixed point properties of representations of semitopological semigroups of nonexpansive mappings on weak and weak* compact convex sets in Banach or dual spaces. More particularly, we study the following problems : Problem 1 : Let F be any commuting family of nonexpansive mappings on a nonempty weakly compact convex subset of a Banach space such that for each f in F there is an x whose forbit has a cluster point (in the norm topology). Does F possess a common fixed point ? Problem 2 : What amenability properties of a semitopological semigroup do ensure the existence of a common fixed point for any jointly weakly continuous nonexpansive representation on a nonempty weakly compact convex subset of a Banach space ? Problem 3 : Does any left amenable semitopological semigroup S possess the following fixed point property : (F*) : Whenever S defines a weak* jointly continuous nonexpansive representation on a nonvoid weak* compact convex set in the dual of a Banach space E, there is a common fixed point for S ? Problem 4 : Is there a fixed point proof of the existence of a left Haar measure for locally compact groups ? Our approach is essentially based on the use of the axiom of choice through Zorn's lemma, amenability techniques and the concept of an asymptotic center in geometry of Banach spaces. Some positive answers are obtained for problem 1; however, problem 2 is settled affirmatively for three classes of semitopological semigroups. These classes of semigroups together with left amenable and all left reversible semitopological semigroups possess a fixed point property which is a weak version of (F*). We show that nextremely left amenable discrete semigroups satisfy a fixed point property much more stronger than (F*); whereas, nextremely left amenable semitopological semigroups possess the fixed point property (F*). Among other things, results in Browder [10], Belluce and Kirk [3,4], and Kirk [37] are generalized to noncommutative families. A result of Hsu [32, theorem 4] is extended to semitopological semigroups. Furthermore, some results related to the work of Lim (cf. [47],[49]) are obtained. A positive answer to question 4 is established for amenable locally compact groups.

 Subjects / Keywords

 Graduation date
 201606

 Type of Item
 Thesis

 Degree
 Doctor of Philosophy

 License
 This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for noncommercial 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.