Amenability and fixed point properties of semi-topological semigroups of non-expansive mappings in Banach spaces

• Author / Creator

  Salame, Khadime

• In this thesis we are interested in fixed point properties of representations of semi-topological semigroups of non-expansive 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 non-expansive mappings on a non-empty weakly compact convex subset of a Banach space such that for each f in F there is an x whose f-orbit has a cluster point (in the norm topology). Does F possess a common fixed point ? Problem 2 : What amenability properties of a semi-topological semigroup do ensure the existence of a common fixed point for any jointly weakly continuous non-expansive representation on a non-empty weakly compact convex subset of a Banach space ? Problem 3 : Does any left amenable semi-topological semigroup S possess the following fixed point property : (F) : Whenever S defines a weak jointly continuous non-expansive representation on a non-void 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 semi-topological semigroups. These classes of semigroups together with left amenable and all left reversible semi-topological semigroups possess a fixed point property which is a weak version of (F). We show that n-extremely left amenable discrete semigroups satisfy a fixed point property much more stronger than (F); whereas, n-extremely left amenable semi-topological 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 non-commutative families. A result of Hsu [32, theorem 4] is extended to semi-topological 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

  • Amenability
  • Fixed point properties
  • Non-expansive mappings

• Graduation date

  Fall 2016

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R35T3G736

• 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.