Amenability properties and related problems of Banach algebras associated to groups and semigroups

点击翻译按钮获取中文摘要

The proposed research deals with semitopological semigroups, topological groups and Banach algebras associated to them. We investigate different amenability properties of these objects and study various group/semigroup actions on subsets of a Banach space or a locally convex space. Amenability theory for groups may trace back to 1920's when J. von Neumann investigated the Banach-Taski paradox and raised the general question of whether there is an invariant measure for a group acting on certain sets. M. M. Day laid down the foundation of the theory for semigroups in 1950's. Since then the amenability theory for groups and semigroups has interacted fruitfully with Banach algebra theory, giving rise to many beautiful and deep results regarding the structure of groups/semigroups and the property of related spaces/algebras. B.E. Johnson discovered the relation between amenability of a group and the cohomology property of the corresponding group algebra. He then established the amenability theory for Banach algebras in 1970's. After his pioneer work, weak amenability, operator amenability, weak operator amenability and generalized amenability for Banach algebras have been established and extensively investigated. How these amenabilities (for Banach algebras) reflect properties of related groups and semigroups is a profound question in the area. Centered around this question there is a list of open problems that involve various Banach algebras associated to groups or semigroups. We will focus on weighted group algebras, weighted semigroups algebras and F-algebras to investigate these amenabilities. The topics on Banach algebras associated to groups and semigroups are closely related to the theory of group/semigroup actions on subsets of Banach or, more generally, locally convex topological spaces. There are variety types of group/semigroup actions on a set of a locally convex space. Among them affine actions and non-expansive actions are of extreme importance to many analysis areas. Studying these actions provides keys to better understanding of the spaces on which the groups/semigroups act. We will concentrate on fixed point properties for affine or non-expansive semigroup actions on two types of sets: (1) weakly or weak* compact sets of a Banach or a dual Banach space, and (2) subsets of a strictly convex Banach space or a Hilbert space. In addition to the expected theoretical contributions to Banach algebra, harmonic analysis and fixed point theories, the research will have applications in dynamic systems, ergodic theory and approximation theory. The program provides a great opportunity for graduate students at both PhD and Master's levels to choose topics for their thesis. It is also suitable for a postdoctoral fellow who wishes to do significant research. We plan to train a few graduate students under the program, and we will provide postdoctoral positions in carrying out the program.