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

拟研究的内容涉及拓扑半群、拓扑群以及与之相关的Banach代数。我们调查这些对象的不同顺从性的性质，并研究了各种群/半群的Banach空间或局部凸空间的子集上的行动。 群的顺从性理论可以追溯到20世纪20年代，当时J·冯·诺依曼研究了巴拿赫-塔斯基悖论，并提出了一个一般性的问题，即是否存在作用于某些集合上的群的不变测度。M. M.戴在20世纪50年代奠定了半群理论的基础。从那时起，群和半群的顺从性理论与Banach代数理论产生了富有成效的相互作用，产生了许多关于群/半群结构和相关空间/代数性质的美丽而深刻的结果。 B.E.约翰逊发现了群的顺从性与相应群代数的上同调性之间的关系。70年代，他建立了Banach代数的顺从性理论。在他的开创性工作之后，Banach代数的弱顺从性、算子顺从性、弱算子顺从性和广义顺从性得到了广泛的研究。Banach代数的这些顺从性如何反映相关群和半群的性质是该领域的一个深刻问题。围绕这个问题有一个开放的问题，涉及到各种Banach代数群或半群。我们将集中在加权群代数，加权半群代数和F-代数来研究这些顺应性。 主题的Banach代数相关的群体和半群是密切相关的理论组/半群行动的子集的Banach或更普遍的是，局部凸拓扑空间。在局部凸空间的集合上有各种类型的群/半群作用。其中，仿射作用和非扩张作用在许多分析领域都具有极其重要的意义。研究这些作用为更好地理解群/半群作用的空间提供了关键。我们将集中讨论仿射或非扩张半群作用在两类集合上的不动点性质：（1）Banach或对偶Banach空间的弱或弱 * 紧集，以及（2）严格凸Banach空间或Hilbert空间的子集。 除了预期的理论贡献，Banach代数，调和分析和不动点理论，研究将在动力系统，遍历理论和逼近理论的应用。该计划为博士和硕士水平的研究生提供了一个很好的机会来选择论文主题。它也适合希望做重要研究的博士后研究员。我们计划在该项目下培养一些研究生，我们将提供博士后职位来实施该项目。

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.