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

所提出的研究内容涉及半拓扑半群、拓扑群和与之相关的Banach代数。我们研究了这些对象的不同顺从性，并研究了Banach空间或局部凸空间的子集上的各种群/半群作用。群的可释性理论可以追溯到1920年的S，当时J.von Neumann研究了Banach-TASKI悖论，并提出了群在某些集合上是否存在不变测度的一般问题。S在1950年奠定了半群理论的基础，此后群和半群的顺从性理论与Banach代数理论进行了卓有成效的互动，在群/半群的结构和相关空间/代数的性质方面产生了许多美丽而深刻的结果。*B.E.Johnson发现了群的顺从性与相应群代数的上同调性质之间的关系。S在1970年建立了Banach代数的顺从性理论，经过他的开创性工作，建立了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，并得到了广泛的研究。这些顺从性(对于Banach代数)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，有一系列公开问题，涉及到与群或半群有关的各种Banach代数。我们将集中于加权群代数，加权半群代数和F-代数来研究这些顺序性。*与群和半群相关的Banach代数的主题与Banach子集上的群/半群作用理论密切相关，或者更一般地，局部凸拓扑空间。在一组局部凸空间上存在各种类型的群/半群作用。其中，仿射作用和非扩张作用对许多分析领域都具有极其重要的意义。研究这些行为有助于更好地理解群/半群所作用的空间。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空间或对偶Banach空间的弱或弱*紧集；(2)严格凸Banach空间或Hilbert空间的子集。*除了对Banach代数、调和分析和不动点理论的理论贡献外，该研究还将在动力系统、遍历理论和逼近理论中得到应用。该项目为博士和硕士研究生提供了一个很好的机会来为他们的论文选择主题。它也适合希望做有意义的研究的博士后研究员。我们计划在该计划下培养一些研究生，并将提供博士后职位来实施该计划。**

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.**