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

与群和半群相关的 Banach 代数的顺应性性质和相关问题

• 批准号: RGPIN-2016-05987
• 负责人: Zhang, Yong
• 金额: $ 1.09万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=709839
• 关键词: Amenability; properties; related; problems; Banach

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.

拟议的研究涉及半拓扑半群、拓扑群以及与之相关的巴纳赫代数。我们研究这些对象的不同顺应性属性，并研究巴拿赫空间或局部凸空间子集上的各种群/半群行为。 群体的顺从性理论可以追溯到 1920 年代，当时 J. von Neumann 研究了 Banach-Taski 悖论，并提出了一个普遍问题：对于作用于某些集合的群体是否存在不变测度。 M. M. Day 在 1950 年代奠定了半群理论的基础。从那时起，群和半群的顺应性理论与巴拿赫代数理论进行了富有成效的相互作用，在群/半群的结构以及相关空间/代数的性质方面产生了许多美丽而深刻的结果。 是。约翰逊发现了群的服从性与相应群代数的上同调性质之间的关系。随后，他于 1970 年代建立了 Banach 代数的顺应性理论。经过他的开创性工作，Banach 代数的弱服从性、算子服从性、弱算子服从性和广义服从性已被建立并得到广泛研究。这些便利性（对于巴纳赫代数）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开放问题，涉及与群或半群相关的各种巴拿赫代数。我们将重点关注加权群代数、加权半群代数和 F 代数来研究这些便利性。 与群和半群相关的巴拿赫代数主题与巴拿赫子集或更一般的局部凸拓扑空间上的群/半群作用理论密切相关。一组局部凸空间上存在多种类型的群/半群作用。其中仿射作用和非扩张作用对于许多分析领域极其重要。研究这些行为为更好地理解群体/半群体行为的空间提供了关键。我们将集中研究两种类型集合上的仿射或非扩张半群作用的不动点性质：（1）Banach 或对偶 Banach 空间的弱或弱*紧集，以及（2）严格凸 Banach 空间的子集或希尔伯特空间。 除了对巴拿赫代数、调和分析和不动点理论的预期理论贡献外，该研究还将在动态系统、遍历理论和逼近理论中得到应用。该计划为博士和硕士水平的研究生提供了选择论文主题的绝佳机会。它也适合希望进行重大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，并在该项目的实施过程中提供博士后职位。