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.

拟议的研究涉及与它们相关的半决赛半群，拓扑组和BANACH代数。我们研究了这些物体的不同舒适性特性，并研究了在Banach空间或局部凸出空间的子集上的各种组/半群。 当J. von Neumann调查了Banach-Taski悖论并提出了一个普遍的问题，即是否存在对某些集合作用的群体的衡量标准，这可能会追溯到1920年代，这可能会追溯到1920年代。 M. M. Day奠定了1950年代的分数理论的基础。从那时起，群体和半群的理论与Banach代数理论有效地相互作用，从而产生了许多关于群体/半群的结构以及相关空间/代数的特性的许多美丽而深刻的结果。 是。约翰逊发现了一个群体的合理性与相应组代数的共同学性质之间的关系。然后，他在1970年代建立了Banach代数的理论理论。在他的开拓性工作，弱势症，操作员的舒适性，弱操作员的舒适性和对Banach代数的普遍性的舒适性之后，已建立并进行了广泛的研究。这些不适应（对于Banach代数）如何反映相关群体和半群的特性是该地区的一个深刻问题。以这个问题为中心，有一系列开放问题列表涉及与群体或半群相关的各种Banach代数。我们将专注于加权组代数，加权半群代数和F-Elgebras，以研究这些不适。 与群体和半群相关的BANACH代数的主题与对Banach或更一般而言的本地凸出拓扑空间的群体/半集团行动的理论密切相关。在一组本地凸空间中，有多种类型的组/半群动作。其中，对于许多分析领域，仿射行动和非庞大的行动至关重要。研究这些动作为更好地理解小组/半群制作的空间提供了关键。我们将集中精力针对两种类型集的仿期或非表达半群的固定点特性：（1）Banach或双Banach空间的弱或弱*紧凑的集合，以及（2）严格凸出Banach空间或希尔伯特空间的子集。 除了对BANACH代数的预期理论贡献，谐波分析和固定点理论外，该研究还将在动态系统，千古理论和近似理论中应用。该计划为博士学位和硕士学位的研究生提供了一个很好的机会，可以为论文选择主题。它也适合希望进行大量研究的博士后研究员。我们计划根据该计划培训一些研究生，我们将在执行该计划时提供博士后职位。