Amenability properties of semitopological semigroups and related Banach algebras

半拓扑半群和相关巴纳赫代数的顺应性性质

• 批准号: RGPIN-2022-04137
• 负责人: Zhang, Yong
• 金额: $ 1.31万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758608
• 关键词: Amenability; properties; semitopological; semigroups; related

My team and I propose to investigate amenability properties of semi-topological semigroups, centering on how they determine the common fixed point property of the semigroup and how they decide the structure of a related Banach algebra. This is a constitutive component of our long-term objective. For the latter, we aim to integrate ideas/methods in developing the topological group/semigroup theory and the Banach algebra theory. We engage in revealing deep links between the two areas. Amenability theory for groups may trace back to the 1920s when J. von Neumann, studying the Banach-Tarski paradox, raised the 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 in the 1950s. During the past 70 years, the investigation has been interacting fruitfully with harmonic analysis and Banach algebra theory, giving rise to many beautiful and profound results in the area. The amenability property of a semi-topological semigroup S is expounded in terms of the existence of a left-invariant mean on a subspace of continuous functions on S. When S acts on a topological space, its amenability property determines many features of the action. There are various types of semigroup actions. Among them, affine actions and non-expansive actions are of extreme importance to many analysis areas. These types of actions of S on a weakly compact or weak* compact set of a Banach space are of particular interest, with several long-standing open problems in the background. We will focus on the common fixed point properties of S for such actions, investigating intrinsic relations linking these to the amenability properties of S. After a pioneering work of B.E. Johnson in the 1970s, amenability theory has been extended to the area of Banach algebras, providing inspiring ideas to probe the structure of a Banach algebra. On the other hand, amenability is a very restrictive condition for a Banach algebra. In recent years, generalized/weak versions of amenability for Banach algebras have been introduced in the literature. Regarding them, many crucial problems are pending an intensive investigation. We will concentrate on the problems concerning Banach algebras related to groups and semigroups in this proposal. In particular, we will investigate the weighted group and measure algebras and their center algebras, and weighted measure algebras of semigroups. We will further investigate the class of F-algebras, which includes many important Banach algebras arising from the harmonic analysis. The proposed research will contribute to developing the theory of Banach algebras, harmonic analysis, and general functional analysis. The outcome will have foreseeable applications in dynamic systems, ergodic theory, and approximation theory. Participating in the research, trainees will gain fundamental knowledge and skills in functional analysis that will greatly benefit their future research-related careers.

我和我的团队提议研究半主流半群的舒适性特性，以他们如何确定半群的共同固定点特性以及他们如何确定相关Banach代数的结构。这是我们长期目标的组成部分。对于后者，我们旨在将思想/方法整合到发展拓扑组/半群理论和Banach代数理论中。我们参与了这两个领域之间的深层联系。群体的不及性理论可以追溯到1920年代，当时研究Banach-Tarski悖论的J. von Neumann提出了一个问题，即是否存在对某些集合作用的群体的不变度量。毫米。一天奠定了1950年代理论的基础。在过去的70年中，该调查一直在与谐波分析和Banach代数理论中进行有效的互动，从而在该地区带来了许多美丽而深刻的结果。半流行病学的符合性特性是根据存在于连续函数的子空间上的剩余平均值的存在来阐述的。有多种类型的半群动作。其中，对许多分析领域的仿射行动和非表达行为至关重要。 S在弱紧凑或弱*紧凑的Banach空间集中的这些类型的动作特别感兴趣，背景中有几个长期存在的开放问题。我们将重点介绍S用于此类作用的常见固定点特性，并研究将这些固有关系与S.的固有关系联系起来。约翰逊（Johnson）在1970年代，舒适性理论已扩展到巴纳克代数的地区，提供了鼓舞人心的想法来探究巴纳克代数的结构。另一方面，对BANACH代数是非常限制的条件。近年来，文献中引入了Banach代数的普遍/弱版本。关于他们，许多关键问题正在进行深入的调查。我们将集中精力于本提案中与群体和半群有关的Banach代数的问题。特别是，我们将研究加权组并测量代数及其中心代数，并加权测量半群的代数。我们将进一步研究F-Elgebras的类别，其中包括许多由谐波分析引起的重要BANACH代数。拟议的研究将有助于发展Banach代数，谐波分析和一般功能分析的理论。该结果将在动态系统，千古理论和近似理论中具有可预见的应用。参与研究，学员将获得基本知识和技能，以极大地使他们未来的研究相关职业受益。