Amenability properties of semitopological semigroups and related Banach algebras

我和我的团队建议研究半拓扑半群的顺从性，重点是它们如何确定半群的公共不动点性质以及它们如何决定相关的Banach代数的结构。这是我们长期目标的组成部分。对于后者，我们的目标是将拓扑群/半群理论和Banach代数理论的思想/方法结合起来。我们致力于揭示这两个领域之间的深刻联系。群的顺从性理论可以追溯到20世纪20年代，当时J.von Neumann在研究Banach-Tarski悖论时提出了这样一个问题：对于作用在特定集合上的群，是否存在不变度量。M.M.戴在20世纪50年代奠定了这一理论的基础。在过去的70年里，调和分析和Banach代数理论的研究取得了丰硕的成果，在这个领域产生了许多美丽而深刻的结果。利用S上连续函数子空间上左不变平均的存在性，证明了半拓扑半群S的顺从性.当S作用在一个拓扑空间上时，它的顺从性决定了作用的许多特征.有各种类型的半群作用。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。S在Banach空间的弱紧或弱*紧集合上的这类作用引起了特别的兴趣，其中有几个长期悬而未决的问题。我们将重点研究S在这种作用下的公共不动点性质，研究这些性质与S的顺从性之间的内在联系。在B.E.Johnson于20世纪70年代的一项开创性工作之后，顺从性理论已经扩展到Banach代数领域，为探索Banach代数的结构提供了灵感。另一方面，对于Banach代数来说，顺从性是一个非常严格的条件。近年来，已有文献介绍了Banach代数的广义/弱形式的顺从性。关于这些问题，许多关键问题有待深入调查。在这个方案中，我们将集中讨论与群和半群有关的Banach代数的问题。特别地，我们将研究半群的加权群和测度代数及其中心代数，以及加权测度代数。我们将进一步研究F-代数类，其中包括调和分析产生的许多重要的Banach代数。所提出的研究将有助于发展Banach代数理论、调和分析和一般泛函分析。这一结果将在动力系统、遍历理论和近似理论中有可预见的应用。参与研究，学员将获得泛函分析的基本知识和技能，这将极大地帮助他们未来的研究相关职业。

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.