Amenability properties of semitopological semigroups and related Banach algebras

我的团队和我提议研究半拓扑半群的适应性性质，集中在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巴拿赫代数的结构。这是我们长期目标的一个组成部分。对于后者，我们的目标是整合发展拓扑群/半群理论和Banach代数理论的思想/方法。我们致力于揭示这两个领域之间的深层联系。群体的适应性理论可以追溯到20世纪20年代，当时冯·诺伊曼（J. von Neumann）在研究巴拿赫-塔斯基悖论时，提出了一个问题，即是否存在一个作用于特定集合的群体的不变测度。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，这一研究与调和分析和巴拿赫代数理论有着丰硕的成果，在这一领域产生了许多美丽而深刻的成果。利用S上连续函数的子空间上左不变均值的存在性，阐述了半拓扑半群S的易受性。当S作用于拓扑空间时，其易受性决定了该作用的许多特征。有各种类型的半群动作。其中，仿射作用和非扩张性作用在许多分析领域都是极为重要的。S在Banach空间的弱紧或弱*紧集合上的这些类型的作用是特别有趣的，在背景中有几个长期存在的开放问题。我们将重点讨论S对于这些行为的公共不动点性质，并研究这些性质与S的易受性之间的内在联系。在20世纪70年代B.E. Johnson的开创性工作之后，易受性理论被扩展到Banach代数领域，为探索Banach代数的结构提供了鼓舞人心的思想。另一方面，适性是Banach代数的一个非常严格的条件。近年来，在文献中引入了Banach代数适应性的广义/弱版本。关于这些问题，许多关键问题有待深入调查。本文将集中讨论与群和半群有关的巴拿赫代数问题。特别地，我们将研究权群和权测度代数及其中心代数，以及半群的权测度代数。我们将进一步研究一类f -代数，其中包括许多由调和分析产生的重要的巴拿赫代数。本研究将有助于巴拿赫代数理论、调和分析和泛函分析的发展。结果将在动态系统、遍历理论和近似理论中有可预见的应用。参与研究，学员将获得功能分析的基本知识和技能，这将大大有利于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.