The Role of Representation Theory in Problems of Harmonic Analysis Related to Amenability and Locally Compact Groups

表示论在与顺应性和局部紧群相关的调和分析问题中的作用

• 批准号: RGPIN-2015-04007
• 负责人: SanganiMonfared, Mehdi
• 金额: $ 0.8万
• 依托单位: University of Windsor
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612713
• 关键词: Role; Representation; Theory; Problems; Harmonic

My research involves ideas and problems from a part of mathematics, which is known as functional and harmonic analysis. My colleagues and I have developed several new ideas in representation theory of Banach algebras and applied them in diverse questions such as ideal structure of the second duals, topological centers, character amenability, and topologically introverted spaces. The following is summary of my research program: (1) Introverted spaces: Given a Banach algebra A and a topologically introverted subspace X of A*, Filali, Neufang, and I introduced the concept of subordinate representations and established a one-to-one correspondence between representations of A which are subordinate to X, and, the representations of the dual space X*. This phenomenon was previously only known in special cases. Our contribution established a general pattern, which can be applied to many different cases. We intend to characterize those representations of A that are subordinate to various introverted spaces X. This will help to obtain a better understanding of such spaces with the help of the coordinate functions of representations. (2) Amenability and relative boundaries: For uniform algebras, various types of boundaries have been subject of extensive studies in the literature. However, it is only very recently that relative boundaries and special points within them (strong boundary points, peak points, etc) have been introduced for normed spaces. Recently, I have been able to show some applications of these ideas to the theory of character amenable Banach algebras. The nature of relative boundaries, their relation with introverted spaces, and coordinate functions are natural questions awaiting further studies. (3) Topological centers: Filali, Neufang and I have developed a technique to obtain factorizations of operators in the group von Neumann algebras. Using this technique, we have been able to determine the topological centers of the group von Neumann algebras and the space of uniformly continuous functionals for certain groups. These ideas are very promising, and we intend to further develop the technique for a much wider range of spaces and their quotients, such as the space of p-pseudo measures and their quotients. This research will be of interest to a wide group of mathematicians working in modern analysis. It will provide projects for graduate student’s master and PhD thesis. And it will benefit the advancement of a highly active area of research in modern mathematics.

我的研究涉及数学部分的思想和问题，即泛函分析和调和分析。我和我的同事在巴拿赫代数表示论中提出了一些新思想，并将它们应用于各种问题，例如二阶对偶的理想结构、拓扑中心、特征顺应性和拓扑内向空间。 以下是我的研究计划的摘要： (1) 内向空间：给定一个 Banach 代数 A 和 A* 的拓扑内向子空间 X，Filali、Neufang 和 I 引入了从属表示的概念，并建立了从属于 X 的 A 的表示与对偶空间 X* 的表示之间的一一对应关系。这种现象以前只在特殊情况下才知道。我们的贡献建立了一个通用模式，可以应用于许多不同的情况。我们打算表征那些从属于各种内向空间 X 的 A 表示。这将有助于借助表示的坐标函数更好地理解这些空间。 (2) 适应性和相对边界：对于一致代数，各种类型的边界已成为文献中广泛研究的主题。然而，直到最近，相对边界和其中的特殊点（强边界点、峰值点等）才被引入规范空间。最近，我已经能够展示这些思想在特征巴拿赫代数理论中的一些应用。相对边界的性质、它们与内向空间的关系以及坐标函数是有待进一步研究的自然问题。 (3) 拓扑中心：Filali、Neufang 和我开发了一种获得冯诺依曼代数群中算子因式分解的技术。使用这种技术，我们已经能够确定冯诺依曼代数群的拓扑中心以及某些群的一致连续泛函空间。这些想法非常有前途，我们打算进一步开发适用于更广泛的空间及其商的技术，例如 p-伪测度的空间及其商。 这项研究将引起广大从事现代分析工作的数学家的兴趣。它将为研究生的硕士和博士论文提供项目。它将有利于现代数学这一高度活跃的研究领域的进步。