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.

我的研究涉及数学的一部分中的思想和问题，这被称为功能和谐波分析。我和我的同事在代表理论的代表理论中开发了几个新想法，并将其应用于潜水员问题，例如第二对偶，拓扑中心，性格舒适性和拓扑内向的空间的理想结构。 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*，Filali，Neufang的拓扑内向的子空间X，我介绍了下属表示的概念，并在X的a的表示形式中建立了一对一的对应关系。这种现象以前仅在特殊情况下才知道。我们的贡献建立了一种一般模式，可以应用于许多不同的情况。我们打算表征从属于各种内向空间X的A的那些表示。这将有助于借助表示的坐标函数，以更好地了解此类空间。 （2）舒适性和相对边界：对于统一的代数，各种类型的边界已经在文献中进行了广泛的研究。但是，直到最近才引入了它们内部的相对边界和特殊点（强边界点，峰值点等）。最近，我已经能够将这些想法的某些应用显示在特征理论的Banach代数。相对边界的性质，它们与内向空间和协调功能的关系是等待进一步研究的自然问题。 （3）拓扑中心：Filali，Neufang和我开发了一种技术来获取von Neumann代数组中运营商的因素。使用此技术，我们能够确定von Neumann代数的拓扑中心以及某些组的均匀连续功能的空间。这些想法是非常承诺的，我们打算进一步开发用于更广泛的空间及其引号的技术，例如p-pseudo测量的空间及其报价。 这项研究将吸引一群从事现代分析的数学家。它将为研究生的硕士和博士学位论文提供项目。这将使现代数学研究高度活跃的研究领域有益。