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，这是众所周知的功能和谐波分析的想法和问题。我和我的同事们在Banach代数的表示论中提出了几个新的思想，并将它们应用于不同的问题，如第二类代数的理想结构、拓扑中心、特征顺从性和拓扑内倾空间。 以下是我的研究计划摘要： (1)内倾空间：给定一个Banach代数A和A* 的拓扑内倾子空间X，Filali，Neufang和我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。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一般模式，可以适用于许多不同的情况。我们打算刻画从属于各种内倾空间X的A的那些表示。这将有助于获得更好地了解这种空间的坐标功能的帮助下表示。 (2)顺从性和相对边界：对于一致代数，各种类型的边界已经在文献中得到了广泛的研究。然而，直到最近才在赋范空间中引入了相对边界和其中的特殊点（强边界点，峰值点等）。最近，我已经能够显示一些应用这些想法的理论字符服从巴拿赫代数。相对边界的性质、相对边界与内倾空间的关系以及相对边界的坐标函数是有待进一步研究的自然问题。 (3)拓扑中心：菲拉利，纽方和我已经开发了一种技术，以获得因子分解的运营商在集团冯诺依曼代数。利用这种技术，我们已经能够确定的拓扑中心的群冯诺依曼代数和空间的一致连续泛函的某些群体。这些想法是非常有前途的，我们打算进一步发展的技术，更广泛的空间和他们的子空间，如空间的p-伪措施和他们的子空间。 这项研究将是感兴趣的一个广泛的数学家群体的工作在现代分析。它将为研究生的硕士和博士论文提供项目。这将有利于现代数学中一个高度活跃的研究领域的发展。