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

我的研究涉及到数学中一个部分的思想和问题，这个部分被称为泛函和谐波分析。我和我的同事们在巴拿赫代数的表示理论中发展了一些新的思想，并将它们应用于诸如第二对偶的理想结构、拓扑中心、字符可调性和拓扑内向空间等不同的问题。(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，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，它可以应用于许多不同的情况。我们打算描述那些隶属于各种内向空间x的A的表征，这将有助于借助表征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，各种类型的边界已经在文献中被广泛研究。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，我已经能够展示这些思想的一些应用到特征可服从巴拿赫代数的理论。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和我已经开发了一种技术来获得群von Neumann代数中的算子的因数分解。利用这种方法，我们已经能够确定群的von Neumann代数的拓扑中心和某些群的一致连续泛函的空间。这些想法是非常有前途的，我们打算进一步发展更广泛的空间及其商的技术，例如p伪测度及其商的空间。******这项研究将引起从事现代分析的大批数学家的兴趣。为研究生提供硕士、博士学位论文立项。它将有利于现代数学中一个高度活跃的研究领域的发展。**

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. **