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的一个部分，即泛函和调和分析，涉及到思想和问题。我和我的同事们在Banach代数的表示理论中发展了几个新的想法，并将它们应用于各种问题，如第二对偶的理想结构、拓扑中心、特征随和和拓扑内倾空间。*以下是我的研究计划的总结：*(1)内倾空间：给定一个Banach代数A和A*的一个拓扑内倾子空间X，Filali、Neufang和我引入了从属表示的概念，并在A的从属于X的表示和对偶空间X*的表示之间建立了一一对应关系。这种现象以前只在特殊情况下才为人所知。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。我们打算刻画A的从属于各种内倾空间X的表示，这将有助于借助表示的坐标函数更好地理解这类空间。*(2)可寻性和相对边界：对于一致代数，各种类型的边界在文献中一直是广泛研究的主题。然而，直到最近，赋范空间的相对边界和其中的特殊点(强边界点、峰点等)才被引入。最近，我已经能够展示这些思想在特征标服从Banach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，都是有待进一步研究的自然问题。*(3)拓扑中心：Filali，Neufang和我发展了一种技术来获得群von Neumann代数中算子的因子分解。利用这一技巧，我们已经能够确定群von Neumann代数的拓扑中心和某些群的一致连续泛函空间。这些想法是非常有前途的，我们打算将这一技术进一步发展到更广泛的空间及其商，如p-伪测度空间及其商。*这项研究将会引起从事现代分析工作的数学家的广泛兴趣。它将提供研究生硕士和博士论文的项目。这将有助于推动现代数学这一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. **