Structure of Banach algebras over locally compact groups

局部紧群上的 Banach 代数的结构

• 批准号: RGPIN-2014-05354
• 负责人: Samei, Ebrahim
• 金额: $ 0.8万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567111
• 关键词: Structure; Banach; algebras; over; locally

My main field of research is harmonic analysis with an emphasis on non-commutative groups such as amenable groups and Lie groups. Harmonic analysis is a branch of mathematics concerned with the representation of functions, signals, or waves using Fourier series and Fourier transforms. It has become a vast subject with applications in areas such as signal processing and quantum mechanics over the past two centuries. The research projects that I pursue primarily relate to more modern branches of harmonic analysis and is about the understanding of the structure of various Banach algebras related to locally compact groups, including group algebras and Fourier algebras. My objective is to study in a deeper level the structure of these algebras and their relation to the representations of the groups. One aspect of the structure theory of Banach algebras over locally compact groups I am interested in deals with cohomological properties of these algebras. The seminal paper in this respect is certainly B. E. Johnson’s memoir in 1972 in which he introduces and explores the concept of amenability for a Banach algebra. Z-J Ruan’s fundamental work in 1995 of introducing the concept of operator amenability, through the theory of operator spaces, and applying it in non-commutative harmonic analysis has also had a significant influence on this area. An important open problem in this area is the characterization of the weak amenability of the Fourier algebra on non-compact connected groups (the compact and some non-compact cases have already been established). One research project I plan to pursue is to investigate a more general property, namely the failure of spectral synthesis of the antidiagonal for non-abelian, non-compact connected groups. It is expected that this will imply the failure of the weak amenability and provide more such examples. I will also look at the “local behavior” of their Hochschild cohomology. The overall goal is to get more information about the structure of the algebras or the groups using these results. Another research project of mine deals with "non-communicative" analogue of Beurling algebras. These algebras have played important roles in different areas of harmonic analysis. They are L^1-algebras associated to locally compact groups when we put extra "weights" on the groups. With my co-author H. H. Lee, we developed the corresponding "dual theory" and created the Beurling-Fourier algebras (shortly BF algebras). In several co-authored research projects, I have shown that in some aspect, they behave similar to the classical Beurling algebras whereas in others, they behave very differently. However, this theory is very recent and subsequently there is still much research to be completed concerning them. From one direction, more detailed study of BF algebras on compact groups must be conducted. We also need to investigate BF algebras on non-compact, non-abelian groups e.g. Heisenberg groups and the ax+b-group. The research I am planning to pursue using my NSERC Discovery grant is all a continuation of my work in the past 10 years. I am happy to report that the results obtained so far have been substantial and have received positive reviews from other experts in the fields. Canada is growing as one of the research-intensive countries in the world, and to make sure we continue as such, we need to not only maintain our research programs but also grow and expand them. One necessary means, among many others, is to have faculties with strong research initiatives and programs in all areas of science and technologies including mathematics. The research projects I have outlined above will be part of many streams of research being conducted across the country. This will have a great and positive impact in our future.

我的主要研究领域是调和分析，重点是非交换群，例如服从群和李群。谐波分析是数学的一个分支，涉及使用傅里叶级数和傅里叶变换来表示函数、信号或波。在过去的两个世纪中，它已成为一门广泛的学科，应用于信号处理和量子力学等领域。我从事的研究项目主要涉及调和分析的更现代分支，并且是关于理解与局部紧群相关的各种巴纳赫代数的结构，包括群代数和傅立叶代数。我的目标是更深入地研究这些代数的结构及其与群表示的关系。我感兴趣的是巴拿赫代数在局部紧群上的结构理论的一个方面，即处理这些代数的上同调性质。这方面的开创性论文无疑是 B. E. Johnson 1972 年的回忆录，他在其中介绍并探索了巴拿赫代数的顺应性概念。阮志坚在1995年通过算子空间理论引入算子顺应性概念并将其应用于非交换调和分析的基础工作也对该领域产生了重大影响。该领域的一个重要的开放问题是表征傅立叶代数对非紧连通群的弱适应性（紧和一些非紧情况已经建立）。我计划进行的一个研究项目是研究一个更普遍的性质，即非阿贝尔、非紧连通群的反对角谱合成的失败。预计这将意味着弱服从性的失败，并提供更多这样的例子。我还将研究霍克希尔德上同调的“局部行为”。总体目标是使用这些结果获得有关代数或群的结构的更多信息。我的另一个研究项目涉及伯林代数的“非交流”类似物。这些代数在调和分析的不同领域发挥了重要作用。当我们对群施加额外的“权重”时，它们是与局部紧群相关的 L^1 代数。我们与我的合著者 H. H. Lee 开发了相应的“对偶理论”并创建了 Beurling-Fourier 代数（简称 BF 代数）。在几个共同撰写的研究项目中，我已经证明，在某些方面，它们的行为类似于经典的 Beurling 代数，而在其他方面，它们的行为却非常不同。然而，这个理论是最近才出现的，随后仍然有很多关于它们的研究有待完成。从一个方向来说，必须对紧群上的 BF 代数进行更详细的研究。我们还需要研究非紧、非阿贝尔群的 BF 代数，例如海森堡群和 ax+b 群。我计划利用 NSERC Discovery 资助进行的研究是我过去 10 年工作的延续。我很高兴地向大家报告，迄今为止取得的成果非常丰硕，并得到了该领域其他专家的积极评价。加拿大正在成长为世界上研究密集型国家之一，为了确保我们继续这样的发展，我们不仅需要维持我们的研究项目，还需要发展和扩大它们。其中一个必要的手段是拥有在包括数学在内的所有科学技术领域拥有强大研究主动性和项目的教师队伍。我上面概述的研究项目将是全国各地正在进行的许多研究项目的一部分。这将对我们的未来产生巨大而积极的影响。