Algebras of abstract harmonic analysis

抽象调和分析的代数

• 批准号: RGPIN-2015-04024
• 负责人: Spronk, Nicolaas
• 金额: $ 1.02万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=665740
• 关键词: Algebras; abstract; harmonic; analysis

The mathematical concept of a group is the most basic expression of symmetry. An object like a square admits only finitely many symmetries, whereas a sphere or plane admits infinitely many. In order to keep the analysis of such a group of symmetries more tractable, and to remember more of the spatial features of the underlying sphere or plane, say, one needs to deal with the spatial properties of the symmetry group, a concept we call topology. Hence we obtain topological groups. Furthermore, most natural examples, such as those above, suggest a certain finiteness near individual points, a condition we call local compactness. *******For these locally compact groups it is important to understand their representations, i.e. how they act on other sets with topology. The use of Banach algebras is a tool to encapsulate and help to understand these actions. Such algebras, which include group and measure algebras, weakly almost periodic functions, Fourier and Fourier-Steiltjes algebras, and associated Segal and Beurling algebras all play interesting and distinctive roles in this analysis. My research focusses on the interplay of the structures of locally compact groups and structures of the associated Banach algebras. My most basic question: given a property of a class of locally compact groups, how is that property witnessed in the associated Banach algebras, and vice versa. The resolutions of these questions often force us to use inobvious structures on these Banach algebras, such as operator space structures. For example, amenability,  a fundamental averaging property on groups, is witnessed only in Fourier algebras with an operator space averaging property, and not by a property which can be seen only with Banach algebra techniques.*******Some symmetries observable to physicists require much more subtle symmetries. Remarkably, the mathematical formulation of these, known as quantum groups, also encapsulates generalized Pontryagin duality, which is a framework for understanding the group algebra and Fourier algebra in relation to one another. Hence this more general framework also catches some of my research attention.**

群的数学概念是对称的最基本表达。像正方形这样的物体只允许有有限种对称，而球面或平面则允许有无限种对称。为了使对这样一组对称的分析更容易处理，并记住更多的球面或平面的空间特征，比如说，我们需要处理对称群的空间特性，一个我们称之为拓扑的概念。因此我们得到拓扑群。此外，大多数自然的例子，如上面那些，表明在个别点附近有一定的有限性，我们称之为局部紧性。*******对于这些局部紧群，重要的是要了解它们的表示，即它们如何作用于其他具有拓扑的集合。巴拿赫代数的使用是一个封装和帮助理解这些动作的工具。这些代数，包括群代数和测度代数，弱概周期函数，傅里叶代数和傅里叶- steiltjes代数，以及相关的Segal代数和Beurling代数，都在这个分析中发挥了有趣而独特的作用。我的研究重点是局部紧群的结构和相关巴拿赫代数的结构的相互作用。我最基本的问题：给定一类局部紧群的性质，该性质如何在相关的巴拿赫代数中被证明，反之亦然。这些问题的解决常常迫使我们在这些巴拿赫代数上使用不明显的结构，例如算子空间结构。例如，群上的一个基本平均性质——可调和性，只在具有算子空间平均性质的傅里叶代数中被证明，而不是在只有巴拿赫代数技术才能看到的性质中被证明。*******物理学家观察到的一些对称性需要更微妙的对称性。值得注意的是，这些被称为量子群的数学公式，也概括了广义庞特里亚金对偶，这是理解群代数和傅里叶代数相互关系的框架。因此，这个更通用的框架也引起了我的一些研究注意