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: 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=612726
• 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性的最基本表达。像正方形这样的物体只允许有200多种对称性，而球面或平面则允许有无限多种对称性。为了使对这样一组对称性的分析更容易处理，并记住更多的基础球面或平面的空间特征，比如说，我们需要处理对称性群的空间性质，我们称之为拓扑学。因此，我们得到拓扑群。此外，大多数自然的例子，比如上面的例子，暗示了在单个点附近的某种有限性，我们称之为局部紧性。 对于这些局部紧群，重要的是要理解它们的表示，即它们如何作用于具有拓扑的其他集合。Banach代数的使用是封装并帮助理解这些动作的工具。这样的代数，其中包括群和测度代数，弱概周期函数，傅立叶和傅立叶-Steiltjes代数，以及相关的Segal和Beurling代数都发挥有趣的和独特的作用，在此分析。我的研究重点是相互作用的结构的局部紧群和结构的相关Banach代数。我最基本的问题是：给定一类局部紧群的一个性质，该性质如何在相关的Banach代数中得到证明，反之亦然。这些问题的解决往往迫使我们在这些Banach代数上使用不明显的结构，如算子空间结构。例如，顺从性，一个基本的平均性质的群体，是见证了只有在傅立叶代数与算子空间平均性质，而不是一个性质，可以看到只有与巴拿赫代数技术。 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包含了广义庞特里亚金对偶，这是理解群代数和傅立叶代数相互关系的框架。因此，这个更一般的框架也引起了我的一些研究注意。