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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675381
• 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限多个对称性，而球体或平面则允许无限多个对称性。 为了使对这样一组对称性的分析更容易处理，并记住更多底层球体或平面的空间特征，例如，我们需要处理对称性群的空间属性，我们称之为拓扑的概念。 因此我们得到了拓扑群。 此外，大多数自然示例（例如上面的示例）表明各个点附近存在一定的有限性，我们将这种情况称为局部紧凑性。 ******对于这些局部紧凑的群，了解它们的表示非常重要，即它们如何作用于具有拓扑的其他集合。 巴拿赫代数的使用是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、弱几乎周期函数、傅里叶和傅里叶-斯泰尔切斯代数，以及相关的 Segal 和 Beurling 代数，都在这个分析中发挥着有趣和独特的作用。 我的研究重点是局部紧群结构和相关巴拿赫代数结构之间的相互作用。 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得到体现，反之亦然。 这些问题的解决常常迫使我们在这些 Banach 代数上使用不明显的结构，例如算子空间结构。 例如，服从性（群上的基本平均性质）仅在具有算子空间平均性质的傅立叶代数中可见，而不是仅通过巴纳赫代数技术才能看到的性质。********物理学家可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封装了广义庞特里亚金对偶性，这是理解群代数和傅立叶代数相互关系的框架。 因此，这个更通用的框架也引起了我的一些研究关注。**