Banach algebras in abstract harmonic analysis

抽象调和分析中的巴纳赫代数

• 批准号: RGPIN-2015-05044
• 负责人: Stokke, Ross
• 金额: $ 0.8万
• 依托单位: University of Winnipeg
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708569
• 关键词: Banach; algebras; abstract; harmonic; analysis

This proposal lies in the area of abstract harmonic analysis, which is largely the study of locally compact (topological) groups and their associated Banach algebras. Locally compact groups are abstract mathematical objects, which may be associated with the symmetries of a geometric figure, or the rigid motion of objects in nature. They have many applications, both to the mathematical and physical sciences. A Banach algebra is a collection of elements such that each individual element has a length, and such that any two elements can be added or multiplied. These Banach algebras are often comprised of mappings into the set of real, or complex, numbers, and they are also often comprised of operators, which may be viewed as finite, or infinite, arrays of numbers. There are many interesting and useful Banach algebras naturally associated with any locally compact group, and a general theme in this area of research is to study the relationship between various properties of locally compact groups and their associated Banach algebras. In many ways, this theme is prominent in this research program. One purpose of mathematics is to find simple descriptions of complex phenomena. In particular, the search for simple descriptions of (complicated) structure-preserving maps between similar mathematical objects is a fundamental objective in virtually every area of mathematics. For many years, abstract harmonic analysts have searched for attractive descriptions of the structure-preserving maps between various Banach algebras built over locally compact groups, and this is one of the purposes of this research program. In recent years, modern tools have generated significant progress with this endeavour, and this proposal includes new approaches to solving some of these challenging problems.

该建议在于抽象谐波分析领域，这在很大程度上是对局部紧凑（拓扑）组及其相关的Banach代数的研究。局部紧凑的组是抽象的数学对象，可能与几何图形的对称性或物体在自然界中的刚性运动有关。它们在数学和物理科学方面都有许多应用。 Banach代数是一个元素的集合，使每个元素具有长度，并且可以添加或乘以任何两个元素。 这些BANACH代数通常由映射到真实或复杂数字的集合中，它们通常也由操作员组成，这些操作员可能被视为有限的或无限的数字阵列。 有许多有趣且有用的Banach代数与任何本地紧凑的群体自然相关，并且该研究领域的一般主题是研究本地紧凑型组的各种特性及其相关的Banach代数之间的关系。在许多方面，这个主题在该研究计划中是突出的。 数学的目的之一是找到复杂现象的简单描述。特别是，在几乎每个数学领域中，搜索相似数学对象之间的（复杂）结构保护图的简单描述是一个基本目标。多年来，抽象的谐波分析师一直在搜索在本地紧凑型组建立的各种Banach代数之间具有结构性保护图的有吸引力的描述，这是本研究计划的目的之一。近年来，现代工具在这项工作中取得了重大进展，该提案包括解决其中一些具有挑战性的问题的新方法。