Banach algebras in abstract harmonic analysis

抽象调和分析中的巴纳赫代数

基本信息

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：
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=613324
关键词：
Banach algebras abstract harmonic analysis

项目摘要

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.

该提议属于抽象调和分析领域，主要是局部紧致（拓扑）群及其相关巴拿赫代数的研究。局部紧群是抽象的数学对象，它可能与几何图形的对称性或自然界中物体的刚性运动有关。它们在数学和物理科学方面都有很多应用。巴纳赫代数是元素的集合，其中每个单独的元素都有长度，并且任意两个元素可以相加或相乘。这些巴拿赫代数通常由实数或复数集合的映射组成，并且它们也通常由运算符组成，这些运算符可以被视为有限或无限的数字数组。有许多有趣且有用的巴拿赫代数与任何局部紧群自然相关，该研究领域的一个总主题是研究局部紧群的各种性质与其相关的巴拿赫代数之间的关系。从很多方面来说，这个主题在这个研究项目中都很突出。数学的目的之一是找到复杂现象的简单描述。特别是，寻找相似数学对象之间（复杂）结构保持映射的简单描述几乎是数学每个领域的基本目标。多年来，抽象调和分析师一直在寻找对在局部紧群上构建的各种 Banach 代数之间的结构保持映射的有吸引力的描述，这也是本研究项目的目的之一。近年来，现代工具在这一努力中取得了重大进展，该提案包括解决其中一些挑战性问题的新方法。