Banach algebras in abstract harmonic analysis

这一建议属于抽象调和分析领域，抽象调和分析主要是研究局部紧(拓扑)群及其相关的Banach代数。局部紧群是抽象的数学对象，可以与几何图形的对称性或自然界中对象的刚性运动联系在一起。它们在数学和物理科学中都有很多应用。Banach代数是元素的集合，使得每个单独的元素都有长度，并且使得任意两个元素可以相加或相乘。这些Banach代数通常由到实数或复数集的映射组成，它们还经常由运算符组成，这些运算符可以被视为有限的或无限的数组。有许多有趣和有用的Banach代数自然地与任何局部紧群联系在一起，这一研究领域的一个总的主题是研究局部紧群的各种性质与它们相关的Banach代数之间的关系。在许多方面，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对复杂现象的简单描述。特别是，寻找相似数学对象之间(复杂的)结构保持映射的简单描述是几乎每个数学领域的基本目标。多年来，抽象调和分析者一直在寻找建立在局部紧群上的各种Banach代数之间的结构保持映射的有吸引力的刻画，这也是本研究计划的目的之一。近年来，现代工具在这一努力方面取得了重大进展，该提案包括了解决其中一些具有挑战性的问题的新方法。

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.