Banach algebras, operator spaces and their applications to locally compact quantum groups

Banach代数、算子空间及其在局部紧量子群中的应用

• 批准号: RGPIN-2019-04579
• 负责人: Runde, Volker
• 金额: $ 1.09万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737712
• 关键词: Banach; algebras; operator; spaces; their

Fourier analysis is named after (and was initiated by) 17th and 19th century French mathematician and physicist Jean-Baptiste Fourier who studied heat transfer and vibration. Its starting point is the Fourier series, i.e., a way to express a giving periodic function through overlaying sine and cosine waves. The Fourier transform is, to this day, an important tool to solve differential equations. In the 20th century, it became clear that locally compact abelian (LCA) groups are the appropriate setting to develop Fourier analysis. It allows to define a general Fourier transform that encompasses both Fourier series as well as the classical Fourier transform. The crucial concept here is Pontryagin duality: every LCA group G has a dual group G^. For instance, the dual group of the real line is the real line again, and the dual group of the integers is the unit circle. Moreover, G^^ = G always holds. The study of (not necessarily abelian) locally compact groups is called abstract harmonic analysis, a discipline that has been traditionally strong in Canada since the mid 20th century. A key approach in abstract harmonic analysis is to study not the groups themselves, but the various Banach algebras and spaces associated with them. Since the 1960s - in particular, during the past quarter of a century -, quantization has become more and more important: this refers to replacing commutative objects, such as spaces and algebras of functions, by non-commutative ones, i.e., spaces and algebras of operators. Since the turn of the century, locally compact quantum groups have gained significance: unlike non-abelian locally compact groups, they allow for a Pontryagin style duality that extends the one for LCA groups. The proposed research focuses on three main topics: 1. Locally compact quantum groups: amenability properties and duality. This project is intended to deepen our understanding between the various notions of amenability that exist for locally compact quantum groups. 2. Quantizing functional analysis. The theory of operator spaces, i.e., of spaces of bounded linear operators on Hilbert space, is often referred to as quantized functional analysis. Many concepts and results of classical functional analysis have quantized analogs, but still there is a lot still unclear. We hope to contribute to a further understanding of quantization in functional analysis. 3. Amenability properties of quantized Banach algebras. In the past (and ongoing), my research has been concerned with the amenability properties of quantized Banach algebras. The proposed research will continue along these lines. Overall, the proposed research will continue my work over the past five to ten years and contribute to a deeper understanding of quantization in functional and abstract harmonic analysis.

傅立叶分析是以17世纪和19世纪法国数学家和物理学家让-巴蒂斯特·傅立叶（Jean-Baptiste Fourier）的名字命名的，他研究了传热和振动。它的起点是傅立叶级数，即，一种通过叠加正弦和余弦波来表示给定周期函数的方法。傅里叶变换是，直到今天，一个重要的工具来解决微分方程。在20世纪，很明显，局部紧阿贝尔（LCA）群是发展傅立叶分析的合适背景。它允许定义一个包含傅里叶级数和经典傅里叶变换的一般傅里叶变换。这里的关键概念是庞特里亚金对偶：每个LCA群G都有一个对偶群G^。例如，真实的直线的对偶群又是真实的直线，整数的对偶群是单位圆。此外，G^^ = G总是成立。对局部紧群（不一定是阿贝尔群）的研究被称为抽象调和分析，这是自世纪中期以来加拿大传统上很强的一门学科。抽象调和分析的一个关键方法不是研究群本身，而是研究与之相关的各种Banach代数和空间。自20世纪60年代以来，特别是在过去的四分之一个世纪，量子化变得越来越重要：这是指用非交换对象取代交换对象，如空间和函数代数，即，空间和算子代数。自世纪以来，局部紧量子群获得了重要性：与非阿贝尔局部紧群不同，它们允许庞特里亚金式对偶，扩展了LCA群的对偶。本文的研究主要集中在三个方面：1.局部紧量子群：顺从性和对偶性。这个项目旨在加深我们对局部紧量子群中存在的各种顺从性概念之间的理解。2.量化功能分析。算子空间的理论，即，希尔伯特空间上有界线性算子空间的量子化，通常被称为量子化泛函分析。经典泛函分析的许多概念和结果都有量化的类比，但仍有许多不清楚之处。我们希望有助于进一步了解量化的功能分析。3.量子化Banach代数的顺从性。在过去（和正在进行的），我的研究一直关注的顺从性性质的量化Banach代数。拟议的研究将继续沿着这些路线沿着进行。总的来说，拟议的研究将继续我在过去五到十年的工作，并有助于更深入地了解量化的功能和抽象谐波分析。