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.

傅里叶分析以 17 世纪和 19 世纪法国数学家和物理学家 Jean-Baptiste Fourier 的名字命名（并由他发起），他研究传热和振动。它的起点是傅立叶级数，即通过叠加正弦波和余弦波来表达给定周期函数的方法。迄今为止，傅里叶变换仍然是求解微分方程的重要工具。在 20 世纪，人们逐渐认识到局部紧致阿贝尔 (LCA) 群是发展傅立叶分析的合适环境。它允许定义包含傅里叶级数和经典傅里叶变换的通用傅里叶变换。这里的关键概念是庞特里亚金对偶性：每个 LCA 群 G 都有一个对偶群 G^。例如，实线的对偶群又是实线，整数的对偶群又是单位圆。此外，G^^ = G 始终成立。对（不一定是阿贝尔）局部紧群的研究被称为抽象调和分析，这是一门自 20 世纪中叶以来在加拿大传统上很强大的学科。抽象调和分析的一个关键方法不是研究群本身，而是研究与它们相关的各种巴拿赫代数和空间。自 20 世纪 60 年代以来，特别是在过去的四分之一个世纪中，量化变得越来越重要：这是指用非交换对象（例如空间和代数）取代可交换对象（例如空间和函数代数）的运营商。自世纪之交以来，局部紧致量子群变得越来越重要：与非阿贝尔局部紧群不同，它们允许庞特里亚金式对偶性，从而扩展了 LCA 群的对偶性。拟议的研究重点关注三个主要主题： 1. 局部紧量子群：顺应性和对偶性。该项目旨在加深我们对局部紧量子群存在的各种顺应性概念的理解。 2.量化泛函分析。算子空间理论，即希尔伯特空间上有界线性算子的空间，通常被称为量化泛函分析。经典泛函分析的许多概念和结果都有量化的类似物，但仍有很多不清楚的地方。我们希望有助于进一步理解泛函分析中的量化。 3. 量化巴纳赫代数的顺应性性质。在过去（和正在进行中），我的研究一直关注量化巴拿赫代数的顺应性性质。拟议的研究将沿着这些思路继续进行。总的来说，拟议的研究将继续我过去五到十年的工作，并有助于更深入地理解泛函和抽象谐波分析中的量化。