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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675492
• 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.**

傅立叶分析以17和19世纪的法国数学家和物理学家让·巴蒂斯特·傅里耶尔（Jean-Baptiste Fourier）的名字命名，并研究了传热和振动。它的起点是傅立叶系列，即通过覆盖正弦和余弦波来表达给定定期功能的一种方法。到目前为止，傅立叶变换是解决微分方程的重要工具。******在20世纪，很明显，当地紧凑的Abelian（LCA）组是开发傅立叶分析的合适设置。它允许定义一般的傅立叶变换，该变换均包含傅立叶系列以及经典的傅立叶变换。这里的关键概念是双重二元性：每个LCA组G组都有一个双重组G^。例如，真实行的双组再次是真实行，整数的双组是单位圆。此外，g ^^ = g总是成立。抽象谐波分析的关键方法是不是研究群体本身，而是研究与它们相关的各种Banach代数和空间。自1960年代（尤其是在过去的四个世纪中）以来，量化变得越来越重要：这是指通过非交互性的函数空间和功能代数等交换对象，即操作员的空间和代数。自本世纪之交以来，局部紧凑的量子群已经获得了重要意义：与非亚伯族局部紧凑的群体不同，它们允许pontryagin风格的双重性扩展了LCA群体。局部紧凑型量子组：不合理性和二元性。该项目旨在加深我们在本地紧凑型量子组存在的各种舒适性概念之间的理解。****** 2。量化功能分析。运算符空间的理论，即希尔伯特空间上有界线性运算符的空间的理论，通常称为量化功能分析。经典功能分析的许多概念和结果已经量化了类似物，但仍然不清楚。我们希望在功能分析中进一步了解量化。****** 3。量化Banach代数的不舒服性能。在过去（和正在进行的）中，我的研究一直关注量化Banach代数的舒适性。拟议的研究将继续沿着这些路线。******总体而言，拟议的研究将在过去五到十年中继续我的工作，并有助于更深入地了解功能和抽象的谐波分析中的量化。**