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

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·傅里叶命名的(并由他发起)，他研究了热传递和振动。它的起点是傅里叶级数，即通过叠加正弦波和余弦波来表示给定周期函数的一种方法。时至今日，傅里叶变换仍是解微分方程的重要工具。在20世纪，很明显局部紧交换(LCA)群是发展傅立叶分析的合适环境。它允许定义既包括傅里叶级数又包括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，实线的对偶群又是实线，整数的对偶群是单位圆。此外，G^^=G总是成立的。对局部紧群(不一定是阿贝尔的)的研究被称为抽象调和分析，这是自20世纪中叶以来在加拿大一直很强大的一门学科。抽象调和分析的一个关键方法不是研究群本身，而是研究各种Banach代数及其相关的空间。自20世纪60年代以来，尤其是在过去的四分之一个世纪中，量子化变得越来越重要：这指的是用非交换对象，即算子空间和代数来代替交换对象，如空间和函数代数。自世纪之交以来，局部紧量子群已变得重要起来：与非阿贝尔局部紧量子群不同，它们允许庞特里亚金式的对偶，扩展了LCA群的对偶。提出的研究主要集中在三个方面：1.局部紧量子群：顺从性、性质和对偶性。这个项目旨在加深我们对局部紧致量子群存在的各种顺从性概念之间的理解。2.量化泛函分析。算子空间理论，即Hilbert空间上的有界线性算子空间，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子化的类比，但仍有许多不清楚之处。我们希望对泛函分析中量子化的进一步理解有所贡献。3.量化Banach代数的顺从性。在过去(和正在进行的)中，我的研究一直关注于量化Banach代数的顺从性。拟议的研究将继续沿着这些方向进行。总体而言，拟议的研究将继续我过去五到十年的工作，并有助于加深对泛函和抽象调和分析中的量子化的理解。

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.