Fourier Multipliers on Noncommutative Lp Spaces

数学家使用“函数”来描述和模拟我们的真实世界。理解“函数”的一个有用的方法是将它们分解成频率，就像音乐和弦可以用其组成音符的频率(或音高)来表示一样。这被称为傅里叶变换，是所谓的傅里叶分析方法的一部分。对非对易物体的研究为反映我们日常生活的数学中的许多问题提供了一个新的观点，并可能为量子力学提供“正确的”语言。在现实生活中，执行某些操作的顺序可能会有很大的不同。例如，先煮水加油与先煮油再加水有很大不同。这是一个非对易过程的例子。非对易傅立叶分析是关于函数及其在非对易变量领域的性质的分析。在数学中，最重要的例子是矩阵值函数。这个项目致力于与von Neumann代数相关的非交换LP空间的傅立叶分析，包括(非径向)傅立叶乘子，Mikhlin-乘子理论，Dirac算子，以及群von Neumann代数的无条件序列。一个典型的对象是自由群von Neumann代数上的Hilbert变换。提出的研究中的主要挑战是缺乏几何/度量结构，以及抽象背景下缺乏可交换的乘积。拟议的研究计划将加强调和分析和泛函分析之间的现有联系。非对易调和分析受到量子力学和预测理论的启发，将在这些领域以及金融建模和信号处理等更多的应用课题中做出有价值的贡献。

Mathematicians use "functions" to describe and simulate our real world. A useful method to understand "functions" is to decompose them into frequencies, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes. This is called the Fourier transform and is a part of the so-called Fourier analysis method. The study of noncommutative objects offers a new point of view on many topics in mathematics reflecting our daily life and offers possibly the "right" language for quantum mechanics. In real life, the order in which certain operations are executed can make a big difference. For example, first boiling water and adding oil is very different from first boiling oil and then adding water. This is an example of a noncommutative process. Noncommutative Fourier analysis is about functions and their properties in the realm of non-commuting variables. In mathematics, the most important examples are matrix-valued functions. This project is devoted to the Fourier analysis on noncommutative Lp spaces associated with von Neumann algebras, including (non-radial) Fourier multipliers, the Mikhlin-multiplier theory, Dirac Operators, and unconditional sequences of group von Neumann algebras. A typical object is the Hilbert transform on free group von Neumann algebras. Major challenges in the proposed research are the lack of geometric/metric structure and the lack of a commutative product in the abstract setting. The proposed research program will strengthen the existing link between Harmonic Analysis and Functional Analysis. Noncommutative harmonic analysis is motivated by quantum mechanics and prediction theory and will make valuable contributions to these areas and more applied topics such as financial modeling and signal processing.