Hardy/BMO spaces and Fourier Multipliers in the Noncommutative Setting

该项目旨在发展Hardy/BMO空间和傅里叶乘数的抽象理论，特别是在非交换框架内。半群算子将被重要地用于弥补抽象环境中缺乏几何/度量结构和缺乏交换积的缺陷。从这项工作中获得的新见解也应该反映回非交换几何，算子空间理论和经典谐波分析。对非交换对象的研究为反映日常生活的许多数学主题提供了一个新的视角，并且可以说它代表了量子力学的“正确”语言。在现实生活中，执行某些操作的顺序可能会产生很大的不同。比如先烧开水再加油和先烧开油再加水就有很大的不同。非交换分析是关于非交换变量领域中的函数及其性质的分析。最重要的例子是矩阵值函数。傅里叶分析或谐波分析的理论有很多关于函数的内容，正如这个项目所要展示的，也有关于矩阵值函数的内容。该项目还将对控制理论以及信号和图像处理做出有价值的贡献，其中矩阵值函数是自然发生的。

The project aims to develop an abstract theory of Hardy/BMO spaces and Fourier multipliers, particularly within a noncommutative framework. Semigroups of operators will be used crucially to make up for the lack of geometric/metric structure and the lack of a commutative product in the abstract setting. The new insights gained from this work should also reflect back on noncommutative geometry, operator space theory, and classical harmonic analysis. The study of noncommutative objects offers a new point of view on many topics in mathematics that reflect daily life, and it also arguably represents 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 afterwards adding oil is very different from first boiling oil and then adding water. Noncommutative analysis is about functions and their properties in the realm of noncommuting variables. The most important example is matrix-valued functions. The theory of Fourier analysis or harmonic analysis has a lot to say about functions, and as this project intends to show, also about matrix-valued functions. The project will also make valuable contributions to control theory and signal and image processing, where matrix-valued functions are natural occurences.