Hardy/BMO spaces and Fourier Multipliers in the Noncommutative Setting

该项目的目的是发展一个抽象的理论哈代/BMO空间和傅立叶乘数，特别是在一个非交换的框架。算子半群将被重要地用来弥补几何/度量结构的缺乏和抽象设置中交换积的缺乏。 从这项工作中获得的新见解也应该反映在非交换几何，算子空间理论和经典调和分析。对非对易对象的研究为反映日常生活的数学中的许多主题提供了一个新的观点，它也可以说是量子力学的“正确”语言。 在真实的生活中，某些操作的执行顺序会产生很大的不同。例如，先煮水，然后加入油，这与先煮油，然后加入水是非常不同的。非对易分析是关于非对易变量领域中的函数及其性质。最重要的例子是矩阵值函数。傅立叶分析或调和分析的理论有很多关于函数的内容，正如这个项目打算展示的那样，也有很多关于矩阵值函数的内容。该项目还将对控制理论和信号与图像处理做出有价值的贡献，其中矩阵值函数是自然职业。

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.