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.
