Fourier Multipliers on Noncommutative Lp Spaces

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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.