Noncommutative Hardy Spaces and Littlewood-Paley Theory

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI's research is centered on functional analytic aspects of harmonic analysis. This project is devoted to understanding real-Hardy spaces and the Littlewood-Paley theory in the noncommutative setting. The PI will apply the results to operator algebra/noncommutative geometry as well as to classical analysis. One of the goals is to understand Fourier multipliers on noncommutative groups. It requires a combination of different tools from Hp theory, operator space theory, and noncommutative Lp-spaces. A major challenge is to find noncommutative techniques replacing the use of certain crucial geometric properties of Euclidean spaces. The PI's work on operator-valued Hardy spaces took an initial step in the direction of the proposed research. The proposed project will provide a theoretical counterpart of the recent work by Pisier/Xu, Junge, and Junge/Xu of noncommutative martingales and will complement Arveson's work on noncommutative analytic-Hardy spaces. It will also improve the understanding of the semigroups of (completely) positive operators on von Neumann algebras.Quantum mechanics and Heisenberg's uncertainty principle allow for many noncommutative generalizations (=quantization) of classical mathematical theories following Von Neumann and Murray's pioneering work on noncommutative integration theory. Very interesting new phenomena and difficulties arise in the effort of adopting classical concepts to the noncommutative framework. For example, using the language of von Neumann algebras, it is now possible to talk about the expected exit time for a noncommutative domain although we can never see the "points" of this domain. This project will continue this long term quantization effort and will explore and strengthen the deep links between different branches of mathematics and physics. In turn, it will make valuable contributions to prediction theories, H-infinity control, signal and image processing, statistics, and quantum mechanics as well.