Noncommutative Hardy Spaces and Littlewood-Paley Theory

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI的研究集中在谐波分析的泛函分析方面。这个项目致力于理解非交换环境下的实-哈代空间和Littlewood-Paley理论。PI将把结果应用于算子代数/非交换几何以及经典分析。目标之一是理解非交换群上的傅里叶乘数。它需要结合Hp理论、算子空间理论和非交换lp空间等不同的工具。一个主要的挑战是找到非交换技术来取代欧几里得空间某些关键几何性质的使用。PI在算子值Hardy空间上的工作在提出的研究方向上迈出了第一步。该项目将为Pisier/Xu、Junge和Junge/Xu最近关于非交换鞅的工作提供理论上的对应，并将补充Arveson关于非交换分析hardy空间的工作。它也将提高对von Neumann代数上（完全）正算子的半群的理解。量子力学和海森堡的测不准原理允许经典数学理论的许多非对易推广（=量子化），遵循冯·诺伊曼和默里在非对易积分理论方面的开创性工作。在将经典概念引入非交换框架的过程中，出现了非常有趣的新现象和困难。例如，使用冯·诺伊曼代数的语言，现在可以讨论非交换域的预期退出时间，尽管我们永远无法看到该域的“点”。该项目将继续这一长期的量化工作，并将探索和加强数学和物理不同分支之间的深层联系。反过来，它也将对预测理论、h∞控制、信号和图像处理、统计学和量子力学做出有价值的贡献。

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.