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功能分析方面。这个项目致力于理解实Hardy空间和Littlewood-Paley理论在非交换的设置。PI将把结果应用于算子代数/非交换几何以及经典分析。目标之一是了解非交换群上的傅立叶乘子。它需要Hp理论、算子空间理论和非交换Lp-空间的不同工具的组合。一个主要的挑战是找到非交换的技术，取代使用某些关键的几何性质的欧几里德空间。PI的工作算子值的哈代空间了初步的一步方向提出的研究。拟议的项目将提供一个理论对应的最近的工作由Pisier/徐，荣格，和荣格/徐的非交换鞅，并将补充Arveson的工作非交换分析哈代空间。量子力学和海森堡的不确定性原理允许许多经典数学理论的非对易推广（=量子化）以下冯·诺依曼和默里的开创性工作的非对易积分理论。在将经典概念引入非对易框架的努力中，出现了非常有趣的新现象和困难。例如，使用冯诺依曼代数的语言，现在可以讨论一个非交换域的期望退出时间，尽管我们永远看不到这个域的“点”。该项目将继续这一长期的量化努力，并将探索和加强数学和物理学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.