Noncommutative Hardy Spaces and Littlewood-Paley Theory

非交换 Hardy 空间和 Littlewood-Paley 理论

• 批准号: 0901009
• 负责人: Marius Junge
• 金额: $ 11.1万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-15 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901009&HistoricalAwards=false
• 关键词: Noncommutative; Hardy; Spaces; Littlewood; Paley

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.

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资助。PI 的研究集中在调和分析的函数分析方面。该项目致力于理解非交换环境下的实哈迪空间和利特尔伍德佩利理论。 PI 将把结果应用于算子代数/非交换几何以及经典分析。目标之一是了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 空间的不同工具。一个主要的挑战是找到非交换技术来代替欧几里得空间的某些关键几何特性的使用。 PI 在算子值 Hardy 空间方面的工作朝着拟议研究的方向迈出了第一步。拟议的项目将提供 Pisier/Xu、Junge 和 Junge/Xu 最近关于非交换鞅的工作的理论对应，并将补充 Arveson 在非交换解析哈代空间方面的工作。它还将提高对冯·诺依曼代数上（完全）正算子半群的理解。继冯·诺依曼和默里关于非交换积分理论的开创性工作之后，量子力学和海森堡的不确定性原理允许经典数学理论的许多非交换概括（=量化） 。在将经典概念应用于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，使用冯诺依曼代数的语言，现在可以谈论非交换域的预期退出时间，尽管我们永远看不到该域的“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学不同分支之间的深层联系。反过来，它将为预测理论、H-无穷控制、信号和图像处理、统计学和量子力学做出宝贵的贡献。