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.

该奖项是根据2009年的《美国回收与再投资法》（公法111-5）资助的。PI的研究以谐波分析的功能分析方面为中心。该项目致力于理解现实的空间和非交通环境中的利特伍德 - 帕利理论。 PI将将结果应用于操作员代数/非交通性几何形状以及经典分析。目标之一是了解非交通群体的傅立叶乘数。它需要从HP理论，操作员空间理论和非交通型LP空间的不同工具组合。一个主要的挑战是找到取代欧几里得空间某些至关重要的几何特性的非交换技术。 PI在拟议研究的方向上朝着操作员价值的Hardy空间迈出了第一步。拟议的项目将提供非交通性群众Pisier/Xu，Junge和Junge/Xu最近工作的理论上，并将补充Arveson在非交通性分析性空间上的工作。它还将提高对von Neumann代数（Quantum Mechanics）和海森伯格（Heisenberg）的不确定性原则（完全）正运算符的半群的理解，从而允许在冯·诺伊曼（Von Neumann）和默里（Murray）在非共同整合理论方面的先导工作的经典数学理论的许多非交易性概括（=量化）。非常有趣的新现象和困难是在将经典概念采用非共同框架的努力中出现的。例如，使用von Neumann代数的语言，尽管我们永远看不到该领域的“点”，但现在可以谈论非交通域的预期退出时间。该项目将继续进行长期的量化工作，并将探索和加强数学和物理学不同分支之间的深厚联系。反过来，它将为预测理论，h-侵权控制，信号和图像处理，统计和量子力学做出宝贵的贡献。