Lp Estimates in Non-commutative Probability and Analysis

非交换概率和分析中的 Lp 估计

• 批准号: 0301116
• 负责人: Marius Junge
• 金额: $ 12.52万
• 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: 2003
• 资助国家: 美国
• 起止时间: 2003-05-15 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301116&HistoricalAwards=false
• 关键词: Lp; Estimates; Non; commutative; Probability

AbstractJungeThe starting point of this project on 'Lp estimates in noncommutative probability and analysis' are recent results and techniques from noncommutative martingales inequalities obtained by Pisier/Xu, Randrianantoanina, Junge and Junge/Xu. This new insight enables us to show a noncommutative analog the maximal ergodic theorem for completely positive maps and study square function inequalities in this setting (joint work with LeMerdy/Xu). The formulation and properties are motivated by the theory of Operator Spaces. On the other hand martingale inequalities are crucial in understanding independent, indiscernable and exchangeable sequences in noncommutative Lp spaces and properties of almost uniform convergence. Martingale inequalities and noncommutative probability are also fundamental tools in analyzing the operator space OH and its realization in the predual of type III von Neumann algebras. These techniques are similar to those used by Pisier/Shlyaktenko in proving the noncommutative version of Grothendieck's inequality. Surprising the analog of the 'little Grothendieck inequality' in the context of operator spaces only holds up to a logarithmic factor.Quantum mechanics and Heisenberg's uncertainty principle and mathematical models realizing these phenomena changed not only our perception of the world but also the mathematical discipline. Many noncommutative (=quantum) generalizations of classical mathematical theories for example the theory of quantum groups and noncommutative (=quantum) probability theory. Very interesting new phenomena and difficulties arise when adopting classical concepts to this noncommutative framework. Noncommutative measure theory and the theory of von Neumann algebras provide plenty of examples of genuinely new phenoma. Indeed, von Neumann's motivation for his work on operator algebras (now called von Neumann algebras) was to provide a good mathematical foundation for quantum mechanics. In this tradition the theory of Operator Spaces provides the right language for quantizing Banach spaces, a notion developed to describe the spaces of solutions of differential equations. For example, using this language 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. As a long term perspective these mathematical theories provide new features which may be used to understand phenomena in physics and other natural sciences.

该项目对“非交通概率和分析中的LP估计值”的起点是来自Pisier/Xu，Randrianantoanina，Junge和Junge/Junge/Xu获得的非交通性martingales不平等的最新结果和技术。 这个新的见解使我们能够在这种情况下显示出非交换性类似物的最大千古定理和学习方形函数的不等式（与Lemerdy/XU的联合工作）。 公式和属性是由操作员空间理论的动机。 另一方面，Martingale的不平等对于在非交通性LP空间和几乎均匀收敛的性质中独立，不可分辨和可交换序列至关重要。 Martingale的不平等和非公共概率也是分析操作员空间OH及其在III III型Von Neumann代数中实现的基本工具。 这些技术类似于Pisier/Shlyaktenko在证明Grothendieck不平等的非共同版本中使用的技术。 令人惊讶的是，在操作员空间的背景下，“小格伦迪克不平等”的类似物只能构成对数因素。QuantumMechanics和Heisenberg的不确定性原理和数学模型实现了这些现象，这不仅改变了我们对世界的看法，还改变了数学学科。经典数学理论的许多非交换性（=量子）概括（例如量子群和非交通性（=量子）概率理论的理论。当对这个非交通框架采用经典概念时，出现了非常有趣的新现象和困难。非交通量度理论和冯·诺伊曼代数理论提供了许多真正的新现象的例子。确实，冯·诺伊曼（Von Neumann）在操作员代数（现称von Neumann代数）上的工作是为量子力学提供良好的数学基础。 在这种传统中，操作员空间的理论提供了量化Banach空间的正确语言，这是一个旨在描述微分方程解决方案空间的概念。例如，使用这种语言，现在可以谈论非交通域的预期退出时间，尽管我们永远看不到该域的“点”。从长远来看，这些数学理论提供了新功能，可用于了解物理和其他自然科学中的现象。