Free Probability Theory and its Applications in Operator Algebras

自由概率论及其在算子代数中的应用

• 批准号: 0901344
• 负责人: Junhao Shen
• 金额: $ 13.52万
• 依托单位: University of New Hampshire
• 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 2012-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901344&HistoricalAwards=false
• 关键词: Free; Probability; Theory; its; Applications

AbstractShenThe PI will apply Voiculescu?s "non-commutative" (or "free") probability theory to the subject of operator algebras, which includes von Neumann algebras and C*-algebras. The study of operator algebras is often viewed as the study of "non-commutative measure theory" or ?non-commutative geometry?. Over the last seventy years, this has turned out to be a rich and fertile field of study. Non-commutative probability theory (or free probability theory) in the context of non-commutative measure spaces was developed by D. Voiculescu in the 1980's. Several important problems in the area of operator algebras were solved by the tools from non-commutative probability theory. In this project, the PI wishes to further develop free probability theory; with a view to attacking other problems in the area of operator algebras. In particular, he will study Voiculescu?s topological free entropy theory for unital C*-algebras; to search C*-algebras whose BDF-extension semigroups are not groups; to compute Voiculescu?s free entropy dimension for more finite von Neumann algebras; and to study the generator problems for von Neumann algebras.The theory of operator algebras was introduced to obtain a more rigorous mathematical formulation of the basics of quantum mechanics. From a probabilistic point view, free probability theory has surprising applications in the area of operator algebras. This project aims to develop new tools in free probability theory aiming at problems in the area operator algebras. The solutions to these problems will provide us with new ways to classify von Neumann algebras and C* algebras. The significance of the project lies in the fact that new developments in the theory of free probability and operator algebras have always had profound applications to several fields in mathematics and physics such as statistics and quantum mechanics. Intellectual Merit of the Proposed Activities: The project is the continuation of principal investigator's prior work in free probability theory, in BDF-extension semigroups, and in generator problem of von Neumann algebras. Broader Impact of the Proposed Activities: The project will take place at a location that strengthens the broader impacts of research development in the University of New Hampshire. The research group of Operator Algebra and Operator Theory in the University of New Hampshire consists of several well-established mathematicians. Summer support for graduate students will help graduate students to learn the subjects.This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5.

沈阳等人将把伏尔库列斯库·S的“非对易”(或“自由”)概率论应用于算子代数的研究，包括冯·诺依曼代数和C~*-代数。算子代数的研究通常被视为“非交换测度论”或“非交换几何”的研究。在过去的70年里，这被证明是一个丰富多彩的研究领域。非对易度量空间中的非对易概率论(或称自由概率论)是由D·沃库列斯库在20世纪80年代末由S发展起来的，用非对易概率论的工具解决了算子代数领域中的几个重要问题。在这个项目中，PI希望进一步发展自由概率理论，以期解决算子代数领域的其他问题。特别是，他将研究单位C*-代数的Vocerescu？S拓扑自由熵理论；寻找BDF-扩张半群不是群的C*-代数；计算更有限的von Neumann代数的Voulescu？S自由熵维度；研究von Neumann代数的生成元问题。引入算符代数理论，得到量子力学基本知识的更严格的数学表述。从概率的角度来看，自由概率理论在算子代数领域有着惊人的应用。这个项目旨在开发自由概率理论中的新工具，以解决面积算子代数中的问题。这些问题的解决将为我们分类von Neumann代数和C*代数提供新的途径。该项目的意义在于，自由概率理论和算子代数的新发展一直在统计学和量子力学等数学和物理领域有着深刻的应用。拟议活动的智力价值：该项目是主要研究人员在自由概率论、BDF-扩张半群和von Neumann代数的生成元问题方面先前工作的继续。拟议活动的更广泛影响：该项目将在新汉普郡大学加强研究发展的更广泛影响的地点进行。新汉普郡大学的算子代数和算子理论研究小组由几位著名的数学家组成。对研究生的暑期支持将帮助研究生学习这些科目。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。