Project in Operator Algebra

算子代数项目

• 批准号: 9970486
• 负责人: Florin Radulescu
• 金额: $ 9.77万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970486&HistoricalAwards=false
• 关键词: Project; Operator; Algebra

AbstractRadulescuThe aim of this project is to investigate a range of problems in the structure theory of von Neumann algebras, concerning the harmonic analysis of such algebras arising in connection with discrete groups and in deformation quantization theory and on the convex analysis of these von Neumann algebras. The goal is to determine the structure of the von Neumann algebras that reflect the properties of (quantum) non commutative probability, and to relate this structure to the representation theory of Lie groups and their discrete subgroups. An invariant for such von Neumann algebras, which will be investigated in this project, comes from the analysis of the structure of convex sets, consisting of noncommutative moments associated with these von Neumann algebras. This approach is related to Voiculescu's free entropy, where the (renormalized) asymptotic measure of such sets plays an essential role. The operator algebras studied in this project are related to the atom model introduced by Wigner. His approach, while philosophically relying on Heisenberg's Uncertaneity Principle, was based on the theory of random matrices. The random matrices have also numerous other applications in prediction theory or atmospheric science. A surprising, recent, discovery, by Voiculescu, shows that the random matrices are, the asymptotic generators for the von Neumann algebras associated to quantum probability. The aim of this project is to develop a conceptual relation between the random matrices in von Neumann algebras and the quantization theory. Such a development would probably explain the complementarity between the two theories (random matrices versus infinite matrices) and hence would hopefully give new tools to solve a number of long-standing problems in these areas.

本课题的目的是研究von Neumann代数结构理论中的一系列问题，涉及与离散群和形变量子化理论有关的此类代数的调和分析以及von Neumann代数的凸性分析。目的是确定反映(量子)非对易概率性质的von Neumann代数的结构，并将这种结构与李群及其离散子群的表示理论联系起来。这类von Neumann代数的一个不变量来自于对凸集结构的分析，凸集由与这些von Neumann代数相关的非对易矩组成。这一方法与Voulescu的自由熵有关，其中此类集合的(重整化)渐近度量起着关键作用。本课题所研究的算子代数与Wigner提出的原子模型有关。虽然他的方法在哲学上依赖于海森伯格的不确定性原理，但它是基于随机矩阵理论的。随机矩阵在预测理论或大气科学中也有许多其他应用。沃库列斯库最近的一项惊人发现表明，随机矩阵是与量子概率有关的冯·诺依曼代数的渐近生成器。这个项目的目的是在von Neumann代数中的随机矩阵和量子化理论之间建立一个概念上的联系。这样的发展可能会解释这两种理论(随机矩阵和无限矩阵)之间的互补性，因此有望提供新的工具来解决这些领域中的一些长期存在的问题。