Project in Operator Algebra

摘要：本课题的目的是研究von Neumann代数结构理论中的一系列问题，包括离散群和变形量化理论中该类代数的调和分析以及这些von Neumann代数的凸分析。目标是确定反映（量子）非交换概率性质的冯·诺伊曼代数的结构，并将这种结构与李群及其离散子群的表示理论联系起来。这种von Neumann代数的不变量，将在本项目中进行研究，来自于对凸集结构的分析，凸集由与这些von Neumann代数相关的非交换矩组成。这种方法与Voiculescu的自由熵有关，其中这些集合的（重整化的）渐近度量起着至关重要的作用。本课题研究的算子代数与Wigner引入的原子模型有关。他的方法虽然在哲学上依赖于海森堡的测不准原理，但却是基于随机矩阵理论。随机矩阵在预测理论或大气科学中也有许多其他应用。Voiculescu最近的一个令人惊讶的发现表明，随机矩阵是与量子概率相关的冯·诺伊曼代数的渐近生成器。本课题的目的是发展冯·诺依曼代数中的随机矩阵与量子化理论之间的概念关系。这样的发展可能会解释两种理论（随机矩阵与无限矩阵）之间的互补性，因此有望提供新的工具来解决这些领域中许多长期存在的问题。

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.