Mathematical Sciences: Projects in Operator Algebras

数学科学：算子代数项目

• 批准号: 9622911
• 负责人: Florin Radulescu
• 金额: $ 8.75万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622911&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Projects; Operator; Algebras

9622911 Florin Radulescu The aim of this proposal is the investigation of a range of problems in the structure theory of von Neumann algebras. The important objects in this research are the harmonic analysis of the von Neumann algebras arising in connection with discrete groups and in deformation quantization theory. The main stream is to determine the structure of the von Neumann algebras that reflect the properties of (quantum) non commutative probability and to relate this structure with the representation theory of Lie groups and their discrete subgroups. The recent advances by Voiculescu in noncommutative probability theory have shown that the asymptotic behavior of random matrices is best represented by elements in the von Neumann algebras associated with free groups, whose spectral distribution is the Wigner semicircular law. In this research an important part is played by the problem of characterizing the factors of discrete groups by computing the invariants, defined by Murray and von Neumann, related to the fundamental groups of such an algebra. This is also based on some analytical aspects of number theory and the von Neumann algebras of discrete groups. The connection is realized by using a new representation for the algebra of the modular group and its subgroups. This representation is obtained by the identification of Toeplitz operators whose symbols are (arithmetic) automorphic forms with intertwining operators between different representations of the modular groups (or its congruence subgroups). In this context, the arithmetic Hecke operators turn out to be completely positive maps for which the associated subfactors (via Connes's correspondence theory) have unexpected higher relative commutant invariants. Recently, in the last twenty five years, the theory of operator algebras and in particular its sub-theory, concerning the von Neumann algebras, has been proven to appear in almost any other branch of mathematics. One possible explanation for this phenomenon is that (as it was certainly first hinted by von Neumann) the operator algebras are concerned with the (hidden) symmetries of nature and in particular the symmetries and motion in the space-time as it was first envisaged by quantum physicists. The von Neumann algebras that are studied in this project have proven to be intimately related to some models for the atoms that were first studied by Wigner. One of the ideas in Wigner's approach was that a possible way to study those models, in view of Heisenberg's uncertainty principle, is realized by the random matrices. A surprising discovery in the last five years was that this amounts to the study of certain properties of the algebras described above. The random matrices in themselves have numerous other applications in other branches of science like prediction theory or atmospheric science, and there is acknowledged hope that the better our understanding is on random matrices and the algebras associated with them-the better will be our understanding on the above natural phenomena.

9622911弗洛林·拉杜列斯库这个建议的目的是研究冯·诺依曼代数结构理论中的一系列问题。本研究的重要对象是与离散群有关的von Neumann代数的调和分析和形变量子化理论。主流是确定反映(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。Voulescu在非对易概率论方面的最新进展表明，随机矩阵的渐近行为最好地由与自由群相关的von Neumann代数中的元素来表示，其谱分布是Wigner半圆定律。在这项研究中，一个重要的部分是通过计算Murray和von Neumann定义的与离散群的基本群有关的不变量来刻画离散群的因子的问题。这也是基于数论和离散群的von Neumann代数的一些分析方面。这种联系是通过模群及其子群的代数的一种新的表示来实现的。这种表示是通过识别Toeplitz算子得到的，Toeplitz算子的符号是(算术)自同构形式，并且在模群(或其同余子群)的不同表示之间交织着算子。在此背景下，算术Hecke算子被证明是完全正的映射，其关联子因子(通过Connes的对应理论)具有意想不到的更高的相对交换不变量。最近，在过去的25年里，关于von Neumann代数的算子代数理论，特别是它的子理论，已经被证明出现在几乎任何其他数学分支中。对这一现象的一种可能的解释是(正如冯·诺伊曼首先暗示的那样)算子代数与自然的(隐藏的)对称性有关，特别是与量子物理学家最初设想的时空中的对称性和运动有关。这个项目中研究的von Neumann代数已经被证明与Wigner最先研究的原子的一些模型密切相关。维格纳方法的一个想法是，根据海森伯格的测不准原理，研究这些模型的一种可能方法是通过随机矩阵实现的。过去五年的一个令人惊讶的发现是，这相当于研究上述代数的某些性质。随机矩阵本身在预测理论或大气科学等其他科学分支中也有许多其他应用，人们普遍希望，我们对随机矩阵及其相关代数的理解越好，我们对上述自然现象的理解就越好。