Mathematical Sciences: Projects in Operator Algebras

[9622911] Florin Radulescu本提案的目的是研究冯·诺依曼代数结构理论中的一系列问题。本文的主要研究对象是离散群和变形量子化理论中von Neumann代数的谐波分析。目前的主流是确定反映（量子）非交换概率性质的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。Voiculescu在非交换概率论方面的最新进展表明，随机矩阵的渐近行为最好用与自由群相关的von Neumann代数中的元素来表示，自由群的谱分布符合Wigner半圆定律。在本研究中，通过计算Murray和von Neumann定义的与此类代数的基本群相关的不变量来表征离散群的因子的问题发挥了重要作用。这也是基于数论的一些分析方面和离散群的冯·诺伊曼代数。这种连接是通过使用模群及其子群的代数的新表示来实现的。这种表示是通过识别Toeplitz算子得到的，这些算子的符号是模群（或其同余子群）的不同表示之间的（算术）自同构形式的纠缠算子。在这种情况下，算术Hecke算子被证明是完全正映射，其相关子因子（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，算子代数理论，特别是它的子理论，关于冯·诺伊曼代数，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，特别是量子物理学家最初设想的时空中的对称性和运动。在这个项目中研究的冯·诺伊曼代数已被证明与维格纳首先研究的一些原子模型密切相关。维格纳方法中的一个想法是，考虑到海森堡的测不准原理，研究这些模型的一种可能方法是通过随机矩阵来实现的。在过去的五年里，一个令人惊讶的发现是，这相当于对上述代数的某些性质的研究。随机矩阵本身在预测理论或大气科学等其他科学分支中有许多其他应用，并且人们公认的希望是，我们对随机矩阵和与之相关的代数的理解越好，我们对上述自然现象的理解就会越好。

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.