Mathematical Sciences: Projects in Operator Algebras

小行星9622911 这个建议的目的是调查的一系列问题的结构理论的冯诺依曼代数。本研究的重要对象是与离散群和形变量子化理论有关的冯诺依曼代数的调和分析。主要的工作是确定冯诺依曼代数的结构，这些代数反映了（量子）非交换概率的性质，并将这种结构与李群及其离散子群的表示论联系起来。Voiculescu在非交换概率论中的最新进展表明，随机矩阵的渐近行为最好由与自由群相关的冯诺依曼代数中的元素表示，其谱分布是维格纳半圆律。在这项研究中的一个重要组成部分是发挥问题的特点因素的离散群体计算的不变量，定义的默里和冯诺依曼，有关的基本群体这样一个代数。这也是基于一些分析方面的数论和冯诺依曼代数的离散群。这种连通是通过模群及其子群的代数的一种新表示来实现的。这种表示是通过识别Toeplitz算子获得的，这些算子的符号是（算术）自守形式，在模群（或其同余子群）的不同表示之间具有交织算子。在这种情况下，算术Hecke运营商原来是完全积极的地图，其相关的子因子（通过康纳斯的对应理论）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，关于冯诺依曼代数，已被证明出现在几乎任何其他分支的数学。对这种现象的一种可能的解释是（正如冯·诺依曼首先暗示的那样）算子代数与自然界的（隐藏的）对称性有关，特别是与量子物理学家首先设想的时空中的对称性和运动有关。冯诺依曼代数是在这个项目中研究已被证明是密切相关的一些模型的原子，首先研究了维格纳。维格纳方法的一个想法是，鉴于海森堡的不确定性原理，研究这些模型的一种可能方法是通过随机矩阵来实现的。一个令人惊讶的发现，在过去五年中，这相当于研究某些性质的代数上述。随机矩阵本身在其他科学分支中有许多其他应用，如预测理论或大气科学，并且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.