Project in Operator Algebra

Radulescu在这个项目中将研究冯诺依曼代数结构理论中的一系列问题。 他将专注于谐波分析的冯诺依曼代数所产生的连接与离散群体，自由概率和变形量子化理论。目标是确定反映非交换概率性质的冯诺依曼代数的结构，并将此结构与李群及其离散子群的表示理论联系起来。拉杜列斯库打算调查问题的特点因素的离散群体计算其不变量，使用的想法默里和冯诺依曼，有关的基本群体这样一个代数，一套所有指数值的子因素或结构的凸集的非交换时刻。他还将调查康纳斯嵌入猜想通过张量积算子代数。这项研究的目的是发现非交换方面的真实的字隐藏在“量子”结构。非交换性是自然界的一个真实方面：例如，两个矩阵相乘时不需要交换。著名的海森堡不确定性原理揭示了自然界同样的非对易性。冯·诺依曼发展了一类“连续”矩阵，它在保持通常矩阵的一些性质的同时，足够微妙地编码量子力学的许多方面。 最近的发展琼斯已经证明，方面的冯诺依曼代数是密切相关的方面的拓扑结构，纽结理论，并最终生物学。同样，Voiculescu最近的工作证明了存在一个由同一类型的对象编码的非交换概率论。这个研究项目的目的是有助于冯·诺依曼代数的一些结构方面的理解和分类。_--

AbstractRadulescuRadulescu will investigate in this project a range of problems in the structure theory of von Neumann algebras. He will concentrate on the harmonic analysis of the von Neumann algebras arising in connection with discrete groups, free probability and in deformation quantization theory. The goal is to determine the structure of the von Neumann algebras that reflect the properties of non commutative probability, and to relate this structure to the representation theory of Lie groups and their discrete subgroups. Radulescu intends to investigate the problem of characterizing the factors of discrete groups by computing their invariants, using ideas of Murray and von Neumann, related to the fundamental groups of such an algebra, the set of all index values for subfactors or the structure of convex sets of non-commutative moments. He will investigate also Connes embedding conjecture via tensor products on operator algebras.The aim of this research is to discover non-commutative aspects of the real word hidden in "quantum" structures. Non-commutativity is a true aspect of the nature: for example two matrices do not necessary commute when multiplied. The famous Heisenberg's Uncertaneity principle reveals the same non-commutative aspect of nature. Von Neumann has developed a class of "continuous" matrices, that, while keeping some of the properties of usual matrices, are subtle enough to encode many of the aspects of quantum mechanics. Recent developments by Jones have proven that aspects of such von Neumann algebras are intimately related to aspects of topology, knot theory and ultimately biology. Likewise recent work by Voiculescu has proven that the there is a non-commutative probability theory encoded by the same type of objects. The aim of this research project is to contribute to the understanding and classification of some of the structural aspects of von Neumann algebras._________________________________________--