Project in Operator Algebra

RadulescuRadulescu将在这个项目中研究von Neumann代数结构理论中的一系列问题。他将专注于von Neumann代数的调和分析，这些分析与离散群、自由概率和形变量子化理论有关。目的是确定反映非对易概率性质的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。Radulescu试图利用Murray和von Neumann的思想，通过计算离散群的因子的不变量来刻画离散群的因子的问题，这些不变量与这样的代数的基本群、子因子的所有指标值的集合或非对易矩的凸集的结构有关。他还将研究Connes通过算子代数上的张量积嵌入猜想。这项研究的目的是发现隐藏在“量子”结构中的实数的非对易方面。非交换性是自然界的一个真实方面：例如，两个矩阵相乘时不一定要交换。著名的海森堡不确定原理揭示了自然界同样不可交换的一面。冯·诺伊曼开发了一类“连续”矩阵，这种矩阵在保留了通常矩阵的一些性质的同时，足够微妙，足以对量子力学的许多方面进行编码。Jones最近的发展已经证明，这种von Neumann代数的方面与拓扑学、纽结理论以及最终的生物学密切相关。同样，沃库列斯库最近的工作证明，存在由相同类型的物体编码的非对易概率理论。这项研究项目的目的是有助于了解和分类冯·诺伊曼algebras._________________________________________--的一些结构方面。

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._________________________________________--