Free Probability and Problems in Operator Algebras

职务名称：自由概率论和算子代数中的问题技术描述：该项目涉及使用自由概率论的技术研究某些冯诺依曼代数和C *-代数。 关于von Neumann代数的一些问题要考虑的是自由群因子的同构问题，以及von Neumann代数的合并自由积和作为关于非迹态的自由积而产生的III型von Neumann代数的相关分类问题。关于与自由积有关的C*-代数，项目是研究简单性、稳定秩和纯无限性等性质，并求出它们的某些自同构的Voiculescu拓扑熵。 在一个不同的方向，该项目涉及到的研究从无限第二型冯诺依曼代数因子的理想ofelements的分解器;特别是，以表征哪些元素是分解器的条款，例如，广义奇异数。非技术说明：在20世纪80年代中期，Voiculescu发现（或创造，取决于你的观点），一种新的概率理论，它与通常的概率理论非常相似，除了通常的独立性概念被自由性取代，which哪一个is exhibited展示by certain某些noncommuting非commuting交换random随机variables变量. 事实上，自由性与自由群的组合学有关，并受到自由群的启发，自由群是具有“最大非交换性”的群。在过去的15年里，自由概率论被证明是一个基本的新理论，它触及了数学和物理学的各个领域，包括随机矩阵、组合数学和算子理论。 例如，自由度对随机矩阵中Wigner的随机律的出现提供了深刻而令人满意的解释，并已被证明对随机矩阵的进一步研究是非常有用的。 自由概率论的自然背景是非交换冯诺依曼代数和C*-代数，因为谱分析的全部力量可以发挥出来。 本项目的主要目的是阐明与自由概率论有关的C ~*-代数和von Neumann代数的结构。 这一研究的成功完成将是自由概率理论的一个应用和阐明，并将加深我们对C ~*-代数和von Neumann代数的理解。 冯诺依曼代数和C*-代数首先考虑冯诺依曼和，分别，Gel'fand和Naimark，在20世纪30年代和40年代。它们是经典分析的非对易类似物的自然背景，因此它们自然地出现在量子力学的量子理论中。 此外，非交换方法越来越多地被用来研究数学中的经典交换问题-见证琼斯多项式用于区分结，和康纳斯的非交换几何方法用于研究分形。

Title: Free probability theory and problems in operator algebrasTechnical description: The project involves study certain von Neumann algebras andC*-algebras using the techniques of free probability theory. Some of theproblems on von Neumann algebras to be considered are the isomorphism problemfor free group factors, as well as related classification problems foramalgamated free products of von Neumannn algebras and for type III von Neumannalgebras arising as free products with respect to non-tracial states.Regarding C*-algebras related to free products, the project is to studyproperties like simplicity, stable rank and pure infiniteness, and to attemptto find Voiculescu's topological entropy of certain automorphisms of them. Ina different direction, the project involves the study of commutators ofelements from ideals of infinite type II von Neumann algebra factors; inparticular, to characterize which elements are commutators in terms of, forinstance, generalized singular numbers.Non-technical description: In the mid 1980's, Voiculescu discovered (orcreated, depending on your perspective), a new sort of probbility theorywhich is quite analogous to usual probability theory, except that theusual notion of independence is replaced by freeness, which is exhibitedby certain noncommuting random variables. In fact, freeness is relatedto and inspired by the combinatorics of free groups, which are groupswith "maximal noncommutativity." In the last decade and a half, freeprobability has been shown to be a fundamental new theory, which toucheson diverse areas of mathematics and physics, including random matrices,combinatorics and operator theory. For example, freeness provides adeep and satisfying explanation of the appearance of Wigner's semicirclelaw in random matrices, and has proven very useful for the further studyof random matrices. The natural context for free probability theory isnoncommutative von Neumann algebras and C*-algebras, because the fullpower of spectral analysis can be brought to bear. The project is toelucidate the structure of C*-algebras and von Neumann algebras relatedto free probability theory. Successful completion of this research willbe both and application and an elucidation of free probability theoryand will deepen our understanding of C*-algebras and von Neumannalgebras. Von Neumann algebras and C*-algebras were first considered byvon Neumann and, respectively, Gel'fand and Naimark, in the 1930s and1940s. They are natural contexts for noncommutative analogues ofclassical analysis, and they thus arise naturally in the mathematicaltheory of quantum mechanics. Moreover, noncommutative methods areincreasingly being used to study classically commutative problems inmathematics --- witness the Jones polynomial used to distinguish knots,and Connes' noncommutative geometric methods used to study fractals.