Free Probability and Problems in Operator Algebras

题目：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技术研究某些von Neumann代数和c *-代数。von Neumann代数的一些问题是自由群因子的同构问题，以及von Neumann代数的合并自由积和III型von Neumann代数在非迹态下作为自由积产生的相关分类问题。对于与自由积相关的C*-代数，本课题主要研究其简单性、稳定秩和纯无穷等性质，并试图找到其某些自同构的Voiculescu拓扑熵。在另一个方向上，该项目涉及无限型冯·诺伊曼代数因子理想中元素的换向子的研究；特别是，用广义奇异数来描述哪些元素是对易子。非技术描述：在20世纪80年代中期，Voiculescu发现了（或者创造了，取决于你的观点）一种新的概率论，它非常类似于通常的概率论，除了通常的独立性概念被自由所取代，这是由某些非交换随机变量所表现出来的。事实上，自由与自由群的组合学有关，并受到其启发，自由群是具有“极大非交换性”的群。在过去的15年里，自由概率已经被证明是一个基本的新理论，它涉及到数学和物理的各个领域，包括随机矩阵、组合学和算子理论。例如，自由性对随机矩阵中维格纳半圆定律的出现提供了深刻而令人满意的解释，并被证明对随机矩阵的进一步研究非常有用。自由概率论的自然背景是非交换的冯·诺伊曼代数和C*-代数，因为谱分析的全部力量可以发挥作用。该项目旨在阐明与自由概率论有关的C*代数和冯·诺伊曼代数的结构。本研究的成功完成将是对自由概率论的阐释和应用，并将加深我们对C*代数和冯·诺伊曼代数的理解。冯·诺伊曼代数和C*-代数分别在20世纪30年代和40年代由冯·诺伊曼和Gel’fand和Naimark首先提出。它们是经典分析的非交换类比的自然背景，因此它们自然地出现在量子力学的数学理论中。此外，非交换方法越来越多地被用于研究数学中的经典交换问题，例如用于区分结的Jones多项式和用于研究分形的Connes非交换几何方法。

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.