Free Probability Theory and Applications to Free Group Factors

自由概率论及其在自由群因子中的应用

• 批准号: 0600814
• 负责人: Kenneth Dykema
• 金额: $ 17.83万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600814&HistoricalAwards=false
• 关键词: Free; Probability; Theory; Applications; Group

The invariant subspace problem and its cousin, the hyperinvariant subspace problem, are fundamental questions about the structure of operators on Hilbert space whose solution, especially if accompanied by further detail about the subspaces and the resulting decomposition of the operator, would be an enormous advance in our understanding of such operators. Recently, there has been great progress in the problem of the existence of invariant subspaces of operators relative to a von Neumann algebra having a finite trace. A von Neumann algebra with a finite trace is an infinite dimensional, noncommutative analogue of a probability space, and many important classes of operators can be found inside von Neumann algebras having finite traces. However, the essential unsolved case and a key class of operators left virtually untouched by this recent progress is the class of quasinilpotent operators in a finite von Neumann algebra. The proposed research seeks to find invariant subspaces for them, relative to their von Neumann algebras. Free probability theory is an analogue of usual probability theory where independence is replaced by freeness, a completely noncommutative notion. Free probability theory and its off spring, free entropy, have been at the heart of much progress over the last decade in understanding certain classes of operators and von Neumann algebras with finite trace, called the free group factors. However, an outstanding open problem on them is whether all free group factors are isomorphic to each other or whether they constitute a diverse family. At the moment, this problem seems intimately bound up with the question of whether free entropy dimension is an invariant for von Neumann algebras having finite trace, or whether it can take different values on different sets of generators of a given von Neumann algebra. A second part of the proposed research will test this invariance question by computing the free entropy dimension of certain recently discovered "exotic" generators of free group factors.

不变子空间问题及其近亲超不变子空间问题是关于希尔伯特空间上算子结构的基本问题，其解决方案，特别是如果伴随着有关子空间的进一步细节和算子的分解，将是一个巨大的进步。我们对此类运营商的理解。最近，关于具有有限迹的冯·诺依曼代数的算子不变子空间的存在性问题取得了很大进展。具有有限迹的冯诺依曼代数是概率空间的无限维、非交换的类似物，并且可以在具有有限迹的冯诺依曼代数中找到许多重要的算子类别。然而，未解决的基本情况和最近进展几乎未触及的一类关键算子是有限冯诺依曼代数中的准幂算子类。拟议的研究旨在找到它们相对于冯诺依曼代数的不变子空间。自由概率论与通常的概率论类似，其中独立性被自由性所取代，自由性是一种完全不可交换的概念。自由概率论及其后代——自由熵——在过去十年中一直是理解某些类算子和有限迹冯诺依曼代数（称为自由群因子）方面取得重大进展的核心。然而，它们的一个突出的悬而未决的问题是，所有自由群体因素是否彼此同构，或者它们是否构成一个多样化的家庭。目前，这个问题似乎与以下问题密切相关：自由熵维数是否是具有有限迹的冯·诺依曼代数的不变量，或者它是否可以在给定冯·诺依曼代数的不同生成元组上取不同的值。拟议研究的第二部分将通过计算某些最近发现的自由群因子的“奇异”生成器的自由熵维度来测试这个不变性问题。