Invariant Subspaces and Free Probability in the Context of von Neumann algebras

AbstractDykema不变的子空间问题和它的表弟，hyperinvariant子空间问题，是基本问题的结构运营商在希尔伯特空间的解决方案，特别是如果伴随着进一步的细节有关的子空间和由此产生的分解的运营商，将是一个巨大的进步，在我们的理解这样的运营商。近年来，关于迹有限的vonNeumann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模拟，并且在具有有限迹的冯诺依曼代数中可以找到许多重要的算子类。然而，基本的未解决的情况下，一个关键类的运营商几乎没有触及这一最近的进展是类拟幂零算子在有限冯诺依曼代数。拟议的研究旨在找到他们的不变子空间，相对于他们的冯诺依曼代数。自由概率论是一种类似于通常的概率论，其中独立性被自由性取代，自由性是一个完全不可交换的概念。自由概率论和它的衍生物自由熵，在过去十年中，在理解某些类算子和具有有限迹的冯·诺依曼代数（称为自由群因子）方面取得了很大进展。然而，一个悬而未决的问题是，是否所有的自由群因子是同构的，或者它们是否构成一个不同的家庭。目前，这个问题似乎与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集合上取不同值的问题密切相关。拟议的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“外来”生成器的自由熵维数来测试这个不变性问题。

AbstractDykemaThe 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.