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

不变子空间问题和它的近亲超不变子空间问题是关于Hilbert空间上算子结构的基本问题，特别是如果伴随着对子空间和由此得到的算子分解的进一步细节，它的解决将是我们对这类算子的理解的巨大进步。最近，关于具有有限迹的von Neumann代数的算子不变子空间的存在性问题取得了很大的进展。具有有限迹的von Neumann代数是概率空间的无限维非对易模拟，在具有有限迹的von Neumann代数中可以找到许多重要的算子类。然而，有限von Neumann代数中的拟幂零算子类是最基本的未解决的情况，也是最近这一进展几乎未触及的一类关键算子。所提出的研究试图找到它们相对于它们的von Neumann代数的不变子空间。自由概率论类似于通常的概率论，其中独立性被自由取代，自由是一个完全不可交换的概念。在过去的十年里，在理解某些算符和具有有限迹的von Neumann代数(称为自由群因子)方面，自由概率理论及其衍生的自由熵一直是许多进步的核心。然而，关于它们的一个突出的悬而未决的问题是，所有的自由基团因子是否彼此同构，或者它们是否构成了一个不同的家庭。目前，这个问题似乎与这样一个问题紧密地联系在一起：对于具有有限迹的von Neumann代数，自由熵维度是否是不变量，或者它是否可以在给定的von Neumann代数的不同生成元集上取不同的值。这项拟议研究的第二部分将通过计算最近发现的某些自由群因子的“奇异”生成器的自由熵维度来测试这个不变性问题。

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.