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

【摘要】不变子空间问题及其近亲——超不变子空间问题，是关于Hilbert空间上算子结构的基本问题，它们的解，特别是伴随子空间的进一步细节和由此产生的算子分解，将极大地促进我们对这类算子的理解。近年来，对于具有有限迹的冯诺依曼代数算子的不变子空间的存在性问题，取得了很大的进展。具有有限迹的冯·诺伊曼代数是一个无限维的、非交换的概率空间的模拟，许多重要的算子类可以在具有有限迹的冯·诺伊曼代数中找到。然而，基本的未解决的情况和一类关键的算子几乎没有被这一最新进展所触及，那就是有限冯诺依曼代数中的一类拟无效算子。提出的研究旨在为他们找到不变的子空间，相对于他们的冯·诺伊曼代数。自由概率论是一般概率论的类似物，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产物——自由熵，在过去十年中，在理解某些类型的算子和冯·诺伊曼代数（称为自由群因子）的有限轨迹方面取得了许多进展。然而，一个悬而未决的突出问题是，所有的自由群体因素是否同构，或者它们是否构成一个多样化的家族。目前，这个问题似乎与自由熵维是否是具有有限迹的冯诺依曼代数的不变量的问题密切相关，或者它是否可以在给定的冯诺依曼代数的不同生成集上取不同的值。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“外来”生成器的自由熵维来检验这个不变性问题。

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.