Free Probability Theory and Applications to Free Group Factors

点击翻译按钮获取中文摘要

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.