Finite Factors, Operators, and Free Probability

Indiana University, Bloomington IN

Investigators

Abstract

This project focuses on several aspects of operator theory and function theory that arise in the study of free random variables and related areas. The tools involved are partly intrinsic to these areas, but also involve input from other areas, such as combinatorics and algebraic geometry. One important theme is the use of the combinatorial Littlewood-Richardson rule in the study of eigenvalue problems for compact operators on a Hilbert space or for self-adjoint elements in a finite von Neumann algebra. This rule (and its continuous extensions) also plays a role in a different direction concerning the classification of invariant subspaces of certain operators. A second major theme is the study of weak and strong limit laws in free probability, as well as regularity questions for free convolutions. There are many problems here where methods of classical function theory yield interesting, and sometimes unexpected results. Other problems of operator theory to be considered treat the spectral Nevanlinna-Pick problem and its analogues, hyperinvariant subspaces, and p-entropies. Of these questions, the Nevanlinna-Pick problem is inspired by control theory questions and may have practical applications, while p-entropies have applications in theoretical computer science. The aim of this project is to solve several problems in functional analysis by highlighting their connections to areas of mathematics that do not seem, at first glance, to be related. An example is the use of methods of algebraic geometry in the solution of analysis problems. Some of the areas of study in this project have potential applications in engineering, especially in control theory (specifically, the control of large structures), and earlier results of the principal investigator have actually been incorporated into engineering projects. Besides their purely scientific merit, the activities supported by this grant are expected to have an impact through the training of doctoral and postdoctoral students and the creation of new course materials that will incorporate some of the research findings from the project. The principal investigator has trained a number of Ph.D. students, of whom three are women and one is Hispanic. The work of these students has had a significant impact on the fields that they entered (operator theory, free probability, control theory, and partial differential equations). The principal investigator has also mentored several postdoctoral associates who went on to successful careers. Finally, the principal investigator has directed two undergraduates in the Indiana University REU program. As indicated, mentoring activities at the undergraduate, graduate, and postdoctoral levels will play an integral role in this project.