Finite Factors, Free Probability, and Combinatorics in Operator Theory

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Problems arising in control theory, in the physical sciences, and in other areas of mathematics have led naturally to the consideration of linear maps between infinite-dimensional spaces, as well as to other objects constructed from collections of such maps. This project is intended to add to our understanding of these concepts, with applications to the areas mentioned before as possibly distant goals. The work on this project involves mathematics from apparently disparate mathematical disciplines, such as combinatorics and probability. More specifically, it engages various newer probabilistic concepts that are appropriate for a noncommutative setting. The principal investigator will involve both graduate and undergraduate students in this research. Indeed, some of the combinatorial problems can be formulated in a sufficiently elementary manner as to be accessible without too much technical background. (Previous work with undergraduate students did result in published material.)The concrete problems in this proposal fall into several categories. The first one concerns the Littlewood-Richardson rule from combinatorics and representation theory. It is hoped that appropriate formulations of this rule will be found that apply to the study of sums and products of compact operators and elements of finite von Neumann algebras. The second category is connected with the free probability theory introduced by Voiculescu. Here the problems revolve around the addition problem for collections of free random variables and its applications to random matrices (e.g., the study of outlying eigenvalues resulting from perturbations). The third category consists of several somewhat loosely connected problems from single operator theory or the closely related non-selfadjoint algebras they generate. The problems range from structure theory for commuting isometries to questions about hyper-reflexivity, invariant subspaces, and numerical ranges. (As far as applications are concerned, the second category of problems has connections to communications theory, while single operator theory is a valuable tool in control theory. These applications are not the focus of the proposed research, but earlier work of the principal investigator has had impact in applied areas.)