Free Probability and Problems in Operator Algebras

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Title: Free probability theory and problems in operator algebras Technical description: The project involves study certain von Neumann algebras and C*-algebras using the techniques of free probability theory. Some of the problems on von Neumann algebras to be considered are the isomorphism problem for free group factors, as well as related classification problems for amalgamated free products of von Neumannn algebras and for type III von Neumann algebras arising as free products with respect to non-tracial states. Regarding C*-algebras related to free products, the project is to study properties like simplicity, stable rank and pure infiniteness, and to attempt to find Voiculescu's topological entropy of certain automorphisms of them. In a different direction, the project involves the study of commutators of elements from ideals of infinite type II von Neumann algebra factors; in particular, to characterize which elements are commutators in terms of, for instance, generalized singular numbers. Non-technical description: In the mid 1980's, Voiculescu discovered (or created, depending on your perspective), a new sort of probbility theory which is quite analogous to usual probability theory, except that the usual notion of independence is replaced by freeness, which is exhibited by certain noncommuting random variables. In fact, freeness is related to and inspired by the combinatorics of free groups, which are groups with "maximal noncommutativity." In the last decade and a half, free probability has been shown to be a fundamental new theory, which touches on diverse areas of mathematics and physics, including random matrices, combinatorics and operator theory. For example, freeness provides a deep and satisfying explanation of the appearance of Wigner's semicircle law in random matrices, and has proven very useful for the further study of random matrices. The natural context for free probability theory is noncommutative von Neumann algebras and C*-algebras, because the full power of spectral analysis can be brought to bear. The project is to elucidate the structure of C*-algebras and von Neumann algebras related to free probability theory. Successful completion of this research will be both and application and an elucidation of free probability theory and will deepen our understanding of C*-algebras and von Neumann algebras. Von Neumann algebras and C*-algebras were first considered by von Neumann and, respectively, Gel'fand and Naimark, in the 1930s and 1940s. They are natural contexts for noncommutative analogues of classical analysis, and they thus arise naturally in the mathematical theory of quantum mechanics. Moreover, noncommutative methods are increasingly being used to study classically commutative problems in mathematics --- witness the Jones polynomial used to distinguish knots, and Connes' noncommutative geometric methods used to study fractals.