Free Probability and Problems in Operator Algebras

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Title: Free probability theory and problems in operator algebrasTechnical description: The project involves study certain von Neumann algebras andC*-algebras using the techniques of free probability theory. Some of theproblems on von Neumann algebras to be considered are the isomorphism problemfor free group factors, as well as related classification problems foramalgamated free products of von Neumannn algebras and for type III von Neumannalgebras arising as free products with respect to non-tracial states.Regarding C*-algebras related to free products, the project is to studyproperties like simplicity, stable rank and pure infiniteness, and to attemptto find Voiculescu's topological entropy of certain automorphisms of them. Ina different direction, the project involves the study of commutators ofelements from ideals of infinite type II von Neumann algebra factors; inparticular, to characterize which elements are commutators in terms of, forinstance, generalized singular numbers.Non-technical description: In the mid 1980's, Voiculescu discovered (orcreated, depending on your perspective), a new sort of probbility theorywhich is quite analogous to usual probability theory, except that theusual notion of independence is replaced by freeness, which is exhibitedby certain noncommuting random variables. In fact, freeness is relatedto and inspired by the combinatorics of free groups, which are groupswith "maximal noncommutativity." In the last decade and a half, freeprobability has been shown to be a fundamental new theory, which toucheson diverse areas of mathematics and physics, including random matrices,combinatorics and operator theory. For example, freeness provides adeep and satisfying explanation of the appearance of Wigner's semicirclelaw in random matrices, and has proven very useful for the further studyof random matrices. The natural context for free probability theory isnoncommutative von Neumann algebras and C*-algebras, because the fullpower of spectral analysis can be brought to bear. The project is toelucidate the structure of C*-algebras and von Neumann algebras relatedto free probability theory. Successful completion of this research willbe both and application and an elucidation of free probability theoryand will deepen our understanding of C*-algebras and von Neumannalgebras. Von Neumann algebras and C*-algebras were first considered byvon Neumann and, respectively, Gel'fand and Naimark, in the 1930s and1940s. They are natural contexts for noncommutative analogues ofclassical analysis, and they thus arise naturally in the mathematicaltheory of quantum mechanics. Moreover, noncommutative methods areincreasingly being used to study classically commutative problems inmathematics --- witness the Jones polynomial used to distinguish knots,and Connes' noncommutative geometric methods used to study fractals.