UNIVERSITY SYSTEM OF NEW HAMPSHIRE
The PI will apply Voiculescus "non-commutative" (or "free") probability theory to the subject of operator algebras, which includes von Neumann algebras and C*-algebras. The study of operator algebras is often viewed as the study of "non-commutative measure theory" or non-commutative geometry. Over the last seventy years, this has turned out to be a rich and fertile field of study. Non-commutative probability theory (or free probability theory) in the context of non-commutative measure spaces was developed by D. Voiculescu in the 1980's. Several important problems in the area of operator algebras were solved by the tools from non-commutative probability theory. In this project, the PI wishes to further develop free probability theory; with a view to attacking other problems in the area of operator algebras. In particular, he will study Voiculescus topological free entropy theory for unital C*-algebras; to search C*-algebras whose BDF-extension semigroups are not groups; to compute Voiculescus free entropy dimension for more finite von Neumann algebras; and to study the generator problems for von Neumann algebras.
The theory of operator algebras was introduced to obtain a more rigorous mathematical formulation of the basics of quantum mechanics. From a probabilistic point view, free probability theory has surprising applications in the area of operator algebras. This project aims to develop new tools in free probability theory aiming at problems in the area operator algebras. The solutions to these problems will provide us with new ways to classify von Neumann algebras and C* algebras. The significance of the project lies in the fact that new developments in the theory of free probability and operator algebras have always had profound applications to several fields in mathematics and physics such as statistics and quantum mechanics. Intellectual Merit of the Proposed Activities: The project is the continuation of principal investigator's prior work in free probability theory, in BDF-extension semigroups, and in generator problem of von Neumann algebras. Broader Impact of the Proposed Activities: The project will take place at a location that strengthens the broader impacts of research development in the University of New Hampshire. The research group of Operator Algebra and Operator Theory in the University of New Hampshire consists of several well-established mathematicians. Summer support for graduate students will help graduate students to learn the subjects.