von Neumann Algebras, Free Probability and Free Entropy

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The research objective of this project is to further develop "noncommutative" (or "free") probability theory and to find its applications to the subject of von Neumann algebras. John von Neumann introduced the concept of a von Neumann algebra in the 1930's to provide a natural framework for the study of noncommutative objects, initially in the context of quantum mechanics. The study of von Neumann algebras is often viewed as the study of "noncommutative measure spaces". Over the last seventy years, this has turned out to be a rich and fertile field of study, with a large number of concrete objects, unexpected properties and challenging and central open problems. Noncommutative probability theory (or free probability theory) in the context of noncommutative measure spaces was developed by Dan Voiculescu in the 1980's. Interesting for its sake, it was able to solve several important problems in the area of von Neumann algebras. We wish to further develop free probability theory; with a view to attacking other problems in the area of von Neumann algebras. In particular, we wish to compute the free entropy dimensions of von Neumann algebras by computing their free orbit dimensions; to search for new von Neumann algebras that have no Cartan subalgebras (Dan Voiculescu showed that free group factors have no Cartan subalgebras); to investigate maximal injective subalgebra problems of von Neumann algebras from the point view of free probability theory; and to find full factors that have Kadison's Similarity Property by using G. Pisier's similarity length. Main motivation for the introduction of von Neumann algebras is to obtain a more rigorous mathematical formulation of the basics of quantum mechanics. From a probabilistic point view, free probability theory has some surprising applications in the area of von Neumann algebras. Free probability theory gives us deep insights into many other open problems in the area of von Neumann algebras, such as the classification problem of von Neumann algebras, the generator question and decomposability. Further investigation of the links between these two fields, free probability theory and von Neumann algebras, is both necessary and urgent. This project aims to develop new tools in free probability theory aiming at problems in the area of von Neumann algebras. The solutions to these problems will provide us with new ways to classify von Neumann algebras. The significance of the project lies in the fact that new developments in the theory of free probability and von Neumann algebra 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 generator problem of von Neumann algebras and in maximal injective subalgebras 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. Joint-workshops will be organized to focus on helping graduate students to learn the subjects.