Free Gibbs States: Von Neumann Algebras, Random Matrices, and Subfactors

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

This project intends to develop mathematical tools that are in common use in classical analysis (e.g., notions of regularity, ideas from the theory of partial differential equations) in the noncommutative context that arises in so-called free probability theory. The underlying mathematics is extremely rich and has many connections with quantum physics. Progress in development of these tools is expected to have a significant impact on von Neumann algebra theory and random matrix theory, as well as on subfactor theory. The principal aim fo the project is to understand the structure of a class of noncommutative states called "free Gibbs states." These objects appear in Voiculescu's free probability theory as analogs of classical-probability Gibbs distributions. Besides free probability, their properties turn out to be important in von Neumann algebra theory, random matrix theory, and subfactor theory. The main new tool useful for understanding these states is that of free transport. Developed by A. Guoinnet and the principal investigator, this tool has already been proved to be useful for answering questions such as isomorphism of certain von Neumann algebras and computations of standard invariants of certain subfactors. More development and understanding is needed. The project will study and develop noncommutative techniques in partial differential equations that would enable one to construct transport maps, as well as to study critical phenomena. Along the way, the principal investigator will investigate questions of regularity of noncommutative maps.