Connections between Free Entropy and Model-Theory of Tracial W*-Algebras

We motivate and develop a version of free entropy for model-theoretic types in tracial von Neumann algebras.  Farah, Hart, and Sherman studied tracial von Neumann algebras from the viewpoint of Ben Yaacov's continuous model theory.  One notion of continuous model theory is the type of a tuple (X_1,...,X_d), which describes the values of all logical formulas evaluated on X_1, ..., X_d.  Ben Yaacov showed that diffuse classical probability spaces admit quantifier elimination, hence the type of a tuple of random variables (X_1,...,X_d) is uniquely determined by its probability distribution (which corresponds to its quantifier-free type).  While much of free probability theory focuses on the quantifier-free type of tuples, we propose to study the full type instead, and thus we develop a version of free entropy theory for full types.  As an application, we show that a matrix ultraproduct is not strongly 1-bounded, and that any W*-algebra such that all embeddings into a matrix ultraproduct are automorphically conjugate must be strongly 1-bounded.