The Bi-Free Analogue of Microstate Free Entropy
In a series of 6+ papers, Voiculescu developed notions of entropy in free probability. One of these notions, known as microstate free entropy, is intimately connected with the asymptotic properties of matrices. Microstate free entropy and its analogues have been instrumental in advancing the theory of von Neumann algebras. In this talk, we will introduce the audience to microstate free entropy. After motivating, defining, and showing some elementary properties of microstate free entropy, we will discuss the applications to von Neumann algebras. Finally, we will discuss the bi-free analogue of microstate free entropy and its potential to solve problems in von Neumann algebras.