Notions of Entropy in Free Probability Theory and Applications to Operator Algebras

The theory of free probability was developed by Voiculescu in the 1980's as an extension of classical probability to the non-commutative setting and aims to study algebras of operators from a probabilistic point of view. The connection of free probability with random matrix theory, as well as the development of notions of entropy in the non-commutative framework, lead to significant breakthroughs regarding the structure of certain von Neumann algebras. More recently, in 2013 Voiculescu laid the foundations of bi-free probability theory, which extends the free setting and involves the simultaneous study of left and right actions of algebras on reduced free product spaces. In this talk, we will give an overview of the contributions of free probability theory to the field of operator algebras and will discuss the development of notions of entropy within the context of bi-free probability theory.