Results on R-Diagonal Operators in Bi-Free Probability Theory and Applications of Set Theory to Operator Algebras

The contents of this dissertation lie in the branch of pure mathematics known as functional analysis and are focused on the theory of bi-free probability and on the interplay between set theory and the field of operator algebras. The material is comprised of two main parts. The first part of this dissertation investigates applications of set theory to operator algebras and is further divided into two chapters. The first chapter is focused on the Calkin algebra Q(H) and explores the class of C*-algebras which embed into it. We prove that under Martin's axiom every C*-algebra of density character less than the caridnality of the continuum embeds into the Calkin algebra and, moreover, we show that the assertion "every C*-algebra of density character less than continuum embeds into Q(H)'' is independent from ZFC. In the second chapter we investigate separably representable AF operator algebras from a descriptive set-theoretic viewpoint. Contrary to the case of separable AF C*-algebras which are classified up to isomorphism by their K-theory, we show that the canonical isomorphism relations for separable, non-self-adjoint AF operator algebras are not classifiable by countable structures. The second part of this dissertation focuses on the theory of bi-free probability. This part is further divided into four chapters, the first of which concerns the development of the theory of R-diagonal operators in the setting of bi-free probability theory. We define bi-R-diagonal pairs based on certain alternating cumulant conditions and give a complete description of their joint distributions in terms of their invariance under multiplication by bi-Haar unitary pairs. The final three chapters of this manuscript concern the development of non-microstate bi-free Fisher information and entropy with respect to completely positive maps. By extending the operator-valued bi-free structures and allowing the implementation of completely positive maps into bi-free conjugate variable expressions, we define notions of Fisher information and entropy which generalize the corresponding notions of entropy in the bi-free setting. As an application we show that minimal values of the bi-free Fisher information and maximal values of the non-microstates bi-free entropy are attained at bi-R-diagonal pairs of operators.