The Theory of Free Probability and Some Applications in the Study of Random Matrices

Abstract

In this thesis, we provide a review of the theories of C*-algebras and free probability, and we offer a survey of some applications of free probability in the study of the spectrum of random matrices. Then, we study the occurence of *-freeness in tensor products of families of noncommutative random variables, with possible applications in the study of the spectrum of tensor products of large random matrices. Main Original Results: In this context, the main original results we obtained are as follows: general sufficient conditions for the *-freeness in tensor products (Corollary 6.11 and Proposition 6.15); a complete characterization for the occurence of *-freeness in the tensor product of two families that contain *-free variables (Theorem 6.20); and a nontrivial necessary condition for the occurence of *-freeness in tensor products in group algebras (Corollary 6.42).