Non-commutative Independences for Pairs of Faces

Abstract

In non-commutative probability theories, many different notions of independence arise in various contexts. According to the early classification work, there are only five such notions that are universal/natural in the sense of Speicher and Muraki, with free independence playing a prominent role. This thesis follows Voiculescu's recent generalization of free independence to bi-free independence to allow the simultanous study of left and right non-commutative random variables. We consider similar generalizations of other types of independence in the literature and show that many of the important concepts such as cumulants, convolutions, transforms, limit theorems, and infinite divisibility have counterparts in the new setting. While many of the results in this thesis were to be expected, some peculiarities and difficulties do occur in this left-right framework as things become much more complicated.