Independences and Partial $R$-Transforms in Bi-Free Probability
Operator Algebras
2016-09-08 v2 Probability
Abstract
In this paper, we examine how various notions of independence in non-commutative probability theory arise in bi-free probability. We exhibit how Boolean and monotone independence occur from bi-free pairs of faces and establish a Kac/Loeve Theorem for bi-free independence. In addition, we prove that bi-freeness is preserved under tensoring with matrices. Finally, via combinatorial arguments, we construct partial -transforms in two settings relating the moments and cumulants of a left-right pair of operators.
Keywords
Cite
@article{arxiv.1410.4265,
title = {Independences and Partial $R$-Transforms in Bi-Free Probability},
author = {Paul Skoufranis},
journal= {arXiv preprint arXiv:1410.4265},
year = {2016}
}