Independences and Partial $R$-Transforms in Bi-Free Probability

TL;DR

This paper explores different notions of independence in bi-free probability, establishing new theorems, showing invariance properties, and constructing partial R-transforms to connect moments and cumulants.

Contribution

It introduces a Kac/Loeve theorem for bi-free independence, demonstrates bi-freeness preservation under tensoring, and constructs partial R-transforms linking moments and cumulants.

Findings

Boolean and monotone independence arise from bi-free pairs

Bi-freeness is preserved under tensoring with matrices

Partial R-transforms relate moments and cumulants in bi-free settings

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 $R$-transforms in two settings relating the moments and cumulants of a left-right pair of operators.