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

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.