Independences and Partial $R$-Transforms in Bi-Free 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 $R$-transforms in two settings relating the moments and cumulants of a left-right pair of operators.

References in corpus (4)

• Free Probability for Pairs of Faces I
• Combinatorics of Bi-Freeness with Amalgamation
• Double-ended queues and joint moments of left-right canonical operators on full Fock space
• On two-faced families of non-commutative random variables

Cited by in corpus (5)

• Conditionally Bi-Free Independence for Pairs of Algebras
• On Operator-Valued Bi-Free Distributions
• Towards a Classification of Multi-Faced Independence: A Representation-Theoretic Approach
• Conditionally Bi-Free Independence with Amalgamation
• Bi-Free Entropy with Respect to Completely Positive Maps