Combinatorics of Bi-Freeness with Amalgamation

Abstract

In this paper, we develop the theory of bi-freeness in an amalgamated setting. We construct the operator-valued bi-free cumulant functions, and show that the vanishing of mixed cumulants is necessary and sufficient for bi-free independence. Further, we develop a multiplicative convolution for operator-valued random variables and explore ways to construct bi-free pairs of B-faces.

arXiv comment35 pages

References in corpus (4)

• Free Probability for Pairs of Faces I
• Double-ended queues and joint moments of left-right canonical operators on full Fock space
• On two-faced families of non-commutative random variables
• Operator-valued distributions: I. Characterizations of freeness

Cited by in corpus (14)

• Conditionally Bi-Free Independence for Pairs of Algebras
• On Operator-Valued Bi-Free Distributions
• Independences and Partial $R$-Transforms in Bi-Free Probability
• A Combinatorial Approach to Voiculescu's Bi-Free Partial Transforms
• An alternating moment condition for bi-freeness
• Bi-Boolean independence for pairs of algebras
• Bi-monotonic independence for pairs of algebras
• Conditionally Bi-Free Independence with Amalgamation
• Analogues of Entropy in Bi-Free Probability Theory: Microstates
• Limit theorems in bi-free probability theory
• On Bi-R-Diagonal Pairs of Operators
• Analytic aspects of the bi-free partial R-transform
• Bi-Free Entropy with Respect to Completely Positive Maps
• Shuffle Algebras and Non-Commutative Probability for Pairs of Faces