Conditionally Bi-Free Independence with Amalgamation

Abstract

In this paper, we introduce the notion of conditionally bi-free independence in an amalgamated setting. We define operator-valued conditionally bi-multiplicative pairs of functions and construct operator-valued conditionally bi-free moment and cumulant functions. It is demonstrated that conditionally bi-free independence with amalgamation is equivalent to the vanishing of mixed operator-valued bi-free and conditionally bi-free cumulants. Furthermore, an operator-valued conditionally bi-free partial $\mathcal{R}$-transform is constructed and various operator-valued conditionally bi-free limit theorems are studied.

References in corpus (2)

• Combinatorics of Bi-Freeness with Amalgamation
• Conditionally Bi-Free Independence for Pairs of Algebras

Cited by in corpus (1)

• Bi-Free Entropy with Respect to Completely Positive Maps