Conditionally Bi-Free Independence with Amalgamation

TL;DR

This paper introduces conditionally bi-free independence with amalgamation, defining related functions and transforms, and establishing key equivalences and limit theorems in operator-valued free probability theory.

Contribution

It develops the concept of conditionally bi-free independence with amalgamation, including new functions, transforms, and limit theorems, advancing operator-valued free probability.

Findings

Conditional bi-free independence with amalgamation is characterized by cumulant vanishing.

Operator-valued conditionally bi-free partial R-transform is constructed.

Various limit theorems in operator-valued conditionally bi-free probability are established.

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.