Combinatorics of Bi-Freeness with Amalgamation

TL;DR

This paper advances the theory of bi-freeness with amalgamation by establishing cumulant functions, characterizing independence, and developing convolution and construction methods for operator-valued random variables.

Contribution

It introduces operator-valued bi-free cumulant functions, characterizes bi-free independence via cumulants, and develops a multiplicative convolution for operator-valued variables.

Findings

Vanishing mixed cumulants characterize bi-free independence.

Constructed a multiplicative convolution for operator-valued variables.

Explored methods to construct bi-free pairs of B-faces.

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.