Combinatorics of Bi-Freeness with Amalgamation
  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. 
Ian Charlesworth, Brent Nelson, and Paul Skoufranis
math.OA math.FA