%0 Journal Article
%T Combinatorics of Bi-Freeness with Amalgamation
%A Ian Charlesworth
%A Brent Nelson
%A Paul Skoufranis
%J Mathematics
%D 2014
%I arXiv
%R 10.1007/s00220-015-2326-8
%X 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.
%U http://arxiv.org/abs/1408.3251v1