Combinatorics of Bi-Freeness with Amalgamation
arXiv:1408.3251 · doi:10.1007/s00220-015-2326-8
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.
35 pages