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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 with amalgamation. Further, we develop a multiplicative convolution for operator-valued random variables and explore ways to construct bi-free pairs of B-faces.

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Correspondence to Paul Skoufranis.

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Communicated by Y. Kawahigashi

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Charlesworth, I., Nelson, B. & Skoufranis, P. Combinatorics of Bi-Freeness with Amalgamation. Commun. Math. Phys. 338, 801–847 (2015).

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