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Combinatorics of Bi-Freeness with Amalgamation

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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 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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Authors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, 90095, USA

    Ian Charlesworth & Brent Nelson

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Paul Skoufranis

Authors
  1. Ian Charlesworth
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  2. Brent Nelson
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  3. Paul Skoufranis
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Corresponding author

Correspondence to Paul Skoufranis.

Additional information

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). https://doi.org/10.1007/s00220-015-2326-8

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  • DOI: https://doi.org/10.1007/s00220-015-2326-8

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