Communications in Mathematical Physics

, Volume 338, Issue 2, pp 801–847 | Cite as

Combinatorics of Bi-Freeness with Amalgamation

  • Ian Charlesworth
  • Brent Nelson
  • Paul Skoufranis


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.


Probability Space Moment Function Multiplicative Function Free Probability Unital Algebra 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Ian Charlesworth
    • 1
  • Brent Nelson
    • 1
  • Paul Skoufranis
    • 2
  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

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