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On Two-faced Families of Non-commutative Random Variables

  • Ian Charlesworth (a1), Brent Nelson (a1) and Paul Skoufranis (a1)

Abstract

We demonstrate that the notions of bi-free independence and combinatorial-bi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.

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[1] Mastnak, M. and Nica, A., Double-ended queues and joint moments of left-right canonical operators on a full Fock space. Internat. J. Math. 26(2015), no. z, 1550016, 34 pp.http://dx.doi.org/10.1142/S0129167X15500160
[2] Nica, A., R-transforms of free joint distributions and non-crossing partitions. J. Funct. Anal. 135(1996), 271–296. http://dx.doi.org/10.1006/jfan.1996.0011
[3] Nica, A. and Speicher, R., A “Fourier transform” for multiplicative functions on non-crossing partitions. J. Algebraic Combin. 6(1997), 141–160.http://dx.doi.org/10.1023/A:1008643104945
[4] Speicher, R., Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 294(1994), 611–628.http://dx.doi.org/10.1007/BF01459754
[5] Voiculescu, D.-V., Free probability for pairs of faces I. Comm. Math. Phys. 332(2014), 955–980.http://dx.doi.org/10.1007/s00220-014-2060-7
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On Two-faced Families of Non-commutative Random Variables

  • Ian Charlesworth (a1), Brent Nelson (a1) and Paul Skoufranis (a1)

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