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On Two-faced Families of Non-commutative Random Variables

Published online by Cambridge University Press:  20 November 2018

Ian Charlesworth ,
Brent Nelson  and
Paul Skoufranis
Ian Charlesworth
Affiliation:
Department of Mathematics, UCLA, Los Angeles, California, 90095, USA. e-mail: ilc@math.ucla.edu, bnelson6@math.ucla.edu, pskoufra@math.ucla.edu
Brent Nelson
Affiliation:
Department of Mathematics, UCLA, Los Angeles, California, 90095, USA. e-mail: ilc@math.ucla.edu, bnelson6@math.ucla.edu, pskoufra@math.ucla.edu
Paul Skoufranis
Affiliation:
Department of Mathematics, UCLA, Los Angeles, California, 90095, USA. e-mail: ilc@math.ucla.edu, bnelson6@math.ucla.edu, pskoufra@math.ucla.edu
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Abstract

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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.

Keywords

46L54free probabilityoperator algebrasbi-free
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 6 , 01 December 2015 , pp. 1290 - 1325
Copyright
Copyright © Canadian Mathematical Society 2015

References

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