Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-15T17:14:49.665Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected], [email protected]
Brent Nelson
Affiliation:
Department of Mathematics, UCLA, Los Angeles, California, 90095, USA. e-mail: [email protected], [email protected], [email protected]
Paul Skoufranis
Affiliation:
Department of Mathematics, UCLA, Los Angeles, California, 90095, USA. e-mail: [email protected], [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[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 Google Scholar
[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 Google Scholar
[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 Google Scholar
[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 Google Scholar
[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 Google Scholar