Advertisement

Archiv der Mathematik

, Volume 107, Issue 4, pp 445–453 | Cite as

Mixtures of classical and free independence

  • Roland SpeicherEmail author
  • Janusz Wysoczański
Article

Abstract

We revive the concept of \({\Lambda}\)-freeness of Młotkowski (Quant Prob Relat Top 7:27–41, 2004), which describes a mixture of classical and free independence between algebras of random variables. In particular, we give a description of this in terms of cumulants; this will be instrumental in the subsequent paper (Speicher and Weber, Quantum groups with partial commutation relations, 2016) where the quantum symmetries underlying these mixtures of classical and free independences will be considered.

Mathematics Subject Classification

46L54 

Keywords

Free independence Lambda-freeness Free cumulants 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ben Ghorbal A., Schürmann M.: Non-commutative notions of stochastic independence, Math. Proc. Cambridge Philos. Soc. 133, 531–561 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Charlesworth I., Nelson B., Skoufranis P.: On two-faced families of non-commutative random variables, Canad. J. Math. 67, 1290–1325 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Charney R.: An introduction to right-angled Artin groups, Geom. Dedicata 125, 141–158 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D. Foata and P. Cartier, Problèmes combinatiores de commutation et réarrengements, Springer-Verlag, Berlin, New York, 1969.Google Scholar
  5. 5.
    E. R. Green, Graph products of groups, PhD thesis, University of Leeds, 1990.Google Scholar
  6. 6.
    Hermiller S., Meier J.: Algorithms and geometry for graph products of groups, J. Algebra 171, 230–257 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kula A., Wysoczański J.: An example of a Boolean-free type central limit theorem, Probab. Math. Statist. 33, 341–352 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Mastnak M., Nica A.: Double-ended queues and joint moments of left–right canonical operators on full Fock space, Int. J. Math. 26, 1550016 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Młotkowski W.: \({\Lambda}\)-free probability, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7, 27–41 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Muraki N.: The five independences as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6, 337–371 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nica A., Speicher R.: Lectures on the combinatorics of free probability, volume 335 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  12. 12.
    R. Speicher, On universal products, Free probability theory (Waterloo, ON, 1995) 12 (1997), 257–266.Google Scholar
  13. 13.
    R. Speicher and M. Weber, Quantum groups with partial commutation relations preprint, arXiv:1603.09192, 2016.
  14. 14.
    Voiculescu D.-V.: Free probability for pairs of faces I, Comm. Math. Phys. 332, 955–980 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wysoczański J.: bm-independence and bm-central limit theorems associated with symmetric cones, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13, 461–488 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Universität des SaarlandesSaarbrückenGermany
  2. 2.Institute of MathematicsWroclaw UniversityWroclawPoland

Personalised recommendations