Communications in Mathematical Physics

, Volume 301, Issue 3, pp 627–659 | Cite as

Asymptotic Infinitesimal Freeness with Amalgamation for Haar Quantum Unitary Random Matrices

  • Stephen CurranEmail author
  • Roland Speicher
Open Access


We consider the limiting distribution of \({U_NA_NU_N^*}\) and B N (and more general expressions), where A N and B N are N × N matrices with entries in a unital C*-algebra \({\mathcal B}\) which have limiting \({\mathcal B}\)-valued distributions as N → ∞, and U N is a N × N Haar distributed quantum unitary random matrix with entries independent from \({\mathcal B}\). Under a boundedness assumption, we show that \({U_NA_NU_N^*}\) and B N are asymptotically free with amalgamation over \({\mathcal B}\). Moreover, this also holds in the stronger infinitesimal sense of Belinschi-Shlyakhtenko.

We provide an example which demonstrates that this result may fail for classical Haar unitary random matrices when the algebra \({\mathcal B}\) is infinite-dimensional.


Joint Distribution Free Product Compact Quantum Group Block Versus Unital Algebra 
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We would like to thank T. Banica, M. Neufang, and D. Shlyakhtenko for several useful discussions. S.C. would like to thank his thesis advisor, D.-V. Voiculescu, for his continued guidance and support while completing this project.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Banica T.: Le groupe quantique compact libre U(n). Commun. Math. Phys. 190, 143–172 (1997)zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Banica T., Collins B.: Integration over compact quantum groups. Publ. Res. Inst. Math. Sci. 43, 277–302 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Banica T., Collins B.: Integration over quantum permutation groups. J. Funct. Anal. 242, 641–657 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Banica, T., Curran, S., Speicher, R.: De Finetti theorems for easy quantum groups. Ann. Probab. (in press)Google Scholar
  5. 5.
    Banica T., Speicher R.: Liberation of orthogonal Lie groups. Adv. Math. 222, 1461–1501 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Belinschi, S.T., Shlyakhtenko, D.: Free probability of type B: Analytic interpretation and applications. [math.OA], 2009
  7. 7.
    Biane P., Goodman F., Nica A.: Non-crossing cumulants of type B. Trans. Amer. Math. Soc. 355, 2263–2303 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Brown, N.P., Dykema, K.J., Jung, K.: Free entropy dimension in amalgamated free products. Proc. Lond. Math. Soc. 97(3), 339–367 (2008), With an appendix by Wolfgang LückGoogle Scholar
  9. 9.
    Chen Q., Przytycki J.H.: The Gram matrix of a Temperley-Lieb algebra is similar to the matrix of chromatic joins. Commun. Contemp. Math. 10, 849–855 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability. Int. Math. Res. Not. 953–982 (2003)Google Scholar
  11. 11.
    Collins B., Śniady P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773–795 (2006)zbMATHCrossRefADSGoogle Scholar
  12. 12.
    Curran S.: Quantum exchangeable sequences of algebras. Indiana Univ. Math. J. 58, 1097–1126 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Curran S.: Quantum rotatability. Trans. Amer. Math. Soc. 362, 4831–4851 (2010)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Fevrier, M., Nica, A.: Infinitesimal non-crossing cumulants and free probability of type B. [math.OA], 2009
  15. 15.
    Kodiyalam, V., Sunder, V.S.: Temperley-Lieb and non-crossing partition planar algebras. In: Noncommutative rings, group rings, diagram algebras and their applications, Vol. 456 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2008, pp. 61–72Google Scholar
  16. 16.
    Köstler C., Speicher R.: A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation. Commun. Math. Phys. 291, 473–490 (2009)zbMATHCrossRefADSGoogle Scholar
  17. 17.
    Nica A., Speicher R.: Lectures on the combinatorics of free probability. Vol. 335 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2006)Google Scholar
  18. 18.
    Speicher R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Amer. Math. Soc. 132, x+88 (1998)MathSciNetGoogle Scholar
  19. 19.
    Voiculescu, D.: Symmetries of some reduced free product C*-algebras. In: Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983), Vol. 1132 of Lecture Notes in Math., Berlin: Springer, 1985, pp. 556–588Google Scholar
  20. 20.
    Voiculescu D.: Limit laws for random matrices and free products. Invent. Math. 104, 201–220 (1991)zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Wang S.: Free products of compact quantum groups. Commun. Math. Phys. 167, 671–692 (1995)zbMATHCrossRefADSGoogle Scholar
  22. 22.
    Wang S.: Quantum symmetry groups of finite spaces. Commun. Math. Phys. 195, 195–211 (1998)zbMATHCrossRefADSGoogle Scholar
  23. 23.
    Woronowicz S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111, 613–665 (1987)zbMATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA
  2. 2.Saarland University, FR 6.1 - MathematikSaarbruckenGermany
  3. 3.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

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