Skip to main content

General De Finetti Type Theorems in Noncommutative Probability

Abstract

We prove general de Finetti type theorems for classical and free independence. The de Finetti type theorems work for all non-easy quantum groups, which generalize a recent work of Banica, Curran and Speicher. We determine maximal distributional symmetries which means the corresponding de Finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one. In addition, we define Boolean quantum semigroups in analogous to the easy quantum groups, by universal conditions on matrix coordinate generators and an orthogonal projection. Then, we show a general de Finetti type theorem for Boolean independence.

This is a preview of subscription content, access via your institution.

References

  1. Arizmendi, O., Hasebe, T., Lehner, F., Vargas, C.: Relations between cumulants in noncommutative probability. Adv. Math. 282, 56–92 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  2. Banica, T., Curran, S., Speicher, R.: Classification results for easy quantum groups. Pac. J. Math. 247(1), 1–26 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  3. Banica, T., Curran, S., Speicher, R.: De Finetti theorems for easy quantum groups. Ann. Probab. 40(1), 401–435 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  4. Bożejko, M., Speicher, R.: \(\psi \)-independent and symmetrized white noises, pp. 219–236. Quantum Probab. Relat. Top, VI (1991)

    MATH  Google Scholar 

  5. Curran, S.: Quantum rotatability. Trans. Am. Math. Soc. 362(9), 4831–4851 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  6. Freedman, D.A.: Invariants under mixing which generalize de Finetti’s theorem: continuous time parameter. Ann. Math. Stat. 34, 1194–1216 (1963)

    MathSciNet  MATH  Article  Google Scholar 

  7. Hayase, T.: De Finetti theorems for a Boolean analogue of easy quantum groups. J. Math. Sci. Univ. Tokyo 24(3), 355–398 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Kallenberg, O.: Spreading-invariant sequences and processes on bounded index sets. Probab. Theory Related Fields 118(2), 211–250 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  9. Kallenberg, O.: Probabilistic Symmetries and Invariance Principles. Probability and Its Applications. Springer, New York (2005)

    MATH  Google Scholar 

  10. Köstler, C., Speicher, R.: A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation. Commun. Math. Phys. 291(2), 473–490 (2009)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  11. Lehner, F.: Cumulants in noncommutative probability theory. I. Noncommutative exchangeability systems. Math. Z. 248(1), 67–100 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  12. Liu, W.: A noncommutative de Finetti theorem for boolean independence. J. Funct. Anal. 269(7), 1950–1994 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  13. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335. Cambridge University Press, Cambridge (2006)

  14. Popa, M.: A new proof for the multiplicative property of the Boolean cumulants with applications to the operator-valued case. Colloq. Math. 117(1), 81–93 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  15. Ryll-Nardzewski, C.: On stationary sequences of random variables and the de Finetti’s equivalence. Colloq. Math. 4, 149–156 (1957)

    MathSciNet  MATH  Article  Google Scholar 

  16. Sołtan, P.M.: Quantum families of maps and quantum semigroups on finite quantum spaces. J. Geom. Phys. 59(3), 354–368 (2009)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  17. Sołtan, P.M.: On quantum semigroup actions on finite quantum spaces. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 12(3), 503–509 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  18. Speicher, R.: On universal products. In: Free probability theory (Waterloo, ON, 1995), volume 12 of Fields Inst. Commun., pp. 257–266. American Mathematical Society, Providence, RI (1997)

  19. Speicher, R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Am. Math. Soc. 132(627), x+88 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Speicher, Roland, Woroudi, Reza: Boolean convolution. In: Free probability theory (Waterloo, ON, 1995), volume 12 of Fields Inst. Commun., pp. 267–279. American Mathematical Society, Providence, RI (1997)

  21. Strătilă, S.: Modular theory in operator algebras. Editura Academiei Republicii Socialiste România, Bucharest; Abacus Press, Tunbridge Wells, (1981). Translated from the Romanian by the author

  22. Voiculescu, D.V., Dykema, K.J., Nica, A.: Free random variables, volume 1 of CRM Monograph Series. American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups

  23. Wang, S.: Free products of compact quantum groups. Commun. Math. Phys. 167(3), 671–692 (1995)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  24. Wang, S.: Quantum symmetry groups of finite spaces. Commun. Math. Phys. 195(1), 195–211 (1998)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  25. Weber, M.: On the classification of easy quantum groups. Adv. Math. 245, 500–533 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  26. Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111(4), 613–665 (1987)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  27. Woronowicz, S.L.: Unbounded elements affiliated with \(C^*\)-algebras and noncompact quantum groups. Commun. Math. Phys. 136(2), 399–432 (1991)

    ADS  MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgement

The author would like to thank his thesis advisor, D.-V. Voiculescu, for his continued guidance and support while completing this project.While working on this paper, the author was supported in part by funds from NSF grant DMS-1301727.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Weihua Liu.

Additional information

Communicated by Y. Kawahigashi

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, W. General De Finetti Type Theorems in Noncommutative Probability. Commun. Math. Phys. 369, 837–866 (2019). https://doi.org/10.1007/s00220-019-03471-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03471-y