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Bi-monotonic Independence for Pairs of Algebras

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Abstract

In this article, the notion of bi-monotonic independence is introduced as an extension of monotonic independence to the two-faced framework for a family of pairs of algebras in a non-commutative space. The associated cumulants are defined, and a moment-cumulant formula is derived in the bi-monotonic setting. In general, the bi-monotonic product of states is not a state and the bi-monotonic convolution of probability measures on the plane is not a probability measure. This provides an additional example of how positivity need not be preserved under conditional bi-free convolutions.

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References

  1. Belinschi, S.T., Bercovici, H., Gu, Y., Skoufranis, P.: Analytic subordination for bi-free convolution. J. Funct. Anal. 275(4), 926–966 (2018)

    Article  MathSciNet  Google Scholar 

  2. Bożejko, M., Leinert, M., Speicher, R.: Convolution and limit theorems for conditionally free random variables. Pac. J. Math. 175(2), 357–388 (1996)

    Article  MathSciNet  Google Scholar 

  3. Charlesworth, I.: An alternating moment condition for bi-freeness (2016). arXiv:1611.01262

  4. Charlesworth, I., Nelson, B., Skoufranis, P.: Combinatorics of bi-freeness with amalgamation. Commum. Math. Phys. 338(2), 801–847 (2015)

    Article  MathSciNet  Google Scholar 

  5. Charlesworth, I., Nelson, B., Skoufranis, P.: On two-faced families of non-commutative random variables. Can. J. Math. 67(6), 1290–1325 (2015)

    Article  MathSciNet  Google Scholar 

  6. Franz, U.: Monotone independence is associative. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 4(3), 401–407 (2001)

    Article  MathSciNet  Google Scholar 

  7. Franz, U.: Multiplicative Monotone Convolutions, vol. 73, pp. 153–166. Banach Center Publications, Warsaw (2006)

    MATH  Google Scholar 

  8. Gerhold, M.: Bimonotone Brownian motion (2017). arXiv:1708.03510

  9. Gu, Y., Skoufranis, P.: Conditionally bi-free independence for pairs of faces. J. Funct. Anal. 273(5), 1663–1733 (2017)

    Article  MathSciNet  Google Scholar 

  10. Gu, Y., Skoufranis, P.: Bi-boolean independence for pairs of algebras. Complex Anal. Oper. Thoery (to appear). arxiv:1703.03072

  11. Hasebe, T., Saigo, H.: Joint cumulants for natural independence. Electron. Commun. Probab. 16(44), 491–506 (2011)

    Article  MathSciNet  Google Scholar 

  12. Hasebe, T., Saigo, H.: The monotone cumulants. Ann. Inst. Henri Poincaré Probab. Stat. 47(4), 1160–1170 (2011)

    Article  MathSciNet  Google Scholar 

  13. Mastnak, M., Nica, A.: Double-ended queues and joint moments of left-right canonical operators on full Fock space. Int. J. Math. 26(2), 1550016 (2015)

    Article  MathSciNet  Google Scholar 

  14. Muraki, N.: Monotonic convolution and monotonic Lévy–Hinčin formula. Unpublished (2000). https://www.math.sci.hokudai.ac.jp/~thasebe/Muraki2000.pdf

  15. Muraki, N.: Monotonic independence, monotonic central limit theorem and monotonic law of small numbers. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 4(1), 39–58 (2001)

    Article  MathSciNet  Google Scholar 

  16. Muraki, N.: The five independences as quasi-universal products. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 5(1), 113–134 (2002)

    Article  MathSciNet  Google Scholar 

  17. Muraki, N.: The five independences as natural products. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 6(3), 337–371 (2003)

    Article  MathSciNet  Google Scholar 

  18. Popa, M.: A combinatorial approach to monotonic independence over a \(C^*\)-algebra. Pac. J. Math. 237, 299–325 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Speicher, R.: Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298(1), 611–628 (1994)

    Article  MathSciNet  Google Scholar 

  21. Speicher, R.: On universal products. Fields Inst. Commun. 12, 257–266 (1997)

    MathSciNet  MATH  Google Scholar 

  22. Speicher, R., Woroudi, R.: Boolean convolution. Fields Inst. Commun. 12, 267–280 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Voiculescu, D.: Addition of certain noncommuting random variables. J. Funct. Anal. 66(3), 323–346 (1986)

    Article  MathSciNet  Google Scholar 

  24. Voiculescu, D.: Operations on certain non-commutative operator-valued random variables. Astérisque 232, 243–275 (1995)

    MathSciNet  MATH  Google Scholar 

  25. Voiculescu, D.: Free probability for pairs of faces I. Commun. Math. Phys. 332(3), 955–980 (2014)

    Article  MathSciNet  Google Scholar 

  26. Voiculescu, D.: Free probability for pairs of faces II: 2-variables bi-free partial \(R\)-transform and systems with rank \(\le 1\) commutation. Ann. Inst. Henri Poincaré Probab. Stat. 52(1), 1–15 (2016)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

T.H. is supported by JSPS Grant-in-Aid for Young Scientists (B) 15K17549 and (A) 17H04823. P.S. is supported by NSERC (Canada) Grant RGPIN-2017-05711. The authors are grateful to Malte Gerhold for pointing out an error in a previous version of manuscript.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston, ON, K7L 3N6, Canada

    Yinzheng Gu

  2. Department of Mathematics, Hokkaido University, North 10, West 8, Kita-ku, Sapporo, 060-0810, Japan

    Takahiro Hasebe

  3. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

    Paul Skoufranis

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  1. Yinzheng Gu
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  2. Takahiro Hasebe
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  3. Paul Skoufranis
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Correspondence to Takahiro Hasebe.

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Gu, Y., Hasebe, T. & Skoufranis, P. Bi-monotonic Independence for Pairs of Algebras. J Theor Probab 33, 533–566 (2020). https://doi.org/10.1007/s10959-019-00884-2

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  • DOI: https://doi.org/10.1007/s10959-019-00884-2

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