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Bi-monotonic Independence for Pairs of Algebras

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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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  1. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 6.

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

  7. 7.

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

  8. 8.

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

  9. 9.

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

  10. 10.

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

  11. 11.

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

  12. 12.

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

  13. 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)

  14. 14.

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 Unpublished (2000).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

  15. 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)

  16. 16.

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

  17. 17.

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

  18. 18.

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

  19. 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)

  20. 20.

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

  21. 21.

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

  22. 22.

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

  23. 23.

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

  24. 24.

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

  25. 25.

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

  26. 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)

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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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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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  • Bi-monotonic independence
  • Bi-monotonic cumulants
  • Bi-monotonic convolution

Mathematics Subject Classification (2010)

  • Primary 46L53
  • Secondary 46L54