In this paper, the notion of bi-Boolean independence for non-unital pairs of algebras is introduced thereby extending the notion of Boolean independence to pairs of algebras. The notion of B-\((\ell , r)\)-cumulants is defined via a bi-Boolean moment-cumulant formula over the lattice of bi-interval partitions, and it is demonstrated that bi-Boolean independence is equivalent to the vanishing of mixed B-\((\ell , r)\)-cumulants. Furthermore, some of the simplest bi-Boolean convolutions are considered, and a bi-Boolean partial \(\eta \)-transform is constructed for the study of limit theorems and infinite divisibility with respect to the additive bi-Boolean convolution. In particular, a bi-Boolean Lévy–Hinčin formula is derived in perfect analogy with the bi-free case, and some Bercovici–Pata type bijections are provided. Additional topics considered include the additive bi-Fermi convolution, some relations between the \((\ell , r)\)- and B-\((\ell , r)\)-cumulants, and bi-Boolean independence in an amalgamated setting.
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Belinschi, S., Nica, A.: \(\eta \)-series and a Boolean Bercovici–Pata bijection for bounded \(k\)-tuples. Adv. Math. 217(1), 1–41 (2008)
Bercovici, H.: On Boolean convolutions, operator theory 20. Theta Ser. Adv. Math. 6, 7–13 (2006)
Bercovici, H., Pata, V.: Stable laws and domains of attraction in free probability theory, with an appendix by P. Biane. Ann. Math. 149(3), 1023–1060 (1999)
Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42(3), 733–773 (1993)
Bożejko, M., Leinert, M., Speicher, R.: Convolution and limit theorems for conditionally free random variables. Pac. J. Math. 175(2), 357–388 (1996)
Bożejko, M., Speicher, R.: \(\psi \)-independent and symmetrized white noises. In: Accardi, L. (ed.) Quantum Probability and Related Topics, vol. VI, pp. 219–236. World Scientific, Singapore (1991)
Charlesworth, I., Nelson, B., Skoufranis, P.: Combinatorics of bi-freeness with amalgamation. Commun. Math. Phys. 338(2), 801–847 (2015)
Charlesworth, I., Nelson, B., Skoufranis, P.: On two-faced families of non-commutative random variables. Can. J. Math. 67(6), 1290–1325 (2015)
Franz, U.: Boolean convolution of probability measures on the unit circle. Anal. Probab. Sémin. Congr. 16, 83–94 (2008)
Gu, Y., Skoufranis, P.: Conditionally bi-free independence for pairs of faces. J. Funct. Anal. 273(5), 1663–1733 (2017)
Gu, Y., Skoufranis, P.: Conditionally bi-free independence with amalgamation. Int. Math. Res. Not. (2018) (to appear). https://doi.org/10.1093/imrn/rnx104
Huang, H.-W., Wang, J.-C.: Analytic aspects of the bi-free partial \(R\)-transform. J. Funct. Anal. 271(4), 922–957 (2016)
Krystek, A.D.: Infinite divisibility for the conditionally free convolution. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(4), 499–522 (2007)
Mastnak, M., Nica, A.: Double-ended queues and joint moments of left-right canonical operators on full Fock space. Int. J. Math. 262, 1550016 (2015)
Muraki, N.: The five independences as quasi-universal products. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(1), 113–134 (2002)
Nica, A., Speicher, R.: On the multiplication of free \(N\)-tuples of noncommutative random variables, with an appendix by D. Voiculescu. Am. J. Math. 118(4), 799–837 (1996)
Oravecz, F.: Fermi convolution. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(2), 235–242 (2002)
Oravecz, F.: Minimality of the Boolean and the Fermi convolutions. Interdiscipl. Inform. Sci. 10(1), 59–67 (2004)
Popa, M.: A new proof for the multiplicative property of the boolean cumulants with applications to operator-valued case. Colloq. Math. 117(1), 81–93 (2009)
Skoufranis, P.: A combinatorial approach to Voiculescu’s bi-free partial transforms. Pac. J. Math. 283(2), 419–447 (2016)
Skoufranis, P.: Independences and partial \(R\)-transforms in bi-free probability. Ann. Inst. Henri Poincaré Probab. Stat. 52(3), 1437–1473 (2016)
Skoufranis, P.: On operator-valued bi-free distributions. Adv. Math. 303, 638–715 (2016)
Speicher, R.: Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298(1), 611–628 (1994)
Speicher, R.: On universal products. Fields Inst. Commun. 12, 257–266 (1997)
Speicher, R., Woroudi, R.: Boolean convolution. Fields Inst. Commun. 12, 267–279 (1997)
Voiculescu, D.: Free probability for pairs of faces I. Commun. Math. Phys. 332(3), 955–980 (2014)
Voiculescu, D.: Free probability for pairs of faces III: 2-variables bi-free partial \(S\)- and \(T\)-transforms. J. Funct. Anal. 270(10), 3623–3638 (2016)
Voiculescu, D.: The bi-free extension of free probability, series. In: Mathematical Analysis, Probability and Applications—Plenary Lectures. Springer Proceedings in Mathematics and Statistics, vol. 177, pp. 217-233. Springer International Publishing (2016)
Wang, J.-C.: Limit theorems for additive conditionally free convolution. Can. J. Math. 63(1), 222–240 (2011)
We would like to express our gratitude to the anonymous referee for a careful reading and detailed comments that improved the quality of the paper.
Communicated by Hari Bercovici.
The work of Yinzheng Gu was partially supported by CIMI (Centre International de Mathématiques et d’Informatique) Excellence Program, ANR-11-LABX-0040-CIMI within the Program ANR-11-IDEX-0002-02, while visiting the Institute of Mathematics of Toulouse. He would like to thank the institute for the generous hospitality and Serban Belinschi for his constant support and valuable advice when this research was conducted.
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Gu, Y., Skoufranis, P. Bi-Boolean Independence for Pairs of Algebras. Complex Anal. Oper. Theory 13, 3023–3089 (2019). https://doi.org/10.1007/s11785-017-0750-9