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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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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).

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