We extend the shuffle algebra perspective on scalar-valued non-commutative probability theory to the operator-valued case. Given an operator-valued probability space with an algebra B acting on it (on the left and on the right), we associate operators in the operad of multilinear maps on B to the operator-valued distribution and free cumulants of a random variable. These operators define a representation of a PROS of non-crossing partitions. Using concepts from higher category theory, specifically 2-monoidal categories, we define a notion of unshuffle Hopf algebra on an underlying PROS. We introduce a PROS of words insertions and show that both the latter and the PROS of non-crossing partitions are unshuffle Hopf algebras. The two relate by mean of a map of unshuffle bialgebra (in a 2-monoidal sense) which we call the splitting map. Ultimately, we obtain a left half-shuffle fixed point equation corresponding to free moment-cumulant relations in a shuffle algebra of bicollection homomorphisms on the PROS of words insertions. Right half-shuffle and shuffle laws are interpreted in the framework of boolean and monotone non-commutative probability theory, respectively.
The amalgamated T -transform of a non-commutative distribution was introduced by K. Dykema. It provides a fundamental tool for computing distributions of random variables in Voiculescu's free probability theory. The T -transform factorizes in a rather non-trivial way over a product of free random variables. In this article, we present a simple graphical proof of this property, followed by a more conceptual one, using the abstract setting of an operad with multiplication.
One can build an operatorial model for freeness by considering either the righthanded either the left-handed representation of algebras of operators acting on the free product of the underlying pointed Hilbert spaces. Considering both at the same time, that is computing distributions of operators in the algebra generated by the left-and right-handed representations, led Voiculescu in 2013 to define and study bifreeness and, in the sequel, triggered the development of an extension of noncommutative probability now frequently referred to as multi-faced (two-faced in the example given above). Many examples of two-faced independences emerged these past years. Of great interest to us are biBoolean, bifree and Type I bimonotone independences. In this paper we extend the preLie calculus prevailing to free, Boolean and monotone moment-cumulant relations initiated by K. Ebrahimi-Fard and F. Patras to their above mentioned two-faced equivalents. M.G. is supported by the German Research Foundation (DFG) grant no. 397960675. J.D. and N.G. are supported by the trilateral (DFG/ANR/JST) grant "EnhanceD Data stream Analysis with the Signature Method" . N.G is supported by ANR "STARS".
One can build an operatorial model for freeness by considering either the right-handed or the left-handed representation of algebras of operators acting on the free product of the underlying pointed Hilbert spaces. Considering both at the same time, that is, computing distributions of operators in the algebra generated by the left- and right-handed representations, led Voiculescu in 2013 to define and study bifreeness and, in the sequel, triggered the development of an extension of noncommutative probability now frequently referred to as multi-faced (two-faced in the example given above). Many examples of two-faced independences emerged these past years. Of great interest to us are biBoolean, bifree and type I bimonotone independences. In this paper, we extend the preLie calculus pertaining to free, Boolean, and monotone moment-cumulant relations initiated by K. Ebrahimi-Fard and F. Patras to their above-mentioned two-faced equivalents.
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