A shuffle algebra point of view on operator-valued probability theory

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 (in a $2$-monoidal sense). The two relate by mean of a map of unshuffle bialgebras (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. Keywords: operator-valued non-commutative probability theory, higher category theory, duoidal categories, operads, properads, PROS, shuffle algebra, half-shuffles