In this talk I will explain the basics of operator-valued free non-commutative probability theory. Then, using operads and 2-monoidal categories, we will extend the shuffle algebras approach to free, boolean and monotone moment-cumulant relations (see Ebrahimi-Fard & Patras, shuffle group laws : applications in free probability) to their operator-valued counterparts. I will end this talk with loose ends related to free multiplicative operator-valued convolution and potential use of shuffle algebras in this context.
arXiv:2005.12049, Gilliers Nicolas, A shuffle algebra point of view on operator-valued probability theory