Title : On conjugate systems with respect to completely positive maps

Abstract : In 2010, Dabrowski showed that a von Neumann algebra generated by self-adjoint operators is a factor when they admit a conjugate system. We extend this to the operator-valued case by defining an operator valued partial derivative and conjugate systems with respect to completely positive maps. We study operator-valued cumulants using the lattice of non-crossing partition and provide a cumulant characterization for the existence of conjugate variables.