We study classes of generalized dynamical maps which are not completely positive. We show that some of these maps correspond to actual, physical systems described in the environmental representation by a unitary dynamics of an extended system and the partial trace. The non complete positivity is due to certain prior correlations of the principal system with the environment. These correlation may have classical nature and no entanglement is necessary to generate a non CP dynamics. We discuss its applications to the dynamical decoupling and quantum channels.