We introduce a framework for the construction of completely positive
dynamical evolutions in the presence of system- environment initial
correlations. The construction relies upon commutativity of the
compatibility domain obtained by considering the marginals with respect
to the environmental degrees of freedom of the considered class of
correlated states, as well as basic properties of completely positive
maps. Our approach allows to consider states that can have finite
discord, though it does not include entangled states, and it explicitly
shows the non-uniqueness of the completely positive extensions of the
obtained dynamical map outside the compatibility domain. The possible
relevance of such maps for the treatment of open quantum system
dynamics is critically discussed, together with the connection to
previous literature.