  We introduce a general 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. Our approach
allows to consider states whose discord is not necessarily zero and explicitly
show the non-uniqueness of the completely positive extensions of the obtained
dynamical map outside the compatibility domain. The relevance of such maps for
the treatment of open quantum system dynamics is discussed and connection to
previous literature is critically assessed.
