Given commuting families of Hermitian matrices {A1, ..., Ak} and {B1, ....,
Bk}, conditions for the existence of a completely positive map L, such that
L(Aj) = Bj for j = 1, ...,k, are studied. Additional properties such as unital
or / and trace preserving on the map ? are also considered. Connections of the
study to dilation theory, matrix inequalities, unitary orbits, and quantum
information science are mentioned.