  By the Choi matrix criteria it is easy to determine if a specific linear
matrix map is completely positive, but to establish whether a linear matrix map
is positive is much less straightforward. In this paper we consider classes of
linear matrix maps, determined by structural conditions on an associated
matrix, for which positivity and complete positivity coincide. The basis of our
proofs lies in a representation of $*$-linear matrix maps going back to work of
R.D. Hill which enables us to formulate a sufficient condition in terms of
surjectivity of certain bilinear maps.
