Linear matrix maps for which positivity and complete positivity coincide

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.