Facial structure of positive linear maps between matrix algebras

Abstract. We give a lattice isomorphism between faces of the convex cone of all completely positive linear maps from Mm into Mn and subspaces of m × n matrices. Using this, we see that every face of the convex cone of all decomposable positive linear maps arises from a pair of subspaces of m × n matrices. Because every positive linear map from M2 into M2 is decomposable, we may determine completely the lattice structure for the faces of the convex cone of all positive linear maps between the 2 × 2 matrix algebras, in terms of pairs of subspaces in M2. 1