On the Convex Set of Completely Positive Linear Maps in Matrix Algebras
by Seung-Hyeok Kye
Math. Proc. Cambridge Philos. Soc. 122 (1997), 45-54

Let $P_I$ (respectively $CP_I$) be the convex compact set of all unital positive (respectively completely positive) linear maps from the matrix algebra $M_m(\Bbb C)$ into $M_n(\Bbb C)$. We show that maximal faces of $CP_I$ correspond to one dimensional subspaces of the vector space $M_{m,n}(\Bbb C)$. Furthermore, a maximal face of $CP_I$ lies on the boundary of $P_I$ if and only if the corresponding subspace is generated by a rank one matrix.