Facial Structures for Positive Linear Maps between Matrix Algebras
by Seung-Hyeok Kye
Canad. Math. Bull. 39 (1996), 74-82

Let $\Cal P$ denote the convex set of all positive linear maps from the matrix algebra $M_n(\Bbb C)$ into itself. We construct a join homomorphism from the complete lattice $\Cal F(\Cal P)$ of all faces of $\Cal P$ into the complete lattice $\Cal J(\Cal V)$ of all join homomorphisms between the lattice $\Cal V$ of all subspaces of $\Bbb C^n$. We also characterize all maximal faces of $\Cal P$.