On the facial structures for positive linear maps between matrix algebras
by Seung-Hyeok Kye
Proceedings of the 4th Operator Algebras International Conference: Operator Algebras and Mathmatical Physics, (Constanta, Romania, 2001), Theta Foundation, 2003, pp. 243-250.

We give a lattice isomorphism between faces of the convex cone of all completelypositive linear maps from $M_m$ into $M_n$ and subspaces of $m\times n$ matrices.Using this, we see that every face of the convex cone of all decomposable positive linear mapsarises from a pair of subspaces of $m\times n$ matrices. Because every positive linear mapfrom $M_2$ into $M_2$ is decomposable, we may determine completely the lattice structure for the faces of theconvex cone of all positive linear maps between the $2\times 2$matrix algebras, in terms of pairs of subspaces in $M_2$.

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