Boundaries of the Cone of Positive Linear Maps and Subcones in Matrix Algebras
by Seung-Hyeok Kye
J. Korean Math. Soc. 33 (1996), 669-677

Let $\Bbb P_s$ be the convex cone of all $s$-positive linear maps from the matrix algebra $M_m(\Bbb C)$ into $M_n(\Bbb C)$. We show that every maximal face of $\Bbb P_s$ corresponds to an $m\times n$ matrix whose rank is less than or equal to $s$. We also discuss the relations between maximal faces of the cones $\Bbb P_s$ and $\Bbb P_t$ for different $s$ and $t$.