Rank properties of exposed positive maps

Abstract

Let $\cK$ and $\cH$ be finite dimensional Hilbert spaces and let $\fP$ denote the cone of all positive linear maps acting from $\fB(\cK)$ into $\fB(\cH)$. We show that each map of the form $\phi(X)=AXA^*$ or $\phi(X)=AX^TA^*$ is an exposed point of $\fP$. We also show that if a map $\phi$ is an exposed point of $\fP$ then either $\phi$ is rank 1 non-increasing or $\rank\phi(P)>1$ for any one-dimensional projection $P\in\fB(\cK)$.