Rank properties of exposed positive maps

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 $ϕ(X)=AXA^*$ or $ϕ(X)=AX^TA^*$ is an exposed point of $\fP$. We also show that if a map $ϕ$ is an exposed point of $\fP$ then either $ϕ$ is rank 1 non-increasing or $\rankϕ(P)>1$ for any one-dimensional projection $P\in\fB(\cK)$.

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