Exposedness of elementary positive maps between matrix algebras

The positive linear maps ${\text{ad}}_s$ which send matrices $x$ to $s^{\ast }xs$ play important roles in quantum information theory as well as matrix theory. It was proved by Marciniak [Linear Multilinear Alg. 61 (2013), 970–975] that the map ${\text{ad}}_s$ generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz’s method, respectively.