A note on the p→qp→q norms of 2-positive maps

King and Ruskai asked whether the p→qp→q norm of a completely positive map ΦΦ, acting between Schatten p and q classes of Hermitian operators,‖Φ‖p→q=supA=A∗‖Φ(A)‖q‖A‖p,is equal to the p→qp→q norm of that map when acting between Schatten classes of general, not necessarily Hermitian, operators. The first proof of this statement has been given by Watrous. Here, we give an alternative proof that is also valid for 2-positive maps ΦΦ.