Still more on norms of completely positive maps
  King and Ruskai asked whether the norm of a completely positive map acting between Schatten classes of operators is equal to that of its restriction to the real subspace of self-adjoint operators. Proofs have been promptly supplied by Watrous and Audenaert. Here we provide one more proof, in fact of a slightly more general fact, under the (slightly weaker) assumption of 2-positivity. The argument is elementary and self-contained. 
Stanislaw J. Szarek