A Note on the p->q norms of Completely Positive Maps

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King and Ruskai asked whether the p→ q norm of a completely positive map Φ, acting between Schatten p and q classes of self-adjoint operators, ||Φ||ₚ→ q = ₐ₌ₐ* ||Φ(A)||q/||A||ₚ, is equal to the p→ q norm of that map when acting between Schatten classes of general, not necessarily self-adjoint, operators. The first proof has been given by Watrous. We give an alternative proof of this statement.