  This paper considers basic properties of super-operator norms induced by
Schatten p-norms. Such super-operator norms arise in various contexts in the
study of quantum information. It is proved that for completely positive
super-operators, the value of any such norm is achieved by a positive
semidefinite input, answering a question recently posed by King and Ruskai.
However, for any choice of p, there exists a super-operator that is the
difference of two completely positive, trace-preserving super-operators such
that the value of the super-operator norm is not even achieved on a Hermitian
input operator. Also considered are the properties of the above norms for
super-operators tensored with the identity super-operator.
