Computing the norms of elementary operators

We provide a direct proof that the Haagerup estimate on the completely bounded norm of elementary operators is best possible in the case of B(H) via a generalisation of a theorem of Stampfli. We show that for an elementary operator T of length, the completely bounded norm is equal to the k-norm for k = l. A C*-algebra A has the property that the completely bounded norm of every elementary operator is the k-norm, if and only if A is either k-subhomogeneous or a k-subhomogeneous extension of an antiliminal C*-algebra.