Computing the norms of elementary operators

Abstract

We provide a direct proof that the Haagerup estimate on the completely bounded norm of elementary operators is best possible in the case of $\mathcal{B}(H)$ via a generalisation of a theorem of Stampfli. We show that for an elementary operator $T$ of length $\ell$, the completely bounded norm is equal to the $k$-norm for $k = \ell$. 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.