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.
Citation
Richard M. Timoney. "Computing the norms of elementary operators." Illinois J. Math. 47 (4) 1207 - 1226, Winter 2003. https://doi.org/10.1215/ijm/1258138100
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