Illinois Journal of Mathematics

Computing the norms of elementary operators

Richard M. Timoney

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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.

Article information

Illinois J. Math., Volume 47, Number 4 (2003), 1207-1226.

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B47: Commutators, derivations, elementary operators, etc.
Secondary: 46L07: Operator spaces and completely bounded maps [See also 47L25] 47A12: Numerical range, numerical radius 47A30: Norms (inequalities, more than one norm, etc.) 47L25: Operator spaces (= matricially normed spaces) [See also 46L07]


Timoney, Richard M. Computing the norms of elementary operators. Illinois J. Math. 47 (2003), no. 4, 1207--1226. doi:10.1215/ijm/1258138100.

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