Computing the norms of elementary operators
Let \(A\) be a \(C^*\)-algebra. Given an elementary operator \(T:A\rightarrow A\), \(Tx=\sum_{i=1}^la_ixb_i\), \(a_i,b_i\in M(A)\), \(i=1,\ldots,l\), the Haagerup estimate of the completely bounded norm \(\| T\| _{CB}\) is \[ \| T\| \leq\| T\| _{CB}\leq\sqrt{\left\| \sum_{j=1}^la_ja_j^*\right\| \,\left\| \sum_{j=1}^lb_j^*b_j\right\| }. \] The author characterizes equality in this inequality. The characterization is given in terms of extremal matrix numerical ranges, and for selfadjoint \(T\) an explicit \(k\) is given so that \(\| T\| _k=\| T\| _{CB}\). The \(C^*\)-algebras for which the equality is realized are characterized.
scientific article; zbMATH DE number 2028868