The numerical radius of a completely bounded map
\textit{V. I. Paulsen} and \textit{C. Y. Suen}, [J. Oper. Theory 13, 87-101 (1985; Zbl 0593.46050)] showed that every completely bounded map \(L\) from a \(C^*\)-algebra \(A\) into \(L(H)\) has a representation as \(L(a) = V^*T \pi (a)V\), where \(\pi\) is a \(*\)-homomorphism, \(T\) is in the commutant of \(\pi(A)\) and \(V\) is an isometry. Such a realization of \(L\) was called a commutant representation with isometry. In this paper the author shows that among all commutant representations with isometry for \(L\), there exist ones for which the numerical radius \(W(T)\) is a minimum with \[ W(T) = \min \bigl\{ \| \varphi \|_{cb} : \varphi \pm \text{Re} \alpha L \text{ is completely positive for all } | \alpha | = 1\bigr\} \] and moreover \(W(T) \leq \| L \|_{cb} \leq 2W (T)\).
scientific article; zbMATH DE number 148673