The numerical radius of a completely bounded map
type: Item
instance of: scholarly article
title: The numerical radius of a completely bounded map
zbMATH Open document ID: 0801.46067
DOI: 10.1007/BF00050890
author: Ching Y. Suen
published in: Acta Mathematica Hungarica
publication date: 1993-04-01T00:00:00Z
review text: \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 \(W(T) \leq \| L \|_{cb} \leq 2W (T)\).
reviewed by: Jafar Zafarani
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 47A12
zbMATH DE Number: 148673
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: \(*\)-homomorphism
zbMATH Keywords: commutant representation
zbMATH Keywords: numerical radius
MaRDI profile type: Publication
cites work: Subalgebras of \(C^ *\)-algebras
cites work: Q3719297
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description: scientific article; zbMATH DE number 148673