Tensor products of non-self-adjoint operator algebras
type: Item
instance of: scholarly article
title: Tensor products of non-self-adjoint operator algebras
zbMATH Open document ID: 0716.46053
author: Vern I. Paulsen
author: S. C. Power
published in: Rocky Mountain Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: An arbitrary complex algebra \({\mathcal A}\) is said to be an operator algebra if there exists a representation \(\rho\) of \({\mathcal A}\) on some Hilbert space such that for every integer \(k\geq 1\) the algebra \({\mathcal M}_ k({\mathcal A})\) of \(k\times k\) matrices with entries from \({\mathcal A}\) is given the norm which is induced by \(\rho\). Such a family of norms on \({\mathcal M}_ k({\mathcal A})\) is called an operator norm. Given two unital operator algebras \({\mathcal A}_ 1\) and \({\mathcal A}_ 2\), a complete operator cross-norm is any operator norm on \({\mathcal A}_ 1\otimes {\mathcal A}_ 2\) which is a cross-norm and for which the natural inclusions of \({\mathcal A}_ i\) \((i=1,2)\) into \({\mathcal A}_ 1\otimes {\mathcal A}_ 1\) \((a_ 1\to a_ 1\otimes 1\), \(a_ 2\to 1\otimes a_ 2)\) induce isometries of \({\mathcal M}_ k({\mathcal A}_ i)\) into \({\mathcal M}_ k({\mathcal A}_ 1\otimes {\mathcal A}_ 2)\) for all k. In the present paper the authors introduce and study three natural complete operator cross-norms, called spatial, minimal and maximal. It is worth mentioning that, in general, the minimal and maximal norms may differ and to establish their equality is usually equivalent to the possibility of making use of Ando's dilation theorem for commuting contractions or the commutant lifting theorem of Sz.-Nagy and Foiaş.
reviewed by: Florian-Horia Vasilescu
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 47L30
zbMATH DE Number: 4180124
zbMATH Keywords: spatial norm
zbMATH Keywords: operator algebra
zbMATH Keywords: operator norm
zbMATH Keywords: complete operator cross- norm
zbMATH Keywords: minimal and maximal norms
zbMATH Keywords: Ando's dilation theorem for commuting contractions
zbMATH Keywords: commutant lifting theorem of Sz.-Nagy and Foiaş
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181073110
OpenAlex ID: W1976335105
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DOI: 10.1216/RMJM/1181073110
description: scientific article; zbMATH DE number 4180124