A completely bounded characterization of operator algebras
type: Item
instance of: scholarly article
title: A completely bounded characterization of operator algebras
zbMATH Open document ID: 0892.47048
DOI: 10.1007/BF01460988
author: David P. Blecher
published in: Mathematische Annalen
publication date: 1996-02-12T00:00:00Z
full work available at URL: https://eudml.org/doc/165366
review text: Let \(B(H)\) denote the Banach algebra of all bounded operators on a Hilbert space \(H\). An operator algebra is a (not necessarily closed) subalgebra of \(B(H)\) for some Hilbert space \(H\). \textit{D. Blecher, Z-j. Ruan} and \textit{A. Sinclair} [J. Funct. Anal. 89, No. 1, 188-201 (1990; Zbl 0714.46043)] have given an abstract characterization of operator algebras up to completely isometric isomorphism as follows. Let \({\mathcal A}\) be a unital algebra which is also an \(L^{\infty}\)-matrix normed space. Then \({\mathcal A}\) may be represented completely isometrically isomorphically as an operator algebra if and only if the multiplication on \({\mathcal A}\) is contractive. An algebra which is also an \(L^{\infty}\)-matrix normed space such that the multiplication is completely bounded is called an \(h\)-algebra. Clearly every operator algebra is an \(h\)-algebra. The author proves that the converse is also true. He shows that every \(h\)-algebra is completely boundedly isomorphic to an operator algebra. Applications and examples are also presented.
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 46H15
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46K50
zbMATH DE Number: 840183
zbMATH Keywords: operator space
zbMATH Keywords: operator algebra
zbMATH Keywords: completely bounded
reviewed by: K. Seddighi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 840183