Dual spaces and isomorphisms of some differential Banach \({}^*\)-algebras of operators
type: Item
instance of: scholarly article
title: Dual spaces and isomorphisms of some differential Banach \({}^*\)-algebras of operators
zbMATH Open document ID: 1031.47049
author: E. Kissin
author: V. S. Shulman
published in: Pacific Journal of Mathematics
publication date: 2000-06-22T00:00:00Z
review text: This paper is a continuation of former investigations done by the authors on the differential Banach *-algebras \({\mathcal A}_S\) and \({\mathcal F}_S\) of operators associated with a symmetric operator \(S\) on a Hilbert space \(H\). The algebra \({\mathcal A}_S\) is the domain of the largest *-derivation \(\delta_S\) of \(B(H)\) implemented by \(S\) and the algebra \({\mathcal F}_S\) is the closure of the set of all finite rank operators in \({\mathcal A}_S\) with respect to the norm \(\|A\|_S=\|A\|+ \|\delta_S(A)\|\). After the necessary preliminaries, in Section 2 the authors study the structure of the algebra \({\mathcal A}_S\) and that of its closure in the operator norm when \(S\) is self-adjoint or \(S\) is symmetric with at least one finite deficiency index. In this latter case, it is shown that the algebras under consideration contain closed ideals of finite codimension. If \(S\) is bounded, then we have \({\mathcal A}_S=B(H)\) and \({\mathcal F}_S=C(H)\). So, in this case \({\mathcal A}_S\) is isometrically isomorphic to the second dual of \({\mathcal F}_S\). In Section 3, the authors investigate the dual and the second dual of \({\mathcal F}_S\) when \(S\) is unbounded. For a self-adjoint \(S\) it is proved that just as in the bounded case, \({\mathcal A}_S\) is isometrically isomorphic to the second dual of \({\mathcal F}_S\). Section 4 is devoted to the study of the problem of classification of the algebras \({\mathcal A}_S\) (resp. \({\mathcal F}_S\)) up to *-isomorphisms. It turns out that the bounded *-isomorphisms between such algebras are all implemented by unitary operators and that two such algebras associated with the symmetric operators \(S\) and \(T\) are isometrically *-isomorphic if and only if there is a unitary operator \(U\) such that \(S-\lambda I=\pm UTU^*\) for some real number \(\lambda\). For bounded but not necessarily isometrical *-isomorphisms some interesting partial results are obtained in the case when \(S\) is self-adjoint.
reviewed by: Lajos Molnár
Mathematics Subject Classification ID: 47L80
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 46K99
zbMATH DE Number: 1463707
zbMATH Keywords: differential Banach *-algebras
zbMATH Keywords: symmetric operators
MaRDI profile type: Publication
full work available at URL: pjm.1999.190.329
OpenAlex ID: W2046259828
DOI: 10.2140/PJM.1999.190.329
description: scientific article; zbMATH DE number 1463707