Dual spaces and isomorphisms of some differential Banach \({}^*\)-algebras of operators (Q1568867)

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.