Indices of unbounded derivations of \(C^*\)-algebras (Q749853)

The paper studies some properties of J-symmetric representations of *-algebras on infinite metric spaces. Making use of this, it defines the index ind(\(\delta,S\)) of a *-derivation \(\delta\) of a \(C^*\)-algebra \({\mathcal A}\) relative to a symmetric implementation S of \(\delta\). The index consists of six integers which characterize the J-symmetric representation \(\pi_ S\) of the domain \(D(\delta)\) of \(\delta\) on the deficiency space \(N(S)\) of the operator \(S\). The paper proves the stability of the index under bounded perturbations of the derivation and that, under certain conditions on \(\delta\), ind(\(\delta,S\)) has the same value for all maximal symmetric implementations \(S\) of \(\delta\). It applies the developed methods to the problem of the classification of symmetric operators with deficiency indices \((1,1)\).