On the uniqueness of representational indices of derivations of \(C^*\)- algebras (Q1318483)

Let \(\mathcal A\) be a \(C^*\)-algebra of bounded operators on a Hilbert space \(H\), let \(\delta\) be a closed unbounded *-derivation from \(\mathcal A\) into \(B(H)\) with the domain \(D(\delta)\) dense in \(\mathcal A\) and let \(S\) be an unbounded symmetric operator on \(H\) which implements \(\delta\), i.e., \[ AD(S)\subseteq D(S)\text{ and }\delta(A)|_{D(S)}= i(SA- AS)|_{D(S)},\;A\in D(\delta). \] The deficiency space \(N(S)\) of \(S\) is a Krein space with respect to a certain indefinite metric and there exists a \(J\)-symmetric representation \(\pi^ \delta_ S\) of \(D(\delta)\) on \(N(S)\). If \(S\) is a maximal symmetric implementation of \(\delta\), then the class of all representations of \(D(\delta)\) equivalent to \(\pi^ \delta_ S\) is called a representational index of \(\delta\) relative to \(S\). The paper shows that if \(\delta\) has a minimal implementation \(S_ 0\) (which is the case, for example, when \(\mathcal A\) contains the ideal of all compact operators) and if the representation \(\pi^ \delta_{S_ 0}\) is finitely \(\Pi_ -\)- or \(\Pi_ +\)-decomposable, then all the representations \(\pi^ \delta_ S\) which correspond to maximal symmetric implementation of \(\delta\) are equivalent, so that \(\delta\) has a unique representational index.