Derivations of covariant representations of \(C^ *\)-algebras (Q1822019)

We give a result with respect to continuity on spectral subspaces of derivations. Let (A,\(\alpha\),G) be a \(C^*\)-dynamical system, where the group G is compact or locally compact abelian, and \(\pi\) a representation of A which induces the \(W^*\)-dynamical system (M,\({\tilde \alpha}\),G), where \(M=\pi (A)''\). Let \(A_ F\) and \(M_ F\) denote the algebras of G-finite elements. If a derivation \(\delta\) defined on \(A_ F\) is bounded on each spectral subspace for a compact subset of G, then there exists a unique derivation \({\tilde \delta}\) defined on \(M_ F\) such that \({\tilde \delta}\circ \pi \supset \pi \circ \delta\) and the restriction of \({\tilde \delta}\) to each spectral subspace for a compact subset is \(\sigma\)-weakly continuous. In relation to this the author constructs the universal \(W^*\)-dynamical system associated with a \(C^*\)-dynamical system [see the \(W^*\)- dynamical system associated with a \(C^*\)-dynamical system, and unbounded derivations, to appear in J. Funct. Anal.]. These papers contain some applications to the generating problem of unbounded derivations.