On action of diffeomorphisms of \(C^{\ast}\)-algebras on derivations (Q2497935)

Let \({\mathbb A}\) be a \(C^{*}\)-algebra. A closed linear map \(\delta\) from a dense \(*\)-subalgebra \(D(\delta )\) of \({\mathbb A}\) into \({\mathbb A}\) is called a closed \(*\)-derivation if \(\delta (AB)=A\delta (B)+\delta (A)B\) and \(\delta (A^{*})=\delta (A)^{*}\) for all \(A,B\in D(\delta )\). Let \({\mathcal A}\) be a dense \(*\)-subalgebra of \(\mathbb A\). Denote by \(\text{Der}({\mathcal A})\) the set of all closed \(*\)-derivations \(\delta\) on \(\mathbb A\) with \({\mathcal A}=D(\delta )\). \(\mathcal A\) is called a domain if \(\text{Der}({\mathcal A})\neq \emptyset\). In the present paper, it is shown that all \(*\)-automorphisms of a domain \(\mathcal A\) of \(\mathbb A\) extend to \(*\)-automorphisms of \(\mathbb A\). These extensions are called diffeomorphisms of \(\mathbb A\); they form a group \(\text{Dif}({\mathcal A})\). Each diffeomorphism \(\phi\) defines a transformation \(T_{\phi}\) of \(\text{Der}({\mathcal A})\). \(T_{\phi}({\delta}):=\phi^{-1}\delta\phi\in \text{Der}({\mathcal A})\). Denote by \({\mathcal Z}(\delta )\) the centralizer of \(\delta\) \({\mathcal Z}(\delta )=\{\phi\in \text{Dif}({\mathcal A}): \delta =T_{\phi}(\delta )\}\) and by \(B(\delta )\) the subgroup of \(\text{Dif}({\mathcal A})\) of diffeomorphisms that define bounded shifts of \(\delta\), \(B(\delta )=\{\phi\in \text{Dif}({\mathcal A})\): the derivation \(T_{\phi}(\delta )-\delta\) is bounded on \({\mathcal A}\}\). In the paper, the subgroups of diffeomorphisms that define bounded shifts of derivations and the subgroups of the stabilizers of derivations are studied.