A note on the injectivity of actions of compact quantum groups on a class of $C^{\ast }$-algebras

We give some sufficient conditions for the injectivity of actions of compact quantum groups on $C^{\ast }$-algebras. As an application, we prove that any faithful smooth action by a compact quantum group on a compact smooth (not necessarily connected) manifold is injective. A similar result is proved for actions on $C^{\ast }$-algebras obtained by Rieffel deformations of compact, smooth manifolds.