  We give some sufficient conditions for the injectivity of actions of compact
quantum groups on $C^{\ast}$-algebra. 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-deformation of compact,
smooth manifolds.
