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We give some sufficient conditions for the injectivity of actions of compactquantum groups on $C^{\ast}$-algebra. As an application, we prove that anyfaithful smooth action by a compact quantum group on a compact smooth (notnecessarily connected) manifold is injective. A similar result is proved foractions on $C^{\ast}$- algebras obtained by Rieffel-deformation of compact,smooth manifolds.

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