Stacey has recently characterised the crossed product A × α N A{ \times _\alpha }{\mathbf {N}} of a C ∗ {C^{\ast }} -algebra A A by an endomorphism α \alpha as a C ∗ {C^{\ast }} -algebra whose representations are given by covariant representations of the system ( A , α ) (A,\alpha ) . Following work of O’Donovan for automorphisms, we give conditions on a covariant representation ( π , S ) (\pi ,S) of ( A , α ) (A,\alpha ) which ensure that the corresponding representation π × S \pi \times S of A × α N A{ \times _\alpha }{\mathbf {N}} is faithful. We then use this result to improve a theorem of Paschke on the simplicity of A × α N A{ \times _\alpha }{\mathbf {N}} .