Faithful Representations of Crossed Products by Endomorphisms

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