Faithful representations of crossed products by endomorphisms

Abstract:

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