Conjugate Dynamical Systems on C*-algebras

Let $(A,\alpha )$ and $(B,\beta )$ be C*-dynamical systems where $\alpha$ and $\beta$ are arbitrary *-endomorphisms. When $\alpha$ is injective or surjective, we show that the semicrossed products $A{\times }_{\alpha }\mathbb{Z}$ and $B{\times }_{\beta }\mathbb{Z}$ are isometrically isomorphic if and only if $(A,\alpha )$ and $(B,\beta )$ are outer conjugate. This conclusion also holds in various other cases as well.