Dilating covariant representations of the non-commutative disc algebras

Let $\phi$ be an isometric automorphism of the non-commutative disc algebra $\fA_n$ for $n \geq 2$. We show that every contractive covariant representation of $(\fA_n, \phi)$ dilates to a unitary covariant representation of $(\O_n, \phi)$. Hence the C*-envelope of the semicrossed product $\fA_n \times_{\phi} \bZ^+$ is $\O_n \times_{\phi} \bZ$.