Dilating covariant representations of the non-commutative disc algebras

Let φ be an isometric automorphism of the non-commutative disc algebra An for n⩾2. We show that every contractive covariant representation of (An,φ) dilates to a unitary covariant representation of (On,φ). Hence the C∗-envelope of the semicrossed product An×φZ+ is On×φZ.