The C*-envelope of a semicrossed product and Nest Representations

Let X be compact Hausdorff, and phi : X -> X a continuous surjection. Let A be the semicrossed product algebra corresponding to the relation fU = U f circle phi or to the relation Uf = f circle phi U. Then the C*-envelope of A is the crossed product of a commutative C*-algebra which contains C(X) as a subalgebra, with respect to a homeomorphism which we construct. We also show there are "sufficiently many" nest representations.