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

Let $X$ be compact Hausdorff, and $\phi: X \to X$ a continuous surjection. Let $\mathcal{A}$ be the semicrossed product algebra corresponding to the relation fU = Uf\circ \phi$or to the relation$Uf = f\circ \phi U.$Then the C$^*$-envelope of$\mathcal{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.