2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/172761Let $X$ be compact Hausdorff, and $ϕ: X \to X$ a continuous surjection. Let $\mathcal{A}$ be the semicrossed product algebra corresponding to the relation fU = Uf\circ ϕ$ or to the relation $Uf = f\circ ϕU.$ 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.Operator AlgebrasDynamical Systems46L52; 37B99text