The C${}^{\ast }$-envelope of a semicrossed product and nest representations

Abstract

Let $X$ be compact Hausdorff, and $\varphi :X\to X$ a continuous surjection. Let $\mathcal{A}$ be the semicrossed product algebra corresponding to the relation $fU=Uf\circ \varphi$. Then the C${}^{\ast }$-envelope of $\mathcal{A}$ is the crossed product of a commutative C${}^{\ast }$-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.

17 pages