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

TL;DR

This paper determines the C$^*$-envelope of certain semicrossed product algebras associated with a continuous surjection on a compact Hausdorff space, and demonstrates the abundance of nest representations.

Contribution

It explicitly identifies the C$^*$-envelope as a crossed product involving a constructed homeomorphism, advancing understanding of semicrossed products.

Findings

C$^*$-envelope is a crossed product of a commutative C$^*$-algebra with a homeomorphism

Constructs a specific homeomorphism related to the algebra

Shows existence of sufficiently many nest representations

Abstract

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.