Semicrossed Products and Reflexivity

Abstract

Given a w*-closed unital algebra $A$ acting on $H_0$ and a contractive w*-continuous endomorphism $\beta$ of $A$, there is a w*-closed (non-selfadjoint) unital algebra $\mathbb{Z}_+\bar{\times}_\beta A$ acting on $H_0\otimes\ell^2({\mathbb{Z}_+})$, called the w*-semicrossed product of $A$ with $\beta$. We prove that the w*-semicrossed product is a reflexive operator algebra provided $A$ is reflexive and $\beta$ is unitarily implemented, and that it has the bicommutant property if and only if so does $A$. Also, we show that the w*-semicrossed product generated by a commutative C*-algebra and a *-endomorphism is reflexive.