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.

References in corpus (2)

• The C${}^{\ast }$-envelope of a semicrossed product and nest representations
• Extensions and Dilations for $C^{\ast }$-dynamical Systems

Cited by in corpus (2)

• Semicrossed products of C*-algebras and their C*-envelopes
• Operator algebras from the discrete Heisenberg semigroup