Journal of Operator Theory
Volume 67, Issue 2, Spring 2012 pp. 379-395.
Semicrossed products and reflexivityAuthors: Evgenios T.A. Kakariadis
Author institution: Department of Mathematics, University of Athens, Panepistimioupolis, GR-157 84, Athens, Greece
Summary: Given a w*-closed unital algebra $\mathcal{A}$ acting on $H_0$ and a contractive w*-continuous endomorphism $\beta$ of $\mathcal{A}$, there is a w*-closed (non-selfadjoint) unital algebra $\mathbb{Z}_+\overline{\times}_\beta \mathcal{A}$ acting on $H_0\otimes\ell^2({\mathbb{Z}_+})$, called the w*-semicrossed product of $\mathcal{A}$ with $\beta$. We prove that $\mathbb{Z}_+\overline{\times}_\beta \mathcal{A}$ is a reflexive operator algebra provided $\mathcal{A}$ is reflexive and $\beta$ is unitarily implemented, and that $\mathbb{Z}_+\overline{\times}_\beta \mathcal{A}$ has the bicommutant property if and only if so does $\mathcal{A}$. Also, we show that the w*-semicrossed product generated by a commutative $C^*$-algebra and a continuous map is reflexive.
Keywords: $C^*$-envelope, reflexive subspace, semicrossed product
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