Journal of Operator Theory

Volume 67, Issue 2, Spring 2012 pp. 379-395.

Semicrossed products and reflexivity

Authors: 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 $\A$ acting on $H_0$ and a contractive w*-continuous endomorphism $\beta$ of $\A$, there is a w*-closed (non-selfadjoint) unital algebra $\mathbb{Z}_+\overline{\times}_\beta \A$ acting on $H_0\otimes\ell^2({\mathbb{Z}_+})$, called the w*-semicrossed product of $\A$ with $\beta$. We prove that $\mathbb{Z}_+\overline{\times}_\beta \A$ is a reflexive operator algebra provided $\A$ is reflexive and $\beta$ is unitarily implemented, and that $\mathbb{Z}_+\overline{\times}_\beta \A$ 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 continuous map is reflexive.

Keywords: $C^*$-envelope, reflexive subspace, semicrossed product

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