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.

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