Semicrossed Products and Reflexivity

Details Semicrossed Products and Reflexivity Semicrossed Products and Reflexivity

2009-07-30

arxiv.org/abs/0907.5314v1

Mathematics

Operator Algebras

Scientific paper

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 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.

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