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Free Multivariate w*-Semicrossed Products: Reflexivity and the Bicommutant Property

Published online by Cambridge University Press:  20 November 2018

Robert T. Bickerton  and
Evgenios T. A. Kakariadis
Robert T. Bickerton
Affiliation:
Newcastle University, Newcastle, NE1 7 RU, United Kingdom, e-mail: r.bickerton@ncl.ac.uk , evgenios.kakariadis@ncl.ac.uk
Evgenios T. A. Kakariadis
Affiliation:
Newcastle University, Newcastle, NE1 7 RU, United Kingdom, e-mail: r.bickerton@ncl.ac.uk , evgenios.kakariadis@ncl.ac.uk
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Abstract

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We study ${{\text{w}}^{*}}$-semicrossed products over actions of the free semigroup and the free abelian semigroup on (possibly non-selfadjoint) ${{\text{w}}^{*}}$-closed algebras. We show that they are reflexive when the dynamics are implemented by uniformly bounded families of invertible row operators. Combining with results of Helmer, we derive that ${{\text{w}}^{*}}$-semicrossed products of factors (on a separableHilbert space) are reflexive. Furthermore, we show that ${{\text{w}}^{*}}$-semicrossed products of automorphic actions on maximal abelian self adjoint algebras are reflexive. In all cases we prove that the ${{\text{w}}^{*}}$-semicrossed products have the bicommutant property if and only if the ambient algebra of the dynamics does also.

Keywords

47A1547L6547L7547L80reflexivitysemicrossed product
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 6 , 01 December 2018 , pp. 1201 - 1235
Copyright
Copyright © Canadian Mathematical Society 2018

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