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Acta Mathematica Hungarica

, Volume 157, Issue 1, pp 158–172 | Cite as

Characterization of reflexive closure of some operator algebras acting on Hilbert \({C^{\star}}\)-modules

  • H. GhahramaniEmail author
  • S. Sattari
Article
  • 25 Downloads

Abstract

Let \({\mathcal M}\) be a Hilbert \({C^{*}}\)-module over a \({C^{*}}\)-algebra \({\mathcal A}\). Suppose that \({\mathcal{K}(\mathcal{M})}\) is the space of compact operators on \({\mathcal M}\) and the bounded anti-homomorphism \({\rho \colon \mathcal{A}\rightarrow \mathcal{B}(\mathcal{M})}\) defined by \({\rho(a)(m)=ma}\) for all \({a\in\mathcal{A}}\) and \({m\in\mathcal{M}}\). In this paper, we first provide some characterizations of module maps on Banach modules over Banach algebras by several local conditions (some of our results are a generalization of previous results) and then apply them to characterize the reflexive closure of \({\mathcal{K}(\mathcal{M})}\) and \({\rho(\mathcal{A})}\), i.e., \({{\rm Alg}{\rm Lat} \mathcal{K}(\mathcal{M})}\) and \({{\rm Alg}{\rm Lat} \rho(\mathcal{A})}\), where we think of \({\mathcal{K}(\mathcal{M})}\) and \({\rho(\mathcal{A})}\) as operator algebras acting on \({\mathcal M}\). As an application of our results on reflexive closure of \({\mathcal{K}(\mathcal{M})}\) and \({\rho(\mathcal{A})}\), a characterization of commutativity for \({C^{*}}\)-algebras is given.

Key words and phrases

Hilbert \({C^{*}}\)-module Banach right module module map reflexive closure 

Mathematics Subject Classification

46L08 47L10 47C15 47B49 46L05 

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Notes

Acknowledgement

The authors express their sincere thanks to the referee(s) for this paper.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KurdistanSanandajIran

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