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.
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Ghahramani, H., Sattari, S. Characterization of reflexive closure of some operator algebras acting on Hilbert \({C^{\star}}\)-modules. Acta Math. Hungar. 157, 158–172 (2019). https://doi.org/10.1007/s10474-018-0877-9
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DOI: https://doi.org/10.1007/s10474-018-0877-9