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On centralizers of reflexive algebras

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Abstract

Let \({\mathcal{L}}\) be a subspace lattice on a complex Banach space X and δ be a linear mapping from \({alg\mathcal{L}}\) into B(X) such that for every \({A \in alg\mathcal{L}, 2\delta(A^2)=\delta(A)A + A\delta(A)}\) or \({\delta(A^3) = A\delta(A)A}\) . We show that if one of the following holds (1) \({\vee\{L : L \in \mathcal{J}(\mathcal{L})\}=X}\) , (2) \({\wedge\{L_-: L \in \mathcal{J}(\mathcal{L})\}=(0)}\) and X is reflexive, then δ is a centralizer. We also show that if \({\mathcal{L}}\) is a CSL and δ is a linear mapping from \({alg\mathcal{L}}\) into itself, then δ is a centralizer.

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  1. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, People’s Republic of China

    Jianbin Guo & Jiankui Li

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  1. Jianbin Guo
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  2. Jiankui Li
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Correspondence to Jiankui Li.

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Guo, J., Li, J. On centralizers of reflexive algebras. Aequat. Math. 84, 1–12 (2012). https://doi.org/10.1007/s00010-012-0137-y

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  • DOI: https://doi.org/10.1007/s00010-012-0137-y

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