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Isomorphisms of standard operator algebras


Author: Peter Šemrl
Journal: Proc. Amer. Math. Soc. 123 (1995), 1851-1855
MSC: Primary 47D30
DOI: https://doi.org/10.1090/S0002-9939-1995-1242104-8
MathSciNet review: 1242104
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Abstract: Let X and Y be Banach spaces, $ \dim X = \infty $, and let $ \mathcal{A}$ and $ \mathcal{B}$ be standard operator algebras on X and Y, respectively. Assume that $ \phi :\mathcal{A} \to \mathcal{B}$ is a bijective mapping satisfying $ \left\Vert {\phi (AB) - \phi (A)\phi (B)} \right\Vert \leq \varepsilon ,A,B \in \mathcal{A}$, where $ \varepsilon $ is a given positive real number (no linearity or continuity of $ \phi $ is assumed). Then $ \phi $ is a spatially implemented linear or conjugate linear algebra isomorphism. In particular, $ \phi $ is continuous.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1242104-8
Article copyright: © Copyright 1995 American Mathematical Society

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