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 $\left \| {\phi (AB) - \phi (A)\phi (B)} \right \| \leq \varepsilon ,A,B \in \mathcal {A}$, 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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© Copyright 1995
American Mathematical Society