Isomorphisms of standard operator algebras

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.