Perturbations of surjective homomorphisms between algebras of operators on Banach spaces

Abstract

A remarkable result of Moln\'ar [Proc. Amer. Math. Soc., 126 (1998), 853-861] states that automorphisms of the algebra of operators acting on a separable Hilbert space is stable under "small" perturbations. More precisely, if $\phi,\psi$ are endomorphisms of $\mathcal{B}(\mathcal{H})$ such that $\|\phi(A)-\psi(A)\|<\|A\|$ and $\psi$ is surjective then so is $\phi$. The aim of this paper is to extend this result to a larger class of Banach spaces including $\ell_p$ and $L_p$ spaces ($1<p<+\infty$). En route to the proof we show that for any Banach space $X$ from the above class all faithful, unital, separable, reflexive representations of $\mathcal B (X)$ which preserve rank one operators are in fact isomorphisms.