Perturbations of surjective homomorphisms between algebras of operators on Banach spaces

Abstract

A remarkable result of Molnár [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 $\varphi ,\psi$ are endomorphisms of $\mathcal{B}(\mathcal{H})$ such that $\Vert \varphi (A)-\psi (A)\Vert <\Vert A\Vert$ and $\psi$ is surjective then so is $\varphi$. 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.

14 pp. To appear in Proceedings of the American Mathematical Society

Cited by in corpus (1)

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