Perturbations of surjective homomorphisms between algebras of operators on Banach spaces.

A remarkable result of Molnár [Proc. Amer. Math. Soc. 126 (1998), pp. 853–861] states that automorphisms of the algebra of operators acting on a separable Hilbert space are 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, where 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. [ABSTRACT FROM AUTHOR]