Perturbations of surjective homomorphisms between algebras of operators on Banach spaces

Horváth, Bence; Tarcsay, Zsigmond

doi:10.1090/proc/15666

Perturbations of surjective homomorphisms between algebras of operators on Banach spaces
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by Bence Horváth and Zsigmond Tarcsay PDF

Proc. Amer. Math. Soc. 150 (2022), 747-761 Request permission

Abstract:

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.

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