PERTURBATIONS OF SURJECTIVE HOMOMORPHISMS BETWEEN ALGEBRAS OF OPERATORS ON BANACH SPACES

Tarcsay, Zsigmond and Horváth, Bence (2021) PERTURBATIONS OF SURJECTIVE HOMOMORPHISMS BETWEEN ALGEBRAS OF OPERATORS ON BANACH SPACES. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. pp. 1-14. ISSN 0002-9939 (In Press)

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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 are stable under “small” perturbations. More precisely, if φ, ψ are endomorphisms of B(H) such that ||φ(A) − ψ(A)|| < ||A|| and ψ is surjective, then so is φ. The aim of this paper is to extend this result to a larger class of Banach spaces including ℓp and Lp spaces, where 1 < p < ∞. En route to the proof we show that for any Banach space X from the above class all faithful, unital, separable, reflexive representations of B(X) which preserve rank one operators are in fact isomorphisms.

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