Surjective Homomorphisms from Algebras of Operators on Long Sequence Spaces are Automatically Injective

Abstract

We study automatic injectivity of surjective algebra homomorphisms from $\mathscr{B}(X)$ , the algebra of (bounded, linear) operators on X, to $\mathscr{B}(Y)$ , where X is one of the following long sequence spaces: c 0 (λ), $\ell_{\infty}^c(\lambda)$ , and $\ell_p(\lambda)$ ( $1 \leqslant p \lt \infty$ ) and Y is arbitrary. En route to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the ‘sequential strong operator topology’.