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 (bounded, linear) operators on X, to $\mathscr{B}(Y)$, where X is one following long sequence spaces: c0(λ), $\ell_{\infty}^c(\lambda)$, and $\ell_p(\lambda)$ ($1 \leqslant p \lt \infty$) Y arbitrary. En route proof that these spaces do indeed enjoy such a property, we classify two-sided ideals any aforementioned Banach are closed with respect ‘sequential strong operator topology’.