Surjective homomorphisms from algebras of operators on long sequence spaces are automatically injective

TL;DR

This paper proves that surjective algebra homomorphisms from operator algebras on certain long sequence spaces are necessarily injective, and classifies specific ideals in these operator algebras.

Contribution

It establishes automatic injectivity of surjective homomorphisms for operators on long sequence spaces and classifies related two-sided ideals.

Findings

Surjective homomorphisms are automatically injective on these spaces.

Classified two-sided ideals closed in the sequential strong operator topology.

Extended understanding of algebraic structure of operators on long sequence spaces.

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 \emph{long} sequence spaces: $c_0(\lambda)$, $\ell_{\infty}^c(\lambda)$, and $\ell_p(\lambda)$ ($1 \leqslant p < \infty$) and $Y$ is arbitrary. \textit{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'.