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arXiv · 2007.14112

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

Bence Horváth·Tomasz Kania
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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'.

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Bence Horváth, Tomasz Kania. 2020-07-28. Surjective homomorphisms from algebras of operators on long sequence spaces are automatically injective. https://doi.org/10.1093/qmath/haaa066

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