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arXiv 2007.141122020-07math.FA

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

Bence Horváth, Tomasz Kania

备注：: 22 pp; to appear in Quart. J. Math. (Oxford)
发表信息：: The Quarterly Journal of Mathematics (Oxford) 72, (2021), pp 1167--1189
DOI：: 10.1093/qmath/haaa066

英文摘要：

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: $c_0(λ)$, $\ell_{\infty}^c(λ)$, and $\ell_p(λ)$ ($1 \leqslant p < \infty$) 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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