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

Abstract

We study automatic injectivity of surjective algebra homomorphisms from $\mathcal{B}(X)$, the algebra of (bounded, linear) operators on $X$, to $\mathcal{B}(Y)$, where $X$ is one of the following \emph{long} sequence spaces: $c_0(\lambda )$, ${\ell }_{\infty }^c(\lambda )$, and ${\ell }_p(\lambda )$ ($1⩽p<\infty$) 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'.

arXiv comment22 pp; to appear in Quart. J. Math. (Oxford)

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• Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support
• Separated sets and Auerbach systems in Banach spaces
• Closed Ideals of Operators between the Classical Sequence Spaces
• When are full representations of algebras of operators on Banach spaces automatically faithful?
• Banach spaces for which the space of operators has $2^{\mathfrak{c}}$ closed ideals

Cited by in corpus (2)

• The SHAI property for the operators on L^p
• Smooth norms in dense subspaces of ${\ell }_p(\Gamma )$ and operator ranges