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arXiv · 1609.02685

A 1-separably injective space that does not contain $\ell_\infty$

Antonio Avilés·Piotr Koszmider
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Abstract

We show that the problem whether every $1$-separably injective Banach space contains an isomorphic copy of $\ell_\infty$ is undecidable. Namely, unlike under the continuum hypothesis, assuming Martin's axiom and the negation of the continuum hypothesis, there is an $1$-separably injective Banach space of the form $C(K)$ (which means that $K$ is an $F$-space) without an isomorphic copy of $\ell_\infty$. This result is a consequence of our study of $ω_2$-subsets of tightly $σ$-filtered Boolean algebras introduced by Koppelberg for which we obtain some general principles useful when transferring properties of Boolean algebras to the level of Banach spaces.

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Antonio Avilés, Piotr Koszmider. 2017-03-20. A 1-separably injective space that does not contain $\ell_\infty$. https://doi.org/10.1112/blms.12134

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