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

TL;DR

This paper demonstrates, under certain set-theoretic assumptions, the existence of a 1-separably injective Banach space that does not contain an isomorphic copy of _, challenging previous conjectures.

Contribution

It constructs a specific example of a 1-separably injective Banach space without _, using Boolean algebra techniques and set-theoretic assumptions.

Findings

Existence of such Banach space under Martin's axiom

Development of principles transferring Boolean algebra properties to Banach spaces

Insight into the undecidability of the problem under different set-theoretic axioms

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 $\omega_2$-subsets of tightly $\sigma$-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.