A 1-separably injective space that does not contain $\ell_\infty$
  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. 
Antonio Avil\'es, Piotr Koszmider
math.FA math.LO