A 1-separably injective space that does not contain ${\ell }_{\infty }$

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.

References in corpus (2)

• Banach spaces of universal disposition
• Applications of another characterization of betaN\N