Let~$\mathcal{ A}$ be an almost disjoint family of subsets of an infinite set~$\Gamma$, and denote by~$X_{ \mathcal{ A}}$ the closed subspace of~$\ell_\infty(\Gamma)$ spanned by the indicator functions of intersections of finitely many sets in~$\mathcal{ A}$. We show that if~$\mathcal{ A}$ has cardinality greater than~$\Gamma$, then the closed subspace of~$X_{ \mathcal{ A}}$ spanned by the indicator functions of sets of the form $\bigcap_{ j=1}^{ n+1}A_j$, where $n\in\N$ and $A_1,\ldots,A_{ n+1}\in\mathcal{ A}$ are distinct, cannot be the kernel of any bounded operator \mbox{ $X_{ \mathcal{ A}}\rightarrow \ell_{ \infty}(\Gamma)$}. As a consequence, we deduce that the subspace \[ \bigl\{ x\in \ell_{ \infty}(\Gamma) : \text{ the set}\ \{ \gamma \in \Gamma : \lvert x(\gamma)\rvert > \varepsilon \}\ \text{ has cardinality smaller than}\ \Gamma\ \text{ for every}\ \varepsilon>0\bigr\} \] of~$\ell_\infty(\Gamma)$ is not the kernel of any bounded operator on~$\ell_\infty(\Gamma)$; this generalises results of Kalton and of Pe\l{ }czy\'{ n}ski and Sudakov. The situation is more complex for the Banach space~$\ell_\infty^c(\Gamma)$ of countably supported, bounded functions defined on an uncountable set~$\Gamma$. We show that it is undecidable in \textsf{ ZFC} whether every bounded operator on~$\ell_\infty^c(\omega_1)$ which vanishes on~$c_0(\omega_1)$ must vanish on a subspace of the form~$\ell_\infty^c(A)$ for some uncountable subset~$A$ of~$\omega_1$.
Categories: math.FA