Kernels of operators on Banach spaces induced by almost disjoint families

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~\(\omega_1\).