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~$\mathrm{\Gamma }$, and denote by~$X_{\mathcal{A}}$ the closed subspace of~${\mathrm{\ell }}_{\mathrm{\infty }}(\mathrm{\Gamma })$ spanned by the indicator functions of intersections of finitely many sets in~$\mathcal{A}$. We show that if~$\mathcal{A}$ has cardinality greater than~$\mathrm{\Gamma }$, 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 $n\in \mathbb{N}$ and $A_1,\dots ,A_{n+1}\in \mathcal{A}$ are distinct, cannot be the kernel of any bounded operator \mbox{$X_{\mathcal{A}}\to {\mathrm{\ell }}_{\mathrm{\infty }}(\mathrm{\Gamma })$}. As a consequence, we deduce that the subspace $$\{x\in {\mathrm{\ell }}_{\mathrm{\infty }}(\mathrm{\Gamma }):\text{the set }\{\gamma \in \mathrm{\Gamma }:\mid x(\gamma )\mid >\epsilon \}\text{ has cardinality smaller than }\mathrm{\Gamma }\text{for every }\epsilon >0\}$$ of~${\mathrm{\ell }}_{\mathrm{\infty }}(\mathrm{\Gamma })$ is not the kernel of any bounded operator on~${\mathrm{\ell }}_{\mathrm{\infty }}(\mathrm{\Gamma })$; this generalises results of Kalton and of Pełczyński and Sudakov. The situation is more complex for the Banach space~${\mathrm{\ell }}_{\mathrm{\infty }}^c(\mathrm{\Gamma })$ of countably supported, bounded functions defined on an uncountable set~$\mathrm{\Gamma }$. We show that it is undecidable in $\mathsf{\text{ZFC}}$ whether every bounded operator on~${\mathrm{\ell }}_{\mathrm{\infty }}^c({\omega }_1)$ which vanishes on~$c_0({\omega }_1)$ must vanish on a subspace of the form~${\mathrm{\ell }}_{\mathrm{\infty }}^c(A)$ for some uncountable subset~$A$ of~${\omega }_1$.