Kernels of operators on Banach spaces induced by almost disjoint families

Abstract

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 $n\in \mathbb{N}$ and $A_1,\dots ,A_{n+1}\in \mathcal{A}$ are distinct, cannot be the kernel of any bounded operator \mbox{$X_{\mathcal{A}}\to {\ell }_{\infty }(\Gamma )$}. As a consequence, we deduce that the subspace \[ \bigl\{ x\in \ell_{\infty}(Γ) : \text{the set}\ \{γ\in Γ: \lvert x(γ)\rvert > \varepsilon \}\ \text{has cardinality smaller than}\ Γ \text{for every}\ \varepsilon>0\bigr\} \] of~${\ell }_{\infty }(\Gamma )$ is not the kernel of any bounded operator on~${\ell }_{\infty }(\Gamma )$; this generalises results of Kalton and of Pełczyński 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~${\omega }_1$.

Major changes in Section 3 (previously Section 4). To appear in \textit{Houston Journal of Mathematics}