Let A be an almost disjoint family of subsets of an infinite set J, and denote by m(J) the Banach space of bounded, scalar-valued functions defined on J and by X_A the closed subspace of m(J) spanned by the indicator functions of intersections of finitely many sets in A. We show that if A has cardinality greater than J, then the closed subspace of X_A spanned by the indicator functions of sets that are finite intersections of at least two distinct sets in A cannot be the kernel of any bounded operator from X_A to m(J). As a consequence, we deduce that the subspace of m(J) consisting of elements x for which the set {j∈J : |x(j)|>ε} has cardinality smaller than J for every ε>0 is not the kernel of any bounded operator on m(J); this generalises results of Kalton and of Pełczyński and Sudakov.

The situation is more complex for the Banach space m_c(J) of countably supported, bounded functions defined on an uncountable set J. We show that it is undecidable in ZFC whether every bounded operator on m_c(ω₁) which vanishes on c₀(ω₁) must vanish on a subspace of the form m_c(A) for some uncountable subset A of ω₁