Kernels of operators on Banach spaces induced by almost disjoint families

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 XA 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 XA 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 XA 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 mc(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 mc(ω1) which vanishes on c0(ω1) must vanish on a subspace of the form mc(A) for some uncountable subset A of ω1