Almost disjoint families of countable sets and separable complementation properties

We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta K A induced by almost disjoint families A of countable subsets of uncountable sets. For these spaces, we prove among other things that C ( K A ) has the controlled variant of the separable complementation property if and only if C ( K A ) is Lindelöf in the weak topology if and only if K A is monolithic. We give an example of A for which C ( K A ) has the SCP while K A is not monolithic and an example of a space C ( K A ) with controlled and continuous SCP which has neither a projectional skeleton nor a projectional resolution of the identity. Finally, we describe the structure of almost disjoint families of cardinality ω 1 which induce monolithic spaces of the form K A : they can be obtained from countably many ladder systems and pairwise disjoint families by applying simple operations.