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 KAKA induced by almost disjoint families AA of countable subsets of uncountable sets. For these spaces, we prove among other things that C(KA)C(KA) has the controlled variant of the separable complementation property if and only if C(KA)C(KA) is Lindelöf in the weak topology if and only if KAKA is monolithic. We give an example of AA for which C(KA)C(KA) has the SCP while KAKA is not monolithic and an example of a space C(KA)C(KA) 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 ω1ω1 which induce monolithic spaces of the form KAKA: they can be obtained from countably many ladder systems and pairwise disjoint families by applying simple operations.