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 others 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 applying simple operations. MSC (2010) Primary: 03E75, 46E15. Secondary: 46B20, 46B26.

For a compact K, a necessary condition for C(K) to have the Controlled Separable Complementation Property is that K be monolithic. In this paper, we prove that when K contains no copy of [0, ω ω ] and the set of points which admit a countable neighborhood base is a cofinite subset of K, then monolithicity of K is sufficient for C(K) to enjoy the Controlled Separable Complementation Property. We also show that, for this type of compacta K, the space C(K) is separably extensible.