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We study the separable complementation property (SCP) and its naturalvariations in Banach spaces of continuous functions over compacta $K_{\mathcalA}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets ofuncountable sets. For these spaces, we prove among others that $C(K_{\mathcalA})$ has the controlled variant of the separable complementation property ifand only if $C(K_{\mathcal A})$ is Lindel\"of in the weak topology if and onlyif $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ forwhich $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is not monolithicand an example of a space $C(K_{\mathcal A})$ with controlled and continuousSCP which has neither a projectional skeleton nor a projectional resolution ofthe identity. Finally, we describe the structure of almost disjoint families ofcardinality $\omega_1$ which induce monolithic spaces of the form $K_{\mathcalA}$: They can be obtained from countably many ladder systems and pairwisedisjoint families applying simple operations.

Almost disjoint families of countable sets and separable complementation properties