Almost disjoint families of countable sets and separable complementation properties

TL;DR

This paper investigates the separable complementation property in Banach spaces of continuous functions over compacta generated by almost disjoint families, establishing conditions for various SCP variants and characterizing the structure of certain almost disjoint families.

Contribution

It characterizes when spaces have the controlled SCP, provides examples with specific properties, and describes the structure of almost disjoint families inducing monolithic spaces.

Findings

Controlled SCP holds iff the space is Lindelöf in the weak topology.

Existence of spaces with SCP but without monolithic compacta.

Structures of almost disjoint families of size ω₁ derived from ladder systems.

Abstract

We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta $K_{\mathcal A}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets of uncountable sets. For these spaces, we prove among others that $C(K_{\mathcal A})$ has the controlled variant of the separable complementation property if and only if $C(K_{\mathcal A})$ is Lindel\"of in the weak topology if and only if $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ for which $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is not monolithic and an example of a space $C(K_{\mathcal A})$ 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 $\omega_1$ which induce monolithic spaces of the form $K_{\mathcal A}$: They can be obtained from countably many ladder systems and pairwise disjoint families applying simple operations.