This is an announcement for the paper "Almost disjoint families of countable sets and separable properties" by Jesus Ferrer, Piotr Koszmider, and Wieslaw Kubis.

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.

Archive classification: math.FA

Mathematics Subject Classification: Primary: 46E15, 03E75. Secondary: 46B20, 46B26

Remarks: 21 pages

Submitted from: kubis@math.cas.cz

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1209.0199

or

http://arXiv.org/abs/1209.0199