Abstract of a paper by Jesus Ferrer, Piotr Koszmider, and Wieslaw Kubis
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
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