Almost disjoint families of countable sets and separable complementation properties

作者：Ferrer Jesus; Koszmider Piotr; Kubis Wieslaw^*

来源：Journal of Mathematical Analysis and Applications, 2013, 401(2): 939-949.

DOI：10.1016/j.jmaa.2013.01.008

• Almost disjoint family
• Projections in Banach spaces
• Separable complementation property
• Ladder system space

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 other things that C(K-A) has the controlled variant of the separable complementation property if and only if C(K-A) is Lindelof 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 omega(1) which induce monolithic spaces of the form K-A: they can be obtained from countably many ladder systems and pairwise disjoint families by applying simple operations.

• 出版日期2013-5