Almost disjoint families of countable sets and separable complementation properties

Abstract

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 others that C(K A ) has the controlled variant of the separable complementation property if and only if C(K A ) is Lindelöf 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 ω 1 which induce monolithic spaces of the form K A : They can be obtained from countably many ladder systems and pairwise disjoint families applying simple operations. MSC (2010) Primary: 03E75, 46E15. Secondary: 46B20, 46B26.

Rich families and elementary submodels

“…We apply the main theorem to clarify and simplify the definion of a projectional skeleton. This notion was introduced by W. Kubiś in [14] and further investigated in [3,4,8,13]. Let us give the original definition.…”

The Separable Complementation Property and Mrówka Compacta

“…In (Ferrer, J., Koszmider, P. & W. Kubiś, 2013) an example of a Mrówka compact K A is given such that C(K A ) has the S CP but not the CS CP, and also in that reference a broad class of Mrówka compacta whose function space enjoys the CS CP but cannot be isomorphic to any c 0 (Γ) space is highlighted, namely the non-trivial ladder systems in ω 1 .…”

“…Ladder systems were originally used by R. Pol, see (Pol, R., 1979), to give the first example of a weakly Lindelöf C(K) space such that K is not a Corson compact. We used ladder systems in (Ferrer, J., Koszmider, P. & W. Kubiś, 2013) to prove that there are Mrówka compact spaces K A which are monolithic, hence its associated function space C(K A ) has the CS CP, but K A is not even a continuous image of a Valdivia compact and so C(K A ) is not isomorphic to any c 0 (Γ) space.…”

“…If X is a Banach space such that it has the CS CP, it is not hard to see, ( Ferrer, J. et al,(2013), that the dual unit ball B X * is monolithic respect to the weak-star topology. Consequently, if C(K) has the CS CP, since being monolithic is hereditary, it follows that K is monolithic.…”

“…& Wójtowicz, M., 2011;Ferrer, J., Koszmider, P. & W. Kubiś, 2013;Ferrer, J., 2014;Ferrer, J., 2009), we studied the controlled version of the separable complementation property (CS CP, for short) for general Banach spaces and in particular for C(K A ) spaces when K A is the Mrówka compact associated to an almost disjoint family A of countable subsets of a given set. After seeing that K being monolithic, see (Arkhangel'skii, A. V., 1992), is a necessary condition in order that the space C(K) enjoys the CS CP, we proved this condition to be sufficient when K is a Mrówka compact and moreover we also showed that this condition suffices in general when K is a scattered compact such that each of its points, except possibly the ones in the top layer, admit a countable neighborhood base.…”