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Abstract: We study the stability properties of the class of weak*-extensible spaces introduced by Wang, Zhao, and Qiang showing, among other things, that weak*-extensibility is equivalent to having a weak*-sequentially continuous dual ball (in short, w*SC) for duals of separable spaces or twisted sums of w*SC spaces. This shows that weak*-extensibility is not a -space property, solving a question posed by Wang, Zhao, and Qiang. We also introduce a restricted form of weak*-extensibility, called separable weak*-extensibility, and show that separably weak*-extensible Banach spaces have the Gelfand–Phillips property, although they are not necessarily w*SC spaces. [Copyright &y& Elsevier]

We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact K A generated by an almost disjoint family A of infinite subsets of ω — we present a solution to two problems and develop previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space C (K A) is uniquely determined up to isomorphism by the cardinality of A whenever | A | < c , while there are 2 c nonisomorphic spaces C (K A) with | A | = c. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of c 0 and some C (K) spaces. [ABSTRACT FROM AUTHOR]