The convex point of continuity property in Banach spaces not containing \(\ell _{1}\)
Recall that a closed, bounded and convex subset \(C\) of a Banach space is said to have the point of continuity property (PCP) (resp., the convex point of continuity property (CPCP), provided that every nonempty closed and bounded subset of \(C\) (resp., every nonempty closed, bounded and convex subset) admits a point of continuity of the identity map from the weak to norm topologies. A Banach space \(X\) satisfies PCP (resp., CPCP) if its closed unit ball has PCP (resp. CPCP). A well-known open problem is if PCP and CPCP are determined by the separable subspaces. The main goal of the paper under review is to prove that, for Banach spaces not containing subspaces isomorphic to \(l_{1}\), CPCP is determined on subspaces with a Schauder basis. In order to prove this, first the authors obtain a local characterization of CPCP for closed, bounded and convex subsets in Banach spaces without \(l_{1}\)-copies. This characterization is an extension of a result in [\textit{S. Argyros, E. Odell} and \textit{H. Rosenthal}, Lect. Notes Math. 1332, 80--111 (1988; Zbl 0658.46012)].
scientific article; zbMATH DE number 5871229
scientific article