Fragmentability in Banach spaces: interaction of topologies
This paper is a survey article aimed to give a relatively concise description of the state-of-the-art of the theory of fragmentability and \(\sigma\)-fragmentability in Banach spaces for a broad audience. Modern directions of investigation, related to renorming of Banach spaces and the Lindelöf property are also discussed. Here, the author concentrates mainly on classical problems of the theory, including different characterizations for a Banach space to have an equivalent Kadec (resp. LUC) norm, the existence of a \(\tau_p\)-Kadec norm on \(C(K)\) when \(K\) is a compact space and the characterization of fragmentability through the Lindelöf property. Most proofs are given and most of them are accompanied with historical references and comments about open questions for future research. A comprehensive bibliography is provided at the end of the paper. The first section, of preliminary nature, is dedicated to presenting the basic results and definitions for the rest of the paper; among other things, the Radon-Nikodým Property (RNP), Asplund spaces and an application to the Ryll-Nardzewski's fixed point theorem are presented. Section 2 is one of the main parts of the paper. Here, it is proved that the fragmentability of a compact subset \(K\) of \([-1,1]^D\) is equivalent to the separability of the metric spaces \((K,d_A)\) for each countable set \(A\subseteq D\). As a consequence, it follows easily that each \(\tau_p\)-compact subset of \(C(K)\) is norm-fragmented for every compact Hausdorff space \(K\) and that each \(w\)-compact subset of a Banach space is norm-fragmented. Radon-Nikodým (RN) compacta are defined as those compact spaces that are homeomorphic to a weak\(^\ast\)-compact subset of a dual Banach space with the RNP. As a consequence, one obtains that Eberlein compact and scattered compact Hausdorff spaces are RN-compact, and that a compact space \(K\) is RN-compact iff \(K\) is fragmented by a lower-semicontinuous metric. From this fact, it follows that \(K\) is RN-compact iff it is homeomorphic to a compact subset of \(([-1,1]^D ,\tau_p)\) for some index set \(D\). Then the author discusses the relationship among these notions and other variations of them. The connection with Gâteaux and Fréchet differentiability is treated using \textit{usco} set-valued maps. As an application, it is shown that if \(E\) is a Banach space, \(B^*\) is the unit ball of the dual space \(E^*\) and \((B^*,w^*)\) is norm-fragmented, then \(E\) is an Asplund space. Section 3 concerns the notion of \(\sigma\)-fragmentability that is better suited to deal with non-compact spaces. In the line of the previous section, the author presents the basic results about \(\sigma\)-fragmented spaces and their applications. For instance, it is proved that \(C_p(\beta \mathbb N)\) does not embed in \(C_p(\{0,1\}^D)\) for an arbitrary index set \(D\), which answers a question by Arkhangelskii. Section 4 is dedicated to presenting recent topological methods (developed by Raja) that apply in renorming of Banach spaces. As an illustration, the author presents in detail the class of compact totally ordered spaces. The relation between fragmentability and the Lindelöf property is addressed in Section 5. Finally, several interesting open questions are collected in Section 6.
scientific article; zbMATH DE number 5925092
scientific article