Rethinking polyhedrality for lindenstrauss spaces

Abstract

We present a Lindenstrauss space with an extreme point that does not contain a subspace linearly isometric to c. This example disproves a result stated by Zippin in a paper published in 1969 and it shows that some classical characterizations of polyhedral Lindenstrauss spaces, based on Zippin’s result, are false, whereas some others remain unproven; then we provide a correct proof for those characterizations. Finally, we also disprove a characterization of polyhedral Lindenstrauss spaces given by Lazar, in terms of the compact norm-preserving extension of compact operators, and we give an equivalent condition for a Banach space X to satisfy this property.

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Correspondence to Emanuele Casini.

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Casini, E., Miglierina, E., Piasecki, Ł. et al. Rethinking polyhedrality for lindenstrauss spaces. Isr. J. Math. 216, 355–369 (2016). https://doi.org/10.1007/s11856-016-1412-8

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