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.
Rethinking polyhedrality for lindenstrauss spaces
CASINI, EMANUELE GIUSEPPE;
2016-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.