Octahedral norms in free Banach lattices
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Octahedral norms in free Banach lattices Sheldon Dantas1 · Gonzalo Martínez-Cervantes2 · José David Rodríguez Abellán2 · Abraham Rueda Zoca3 Received: 11 July 2020 / Accepted: 28 September 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In this paper, we study octahedral norms in free Banach lattices F B L[E] generated by a Banach space E. We prove that if E is an L 1 (μ)-space, a predual of von Neumann algebra, a predual of a JBW∗ -triple, the dual of an M-embedded Banach space, the disc algebra or the projective tensor product under some hypothesis, then the norm of F B L[E] is octahedral. We get the analogous result when the topological dual E ∗ of E is almost square. We finish the paper by proving that the norm of the free Banach lattice generated by a Banach space of dimension ≥ 2 is nowhere Fréchet differentiable. Moreover, we discuss some open problems on this topic. Keywords Banach lattice · Free Banach lattice · Octahedral norms · Almost square · Diameter two properties Mathematics Subject Classification 46B04 · 46B20 · 46B42
1 Introduction Recently, a new concept was introduced by A. Avilés, J. Rodríguez, and P. Tradacete in order to establish strong connections between Banach lattices and Banach spaces [4]. If E is a Banach space, then the free Banach lattice generated by E is a Banach lattice that contains a subspace which is linearly isometric to E and such that the elements of this new space can be seen as lattice-free generators (see Sect. 2 for a precise definition). Since then, it has been
S. Dantas was supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778 and by the Estonian Research Council grant PRG877. G. Martínez-Cervantes and J. D. Rodríguez Abellán were supported by the project MTM2017-86182-P (Government of Spain, AEI/FEDER, EU) and the project 20797/PI/18 by Fundación Séneca, ACyT Región de Murcia. The research of G. Martínez-Cervantes has been co-financed by the European Social Fund (ESF) and the Youth European Initiative (YEI) under the Spanish Seneca Foundation (CARM) (ref. 21319/PDGI/19). J. D. Rodríguez Abellán was supported by FPI contract of Fundación Séneca, ACyT Región de Murcia. The research of A. Rueda Zoca was supported by MICINN (Spain) Grant PGC2018-093794-B-I00 (MCIU, AEI, FEDER, UE), by Junta de Andalucía Grant A-FQM-484-UGR18 and by Junta de Andalucía Grant FQM-0185.
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José David Rodríguez Abellán [email protected]
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intensively studied for the purpose of better understanding how the vector space, the order, and the topological structures are related. In particular, in [4] they answered a question of J. Diestel, providing an example of a Banach lattice which is weakly compactly generated as a lattice but not as a Banach space. Namely, they proved that the free Banach lattice generated by p () contains an isomorphic copy of 1 () whenever 1 < p ≤ 2 and, therefore, it is not weakly compactly generated
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