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Octahedral norms in free Banach lattices

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Abstract

In this paper, we study octahedral norms in free Banach lattices FBL[E] generated by a Banach space E. We prove that if E is an \(L_1(\mu )\)-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 FBL[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 \( \ge 2\) is nowhere Fréchet differentiable. Moreover, we discuss some open problems on this topic.

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Acknowledgements

We thank Pedro Tradacete for the fruitful discussions held on the early phase of this work. We also thank Antonio Avilés and José Rodríguez for pointing out typos and for comments that have improved the exposition of the text. Finally, we thank Johann Langemets for pointing out Remark 4.2.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27, Prague 6, Czech Republic

    Sheldon Dantas

  2. Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100, Murcia, Spain

    Gonzalo Martínez-Cervantes & José David Rodríguez Abellán

  3. Departamento de Análisis Matemático, 18071, Granada, Spain

    Abraham Rueda Zoca

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  1. Sheldon Dantas
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  2. Gonzalo Martínez-Cervantes
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  3. José David Rodríguez Abellán
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  4. Abraham Rueda Zoca
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Correspondence to José David Rodríguez Abellán.

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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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Dantas, S., Martínez-Cervantes, G., Rodríguez Abellán, J.D. et al. Octahedral norms in free Banach lattices. RACSAM 115, 6 (2021). https://doi.org/10.1007/s13398-020-00940-1

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  • DOI: https://doi.org/10.1007/s13398-020-00940-1

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