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arXiv · 2007.05288

Octahedral norms in free Banach lattices

Sheldon Dantas·Gonzalo Martínez-Cervantes·José David Rodríguez Abellán·Abraham Rueda Zoca
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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 $ \geq 2$ is nowhere Fr\'echet differentiable. Moreover, we discuss some open problems on this topic.

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Sheldon Dantas, Gonzalo Martínez-Cervantes, José David Rodríguez Abellán, Abraham Rueda Zoca. 2020-07-10. Octahedral norms in free Banach lattices. https://arxiv.org/abs/2007.05288

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