Octahedral norms in free Banach lattices

TL;DR

This paper investigates conditions under which free Banach lattices have octahedral norms, showing that certain classes of Banach spaces induce octahedrality and that these norms are nowhere Fréchet differentiable.

Contribution

It establishes new classes of Banach spaces whose free Banach lattices possess octahedral norms and analyzes the differentiability properties of these norms.

Findings

Octahedral norms occur in free Banach lattices generated by specific Banach spaces.

The norm of such lattices is nowhere Fréchet differentiable for spaces of dimension ≥ 2.

Conditions like being an $L_1(u)$-space or dual of an M-embedded space imply octahedrality.

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.