Octahedral norms in free Banach lattices

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${}^{\ast }$-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 $E^{\ast }$ 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.