Octahedral norms in free Banach lattices

Abstract

In this paper, we study octahedral norms in free Banach lattices F BL [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 BL[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.