Amalgamation and Injectivity in Banach Lattices

We study distinguished objects in the category BL of Banach lattices and lattice homomorphisms. The free Banach lattice construction introduced by de Pagter and Wickstead [ 8] generates push-outs, and combining this with an old result of Kellerer [ 17] on marginal measures, the amalgamation property of Banach lattices is established. This will be the key tool to prove that L₁([0,1]ᶜ) is separably BL-injective, as well as to give more abstract examples of Banach lattices of universal disposition for separable sublattices. Finally, an analysis of the ideals on C(Δ ,L₁), which is a separably universal Banach lattice as shown by Leung et al. [ 21], allows us to conclude that separably BL-injective Banach lattices are necessarily non-separable.