Amalgamation and Injectivity in Banach Lattices

Abstract We study distinguished objects in the category $\mathcal{B}\mathcal{L}$ of Banach lattices and lattice homomorphisms. The free construction introduced by de Pagter Wickstead [ 8] generates push-outs, combining this with an old result Kellerer 17] on marginal measures, amalgamation property is established. This will be key tool to prove that $L_1([0,1]^{\mathfrak{c}})$ separably $\mathcal{B}\mathcal{L}$-injective, as well give more abstract examples universal disposition for separable sublattices. Finally, analysis ideals $C(\Delta ,L_1)$, which a shown Leung et al. 21], allows us conclude $\mathcal{B}\mathcal{L}$-injective are necessarily non-separable.