Amalgamation and injectivity in Banach lattices

Antonio Avilés, Pedro Tradacete

Published 2020-07-30Version 1

We study distinguished objects in the category $\mathcal{BL}$ of Banach lattices and lattice homomorphisms. The free Banach lattice construction introduced by B. de Pagter and A. W. Wickstead generates push-outs, and combining this with an old result of H. G. Kellerer on marginal measures, the amalgamation property of Banach lattices is established. This will be the key tool to prove that $L_1([0,1]^{\mathfrak{c}})$ is separably $\mathcal{BL}$-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 $C(\Delta,L_1)$, which is a separably universal Banach lattice as shown by D. H. Leung, L. Li, T. Oikhberg and M. A. Tursi, allows us to conclude that separably $\mathcal{BL}$-injective Banach lattices are necessarily non-separable.