Amalgamation and injectivity in Banach lattices

TL;DR

This paper investigates the structure and properties of Banach lattices, establishing their amalgamation property and demonstrating the existence of separably injective Banach lattices, with implications for universality and separability.

Contribution

It proves the amalgamation property for Banach lattices using free constructions and applies this to show the existence of separably -injective Banach lattices and their non-separability.

Findings

Banach lattices have the amalgamation property.

Existence of separably -injective Banach lattices.

Separable -injective Banach lattices are necessarily non-separable.

Abstract

We study distinguished objects in the category $\mathcal{BL}$ of Banach lattices and lattice homomorphisms. The free Banach lattice construction introduced by de Pagter and Wickstead generates push-outs, and combining this with an old result of 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 Leung, Li, Oikhberg and Tursi, allows us to conclude that separably $\mathcal{BL}$-injective Banach lattices are necessarily non-separable.