A Godefroy-Kalton principle for free Banach lattices

TL;DR

This paper explores the lattice-lifting property in Banach lattices, establishing conditions for its presence and analyzing specific classes like spaces with unconditional bases and $C(K)$ spaces.

Contribution

It introduces the lattice-lifting property in Banach lattices, providing necessary conditions and identifying classes that satisfy this property, extending the Lipschitz-lifting concept.

Findings

Banach spaces with a 1-unconditional basis have the lattice-lifting property

Free Banach lattices share the lattice-lifting property

Necessary conditions for a Banach lattice to have the property

Abstract

Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Banach lattices and lattice homomorphisms. Namely, a Banach lattice $X$ satisfies the lattice-lifting property if every lattice homomorphism to $X$ having a bounded linear right-inverse must have a lattice homomorphism right-inverse. In terms of free Banach lattices, this can be rephrased into the following question: which Banach lattices embed into the free Banach lattice which they generate as a lattice-complemented sublattice? We will provide necessary conditions for a Banach lattice to have the lattice-lifting property, and show that this property is shared by Banach spaces with a $1$-unconditional basis as well as free Banach lattices. The case of $C(K)$ spaces will also be analyzed.