Free Banach lattices under convexity conditions

TL;DR

This paper establishes the existence of free Banach lattices within specific convexity-related subcategories, explicitly characterizes their norms, and explores their connection to nonlinear p-summing maps.

Contribution

It proves the existence of free objects in convexity-constrained Banach lattice subcategories and explicitly identifies their norms, extending prior results.

Findings

Existence of free Banach lattices in p-convex and AM-space subcategories.

Explicit norm descriptions for free Banach lattices in these categories.

Connection between these norms and nonlinear p-summing maps.

Abstract

We prove the existence of free objects in certain subcategories of Banach lattices, including $p$-convex Banach lattices, Banach lattices with upper $p$-estimates, and AM-spaces. From this we immediately deduce that projectively universal objects exist in each of these subcategories, extending results of Leung, Li, Oikhberg and Tursi (\emph{Israel J.\ Math.}~2019). In the $p$-con\-vex and AM-space cases, we are able to explicitly identify the norms of the free Banach lattices, and we conclude by investigating the structure of these norms in connection with nonlinear $p$-summing maps.