Banach lattices with upper p-estimates: free and injective objects

• Published in: Mathematische annalen 2024-10
• Main Authors: García-Sánchez, E., Leung, D. H., Taylor, M. A., Tradacete, P.
• Format: Article
• Language: English
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-024-03002-8

We study the free Banach lattice $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] with upper p -estimates generated by a Banach space E . Using a classical result of Pisier on factorization through $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) together with a finite dimensional reduction, it is shown that the spaces $$\ell ^{p,\infty }(n)$$ ℓ p , ∞ ( n ) witness the universal property of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] isomorphically. As a consequence, we obtain a functional representation for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] , answering a question from Oikhberg et al. [Free Banach lattices. J Eur Math Soc (JEMS), 2024]. More generally, our proof allows us to identify the norm of any free Banach lattice over E associated with a rearrangement invariant function space. After obtaining the above functional representation, we take the first steps towards analyzing the fine structure of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] . Notably, we prove that the norm for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] cannot be isometrically witnessed by $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) and settle the question of characterizing when an embedding between Banach spaces extends to a lattice embedding between the corresponding free Banach lattices with upper p -estimates. To prove this latter result, we introduce a novel push-out argument, which when combined with the injectivity of $$\ell ^p$$ ℓ p allows us to give an alternative proof of the subspace problem for free p -convex Banach lattices. On the other hand, we prove that $$\ell ^{p,\infty }$$ ℓ p , ∞ is not injective in the class of Banach lattices with upper p -estimates, elucidating one of many difficulties arising in the study of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] .