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

by E. García-Sánchez and D. Leung and M. Taylor and P. Tradacete

(Report number 2024-10)

Abstract
We study the free Banach lattice \(\text{FBL}^{(p,\infty)}[E]\) with upper \(p\)-estimates generated by a Banach space \(E\). Using a classical result of Pisier on factorization through \(L^{p,\infty}(\mu)\) together with a finite dimensional reduction, it is shown that the spaces \(\ell^{p,\infty}(n)\) witness the universal property of \(\text{FBL}^{(p,\infty)}[E]\) isomorphically. As a consequence, we obtain a functional representation for \(\text{FBL}^{(p,\infty)}[E]\), answering a previously open question. 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 \(\text{FBL}^{(p,\infty)}[E]\). Notably, we prove that the norm for \(\text{FBL}^{(p,\infty)}[E]\) cannot be isometrically witnessed by \(L^{p,\infty}(\mu)\) 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 \(\ell^p\) allows us to give an alternative proof of the subspace problem for free \(p\)-convex Banach lattices. On the other hand, we prove that \(\ell^{p,\infty}\) is not injective in the class of Banach lattices with upper \(p\)-estimates, elucidating one of many difficulties arising in the study of \(\text{FBL}^{(p,\infty)}[E]\).

Keywords: Free Banach lattice; upper $p$-estimate; weak-$L^p$.

@Techreport{GLTT24_1092,
  author = {E. García-Sánchez and D. Leung and M. Taylor and P. Tradacete},
  title = {Banach lattices with upper p-estimates: Free and injective objects},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-10},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-10.pdf },
  year = {2024}
}

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