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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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