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

Abstract

We study the free Banach lattice $FB{L}^{(p,\infty )}[E]$ with upper $p$-estimates generated by a Banach space $E$. Using a classical result of Pisier on factorization through $L^{p,\infty }(\hat{I}¼)$ together with a finite dimensional reduction, it is shown that the spaces ${\ell }^{p,\infty }(n)$ witness the universal property of $FB{L}^{(p,\infty )}[E]$ isomorphically. As a consequence, we obtain a functional representation for $FB{L}^{(p,\infty )}[E]$. 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 $FB{L}^{(p,\infty )}[E]$. Notably, we prove that the norm for $FB{L}^{(p,\infty )}[E]$ cannot be isometrically witnessed by $L^{p,\infty }(\hat{I}¼)$ 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 $FB{L}^{(p,\infty )}[E]$.

37 pages