Free Banach lattices

Abstract

We investigate the structure of the free $p$-convex Banach lattice ${\mathrm{FBL}}^{(p)}[E]$ over a Banach space $E$. After recalling why such a free lattice exists, and giving a convenient functional representation of it, we focus our study on how properties of an operator $T:E\to F$ between Banach spaces transfer to the associated lattice homomorphism $\bar{T}:{\mathrm{FBL}}^{(p)}[E]\to {\mathrm{FBL}}^{(p)}[F]$. Particular consideration is devoted to the case when the operator $T$ is an isomorphic embedding, which leads us to examine extension properties of operators into ${\ell }_p$, and several classical Banach space properties such as being a G.T. space. A detailed investigation of basic sequences and sublattices of free Banach lattices is provided. Among other things, this allows us to settle an a priori unrelated question, providing the first instance of a subspace of a Banach lattice without bibasic sequences. In addition, we begin to build a dictionary between Banach space properties of $E$ and Banach lattice properties of ${\mathrm{FBL}}^{(p)}[E]$. In particular, we characterize the existence of lattice copies of ${\ell }_1$ in ${\mathrm{FBL}}^{(p)}[E]$ and show that $\mathrm{FBL}[E]$ has an upper $p$-estimate if and only if ${\text{id}}_{E^{\ast }}$ is $(q,1)$-summing ($1/p+1/q=1$). We also highlight the significant differences between ${\mathrm{FBL}}^{(p)}$-spaces depending on whether $p$ is finite or infinite. For example, we show that ${\mathrm{FBL}}^{(\infty )}[E]$ is lattice isometric to ${\mathrm{FBL}}^{(\infty )}[F]$ whenever $E$ and $F$ have monotone finite-dimensional decompositions, while, on the other hand, when $p<\infty$ and $E^{\ast }$ is smooth, ${\mathrm{FBL}}^{(p)}[E]$ determines $E$ isometrically.