Free Banach lattices

We investigate the structure of the free $p$-convex Banach lattice $FBL^{(p)}[E]$ 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 $T:E\rightarrow F$ between Banach spaces transfer to the associated lattice homomorphism $\overline{T}:FBL^{(p)}[E]\rightarrow FBL^{(p)}[F]$. 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. In addition, we begin to build a dictionary between Banach space properties of $E$ and Banach lattice properties of $FBL^{(p)}[E]$. In particular, we characterize the existence of lattice copies of $\ell_1$ in $FBL^{(p)}[E]$ and show that $FBL[E]$ has an upper $p$-estimate if and only if $id_{E^*}$ is $(q,1)$-summing ($\frac{1}{p}+\frac{1}{q}=1$). We also highlight the significant differences between $FBL^{(p)}$-spaces depending on whether $p$ is finite or infinite. For example, we show that $FBL^{(\infty)}[E]$ is lattice isometric to $FBL^{(\infty)}[F]$ whenever $E$ and $F$ have monotone finite dimensional decompositions, while, on the other hand, when $p<\infty$ and $E^*$ is smooth, $FBL^{(p)}[E]$ determines $E$ isometrically.