Free dual spaces and free Banach lattices

Abstract

The relation between the free Banach lattice generated by a Banach space and free dual spaces is clarified. In particular, it is shown that for every Banach space $E$ the free $p$-convex Banach lattice generated by $E^{\ast \ast }$, denoted $FB{L}^p[E^{\ast \ast }]$, admits a canonical isometric lattice embedding into $FB{L}^p[E{]}^{\ast \ast }$ and $FB{L}^p[E^{\ast \ast }]$ is lattice finitely representable in $FB{L}^p[E]$. Moreover, we also show that for $p>1$, $FB{L}^p[E{]}^{\ast \ast }$ can actually be considered as the free dual $p$-convex Banach lattice generated by $E$, whereas for $p=1$ this happens precisely when $E$ does not contain complemented copies of ${\ell }_1$.

arXiv comment30 pages; includes new result on lattice finite representability of $FB{L}^p[E^{\ast \ast }]$ in $FB{L}^p[E]$

References in corpus (1)

• On projective Banach lattices of the form C(K) and FBL[E]