Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Free dual spaces and free Banach lattices

Enrique García-Sánchez, Pedro Tradacete

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 $E^{**}$, denoted $FBL^p[E^{**}]$, admits a canonical isometric lattice embedding into $FBL^p[E]^{**}$ and $FBL^p[E^{**}]$ is lattice finitely representable in $FBL^p[E]$. Moreover, we also show that for $p>1$, $FBL^p[E]^{**}$ 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 $\ell_1$.

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 the free -convex Banach lattice generated by , denoted , admits a canonical isometric lattice embedding into and is lattice finitely representable in . Moreover, we also show that for , can actually be considered as the free dual -convex Banach lattice generated by , whereas for this happens precisely when does not contain complemented copies of .
Paper Structure (5 sections, 17 theorems, 70 equations)

This paper contains 5 sections, 17 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminary computations and local reflexivity
  3. ${\mathrm{FBL}}^{(p)}[E^{**}]$ embeds in ${\mathrm{FBL}}^{(p)}[E]^{**}$
  4. Free $p$-convex dual Banach lattices
  5. Free duals of $AL$-spaces

Key Result

Lemma 2.1

For every Banach space $E$, $C_{J_E}:\mathcal{H}_p(E^{**},\mathbb R)\rightarrow \mathcal{H}_p(E,\mathbb R)$ is a surjective lattice homomorphism, and $\kappa_E:\mathcal{H}_p(E,\mathbb R)\rightarrow \mathcal{H}_p(E^{**},\mathbb R)$ is an isometric lattice embedding such that $C_{J_E}\circ \kappa_{E}=

Theorems & Definitions (39)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • ...and 29 more