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Geometry of unit balls of free Banach lattices, and its applications

Timur Oikhberg

Abstract

We begin by describing the unit ball of the free $p$-convex Banach lattice over a Banach space $E$ (denoted by ${\mathrm{FBL}}^{(p)}[E]$) as a closed solid convex hull of an appropriate set. Based on it, we show that, if a Banach space $E$ has the $λ$-Approximation Property, then ${\mathrm{FBL}}^{(p)}[E]$ has the $λ$-Positive Approximation Property. Further, we show that operators $u \in B(E,F)$ (where $E$ and $F$ are Banach spaces) which extend to lattice homomorphisms from ${\mathrm{FBL}}^{(q)}[E]$ to ${\mathrm{FBL}}^{(p)}[F]$ are precisely those whose adjoints are $(q,p)$-mixing. Related results are also obtained for free lattices with an upper $p$-estimate.

Geometry of unit balls of free Banach lattices, and its applications

Abstract

We begin by describing the unit ball of the free -convex Banach lattice over a Banach space (denoted by ) as a closed solid convex hull of an appropriate set. Based on it, we show that, if a Banach space has the -Approximation Property, then has the -Positive Approximation Property. Further, we show that operators (where and are Banach spaces) which extend to lattice homomorphisms from to are precisely those whose adjoints are -mixing. Related results are also obtained for free lattices with an upper -estimate.
Paper Structure (11 sections, 25 theorems, 72 equations)

This paper contains 11 sections, 25 theorems, 72 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Representing unit balls as solid convex hulls
  3. Survey of solid convex hulls
  4. The unit ball of ${\mathrm{FBL}}^{(p)}[E]$
  5. The unit ball of $\mathrm{FBL}^{\uparrow p}[E]$
  6. Positive Bounded Approximation Property
  7. Duals of free Banach lattices
  8. Sums of atoms in $\mathrm{FBL}^{\uparrow p}[E]^*$
  9. Sums of atoms in ${\mathrm{FBL}}^{(p)}[E]^*$
  10. Embeddings of free lattices
  11. Extensions of linear operators to lattice homomorphisms

Key Result

Lemma 2.1

If $B$ is a solid convex subset of $Z$, then so is its norm closure $\overline{B}$.

Theorems & Definitions (49)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 39 more