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Local complementation in Banach spaces and its preservation under free constructions

Antonio Avilés, Gonzalo Martínez-Cervantes, Abraham Rueda Zoca

Abstract

In this work we consider natural generalizations of local complementation in Banach spaces, which include Lipschitz-local complementation. We show that all these notions are indeed equivalent to the classical notion of local complementation of Banach spaces. As an application, we show that local complementation is naturally preserved under certain free constructions in Functional Analysis, including Lipschitz-free spaces and free Banach lattices.

Local complementation in Banach spaces and its preservation under free constructions

Abstract

In this work we consider natural generalizations of local complementation in Banach spaces, which include Lipschitz-local complementation. We show that all these notions are indeed equivalent to the classical notion of local complementation of Banach spaces. As an application, we show that local complementation is naturally preserved under certain free constructions in Functional Analysis, including Lipschitz-free spaces and free Banach lattices.

Paper Structure

This paper contains 4 sections, 5 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Categorical framework
  3. Local complementation relative to a category ${\hbox{\sf Cat}}_1$
  4. Local complementation under a free functor

Key Result

Lemma 2.2

Let ${\hbox{\sf Cat}}_1, {\hbox{\sf Cat}}_2$ be two metric-algebraic categories with ${\hbox{\sf Cat}}_2 \subseteq {\hbox{\sf Cat}}_1$ and let $X$ be an object in ${\hbox{\sf Cat}}_1$. If we have two free objects $(\delta_X,F[X])$ and $(\delta'_X,F'[X])$ in ${\hbox{\sf Cat}}_2$ generated by $X$ then

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 4.1
  • ...and 3 more