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Riesz completions of some spaces of regular operators

Anthony W. Wickstead

Abstract

We describe the Riesz completion (in the sense of van Haandel) of some spaces of regular operators as explicitly identified subspaces of the regular operators into larger range spaces

Riesz completions of some spaces of regular operators

Abstract

We describe the Riesz completion (in the sense of van Haandel) of some spaces of regular operators as explicitly identified subspaces of the regular operators into larger range spaces
Paper Structure (6 sections, 28 theorems, 25 equations)

This paper contains 6 sections, 28 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Regular Operators Defined on $\ell^\infty_0$
  3. The Riesz completion of ${\mathcal{L}}^r(\ell^\infty_0,F)$
  4. Order continuous operators defined on $\ell^\infty_0$
  5. Attempts to extend the domain to be $c$
  6. Miscellaneous observations

Key Result

Theorem 2.1

If $F$ is an Archimedean Riesz space then ${\mathcal{L}}^b(\ell^\infty_0,F)={\mathcal{L}}^r(\ell^\infty_0,F)$ which is a Riesz space if and only if $F$ is Dedekind $\sigma$-complete.

Theorems & Definitions (52)

  • Theorem 2.1: AG, Theorem 4.1 and Proposition 4.2
  • Lemma 2.2: AW, Lemma 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5: AW, Theorems 4.4 and 4.6
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • ...and 42 more