Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Order continuity and regularity on vector lattices and on lattices of continuous functions

Eugene Bilokopytov

Abstract

We give several characterizations of order continuous vector lattice homomorphisms between Archimedean vector lattices. We reduce the proofs of some of the equivalences to the case of composition operators between vector lattices of continuous functions, and so we obtain a characterization of order continuity of such operators. Motivated by this, we investigate various properties of the sublattices of the space $C\left(X\right)$, where $X$ is a Tychonoff topological space. We also obtain several characterizations of a regular sublattice of a vector lattice, and show that the closure of a regular sublattice of a Banach lattice is also regular.

Order continuity and regularity on vector lattices and on lattices of continuous functions

Abstract

We give several characterizations of order continuous vector lattice homomorphisms between Archimedean vector lattices. We reduce the proofs of some of the equivalences to the case of composition operators between vector lattices of continuous functions, and so we obtain a characterization of order continuity of such operators. Motivated by this, we investigate various properties of the sublattices of the space , where is a Tychonoff topological space. We also obtain several characterizations of a regular sublattice of a vector lattice, and show that the closure of a regular sublattice of a Banach lattice is also regular.

Paper Structure

This paper contains 8 sections, 34 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Order continuity of positive operators and homomorphisms
  4. Preliminaries on $\mathcal{C}\left(X\right)$
  5. Sublattices of $\mathcal{C}\left(X\right)$
  6. Some classes of continuous maps between topological spaces.
  7. Order continuity of composition operators
  8. Acknowledgements

Key Result

Proposition 2.3

Let $E$ be a sublattice of $F$ and let $G\subset E$. Then $G^{d}_{E}=G^{d}_{F}\cap E$ and $G^{dd}_{F}\cap E\subset G^{dd}_{E}$. If additionally $G^{dd}_{F}\cap E$ is a band in $E$, then $G^{dd}_{F}\cap E= G^{dd}_{E}$. If $G$ is itself a band in $E$, then $G=G^{dd}_{F}\cap E$.

Theorems & Definitions (83)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • ...and 73 more