Riesz representation theorems for vector lattices and Banach lattices of regular operators

TL;DR

This paper extends Riesz representation theorems to vector and Banach lattices of regular operators, characterizing these operators via E-valued measures and establishing isomorphisms under various conditions.

Contribution

It provides new isomorphism and isometric isomorphism results for vector lattices of regular operators into Dedekind complete and order continuous Banach lattices.

Findings

Vector lattice of operators is isomorphic to E-valued measures.

When E is an order continuous Banach lattice, the isomorphism is an isometric Banach lattice isomorphism.

Regular operators from C(X) into E are norm to order bounded under certain conditions.

Abstract

For a non-empty locally compact Hausdorff space $X$ and a Dedekind complete normal vector lattice $E$, we show that the vector lattice of norm to order bounded operators from ${\text C}_{\text c}(X)$ or ${\text C}_0(X)$ into $E$ is isomorphic to the vector lattice of $E$-valued regular Borel measures on $X$. When $E$ is an order continuous Banach lattice, the isomorphism is an isometric isomorphism between Banach lattices. When $X$ is compact, every regular operator from $\mathrm{C}(X)$ into $E$ is norm to order bounded. For some spaces $E$, such as KB-spaces or the regular operators on a KB-space, every regular operator from ${\mathrm C}_0(X)$ into $E$ is norm to order bounded. Additional results are obtained for the whole space of regular operators from ${\text C}_{\text c}(X)$ into an order continuous Banach lattice. As a preparation, vector lattices and Banach lattices, resp. cones, of measures with values in a Dedekind complete vector lattice $E$, resp. in the extended positive cone of $E$, are investigated, as well as vector and Banach lattices of norm to order bounded operators. When $E$ is the real numbers, our results specialise to the well-known Riesz representation theorems for the order and norm duals of ${\text C}_{\text c}(X)$ and ${\text C}_0(X)$.