Riesz representation theorems for positive linear operators

arXiv (Cornell University)

Abstract

We generalise the Riesz representation theorems for positive linear functionals on $\mathrm{C}_{\mathrm c}(X)$ and $\mathrm{C}_{\mathrm 0}(X)$, where $X$ is a locally compact Hausdorff space, to positive linear operators from these spaces into a partially ordered vector space $E$. The representing measures are defined on the Borel $σ$-algebra of $X$ and take their values in the extended positive cone of $E$; the corresponding integrals are order integrals. We give explicit formulas for the values of the representing measures at open and at compact subsets of $X$. Results are included where the space $E$ need not be a vector lattice, nor a normed space. Representing measures exist for positive linear operators into Banach lattices with order continuous norms, into the regular operators on a KB-space, into the self-adjoint linear operators in a weakly closed complex linear subspace of the bounded linear operators on a complex Hilbert space, and into JBW-algebras.

Authors 2

1. Marcel de Jeu Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute , Leiden University , P.O. Box 9512 , 2300 RA Leiden , The Netherlands

2. Xingni Jiang