Riesz representation theorems for positive linear operators

Abstract

We generalise the Riesz representation theorems for positive linear functionals on \(\text {C}_{\text {c}}(X)\) and \(\text {C}_{\text {0}}(X)\), 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 \(\sigma \)-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 example, for positive linear operators into Banach lattices with order continuous norms, into the regular operators on KB-spaces, into the self-adjoint linear operators on complex Hilbert spaces, and into JBW-algebras.