More on the extension of linear operators on riesz spaces

Abstract

The classical Kantorovich theorem asserts the existence and uniqueness of a linear extension of a positive additive mapping, defined on the positive cone E+ of a Riesz space E taking values in an Archimedean Riesz space F, to the entire space E. We prove that, if E has the principal projection property and f is Dedekind σ-complete then for every e ∈ E+ every positive finitely additive f-valued measure defined on the Boolean algebra Ϝe of fragments of c has a unique positive linear extension to the ideal Ee of E generated by e. If, moreover, the measure is τ-continuous then the linear extension is order continuous.