More on the extension of linear operators on Riesz spaces

Keywords:
positive operator, linear extension, Riesz space, vector latticeAbstract
The classical Kantorovich theorem asserts the existence and uniqueness of a linear extension of a positive additive mapping, defined on the positive cone E+E+ of a Riesz space EE taking values in an Archimedean Riesz space FF, to the entire space EE. We prove that, if EE has the principal projection property and FF is Dedekind σσ-complete then for every e∈E+e∈E+ every positive finitely additive FF-valued measure defined on the Boolean algebra Fe of fragments of e 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.