Localization in pre-Riesz spaces and factorization of multilinear Riesz* homomorphisms

Abstract In Archimedean vector lattices, it is well known that every principal ideal can be represented as a norm dense sublattice of some $$\textrm{C}(K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>C</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> space for K a compact Hausdorff space. Therefore, by restricting to a suitable principal ideal, calculations can be done locally using continuous functions. In this way, many results known for spaces of continuous functions can be transferred to Archimedean vector lattices. By generalizing the notion of sublattices to partially ordered vector spaces, we introduce and investigate a similar localization principle in pre-Riesz spaces. For the principal ideal, we use the functional representation of order unit spaces. Using this localization, we show an extension and factorization result for multilinear Riesz* homomorphisms for suitable pre-Riesz spaces.