On the Biadjoint of Riesz-like homomorphisms on partially ordered vector spaces

Abstract

We generalise to partially ordered vector spaces, with a new technique, Arendt's approach to Kim's characterisation of Riesz homomorphisms.In [8] Kim proved the elegant duality connection between Riesz homomorphisms and interval preserving maps. A particular consequence is the fact, referred to as Kim's Theorem in this paper, that the biadjoint of a Riesz homomorphism is again a Riesz homomorphism. It is the latter consequence that we focus on. Kim's Theorem has interesting applications in spectral theory and there is a need for similar results in the much wider context of partially ordered vector spaces. We denote by L r (E,F) the directed partially ordered vector space of all regular operators E ->• F, that is, the space of all differences of positive operators E -* F. Instead of L T (E,H) we write E~. A map T : E -> F is interval preserving if T is positive and for every x € E+ and y £ F + with y ^ T(x), there exists z 6 E + such that z ^ x and T(z) = y. The choice of an analogue for Riesz homomorphism in the setting of partially ordered vector spaces is not obvious at all. Somewhat surprisingly, the choice of homomorphisms that was made in [3] to describe the enveloping Riesz space does not work well for this duality problem as we shall show in the example following Theorem 10. Instead we have opted for a stronger type of homomorphism that is very