Norm-closed intervals of norm-complete ordered abelian groups

Abstract

Let $(G,u)$ be an archimedean norm-complete dimension group with order unit. Continuing a previous paper, we study intervals (i.e., nonempty upward directed lower subsets) of $G$ which are closed with respect to the canonical norm of $(G,u)$. In particular, we establish a canonical one-to-one correspondence between closed intervals of $G$ and certain affine lower semicontinuous functions on the state space of $(G,u)$, which allows us to solve several problems of K. R. Goodearl about inserting affine continuous functions between convex upper semicontinuous and concave lower semicontinuous functions. This yields in turn new results about analogues of multiplier groups for norm-closed intervals.