One-sided approximation in affine function spaces

Abstract

Let $H$ be a subgroup of a partially ordered abelian group $G$ with order unit $u$, and let $S(G,u)$ denote the convex subset of $\bR^G$ consisting of all traces (states) $\tau$ on $G$ with $\tau(u)=1$. We say that $H$ has property $(B)$ if, for any integer $m\ge 2$, any $h\in H$ and any $\epsilon>0$, there exists $h'\in H$ such that $\tau(h)-m\tau(h')\ge -\epsilon$ for each $\tau\in S(G,u)$. We show that, if $S(G,u)$ is finite-dimensional, this condition is equivalent to asking that $\tau(H)$ is $\{0\}$ or dense in $\bR$ for all $\tau$ in the smallest face of $S(G,u)$ containing all traces that vanish identically on $H$. When $G$ is a simple dimension group and $H$ is a convex subgroup of $G$, we show that $G/H$ is unperforated if and only if $H$ has property $(B)$. We apply both results to provide a criterion for a trace of $G$ to be refinable when $G$ is a simple dimension group with finitely many pure traces.