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 R^G consisting of all traces (states) τ on G with τ(u)=1. We say that H has property (B) if, for any integer m ≥ 2, any h ∈ H and any ε > 0, there exists some h' ∈ H such that τ(h)-mτ(h') ≥ - ε for each τ ∈ S(G,u). We show that, if S(G,u) is finite-dimensional, this condition is equivalent to asking that τ(H) is {0} or dense in R for all τ 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.

Both authors supported in part by NSERC Discovery grants.