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 RG consisting of all traces (states) on G with (u)=1. We say that H has property (B) if, for any integer m 5 2, any h 8 H and any > 0, there exists some h' 8 H such that (h)-m(h') 5 - for each 8 S(G,u). We show that, if S(G,u) is finite-dimensional, this condition is equivalent to asking that (H) is {0} or dense in R for all 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.
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