 

David Handelman and
Damien Roy
Onesided approximation in affine function spaces
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Published: 
April 26, 2019. 
Keywords: 
partially ordered abelian groups, Choquet theory, approximation, trace, Gordan's theorem, Farkas' lemma, unperforation, refinable measure, diophantine inequalities, Kronecker theorem. 
Subject: 
Primary 19K14; Secondary 11J25, 15A39, 06F20, 37A55. 


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 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 finitedimensional, 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. 

Acknowledgements
Both authors supported in part by NSERC Discovery grants.


Author information
David Handelman:
Department of Mathematics and Statistics
University of Ottawa
150 Louis Pasteur
Ottawa, Ontario K1N 6N5, Canada
dehsg@uottawa.ca
Damien Roy:
Department of Mathematics and Statistics
University of Ottawa
150 Louis Pasteur
Ottawa, Ontario K1N 6N5, Canada
droy@uottawa.ca

