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.
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References
Alfsen, E.M.: Compact convex sets and boundary integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete vol. 57, Springer-Verlag, 1971.
Edwards, D. A.: Séparation des fonctions réelles définies sur un simplexe de Choquet, Comptes Rendus de l'Académie des Sciences Paris 261(11 Octobre 1965), (15), 2798-2800.
Goodearl, K. R.: Partially ordered abelian groups with the interpolation property, Mathematical Surveys and Monographs, No. 20, American Mathematical Society, 1986.
Goodearl, K. R.: Extensions of dimension groups and AF C*-algebras, Journal für die reine und angewandte Mathematik 412(1990), 150-219.
Goodearl, K. R.: K0 of multiplier algebras of C*-algebras with real rank zero, K-Theory 10(1996), 419-489.
Pardo, E.: Metric completions of ordered groups and K 0 of exchange rings, to appear in Transactions of the American Mathematical Society.
Wehrung, F.: Bounded countable atomic compactness of ordered groups, Fundamenta Mathematicæ 148(1995), 101-116.
Wehrung, F.: Monoids of intervals of ordered abelian groups, Journal of Algebra 182(1996), 287-328.
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Wehrung, F. Norm-Closed Intervals of Norm-Complete Ordered Abelian Groups. Positivity 1, 271–290 (1997). https://doi.org/10.1023/A:1009712111747
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DOI: https://doi.org/10.1023/A:1009712111747