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Real Dimension Groups

  Published:2012-03-24
 Printed: Sep 2013
  • David Handelman,
    Mathematics Department, University of Ottawa, Ottawa ON K1N 6N5
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Abstract

Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like, i.e., it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over the a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In the Appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.
Keywords: dimension group, simplicial vector space, direct limit, Riesz interpolation dimension group, simplicial vector space, direct limit, Riesz interpolation
MSC Classifications: 46A40, 06F20, 13J25, 19K14 show english descriptions Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42]
Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40]
Ordered rings [See also 06F25]
$K_0$ as an ordered group, traces
46A40 - Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42]
06F20 - Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40]
13J25 - Ordered rings [See also 06F25]
19K14 - $K_0$ as an ordered group, traces
 

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