Real Dimension Groups

Handelman, David

Canadian mathematical bulletin, Tome 56 (2013) no. 3, pp. 551-563

Résumé

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; for instance, it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over 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 an 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.

Détail

DOI : 10.4153/CMB-2012-006-4

Mots-clés : 46A40, 06F20, 13J25, 19K14, dimension group, simplicial vector space, direct limit, Riesz interpolation