Abstract
This paper systematically studies finite rank dimension groups, as well as finite-dimensional ordered real vector spaces with Riesz interpolation. We provide an explicit description and classification of finite rank dimension groups, in the following sense. We show that for each n, there are (up to isomorphism) finitely many ordered real vector spaces of dimension n that have Riesz interpolation, and we give an explicit model for each of them in terms of combinatorial data. We show that every finite rank dimension group can be realized as a subgroup of a finite-dimensional ordered real vector space with Riesz interpolation via a canonical embedding. We then characterize which of the subgroups of a finite-dimensional ordered real vector space have Riesz interpolation (and are therefore dimension groups).
Original language | English |
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Pages (from-to) | 3404-3428 |
Number of pages | 25 |
Journal | Journal of Functional Analysis |
Volume | 260 |
Issue number | 11 |
Early online date | 30 Dec 2010 |
DOIs | |
Publication status | Published - 1 Jun 2011 |
Keywords
- Dimension groups
- Ordered abelian groups
- Riesz interpolation