A classification of finite rank dimension groups by their representations in ordered real vector spaces

G.R. Maloney, A. Tikuisis

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1 Citation (Scopus)
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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 languageEnglish
Pages (from-to)3404-3428
Number of pages25
JournalJournal of Functional Analysis
Volume260
Issue number11
Early online date30 Dec 2010
DOIs
Publication statusPublished - 1 Jun 2011

Keywords

  • Dimension groups
  • Ordered abelian groups
  • Riesz interpolation

ASJC Scopus subject areas

  • Analysis

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