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

G.R. Maloney, A. Tikuisis

Research output: Contribution to journalArticle

1 Citation (Scopus)
4 Downloads (Pure)

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

Fingerprint

Dimension Group
Finite Rank
Vector space
Interpolate
Subgroup
Isomorphism

Keywords

  • Dimension groups
  • Ordered abelian groups
  • Riesz interpolation

ASJC Scopus subject areas

  • Analysis

Cite this

A classification of finite rank dimension groups by their representations in ordered real vector spaces. / Maloney, G.R.; Tikuisis, A.

In: Journal of Functional Analysis, Vol. 260, No. 11, 01.06.2011, p. 3404-3428.

Research output: Contribution to journalArticle

@article{64687ab00fe84d979a6d54f277deb7ba,
title = "A classification of finite rank dimension groups by their representations in ordered real vector spaces",
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).",
keywords = "Dimension groups, Ordered abelian groups, Riesz interpolation",
author = "G.R. Maloney and A. Tikuisis",
year = "2011",
month = "6",
day = "1",
doi = "10.1016/j.jfa.2010.12.026",
language = "English",
volume = "260",
pages = "3404--3428",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press Inc.",
number = "11",

}

TY - JOUR

T1 - A classification of finite rank dimension groups by their representations in ordered real vector spaces

AU - Maloney, G.R.

AU - Tikuisis, A.

PY - 2011/6/1

Y1 - 2011/6/1

N2 - 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).

AB - 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).

KW - Dimension groups

KW - Ordered abelian groups

KW - Riesz interpolation

UR - http://www.scopus.com/inward/record.url?scp=79952485475&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2010.12.026

DO - 10.1016/j.jfa.2010.12.026

M3 - Article

AN - SCOPUS:79952485475

VL - 260

SP - 3404

EP - 3428

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 11

ER -