Skip to main content Accessibility help
×
Home

FINITE-DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS–SHEN’S UNIMODULARITY CONJECTURE

  • AARON TIKUISIS (a1)

Abstract

It is shown that, for any field $\mathbb{F}\subseteq \mathbb{R}$ , any ordered vector space structure of $\mathbb{F}^{n}$ with Riesz interpolation is given by an inductive limit of a sequence with finite stages $(\mathbb{F}^{n},\mathbb{F}_{\geq 0}^{n})$ (where $n$ does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statement, except with $\mathbb{F}$ replaced by the integers, $\mathbb{Z}$ . Indeed, it shows that although Effros and Shen’s conjecture is false, it is true after tensoring with $\mathbb{Q}$ .

Copyright

References

Hide All
[1] Birkhoff, G., ‘Lattice-ordered groups’, Ann. of Math. (2) 43 (1942), 298331.
[2] Effros, E. G., Handelman, D. E. and Shen, C. L., ‘Dimension groups and their affine representations’, Amer. J. Math. 102(2) (1980), 385407.
[3] Effros, E. G. and Shen, C. L., ‘Dimension groups and finite difference equations’, J. Operator Theory 2(2) (1979), 215231.
[4] Goodearl, K. R., Partially Ordered Abelian Groups with Interpolation, Mathematical Surveys and Monographs, 20 (American Mathematical Society, Providence, RI, 1986).
[5] Goodearl, K. R. and Handelman, D. E., ‘Tensor products of dimension groups and K 0 of unit-regular rings’, Canad. J. Math. 38(3) (1986), 633658.
[6] Handelman, D., ‘Real dimension groups’, Canad. Math. Bull. 56(3) (2013), 551563.
[7] Maloney, G. R. and Tikuisis, A., ‘A classification of finite rank dimension groups by their representations in ordered real vector spaces’, J. Funct. Anal. 260(11) (2011), 34043428.
[8] Riedel, N., ‘A counterexample to the unimodular conjecture on finitely generated dimension groups’, Proc. Amer. Math. Soc. 83(1) (1981), 1115.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

FINITE-DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS–SHEN’S UNIMODULARITY CONJECTURE

  • AARON TIKUISIS (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed