A Modification of the Effros-Handelman-Shen Theorem with $\mathbb{Z}_2$ actions

C. R. Math. Rep. Acad. Sci. Canada Vol. 43 (3) 2021, pp. 87-102
(Received: 2021-04-08 )

Bit Na Choi, Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA; e-mail: Bitna.Choi@unh.edu

Andrew J. Dean, Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON, Canada P7B 5E1; e-mail: ajdean@lakeheadu.ca

Abstract/Résumé:

We show that a \(\mathbb{Z}_2\) action on a lattice-ordered dimension group will arise as an inductive limit of \(\mathbb{Z}_2\) actions on simplicial groups. The motivation for this study is the range of invariant problem in Elliott and Su’s classification of AF type \(\mathbb{Z}_2\) actions. We modify the proof of the Effros-Handelman-Shen theorem to include \(\mathbb{Z}_2\) actions.

Nous montrons qu’une action de \(\mathbb{Z}_2\) sur un groupe de dimension ordonné par treillis apparaît comme une limite inductive d’actions de \(\mathbb{Z}_2\) sur des groupes simpliciaux. La motivation de cette étude est le problème de la gamme de l’invariant dans la classification d’Elliott et de Su des actions de \(\mathbb{Z}_2\) de type AF. Nous modifions la preuve du théorème d’Effros-Handelman-Shen pour inclure les actions de \(\mathbb{Z}_2\).

Keywords: Dimension groups, K-theory, classification

AMS Subject Classification: Ordered abelian groups; Riesz groups; ordered linear spaces, $K_0$ as an ordered group; traces 06F20, 19K14

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