Non-direct limit of simple dimension groups with finitely many pure traces

TL;DR

This paper demonstrates that some simple dimension groups with finitely many pure traces cannot be represented as direct limits of simpler groups, revealing limitations in their structural decompositions, while also identifying cases where such representations are possible.

Contribution

It shows the existence of simple dimension groups that defy expression as direct limits of simpler groups with finitely many pure traces, and extends the class of initial objects for AF C*-algebras.

Findings

Certain simple dimension groups cannot be expressed as direct limits of simpler groups.

p-divisible simple dimension groups can be represented as such direct limits.

The class of initial objects for AF C*-algebras is enlarged.

Abstract

There exist simple dimension groups which cannot be expressed as a direct limit of simple, or even approximately divisible dimension groups, each with finitely many pure traces, and we can specify its infinite-dimensional Choquet simplex of traces; a more drastic property is noted. On the other hand, a very easy argument shows that if $G$ is a $p$-divisible simple dimension group (for some integer $p>1$), then it can be expressed as such a direct limit. We also enlarge the class of initial objects for AF (and slightly more general) C*-algebras.