IUMJ

Title: Limits of simple dimension groups and weakly initial objects

Authors: David Handelman

Issue: Volume 63 (2014), Issue 5, 1567-1600

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. Every infinite-dimensional Choquet simplex can be realized as the trace space of such dimension groups. Simple initial objects (in the category of dimension groups) in the sense of Elliott and R\o rdam [G.A. Elliot and M. R\ordam, \textit{Perturbation of Hausdorff moment sequences and an application to the theory of C*-Algebras of real rank zero}, Abel Symp., vol. 1, Springer, Berlin, 2006] satisfy this. Related to this is a drastic property, defined in terms of non-existence of positive homomorphisms from simple dimension groups with finite pure trace space. We also enlarge the class of weakly initial objects for AF (and slightly more general) C*-algebras. On the other hand, if $G$ is a $p$-divisible simple dimension group (for some integer $p>1$),then it can be expressed as such a direct limit.