Strong perforation in infinitely generated $\mathrm{K}_0$-groups of simple $C^*$-algebras

Abstract

Let $(G,G^{+})$ be an ordered abelian group. We say that $G$ has strong perforation if there exists $x \in G$, $x \notin G^{+}$, such that $nx \in G^{+}$, $nx \neq 0$ for some natural number $n$. Otherwise, the group is said to be weakly unperforated. Examples of simple $C^{*}$-algebras whose ordered $\mathrm{K}_0$-groups have this property and for which the entire order structure on $\mathrm{K}_0$ is known have, until now, been restricted to the case where $\mathrm{K}_0$ is group isomorphic to the integers. We construct simple, separable, unital $C^{*}$-algebras with strongly perforated $\mathrm{K}_0$-groups group isomorphic to an arbitrary infinitely generated subgroup of the rationals, and determine the order structure on $\mathrm{K}_0$ in each case.