  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.
