Simple dimension groups that are isomorphic to stationary inductive limits

A dimension group is an ordered abelian that inductive limit of a sequence simplicial groups, and stationary such in which the homomorphism same at every stage. If simple then up to scalar multiplication it admits unique trace (positive real-valued homomorphism), but short exact associated this need not split. In earlier paper, Handelman [J. Operator Theory 6 (1981), pp. 55–74] described these groups concretely case when has trivial kernel—i.e., totally ordered—and free. The main result here concrete description how built from kernel image its trace. Specifically, contains direct sum with copy image, generated by finitely many extra elements. Moreover, any stationary. following interesting fact proved along way result: given positive integer m m square matrix alttext="upper B"> B encoding="application/x-tex">B , there are two distinct powers difference all entries divisible .