Recent results of Edward G.
Effros and the author show that if a dimension group is simple, totally ordered and
with underlying group Zn, then we can construct explicitly an AF C∗-algebra with
the given group as its K0 by using the Jacobi-Perron algorithm. While the
Jacobi-Perron algorithm breaks down for nontotally ordered groups, we study the
construction problem via the consideration of automorphisms of the dimension group.
We find the necessary and sufficient condition for a nontotally ordered simple
dimension group (Z3,P(1,α,β)) being stationary is that both α and β lie in the same
quadratic number field. We also provide an explicit method for constructing
Bratteli diagrams (and hence corresponding AF C∗-algebras) for this type of
groups.
