Vol. 97, No. 2, 1981

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Automorphisms of dimension groups and the construction of AF algebras

Chao-Liang Shen

Vol. 97 (1981), No. 2, 461–469

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.

Mathematical Subject Classification 2000
Primary: 46L05
Secondary: 06F20, 22A25
Received: 10 March 1980
Published: 1 December 1981
Chao-Liang Shen