AT structure of AH algebras with the ideal property and torsion free K-theory ✩

Let A be an AH algebra, that is, A is the inductive limit C∗-algebra of A1 φ1,2 −−→ A2 φ2,3 −−→ A3 −→ ··· −→ An −→··· with An = tn i=1 Pn,iM[n,i](C(Xn,i))Pn,i, where Xn,i are compact metric spaces, tn and [n, i] are positive integers, and Pn,i ∈ M[n,i](C(Xn,i )) are projections. Suppose that A has the ideal property: each closed two-sided ideal of A is generated by the projections inside the ideal, as a closed two-sided ideal. Suppose that supn,i dim(Xn,i) < +∞. (This condition can be relaxed to a certain condition called very slow dimension growth.) In this article, we prove that if we further assume that K∗(A) is torsion free, then A is an approximate circle algebra (or an AT algebra), that is, A can be written as the inductive limit of B1 −→ B2 −→···−→Bn −→···,where Bn = sn i=1M{n,i}(C(S1)). One of the main technical results of this article, called the decomposition theorem, is proved for the general case, i.e., without the assumption that K∗(A) is torsion free. This decomposition theorem will play an essential role in the proof of a general reduction theorem, where the condition that K∗(A) is torsion free is dropped, in the subsequent paper Gong et al. (preprint) [31]—of course, in that case, in addition to space S1, we will also need the spaces TII,k , TIII,k, and S2, as in Gong (2002) [29]. © 2009 Elsevier Inc. All rights reserved.