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 A(1) ->phi 1.2 A(2) ->phi 2.3 A(3) ... -> A(n) -> ... with A(n) = circle times(tn)(i=1) P-n,P-i M-[n,M-i] (C(X-n,X-i)) P-n,P-i, where X-n,X-i are compact metric spaces, t(n) and [n, i ] are positive integers, and P-n,P-i epsilon M-[n,M-i](C(X-n,X-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 sup(n,i) dim(X-n,X-i) < + infinity. (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 B-1 -> B-2 -> ... -> B-n -> ..., where B-n = circle times(sn)(i=1) M-[n,M- i] (C(S-1)). 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 T-II,T-k, T-III,T-k, and S-2, as in Gong (2002) [29]. (C) 2009 Elsevier Inc. All rights reserved.