Gong, Guihua; Jiang, Chunlan; Li, Liangqing; Pasnicu, Cornel
J. Funct. Anal. 258 (2010), no. 6, 2119–2143
Let A be an AH algebra, that is, A is the inductive limit C*-algebra of
with , 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,idim(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
where . 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) —of course, in that case, in addition to space S^1, we will also need the spaces TII,k, TIII,k, and S^2, as in Gong (2002) .