Formula omitted] structure of AH algebras with the ideal property and torsion free K-theory

Abstract

Let A be an AH algebra, that is, A is the inductive limit C ∗ -algebra of A 1 → ϕ 1 , 2 A 2 → ϕ 2 , 3 A 3 → ⋯ → A n → ⋯ with A n = ⊕ i = 1 t n P n , i M [ n , i ] ( C ( X n , i ) ) P n , i , where X n , i are compact metric spaces, t n and [ n , i ] are positive integers, and P n , i ∈ M [ n , i ] ( C ( X n , 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 , 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 A T algebra), that is, A can be written as the inductive limit of B 1 → B 2 → ⋯ → B n → ⋯ , where B n = ⊕ i = 1 s n M { n , 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 S 1 , we will also need the spaces T II , k , T III , k , and S 2 , as in Gong (2002) [29].