A Reduction Theorem for AH Algebras with the Ideal Property

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 plus(tn)(i=1) Pn,iM[n,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. In this article, we prove that A can be written as the inductive limit of B-1 -> B-2 -> ... -> B-n -> ... , where B-n = circle plus(sn)(i=1) Q(n,i)M({n,i})(C(Y-n,Y-i)) Q(n,i), where Y-n,Y-i are {pt}, [0, 1], S-1, T-II,T-k, T-III,T-k and S-2 (all of them are connected simplicial complexes of dimension at most three), s(n) and {n,i} are positive integers and Q(n,i) epsilon M-{n,M-i}(C(Y-n,Y-i)) are projections. This theorem unifies and generalizes the reduction theorem for real rank zero AH algebras due to Dadarlat and Gong [4, 6, 17] and the reduction theorem for simple AH algebras due to Gong (see [19]).