A reduction theorem for AH algebras with the ideal property

Guihua Gong, Chunlan Jiang, Liangqing Li, Cornel Pasnicu

Int. Math. Res. Not. IMRN, to appear

Let A be an AH algebra, that is, A is the inductive limit C∗-algebra of

with An=⨁tni=1Pn,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,idim(Xn,i)<+∞. In this article, we prove that A can be written as the inductive limit of

where Bn=⨁sni=1Qn,iM{n,i}(C(Yn,i))Qn,i, where Yn,i are {pt}, [0,1],S1,TII,k, TIII,k and S2 (all of them are connected simplicial complexes of dimension at most three), sn and {n,i} are positive integers and Qn,i∈M{n,i}(C(Yn,i)) are projections. This theorem unifies and generalizes the reduction theorem for real rank zero AH algebras due to Dadarlat and Gong ([D], [G3] and [DG]) and the reduction theorem for simple AH algebras due to Gong (see [G4]).