**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

*A1−→−ϕ1,2A2−→−ϕ2,3A3⟶⋯⟶An⟶⋯*

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

*B1⟶B2⟶⋯⟶Bn⟶⋯,*

where *Bn=**⨁sn**i*=*1**Qn,i**M*{*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]).

###### Cornel Pasnicu

Professor

cornel.pasnicu@utsa.edu(210) 458-4495

FLN 4.01.32

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