2016/07/26 by Guihua Gong, Chunlan Jiang, Gong, Guihua +5 · 1 citation
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA
paper · pdf · doi:10.48550/arxiv.1607.07575
33 pages, Accept by International Mathematics Research Notices
arxiv created 2017/04/22 · arxiv updated 2017/04/25
Let A be an AH algebra, that is, A is the inductive limit C*-algebra of A1\xrightarrowϕ1,2A2\xrightarrowϕ2,3A3\longrightarrow⋯\longrightarrow An\longrightarrow⋯ with An=\bigoplusi=1^tnPn,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\longrightarrow B2\longrightarrow⋯\longrightarrow Bn\longrightarrow⋯, where Bn=\bigoplusi=1^snQn,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]).