A Reduction Theorem for AH Algebras with the Ideal Property. (6th June 2017)

• Record Type: Journal Article
• Authors: Gong, Guihua; Jiang, Chunlan; Li, Liangqing; Pasnicu, Cornel
• Is Part Of: International mathematics research notices. Volume 2018:Number 24(2018)
• Page Start: 7606
• Page End: 7641
• Publication Date: 2017-06-06
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx100
• Languages: English
• ISSNs: 1073-7928

Abstract

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 $A_{n}=\bigoplus_{i=1}^{t_{n}}P_{n, i}M_{[n, i]}(C(X_{n, i}))P_{n, i}$, where $X_{n, i}$ are compact metric spaces, $t_{n}$ and $[n, i]$ are positive integers, and $P_{n, i}\in M_{[n, i]}(C(X_{n, i}))$ 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 $\sup_{n, i}\dim(X_{n, i})<+\infty$ . In this article, we prove that $A$ can be written as the inductive limit of B 1 ⟶ B 2 ⟶ ⋯ ⟶ B n ⟶ ⋯, where $B_{n}=\bigoplus_{i=1}^{s_{n}}Q_{n, i}M_{\{n, i\}}(C(Y_{n, i}))Q_{n, i}$, where $Y_{n, i}$ are $\{pt\}, [0, 1], S^{1}, T_{II, k}, T_{III, k}$ and $S^{2}$ (all of them are connected simplicial complexes of dimension at most three), $s_{n}$ and $\{n, i\}$ are positive integers and $Q_{n, i}\in M_{\{n, i\}}(C(Y_{n, i}))$ 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 ]).