具有理想性质的$AH$代数的一个约化定理
A Reduction theorem for $AH$ algebras with ideal property
摘要：
设$A$是一个$AH$代数，即$A$是$$A_{1}\xrightarrow{ϕ_{1,2}}A_{2}\xrightarrow{ϕ_{2,3}}A_{3}\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots$$与$A_{n}=\bigoplus_{i=1}^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$的归纳极限$C^{*}$ -代数，其中$X_{n,i}$是紧度量空间，$t_{n}$和$[n,i]$是正整数，$P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$是投影.假设$A$具有理想性质：$A$的每个闭双侧理想都是由理想内部的投影生成的，作为闭双侧理想。假设$\sup_{n,i}dim(X_{n,i})<+\infty$ .本文证明了$A$可以表示为$$B_{1}\longrightarrow B_{2}\longrightarrow\cdots\longrightarrow B_{n}\longrightarrow\cdots,$$的归纳极限，其中$B_{n}=\bigoplus_{i=1}^{s_{n}}Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$，其中$Y_{n,i}$是$\{pt\}$，$[0,1]$，$ S^{1}$，$ T_{II, k},$，$T_{III, k}$和$S^{2}$（它们都是维数不超过3的连通单纯复形），$s_{n}$和$\{n,i\}$是正整数，$Q_{n,i}\in M_{\{n,i\}}(C(Y_{n,i}))$是投影.这个定理统一和推广了Dadarlat和Gong（[D]，[G3]和[DG]）的实秩零$AH$代数的约化定理和Gong（见[G4]）的单$AH$代数的约化定理。
英文摘要：
Let $A$ be an $AH$ algebra, that is, $A$ is the inductive limit $C^{*}$-algebra of $$A_{1}\xrightarrow{ϕ_{1,2}}A_{2}\xrightarrow{ϕ_{2,3}}A_{3}\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots$$ 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 $$B_{1}\longrightarrow B_{2}\longrightarrow\cdots\longrightarrow B_{n}\longrightarrow\cdots,$$ 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 ([D], [G3] and [DG]) and the reduction theorem for simple $AH$ algebras due to Gong (see [G4]).
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