递归次齐次代数直接极限的消去与稳定秩
Cancellation and stable rank for direct limits of recursive subhomogeneous algebras
摘要：
证明了无维数增长的递归次齐次代数的单位元单直极限$A$的下列结果：（1）A有稳定秩1。(2)$M_{\infty} (A)$中的投影满足消去：如果是$e \oplus q \sim f \oplus q$，则是$e \sim f$。(3)$A$满足Blackadar的第二个基本可比性问题：如果$p, q \in M_{\infty} (A)$是投影，使得对于$A$上的所有归一化迹线$τ$，$τ(p) < τ(q)$，则$p$等价于$q$的子投影。(4)$K_0 (A)$是无穿孔的严格顺序：如果$η\in K_0 (A)$和有$n > 0$这样的$n η> 0$，那么$η> 0$。最后三个结果在某些较弱的维数增长条件下成立，并且没有假设简单。我们使用这些结果，以获得以前未知的信息的顺序K-理论的交叉产品$C^* (Z, X, h)$从一个最小同胚的无限有限维紧度量空间$X$。具体地说，$K_0 (C^* (Z, X, h))$对于严格的顺序是未穿孔的，并且满足以下Blackadar的第二基本可比性问题的K理论版本：如果$η\in K_0 (A)$对于$A$上的所有归一化迹$τ$满足$τ_* (\et) > 0$，则存在投影$p \in M_{\infty} (A)$使得$η= [p]$。
英文摘要：
We prove the following results for a unital simple direct limit $A$ of recursive subhomogeneous algebras with no dimension growth: (1) A has stable rank 1. (2) The projections in $M_{\infty} (A)$ satisfy cancellation: if $e \oplus q \sim f \oplus q$, then $e \sim f$. (3) $A$ satisfies Blackadar's Second Fundamental Comparability Question: if $p, q \in M_{\infty} (A)$ are projections such that $τ(p) < τ(q)$ for all normalized traces $τ$ on $A$, then $p$ is equivalent to a subprojection of $q$. (4) $K_0 (A)$ is unperforated for the strict order: if $η\in K_0 (A)$ and there is $n > 0$ such that $n η> 0$, then $η> 0$. The last three of these results hold under certain weaker dimension growth conditions and without assuming simplicity. We use these results to obtain previously unknown information on the ordered K-theory of the crossed product $C^* (Z, X, h)$ obtained from a minimal homeomorphism of an infinite finite dimensional compact metric space $X$. Specifically, $K_0 (C^* (Z, X, h))$ is unperforated for the strict order, and satisfies the following K-theoretic version of Blackadar's Second Fundamental Comparability Question: if $η\in K_0 (A)$ satisfies $τ_* (\et) > 0$ for all normalized traces $τ$ on $A$, then there is a projection $p \in M_{\infty} (A)$ such that $η= [p]$.
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