Cancellation and stable rank for direct limits of recursive subhomogeneous algebras

Details Cancellation and stable rank for direct limits of recursive subhomogeneous algebras Cancellation and stable rank for direct limits of recursive subhomogeneous algebras

2001-01-18

arxiv.org/abs/math/0101157v1

Mathematics

Operator Algebras

27 pages, AMSLaTeX

Scientific paper

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 $\tau (p) < \tau (q)$ for all normalized traces $\tau$ on $A$, then $p$ is equivalent to a subprojection of $q$. (4) $K_0 (A)$ is unperforated for the strict order: if $\eta \in K_0 (A)$ and there is $n > 0$ such that $n \eta > 0$, then $\eta > 0$. 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 $\eta \in K_0 (A)$ satisfies $\tau_* (\et) > 0$ for all normalized traces $\tau$ on $A$, then there is a projection $p \in M_{\infty} (A)$ such that $\eta = [p]$.

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