Cancellation and stable rank for direct limits of recursive subhomogeneous algebras

Abstract

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 $C^{\ast }(Z,X,h)$ obtained from a minimal homeomorphism of an infinite finite dimensional compact metric space $X$. Specifically, $K_0(C^{\ast }(Z,X,h))$ 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 $τ_* (\et) > 0$ for all normalized traces $\tau$ on $A$, then there is a projection $p\in M_{\infty }(A)$ such that $\eta =[p]$.

27 pages, AMSLaTeX