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) $\operatorname {tsr}(A) = 1.$ (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 $p \precsim q.$ (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 $C^* (\mathbf {Z}, X, h)$ obtained from a minimal homeomorphism of a finite-dimensional infinite compact metric space $X.$ Specifically, $K_0 (C^* (\mathbf {Z}, X, h))$ 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 $\tau _* (\eta ) > 0$ for all normalized traces $\tau$ on $A,$ then there is a projection $p \in M_{\infty } (A)$ such that $\eta = [p].$