Cancellation and stable rank for direct limits of recursive subhomogeneous algebras

Abstract

We prove the following results for a unital simple direct limit A A of recursive subhomogeneous algebras with no dimension growth: (1) tsr ⁡ ( A ) = 1. \operatorname {tsr}(A) = 1. (2) The projections in M ∞ ( A ) M_{\infty }(A) satisfy cancellation: if e ⊕ q ∼ f ⊕ q , e \oplus q \sim f \oplus q, then e ∼ f . e \sim f. (3) A A satisfies Blackadar’s Second Fundamental Comparability Question: if p , q ∈ M ∞ ( A ) p, \, q \in M_{\infty }(A) are projections such that τ ( p ) > τ ( q ) \tau (p) > \tau (q) for all normalized traces τ \tau on A , A, then p ≾ q . p \precsim q. (4) K 0 ( A ) K_0 (A) is unperforated for the strict order: if η ∈ K 0 ( A ) \eta \in K_0 (A) and there is n > 0 n > 0 such that n η > 0 , n \eta > 0, then η > 0. \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 ) C^* (\mathbf {Z}, X, h) obtained from a minimal homeomorphism of a finite-dimensional infinite compact metric space X . X. Specifically, K 0 ( C ∗ ( Z , X , h ) ) 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 η ∈ K 0 ( A ) \eta \in K_0 (A) satisfies τ ∗ ( η ) > 0 \tau _* (\eta ) > 0 for all normalized traces τ \tau on A , A, then there is a projection p ∈ M ∞ ( A ) p \in M_{\infty } (A) such that η = [ p ] . \eta = [p].