Cancellation and Stable Rank for Direct Limits of Recursive Subhomogeneous Algebras

We prove the following results for a unital simple direct limit A of recursive subhomogeneous algebras with no dimension growth: (1) tsr(A) = 1. (2) The projections in M∞(A) satisfy cancellation: if e ⊕ q ∼ f ⊕ q, then e ∼ f . (3) A satisfies Blackadar’s Second Fundamental Comparability Question: if p, q ∈ M∞(A) are projections such that τ(p) < τ(q) for all normalized traces τ on A, then p q. (4) K0(A) is unperforated for the strict order: if η ∈ K0(A) and there is n > 0 such that nη > 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, K0(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 ∈ K0(A) satisfies τ∗(η) > 0 for all normalized traces τ on A, then there is a projection p ∈ M∞(A) such that η = [p].