Real Rank and Property (sp) for Direct Limits of Recursive Subhomogeneous Algebras

Let A be a unital simple direct limit of recursive subhomogeneous algebras with no dimension growth. We give criteria which specify exactly when A has real rank zero, and exactly when A has the Property (SP): every nonzero hereditary subalgebra of A contains a nonzero projection. Specifically, A has real rank zero if and only if the image of K0(A) in Aff(T (A)) is dense, and A has the Property (SP) if and only if for every ε > 0 there is η ∈ K0(A) such that the corresponding affine function f on T (A) satisfies 0 < f(τ) < ε for all tracial states τ . By comparison with results for unital simple direct limits of homogeneous C*-algebras with no dimension growth, one might hope that weaker conditions might suffice. We give examples to show that several plausible weaker conditions do not suffice for the results above. If A has real rank zero and at most countably many extreme tracial states, we apply results of H. Lin to show that A has tracial rank zero and is classifiable.