  Let A be a unital simple direct limit of recursive subhomogeneous C*-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 natural map from K_0 (A) to the continuous
affine functions on the tracial state space has dense range, A has the Property
(SP) if and only if the range of this map contains strictly positive functions
with arbitrarily small norm. By comparison with results for unital simple
direct limit 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.
