  Suppose that A is a C*-algebra for which A is isomorphic to A tensor Z, where
Z is the Jiang-Su algebra: a unital, simple, stably finite, separable, nuclear,
infinite dimensional C*-algebra with the same Elliott invariant as the complex
numbers. We show that:
  (i) The Cuntz semigroup W(A) of equivalence classes of positive elements in
matrix algebras over A is weakly unperforated.
  (ii) If A is exact, then A is purely infinite if and only if A is traceless.
  (iii) If A is separable and nuclear, then A is isomorphic to A tensor O_infty
if and only if A is traceless.
  (iv) If A is simple and unital, then the stable rank of A is one if and only
if A is finite.
  We also characterise when A is of real rank zero.
