The stable and the real rank of Z-absorbing C*-algebras
  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. 
Mikael Rordam