A simple C*-algebra with a finite and an infinite projection
  An example is given of a simple, unital C*-algebra which contains an infinite and a non-zero finite projection. This C*-algebra is also an example of an infinite simple C*-algebra which is not purely infinite. A corner of this C*-algebra is a finite, simple, unital C*-algebra which is not stably finite.   Our example shows that the type decomposition for von Neumann factors does not carry over to simple C*-algebras.   Added March 2002: We also give an example of a simple, separable, nuclear C*-algebra in the UCT class which contains an infinite and a non-zero finite projection. This nuclear C*-algebra arises as a crossed product of an inductive limit of type I C*-algebras by an action of the integers. 
Mikael Rordam