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

A simple C*-algebra with a finite and an infinite projection