Topological structure of the unitary group of certain C*-algebras

Let 0 - B (i)-> E (pi)-> A -> 0 be a short exact sequence of C*-algebras where A is a purely infinite simple C*-algebra and B is an essential ideal of E. In the case B is the compacts or a nonunital purely infinite simple C*-algebra we completely determine the homotopy groups of the unitary group of E in terms of K-theory. The result can be viewed as a generalization of the well-known Kuiper's theorem to a new class of C*-algebras (including certain separable C*-algebras).