A classification theorem for nuclear purely infinite simple C^*-algebras

Starting from Kirchberg's theorems announced at the operator algebra conference in Genève in 1994, namely { O}₂ ⊗ A { O}₂ for separable unital nuclear simple A and { O}∞ ⊗ {A} A for separable unital nuclear purely infinite simple A, we prove that KK -equivalence implies isomorphism for nonunital separable nuclear purely infinite simple C^* -algebras. It follows that if A and B are unital separable nuclear purely infinite simple C^* -algebras which satisfy the Universal Coefficient Theorem, and if there is a graded isomorphism from K_* (A) to K_* (B) which preserves the K_0 -class of the identity, then A B.