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 {\cal O}_{2} \otimes A \cong {\cal O}_{2} for separable unital nuclear simple A and {\cal O}_{\infty} \otimes {A} \cong 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 \cong B.

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