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

### Summary

Summary: 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 $A \cong B.$

### Mathematics Subject Classification

46L35, 19K99, 46L80