A classification theorem for nuclear purely infinite simple $C^{\ast }$-algebras

Abstract

Starting from Kirchberg's theorems announced at the operator algebra conference in Genève in 1994, namely ${\mathcal{O}}_2\otimes A\cong {\mathcal{O}}_2$ for separable unital nuclear simple $A$ and ${\mathcal{O}}_{\infty }\otimes A\cong A$ for separable unital nuclear purely infinite simple $A,$ we prove that $KK$-equivalence implies isomorphism for nonunital separable nuclear purely infinite simple $C^{\ast }$-algebras. It follows that if $A$ and $B$ are unital separable nuclear purely infinite simple $C^{\ast }$-algebras which satisfy the Universal Coefficient Theorem, and if there is a graded isomorphism from $K_{\ast }(A)$ to $K_{\ast }(B)$ which preserves the $K_0$-class of the identity, then $A\cong B.$