#### DOCUMENTA MATHEMATICA, Vol. 5 (2000), 49-114

N. Christopher Phillips

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 $A \cong B.$

Our main technical results are, we believe, of independent interest. We say that two asymptotic morphisms $t \mapsto \ph_t$ and $t \mapsto \ps_t$ from $A$ to $B$ are asymptotically unitarily equivalent if there exists a continuous unitary path $t \mapsto u_t$ in the unitization $B^+$ such that $\| u_t \ph_t (a) u_t^* - \ps_t (a) \| \to 0$ for all $a$ in $A.$ We prove the following two results on deformations and unitary equivalence. Let $A$ be separable, nuclear, unital, and simple, and let $D$ be unital. Then any asymptotic morphism from $A$ to $\Kt {\cal O}_{\infty} \otimes {D}$ is asymptotically unitarily equivalent to a homomorphism, and two homotopic homomorphisms from $A$ to $\Kt {\cal O}_{\infty} \otimes {D}$ are necessarily asymptotically unitarily equivalent. We also give some nonclassification results for the nonnuclear case.

1991 Mathematics Subject Classification: Primary 46L35; Secondary 19K99, 46L80.

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