  Starting from Kirchberg's theorems announced in 1994, namely O_2 tensor A is
isomorphic to O_2 for separable unital nuclear simple A and O_infinity tensor A
is isomorphic to A if in addition A is purely infinite, 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 class of the identity, then A is isomorphic to B.
  Our main technical results are, we believe, of independent interest. We say
that two asymptotic morphisms t ---> \phi_t and t ---> \psi_t from A to B are
asymptotically unitarily equivalent if there exists a continuous unitary path t
---> u_t in the unitization B^+ such that || u_t \phi_t (a) u_t^* - \psi_t (a)
|| ---> 0 for all a in A. Let A be separable, nuclear, unital, and simple, and
let D be unital. We prove that any asymptotic morphism from A to K tensor
O_infinity tensor D is asymptotically unitarily equivalent to a homomorphism,
and two homotopic homomorphisms from A to K tensor O_infinity tensor D are
asymptotically unitarily equivalent.
