ar X iv : f un ct - a n / 95 06 01 0 v 2 1 8 D ec 1 99 7 A Classification Theorem for Nuclear

Starting from Kirchberg’s theorems announced at the operator algebra conference in Genève in 1994, namely O2 ⊗A ∼= O2 for separable unital nuclear simple A and O∞ ⊗A ∼= 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 K0-class of the identity, then A ∼= B. Our main technical results are, we believe, of independent interest. We say that two asymptotic morphisms t 7→ φt and t 7→ ψt from A to B are asymptotically unitarily equivalent if there exists a continuous unitary path t 7→ ut in the unitization B + such that ‖utφt(a)u ∗ t − ψt(a)‖ → 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 K⊗O∞⊗D is asymptotically unitarily equivalent to a homomorphism, and two homotopic homomorphisms from A to K ⊗O∞ ⊗D are necessarily asymptotically unitarily equivalent. We also give some nonclassification results for the nonnuclear case. Research partially supported by NSF grant DMS 94-00904, and by the Fields Institute for Research in Mathematical Sciences. AMS 1991 subject classification numbers: Primary 46L35; Secondary 19K99, 46L80.