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

Starting from Kirchberg's theorems announced at the operator algebra conference in Gen eve in 1994, namely O2 A = O2 for separable unital nuclear simple A and O1 A = A for separable unital nuclear purely innite simple A; we prove that KK-equivalence implies isomorphism for nonunital separable nuclear purely innite simple C -algebras. It follows that if A and B are unital separable nuclear purely innite simple C -algebras which satisfy the Universal Coecien t Theorem, and if there is a graded isomorphism fromK (A) to K (B) which preserves the K0-class of the identity, then A = B: Our main technical results are, we believe, of independent in- terest. We say that two asymptotic morphisms t 7! 't and t 7! t from A to B are asymptotically unitarily equivalent if there exists a continuous unitary path t 7! ut in the unitization B + such that kut't(a)ut t(a)k ! 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 O1 D is asymptotically unitarily equiv- alent to a homomorphism, and two homotopic homomorphisms from A to K O1 D are necessarily asymptotically unitarily equivalent. We also give some nonclassication results for the nonnuclear case. 1991 Mathematics Subject Classication: Primary 46L35; Secondary 19K99, 46L80.