A Classification Theorem for Nuclear Purely Infinite Simple C*-Algebras

Abstract

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 ---> ϕ_t and t ---> ψ_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 ϕ_t (a) u_t^* - ψ_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.