A Classification Theorem for Nuclear Purely Infinite Simple C*-Algebras
Abstract
Description
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.
LaTeX, 48 pages (in 10pt type). This revision incorporates four modifications. First, there is now a preprint containing the theorems from which the present paper starts (funct-an/9712002). Second, the proofs have been slightly simplified (eliminating "local asymptotic morphisms"). Third, a section containing some nonclassification results in the nonnuclear case has been added. Fourth, some misprints have been corrected
LaTeX, 48 pages (in 10pt type). This revision incorporates four modifications. First, there is now a preprint containing the theorems from which the present paper starts (funct-an/9712002). Second, the proofs have been slightly simplified (eliminating "local asymptotic morphisms"). Third, a section containing some nonclassification results in the nonnuclear case has been added. Fourth, some misprints have been corrected