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A classification theorem for nuclear purely infinite simple \(C^*\)-algebras. (English) Zbl 0943.46037
Starting from Kirchberg’s theorems announced at the operator algebra conference in Genève in 1994, namely \({\mathcal O}_{2} \otimes A \cong {\mathcal O}_{2}\) for separable unital nuclear simple \(A\) and \({\mathcal O}_{\infty} \otimes {A} \cong A\) 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 \(K_0\)-class of the identity, then \(A \cong B.\)
Our main technical results are, we believe, of independent interest. We say that two asymptotic morphisms \(t \mapsto \varphi_t\) and \(t \mapsto \psi_t\) from \(A\) to \(B\) are asymptotically unitarily equivalent if there exists a continuous unitary path \(t \mapsto u_t\) in the unitization \(B^+\) such that \(\|u_t \varphi_t (a) u_t^* - \psi_t (a) \|\to 0\) 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 \(K\otimes {\mathcal O}_{\infty} \otimes {D}\) is asymptotically unitarily equivalent to a homomorphism, and two homotopic homomorphisms from \(A\) to \(K\otimes {\mathcal O}_{\infty} \otimes {D}\) are necessarily asymptotically unitarily equivalent. We also give some nonclassification results for the nonnuclear case.

46L35 Classifications of \(C^*\)-algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K99 \(K\)-theory and operator algebras
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