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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.

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