The classification of real purely infinite simple C*-algebras

We classify real Kirchberg algebras using united K -theory. Precisely, let A and B be real simple separable nuclear purely infinite C*-algebras that satisfy the universal coefficient theorem such that A_{\mathbb{C}} and B_{\mathbb{C}} are also simple. In the stable case, A and B are isomorphic if and only if K^{CRT}(A) \cong K^{CRT}(B) . In the unital case, A and B are isomorphic if and only if (K^{CRT}(A), [1_A]) \cong (K^{CRT}(B), [1_B]) . We also prove that the complexification of such a real C*-algebra is purely infinite, resolving a question left open from [43]. Thus the real C*-algebras classified here are exactly those real C*-algebras whose complexification falls under the classification result of Kirchberg [26] and Phillips[35]. As an application, we find all real forms of the complex Cuntz algebras {\mathcal{O}}_n for 2 \leq n \leq \infty .

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