The classification of real purely infinite simple C*-algebras

Abstract

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 $2\le n\le \infty$.