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 $A\sc$ and $B\sc$ 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 \cite{stacey03}. Thus the real C*-algebras classified here are exactly those real C*-algebras whose complexification falls under the classification result of Kirchberg \cite{kirchberg94} and Phillips \cite{phillips00}. As an application, we find all real forms of the complex Cuntz algebras $Ø_n$ for $2 <= n leq \infty$.

2010 Mathematics Subject Classification: 46L35, 46L80, 19K99

Keywords and Phrases: Real C*-algebras, $K$-theory, classification

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