Real \(C^*\)-algebras, united \(K\)-theory, and the Künneth formula (Q1866111)

The author defines united \(K\)-theory for real \(C^*\)-algebras, which generalizes Bousfield's topological united \(K\)-theory and incorporates real, complex and self-conjugate \(K\)-theory and the natural transformations among these functors. A Künneth-type short exact sequence is constructed for the united \(K\)-theory of the tensor product of two real \(C^*\)-algebras, which holds when the complexification of one of the \(C^*\)-algebras is in the bootstrap category. As an application, the united \(K\)-theory of the tensor product of two real Cuntz algebras is computed. Unlike the complex case, the isomorphism class of \({\mathcal O}^{\mathbb R}_{k+1}\otimes{\mathcal O}^{\mathbb R}_{l+1}\) is not determined solely by the greatest common divisor of \(k\) and \(l\). This gives examples of nonisomorphic simple purely infinite real \(C^*\)-algebras whose complexifications are isomorphic.