A ug 2 00 2 Real C * - Algebras , United K - Theory , and the Künneth Formula Jeffrey

We define united K-theory for real C*-algebras, generalizing Bousfield’s topological united K-theory. United K-theory incorporates three functors — real K-theory, complex K-theory, and self-conjugate K-theory — and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Künneth-type, non-splitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C*-algebras A and B which holds as long as the complexification of A is in the bootstrap category N . Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product O k+1 ⊗ O R l+1 is not determined solely by the greatest common divisor of k and l. Hence, we have examples of non-isomorphic, simple, purely infinite, real C*-algebras whose complexifications are isomorphic.