Axiomatic Kk-theory for Real C*-algebras

We establish axiomatic characterizations of K-theory and KK-theory for real C*-algebras. In particular, let F be an abelian group-valued functor on separable real C*-algebras. We prove that if F is homotopy invariant, stable, and split exact, then F factors through the category KK. Also, if F is homotopy invariant, stable, half exact, continuous, and satisfies an appropriate dimension axiom, then there is a natural isomorphism K(A) → F (A) for a large class of separable real C*-algebras A. Furthermore, we prove that a natural transformation F (A) → G(A) of homotopy invariant, stable, half-exact functors which is an isomorphism when A is complex is necessarily an isomorphism when A is real.