A descent theorem in topological K-theory
  Let A be a Banach algebra and A' its complexification. In this paper we show that the homotopy fixed point set of K(A'), the topological K-theory space of A', under complex conjugation is just K(A), the topological K-theory space of A. This result generalizes the well known fact that BO is BU^hZ/2. The proof uses in an essential way Atiyah's KR theory and the Clifford algebra definition of higher K-groups. 
Max Karoubi