  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.
