A descent theorem in topological \(K\)-theory (Q1598939)

Let \(A\) be a real Banach algebra and denote by \(K(A)\) its \(K\)-theory space. If \(A'=A\otimes_\mathbb{R} \mathbb{C}\) is the complexification of \(A\), the group \(G=\mathbb{Z}/2\) acts on \(K(A')\) by complex conjugation, and there is a natural map \(\sigma: K(A)\to K(A')^{hG}\) where \(Y^{hG}\) is the homotopy fixed point set of the \(G\)-space \(Y\). The main result of the paper is the ``descent theorem'': the map \(\sigma\) defined above is a homotopy equivalence. In other words, real \(K\)-theory can be deduced from complex \(K\)-theory via the usual descent spectral sequence. In the proof of this theorem, the author uses Atiyah's \(KR\)-theory and the definition of higher \(K\)-groups via Clifford algebras. Also, new applications are stated.