A \(\mathbb{Z}/2\) descent theorem for the algebraic \(K\)-theory of a semisimple real algebra (Q1386679)

Let \(G\) be a topological group and \(EG\) a free contractible \(G\)-space such that \(EG\to BG= EG/G\) is a fibre bundle. For a \(G\)-space \(X\) the set \(X^{hG}\) of equivariant maps from \(EG\) to \(X\) is then called the homotopy fixed point set of \(X\). Since the fixed point set \(X^G\) of \(X\) is the space of equivariant maps from a point to \(X\), the projection \(p: EG\to *\) induces a map \(p^*: X^G\to X^{hG}\). According to \textit{R. W. Thomason} [``The homotopy limit problem'', Contemp. Math. 19, 407-419 (1983; Zbl 0528.55008)] we have a problem studying whether \(p^*\) is an equivalence of some kind. In this paper the author gives an answer to this problem for the algebraic \(K\)-theory \(\widetilde K(R)= BGL(R)^+\) of an antistructure \((R,\alpha,u)\). By the antistructure one means a finite-dimensional semisimple real algebra \(R\) with an anti-involution \(\alpha\) with respect to a unit \(u\in R\), with \(u\alpha (u)=1\). Making use of this structure the author introduces an involution on \(\widetilde K(R)\), and so \(\widetilde K(R)\) can be viewed as a \({\mathbb{Z}}/2\)-space. The main result (Theorem A) of the paper is that \((\widetilde K(R)^{{\mathbb{Z}}/2})_2^{\widehat{ }} \simeq (\widetilde K(R)_0^{h {\mathbb{Z}}/2})_2^{\widehat{ }}\). Here the index number 0 indicates the component of the homotopy fixed point set which contains the constant map, and the 2-adic completions are in the sense of Bousfield and Kan. Also the author shows that when \(u\) is in the center of \(R\), \(\widetilde K(R)^{{\mathbb{Z}}/2}\) becomes the classifying space for unitary algebraic \(K\)-theory. One of the steps in the author's proof consists in presenting 11 kinds of prototypes \((R_i, \alpha_i, u_i)\) of antistructures where \(R_i= \mathbb{R}\), \(\mathbb{C}, \mathbb{H}, \mathbb{R} \times \mathbb{R}\), \(\mathbb{C} \times \mathbb{C}\) or \(\mathbb{H} \times \mathbb{H}\) and in showing that any \(\widetilde K(R)\) is equivariantly homotopic to a direct product of \(\widetilde K(R_i)\)'s. It turns out that the problem for \(\widetilde K(R)\) can be referred to that for each \(\widetilde K(R_i)\). The proof for \(\widetilde K(R_i)\)'s is heavily based on Suslin's results on the connection between algebraic \(K\)-theory and topological \(K\)-theory and is done by applying the completion theorems of Atiyah and Segal for topological \(K\)-theories and their modifications. Further the author calculates the number of components for each \(\widetilde K(R_i)\), and the paper contains an appendix in which the author proves some auxiliary theorems needed in the preceding sections.