Date: Sun, 2 Jan 2000 10:09:48 +0100
From: Max Karoubi <karoubi@math.jussieu.fr>
Subject: homotopy fixed point set related to complex conjugation

Let A be a real Banach algebra and A' its complexification. I have recently
proved a "descent theorem" comparing the topological K-theory of A and A' :
we have K(A) = K(A') ^hZ/2 ; which means the homotopy fixed point set of
Z/2 acting on the K-theory space K(A') of A' via the complex conjugation
[in general, the K-theory space of a Banach algebra C is K _0(C) x BGL(C)
]. This result seems to be known by the experts for A = R and H (fields of
reals and quaternions respectively], in which case the theorem reduces to
BO = BU ^hZ/2 and BSp = BU ^hZ/2 (for ANOTHER involution of BU). Does
somebody know references in the litterature for these homotopy equivalences
?