Subject:
Question for the list
From:
"John R. Klein" <klein@math.wayne.edu>
Date:
Mon, 07 Mar 2005 09:47:03 +0200
To:
dmd1@lehigh.edu


In the year 2000, Max Karoubi posted a question to the list (see below) about the
the homotopy fixed set of Galois action of Z_2 on the K-theory of Banach Algebras.
There he mentions a "known" result that the inclusion

            BO   -->  (BU)^{hZ_2}

is a homotopy equivalence. I'm trying to find a homotopy theoretic proof of this statement
(i.e., no operator algebras permitted). Does anybody know of one?

Incidentally, is there a published reference for this result besides Karoubi's 2000 paper in the
K-theory journal?

jk

> 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
> ?
>