Subject: Re: two reponses From: Max Karoubi Date: Thu, 10 Mar 2005 09:03:12 +0100 Dear John (Klein), I gave a complete proof of my question, using essentially the KR-theory of Atiyah and Clifford bundles in my paper A descent theorem in topological K-theory. K-theory 24, p. 109-114 (2001). The theorem is stated in terms of the classifying space of the topological K-theory for ANY Banach algebra (not just the field of real numbers), as I said in my previous message of 2000. I take this opportunity to remark that the classifying space of a Banach algebra A (same type for algebraic K-theory) is NOT K(A) x BGL(A) in a functorial way, as I wrote too fast in my 2000 question (this is important since a finite group may act on A). The "correct" definition is the loop space of BGL(SA), where SA is the topological suspension of the Banach algebra A ; one may find the details in a recent paper with J. Berrick (see the K-theory archives or my Web page) which is going to appear in the Amer. J. of Math. Max KAROUBI http://www.math.jussieu.fr/~karoubi/ > On Wed, 9 Mar 2005, Don Davis wrote: > >> Subject: Question for the list > >> From: "John R. Klein" > >> Date: Mon, 07 Mar 2005 09:47:03 +0200 > >> > >> 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? > Max KAROUBI Professeur ŕ l'Université Paris 7 http://www.math.jussieu.fr/~karoubi/ _____________________________________________________ Subject: Re: two reponses From: Lisbeth Fajstrup Date: Fri, 11 Mar 2005 13:08:42 +0100 Dear All. Here is yet another reference to the descent result asked for by John Klein. As both John R. and John G. remark, this is a consequence of well known results for Atiyah Real K-theory, and it is just an exercise in fitting it all together. I elaborated on that in " Tate Cohomology of Periodic Real K-theory is Trivial", Trans. Amer. Math. Soc. Vol 347 No 5, May 1995. The main point is, that the generator of KR(RP1) is nilpotent. Lisbeth Fajstrup.