process

68 A. Bartels, S. Echterhoff, and W. LückL hn1i.R.y/ÌG/ is the algebraic L-theory with decoration h 1i of the twistedgroup ring with involution R.y/ Ì G, K n .A.y/ Ì l 1 G/ is the topological K-theory of the crossed product Banach algebra A.y/ Ì l 1 G, K n .A.y/ Ì r G/ isthe topological K-theory of the reduced crossed product C -algebra A.y/ Ì r G,and K n .A.y/ Ì m G/ is the topological K-theory of the maximal crossed productC -algebra A.y/ Ì m G.The natural transformations I 1 , I 2 , I 3 and I 4 become under this identificationsthe obvious change of rings and theory homomorphisms.(iii) These constructions are in the obvious sense natural in R and A respectively andin G .We defer the details of the proof of Theorem 8.1 in [2]. Its proof requires some workbut there are many special cases which have already been taken care of. If we wouldnot insist on groupoids but only on groups as input, these are the standard algebraic K-and L-theory spectra or topological K-theory spectra associated to group rings, groupBanach algebras and group C -algebras. The construction for the algebraic K- andL-theory and the topological K-theory in the case, where G acts trivially on a ring Ror a C -algebra are already carried out or can easily be derived from [4], [13], and [24]except for the case of a Banach algebra. The case of the K-theory spectrum associatedto an additive category with G-action has already been carried out in [7]. The mainwork which remains to do is to treat the Banach case and to construct the relevantnatural transformation from KH to K topA;l 1References[1] J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Math. Universityof Chicago Press, Chicago, Ill., 1974.[2] A. Bartels, M. Joachim, and W. Lück, From algebraic to topological K-theory spectra, inpreparation, 2007.[3] A. Bartels and W. Lück, Induction theorems and isomorphism conjectures for K- and L-theory, Forum Math. 19 (2007), 379–406.[4] A. Bartels and W. Lück, Isomorphism conjecture for homotopy K-theory and groups actingon trees, J. Pure Appl. Algebra 205 (2006),660–696.[5] A. Bartels, W. Lück, and H. Reich, The K-theoretic Farrell-Jones Conjecture for hyperbolicgroups, Invent. Math. 172 (2008), 29–70.[6] A. Bartels, W. Lück, and H. Reich, On the Farrell-Jones Conjecture and its applications,J. Topol. 1 (2008), 57–86.[7] A. Bartels and H. Reich, Coefficients for the Farrell-Jones conjecture, Adv. Math.,209(1):337–362, 2007.[8] H. Bass, Algebraic K-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968.[9] P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and K-theory ofgroup C -algebras, in C -algebras: 1943–1993 (San Antonio, TX, 1993), Amer. Math.Soc., Providence, RI, 1994, 240–291.