process

Coarse and equivariant co-assembly maps 75Whereas [7] studies the invertibility of (7) by relating it to (2), here we are going tostudy the map (7) itself.It is shown in [11] that the localisation of the category KK G at the weak equivalencesis isomorphic to the category RKK G .EG/ whose morphism spaces are the groupsRKK G .EGI A; B/ as defined by Kasparov in [10]. This statement is equivalent to theexistence of a Poincaré duality isomorphismKK G .A ˝ P;B/Š RKKG .EGI A; B/ (8)for all G-C -algebras A and B (this notion of duality is analysed in [6]). The canonicalfunctor from KK G to the localisation becomes the obvious functorp EG W KKG .A; B/ ! RKK G .EGI A; B/:Since D is a weak equivalence, p EG.D/ is invertible. Hence the maps in the followingcommuting square are isomorphisms for all G-C -algebras A and B:RKK G .EGI A; B ˝ P/ŠD RKK G .EGI A; B/ŠD RKK G .EGI A ˝ P;B ˝ P/ŠTogether with (8) this impliesD ŠD RKKG .EGI A ˝ P;B/.KK G .A ˝ P;B/Š KKG .A ˝ P;B ˝ P/:In the following, it will be useful to turn the isomorphismKtop .G; A/ Š K .A ˝ P/ Ì r G in Theorem 1.(c) into a definition.2.2 Group actions on coarse spaces. Let G be a locally compact group and let X bea right G-space and a coarse space. We always assume that G acts continuously andcoarsely on X, that is, the set f.xg; yg/ j g 2 K; .x; y/ 2 Eg is an entourage for anycompact subset K of G and any entourage E of X.Definition 2. We say that G acts by translations on X if f.x; gx/ j x 2 X; g 2 Kgis an entourage for all compact subsets K G. We say that G acts by isometries ifevery entourage of X is contained in a G-invariant entourage.Example 3. Let G be a locally compact group. Then G has a unique coarse structurefor which the right translation action is isometric; the corresponding coarse space isdenoted jGj. The generating entourages are of the form[Kg Kg Df.xg; yg/ j g 2 G; x; y 2 Kgg2Gfor compact subsets K of G. The left translation action is an action by translations forthis coarse structure.