process

Coarse and equivariant co-assembly maps 77where we use the obvious action of H on D ˝ K H and its multiplier algebra. Theaction of H on xB red .X; D ˝ K H / need not be continuous; we let xB redH.X; D/ be thesubalgebra of H -continuous elements in xB red .X; D ˝ K H /. We let Nc redH.X; D/ bethe subalgebra of vanishing variation functions in xB redH.X; D/. Both algebras containC 0 .X; D ˝ K H / as an ideal. The corresponding quotients are denoted by B redH.X; D/and c redH .X; D/. By construction, we have a natural morphism of extensions of H -C -algebrasC 0 .X; D ˝ K H / NcredH .X; D/ c redH.X; D/C 0 .X; D ˝ K H / x B redH .X; D/ Bred(9)H .X; D/.Concerning the extension of this construction to -coarse spaces, we only mentionone technical subtlety. We must extend the functor Ktop .H; / from C -algebras to-H -C -algebras. Here we use the definitionK top .H; A/ Š K .A ˝ P/ Ì r H ; (10)where D 2 KK H .P; C/ is a Dirac morphism for H . The more traditional definitionas a colimit of KK G .C 0.X/; A/, where X EG is G-compact, yields a wrong resultif A is a -H -C -algebra because colimits and limits do not commute.Let H be a locally compact group, let X be a coarse space with an isometric,continuous, proper action of H , and let D be an H -C -algebra. The H -equivariantcoarse K-theory KXH .X; D/ of X with coefficients in D is defined in [7] byKXH .X; D/ WD Ktop H; C 0 .P .X/; D/ : (11)As observed in [7], we have Ktop H; C 0 .P .X/; D/ Š K .C 0 .P .X/; D/ Ì H/ becauseH acts properly on P .X/.For most of our applications, X will be equivariantly uniformly contractible forall compact subgroups K H , that is, the natural embedding X ! P .X/ is aK-equivariant coarse homotopy equivalence. In such cases, we simply haveKXH .X; D/ Š Ktop H; C 0 .X; D/ : (12)In particular, this applies if X is an H -compact universal proper H -space (again, recallthat the coarse structure is determined by requiring H to act isometrically).The H -equivariant coarse co-assembly map for X with coefficients in D is a certainmap W K topC1H; credH .X; D/ ! KXH .X; D/defined in [7]. In the special case where we have (12), this is simply the boundarymap for the extension C 0 .X; D ˝ K H / Nc redH.X; D/ credH.X; D/. We areimplicitly using the fact that the functor Ktop .H; / has long exact sequences for arbitraryextensions of H -C -algebras, which is proved in [7] using the isomorphismKtop .H; B/ WD K .B ˝ P/ Ì r H Š K .B ˝max P/ Ì H