process

Coarse and equivariant co-assembly maps 85for all A; D, where we use the G-equivariant Morita-Rieffel equivalence betweenC 0 .jGj;D/Ì G and D. It fits into a commuting diagramK C1 .c redG .jGj;D/˝max A/ Ì G @K .C 0 .EG;D ˝max A/ Ì G/‰ D;AKK G .C;D˝max A/p EGŠ RKKG .EGI C;D˝max A/.Lemma 9. The class of G-C -algebras A for which ‰ D;A is an isomorphism for all Dis triangulated and thick and contains all G-C -algebras of the form C 0 .G=H / forcompact open subgroups H of G.Proof. The fact that this category of algebras is triangulated and thick means that it isclosed under suspensions, extensions, and direct summands. These formal propertiesare easy to check.Since H G is open, there is no difference between H -continuity and G-continuity.Hencec redG.jGj;D/˝max C 0 .G=H / Ì G Š c red .jGj;D/˝max C 0 .G=H / Ì GH c redH.jGj;D/Ì H;where means Morita-Rieffel equivalence. Similar simplifications can be made inother corners of the square. Hence the diagram for A D C 0 .G=H / and G acting on theright is equivalent to a corresponding diagram for trivial A and H acting on the right.The latter case is contained in Lemma 8.Theorem 10. Let G be an almost totally disconnected group with G-compact EG.Then for every B 2 KK G , the mapis an isomorphism and the diagram‰ W K topC1 G; cred .jGj;B/ ! KK G .C;B ˝ P/C1 G; cred .jGj;B/ KX G .jGj;B/K topŠ‰ KK G .C;B ˝ P/ pEG RKKG .EGI C;B ˝ P/commutes. In particular, K topC1 G; cred .jGj/ is naturally isomorphic to KK G .C; P/.Proof. It is shown in [7] that for such groups G, the algebra P belongs to the thick triangulatedsubcategory of KK G that is generated by C 0 .G=H / for compact subgroups Hof G. Hence the assertion follows from Lemma 9 and our definition of K top .Š