process

Coarse and equivariant co-assembly maps 79and similarly for P .X/ instead of X. There are extensions B X E X E X =B Xand B P .X/ E P .X/ E P .X/ =B P .X/ withE P .X/ =B P .X/ Š E X =B X Š .c redH .X; D/ ˝max A/ Ì H;and a natural commuting diagramK C1 .E X =B X /KK G .C;B X/@K .B P .X/ /p EG RKKG .EGI C;B X /.(13)Proof. The quotients E X =B X and E P .X/ =B P .X/ are as asserted and agree becauseX ! P .X/ is a coarse equivalence and because maximal tensor products and fullcrossed products are exact functors in complete generality, unlike spatial tensor productsand reduced crossed products. We let @ be the K-theory boundary map for the extensionB P .X/ E P .X/ E X =B X .Since we have a natural map c redH.X; D/ ˝max A ! c redH.X; D ˝max A/, we mayreplace the pair .D; A/ by .D ˝max A; C/ and omit A if convenient. Stabilising Dby K H , we can further eliminate the stabilisations.First we lift the K-theory boundary map for the extension B X E X E X =B Xto a map W K C1 .E X =B X / ! KK G .C;B X/. The G-equivariance of the resultingKasparov cycles follows from the assumption that G acts on X by translations.We have to distinguish between the cases D0 and D1. We only write downthe construction for D0. Since the algebra E X =B X is matrix-stable, K 1 .E X =B X /is the homotopy group of unitaries in E X =B X without further stabilisation. A cyclefor KK G 0 .C;B X/ is given by two G-equivariant Hilbert modules E˙ over B X and aG-continuous adjointable operator F W E C ! E for which 1 FF , 1 F F andgF F for g 2 G are compact; we take E˙ D B X and let F 2 E X M.B X / bea lifting for a unitary u 2 E X =B X . Since G acts on X by translations, the inducedaction on Nc red .X; D/ and hence on E X =B X is trivial. Hence u is a G-invariant unitaryin E X =B X . For the lifting F , this means that1 FF ; 1 F F; gF F 2 B X :Hence F defines a cycle for KK G 0 .C;B X/. We get a well-defined map Œu 7! ŒF fromK 1 .E X =B X / to KK G 0 .C;B X/ because homotopic unitaries yield operator homotopicKasparov cycles.Next we have to factor the map p EG ı in (13) through K 0.B P .X/ /. The mainingredient is a certain continuous map Nc W EG X ! P .X/. We use the same descriptionof P .X/ as in [7] as the space of positive measures on X with 1=2 < .X/ 1;this is a -coarse space in a natural way, we write it as P .X/ D S P d .X/.There is a function c W EG ! R C for which R EGc.g/dg D 1 for all 2 EGand supp c \ Y is compact for G-compact Y EG. If 2 EG, x 2 X, then the