process

80 H. Emerson and R. MeyerconditionZhNc.;x/;˛i WDGc.g/˛.g 1 x/dgfor ˛ 2 C 0 .X/ defines a probability measure on X. Since such measures are containedin P .X/, Nc defines a map Nc W EG X ! P .X/. This map is continuous and satisfiesNc.g;g 1 xh/ DNc.;x/h for all g 2 G, 2 EG, x 2 X, h 2 H .ForaC -algebra Z, let C.EG;Z/ be the -C -algebra of all continuous functionsf W EG ! Z without any growth restriction. Thus C.EG;Z/ D lim C.K;Z/,where K runs through the directed set of compact subsets of EG.We claim that . Nc f /./.x/ WD f Nc.;x/ for f 2 C 0 .P .X/; D/ defines a continuous-homomorphismNc W C 0 .P .X/; D/ ! C EG;C 0 .X; D/ :If K EG is compact, then there is a compact subset L G such that c. g/ D 0for 2 K and g … L. Hence Nc.;x/ is supported in L 1 x for 2 K. Since G actson X by translations, such measures are contained in a filtration level P d .X/. HenceNc .f / restricts to a C 0 -function K X ! D for all f 2 C 0 .P .X/; D/. This provesthe claim. Since Nc is H -equivariant and G-invariant, we get an induced mapB P .X/ D C 0 .P .X/; D/ Ì H ! .C EG;C 0 .X; D/ Ì H/ G D C.EG;B X / G ;where Z G Z denotes the subalgebra of G-invariant elements. We obtain an induced-homomorphism between the stable multiplier algebras as well.An element of K 0 B P .X/ / is represented by a self-adjoint bounded multiplier F 2M.B P .X/ ˝ K/ such that 1 FF and 1 F F belong to B P .X/ ˝ K. NowFQWDNc .F / is a G-invariant bounded multiplier of C.EG;B X ˝ K/ and hence a G-invariantmultiplier of C 0 .EG;B X ˝ K/, such that ˛ .1 FQFQ / and ˛ .1 FQ F/ Q belongto C 0 .EG;B X ˝ K/ for all ˛ 2 C 0 .EG/. This says exactly that FQis a cycle forRKK G 0 .EGI C;B X/. This construction provides the natural map W K 0 .B P .X/ / ! RKK G 0 .EGI C;B X/:Finally, a routine computation, which we omit, shows that the two images of a unitaryu 2 E X =B X differ by a compact perturbation. Hence the diagram (13) commutes.We are mainly interested in the case where A is the source P of a Dirac morphismfor H . Then K C1 .E X =B X / D Ktop H; c redH.X; D/ , and the top row in (13) is theH -equivariant coarse co-assembly map for X with coefficients D. Since we assume Hto act properly on X,wehaveaKK G -equivalence B X C 0 .X; D/ Ì H , and similarlyfor P .X/. Hence we now get a commuting squareK topC1H; credH .X; D/ @KK G .C;C 0.X; D/ Ì H/p EGKX H.X; D/ RKKG .EGI C;C 0 .X; D/ Ì H/.(14)