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84 H. Emerson and R. MeyerLet h and k be the Lie algebras of H and K. There is a K-equivariant homeomorphismh=k Š H=K, where K acts on h=k by conjugation. Now we need to know whetherthere is a K-equivariant spin structure on h=k. One can check that this is the case ifk 0; 1 mod 4. Since we can choose k as large as we like, we can always assumethat this is the case. The spin structure allows us to use Bott periodicity to identifyKK .H=K/ Š KH N.C/, which is the representation ring of K in degree N , whereN D dim h=k. Using our construction, the trivial representation of K yields a canonicalelement in KK G N C;C 0.X k / .This construction is much shorter than the corresponding one in [3] because we useKasparov’s result about dual-Dirac morphisms for almost connected groups. Much ofthe corresponding argument in [3] is concerned with proving a variation on this resultof Kasparov.4 Computation of KK G .C; P/So far we have merely used the diagram (13) to construct certain elements in KK G .C;B/.Now we show that this construction yields an isomorphic description of KK G .C; P/.This assertion requires G to be a totally disconnected group with a G-compact universalproper G-space. We assume this throughout this section.Lemma 8. In the situation of Lemma 7, suppose that X DjGj with G acting by lefttranslations and that H G is a compact subgroup acting on X by right translations;here jGj carries the coarse structure of Example 3. Then the maps and areisomorphisms.Proof. We reduce this assertion to results of [7]. The C -algebras c redH.jGj;D/Ì Hand c redH.jGj;D/H are strongly Morita equivalent, whence have isomorphic K-theory.It is shown in [7] thatK C1 c red H .jGj;D/H Š KK G .C; IndG H D/: (16)Finally, IndH G .D/ D C 0.G; D/ HC 0 .G; D/ Ì H . Hence we getis G-equivariantly Morita–Rieffel equivalent toK C1 .c redH.jGj;D/Ì H/ Š K C1 c red H .jGj;D/H Š KK G C; IndG H .D/Š KK G C;C 0.G; D/ Ì H :(17)It is a routine exercise to verify that this composition agrees with the map in (13).Similar considerations apply to the map .We now set X DjGj, and let G D H . The actions of G on jGj on the left and rightare by translations and isometries, respectively. Lemma 7 yields a map‰ W K C1 .c red .jGj;D/˝max A/ Ì G ! KK G C;C 0.jGj;D˝max A/ Ì G : (18)