process

26 R. Meyerconstruction is certainly natural for G-equivariant -homomorphisms and hence forKK G -morphisms by the uniqueness part of the universal property of KK G .Let e W C ! C .G/ be the embedding that corresponds to the trivial representationof G. Recall that G Ë .A/ Š C .G/ ˝ A. Hence the exterior product of the identitymap on A and KK.e / provides ˛A 2 KK 0 A; G Ë .A/ . Again, naturality for -homomorphisms is clear and implies naturality for morphisms in KK.Finally, it remains to check that.A/ .˛A/! G Ë .A/ ˇ.A/! .A/;G Ë B ˛GËB! G Ë .G Ë B/ GˡB! G Ë Bare the identity morphisms in KK G . Then we get the desired adjointness using a generalconstruction from category theory (see [35]). In fact, both composites are equal to theidentity already as correspondences, so that we do not have to know anything aboutKasparov theory except its C -stability to check this.A similar argument yields an adjointness relationKK G 0 A; .B/ Š KK 0 .G Ë A; B/ (9)for a discrete group G. More conceptually, (9) corresponds via Baaj–Skandalis duality[3] to the Green–Julg Theorem for the dual quantum group of G, which is compactbecause G is discrete. But we can also write down unit and counit of adjunction directly.The trivial representation C .G/ ! C yields natural -homomorphismsG Ë .B/ Š C .G/ ˝max B ! Band hence ˇB 2 KK 0 .G Ë .B/;B/. The canonical embedding A ! G Ë A isG-equivariant if we let G act on G Ë A by conjugation; but this action is inner, sothat G Ë A and .G Ë A/ are G-equivariantly Morita–Rieffel equivalent. Thus thecanonical embedding A ! G Ë A yields a correspondence A Ü .G Ë A/ and˛A 2 KK G 0 A; .G Ë A/ . We must check that the compositesG Ë A G˲A! G Ë .G Ë A/ ˇGËA! G Ë A;.B/ ˛.B/! G Ë .B/ .ˇB /! .B/are identity morphisms in KK and KK G , respectively. Once again, this holds alreadyon the level of correspondences.4.2 Triangulated category structure. We turn KK G into a triangulated category byextending standard constructions for topological spaces [38]. Some arrows changedirection because the functor C 0 from spaces to C -algebras is contravariant. We havealready observed that KK G is additive. The suspension is given by † 1 .A/ WD Sus.A/.Since Sus 2 .A/ Š A in KK G by Bott periodicity, we have † D † 1 . Thus we do notneed formal desuspensions as for the stable homotopy category.