process

Categorical aspects of bivariant K-theory 25for any G-C -algebra A. Since these extensions are natural, we even get bifunctors˝min ; ˝max W KK G KK G ! KK G :The associativity, commutativity, and unit constraints in G-C alg induce correspondingconstraints in KK G , so that both ˝min and ˝max turn KK G into a symmetricmonoidal category.Another example is the functor W C alg ! G-C alg that equips a C -algebrawith the trivial G-action; it extends to a functor W KK ! KK G .The universal property also allows us to prove identities between functors. Forinstance, we have natural isomorphisms G Ë r ..A/ ˝min B/ D A ˝min .G Ë r B/ forall G-C -algebras B. Naturality means, to begin with, that the diagramG Ë r ..A 1 / ˝min B 1 /ŠA 1 ˝min .G Ë r B 1 /GË r .˛/˝minˇG Ë r ..A 2 / ˝min B 2 /˛˝min .GË rˇ/Š A 2 ˝min .G Ë r B 2 /commutes if ˛ W A 1 ! A 2 and ˇ W B 1 ! B 2 are a -homomorphism and a G-equivariant -homomorphism, respectively. Two applications of the uniqueness part ofthe universal property show that this diagram remains commutative in KK if ˛ 2KK 0 .A 1 ;A 2 / and ˇ 2 KK G 0 .B 1;B 2 /. Similar remarks apply to the natural isomorphismG Ë ..A/ ˝max B/ Š A ˝max .G Ë B/ and hence to the isomorphismsG Ë .A/ Š C .G/ ˝max A and G Ë r .A/ Š C red .G/ ˝min A.Adjointness relations in Kasparov theory are usually proved most easily by constructingthe unit and counit of the adjunction. For instance, if G is a compact groupthen the functor is left adjoint to G Ë D G Ë r , that is, for all A 22 KK andB 22 KK G , we have natural isomorphismsKK G ..A/; B/ Š KK .A; G Ë B/: (8)This is also known as the Green–Julg Theorem. ForA D C, it specialises to a naturalisomorphism K G .B/ Š K .G Ë B/; this was one of the first appearances of noncommutativealgebras in topological K-theory.Proof of (8). We already know that and G Ë are functors between KK and KK G .It remains to construct natural elements˛A 2 KK 0 A; G Ë .A/ ;ˇB 2 KK G 0 ..G Ë B/;B/for all A 22 KK, B 22 KK G that satisfy the conditions for unit and counit of adjunction[35].The main point is that .G Ë B/ is the G-fixed point subalgebra of B K D B ˝K.L 2 G/. The embedding .G Ë B/ ! B K provides a G-equivariant correspondenceˇB from .G Ë B/ to B and thus an element of KK G 0 ..G Ë B/;B/. This