process

22 R. Meyernot know the equivalence relation to put on these cycles. Brown–Douglas–Fillmorestudied extensions of C 0 .X/ (and more general C -algebras) by the compact operatorsand found that the resulting structure set is naturally isomorphic to a K-homology group.Kasparov unified and vastly generalised these two results, defining a bivariant functorKK .A; B/ that combines K-theory and K-homology and that is closely related tothe classification of extensions B ˝ K E A (see [29], [30]). A deep theorem ofKasparov shows that two reasonable equivalence relations for these cycles coincide; thisclarifies the homotopy invariance of the extension groups of Brown–Douglas–Fillmore.Furthermore, he constructed an equivariant version of his theory in [31] and applied itto prove the Novikov conjecture for discrete subgroups of Lie groups.The most remarkable feature of Kasparov theory is an associative product on KKcalled Kasparov product. This generalises various known product constructions inK-theory and K-homology and allows to view KK as a category.In applications, we usually need some non-obvious KK-element, and we mustcompute certain Kasparov products explicitly. This requires a concrete description ofKasparov cycles and their products. Since both are somewhat technical, we do notdiscuss them here and merely refer to [5] for a detailed treatment and to [56] for a veryuseful survey article. Instead, we use Higson’s characterisation of KK by a universalproperty [23], which is based on ideas of Cuntz ([10], [9]). The extension to theequivariant case is due to Thomsen [58]. A simpler proof of Thomsen’s theorem andvarious related results can be found in [36].We do not discuss KK G for Z=2-graded G-C -algebras here because it does not fit sowell with the universal property approach, which would simply yield KK GZ=2 becauseZ=2-graded G-C -algebras are the same as G Z=2-C -algebras. The relationshipbetween the two theories is explained in [36], following Ulrich Haag [22]. The gradedversion of Kasparov theory is often useful because it allows us to treat even and oddKK-cycles simultaneously.Fix a locally compact group G. The Kasparov groups KK G 0 .A; B/ for A; B 2G-C sep form the morphism sets A ! B of a category, which we denote by KK G ;the composition in KK G is the Kasparov product. The categories G-C sep and KK Ghave the same objects. We have a canonical functorKK G W G-C sep ! KK Gthat acts identically on objects. This functor contains all information about equivariantKasparov theory for G.Definition 49. A G-equivariant -homomorphism f W A ! B is called a KK G -equivalence if KK G .f / is invertible in KK G .Theorem 50. Let S be a full subcategory of G-C sep that is closed under suspensions,G-equivariantly cp-split extensions, and Morita–Rieffel equivalence. Let KK G .S/ bethe full subcategory of KK G with object class S and let KK G S W S ! KKG .S/ be therestriction of KK G .