process

Categorical aspects of bivariant K-theory 35of duality due to Kasparov [31] that is studied more systematically in [17]. It requiresyet another version RKK G .XI A; B/ of Kasparov theory that is defined for a locallycompact space X and two G-C -algebras A and B. Roughly speaking, the cycles forthis theory are G-equivariant families of cycles for KK .A; B/ parametrised by X. Thegroups RKK G .XI A; B/ are contravariantly functorial and homotopy invariant in X (forG-equivariant continuous maps).We have RKK G .?I A; B/ D KKG .A; B/ and, more generally, RKKG .XI A; B/ ŠKK G A; C.X; B/ if X is compact. The same statement holds for non-compact X,but the algebra C.X; B/ is not a C -algebra any more: it is an inverse system ofC -algebras.Definition 71 ([17]). A G-C -algebra P X is called an abstract dual for X if, forall second countable locally compact G-spaces Y and all separable G-C -algebras Aand B, there are natural isomorphismsRKK G .X Y I A; B/ Š RKK G .Y I P X ˝ A; B/that are compatible with tensor products.Abstract duals exist for many spaces. For trivial reasons, C is an abstract dualfor the one-point space. For a smooth manifold X with an isometric action of G, bothC 0 .T X/and the algebra of C 0 -sections of the Clifford algebra bundle on X are abstractduals for X; ifX has a G-equivariant Spin c -structure – as in the example of Z actingon R – we may also use a suspension of C 0 .X/. For a finite-dimensional simplicialcomplex with a simplicial action of G, an abstract dual is constructed by GennadiKasparov and Georges Skandalis in [32] and in more detail in [17]. It seems likelythat the construction can be carried over to infinite-dimensional simplicial complexesas well, but this has not yet been written down.There are also spaces with no abstract dual. A prominent example is the Cantor set:it has no abstract dual, even for trivial G (see [17]).Let D be the class of all G-spaces that admit a dual. Recall that X 7! RKK G .X Y I A; B/ is a contravariant homotopy functor for continuous G-maps. Passing to corepresentingobjects, we get a covariant homotopy functorD ! KK G ; X 7! P X :This functor is very useful to translate constructions from homotopy theory to bivariantK-theory. An instance of this is the comparison of the Baum–Connes assembly mapsin both setups:Theorem 72. Let F be the family of finite subgroups of a discrete group G, and letE.G; F / be the universal .G; F /-CW-complex. Then E.G; F / has an abstract dual P ,and the map E.G; F / ! ? induces a Dirac morphism in KK G 0 .P; C/.Theorem 72 should hold for all families of subgroups F , but only the above specialcase is treated in [17], [38].