process

Categorical aspects of bivariant K-theory 3Definition 2. We write A 22 C to denote that A is an object of the category C. Thenotation f 2 C means that f is a morphism in C; but to avoid confusion we alwaysspecify domain and target and write f 2 C.A; B/ instead of f 2 C.2.2 Group actions, and crossed products. For any locally compact group G, wehave a reduced group C -algebra C red .G/ and a full group C -algebra C .G/. Both aredefined as completions of the group Banach algebra .L 1 .G/; / for suitable C -norms.They are related by a canonical surjective -homomorphism C .G/ ! C red.G/, whichis an isomorphism if and only if G is amenable.The norm on C .G/ is the maximal C -norm, so that any strongly continuousunitary representation of G on a Hilbert space induces a -representation of C .G/.The norm on C red .G/ is defined using the regular representation of G on L2 .G/; hence arepresentation of G only induces a -representation of C red.G/ if it is weakly containedin the regular representation. For reductive Lie groups and reductive p-adic groups,these representations are exactly the tempered representations, which are much easierto classify than all unitary representations.Definition 3. A G-C -algebra isaC -algebra A with a strongly continuous representationof G by C -algebra automorphisms. The category of G-C -algebras is the categoryG-C alg whose objects are the G-C -algebras and whose morphisms A ! Bare the G-equivariant -homomorphisms A ! B; we denote this morphism set byHom G .A; B/.Example 4. If G D Z, then a G-C -algebra is nothing but a pair .A; ˛/ consisting ofaC -algebra A and a -automorphism ˛ W A ! A: let ˛ be the action of the generator1 2 Z.Equipping C -algebras with a trivial action provides a functor W C alg ! G-C alg; A 7! A : (1)Since C has only the identity automorphism, the trivial action is the only way to turn Cinto a G-C -algebra.The full and reduced C -crossed products are versions of the full and reduced groupC -algebras with coefficients in G-C -algebras (see [44]). They define functorsG Ë ;GË r W G-C alg ! C alg; A 7! G Ë A; G Ë r A;such that G Ë C D C .G/ and G Ë r C D C red .G/.Definition 5. A diagram I ! E ! Q in C alg is an extension if it is isomorphic tothe canonical diagram I ! A ! A=I for some ideal I inaC -algebra A; extensionsin G-C alg are defined similarly, using G-invariant ideals in G-C -algebras. We writeI E Q to denote extensions.Although C -algebra extensions have some things in common with extensions of,say, modules, there are significant differences because C alg is not Abelian, not evenadditive.