process

Categorical aspects of bivariant K-theory 5is a retraction from G Ë r E onto the kernel of G Ë r p by construction. Furthermore, itmaps the dense subspace L 1 .G; E/ into L 1 .G; K/. Hence it maps all of G Ë r E intoG Ë r K. This implies G Ë r K D ker.G Ë r p/ as desired.2.3 Tensor products and nuclearity. Most results in this section are proved in detailin [42], [60]. Let A 1 and A 2 be two C -algebras. Their (algebraic) tensor productA 1 ˝ A 2 is still a -algebra. A C -tensor product of A 1 and A 2 is a C -completionof A 1 ˝ A 2 , that is, a C -algebra that contains A 1 ˝ A 2 as a dense -subalgebra. AC -tensor product is determined uniquely by the restriction of its norm to A 1 ˝ A 2 .Anorm on A 1 ˝ A 2 is allowed if it is a C -norm, that is, multiplication and involutionhave norm 1 and kx xkDkxk 2 for all x 2 A 1 ˝ A 2 .There is a maximal C -norm on A 1 ˝ A 2 . The resulting C -tensor product iscalled maximal C -tensor product and denoted A 1 ˝max A 2 . It is characterised by thefollowing universal property:Proposition 10. There is a natural bijection between non-degenerate -homomorphismsA 1 ˝max A 2 ! B.H/ and pairs of commuting non-degenerate -homomorphismsA 1 ! B.H/ and A 2 ! B.H/; here we may replace B.H/ by any multiplier algebraM.D/ of a C -algebra D.A -representation A ! B.H/ is non-degenerate if A H is dense in H; we needthis to get representations of A 1 and A 2 out of a representation of A 1 ˝max A 2 because,for non-unital algebras, A 1 ˝max A 2 need not contain copies of A 1 and A 2 .The maximal tensor product is natural, that is, it defines a bifunctor˝max W C alg C alg ! C alg:If A 1 and A 2 are G-C -algebras, then A 1 ˝max A 2 inherits two group actions of G bynaturality; these are again strongly continuous, so that A 1 ˝max A 2 becomes a G G-C -algebra. Restricting the action to the diagonal in G G, we turn A 1 ˝max A 2 intoa G-C -algebra. Thus we get a bifunctor˝max W G-C alg G-C alg ! G-C alg:The following lemma asserts, roughly speaking, that this tensor product has thesame formal properties as the usual tensor product for vector spaces:Lemma 11. There are canonical isomorphisms.A ˝max B/ ˝max C Š A ˝max .B ˝max C/;A ˝max B Š B ˝max A;C ˝max A Š A Š A ˝max Cfor all objects of G-C alg (and, in particular, of C alg). These define a structure ofsymmetric monoidal category on G-C alg (see [35], [52]).