process

8 R. MeyerProposition 18. The functor ˝max is exact in each variable, that is, ˝max D mapsextensions in G-C alg again to extensions for each D 22 G-C alg.Both ˝min and ˝max map split extensions to split extensions and (equivariantly)cp-split extensions again to (equivariantly) cp-split extensions.The proof is similar to the proofs of Propositions 6 and 9.If A or B is nuclear, we simply write A ˝ B for A ˝max B Š A ˝min B.2.4 Limits and colimitsProposition 19. The categories C alg and G-C alg are bicomplete, that is, any(small) diagram in these categories has both a limit and a colimit.Proof. To get general limits and colimits, it suffices to construct equalisers and coequalisersfor pairs of parallel morphisms f 0 ;f 1 W A B, direct products and coproductsA 1 A 2 and A 1 t A 2 for any pair of objects and, more generally, for arbitrary setsof objects.The equaliser and coequaliser of f 0 ;f 1 W A B areker.f 0f 1 / Dfa 2 A j f 0 .a/ D f 1 .a/g Aand the quotient of A 1 by the closed -ideal generated by the range of f 0 f 1 , respectively.Here we use that quotients of C -algebras by closed -ideals are againC -algebras. Notice that ker.f 0 f 1 / is indeed a C -subalgebra of A.The direct product A 1 A 2 is the usual direct product, equipped with the canonicalC -algebra structure. We can generalise the construction of the direct product to infinitedirect products: let Q i2I A i be the set of all norm-bounded sequences .a i / i2I witha i 2 A i for all i 2 I ; this is a C -algebra with respect to the obvious -algebra structureand the norm k.a i /k WD sup i2I ka i k. It has the right universal property because any -homomorphism is norm-contracting. (A similar construction with Banach algebraswould fail at this point.)The coproduct A 1 tA 2 is also called free product and denoted A 1 A 2 ; its constructionis more involved. The free C-algebra generated by A 1 and A 2 carries a canonicalinvolution, so that it makes sense to study C -norms on it. It turns out that there is amaximal such C -norm. The resulting C -completion is the free product C -algebra.In the equivariant case, A 1 t A 2 inherits an action of G, which is strongly continuous.The resulting object of G-C alg has the correct universal property for a coproduct.An inductive system of C -algebras .A i ;˛ji / i2I is called reduced if all the maps˛ji W A i ! A j are injective; then they are automatically isometric embeddings. Wemay as well assume that these maps are identical inclusions of C -subalgebras. Thenwe can form a -algebra S A i , and the given C -norms piece together to a C -normon S A i . The resulting completion is lim .A ! i ;˛ji/. In particular, we can construct aninfinite coproduct as the inductive limit of its finite sub-coproducts. Thus we get infinitecoproducts.