process

14 R. MeyerProof. We only mention two facts that are needed for the proof. First, we haveev t ı const D id A and const ı ev t id C.Œ0;1;A/ for all A 22 S and all t 2 Œ0; 1.Secondly, an isomorphism has a unique left and a unique right inverse, and these areagain isomorphisms.The equivalence of (d) and (f) in Lemma 28 says that Ho.S/ is the localisation of Sat the family of homotopy equivalences.Since C Œ0; 1; C 0 .X/ Š C 0 .Œ0; 1 X/ for any locally compact space X, ournotion of homotopy restricts to the usual one for pointed compact spaces. Hence theopposite of the homotopy category of pointed compact spaces is equivalent to a fullsubcategory of Ho.C alg/.The sets ŒA; B inherit a base point Œ0 and a quotient topology from Hom.A; B/;thus Ho.S/ is enriched over pointed topological spaces as well. This topology onŒA; B is not so useful, however, because it need not be Hausdorff.A similar topology exists on Kasparov groups and can be defined in various ways,which turn out to be equivalent [12].Let F W G-C alg ! H -C alg be a functor with natural isomorphismsF C.Œ0; 1; A/ Š C.Œ0; 1; F .A/ that are compatible with evaluation maps for all A. The universal property implies that Fdescends to a functor Ho.G-C alg/ ! Ho.H -C alg/. In particular, this applies tothe suspension, cone, and cylinder functors and, more generally, to the functors A˝maxand A˝min on G-C alg for any A 22 G-C alg because both tensor product functorsare associative and commutative andC.Œ0; 1; A/ Š C.Œ0; 1/ ˝max A Š C.Œ0; 1/ ˝min A:The same reasoning applies to the reduced and full crossed product functors G-C alg !C alg.We may stabilise the homotopy category with respect to the suspension functor andconsider a suspension-stable homotopy category with morphism spacesfA; Bg WD lim !k!1ŒSus k A; Sus k Bfor all A; B 22 G-C alg. We may also enlarge the set of objects by adding formaldesuspensions and generalising the notion of spectrum. This is less interesting forC -algebras than for spaces because most functors of interest satisfy Bott Periodicity,so that suspension and desuspension become equivalent.3.2 Morita–Rieffel equivalence and stable isomorphism. One of the basic ideas ofnon-commutative geometry is that G Ë C 0 .X/ (or G Ë r C 0 .X/) should be a substitutefor the quotient space GnX, which may have bad singularities. In the special case of afree and proper G-space X, we expect that G Ë C 0 .X/ and C 0 .GnX/ are “equivalent”