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30 R. MeyerDefinition 61. Let A and B be two G-C -algebras for a locally compact group G. AG-equivariant asymptotic morphism from A to B is a G-equivariant -homomorphismf W A ! Asymp.B/. We write A; B for the set of homotopy classes of G-equivariantasymptotic morphisms from A to B. Here a homotopy is a G-equivariant -homomorphism A ! Asymp C.Œ0; 1; B/ .The asymptotic algebra fits, by definition, into an extensionC 0 .R C ;B/ C b .R C ;B/ Asymp.B/:Notice that C 0 .R C ;B/ Š Cone.B/ is contractible. If f W A ! Asymp.B/ is aG-equivariant asymptotic morphism, then we can use it to pull back this extensionto an extension Cone.B/ E A in G-C alg; the G-action on E is automaticallystrongly continuous. If F is exact and homotopy invariant, then F Sus n .E/ !F Sus n .A/ is an isomorphism for all n 1 by Proposition 48. The evaluationmap C b .R C ;B/ ! B at some t 2 R C pulls back to a morphism E ! B, andthese morphisms for different t are all homotopic. Hence we get a well-defined mapF Sus n .A/ Š F Sus n .E/ ! F Sus n .B/ for each asymptotic morphism A ! B.This explains how asymptotic morphisms are related to exact homotopy functors. Thisobservation leads to the following theorem:Theorem 62. There are natural bijectionsE0 G .A; B/ Š Sus A K ˝ K.`2N/ ; Sus B K ˝ K.`2N/ for all separable G-C -algebras A; B.An important step in the proof of Theorem 62 is the Connes–Higson construction,which to an extension I E Q in C sep associates an asymptotic morphismSus.Q/ ! I .AG-equivariant generalisation of this construction is discussed in [59].Thus any extension in G-C sep gives rise to an exact triangle Sus.Q/ ! I ! E ! Qin E G .This also leads to the triangulated category structure of E G . As for KK G , we candefine it using mapping cone triangles or extension triangles – both approaches yieldthe same class of exact triangles. The canonical functor KK G ! E G is exact becauseit evidently preserves mapping cone triangles.Now that we have two bivariant homology theories with apparently very similarformal properties, we must ask which one we should use. It may seem that the betterexactness properties of E-theory raise it above KK-theory. But actually, these strongexactness properties have a drawback: for a general group G, the reduced crossedproduct functor need not be exact, so that there is no guarantee that it extends to afunctor E G ! E. Only full crossed products exist for all groups by Proposition 6; theconstruction of G Ë W E G ! E is the same as in KK-theory. Similar problems occurwith ˝min but not with ˝max .Furthermore, since KK G has a weaker universal property, it acts on more functors,so that results about KK G have stronger consequences. A good example of a functor