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4 R. MeyerProposition 6. The full crossed product functor G Ë W G-C alg ! C alg is exactin the sense that it maps extensions in G-C alg to extensions in C alg.Proof. This is Lemma 4.10 in [21].Definition 7. A locally compact group G is called exact if the reduced crossed productfunctor G Ë r W G-C alg ! C alg is exact.Although this is not apparent from the above definition, exactness is a geometricproperty of a group: it is equivalent to Yu’s property (A) or to the existence of anamenable action on a compact space [43].Most groups you know are exact. The only source of non-exact groups known at themoment are Gromov’s random groups. Although exactness might remind you of thenotion of flatness in homological algebra, it has a very different flavour. The differenceis that the functor G Ë r always preserves injections and surjections. What may gowrong for non-exact groups is exactness in the middle (compare the discussion beforeProposition 18). Hence we cannot study the lack of exactness by derived functors.Even for non-exact groups, there is a class of extensions for which reduced crossedproducts are always exact:Definition 8. A section for an extensionI i E p Q (2)in G-C alg is a map (of sets) s W Q ! E with p ıs D id Q . We call (2) split if there is asection that is a G-equivariant -homomorphism. We call (2) G-equivariantly cp-splitif there is a G-equivariant, completely positive, contractive, linear section.Sections are also often called lifts, liftings,orsplittings.Proposition 9. Both the reduced and the full crossed product functors map split extensionsin G-C alg again to split extensions in C alg and G-equivariantly cp-splitextensions in G-C alg to cp-split extensions in C alg.Proof. Let K i E p Q be an extension in G-C alg. Proposition 6 shows thatG Ë K G Ë E G Ë Q is again an extension. Since reduced and full crossedproducts are functorial for equivariant completely positive contractions, this extensionis split or cp-split if the original extension is split or equivariantly cp-split, respectively.This yields the assertions for full crossed products.Since a -homomorphism with dense range is automatically surjective, the inducedmap G Ë r p W G Ë r E ! G Ë r Q is surjective. It is evident from the definition ofreduced crossed products that G Ë r i is injective. What is unclear is whether therange of G Ë r i and the kernel of G Ë r p coincide. As for the full crossed product, aG-equivariant completely positive contractive section s W Q ! E induces a completelypositive contractive section G Ë r s for G Ë r p. The linear map' WD id GËr E.G Ë r s/ ı .G Ë r p/W G Ë r E ! G Ë r E