process

18 R. Meyercorrespondence: the isomorphism is given by left multiplication with u, which definesa G-equivariant unitary operator from B to the closure of uBu B D u B.Hence inner endomorphisms act trivially on C -stable functors. Actually, this isone of the computations that we have omitted in the proof above; the argument can befound in [11].Now we make the definition of a correspondence more concrete if A is unital. Wehave an essential -homomorphism ' W A ! K.E/ for some G-equivariant HilbertB-module E. Since A is unital, this means that K.E/ is unital and ' is a unital -homomorphism. Then E is finitely generated. Thus E D B 1 p for some projectionp 2 M 1 .B/ and ' is a -homomorphism ' W A ! M 1 .B/ with '.1/ D p. Two -homomorphisms ' 1 ;' 2 W A M 1 .B/ yield isomorphic correspondences if andonly if there is a partial isometry v 2 M 1 .B/ with v' 1 .x/v D ' 2 .x/ and v ' 2 .a/v D' 1 .a/ for all a 2 A.Finally, we combine homotopy invariance and C -stability and consider the universalC -stable homotopy-invariant functor. This functor is much easier to characterise:the morphisms in the resulting universal category are simply the homotopy classes ofG-equivariant -homomorphisms K L 2 .G N/ ˝ A ! K L 2 .G N/ ˝ B/ (see[36], Proposition 6.1). Alternatively, we get the same category if we use homotopyclasses of correspondences A Ü B instead.3.4 Exactness properties. Throughout this subsection, we consider functors F W S !C with values in an exact category C. If C is merely additive to begin with, we canequip it with the trivial exact category structure for which all extensions split. We alsosuppose that S is closed under the kinds of C -algebra extensions that we consider;depending on the notion of exactness, this means: direct product extensions, split extensions,cp-split extensions, or all extensions, respectively. Recall that split extensionsin G-C alg are required to split by a G-equivariant -homomorphism.3.4.1 Additive functors. The most trivial split extensions in G-C alg are the productextensions A A B B for two objects A; B. In this case, the coordinateembeddings and projections provide mapsA A B B: (7)Definition 41. We call F additive if it maps product diagrams (7) in S to direct sumdiagrams in C.There is a partially defined addition on -homomorphisms: call two parallel -homomorphisms'; W A B orthogonal if '.a 1 / .a 2 / D 0 for all a 1 ;a 2 2 A. Equivalently,' C W a 7! '.a/ C .a/ is again a -homomorphism.Lemma 42. The functor F is additive if and only if, for all A; B 22 S, the mapsHom.A; B/ ! C F .A/; F .B/ satisfy F.' C / D F.'/ C F. / for all pairs oforthogonal parallel -homomorphisms '; .