process

16 R. MeyerTheorem 34 ([7]). Two -unital G-C -algebras are G-equivariantly Morita–Rieffelequivalent if and only if they are stably isomorphic.In the non-equivariant case, this theorem is due to Brown–Green–Rieffel [7]. Asimpler proof that carries over to the equivariant case appeared in [41].3.3 C -stable functors. The definition of C -stability is more intuitive in the nonequivariantcase:Definition 35. Fix a rank-one projection p 2 K.`2N/. The resulting embeddingA ! A ˝ K.`2N/, a 7! a ˝ p, is called a corner embedding of A.A functor F W C alg ! C is called C -stable if any corner embedding induces anisomorphism F .A/ Š F A ˝ K.`2N/ .The correct equivariant generalisation is the following:Definition 36 ([36]). A functor F W G-C alg ! C is called C -stable if the canonicalembeddings H 1 ! H 1 ˚ H 2 H 2 induce isomorphismsF A ˝ K.H 1 / Š! F A ˝ K.H1 ˚ H 2 / ŠF A ˝ K.H2 / for all non-zero G-Hilbert spaces H 1 and H 2 .Of course, it suffices to require F A ˝ K.H 1 / Š! F A ˝ K.H1 ˚ H 2 / .Itisnot hard to check that Definitions 35 and 36 are equivalent for trivial G.Remark 37. We have argued in §3.2 why C -stability is an essential property for anydecent homology theory for C -algebras. Nevertheless, it is tempting to assume lessbecause C -stability together with split-exactness has very strong implications.One reasonable way to weaken C -stability is to replace K.`2N/ by M n for n 2 Nin Definition 35 (see [57]). If two unital C -algebras are Morita–Rieffel equivalent,then they are also Morita equivalent as rings, that is, the equivalence is implementedby a finitely generated projective module. This implies that a matrix-stable functor isinvariant under Morita–Rieffel equivalence for unital C -algebras.Matrix-stability also makes good sense in G-C alg for a compact group G: simplyrequire H 1 and H 2 in Definition 36 to be finite-dimensional. But we seem to runinto problems for non-compact groups because they may have few finite-dimensionalrepresentations and we lack a finite-dimensional version of the equivariant stabilisationtheorem.Our next goal is to describe the universal C -stable functor. We abbreviate A K WDK.L 2 G/ ˝ A.Definition 38. A correspondence from A to B (or A Ü B) isaG-equivariantHilbert B K -module E together with a G-equivariant essential (or non-degenerate) -homomorphism f W A K ! K.E/.