process

Twisted K-theory – old and new 125Proof. As we have seen above, the two spaces .X;Fredh.H // and .X;.B=A/ /have the same homotopy type. On the other hand, it is a well known consequence ofKuiper’s theorem [34] than the (non unital) ring map .X;B=A/ ! .X;M r .B=A//induces a bijection between the path components of the associated groups of invertibleelements (see for instance [32], p. 93). Therefore, 0 ..X; Fredh.H /// may beidentified withlim 0 ..X; GL r .B=A/// D K 1 .B=A/!rand therefore with K.A/, as we already mentioned in 2.6.Remark 2.5. We may also consider the following “stabilized” bundleFredh s .H / D lim! nFredh.H n /and, without Kuiper’s theorem, prove in the same way that the set of connected componentsof the space of sections of this bundle is isomorphic to K .A/ .X/.There is an obvious ring homomorphism (since the Hilbert tensor product H ˝ His isomorphic to H )K ˝ K KIf A and A 0 are bundles of algebras on X modelled on K, we may use this homomorphismto get a new algebra bundle on X, which we denote by A ˝ A 0 . From thecocycle point of view, we have a commutative diagram, where the top arrow is inducedby complex multiplicationS 1 S 1U.H/ U.H/PU.H / PU.H / S1U.H ˝ H/ PU.H ˝ H/.It follows that W 3 .A ˝ A 0 / D W 3 .A/ C W 3 .A 0 / in H 3 .XI Z/ and one gets a “cupproduct”K .˛/ .X/ K .˛0/ .X/ K .˛C˛0/ .X/.(well defined up to non canonical isomorphism: see 2.1). This is a particular case of a“graded cup-product” which will be introduced in the next section.3 Graded twisted K-theory in the finite and infinite-dimensionalcasesWe are going to change our point of view and now consider Z=2-graded finite-dimensionalalgebras which are central and simple (in the graded sense). We are only inter-