process

Twisted K-theory – old and new 119“cup-product”K˛.X/ K˛0.X/ ! K˛C˛0.X/as mentioned earlier (even if ˛ and ˛0 are in the much smaller group Z=2).To correct this defect, a second definition may be given in terms of Fredholmoperators in a Hilbert space. More precisely, we consider graded Hilbert bundles Ewhich are also graded A-modules in an obvious sense, together with a continuousfamily of Fredholm operatorsD W E ! Ewith the following properties:1. D is self-adjoint of degree 1,2. D commutes with the action of A (in the graded sense).One gets an abelian semi-group from the homotopy classes of such pairs .E; D/, withthe addition rule.E; D/ C .E 0 ;D 0 / D .E ˚ E 0 ;D˚ D 0 /:The associated group gives the second definition of K˛.X/ which is equivalent to thefirst one (see [19], p. 18, and [29], p. 88). We use also the notation K A .X/ instead ofK˛.X/ where ˛ is the class of A in GBr.X/ when we want to be more explicit.With this new viewpoint, the cup-product alluded to above becomes obvious. It isdefined by the following formula 5 :.E; D/ Y .E 0 ;D 0 / D .E y˝ E 0 ;D y˝ 1 C 1 y˝ D 0 /where the symbol y˝ denotes the graded tensor product of bundles or morphisms. It is amap from K A .X/ K A0 .X/ to K A y˝A 0 .X/ and therefore it induces a (non canonical)map from K˛.X/ K˛0.X/ to K˛C˛0.X/.For simplicity’s sake, we have only considered complex K-theory. We could aswell study the real case: one has to replace GBr.X/ byGBrO.X/ D Z=8 H 1 .XI Z=2/ H 2 .XI Z=2/:If we take for the coefficient system ˛ D n to be in Z=8, we get the usual groupsKO n .X/ as defined using Clifford algebras in [28] and [29], p. 88 (these groups beingwritten xK n in the later reference).In this paper, we essentially follow the same pattern, but with bundles of infinitedimension in the spirit of [38]. As a matter of fact, all the technical tools are alreadypresent in [19], [29] and [38], for instance the Fredholm operator machinery which isnecessary to define the cup product. However, this paper is not a rewriting of thesepapers, since we take a more synthetic view point and have other applications in mind.For instance, the K-theory of Banach algebras K n .A/ and its graded version, denoted5 See the appendix about the origin of such a formula.