process

124 M. Karoubithe class ˛ of A in H 3 .XI Z/ and we shall also call it K .˛/ .X/ [Due to 2.4, this is ageneralization of definition 2.2].Before treating the graded case in the next section, we would like to give an equivalentdefinition of K.A/ in terms of Fredholm operators, as was done in [19] for thetorsion elements in H 3 .XI Z/ and in [8] for the general case. The basic idea is to remarkthat PU.H / acts not only on the C*-algebra of compact operators in H , but alsoon the ring of bounded operators End.H / and on the Calkin algebra End.H /=K (withthe norm topology 9 )). Let us call B the algebra bundle with fiber End.H / associatedto the cocycle defined in 2.1 and B=A the quotient algebra bundle. Therefore, we havean exact sequence of C*-algebras bundles0 A B B=A 0.which induces an exact sequence for the associated rings of sections (thanks to a partitionof unity again)0 .X;A/ .X;B/ .X;B=A// 0.If B is trivial, it is well known that the algebra of continuous maps from X to End.H /has trivial K n -groups because this algebra is flabby 10 . By a Mayer–Vietoris argument,it follows that K n ..X; B// is also trivial. Therefore the connecting homomorphismK 1 .B=A/ D K 1 ..X; B=A// ! K 0 ..X; A// K .A/ .X/ D K.A/is an isomorphism, a well-known observation in index theory.Let us now consider the elements of B which map onto .B=A/ via the map .These elements form a bundle of Fredholm operators on H (the twist comes from theaction of PU.H /). This subbundle of B will be denoted by Fredh.H /. Therefore, wehave a principal fibration.X;A/ .X;Fredh.H // .X;.B=A/ /with contractible fiber the Banach space .X;A/ (this fibration admits a local sectionthanks to Michael’s theorem [37]). Therefore, the space of sections of Fredh.H /has the homotopy type of .X;.B=A/ /. In particular, the path components are inbijective correspondence via the map . The following theorem is a generalization ofa well-known theorem of Atiyah and Jänich [2], [27]:Theorem 2.4 ([8]). The set of homotopy classes of continuous sections of the fibrationFredh.H / Xis naturally isomorphic to K .A/ .X/.9 other topologies may be also considered, see [8].10 A unital Banach algebra ƒ is called flabby if there exists a continuous functor from P .ƒ/ to itselfsuch that C Id is isomorphic to . For instance, ƒ D End.H / is flabby since P .A/ is equivalent tothe category of Hilbert spaces which are isomorphic to direct factors in H ; is then defined by the infiniteHilbert sum .E/ D E ˚˚E ˚ :