process

Twisted K-theory – old and new 133K n .A/ with n D q p), for instance Bott periodicity. The following key lemmaenables us to translate many general theorems of K-theory into the equivariant framework:Lemma 5.1. Let E be an object of the category P G .A/. Then E is a direct summandof an object of the type A ˝C M where M is a finite-dimensional G-module.Proof. ([39], p. 134). Let us consider the union of all finite-dimensional invariantsubspaces of the G-Banach space E. According to a version of the Peter-Weyl theoremquoted in [39], this union is dense in A. We now consider a set e 1 ;:::;e n of generatorsof E as an A-module. Since is dense in A and E is projective, one may choosethese generators in the subspace . Let M 1 ;:::;M n be finite-dimensional invariantsubspaces of E containing e 1 ;:::;e n respectively and let M be the following directsum:M D M 1 ˚˚M nWe define an equivariant surjection W A ˝C M Š .A ˝C M 1 / ˚˚.A ˝C M n / ! E:between projective left A-modules by the formula. 1 ˝ m1;:::; n ˝ m n / D 1 m 1 CC n m n :This surjection admits a section, which we can average out thanks to a Haar measurein order to make it equivariant. Therefore, E is a direct summand in A ˝C M as statedin the lemma.As for usual K-theory, one may also define equivariant K-theory for non unitalrings and, using Lemma 5.2 above, prove that any equivariant exact sequence of ringson which G acts0 A0 A A00 0induces an exact sequence of equivariant K-groupsK n 1G .A/ K n 1G .A00 / K n G .A0 / K n G .A/ K n G .A00 /and an analogous exact sequence in the graded equivariant framework.After these generalities, let us assume that G acts on a compact space X and let Abe a bundle of algebras modelled on K. We define the (ungraded) equivariant twistedK-group K .A/G.X/ as K G.A/, where A is the Banach algebra of sections of the bundleA. Similarly, if A is a bundle of graded algebras modelled on K K or M 2 .K/, wedefine the (graded) equivariant twisted K-group KG A.X/ as GrK G.A/, where A is thegraded Banach algebra of sections of A.As seen in §3, it is also natural to consider free graded B-modules together with acontinuous action of G (compatible with the action on X) and a family of Fredholmoperators D which are self-adjoint of degree 1, commuting with the action of G. With