process

120 M. Karoubihere by GrK n .A/, are more systematically used. On the other hand, since the equivariantK-theory and its relation with cohomology have been studied carefully in [17],[35] and [9], we limit ourselves to the applications in this case. One of them is thedefinition of operations in twisted K-theory in the graded and ungraded situations.Here is the contents of the paper:1. K-theory of Z=2-graded Banach algebras2. Ungraded twisted K-theory in the finite and infinite-dimensional cases3. Graded twisted K-theory in the finite and infinite-dimensional cases4. The Thom isomorphism5. General equivariant K-theory6. Some computations in the equivariant case7. Operations in twisted K-theoryAppendix. A short historical survey of twisted K-theoryAcknowledgments. I would like to thank A. Adem, A. Carey, P. Hu, I. Kriz, C. Leruste,V. Mathai, J. Rosenberg, J.-L. Tu, P. Xu and the referee for their remarks and suggestionsafter various drafts of this paper.1 K-theory of Z=2-graded Banach algebrasHigher K-theory of real or complex Banach algebras A is well known (cf. [28] or [12]for instance). Starting from the usual Grothendieck group K.A/ D K 0 .A/, there aremany equivalent ways to define “derived functors” K n .A/, for n 2 Z, such that anyexact sequence of Banach algebras0 A0 A A00 0induces an exact sequence of abelian groups::: K nC1 .A/ K nC1 .A"/ K n .A 0 / K n .A/ K n .A"/ ::::Moreover, by Bott periodicity, these groups are periodic of period 2 in the complexcase and 8 in the real case.The K-theory of Z=2-graded Banach algebras A (in the real or complex case) isless well known 6 and for the purpose of this paper we shall recall its definition whichis already present but not systematically used in [28] and [19]. We first introduce C p;qas the Clifford algebra of R pCq with the quadratic form.x 1 / 2 .x p / 2 C .x pC1 / 2 CC.x pCq / 2 :6 This is of course included in the general KK-theory of Kasparov which was introduced later than ourbasic references, at least for C*-algebras.