process

144 M. KaroubiZ=n identified with the group of n th roots of the unity. Let us call F r the subbundle ofF where the action of Z=n is given by ! r , ! being a fixed primitive root of the unity.Then ‰ n .E; D/ is defined by the following sum:Xn 1‰ n .E; D/ D F r ! r :It belongs formally to K .A/˝n .X/˝Z n where n is the ring of n-cyclotomic integers.However, if n is prime, using the action of the symmetric group S n , it is easy to checkthat F r is isomorphic to F 1 if r ¤ 1. Therefore we end up in K .A/˝n .X/, consideredas a subgroup of K .A/˝n .X/ ˝ n , as it was expected. It is essentially proved in [2]Zthat this definition of ‰ n agrees with the classical one.There is another operation in twisted K-theory which is “complex conjugation”,classically denoted by ‰ 1 , which maps K .A/ .X/ to K .A/ 1 .X/ (if we write multiplicativelythe group law in Br.X/). It is shown in [9], §10, how we can combinethis operation with the previous ones in order to get “internal” operations, i.e. mappingK .A/ .X/ to itself.It is more tricky to define operations in graded twisted K-theory. If ƒ is a Z=2-graded algebra, it is no longer true in general that a graded involution of ƒ is induced byan inner automorphism with an element of degree 0 and of order 2. A typical exampleis the Clifford algebra.C ˚ C/ y˝2 D .C 0;1 / y˝2which may be identified with the graded algebra M 2 .C/. Ifweput 0 10 ie 1 D and e1 02 Di 0we see that there is no inner automorphism by an element of order 2 and degree 0permuting e 1 and e 2 .In order to define such operations in the graded case, we may proceed in at least twoways.We first remark that if A 0 is an oriented bundle modelled on M 2 .K/, the groupsK .A0/ .X/ and K A0 .X/ are isomorphic (see 3.5). Moreover, the ungraded tensor productA 0˝n is isomorphic to the graded one A 0 y˝n (since M 2 .C/ y˝ M 2 .C/ is isomorphic toM 4 .C/ D M 2 .C/ ˝ M 2 .C//. Let us now assume that the bundle of algebras A ismodelled on K K but not necessarily oriented and let L be the orientation real linebundle of A. Then A 0 D A y˝ C.L/ is oriented modelled on M 2 .K/ (C.V / denotesin general the Clifford bundle associated to V ). We may apply the previous methodto define operations from K .A0/ .Y / D K A0 .Y / to Ky˝n.Y A0 /. If we apply the Thomisomorphism to the vector bundle Y D L with basis X (see §4), we deduce (for nodd) operations from K A .X/ to K A y˝n.X/. However, if A is oriented, we loose someinformation since what we get are essentially the previous operations applied to thesuspension of X. Note also that the same method may be applied to K A .X/, when Ais modelled on M 2 .K/.0