process

Twisted K-theory – old and new 143Remark 7.2. If we take a bundle of finite dimensional algebras modelled on A DEnd.E/ where E D C r , there is another way to check (functorially) this untwisting ofthe action of S n on A˝n : we identify A˝n with End.E˝n / and .A˝n / with Aut.E˝n /.We have the following commutative diagram:S nEnd.E˝n / D Aut.E˝n / End.E˝n / D End.E/˝n .The vertical map sends the invertible element ˛ to the automorphism .u 7! ˛u˛ 1/.If is a permutation, the horizontal map sends to the automorphismu 1 ˝˝u n 7! u .1/ ˝˝u .n/while the map sends to the automorphism of E˝n defined byx 1 ˝˝x n 7! x .1/ ˝˝x .n/ :Finally, the composition , computed on a decomposable tensor of E˝n , gives therequired result:x 1 ˝˝x n x .1/ ˝˝x .n/ u u 1 .x .1/ / ˝˝u n .x .n/ /As was shown in [2], a Z-module map 1 u .1/ .x 1 / ˝˝u .n/ .x n /.R.S n / Zdefines an operation in twisted K-theory by taking the composite of the following maps:K .A/ .X/ K .A˝n /S n.X/ Š K .A˝n / 0S n.X/ Š K .A/˝n .X/ ˝ R.S n / K .A/˝n .X/.This is essentially 27 what was done in [9], §10, in order to define the n operation ofGrothendieck in this context for instance.Let us call .F; r/ or even F (for short) a representative of the image of .E; D/ bythe composite of the maps (we now use the Fredholm description of twisted K-theory)K .A/ .X/ K .A˝n /S n.X/ K .A˝n / 0.X/.We can define the Adams operations ‰ n with the method described in [19] (which weintend to generalize later on). For this, we restrict the action of S n to the cyclic group27 The second homomorphism was not explicitly given however.A n