process

142 M. KaroubiWe would like to point out also that the theory K˙.X/ introduced recently byAtiyahand Hopkins [7] is a particular case of twisted equivariant K-theory. As a matter offact, it was explicitly present in [28], §3, 40 years ago, before the formal introductionof twisted K-theory. Here is a detailed explanation of this identification.According to [7], the definition of K˙.X/ (in the complex or real case) is thegroup K Z=2 .X R 8 /, where Z=2 acts on X and also on R 8 D R R 7 by .; / 7!. ;/. According to the Thom isomorphism in equivariant K-theory (proved inthe non spinorial case in [30]), it coincides with an explicit graded twisted K-groupK A Z=2 .X/, as defined in [28]. Here A is the Clifford algebra C.R2 / D C 1;1 of R 2provided with the quadratic form x 2 y 2 and where Z=2 acts via the involution.; / ! . ;/ on R R (this is also mentioned briefly in [7], p. 2, footnote 1).This identification is valid as well in the real framework, where we have 8-periodicity.These groups K A .X/ were considered in [28], §3.3, in a broader context: A mayZ=2be any Clifford algebra bundle C.V / (where V is a real vector bundle provided witha non degenerate quadratic form) and Z=2 may be replaced by any compact Lie groupacting in a coherent way on X and V . The paper [31] gives a method to computethese equivariant twisted K-groups. As quoted in the appendix, the real and complexself-adjoint Fredholm descriptions (for the non twisted case) which play an importantrole in [7] were considered independently in [10] and [32].7 Operations on twisted K-groupsThis section is a partial synthesis of [19] and [9].Let us start with the simple case of bundles of (ungraded) infinite C*-algebrasmodelled on K, like in [9]. As it was shown in [2] and [19], we have a n th power mapP W K A .X/ ! K .A˝n /S n.X/where the symmetric group S n acts on A˝n by permutation of the factors.Lemma 7.1. 26 The group K .A˝n /S n.X/ is isomorphic to the group K .A˝n / 0S n.X/ wherethe symbol 0 means that S n is acting trivially on A˝nProof. As we have shown many times in §6, this “untwisting” of the action of thesymmetric group on A˝n is due to the following fact: the standard representationS n ! PU.H ˝n /can be lifted into a representation W S n ! U.H˝n / in a way compatible with thediagonal action of elements of PU.H /, a fact which is obvious to check.26 We should note that this lemma is not true for K Sn .A˝n / for a general noncommutative ring A.Therefore, it is not possible to define -operations in this case. Twisted K-theory is somehow intermediarybetween the commutative case and the noncommutative one.