process

Twisted K-theory – old and new 135We now say that two such algebra bundles A and A 0 are equivalent if there existtwo trivial algebra bundles T and T 00 such that A ˝ T and A 0 ˝ T 00 are isomorphicas G-bundles of algebras. The quotient is a group since the dual of A is its inverse viathe tensor product of principal PU.H /-bundles. This is the “equivariant Brauer group”Br G .X/.A closely related definition (in the framework of C*-algebras and for a locallycompact group G) is given in [17]. It is very likely that it coincides with this onefor compact Lie groups, in the light of an interesting filtration described in this paper,probably associated to a spectral sequence.Theorem 5.2 (cf. [8], Proposition 6.3). Let X Gassociated to X. Then the natural mapD EG G X be the Borel spaceBr G .X/ ! Br.X G /is an isomorphism.The interest of this theorem lies in the fact that the equivariant K-theoriesK G ..X; A// and K G ..X; A 0 // are isomorphic if A and A 0 are equivalent. Thisfollows from the well-known Morita invariance in operator K-theory. We shall studyconcrete applications of this principle in the next section.6 Some computations of twisted equivariant K-groupsLet us look at the particular case of the ungraded twisted K-groups K .A/G.X/ where Gis a finite group acting on the trivial bundle of algebras A D X M n .C/ via a grouphomomorphism G ! PU.n/. 24 We define zG as the pull-back diagramzGGSU.n/ PU.n/Therefore, zG is a central covering of G with fiber n (whose elements are denoted byGreek letters such as ). The following definition is already present in [31], §2.5 (forn D 2):Definition 6.1. A finite-dimensional representation of zG is of “linear type” if .u/ D.u/ for any 2 n . We now consider the category E A QG .X/ whose objects arelzG-A-modules as before, except that we request that the zG-action be of linear type andcommutes with the action of A. By Morita invariance, E A QG .X/ is equivalent to thelcategory EG Q .X/ l of finite-dimensional zG-bundles on X, the action of zG on the fibersbeing of linear type.24 However, we don’t assume that G acts trivially on X in general.