process

Twisted K-theory – old and new 123Now, a bigger group PU.H / D U.H /=S 1 is acting on K by inner automorphisms. Ifwe take a Čech cocycleg ji W U i \ U j ! PU.H /we may use it to construct a bundle A of (non unital) C*-algebras with fiber K.Let us now consider the commutative diagramS 1 S 1 U.n/U.H/PU.n/ PU.H /.Thanks to Kuiper’s theorem [34], we remark that the classifying space of U.H/ is contractible.Therefore, the classifying space BPU.H / of the topological group PU.H /,isa nice model of the Eilenberg–Mac Lane space K.Z;3/(compare with the well-knownpaper of Dixmier and Douady [18]). Moreover, if we start with a finite-dimensionalalgebra bundle A over X with fiber M n .C/, the diagram above shows how to associateto A another bundle of algebras A 0 with fiber K, together with a C*-inclusionfrom A to A 0 . We note that the invariant W 3 .A/ in Br.X/ D Tors.H 3 .XI Z// definedin [25] is simply induced by the classifying map from X to BPU.H / (which factorsthrough BPU.n/). In this finite example, it is an n-torsion class since one has anothercommutative diagram n S1SU.n/PU.n/ U.n/PU.n/.Theorem 2.2. The inclusion from A to A 0 induces an isomorphismK r .A/ D K r ..X; A// ! K r ..X; A 0 // D K r .A 0 /where the K r define the classical topological K-theory of C*-algebras.Proof. The proof is classical for a trivial algebra bundle, since C is Morita equivalentto K (in the C*-algebra sense). It extends to the general case by a no less classicalMayer–Vietoris argument.Definition 2.3. Let now A be an algebra bundle with fiber K on a compact space Xwith structural group PU.H /. We define K .A/ .X/ (also denoted by K.A/) astheK-theory of the (non unital) Banach algebra .X;A/. This K-theory only depends of