An Equivariant Brauer Group and Actions of Groups onC*-Algebras

AbstractSuppose that (G, T) is a second countable locally compact transformation group given by a homomorphism ℓ:G→Homeo(T), and thatAis a separable continuous-traceC*-algebra with spectrumT. An actionα:G→Aut(A) is said to cover ℓ if the induced action ofGonTcoincides with the original one. We prove that the set BrG(T) of Morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product: [A, α][B, β]=[A⊗C0(T)B, α⊗β], and we refer to BrG(T) as the Equivariant Brauer Group. We give a detailed analysis of the structure of BrG(T) in terms of the Moore cohomology of the groupGand the integral cohomology of the spaceT. Using this, we can characterize the stable continuous-traceC*-algebras with spectrumTwhich admit actions covering ℓ. In particular, we prove that ifG=R, then every stable continuous-traceC*-algebra admits an (essentially unique) action covering ℓ, thereby substantially improving results of Raeburn and Rosenberg