An Equivariant Brauer Group and Actions of Groups on C*-algebras

Suppose that $(G,T)$ is a second countable locally compact transformation group given by a homomorphism $\mathrm{\ell }:G\to \text{Homeo}(T)$, and that $A$ is a separable continuous-trace \cs-algebra with spectrum $T$. An action $\alpha :G\to \text{Aut}(A)$ is said to cover $\mathrm{\ell }$ if the induced action of $G$ on $T$ coincides with the original one. We prove that the set $\text{brgt}$ of Morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product: $[A,\alpha ][B,\beta ]=[A\text{Ttensor}B,\alpha \text{tensor}\beta ]$, and we refer to $\text{brgt}$ as the Equivariant Brauer Group. We give a detailed analysis of the structure of $\text{brgt}$ in terms of the Moore cohomology of the group $G$ and the integral cohomology of the space $T$. Using this, we can characterize the stable continuous-trace \cs-algebras with spectrum $T$ which admit actions covering $\mathrm{\ell }$. In particular, we prove that if $G=\mathbb{R}$, then every stable continuous-trace \cs-algebra admits an (essentially unique) action covering~$\mathrm{\ell }$, thereby substantially improving results of Raeburn and Rosenberg. Versions of this paper in *.dvi and *.ps form are available via World wide web servers at http://coos.dartmouth.edu/~dana/dana.html