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

Abstract

Suppose that $(G,T)$ is a second countable locally compact transformation group given by a homomorphism $\ell:G\to\Homeo(T)$, and that $A$ is a separable continuous-trace \cs-algebra with spectrum $T$. An action $\alpha:G\to\Aut(A)$ is said to cover $\ell$ if the induced action of $G$ on $T$ coincides with the original one. We prove that the set $\brgt$ of Morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product: $[A,\alpha][B,\beta] = [A\Ttensor B,\alpha\tensor\beta]$, and we refer to $\brgt$ as the Equivariant Brauer Group. We give a detailed analysis of the structure of $\brgt$ 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 $\ell$. In particular, we prove that if $G=\R$, then every stable continuous-trace \cs-algebra admits an (essentially unique) action covering~$\ell$, 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