The equivariant brauer group and twisted transformation group $C^{\ast}$-algebras

Abstract

Twisted transformation group $C^{\ast}$-algebras associated to locally compact dynamical systems $(X = Y/N,G)$ are studied, where $G$ is abelian, $N$ is a closed subgroup of $G$, and $Y$ is a locally trivial principal $G$-bundle over $Z = Y/G$. An explicit homomorphism between $H^{2}(G,C(X,\mathbb{T}))$ and the equivariant Brauer group of Crocker, Kumjian, Raeburn and Williams, $Br_{N}(Z)$, is constructed, and this homomorphism is used to give conditions under which a twisted transformation group $C^{\ast}$-algebra $C_{0}(X) \times_{\tau,\omega}G$ will be strongly Morita equivalent to another twisted transformation group $C^{\ast}$-algebra $C_{0}(Z) \times_{Id,\omega}N$. These results are applied to the study of twisted group $C^{\ast}$-algebras $C^{\ast}(\Gamma,\mu)$ where $\Gamma$ is a finitely generated torsion free two-step nilpotent group.