On the structure of twisted group $C^ *$-algebras

Abstract:

We first give general structural results for the twisted group algebras ${C^{\ast } }(G,\sigma )$ of a locally compact group $G$ with large abelian subgroups. In particular, we use a theorem of Williams to realise ${C^{\ast }}(G,\sigma )$ as the sections of a ${C^{\ast }}$-bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when $\Gamma$ is a discrete subgroup of a solvable Lie group $G$, the $K$-groups ${K_ {\ast } }({C^{\ast } }(\Gamma ,\sigma ))$ are isomorphic to certain twisted $K$-groups ${K^{\ast } }(G/\Gamma ,\delta (\sigma ))$ of the homogeneous space $G/\Gamma$, and we discuss how the twisting class $\delta (\sigma ) \in {H^3}(G/\Gamma ,\mathbb {Z})$ depends on the cocycle $\sigma$. For many particular groups, such as ${\mathbb {Z}^n}$ or the integer Heisenberg group, $\delta (\sigma )$ always vanishes, so that ${K_ {\ast } }({C^{\ast } }(\Gamma ,\sigma ))$ is independent of $\sigma$, but a detailed analysis of examples of the form ${\mathbb {Z}^n} \rtimes \mathbb {Z}$ shows this is not in general the case.