On the structure of twisted group 𝐶*-algebras

Abstract

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