Selflessness for twisted group C*-algebras of amenable groups and their inclusions
For a discrete amenable group $G$ with a two-cocycle $σ$, we record a few results on when the twisted group $C^*$-algebra $C^*_r(G,σ)$ is selfless, in the sense of Robert. In particular, for an infinite finitely generated virtually nilpotent $G$, this holds exactly when $(G,σ)$ satisfies Kleppner's condition. For countably infinite FC-hypercentral groups, selflessness is equivalent to Kleppner's condition together with either $\mathcal{Z}$-stability or, equivalently, finite nuclear dimension. We also settle one case outside the finitely generated setting, namely the free abelian group of countably infinite rank with a particular root-of-unity-valued cocycle, where the associated algebra is selfless and has nuclear dimension one. Further, using the relative Kleppner condition, we obtain corresponding selflessness results for inclusions $C^*_r(H,σ')\subseteq C^*_r(G,σ)$ when $H$ is a normal subgroup of $G$. For amenable $G$, such an inclusion is selfless precisely when $C^*_r(H,σ')$ is selfless and $(H\leq G,σ)$ satisfies the relative Kleppner condition. As a consequence, for an infinite finitely generated virtually nilpotent $G$, selflessness of the inclusion $C^*_r(H,σ')\subseteq C^*_r(G,σ)$ is equivalent to the relative Kleppner condition. As applications, we obtain selfless twisted group $C^*$-algebras of the lamplighter group and of the wreath product $\mathbb{Z}\wr\mathbb{Z}$.