Selflessness for twisted group C*-algebras of amenable groups and their inclusions

Abstract

For a discrete amenable group $G$ with a two-cocycle $σ$ we first 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 the larger class of FC-hypercentral groups the same holds modulo $\mathcal{Z}$-stability, equivalently finite nuclear dimension. 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. Thus, 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.