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

Abstract

For a discrete amenable group $G$ with a two-cocycle $\sigma$, we record a few results on when the twisted group $C^{\ast }$-algebra $C_r^{\ast }(G,\sigma )$ is selfless, in the sense of Robert. In particular, for an infinite finitely generated virtually nilpotent $G$, this holds exactly when $(G,\sigma )$ 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 $C_r^{\ast }(H,{\sigma }^{\prime })\subseteq C_r^{\ast }(G,\sigma )$ when $H$ is a normal subgroup of $G$. For amenable $G$, such an inclusion is selfless precisely when $C_r^{\ast }(H,{\sigma }^{\prime })$ is selfless and $(H\le G,\sigma )$ satisfies the relative Kleppner condition. As a consequence, for an infinite finitely generated virtually nilpotent $G$, selflessness of the inclusion $C_r^{\ast }(H,{\sigma }^{\prime })\subseteq C_r^{\ast }(G,\sigma )$ is equivalent to the relative Kleppner condition. As applications, we obtain selfless twisted group $C^{\ast }$-algebras of the lamplighter group and of the wreath product $\mathbb{Z}\wr \mathbb{Z}$.

v2: new results outside the finitely generated setting, new examples of selfless inclusions, expanded preliminaries and updated references; 15 pages