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

For a discrete amenable group $G$ with a two-cocycle $\sigma$ we first record a few results on when the twisted group $C^*$-algebra $C^*_r(G,\sigma)$ 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 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 $C^*_r(H,\sigma')\subseteq C^*_r(G,\sigma)$, when $H$ is a normal subgroup of $G$.

For amenable $G$ such an inclusion is selfless precisely when $C^*_r(H,\sigma')$ is selfless and $(H\leq G,\sigma)$ satisfies the relative Kleppner condition.

Thus, for an infinite finitely generated virtually nilpotent $G$, selflessness of the inclusion $C^*_r(H,\sigma')\subseteq C^*_r(G,\sigma)$ is equivalent to the relative Kleppner condition.