Abstract: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.
From: Tron Omland [view email]
[v1]
Thu, 25 Jun 2026 15:58:37 UTC (10 KB)
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