Year: 2026
Type: preprint
Abstract: For a discrete amenable group $G$ with a two-cocycle $σ$, we study when the twisted group $C^*$-algebra $C^*_r(G,σ)$ is selfless, in the sense of Robert. For a countably infinite abelian group $G$, this holds exactly when the commutator bicharacter of $σ$ is nondegenerate, that is, under Kleppner's condition. The nuclear dimension is then zero if $G$ is torsion and one otherwise. We show that root-of-unity-valued cocycles can give selfless algebras of nuclear dimension one, using slow dimension growth. The characterization extends to countably infinite virtually abelian or locally finite FC-hypercentral groups and to infinite finitely generated virtually nilpotent groups. Further, if $G$ is countable and $H\trianglelefteq G$ is infinite, the inclusion $C^*_r(H,σ')\subseteq C^*_r(G,σ)$ is selfless precisely when $C^*_r(H,σ')$ is selfless and $(H\leq G,σ)$ satisfies the relative Kleppner condition. As applications, we obtain selfless twisted group $C^*$-algebras of the lamplighter group and of the wreath product $\mathbb{Z}\wr\mathbb{Z}$.
Source: arXiv (Cornell University)
Author: Omland, Tron
Institutions: University of Oslo, National Security Authority
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166184934