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 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 Htrianglelefteq G is infinite, the inclusion C^*_r(H,σ')⊆ C^*_r(G,σ) is selfless precisely when C^*_r(H,σ') is selfless and (H≤ 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 ℤwrℤ.

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BibTeX RIS

BibTeX

@misc{omland2026selflessness,
  title = {{Selflessness for twisted group C*-algebras of amenable groups and their inclusions}},
  author = {Tron Omland},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.27198},
  eprint = {2606.27198},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.27198},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Selflessness for twisted group C*-algebras of amenable groups and their inclusions
AU  - Tron Omland
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2606.27198
UR  - https://arxiv.org/abs/2606.27198
AB  - 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 Htrianglelefteq G is infinite, the inclusion C^*_r(H,σ')⊆ C^*_r(G,σ) is selfless precisely when C^*_r(H,σ') is selfless and (H≤ 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 ℤwrℤ.
ER  -

Abstract

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