PreprintCC BY 4.0arXiv
Nonamenable groups whose reduced group C*-algebras are not pure
Abstract
We exhibit nonamenable groups whose reduced group C*-algebras are not pure. More precisely, if Γ is any countably infinite discrete group, then the reduced group C*-algebra of the restricted wreath product (ℤ/2ℤ) wr Γ has an ideal-quotient isomorphic to 𝒦(ℓ2(Γ)). It therefore fails to be nowhere scattered and, in particular, is not pure. Taking Γ to be any nonamenable group, we thereby obtain a negative answer to a question of Thiel.
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BibTeX
@misc{bell2026nonamenable,
title = {{Nonamenable groups whose reduced group C*-algebras are not pure}},
author = {Jamie Bell},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10653},
eprint = {2609.10653},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10653},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nonamenable groups whose reduced group C*-algebras are not pure
AU - Jamie Bell
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.10653
UR - https://arxiv.org/abs/2609.10653
AB - We exhibit nonamenable groups whose reduced group C*-algebras are not pure. More precisely, if Γ is any countably infinite discrete group, then the reduced group C*-algebra of the restricted wreath product (ℤ/2ℤ) wr Γ has an ideal-quotient isomorphic to 𝒦(ℓ^2(Γ)). It therefore fails to be nowhere scattered and, in particular, is not pure. Taking Γ to be any nonamenable group, we thereby obtain a negative answer to a question of Thiel.
ER -