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Ranks of operators and pureness of C*-algebras

  • Hannes Thiel
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Abstract

For simple, nonelementary C*-algebras, we show uniform density of functions realized as ranks of operators. It follows that strict comparison by traces forces almost divisibility, and hence pureness. As an application, we show that strict comparison implies 𝒡-stability for separable, simple, nonelementary C*-algebras with locally finite nuclear dimension. In particular, the Toms--Winter conjecture holds for approximately subhomogeneous C*-algebras. More generally, we obtain uniform density of ranks for C*-algebras with the Global Glimm Property. Consequently, an exact C*-algebra is pure if and only if it has strict comparison together with the Global Glimm Property.

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Cite

BibTeXRIS
BibTeX
@misc{thiel2026ranks,
  title = {{Ranks of operators and pureness of C*-algebras}},
  author = {Hannes Thiel},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06761},
  eprint = {2610.06761},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06761},
}
RIS
TY  - UNPB
TI  - Ranks of operators and pureness of C*-algebras
AU  - Hannes Thiel
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06761
UR  - https://arxiv.org/abs/2610.06761
AB  - For simple, nonelementary C*-algebras, we show uniform density of functions realized as ranks of operators. It follows that strict comparison by traces forces almost divisibility, and hence pureness. As an application, we show that strict comparison implies 𝒡-stability for separable, simple, nonelementary C*-algebras with locally finite nuclear dimension. In particular, the Toms--Winter conjecture holds for approximately subhomogeneous C*-algebras. More generally, we obtain uniform density of ranks for C*-algebras with the Global Glimm Property. Consequently, an exact C*-algebra is pure if and only if it has strict comparison together with the Global Glimm Property.
ER  -