C*-algebras of Essentially Tracially Rank One

Abstract

We introduce the notion of essential tracial rank at most one for simple C^*-algebras. We show that separable simple C^*-algebras with essential tracial rank at most one have several regularity properties, including stable rank one, strict comparison, and tracial approximate divisibility. A finite amenable simple C^*-algebras with this property is 𝒵-stable. We also prove the converse. Thus this property gives a new characterization of separable simple finite amenable 𝒵-stable C^*-algebras and shows that these algebras admit a one-dimensional tracial approximation. In the unital real-rank-zero case, this approximation can be strengthened to show that the algebra has tracial rank zero. This completes a long-standing project. As a byproduct, we prove that every tracial state on a unital separable simple finite amenable C^*-algebra is quasidiagonal and that the algebra itself is quasidiagonal. Consequently, every separable simple finite amenable C^*-algebra of finite nuclear dimension has finite decomposition rank. None of these results requires the UCT.

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BibTeX RIS

BibTeX

@misc{lin2026calgebras,
  title = {{C*-algebras of Essentially Tracially Rank One}},
  author = {Huaxin Lin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14711},
  eprint = {2610.14711},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14711},
}

RIS

TY  - UNPB
TI  - C*-algebras of Essentially Tracially Rank One
AU  - Huaxin Lin
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14711
UR  - https://arxiv.org/abs/2610.14711
AB  - We introduce the notion of essential tracial rank at most one for simple C^*-algebras. We show that separable simple C^*-algebras with essential tracial rank at most one have several regularity properties, including stable rank one, strict comparison, and tracial approximate divisibility. A finite amenable simple C^*-algebras with this property is 𝒵-stable. We also prove the converse. Thus this property gives a new characterization of separable simple finite amenable 𝒵-stable C^*-algebras and shows that these algebras admit a one-dimensional tracial approximation. In the unital real-rank-zero case, this approximation can be strengthened to show that the algebra has tracial rank zero. This completes a long-standing project. As a byproduct, we prove that every tracial state on a unital separable simple finite amenable C^*-algebra is quasidiagonal and that the algebra itself is quasidiagonal. Consequently, every separable simple finite amenable C^*-algebra of finite nuclear dimension has finite decomposition rank. None of these results requires the UCT.
ER  -

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