Regular simple nuclear C*-algebras
Abstract
We prove Z-stability for all simple, separable, pure, nuclear C*-algebras. Combining this with very recent work of Thiel completes the proof of the Toms-Winter conjecture. We do this through the framework of II_1 factorial tracially complete C*-algebras. We characterise regularity for these algebras in various terms. In particular they are regular if and only if they have real rank zero and the Murray-von Neumann equivalence classes of projections correspond to continuous affine functions from the designated traces to [0,1]. As an application of our analysis of ultrapowers of tracial completions, we show simple pure C*-algebras whose quasitraces are traces are K-stable. In the amenable setting we give a local quantisation result, a la Popa, for II_1 factorial amenable tracially complete C*-algebras, which obtains hyperfiniteness from regularity conditions. Our Z-stability results follow from this together with a theorem of Matui and Sato. An initial version of an essential lemma in the Bauer trace simplex case was obtained from ChatGPT Astra.
Related
- Weak central polynomials for algebras of multiplications of simple algebras
- Binomial Artin-Schelter Regular Algebras and Twisted Monoid Algebras
- A characterization of generalized derivations and biderivations on some simple algebras
- A single generation theorem for simple unital C*-algebras with CPoU and comparison
- simple cartan-free modules of finite rank over the super-virasoro algebras
BibTeX
@misc{evington2026regular,
title = {{Regular simple nuclear C*-algebras}},
author = {Samuel Evington and Shanshan Hua and Christopher Schafhauser and Apurva Seth and Aaron Tikuisis and Stuart White},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09390},
eprint = {2610.09390},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09390},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regular simple nuclear C*-algebras
AU - Samuel Evington
AU - Shanshan Hua
AU - Christopher Schafhauser
AU - Apurva Seth
AU - Aaron Tikuisis
AU - Stuart White
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09390
UR - https://arxiv.org/abs/2610.09390
AB - We prove Z-stability for all simple, separable, pure, nuclear C*-algebras. Combining this with very recent work of Thiel completes the proof of the Toms-Winter conjecture. We do this through the framework of II_1 factorial tracially complete C*-algebras. We characterise regularity for these algebras in various terms. In particular they are regular if and only if they have real rank zero and the Murray-von Neumann equivalence classes of projections correspond to continuous affine functions from the designated traces to [0,1]. As an application of our analysis of ultrapowers of tracial completions, we show simple pure C*-algebras whose quasitraces are traces are K-stable. In the amenable setting we give a local quantisation result, a la Popa, for II_1 factorial amenable tracially complete C*-algebras, which obtains hyperfiniteness from regularity conditions. Our Z-stability results follow from this together with a theorem of Matui and Sato. An initial version of an essential lemma in the Bauer trace simplex case was obtained from ChatGPT Astra.
ER -