Skip to content
DraconiticResearch
Search Sources API
194,161 studies31 sourcesUpdated 13 min ago
PreprintarXiv licencearXiv

Equivariant 𝒡-stability for amenable actions

  • Eusebio Gardella
Open at sourcePDFDOICite

Abstract

We prove that every amenable action of a countable discrete group on a separable, simple, unital, nuclear, 𝒡-stable C*-algebra absorbs the trivial action on 𝒡 up to cocycle conjugacy. The group may be nonamenable, as in Suzuki's recent examples, and no assumptions on outerness or on the tracial state space are needed. This extends OpenAI's result for amenable groups to amenable actions. Amenability of the action cannot be omitted in general, as examples of the author and Lupini show. More generally, we show that every tracially amenable action on a separable, unital, 𝒡-stable C*-algebra with traces admits invariant central copies of all matrix algebras, without simplicity or nuclearity. For amenable actions on simple nuclear algebras, equivariant property (SI) allows one to deduce 𝒡-stability from this. As applications, we obtain absorption of 𝒡 for intermediate subalgebras in the crossed product, and also pure infiniteness for outer amenable actions of nonamenable groups. The basic techniques involving Gaussian variables originate in OpenAI's preprint, and are also the main new tool in its proposed solution to the Toms--Winter conjecture. We extend these techniques in order to deal with the weight functions coming from amenability, and one of the main new ingredients is the fact that one can replace an amenable action by an auxiliary one for which the amenability weights are central. In order to make the probabilistic tools accessible to the community beyond what is used here, we include an appendix where the theory is developed assuming only an elementary background, and where the relevant estimates are proved in detail.

Related

  1. An Approximate-Diagonal Characterization of Amenable Actions on C*Algebras
    Zhengyu Fu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. Generic Sharp Shift Embedding for Amenable Group Actions with the Uniform Rokhlin Property and Comparison
    Yonatan Gutman, Chunlin Liu
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  3. Ergodic measures of intermediate entropies for amenable group actions with an approximate product property
    Wenda Zhang, Xiankun Ren
    arXivPreprintCC BY 4.0Full textDynamical systemsarXiv
  4. Comparison and Almost Finiteness for Actions of Amenable Groups
    Eli Glasner, Chunlin Liu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsDynamical systemsarXiv
  5. Equivariant Morita equivalence for finite cyclic group actions on noncommutative tori
    Sayan Chakraborty, Pratik Kumar Kundu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{gardella2026equivariant,
  title = {{Equivariant $\mathcalZ$-stability for amenable actions}},
  author = {Eusebio Gardella},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00995},
  eprint = {2611.00995},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00995},
}
RIS
TY  - UNPB
TI  - Equivariant 𝒡-stability for amenable actions
AU  - Eusebio Gardella
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00995
UR  - https://arxiv.org/abs/2611.00995
AB  - We prove that every amenable action of a countable discrete group on a separable, simple, unital, nuclear, 𝒡-stable C*-algebra absorbs the trivial action on 𝒡 up to cocycle conjugacy. The group may be nonamenable, as in Suzuki's recent examples, and no assumptions on outerness or on the tracial state space are needed. This extends OpenAI's result for amenable groups to amenable actions. Amenability of the action cannot be omitted in general, as examples of the author and Lupini show. More generally, we show that every tracially amenable action on a separable, unital, 𝒡-stable C*-algebra with traces admits invariant central copies of all matrix algebras, without simplicity or nuclearity. For amenable actions on simple nuclear algebras, equivariant property (SI) allows one to deduce 𝒡-stability from this. As applications, we obtain absorption of 𝒡 for intermediate subalgebras in the crossed product, and also pure infiniteness for outer amenable actions of nonamenable groups. The basic techniques involving Gaussian variables originate in OpenAI's preprint, and are also the main new tool in its proposed solution to the Toms--Winter conjecture. We extend these techniques in order to deal with the weight functions coming from amenability, and one of the main new ingredients is the fact that one can replace an amenable action by an auxiliary one for which the amenability weights are central. In order to make the probabilistic tools accessible to the community beyond what is used here, we include an appendix where the theory is developed assuming only an elementary background, and where the relevant estimates are proved in detail.
ER  -