English

$\mathcal Z$-stability of $\mathrm{C}(X)\rtimes\Gamma$

Operator Algebras 2020-08-11 v1 Dynamical Systems

Abstract

Let (X,Γ)(X, \Gamma) be a free and minimal topological dynamical system, where XX is a separable compact Hausdorff space and Γ\Gamma is a countable infinite discrete amenable group. It is shown that if (X,Γ)(X, \Gamma) has the Uniform Rokhlin Property and Cuntz comparison of open sets, then mdim(X,Γ)=0\mathrm{mdim}(X, \Gamma)=0 implies that (C(X)Γ)ZC(X)Γ(\mathrm{C}(X) \rtimes\Gamma)\otimes\mathcal Z \cong \mathrm{C}(X) \rtimes\Gamma, where mdim\mathrm{mdim} is the mean dimension and Z\mathcal Z is the Jiang-Su algebra. In particular, in this case, mdim(X,Γ)=0\mathrm{mdim}(X, \Gamma)=0 implies that the C*-algebra C(X)Γ\mathrm{C}(X) \rtimes\Gamma is classified by the Elliott invariant.

Keywords

Cite

@article{arxiv.2008.03357,
  title  = {$\mathcal Z$-stability of $\mathrm{C}(X)\rtimes\Gamma$},
  author = {Zhuang Niu},
  journal= {arXiv preprint arXiv:2008.03357},
  year   = {2020}
}
R2 v1 2026-06-23T17:42:53.817Z