𝒵-stability of transformation group C*-algebras

Abstract

Let ( X , Γ ) (X, \Gamma ) be a free and minimal topological dynamical system, where X X 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 (URP) and Cuntz comparison of open sets (COS), then m d i m ( X , Γ ) = 0 \mathrm {mdim}(X, \Gamma )=0 implies that ( C ( X ) ⋊ Γ ) ⊗ Z ≅ C ( X ) ⋊ Γ (\mathrm {C}(X) \rtimes \Gamma )\otimes \mathcal Z \cong \mathrm {C}(X) \rtimes \Gamma , where m d i m \mathrm {mdim} is the mean dimension of ( X , Γ ) (X, \Gamma ) , Z \mathcal Z is the Jiang-Su algebra, and C ( X ) ⋊ Γ \mathrm {C}(X) \rtimes \Gamma is the transformation group C*-algebra of ( X , Γ ) (X, \Gamma ) . In particular, in this case, m d i m ( 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.