Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneous C-algebras - Niu, Zhuang (Journal d'analyse mathématique (Jerusalem), 2022)
Let ( X , Γ) be a free minimal dynamical system, where X is a compact separable Hausdorff space and Γ is a discrete amenable group. It is shown that, if ( X , Γ) has a version of Rokhlin property (uniform Rokhlin property) and if C( X ) ⋊ Γ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra C ( X ) ⋊ Γ is at most half of the mean topological dimension of ( X , Γ). These two conditions are shown to be satisfied if Γ = ℤ or if ( X , Γ) is an extension of a free Cantor system and Γ has subexponential growth. The main tools being used are Cuntz comparison of diagonal elements of a subhomogeneous C*-algebra and small subgroupoids.