Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneous C-algebras

• Published in: Journal d'analyse mathématique (Jerusalem) Vol. 146; no. 2; pp. 595 - 672
• Main Author: Niu, Zhuang
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 01.08.2022; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Algebra; Analysis; Dynamical systems; Dynamical Systems and Ergodic Theory; Functional Analysis; Mathematics; Mathematics and Statistics; Partial Differential Equations; Topology
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-022-0205-8

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.