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

## TL;DR

This paper establishes a relationship between the comparison radius of crossed product C*-algebras and the mean topological dimension of free minimal dynamical systems, under certain Rokhlin and comparison conditions.

## Contribution

It proves that the comparison radius is at most half the mean topological dimension when the system has a uniform Rokhlin property and Cuntz comparison on open sets.

## Key findings

- Comparison radius is at most half the mean topological dimension.
- Conditions are satisfied for $ ext{Z}$ actions and certain extensions.
- Uses Cuntz comparison and subgroupoid techniques.

## Abstract

Let $(X, \Gamma)$ be a free minimal dynamical system, where $X$ is a compact separable Hausdorff space and $\Gamma$ is a discrete amenable group. It is shown that, if $(X, \Gamma)$ has a version of Rokhlin property (uniform Rokhlin property) and if $\mathrm{C}(X)\rtimes\Gamma$ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes \Gamma$ is at most half of the mean topological dimension of $(X, \Gamma)$. These two conditions are shown to be satisfied if $\Gamma = \mathbb Z$ or if $(X, \Gamma)$ is an extension of a free Cantor system and $\Gamma$ has subexponential growth. The main tools being used are Cuntz comparison of diagonal elements of a subhomogeneous C*-algebra and small subgroupoids.