Comparison radius and mean topological dimension: $\mathbb{Z}^d$-actions

Abstract

Consider a minimal free topological dynamical system $(X, T, \mathbb{Z}^d)$. It is shown that the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes \mathbb{Z}^d$ is at most the half of the mean topological dimension of $(X, T, \mathbb{Z}^d)$. As a consequence, the C*-algebra $\mathrm{C}(X) \rtimes \mathbb{Z}^d$ is classifiable if $(X, T, \mathbb{Z}^d)$ has zero mean dimension.