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 $\mathrm{C}(X)⋊{\mathbb{Z}}^d$ is at most the half of the mean topological dimension of $(X,T,{\mathbb{Z}}^d)$. As a consequence, the C*-algebra $\mathrm{C}(X)⋊{\mathbb{Z}}^d$ is classifiable if $(X,T,{\mathbb{Z}}^d)$ has zero mean dimension.