Mean dimension and radius of comparison

Abstract

Consider a minimal and free topological dynamical system $(X,\Gamma )$, where $\Gamma$ is a discrete amenable group. Assume $(X,\Gamma )$ has the uniform Rokhlin property (URP) and the Cuntz comparison of open sets (COS), then it is shown that $$\mathrm{rc}(\mathrm{C}(X)⋊\Gamma )=\frac{1}{2}\mathrm{mdim}(X,\Gamma ).$$ The same statement also holds for unital simple AH algebras with diagonal maps. The arguments in both cases are based on the recent preprint of OpenAI dealing with $\mathbb{Z}$-actions and the topological ideas used in that paper.

44 Pages. The work was carried out using ChatGPT. The arguments are based on the recent preprint of OpenAI dealing with $\mathbb{Z}$-actions and the topological ideas used in that paper