The radius of comparison and mean dimension

摘要

This dissertation is a collection of results and examples designed to support a single conjecture, namely, that two dimensional invariants (the mean dimension for topological dynamical systems and the radius of comparison for C*-algebras) are related by a simple equation. Specifically, we conjecture that the mean dimension of a minimal system is twice the radius of comparison of the associated crossed product. With three main Theorems and many examples and supporting results, we verify that this Conjecture is true in many cases. In the case where the dynamical system is not minimal, we show with several important examples that the radius of comparison provides a more nuanced measurement than the mean dimension. This naturally leads to the introduction of a new dimension theory for topological dynamical systems: the dynamic dimension, which extends the mean dimension in the minimal case and improves it in the non-minimal case. Furthermore, our results give an essential tool for computing the radius of comparison of certain crossed product C*-algebras, with important consequences for the Elliott classification program. In particular, our results show that there exist infinite-dimensional dynamical systems who's crossed products satisfy the Toms-Winter conjecture, and are hence amenable to classification by K-theoretic invariants.