The mean dimension of a homeomorphism and the radius of comparison of its C*-algebra

We describe a striking conjectured relation between ``dimensions’’ in topological dynamics and C-algebras. (No previous knowledge of C-algebras or dimension theory will be assumed.) Let $X$ be a compact metric space, and let $h \colon X \to X$ be a minimal homeomorphism (no nontrivial invariant closed subsets). The mean dimension ${\mathit{mdim}} (h)$ of $h$ is a dynamical invariant, which I will describe in the talk, and which was invented for purposes having nothing to do with C-algebras. One can also form a crossed product C-algebra $C^* ({\mathbb{Z}}, X, h)$. It is simple and unital, and there is an explicit description in terms of operators on Hilbert space, which I will give in the talk. The radius of comparison ${\mathit{rc}} (A)$ of a simple unital C-algebra $A$ is an invariant introduced for reasons having nothing to do with dynamics; I will give the motivation for its definition in the talk (but not the definition itself). It has been conjectured, originally on very thin evidence, that the radius of comparison of $C^({\mathbb{Z}},X,h)$ is equal to half the mean dimension of $h$ for any minimal homeomorphism $h$.

In this talk, I will give elementary introductions to mean dimension, the crossed product construction, and the ideas behind the radius of comparison. I will then describe the motivation for the conjecture and some partial results towards it.