Relations between dynamics and C*-algebras: Mean dimension and radius of comparison

This is joint work with Ilan Hirshberg.

For an action of an amenable group G on a compact metric space X, the mean dimension mdim (G, X) was introduced by Lindenstrauss and Weiss. It is designed so that the mean dimension of the shift on ([0, 1]^d)^G is d. Its motivation was unrelated to C*-algebras.

The radius of comparison rc (A) of a C*-algebra A was introduced by Toms to distinguish counterexamples in the Elliott classification program. The algebras he used have nothing to do with dynamics.

A construction called the crossed product C^* (G, X) associates a C-algebra to a dynamical system. Despite the apparent lack of connection between these concepts, there is significant evidence for the conjecture that rc ( C^ (G, X) ) = (1/2) mdim (G, X) when the action is free and minimal. We will explain the concepts above; no previous knowledge of mean dimension, C-algebras, or radius of comparison will be assumed. Then we describe some of the evidence. In particular, we give the first general partial results towards the direction rc ( C^ (G, X) ) \geq (1/2) mdim (G, X). We don’t get the exact conjectured bound, but we get nontrivial results for many of the known examples of free minimal systems with mdim (G, X) > 0.