Radius of comparison and mean cohomological independence dimension

TL;DR

This paper introduces a new mean cohomological independence dimension for group actions on spaces and uses it to establish lower bounds on the radius of comparison of associated crossed product C*-algebras, with applications to specific subshifts.

Contribution

It defines a novel invariant called mean cohomological independence dimension and applies it to derive bounds on the radius of comparison for crossed products, extending previous results.

Findings

Lower bounds for the radius of comparison depending on mean dimension and density parameters.

Explicit bounds for subshifts constructed by Dou in 2017.

Conditions under which the radius exceeds certain thresholds based on cohomology.

Abstract

We introduce a notion of mean cohomological independence dimension for actions of discrete amenable groups on compact metrizable spaces, as a variant of mean dimension, and use it to obtain lower bounds for the radius of comparison of the associated crossed product C*-algebras. Our general theory gives the following for the minimal subshifts constructed by Dou in 2017. Let G be a countable amenable group, let Z be a polyhedron, and let T be Dou's subshift of Z^G (which also depends on a density parameter). Then the radius of comparison of the crossed product is greater than r (1/2) mdim (T) - 2, in which r depends on the density parameter and is close to 1 when the density parameter is close to 1. If Z is even dimensional and has nonvanishing rational cohomology in degree dim (Z), then the radius of comparison of the crossed product is greater than (1/2) mdim (T) - 1, regardless of what the density parameter is.