Radius of comparison and mean cohomological independence dimension

DOI10.1016/J.AIM.2022.108563arXiv2009.13045MaRDI QIDQ6349996

N. Christopher Phillips, Ilan Hirshberg

Publication date: 27 September 2020

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.

Mathematics Subject Classification ID

Noncommutative dynamical systems (46L55) Dynamics induced by group actions other than (mathbb{Z}) and (mathbb{R}), and (mathbb{C}) (37C85) Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) (37B05) ?ech types (55N05) Dimension theory of smooth dynamical systems (37C45) Dynamical systems and the theory of (C^*)-algebras (37A55)

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