Ordinary and dynamical comparison in C*-dynamics
Year: 2026
Type: preprint
Abstract: For a free minimal action of a countable amenable group on a compact metrizable space, we prove that the dynamical radius of comparison is at most half the mean dimension when the action has the uniform Rokhlin property and comparison of sets. We establish these dynamical properties for free marker actions of groups with infinite FC-center, giving a partial positive answer to the marker-to-FCSB question. In particular, the bound holds for every free minimal action of every countable virtually nilpotent group, without a finite-generation assumption. This answers that case of a question of Asadi--Vasfi and Phillips from 2026. A joint diagonal approximation of two positive matrix fields gives a second comparison estimate, by half of Meyerovitch's dynamical dimension, under full diagonal strict comparison. The source enlargement and target rank loss share one half-dimension allowance. We verify the diagonal comparison hypothesis for actions of countable locally finite groups. Finally, we compute the ordinary and dynamical radii in a positive-product version of Hirshberg's AH system, obtaining a positive ordinary radius and zero dynamical radius and answering another question of Asadi--Vasfi and Phillips. We also give an explicit Klein-four realization of their suggested combination, with three strictly different comparison radii. GitHub repository: https://github.com/eulogica/ordinary-dynamical-comparison
Author: Lupini, Martino
Institution: University of Bologna
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225448489