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Abstract

We extend the Gromov–Hausdorff propinquity to a metric on Lipschitz dynamical systems which are given by strongly continuous actions of proper monoids on quantum compact metric spaces via Lipschitz morphisms. We prove that the resulting distance between two Lipschitz dynamical systems is zero if and only if there exists an equivariant full quantum isometry between them. We apply our results to convergence of the dual actions on fuzzy tori to the dual actions on quantum tori. Our framework is general enough to also allow for the study of the convergence of continuous semigroups of positive linear maps and other actions of proper monoids.