Ore localization of amenable monoid actions and applications toward entropy—addition formulas and the bridge theorem

Abstract

For a left action $S\overset{\lambda}{\curvearrowright}X$ of a cancellative right amenable monoid $S$ on a discrete Abelian group $X$, we construct its Ore localization $G\overset{\lambda^*}{\curvearrowright}X^*$, where $G$ is the group of left fractions of $S$; analogously, for a right action $K\overset{\rho}\curvearrowleft S$ on a compact space $K$, we construct its Ore colocalization $K^*\overset{\rho^*}{\curvearrowleft} G$. Both constructions preserve entropy, i.e., for the algebraic entropy $h_{\mathrm{alg}}$ and for the topological entropy $h_{\mathrm{top}}$ one has $h_{\mathrm{alg}}(\lambda)=h_{\mathrm{alg}}(\lambda^*)$ and $h_{\mathrm{top}}(\rho)=h_{\mathrm{top}}(\rho^*)$, respectively. Exploiting these constructions and the theory of quasi-tilings, we extend the Addition Theorem for $h_{\mathrm{top}}$, known for right actions of countable amenable groups on compact metrizable groups, to right actions $K\overset{\rho}{\curvearrowleft} S$ of cancellative right amenable monoids $S$ (with no restrictions on the cardinality) on arbitrary compact groups $K$. When the compact group $K$ is Abelian, we prove that $h_{\mathrm{top}}(\rho)$ coincides with $h_{\mathrm{alg}}(\hat{\rho})$, where $S\overset{\hat{\rho}}\curvearrowright X$ is the dual left action on the discrete Pontryagin dual $X=\hat{K}$, that is, a so-called Bridge Theorem. From the Addition Theorem for $h_{\mathrm{top}}$ and the Bridge Theorem, we obtain an Addition Theorem for $h_{\mathrm{alg}}$ for left actions $S\overset{\lambda}\curvearrowright X$ on discrete Abelian groups, so far known only under the hypotheses that either $X$ is torsion or $S$ is locally monotileable. The proofs substantially use the unified approach towards entropy based on the entropy of actions of cancellative right amenable monoids on appropriately defined normed monoids.