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

For a left action (formula presented) of a cancellative right amenable monoid S on a discrete Abelian group X, we construct its Ore localization (formula presented), where G is the group of left fractions of S; analogously, for a right action (formula presented) on a compact space K, we construct its Ore colocalization (formula presented). Both constructions preserve entropy—that is, for the algebraic entropy halg and for the topological entropy htop one has, respectively, (formula presented). Exploiting these constructions and the theory of quasi-tilings, we extend the addition theorem for htop, known for right actions of countable amenable groups on compact metrizable groups, to right actions (formula presented) of can cellative 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 htop (ρ) coincides with halg. (ρ^), where (formula presented) is the dual left action on the discrete Pontryagin dual X = K^ (that is, a so-called bridge theorem. From the addition theorem for htop and the bridge theorem, we obtain an addition theorem for halg for left actions (formula presented) 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 toward entropy based on the entropy of actions of cancellative right amenable monoids on appropriately defined normed monoids.