Ore localization of amenable monoid actions and applications toward entropy -- addition formulas and the bridge theorem (Q6183834)

Almost one third of this long paper is devoted to introductory and preliminary matters making the paper more user friendly. The original sections deal with Ore localizations of actions on discrete abelian groups and of actions on compact spaces; the topological addition theorem; the bridge theorem and the algebraic addition theorem along with the appendix which is dealing with the bridge theorem for the entropies of amenable group actions. \(\lambda\) denotes a left action of cancellative right amenable monoid \(S\) on a (discrete) abelian group \(X\); \(\rho\) denotes a right action of \(S\) on a compact space \(K\). Some results are as follows: Theorem 2.9. The algebraic entropy under Ore localization is invariant, symbolically \(h_{alg}(\lambda)=h_{alg}(\lambda^*)\). Theorem 3.15. The topological entropy is invariant under Ore colocalization, symbolically \(h_{top}(\rho)=h_{top}(\rho^*)\). This is proved using the fact that the computation of topological entropy can be reduced to actions by surjective self-maps. The authors also generalize Li's topological addition theorem [\textit{H. Li}, Ann. Math. (2) 176, No. 1, 303--347 (2012; Zbl 1250.22006)] as follows: Assume that \(S\) is a cancellative and right amenable monoid, \(K\) a compact group, \(\rho\) is a right \(S\)-action on \(K\) and assume that \(H\) is closed \(S\)-invariant subgroup of \(K\). Then the topological addition theorem holds: \(h_{top}(\rho)=h_{top}(\rho_H)+h_{top}(\rho_{K/H})\). A generalization of the so-called Bridge theorem is also proved: Let \(S\) be a cancellative and right amenable monoid, \(K\) a compact abelian group with a right \(S\)-action \(\rho\) on \(K\) and \(X\) a discrete abelian group with a left \(S\)-action \(\lambda\) on \(X\). Then the following hold: \(h_{top}(\rho)=h_{alg}(\hat\rho)\) and \(h_{alg}(\lambda)=h_{top}(\hat\lambda)\) ( \(\hat{\,}\) stands for Pontryagin duality functor). This theorem and the topological addition theorem are used to prove the algebraic addition theorem: Let \(S\) be a cancellative and right amenable monoid, \(X\) an abelian group, \(\lambda\) a left \(S\)-action on \(X\), and \(Y\) an \(S\)-invariant subgroup of \(X\). Then \(h_{alg}(\lambda)=h_{alg}(\lambda_Y)+h_{alg}(\lambda_{X/Y})\), where \(\lambda_Y\) and \(\lambda_{X/Y}\) are the left \(S\)-actions induced by \(\lambda\) on \(Y\) and \(X/Y\) respectively. The bridge theorem for amenable group actions reads as follows: Theorem A.27. Let \(G\) be an amenable group, \(K\) a compact abelian group, \(X\) a discrete abelian group, and \(\rho\) and \(\lambda\) be a right and a left \(G\)-actions, respectively; then \(h_{top}(\rho)=h_{alg}(\hat\rho)\) and \(h_{alg}(\lambda)=h_{top}(\hat\lambda)\). The authors pose (open) questions whether it is possible to prove the general version of topological additive theorem without use of Ore colocalization and likewise whether it is possible to give a direct proof of the algebraic additive theorem, rather than use the topological additive theorem and the bridge theorem. The reviewer notes that, if these theorems are not equivalent to the respective theorems desired to dispense with, then it should be possible to give alternative proof of the respective theorems.