Expansivity of algebraic semigroup actions

TL;DR

This paper investigates the conditions under which algebraic actions of semigroups are expansive, providing a complete characterization for certain classes of semigroups and extending known results to non-unital cases.

Contribution

It characterizes expansivity of algebraic semigroup actions via algebraic and geometric conditions, extending previous results to broader classes including non-unital semigroups.

Findings

Characterization of expansivity via triviality of orthogonal complements.

Identification of semigroups satisfying the key finite subset condition.

Extension of invertibility conditions for finitely generated modules.

Abstract

For a semigroup $S$ and a right $\mathbb{Z}[S]$-submodule $J\leq \mathbb{Z}[S]^n$, we study expansivity of the algebraic action of $S$ induced on the Pontryagin dual of $\mathbb{Z}[S]^n/J$. We completely determine the class of semigroups for which expansivity of this action is characterized by the triviality of $J^\perp$, in terms of the existence of a finite subset $K\subseteq S$ such that $S=KS$. This condition is satisfied in particular by every monoid, and more generally by every semigroup with a left unital convolution Banach algebra, in which case we are able to extend the characterization of expansivity in terms of an invertibility condition when $J$ is finitely generated. We also exhibit examples of non-unital semigroups satisfying the hypotheses of our results.