Expansivity of algebraic semigroup actions
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918237772447744 |
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| author | Donoso-Echenique, Miguel |
| author_facet | Donoso-Echenique, Miguel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07445 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Expansivity of algebraic semigroup actions Donoso-Echenique, Miguel Dynamical Systems Functional Analysis Group Theory 20M30, 43A20 (Primary), 20M25, 43A15 (Secondary) 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. |
| title | Expansivity of algebraic semigroup actions |
| topic | Dynamical Systems Functional Analysis Group Theory 20M30, 43A20 (Primary), 20M25, 43A15 (Secondary) |
| url | https://arxiv.org/abs/2512.07445 |