Expansivity of algebraic semigroup actions

Abstract

For a semigroup S and a right

-submodule

J ≤ Z [ S ] n $J\leq \mathbb {Z}[S]^n$ upper J less than or equals double struck upper Z left bracket upper S right bracket Superscript n

, we study expansivity of the algebraic action of S induced on the Pontryagin dual of

Z [ S ] n / J $\mathbb {Z}[S]^n/J$ double struck upper Z left bracket upper S right bracket Superscript n divided by upper J

. We completely determine the class of semigroups for which expansivity of this action is characterized by the triviality of

J ⊥ $J^\perp $ upper J Superscript up tack

, in terms of the existence of a finite subset

K ⊆ S $K\subseteq S$ upper K subset of or equal to upper S

such that

S = K S $S=KS$ upper S equals upper K upper S

. 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.