An analytic criterion for the local finiteness of a countable semigroup

TL;DR

This paper establishes a precise criterion linking the local finiteness of a countable semigroup to the topological properties of its semigroup algebra's Arens-Michael envelope, expanding understanding of algebra-topology relationships.

Contribution

It provides a new analytic criterion characterizing when a countable semigroup is locally finite based on the structure of its semigroup algebra's Arens-Michael envelope.

Findings

Countable semigroup is locally finite iff its algebra's envelope is a (DF)-space.

Extends previous work relating finite generation to Fréchet spaces.

Offers a topological characterization of semigroup properties.

Abstract

We prove that a countable semigroup $S$ is locally finite if and only if the Arens-Michael envelope of its semigroup algebra is a $(DF)$-space. This is a counterpart to a recent result of the author, which asserts that $S$ is finitely generated if and only if the Arens-Michael envelope is a Fr\'echet space.