Amenability of semigroups and common multiples in $\ell^1_+$

TL;DR

This paper characterizes the left amenability of semigroups through the existence of common nonzero right multiples in the positive Banach algebra ^1_+(S), providing a novel criterion even for groups.

Contribution

It introduces a new characterization of semigroup amenability based on common multiples in ^1_+(S), linking algebraic and measure-theoretic properties.

Findings

Semigroup S is left amenable iff every two nonzero elements in ^1_+(S) have a common nonzero right multiple.

The characterization applies to groups and semigroups, offering a new perspective on amenability.

Provides a measure-theoretic criterion for amenability in the context of Banach algebras.

Abstract

In this note, we prove that a semigroup $S$ is left amenable if and only if every two nonzero elements of $\ell^1_+(S)$ have a common nonzero right multiple, where $\ell^1_+(S)$ is the positive part of the Banach algebra $\ell^1(S)$, or equivalently the semiring of finite measures on $S$. This characterization of amenability is new even for groups.