Syndetic Sets and Amenability

TL;DR

This paper establishes a link between syndetic set properties and amenability in infinite discrete semigroups, showing that certain syndetic conditions imply the existence of a left invariant mean.

Contribution

It introduces new conditions on syndetic sets that guarantee the amenability of infinite discrete semigroups, using algebraic properties of the Stone-Cech compactification.

Findings

If every right syndetic set is left syndetic, the semigroup has a left invariant mean.

The convex hull of two-sided translates of bounded functions contains a constant function.

The proofs leverage algebraic properties of the Stone-Cech compactification.

Abstract

We prove that if an infinite, discrete semigroup has the property that every right syndetic set is left syndetic, then the semigroup has a left invariant mean. We prove that the weak*-closed convex hull of the two-sided translates of every bounded function on an infinite discrete semigroup contains a constant function. Our proofs use the algebraic properties of the Stone-Cech compactification.