Amenability constants for semilattice algebras

TL;DR

This paper calculates the amenability constants for semigroup algebras of finite unital commutative idempotent semigroups, revealing a specific form and establishing bounds for related Banach algebras.

Contribution

It provides a method to compute amenability constants for semilattice algebras and explores their bounds and implications for Banach algebras graded over semilattices.

Findings

Amenability constants are of the form 4n+1 for these algebras.

No commutative semilattice has an amenability constant between 5 and 9.

Lower bounds for Banach algebra amenability constants are established.

Abstract

For any finite unital commutative idempotent semigroup S, a unital semilattice, we show how to compute the amenability constant of its semigroup algebra l^1(S), which is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We show that there is no commutative semilattice with amenability constant between 5 and 9.