Amenability constants of central Fourier algebras of finite groups

TL;DR

This paper investigates the amenability constants of the central Fourier algebra of finite groups, revealing new properties and counterexamples that differentiate it from related algebras, especially regarding quotient groups.

Contribution

It introduces new results on the amenability constants of $ZA(G)$, compares them with $ZL^1(G)$, and provides counterexamples illustrating their distinct behaviors.

Findings

AM(ZA(G)) equals AM(ZL^1(G)) for certain classes of groups.

AM(ZA(G)) does not always respect quotient groups, unlike AM(ZL^1(G)).

Groups with two conjugacy class sizes have specific amenability constant behaviors.

Abstract

We consider amenability constants of the central Fourier algebra $ZA(G)$ of a finite group $G$. This is a dual object to $ZL^1(G)$ in the sense of hypergroup algebras, and as such shares similar amenability theory. We will provide several classes of groups where $AM(ZA(G)) = AM(ZL^1(G))$, and discuss $AM(ZA(G))$ when $G$ has two conjugacy class sizes. We also produce a new counterexample that shows that unlike $AM(ZL^1(G))$, $AM(ZA(G))$ does not respect quotient groups, however the class of groups that does has $\frac{7}{4}$ as the sharp amenability constant bound.