Approximate Amenability of Segal algebras

TL;DR

This paper investigates the approximate amenability of Segal algebras on various groups, showing that proper Segal algebras generally lack this property across several classes of groups, including abelian, compact, and specific non-abelian groups.

Contribution

It establishes the non-approximate amenability of proper Segal algebras on a broad class of groups, extending previous results to new group structures using hypergroup techniques.

Findings

Proper Segal algebras on abelian groups are not approximately amenable.

Proper Segal algebras on certain compact groups, including SU(2), are not approximately amenable.

The results apply to a wide class of groups using hypergroup methods.

Abstract

In this paper we first show that for a locally compact amenable group $G$, every proper abstract Segal algebra of the Fourier algebra on $G$ is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the $2\times 2$ special unitary group, SU(2), are not approximately amenable.