Amenability properties of the centres of group algebras

TL;DR

This paper investigates the conditions under which the centre of the group algebra of a locally compact group is amenable, revealing non-amenability in many cases and exploring specific conditions for amenability in non-compact groups.

Contribution

It provides new results on the non-amenability of ZL1(G) for certain classes of compact groups and studies conditions for amenability in some non-compact groups.

Findings

ZL1(G) is not amenable for nonabelian connected compact groups

ZL1(G) is not amenable for products of infinitely many finite nonabelian groups

Certain conditions imply amenability and hyper-Tauberian property for non-compact groups

Abstract

Let G be a locally compact group, and ZL1(G) be the centre of its group algebra. We show that when $G$ is compact ZL1(G) is not amenable when G is either nonabelian and connected, or is a product of infinitely many finite nonabelian groups. We also, study, for some non-compact groups G, some conditions which imply amenability and hyper-Tauberian property, for ZL1(G).