Amenability properties of the centres of group algebras

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 non-abelian and connected, or is a product of infinitely many finite non-abelian groups. We also, study, for some non-compact groups G, some conditions which imply amenability and hyper-Tauberian property, for ZL1(G).