  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).
