ZL-amenability and characters for the restricted direct products of finite groups

Let G   be a restricted direct product of finite groups {Gi}i∈I{Gi}i∈I, and let Zℓ1(G)Zℓ1(G) denote the centre of its group algebra. We show that Zℓ1(G)Zℓ1(G) is amenable if and only if GiGi is abelian for all but finitely many i  , and characterize the maximal ideals of Zℓ1(G)Zℓ1(G) which have bounded approximate identities. We also study when an algebra character of Zℓ1(G)Zℓ1(G) belongs to c0c0 or ℓpℓp and provide a variety of examples.