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

Let G be a restricted direct product of finite groups { G i } i ∈ I , and let Z ℓ 1 ( G ) denote the centre of its group algebra. We show that Z ℓ 1 ( G ) is amenable if and only if G i is abelian for all but finitely many i, and characterize the maximal ideals of Z ℓ 1 ( G ) which have bounded approximate identities. We also study when an algebra character of Z ℓ 1 ( G ) belongs to c 0 or ℓ p and provide a variety of examples.