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Let G be a restricted direct product of finite groups {Gi}i∈I, and let Zℓ1(G) denote the centre of its group algebra. We show that Zℓ1(G) is amenable if and only if Gi is abelian for all but finitely many i, and characterize the maximal ideals of Zℓ1(G) which have bounded approximate identities. We also study when an algebra character of Zℓ1(G) belongs to c0 or ℓp and provide a variety of examples.
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