  Let $G$ be a restricted direct product of finite groups $\{G_i \}_{i\in I}$,
and let $\Zl^1(G)$ denote the centre of its group algebra. We show that
$\Zl^1(G)$ is amenable if and only if $G_i$ is abelian for all but finitely
many $i$, and characterize the maximal ideals of $\Zl^1(G)$ which have bounded
approximate identities. We also study when an algebra character of $\Zl^1(G)$
belongs to $c_0$ or $\ell^p$ and provide a variety of examples.
