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

Abstract

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.

v3: AMS-LaTeX, 22 pages. Title changed from v2; small changes made to abstract; incorporates suggestions made by the referee. We have updated some non-standard terminology to agree with existing terminology in the literature. v4: cleaned up some typos and tidied some formatting. May differ from published version in matters of house style. To appear in J. Math. Anal. Appl