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

TL;DR

This paper investigates the amenability of the center of the group algebra for restricted direct products of finite groups, establishing conditions for amenability and character properties, with implications for algebraic structure and analysis.

Contribution

It characterizes when the center of the group algebra is amenable for restricted direct products and describes the properties of algebra characters in this context.

Findings

$ ext{Z}_ ext{l}^1(G)$ is amenable iff all but finitely many $G_i$ are abelian.

Characterizes maximal ideals with bounded approximate identities.

Provides examples of algebra characters in $c_0$ and $ ext{ell}^p$ spaces.

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.