Amenability properties for the centres of certain discrete group algebras

Details Amenability properties for the centres of certain discrete group algebras Amenability properties for the centres of certain discrete group algebras

2011-10-31

arxiv.org/abs/1110.6683v1

Mathematics

Functional Analysis

AMS-LaTeX, 34 pages, submitted

Scientific paper

Let $G$ be a discrete group with finite conjugacy classes, and let $\Zl^1(G)$ denote the centre of its group algebra. In this article, we study various properties of $\Zl^1(G)$, such as amenability and its spectrum, for the case where $G$ is a restricted direct product of finite groups ${G_i}_{i\in I}$. Among other things, we show that $\Zl^1(G)$ is amenable if and only if $G_i$ is abelian for all but finitely many $i$, and characterize maximal ideals of $\Zl^1(G)$ with 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. Lastly, we calculate the exact amenability constant of $\Zl^1(G)$ for certain finite metabelian groups $G$.

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