Amenability and weak amenability of the Fourier algebra

Abstract

Let $G$ be a locally compact group. We show that its Fourier algebra $A(G)$ is amenable if and only if $G$ has an abelian subgroup of finite index, and that its Fourier-Stieltjes algebra $B(G)$ is amenable if and only if $G$ has a compact, abelian subgroup of finite index. We then show that $A(G)$ is weakly amenable if the component of the identity of $G$ is abelian, and we prove some partial results towards the converse.

16 pages; some, hopefully clarifying revisions

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