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.
Let G be a locally compact group. Let A P (G) be the Herz algebra of G associated with 1 < p < oo. We show that if A P (G) is Arens regular, then G is discrete. We also exhibit a number of sufficient conditions for such a group to be finite.
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