To each locally compact group G, one may associate its Fourier algebra A(G); this is a Banach function algebra on G, whose norm encodes the group structure on G and not just its topological structure. From a modern perspective, one may view A(G) as the L¹-convolution algebra of the quantum dual of G.
The notion of amenability for Banach algebras, which has its roots in the study of L¹-convolution algebras of groups, admits a quantitative variant. For L¹-group algebras and C*-algebras, the amenability constant yields no extra information. For Fourier algebras the situation is very different, and in this talk I will give an overview of what is (un)known about the possible values of amenability constants for this class of algebras. In particular, I will discuss a conjecture that the amenability constant of A(G) coincides with another invariant that has better functorial properties; this is work in progress.