Multipliers and asymptotic behaviour of the Fourier algebra of nonamenable groups

Abstract

Let G G be a locally compact group and A ( G ) A(G) the algebra of matrix coefficients of the regular representation. We prove that G G is amenable if and only if there exist functions u ∈ A ( G ) u \in A(G) which vanish at infinity at any arbitrarily slow rate. The "only if" part of the result was essentially known. With the additional hypothesis that G G be discrete, we deduce that G G is amenable if and only if every multiplier of the algebra A ( G ) A(G) is a linear combination of positive definite functions. Again, the "only if" part of this result was known.