Positive definite functions which vanish at infinity

Abstract

Let G be a separable noncompact locally compact group. Let A(G) and B(G), respectively, be the Fourier algebra and the Fourier-Stieltjes algebra of G as defined by P. Eymard. We prove that if G is unimodular and satisfies an additional hypothesis, which implies noncompactness, there exists an element of B(G), indeed a positive definite function, which vanishes at infinity but is not in A(G). This function actually belongs to B? (G), that is, it defines a unitary representation of G which is weakly contained in the regular representation.