On a theorem of Figà-Talamanca

Abstract

We give an example of a noncompact, unimodular group G with the property that B(G) n Co(G) = A(G), where A(G) is the Fourier algebra of G, B (G) is the Fourier-Stieltjes algebra of G and Co(G) is the set of all complex, continuous functions on G vanishing at infinity. This example answers negatively a question raised by A. Figa-Talamanca. 1. In a paper entitled Positive definite functions which vanish at infinity Alessandro Figa-Talamanca constructs a singular continuous, positive defi- nite function, i.e., one not in the Fourier algebra of G, which vanishes at infinity for any locally compact unimodular group that satisfies the condi- tion: (H) The von Neumann algebra, M (X), generated by the left regular repre- sentation A of G is not purely atomic. Groups satisfying (H) are of necessity not compact. In the aforementioned preprint, A.Figa-Talamanca notes that it is unknown whether or not unimod- ularity alone is sufficient (for a noncompact, locally compact group) to guarantee the existence of a singular continuous, positive definite function which vanishes at infinity. We show by means of an example in the following section that unimodularity alone is not sufficient to guarantee the existence of a positive definite function with the aforementioned properties. 2. In the fourth section of (3) the following group and its representation theory are discussed. We will briefly discuss this example with sufficiently many references so that the interested reader can fill in the details. Namely,