A Separation Property of Positive Definite Functions on Locally Compact Groups and Applications to Fourier Algebras

For a closed subgroup H of a locally compact group G consider the property that the continuous positive definite functions on G which are identically one on H separate points in G\H from points in H. We prove a structure theorem for almost connected groups having this separation property for every closed subgroup. Also, when a pair (G, H) has this separation property, there are interesting consequences in the ideal theory of the Fourier algebra of G.