On Continuity and Decomposition of Positive Definite Functions

Let G be a locally compact commutative group and let g and h be positive definite functions on G , which are not identically zero. We show that continuity of gh̄ implies the existence of a character γ of G d (the discrete version of G ) such that γ g and γ h are continuous. As corollary we get a special case of a result of K. de Leeuw and I. Glicksberg concerning almost continuous group representations. In the second part of the paper we prove decomposition theorems for positive definite functions defined on a neighbourhood of the zero.