On a conjecture on the uniform convergence of a sequence of weighted bounded positive definite functions on foundation semigroups

Abstract

In the present paper, we shall establish one of our earlier conjectures by proving that on compact subsets of a *-foundation semigroup S with identity and with a locally bounded Borel measurable weight function w, the pointwise convergence and the uniform convergence of a sequence of w-bounded positive definite functions on S which are also continuous at the identity are equivalent.