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..