The coincidence of some topologies on the unit ball of the Fourier - Stieltjes algebra of weighted foundation semigroups

Abstract

In our earlier paper [{\bf 10}], for a foundation $*-$semigroup $S$ with an identity and with a Borel-measurable weight function $w\leq 1$, we proved that on the unit ball of ${\cal P} (S,w)$, the cone of $w-$bounded continuous positive definite functions on $S$, the weak topology coincides with the compact open topology. In the present paper, through some $C^*-$algebras techniques, we shall extend this result to the unit ball of the Fourier-Stieltjes algebra ${\cal F} (S,w)$ of a foundation semigroup $S$ with a Borel measurable weight function $w$. Indeed, we shall establish our conjecture in [{\bf 10}] even in the more general setting of the Fourier-Stieltjes algebra ${\cal F}(S,w)$ for any Borel measurable weight function $w$. It should be noted that the family of foundation semigroups is quite extensive, for which locally compact groups and discrete semigroups are elementary examples. For further examples we refer to Appendix B of [{\bf 13}].