## Bulletin of the Belgian Mathematical Society - Simon Stevin

### 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}].

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 4 (2005), 535-542.

Dates
First available in Project Euclid: 5 December 2005

https://projecteuclid.org/euclid.bbms/1133793341

Digital Object Identifier
doi:10.36045/bbms/1133793341

Mathematical Reviews number (MathSciNet)
MR2205997

Zentralblatt MATH identifier
1147.46034

#### Citation

Bami, M. Lashkarizadeh. The coincidence of some topologies on the unit ball of the Fourier - Stieltjes algebra of weighted foundation semigroups. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 4, 535--542. doi:10.36045/bbms/1133793341. https://projecteuclid.org/euclid.bbms/1133793341