Representation Theorems for d-Multiplications on Archimedean Unital f-Rings

Abstract

Let C(Ω) be the f-ring of all real-valued continuous functions on a completely regular topological space Ω. The present paper presents a complete description of d-multiplications in C(Ω) as weighted composition multiplications. In this regard, it is proven that if C(Ω) is a d-ring with respect to a multiplication * then there exist a “weight” positive function w in C(Ω), and two functions σ and τ from Ω into Ω such that

A generalization of the latter representation formula to a more general setting of f-rings is obtained. Indeed, it is shown that if ℛ is an Archimedean f-ring with identity and if * is a d-multiplication in ℛ then there exist a positive element w in ℛ, and two ℓ-ring homomorphisms ϕ and ψ from ℛ into the maximal ring of quotients Q(ℛ) of ℛ such that

Key Words:

• Continuous functions
• d-multiplication
• d-ring
• f-ring

Mathematics Subject Classification:

• 06F25
• 46E25

Acknowledgments

Notes

^#Communicated by K. Rangaswamy.