Strict topology over the space of measures with continuous translations on a locally compact foundation semigroup

Let $ mathfrak{S} $ mathfrak{S} be a locally compact semigroup, ω be a weight function on $ mathfrak{S} $ mathfrak{S} , and M _a ($ mathfrak{S} $ mathfrak{S} , ω) be the weighted semigroup algebra of $ mathfrak{S} $ mathfrak{S} . Let L ₀^∞ ($ mathfrak{S} $ mathfrak{S} ; M _a ($ mathfrak{S} $ mathfrak{S} , ω)) be the C*-algebra of all M _a ($ mathfrak{S} $ mathfrak{S} , ω)-measurable functions g on $ mathfrak{S} $ mathfrak{S} such that g/ω vanishes at infinity. We introduce and study a strict topology β ¹($ mathfrak{S} $ mathfrak{S} , ω) on M _a ($ mathfrak{S} $ mathfrak{S} , ω) and show that the Banach space L ₀^∞ ($ mathfrak{S} $ mathfrak{S} ; M _a ($ mathfrak{S} $ mathfrak{S} , ω)) can be identified with the dual of M _a ($ mathfrak{S} $ mathfrak{S} , ω) endowed with β ¹($ mathfrak{S} $ mathfrak{S} , ω). We finally investigate some properties of the locally convex topology β ¹($ mathfrak{S} $ mathfrak{S} , ω) on M _a ($ mathfrak{S} $ mathfrak{S} , ω).