Ideals and Representations of CertainSemigroup Algebras

We characterize various types of ideals in ¹-Munn algebras with a boundedapproximate identity, in terms of the ideals of their underlying algebras. Moreover,we show the correspondence between different types of *-representationsof ¹-Munn algebras and *-representations of their underlying algebras.In the context of semigroup algebras, .rst we prove the conjecture whichwas proposed in [8]. Then we apply the results of the first part to build variousrepresentations of unital semigroup algebras of completely 0-simple inversesemigroups. This generalizes some of the Barnes results [1] with different andmuch shorter proofs.