On homomorphisms from semigroup algebras: generation, characters, and isolated points

We investigate the problem of identifying isolated points in spaces of continuous homomorphisms from ordered AL-algebras to normed algebras. Building on recent work of Bobrowski and the author [Dissertationes Math. (Rozprawy Mat.) 577 (2022), 1–69], we consider homomorphisms from semigroup algebras associated with non-locally-null, Haar measurable subsemigroups of locally compact groups. A key novelty is the removal of the assumption that the semigroup algebra admits an approximate identity of bound 1, which was essential in prior work. Our main result shows that for any continuous character σ σ on a suitable semigroup S , the homomorphism "Equation missing" is isolated in the space of all continuous homomorphisms from L¹(S) L 1 ( S ) to a unital normed algebra A A . Here L¹(S) L 1 ( S ) denotes the semigroup algebra of S , eA e A is the unit of A A , and "Equation missing" is the character on L¹(S) L 1 ( S ) determined by σ σ . This follows from a quantitative rigidity theorem: any continuous homomorphism within operator-norm distance less than 1 from "Equation missing" must coincide with it. Along the way, we develop a general framework for constructing and analysing such homomorphisms, including integral and associated representations, which serve not only to establish the main result, but also to characterise characters on semigroup algebras.