2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68391The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. This is the first in a series of papers in which we have investigated a similar relation on inverse semigroups. We use a new concept of "restricted" representations and study the restricted semigroup algebras and corresponding $C^*$-algebras.13 pagesOperator AlgebrasRepresentation Theory43A35, 43A20text