Constructive Gelfand duality for non-unital commutative C*-algebras

Abstract: We prove constructive versions of various usual results related to the Gelfand duality. Namely, that the constructive Gelfand duality extend to a duality between commutative nonunital C*-algebras and locally compact completely regular locales, that ideals of a commutative C*-algebras are in order preserving bijection with the open sublocales of its spectrum, and a purely constructive result saying that a commutative C*-algebra has a continuous norm if and only its spectrum is open. We also extend all these results to the case of localic C*-algebras. In order to do so we develop the notion of one point compactification of a locally compact regular locale and of unitarization of a C*-algebra in a constructive framework.