Localic metric spaces and the localic Gelfand duality

Abstract

In this paper we prove, as conjectured by B. Banachewski and C.J. Mulvey, that the constructive Gelfand duality can be extended into a duality between compact regular locales and unital abelian localic C⁎-algebras. In order to do so we develop a constructive theory of localic metric spaces and localic Banach spaces, we study the notion of localic completion of such objects and the behavior of these constructions with respect to pull-back along geometric morphisms.