Localic metric spaces and the localic Gelfand duality

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⁎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.