Groupoid models of C*-algebras and the Gelfand functor

We construct a large class of morphisms, which we call partial morphisms, of groupoids that induce *-morphisms of maximal and minimal groupoid C*-algebras. We show that the assignment of a groupoid to its maximal (minimal) groupoid C*-algebra and the assignment of a partial morphism to its induced morphism are functors (both of which extend the Gelfand functor). We show how to geometrically visualize lots of *-morphisms between groupoid C*-algebras. As an application, we construct, without any use of the classification theory, groupoid models of the entire inductive systems used in the original constructions of the Jiang-Su algebra Z and the Razak-Jacelon algebra W. Consequently, the inverse limit of the groupoid models for the aforementioned systems are models for Z and W, respectively.