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 geomet-rically visualize lots of ∗-morphisms between groupoid C^∗-algebras. As an application, we construct, without any use of the classification the-ory, 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.