Groupoid Models of $C^{\ast }$-algebras and Gelfand Duality

Abstract

We construct a large class of morphisms, which we call partial morphisms, of groupoids that induce $\ast$-morphisms of maximal and minimal groupoid $C^{\ast }$-algebras. We show that the association of a groupoid to its maximal (minimal) groupoid $C^{\ast }$-algebra and the association 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 $\ast$-morphisms between groupoid $C^{\ast }$-algebras. As an application, we construct a groupoid models of the entire inductive systems of the Jiang-Su algebra $\mathcal{Z}$ and the Razak-Jacelon algebra $\mathcal{W}$.

Based on lots of feedback, we decided to change the title. We also added the case of reduced (or minimal) groupoid $C^{\ast }$-algebras. Lastly, we added a subsection (subsection 4.4) which completes the Gelfand duality theorem for commutative $C^{\ast }$-algebras will all $\ast$-morphisms. We updated some of our proofs and gave lots of examples that did not exist in the previous versions