Variations on a theme of Gelfand and Naimark

C*-algebras are widely used in mathematical physics to represent the observables of physical systems, and are sometimes taken as the starting point for rigorous formulations of quantum mechanics and classical statistical mechanics. Nevertheless, in many cases the naive choice of an algebra of observables does not admit a C*-algebra structure, and some massaging is necessary. In this paper we investigate what properties of C*-algebras carry over to more general algebras and what modifications of the Gelfand theory of normed algebras are necessary. We use category theory as a guide and, by replacing the ordinary definition of the Gelfand spectrum with a manifestly functorial definition, we succeed in generalizing the Gelfand-Naimark theorem to locally convex *-algebras. We also recall a little-known but potentially very useful generalization of the Stone-Weierstrass theorem to completely regular, Hausdorff spaces.