  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.
