Melvin Henriksen - Embedding a ring of continuous functions in a regular ring: preliminary report

C(X) denotes the ring of continuous real-valued functions on a Tychonoff space X. If , let when f(x) = 0, and let f(x) = 0 otherwise. Let G(X) denote the subalgebra of generated by C(X) and . Then G(X) is the smallest von Neumann regular sublagebra of containing C(X) and is closed under inversion, i.e., any element of G(X) that never vanishes has an inverse in G(X). Let Ba₁(X) denote that family of pointwise limits and of sequences of elements of C(X), G^u(X) the family of uniform limits of sequences of elements of G(X), and the topological space obtained by taking the zerosets of C(X) as a base for a topology on X. Then . Each of these inclusions can be proper, each of these families are vector lattices under the usual pointwise operations, and each of them are algebras with the possible exception of G^u(X). Each is continuous on an open dense subspace of X, and if X is a Baire space then each is continuous on a dense of X. The bounded elements of G^u(X) are closed under multiplication, each in G^u(X) is invertible, and G^u(X) is an algebra iff whenever . It is not known whether G^u(Q) (where Q is the space of rational numbers) is an algebra, but this latter is not closed under inversion. The relationship between some of these vector lattices and the complete ring of quotients and the epimorphic hull of C(X) is studied by making use of results on the latter due to R. Raphael and R. G. Woods that are not as yet published. Indeed, the present research is joint work with these two authors.