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



MELVIN HENRIKSEN, Harvey Mudd College, Claremont, California  91711, USA
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 $f \in C(X)$, let $f'(x) = \frac1f(x)$ when f(x) = 0, and let f(x) = 0 otherwise. Let G(X) denote the subalgebra of $\bbd R_X$ generated by C(X) and $\{f':f \in C(X)\}$. Then G(X) is the smallest von Neumann regular sublagebra of $\bbd R_X$ 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), Gu(X) the family of uniform limits of sequences of elements of G(X), and $X\delta$ the topological space obtained by taking the zerosets of C(X) as a base for a topology on X. Then $C(X) \subset G(X) \subset G^u(X) \subset Ba_1(X) \subset C(X\delta)$. 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 Gu(X). Each $f \in G(X)$is continuous on an open dense subspace of X, and if X is a Baire space then each $f \in G^u(X)$ is continuous on a dense $G\delta$ of X. The bounded elements of Gu(X) are closed under multiplication, each $f \geq 1$ in Gu(X) is invertible, and Gu(X) is an algebra iff $f^2 \vee 1 \in G^u(X)$ whenever $f \in G^u(X)$. It is not known whether Gu(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.


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