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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
, 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), Gu(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 Gu(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 Gu(X) are closed under multiplication,
each
in Gu(X) is invertible, and Gu(X) is an algebra iff
whenever
. 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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