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For a completely
regular space T, topological algegbra A and algebra X, both commutative
and having identity, let C(T,A) = {f : T → A : f is continous},
C∗(T,A) = {f ∈ C(T,A) : f(T) is relatively compact} and ℳ(X) be the set of all
maximal ideals of codimension one in X endowed with the Gelfand topology (i.e., the
weak topology generated by {x : x ∈ X}, where x(m) = x + m). When A is the
real numbers, the spaces ℳ(C(T,A))(= vT) and ℳ(C∗(T,A))(= βT) are
well known. If A is any topological algebra, t ∈ T and m ∈ℳ(A), then
Mt,m = {f ∈ C(T,A) : f(t) ∈ m}∈ℳ(C(T,A)), and (t,m) → Mt,m is an injection
of T ×ℳ(A) into ℳ(C(T,A)). It is shown that if T is realcompact, A is a Q algebra
with continuous inversion and either ℳ(A) is locally equicontinuous or T is
discrete, then this injection is a homeomorphism. It is further shown that if the
assumption about T is reduced to complete regularity, then ℳ(C∗(T,A)) is
homeomorphic to (βT) ×ℳ(A), and if A is also realcompact, then ℳ(C(T,A)) is
homeomorphic to (vT) ×ℳ(A). These results are obtained for topological
algebras over the reals, the complexes and certain ultraregular topological fields
(including all nonarchimedean valued fields) with no assumptions of local
convexity.
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