Vol. 65, No. 2, 1976

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Maximal ideals in algebras of topological algebra valued functions

William Hery

Vol. 65 (1976), No. 2, 365–373

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 (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 (C(T,A))(= vT) and (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 (C(T,A)). It is shown that if T is realcompact, A is a Q algebra with continuous inversion and either (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 (C(T,A)) is homeomorphic to (βT) ×ℳ(A), and if A is also realcompact, then (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.

Mathematical Subject Classification 2000
Primary: 46J99
Secondary: 54C40
Received: 17 June 1975
Published: 1 August 1976
William Hery