Abstract
One main result of this article is a characterization of all
(G 1) as topological algebras withG 1 open inℂ. For this and for similar results of Arens [4], Carpenter [6] and Brooks [5] Runge's approximation theorem is an important tool. It is extended to a characterization of all
(G), whereG is a polynomially convex, open subset of aℂ m.There is stated a similar characterization of allC(G) withG open in aℂ m,which is based on approximation by polynomials in theZ j . and\(\overline Z _j \). A second main result is a characterization of allC(M),whereM is a paracompact manifold of even dimension and which proceeds from ideas in the article [3]. MoreoverC(P 1(ℂ)) is characterized as a top. algebra. All these characterizations base upon the theory of the Gelfand representation of seminormedℂ-algebras.
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Fritz, G. Charakterisierungen einiger Funktionenalgebren. Manuscripta Math 24, 97–113 (1978). https://doi.org/10.1007/BF01168565
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DOI: https://doi.org/10.1007/BF01168565