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Dimension of dense subalgebras of $ C(X)$

Authors: Juan B. Sancho de Salas and Ma. Teresa Sancho de Salas
Journal: Proc. Amer. Math. Soc. 105 (1989), 491-499
MSC: Primary 54C40; Secondary 46J10, 54F45
MathSciNet review: 929426
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Abstract: The real spectrum of any $ {\mathbf{R}}$-algebra $ A$ is the set of all maximal ideals of $ A$ with residue field $ {\mathbf{R}}$, endowed with the initial topology for the functions induced by the elements of $ A$. We prove that a compact metric space $ X$ has dimension $ \leq n$ if and only if $ X$ is the real spectrum of an algebra of Krull dimension $ \leq n$; so that the dimension of $ X$ is the minimum of the Krull dimensions of all dense subalgebra of $ C(X)$. Moreover, we prove that a compact Hausdorff space $ X$ has covering dimension $ \leq n$ if and only if every countably generated subalgebra of $ C(X)$ is contained in the closure of a subalgebra of Krull dimension $ \leq n$.

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Keywords: Dense subalgebras of $ C(X)$, Krull dimension, real spectrum, inverse limits of polyhedra, covering dimension
Article copyright: © Copyright 1989 American Mathematical Society