Dimension of dense subalgebras of $C(X)$

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 $\leq n$.