StoneWeierstrass theorem for locally compact spaces
The following results generalize the StoneWeierstrass theorem (and its complex version (http://planetmath.org/StoneWeierstrassTheoremComplexVersion)) for locally compact spaces. The cost of this generalization^{} is that one no longer deals with all continuous functions^{}, but only those that vanish at infinity.
Real version
Theorem  Let $X$ be a locally compact space and ${C}_{0}(X,\mathbb{R})$ the algebra of continuous functions $X\to \mathbb{R}$ that vanish at infinity (http://planetmath.org/ VanishAtInfinity), endowed with the sup norm $\parallel \cdot {\parallel}_{\mathrm{\infty}}$. Let $\mathcal{A}$ be a subalgebra of ${C}_{0}(X;\mathbb{R})$ for which the following conditions hold:

1.
$\forall x,y\in X,x\ne y,\exists f\in \mathcal{A}:f(x)\ne f(y)$, i.e. $\mathcal{A}$ separates points.

2.
For each $x\in X$ there exists $f\in \mathcal{A}$ such that $f(x)\ne 0$.
Then $\mathcal{A}$ is dense in ${C}_{0}(X;\mathbb{R})$.
Complex version
Theorem  Let $X$ be a locally compact space and ${C}_{0}(X)$ the algebra of continuous functions $X\to \u2102$ that vanish at infinity, endowed with the sup norm $\parallel \cdot {\parallel}_{\mathrm{\infty}}$. Let $\mathcal{A}$ be a subalgebra of ${C}_{0}(X)$ for which the following conditions hold:

1.
$\forall x,y\in X,x\ne y,\exists f\in \mathcal{A}:f(x)\ne f(y)$, i.e. $\mathcal{A}$ separates points.

2.
For each $x\in X$ there exists $f\in \mathcal{A}$ such that $f(x)\ne 0$.

3.
If $f\in \mathcal{A}$ then $\overline{f}\in \mathcal{A}$, i.e. $\mathcal{A}$ is a selfadjoint subalgebra of $C(X)$.
Then $\mathcal{A}$ is dense in ${C}_{0}(X)$.
Title  StoneWeierstrass theorem for locally compact spaces 

Canonical name  StoneWeierstrassTheoremForLocallyCompactSpaces 
Date of creation  20130322 18:41:09 
Last modified on  20130322 18:41:09 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  5 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 46J10 