# Stone-Weierstrass theorem for locally compact spaces

The following results generalize the Stone-Weierstrass 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

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 $\|\cdot\|_{\infty}$. Let $\mathcal{A}$ be a subalgebra of $C_{0}(X;\mathbb{R})$ for which the following conditions hold:

1. 1.

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

2. 2.

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

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 $X\to\mathbb{C}$ that vanish at infinity, endowed with the sup norm $\|\cdot\|_{\infty}$. Let $\mathcal{A}$ be a subalgebra of $C_{0}(X)$ for which the following conditions hold:

1. 1.

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

2. 2.

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

3. 3.

If $f\in\mathcal{A}$ then $\overline{f}\in\mathcal{A}\;$, i.e. $\mathcal{A}$ is a self-adjoint subalgebra of $C(X)$.

Then $\mathcal{A}$ is dense in $C_{0}(X)$.

Title Stone-Weierstrass theorem for locally compact spaces StoneWeierstrassTheoremForLocallyCompactSpaces 2013-03-22 18:41:09 2013-03-22 18:41:09 asteroid (17536) asteroid (17536) 5 asteroid (17536) Definition msc 46J10