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

Theorem - Let $X$ be a locally compact space and $C_0(X,ℝ)$ the algebra of continuous functions $X\to ℝ$ that vanish at infinity (http://planetmath.org/ VanishAtInfinity), endowed with the sup norm $\parallel \cdot {\parallel }_{\mathrm{\infty }}$. Let $𝒜$ be a subalgebra of $C_0(X;ℝ)$ for which the following conditions hold:

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

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

Then $𝒜$ is dense in $C_0(X;ℝ)$.

Complex version

Theorem - Let $X$ be a locally compact space and $C_0(X)$ the algebra of continuous functions $X\to ℂ$ that vanish at infinity, endowed with the sup norm $\parallel \cdot {\parallel }_{\mathrm{\infty }}$. Let $𝒜$ be a subalgebra of $C_0(X)$ for which the following conditions hold:

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

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

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

Then $𝒜$ is dense in $C_0(X)$.