Stone-Weierstrass Theorem/Lemma
Lemma
Let $T = \struct {X, \tau}$ be a compact topological space.
Let $\map C {X, \R}$ be the set of real-valued continuous functions on $T$.
Let $\times$ be the pointwise multiplication on $\map C {X, \R}$.
Let $\struct {\map C {X, \R}, \times}$ be the Banach algebra with respect to $\norm \cdot_\infty$.
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Let $\AA$ be a unital subalgebra of $\map C {X, \R}$.
Suppose that $\AA$ separates points of $X$, that is:
- for distinct $p, q \in X$, there exists $h_{p q} \in \AA$ such that $\map {h_{p q} } p \ne \map {h_{p q} } q$.
Let $\struct {C', \norm {\,\cdot\,}_{C'} }$ be the dual space of $\struct {\map C {X, \R}, \norm {\,\cdot\,}_\infty }$.
Let $B' \subseteq C'$ be the closed unit ball, that is:
- $B' := \set {\ell \in C' : \norm \ell_{C'} \le 1}$
Let:
- $U := \set {\ell \in B' : \ell \restriction_\AA = 0}$
Let $\map E U$ be the set of extreme points of $U$.
Then:
- $\map E U \setminus \set 0 = \O$
Proof
Aiming for a contradiction, suppose there is an $\ell \in \map E U$ such that:
- $(1):\quad \ell \ne 0$
Then:
- $\norm \ell_{C'} = 1$
since:
- $\ell = \norm \ell_{C'} \underbrace { \norm \ell_{C'}^{-1} \ell}_{\in U} + \paren {1 - \norm \ell_{C'} } \underbrace {0}_{\in U}$
Note that $X$ is Hausdorff by Topological Space Separated by Mappings is Hausdorff.
Thus we can apply Riesz-Kakutani Representation Theorem.
That is, there exists a signed measure $\mu$ on $X$ such that:
- $\ds \forall f \in \map C {X, \R} : \map \ell f = \int_X f \rd \mu$
and:
- $\map {\size \mu} X = 1$
Let $\map \supp \mu$ denote the support of $\mu$.
Consider any $g \in \AA$ such that:
- $\forall x \in X : 0 < \map g x < 1$
Then we have:
- $g \mu \in U$
Let:
- $\ds a := \int_X g \rd \size {\mu}$
and:
- $\ds b := \int_X \paren {1 - g} \rd \size {\mu}$
so that:
- $a + b = 1$
As:
- $\ds \mu = a \underbrace{\frac {g \mu} a}_{\in U} + b \underbrace {\frac {\paren {1 - g} \mu} b}_{\in U}$
we have:
- $\ds \mu = \frac {g \mu} a$
Especially:
- $\forall x \in \map \supp \mu : \map g x = a$
We claim that:
- $\exists x_0 \in X : \map \supp \mu = \set {x_0}$
Aiming for a contradiction, suppose there are distinct $p, q \in \map \supp \mu$.
Then by hypothesis there exists $h_{p q} \in \AA$ such that:
- $\map {h_{p q} } p \ne \map {h_{p q} } q$
Let $C,D > 0$ such that:
- $C > \norm {h_{p q} }_\infty $
and:
- $D > \norm {h_{p q} }_\infty + C$
Then:
- $\ds g := \frac {h_{p q} + C} D$
must be constant on $\map \supp \mu$.
This is a contradiction.
$\Box$
Finally, we have:
\(\ds \forall f \in \map C {X, \R}: \, \) | \(\ds \map \ell f\) | \(=\) | \(\ds \map \mu {\set {x_0} } \map f {x_0}\) | as shown above | ||||||||||
\(\ds \) | \(=\) | \(\ds \map \ell 1 \map f {x_0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | since the constant $1 \in \AA$ |
That is $\ell = 0$.
This is a contradiction to $(1)$.
$\blacksquare$