I am currently trying to understand the proof of Proposition 4.3.18 in Pedersen's Analysis now, which reads
To each Tychonoff space $X$ there is a Hausdorff compactification $\beta(X)$, with the property that every continuous function $\Phi: X \to Y$, where $Y$ is a compact Hausdorff space, extends to a continuous function $\beta \Phi: \beta(X) \to Y$.
The proof starts by noting that $C_b(X)$ is a commutative unital C$^*$-algebra, and is therefore isometrically isomorphic to a (commutative and unital) C$^*$-algebra of the form $C(\beta(X))$, where $\beta(X)$ is a compact Hausdorff space.
By the Gelfand duality between the category of commutative and unital C$^*$-algebras and the category of compact Hausdorff spaces, we can take $\beta(X) = \Omega(C_b(X))$, the space of characters on $C_b(X)$.
Then we can define a map $\iota: X \to \beta(X)$, where $\iota(x)(\phi) := \phi(x)$ for all $x \in X$ and $\phi \in \beta(X)$.
The particular part of the proof that I am struggling to understand is the proof that $\iota(X)$ is dense in $\beta(X)$.
He argues that if $\iota(X)$ is not dense in $\beta(X)$, then there is a non-zero continuous map $f: \beta(X) \to \mathbb{C}$ vanishing on $\iota(X)$. This I understand. He then says that under the identification $C_b(X) = C(\beta(X))$, this is impossible. This is the sentence I am stuck on. Why is it impossible under this identification?
We have that $C_b(X)$ is isometrically isomorphic to $C(\Omega(C_b(X)))$ via the map $\delta: g \mapsto (\delta_g: \Omega(C_b(X)) \to \mathbb{C}, \phi \mapsto \phi(g))$. I am pretty sure what Pedersen is getting at is that the map $\delta^{-1}(f)$ is zero, but I am not able to show that this is the case. This answer also claims that a similar map is zero.
In summary, my question is:
Can we show that $\iota(X)$ is dense in $\beta(X)$ by showing that $\delta^{-1}(f) = 0$? If so, how do we do this?
Best Answer
Recently, I wrote all this out in detail for myself, so here I share my notes with you. Note that the assumption that $X$ is Tychonoff can be ommitted. The construction works for every topological space. The Tychnoff assumption is only there to ensure that the canonical inclusion is injective.
Recall that if $A$ is a commutative $C^*$-algebra, then we can consider the space of characters $\Omega(A)$ . If $A$ is a unital $C^*$-algebra, then this becomes a compact Hausdorff space for the weak$^*$-topology. Note that we have a natural map $$i_X: X \to \Omega(C_b(X)): x \mapsto \text{ev}_x.$$ Clearly this is a continuous map, as an easy argument with nets shows.
Lemma: The map $i_X$ has dense image.
Proof: Assume to the contrary that $\overline{i_X(X)}\subsetneq \Omega(C_b(X))$. Then Urysohn's lemma applied to the compact Hausdorff space $\Omega(C_b(X))$ gives a non-zero continuous function $f: \Omega(C_b(X))\to \mathbb{C}$ that is zero on $i_X(X)$. Consider the canonical isomorphism $$\Psi: C_b(X) \to C(\Omega(C_b(X))): \omega \mapsto \text{ev}_\omega.$$ Choose $\omega \in C_b(X)$ with $\text{ev}_\omega = f$. Then for all $x \in X$, we have $$\omega(x) = \text{ev}_x(\omega) = \text{ev}_\omega(\text{ev}_x) = f(i_X(x)) = 0$$ so $\omega = 0$, which is a contradiction. $\quad \square$
Theorem: If $X$ is a topological space, then $(\Omega(C_b(X)), i_X)$ is a Stone-Čech compactification of $X$.
Proof: Let $K$ be a compact Hausdorff space and let $f: X \to K$ be a continuous map. This induces a $*$-morphism $$C(f): C(K) \to C_b(X): g \mapsto g \circ f$$ and this then induces a continuous map $$\Omega(C(f)): \Omega(C_b(X)) \to \Omega(C(K)): \chi \mapsto \chi \circ C(f)$$ Consider the homeomorphism $$i_K: K \to \Omega(C(K)): k \mapsto \text{ev}_k.$$
Then we define the continuous map $F:= i_K^{-1}\circ \Omega(C(f)): \Omega(C_b(X)) \to K$. Moreover, we have $F\circ i_X= f$. Indeed, if $x \in X$, then $$i_K(F \circ i_X(x)) = i_K (F(\text{ev}_x)) = \Omega(C(f))(\text{ev}_x) = \text{ev}_x \circ C(f)= \text{ev}_{f(x)}= i_K(f(x))$$ so that by injectivity of $i_K$ we obtain $F \circ i_X = f$.
The condition $F \circ i_X = f$ determines $F$ uniquely on $i_X(X)$, which is dense in $\Omega(C_b(X))$ by the preceding lemma. Thus $F$ is unique. $\quad \square$