Given a $C^\*$-algebra $A$, we write $\Omega(A)$ for its space of characters, i.e. its non-zero algebra homomorphisms $A \to \mathbb{C}$. If $X$ is a compact Hausdorff space, it is well-known that
$$X \to \Omega(C(X)): x \mapsto \operatorname{ev}\_x$$
is a homeomorphism of topological spaces.




---


Let $X,Y$ be compact Hausdorff spaces.


Let a $\*$-homomorphism $\pi: C(X) \to C(Y)$ be given. Then we can associate a continuous map $Y \to X$ by the composition
$$Y \cong \Omega(C(Y)) \to \Omega(C(X)) \cong X$$
where the middle map sends $\chi \mapsto \chi\circ \pi.$




---


Question: Is it possible to explicitly write down a formula for this map $Y \to X$? I guess this can be answered if we can write down an explicit inverse for the homeomorphism
$$X \to \Omega(C(X)).$$



A $*$-homomorphism $C(X) \to C(Y)$ gives a continuous map $Y \to X$
Given a maximal ideal $\mathfrak{m} \subseteq C(X)$, the corresponding point $x \in X$ or rather its singleton $\{x\}$ is the intersection of all zero sets $Z(f)$ of all functions $f \in \mathfrak{m}$.


So the map $\varphi : Y \to X$ associated to $\pi : C(X) \to C(Y)$ is defined by
$$\{\varphi(y)\} = \bigcap\_{f \in C(X),\, \pi(f)(y)=0} Z(f).$$
But as Yemon Choi points out, $\pi(f)=f \circ \varphi$ can also be seen as the definition of $\varphi$!


