Exercise 13, § 4

Let $X$ be a compact space and let $A$ denote the normed algebra $\mathscr{C}(X; \mathbf{R})$. For each subset $M$ of $A$, let $V(M)$ denote the set of all $x \in X$ such that $u(x) = 0$ for all $u \in M$. For each subset $Y$ of $X$, let $\mathfrak{J}(Y)$ denote the set of all $u \in A$ such that $u(x) = 0$ for all $x \in Y$.

a) $V(M)$ is a closed subspace of $X$ and $\mathfrak{J}(Y)$ is a closed ideal of $A$. If $a$ is the ideal of $A$ generated by a subset $M$ of $A$, then $V(M) = V(a)$; $V$ and $I$ are order-reversing mappings (with respect to the order relation of inclusion), and we have $V\{0\} = X; \ V(A) = \varnothing; \ V(\{u\}) = \varnothing$ for every unit $u$ of $A$;

for every family $(M_\lambda)_{\lambda \in L}$ of subsets of $A$; $V(M.M') = V(M) \cup V(M')$; $\mathfrak{J}(X) = \{0\}; \ \mathfrak{J}(\mathscr{C}) = A; \ \mathfrak{J}\left( \bigcup_{\lambda \in L} Y_\lambda \right) = \bigcap_{\lambda \in L} \mathfrak{J}(Y_\lambda)$ for every family $(Y_\lambda)_{\lambda \in L}$ of subsets of $X$.

b) Show that if $a$ is any ideal of $A$, then $\mathfrak{J}(V(a)) = \overline{a}$, and that if $Y$ is any subset of $X$, then $V(\mathfrak{J})(Y) = \overline{Y}$. (For the first of these relations, observe that $V(\overline{a}) = V(a)$ and that we may therefore suppose $a$ closed; note that if $x, y$ are two distinct points of $X - V(a)$, there exists $u \in a$ such that $u(x) \neq u(y)$, and use Proposition 4 of no. 2. To prove the second relation, use Urysohn's theorem.)

c) Deduce from b) that $x \to \mathfrak{J}(\{x\})$ is a bijection of $X$ onto the set of maximal ideals of $A$, which are therefore closed. An ideal of $A$ is closed if and only if it is an intersection of maximal ideals. The ring $A$ has no radical.

d) Show that the mapping $e \to V(\{e\})$ is a bijection of the set of idempotents of $A$ onto the set of subsets of $X$ which are both open and closed.

A point $x \in X$ is isolated if and only if the maximal ideal $\mathfrak{J}(\{x\})$ is principal.