I am looking for a hint or a methodology to approach this problem, showing in Arveson's A short course on spectral theory:
Let $X$ be a compact Hausdorff space and let $F$ be a proper closed subset of $X$. Let $A$ be the ideal of all functions $f\in C(X)$ that vanish throughout $F$ ($f(p)= 0\ \forall p ∈ F$). Note that $A$ is a $C^∗$-algebra in its own right.
a/ Show that $A$ has a unit if and only if $F$ is both closed and open.
$A$ is unital is equivalent to $\exists e\in A: fe=ef= f, \ \forall f\in A$
then we need $e$ to be satisfying $e(x)= 0\ \forall x\in F$ but $e(x)= 1 \ \forall x\in F^c$ then this implies that $F$ should be empty.
Is that not correct? if otherwise $F$ is not empty then there should be some $x\in F^c$ for which $e(x)= 1/2$ or how would a continuous function jump from $0$ to $1$, by Urysohn lemma, in fact $X$ is compact Hausdorff hence a normal space.
Actually if $X$ is not a discrete space, here a finite set, then the only open and closed subsets of $X$ are $\emptyset$ and $X$ itself, no ?
b/ Assuming that $F$ is not open, identify the unitalization of $A$ in concrete terms by exhibiting a compact Hausdorff space $Y$ such that $\tilde{A}\cong C(Y)$
How would I even approach this ? I assume that $A\cong C(sp(A))$ and that $sp(A)=\{\delta_x|\ x\in F^c\}$ the evaluation functionals