I would like to show that: "every hereditary subalgebra $U$ of a $C^*$-algebra $C_0(X)$ for a locally compact Hausdorff Space $X$ has the form $J_E := \{f \in C_0(X) : f|_E=0 \}$ for a closed subset $E$ of $X$."
Any idea about how to show it concretely? (maybe using the Stone–Weierstrass Theorem?)
This is Proposition 6.4.3 from my book Mathematical Quantization. The point is that any hereditary C${}^\ast$-subalgebra of $C_0(X)$ is an ideal. To see this, let $B$ be a hereditary C${}^\ast$-subalgebra of $C_0(X)$ and let $f \in C_0(X)$ and $g \in B$ be positive. Then for $\epsilon = \frac{1}{\|f\|_\infty}$ we have $0 \leq \epsilon fg \leq g$, and so $fg \in B$ by the hereditary property. Taking linear combinations, we get that $fg \in B$ for any $f \in C_0(X)$ and $g \in B$. So $B$ is a C${}^*$-ideal and therefore has the desired form.
