Exercise 28, § 3

Let X be a compact topological space and $n$ an integer $\geqslant 1$. Let I be a two-sided ideal of the Banach algebra $\mathscr{C}(X,M_n(\mathbf{C}))$. Now prove that there exists a unique closed subset $S\subset$ X such that I is the set of the $f\in$ $\mathscr{C}(X,M_n(\mathbf{C}))$ whose restriction to S is zero.