Let $X$ be a locally compact Hausdorff space and let $U,V$ be open subsets of $X$. Then $C_0(U)$ and $C_0(V)$ are ideals in $C_0(X)$. Sum of ideals in a $C^*$-algebra is again an ideal (without even taking the closure), so $C_0(U)+C_0(V)$ is an ideal in $C_0(X)$ and therefore corresponds to some open subset $W\subseteq X$, i.e. $C_0(U)+C_0(V)=C_0(W)$. I think that $W=U\cup V$.
It is clear that $C_0(U)+C_0(V)\subseteq C_0(U\cup V)$ as an ideal. However, I couldn't show that this inclusion map is surjective.
I have tried to consider the character space of $C_0(U)+C_0(V)$. It corresponds to kernels of irreducible representations of the algebea. Then one can first consider the irredicible representations that vanish on the ideal $C_0(U)$ (and correspond to irreps that factor through the quotient space) and those which doesn't vanish on the ideal and correspond to irreps of the ideal.
But, I would like to show it more directly, that is, to take a fumction in $C_0(U\cup V)$ and approximate it with sums of functions from $C_0(U), C_0(V)$.
Thank you for any help!
Let $f\in C_0(U\cup V)$. Then $f$ has compact support $A\subseteq U\cup V$. Let $A_0=A\setminus V$ and $A_1=A\setminus U$. Then $A_0$ and $A_1$ are compact and disjoint. By Urysohn's lemma, there is a continuous $f:X\to[0,1]$ which is zero on $A_0$ and $1$ on $A_1$. Then $fh$ is zero on $A_0$, and has compact support in $V$. Likewise $f(1-h)$ has compact support in $U$. Then $f=fh+f(1-h)\in C_0(U)+C_0(V)$.