Need reference for ideals and representations of $C_0(X,A)$

Question

Let $A$ be $C^{\ast}$- Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\ast}$ (for $t\in X$). It is well known that $C_0(X,A)$ is $C^{\ast}-$ Algebra.

What’s known about ideals and representations of $C_0(X,A)$?

My guess is that it must be related with ideals and representations of $A$. Can someone give a reference or some ideas?

P.S: The same question was first posted on MSE but unfortunately I dint not get any answer so I am posting it here.

Answer 1

For each $x \in X$ let $I_x$ be a closed ideal of $A$. Then the set of $f \in C_0(X,A)$ satisfying $f(x) \in I_x$ for all $x$ is clearly an ideal of $C_0(X,A)$, and it shouldn't be too hard to show that every closed ideal has this form.

I assume this is "well known" but I don't have a reference.

As for representations, use the fact that $C_0(X,A)$ is $*$-isomorphic to the C*-algebra tensor product $C_0(X) \otimes A$. (Since $C_0(X)$ is abelian there is only one tensor product.) Thus the representations of $C_0(X,A)$ correspond to pairs of representations of $C_0(X)$ and $A$ on the same Hilbert space and whose ranges commute.