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?
As far as I can tell it is $C_0(X,A)\cong C_0(X)\otimes A$, where $\otimes$ denotes any $C^*$-tensor product you want, since $C_0(X)$ is nuclear (as it is abelian). This can be found for example as an exercise in Karen Strung's notes (look for Strung C* algebras on any search engine). For general results on ideals of tensor products, I recommend N. Brown and N. Ozawa's book "C*-algebras and finite dimensional approximations".