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

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.

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.
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.
• I am mainly interested in seeing connection between representations of $A$ and of $C0(X,A)$. May 22 '20 at 10:45
• Well, if $\pi$ is a representation of $A$ and $x \in X$ then $f \mapsto \pi(f(x))$ is a representation of $C_0(X,A)$. What else are you looking for? May 22 '20 at 11:30
• What about the converse i.e.all representations of $C0(X,A)$ are of the form you mentioned? May 23 '20 at 5:00