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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.

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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.

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  • $\begingroup$ Thank you. Do you know about representations too? $\endgroup$
    – Math Lover
    May 22 '20 at 6:12
  • $\begingroup$ I'm not sure what you want in regards to representations ... $\endgroup$
    – Nik Weaver
    May 22 '20 at 10:28
  • $\begingroup$ I am mainly interested in seeing connection between representations of $A$ and of $C0(X,A)$. $\endgroup$
    – Math Lover
    May 22 '20 at 10:45
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    $\begingroup$ 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? $\endgroup$
    – Nik Weaver
    May 22 '20 at 11:30
  • $\begingroup$ What about the converse i.e.all representations of $C0(X,A)$ are of the form you mentioned? $\endgroup$
    – Math Lover
    May 23 '20 at 5:00

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