# Modular and primitive ideals 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 ideals of $$C_{0}(X,A)$$ are of the form $$\{ f\in C_{0}(X,A): f(x) \in I_x \; \forall x\in X \}$$ where $$I_x$$ is closed ideal in $$A$$ for all $$x$$.

What’s known about the modular and primitive ideals of $$C_{0}(X,A)$$.

Any references or ideas?

The primitive ideals $$P$$ of $$C_0(X,A)$$ are of the form $$P=\{f\in C_0(X,A): f(x)\in Q\}$$ where $$x\in X$$ and $$Q\in {\rm Prim}(A)$$.
The modular ideals are more difficult to describe. They have to have the form $$I=\{ f\in C_0(X,A): f(x_i)\in I_i\}$$ where $$\{x_i\}$$ is a compact subset of $$X$$ and $$I_i$$ is a modular ideal in $$A$$ for each $$i$$. But probably not every such ideal will be modular.
The example of the C$$^*$$-algebra of continuous functions $$f$$ from $$[0,1]$$ to the $$2\times 2$$ matrices for which $$f(1)={\rm diag}(\lambda (f), 0)$$ shows the sort of problem that can crop up: every primitive quotient is modular but the ideal $$\{0\}$$ is not modular. One can probably build such an example into a $$C_0(X(A)$$.