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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 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?

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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 $C_0(X(A)$.

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  • $\begingroup$ Dear Douglas, Thank you! Do you have any reference for the results you mentioned? $\endgroup$
    – Math Lover
    Jul 31, 2020 at 14:39
  • $\begingroup$ Not really, I am afraid. The description of the primitive ideals is standard. Modular ideals are rather seldom referred to in the world of C*-algebras. $\endgroup$ Jul 31, 2020 at 16:14

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