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There are various examples of (non-separable) prime C$^*$-algebras $A$ which are not primitive. Is there an example for which $A$ is $\sigma$-unital but non-unital?

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It seems to me you could just tensor a unital example with the compacts. Any closed ideal of $A \otimes K$ has the form $I\otimes K$ where $I$ is a closed ideal of $A$, etc.

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