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?



Is there a non-unital $\sigma$-unital prime C$^*$-algebra which is not primitive?
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.


