It is known that for a postliminal/GCR $C^*$-algebra the map $\pi\mapsto\ker\pi$ from (equivalence classes of) irreducible representations to their kernels is injective. If the algebra is separable, then the converse is also known to be true.

Questions:

1. Does there exist a non-separable non-postliminal $C^*$-algebra with injective $\pi\mapsto\ker\pi$?

2. Does there exist a simple such algebra?

Thank you.