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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.

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    $\begingroup$ Under additional set-theoretic assumptions, Akemann and Weaver demonstrated the existence of a unital Cstar algebra that has only one irrep up to unitary equivalence; it seems to me that such a beast must be simple, and it can't be separable or Type I by classical results $\endgroup$
    – Yemon Choi
    Commented Feb 5, 2020 at 21:43
  • $\begingroup$ (This is merely a comment rather than an answer since I expect Nik Weaver can turn up to give a better answer :) ) $\endgroup$
    – Yemon Choi
    Commented Feb 5, 2020 at 21:44
  • $\begingroup$ Yes, I can't find a simplicity argument in that paper. $\endgroup$
    – Bedovlat
    Commented Feb 5, 2020 at 22:30
  • $\begingroup$ The converse of what? the main statement is not written as an implication. You mean that for a C$^*$-algebra, postliminal implies the given injectivity property? Also, a simple C$^*$-algebra with the given injectivity property means a C$^*$-algebra with only 1 or 2 irreducible representations up to equivalence, if I'm correct...? $\endgroup$
    – YCor
    Commented Feb 5, 2020 at 22:56
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    $\begingroup$ It appears that the answer to your first question is unknown. In the (recent) second edition of Pedersen's "C*-algebras and their automorphism groups" the editors write that your question 1 "remains mysterious." This is in Section 6.8.9. They expand on their statement in the following section 6.9. $\endgroup$
    – Caleb Eckhardt
    Commented Feb 6, 2020 at 19:23

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