Timeline for Non-separable non-postliminal $C^*$-algebra with injective $\pi\mapsto\ker\pi$
Current License: CC BY-SA 4.0
8 events
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| Feb 6, 2020 at 19:23 | comment | added | Caleb Eckhardt | 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. | |
| Feb 5, 2020 at 23:09 | comment | added | Bedovlat | YCor, 2 irreducibles? Where from? If there is exactly 1 primitive ideal then the injectivity implies that there is 1 equivalence class of irreducibles. | |
| Feb 5, 2020 at 22:56 | comment | added | YCor | 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...? | |
| Feb 5, 2020 at 22:52 | history | edited | YCor |
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| Feb 5, 2020 at 22:30 | comment | added | Bedovlat | Yes, I can't find a simplicity argument in that paper. | |
| Feb 5, 2020 at 21:44 | comment | added | Yemon Choi | (This is merely a comment rather than an answer since I expect Nik Weaver can turn up to give a better answer :) ) | |
| Feb 5, 2020 at 21:43 | comment | added | Yemon Choi | 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 | |
| Feb 5, 2020 at 21:01 | history | asked | Bedovlat | CC BY-SA 4.0 |