Timeline for Ideals of maximal tensor product of $C^{\ast}$-algebras
Current License: CC BY-SA 4.0
12 events
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| Jul 20, 2020 at 6:34 | comment | added | Robert Furber | @MathLover Of course, I meant "isomorphism classes of irreducible representations" in my previous comment. The statement I was referring to is the theorem at the start of Chapter 9 of Dixmier's C$^\ast$-algebras. | |
| Jul 20, 2020 at 5:51 | comment | added | Robert Furber | @MathLover I did not intend such a statement. Rather, the kernels of irreducible representations are known as primitive ideals. If a C$^\ast$-algebra is type I, meaning that in every representation the bicommutant of its image is a type I von Neumann algebra, then the map from irreducible representations to primitive ideals is a bijection (because it's an injection). For type I C$^*$-algebras the maximal and minimal tensors coincide, and you may get a good answer to your question, if restricted to primitive ideals. Some information on this is in Dixmier's book on C$^\ast$-algebras. | |
| Jul 16, 2020 at 5:52 | comment | added | Math Lover | @JamieGabe: I'll have a look at that paper. | |
| Jul 16, 2020 at 5:51 | comment | added | Math Lover | @RobertFurber: Are you saying that every ideal would be kernel of some irreducible representations? I am not sure about it. I would love to know if there is anything related to this. Would love to know references. | |
| Jul 16, 2020 at 2:17 | comment | added | Jamie Gabe | Clearly the kernel $\ker (A \otimes_{\max{}} B \to A\otimes_{\min{}} B)$ contains no non-zero elementary tensors. More generally, a two-sided closed ideal in $A\otimes_{\max{}} B$ contains an elementary tensor if and only if it has non-zero image in $A\otimes_{\min{}} B$. This can be shown by tweaking the proof of Kirchberg's Slice Lemma (Lemma 2.15 in Blanchard, Kirchberg "Non-simple purely infinite $C^\ast$-algebras: the Hausdorff case"). | |
| Jul 15, 2020 at 21:30 | comment | added | Robert Furber | @MathLover What is the reason you want to know about ideals rather than representations? | |
| Jul 15, 2020 at 17:25 | comment | added | Nik Weaver | I don't know the answers, but I'd be very surprised if there is any meaningful classification of ideals. | |
| Jul 15, 2020 at 16:29 | comment | added | Math Lover | @NikWeaver: I apologise for the broad question. I am mainly interested in classification of (closed) ideals of max tensor product(if there's any). Also, I'm interested in questions like whether every ideal has elementary tensor or not? As, I have not read anything about ideals of max tensor product so I'm mainly looking for a paper or book which deals with the same topic! | |
| Jul 15, 2020 at 15:15 | comment | added | Nik Weaver | As it stands it sounds a bit like "somebody prove some things for me, I don't know what, you'll have to figure that out"... | |
| Jul 15, 2020 at 15:14 | comment | added | Nik Weaver | You've asked a very similar question before. The question is so broad that it is very hard to think of a meaningful answer. If there is something specific you want to know, ask that. | |
| Jul 15, 2020 at 14:29 | history | edited | Math Lover | CC BY-SA 4.0 |
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| Jul 15, 2020 at 14:14 | history | asked | Math Lover | CC BY-SA 4.0 |