Timeline for Motivation for $C^*$-algebras
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| May 19, 2021 at 21:31 | comment | added | jjcale | One motivation is quantum statistical mechanics, see e.g. "Operator Algebras and Quantum Statistical Mechanics 1", springer.com/gp/book/9783540170938 . | |
| May 19, 2021 at 17:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 20, 2021 at 12:55 | comment | added | user85913 | Not an answer to "why C*-algebras?" but a comment on the more focused "why exotic group C*-algebras?": one perfectly reasonable answer is "to build interesting examples of C*-algebras". Another is that the K-theory of group C*-algebras (and crossed products) is a receptacle for interesting invariants: equivariant indices etc. Different C*-algebras attached to the same group can have different K-theory, and it appears that for some purposes the nicest K-theoretic properties are enjoyed by "exotic" algebras in between the two standard ones. See the work of Baum-Guentner-Willett and others. | |
| Apr 20, 2021 at 9:32 | comment | added | Emiel Lanckriet | I am a student finishing my masters degree, but I have only learned about $C^*$-algebras in the last year. The question came from a fellow student who was also interested in the use of this area of mathematics. It is correct that I don't know a lot about the wider history of the subject, but I would say that the question is more "What did we gain from studying it?", than "Why did we start studying it?" although both are interesting questions. | |
| Apr 19, 2021 at 16:29 | comment | added | Nik Weaver | @MatthewDaws very good point. | |
| Apr 19, 2021 at 15:40 | comment | added | Matthew Daws | Can I read between the lines, and guess you are a student? May I ask: at what point in your education? It could be (I of course cannot know for sure) that the question, in response to a talk you gave, might have been more asking "Do you, as a student, know something of the wider history of this subject?" rather than (as I think people here might be tempted to read) "Why is this area of Mathematics important?" Just a guess... | |
| Apr 19, 2021 at 12:24 | comment | added | Branimir Ćaćić | Seriously, if you’re interested in the original historical motivation for $C^\ast$-algebras, see Nik Weaver’s answer in his second link. Compare the Kadison–Singer problem, whose original formulation comes straight out of quantum mechanics as formulated in terms of $C^\ast$-algebras of bounded observables. [BTW, the recent solution of the Kadison–Singer problem depends crucially on Weaver’s reformulation of Anderson’s reformulation, if I understand the basic history correctly?] | |
| Apr 19, 2021 at 10:58 | comment | added | David Roberts♦ | I changed the notation, $C^*$ is the usual form, not $\mathbb{C}^*$ | |
| Apr 19, 2021 at 10:57 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Mathematical formatting, tag
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| Apr 19, 2021 at 10:49 | review | Close votes | |||
| Apr 19, 2021 at 18:34 | |||||
| Apr 19, 2021 at 10:40 | answer | added | ThiKu | timeline score: 3 | |
| Apr 19, 2021 at 10:39 | comment | added | Nik Weaver | See What are the applications of operator algebras to other areas. My answer to States in C${}^*$-algebra and their origins in physics might help too. | |
| Apr 19, 2021 at 10:27 | history | edited | YCor |
edited tags
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| Apr 19, 2021 at 9:47 | review | First posts | |||
| Apr 19, 2021 at 10:43 | |||||
| Apr 19, 2021 at 9:21 | history | asked | Emiel Lanckriet | CC BY-SA 4.0 |