Timeline for What is a type $\text{II}_1$ factor von Neumann algebra?
Current License: CC BY-SA 4.0
16 events
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| Sep 30 at 3:04 | 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. | |
| Sep 19, 2020 at 0:06 | review | First posts | |||
| Sep 19, 2020 at 0:50 | |||||
| Sep 13, 2020 at 22:39 | comment | added | David Handelman | I’m voting to close this question because not enough effort has been put in by the proposer. | |
| Sep 12, 2020 at 17:29 | comment | added | Yemon Choi | It is still not clear to me what the question here hopes to achieve: the updated version has a link to a short set of notes which gives the basic definitions, and to my eyes as a mathematician those notes are more accurate than the answer you've written for your own question. It seems like you want someone to write a version of the standard definition which uses the language/perspective of quantum information theory; but what you've written below has many imprecisions/errors from a mathematician's point of view | |
| Sep 11, 2020 at 23:58 | history | edited | Victor V Albert | CC BY-SA 4.0 |
added 275 characters in body
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| Sep 11, 2020 at 23:36 | comment | added | Victor V Albert | Sorry all for posting such a vague question. I've looked into it a little, and wrote up something that I think could help the people I had in mind. Feel free to let me know any more thoughts. | |
| Sep 11, 2020 at 23:35 | answer | added | Victor V Albert | timeline score: 0 | |
| Sep 11, 2020 at 5:04 | comment | added | David Roberts♦ | @Yemon I agree. One can at minimum chase the WP definition chain to at least get an idea of what objects are involved, and what examples to look at. I was going to suggest von Neumann's original work, which of course was motivated by physics, but I imagine it's low on intuition based on decades of experience, as a modern survey would be! | |
| Sep 11, 2020 at 2:56 | review | Close votes | |||
| Sep 29, 2020 at 21:14 | |||||
| Sep 11, 2020 at 2:13 | comment | added | Yemon Choi | @DavidRoberts Fair enough. For what it's worth, I am not sure why the OP is linking to a paper of Witten as their example of a formal definition of a two-one factor. The Wikipedia page on von Neumann algebras is actually fairly good at leading up to the definition of a factor and then the classification result of Murray and von Neumann en.wikipedia.org/wiki/Von_Neumann_algebra | |
| Sep 11, 2020 at 1:32 | history | edited | LSpice | CC BY-SA 4.0 |
II is numbering, not multiplication, so should be text
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| Sep 11, 2020 at 0:40 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
filled in reference, added tag
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| Sep 11, 2020 at 0:33 | comment | added | David Roberts♦ | @YemonChoi one could be charitable and guess the OP meant how one could see $II_1$ factors as having canonical representations on certain Hilbert spaces arising from other known structures. But I agree it's not worded that way. | |
| Sep 11, 2020 at 0:17 | comment | added | Yemon Choi | Moreover, it is not clear what "intuition" is supposed to mean here. What algebraic objects of a similar nature have you encountered? Obviously there is no point trying to say that two-one factors are analogues of simple groups, if you are not already familiar with results about simple groups. But if you are familiar with examples of simple and non-simple groups then one can start to explain what von Neumann factors are, and then one can try to say something about what makes two-one factors special | |
| Sep 11, 2020 at 0:15 | comment | added | Yemon Choi | You say that you "have not seen anyone relate such factors to more commonly understood Hilbert spaces". This phrasing suggests you think that factors are examples of Hilbert spaces? If so, that is already a misconception | |
| Sep 11, 2020 at 0:09 | history | asked | Victor V Albert | CC BY-SA 4.0 |