Timeline for Von Neumann algebra associated to the infinite Cuntz algebra
Current License: CC BY-SA 3.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Mar 15, 2015 at 20:08 | comment | added | Marcel Bischoff | If you want the full Fock space, I think the idea still works, see arxiv.org/abs/math/9810003 and msp.org/pjm/1997/177-2/pjm-v177-n2-p07-s.pdf | |
| Jul 29, 2013 at 14:08 | comment | added | André Henriques | I see. Thanks for the clarification. | |
| Jul 29, 2013 at 12:19 | comment | added | Ulrich Pennig | What might have caused the confusion is that the descriptions in terms of faithful representations on Fock space really look very similar, but note that for $\mathcal{O}_{\infty}$ I need to take the full tensor algebra and not just the antisymmetric part. If I only take the latter and take creation and annihilation ops then I get CAR. | |
| Jul 29, 2013 at 12:14 | comment | added | Ulrich Pennig | No, that is not what I said. The CAR-algebra is isomorphic to the UHF-algebra $M_{2^{\infty}}$ (see Example 1.2.6 in Rordam's "Classification of Nuclear, Simple $C^*$-algebras"). I might have said that the methods in our paper apply to the CAR-algebra as well, which is true since both are strongly self-absorbing. | |
| Jul 29, 2013 at 11:04 | comment | added | André Henriques | I'm confused by your remark: didn't you say that $\mathcal O_\infty$ is isomorphic to the $C^*$-algebraic CAR algebra? Now you seem to indicate that they are only similar? | |
| Jul 29, 2013 at 8:47 | comment | added | Ulrich Pennig | This is one of the reasons, I was interested in the question. The Cuntz algebra $\mathcal{O}_{\infty}$ has some properties in common with the ($C^*$-algebraic) CAR-algebra - for example both are strongly self-absorbing. But it is purely infinite and therefore contains no finite projection, which in my head "moves it closer to" type $III_1$-factors. | |
| Jul 28, 2013 at 20:04 | history | answered | André Henriques | CC BY-SA 3.0 |