Timeline for Von Neumann algebra associated to the infinite Cuntz algebra
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Aug 31, 2012 at 7:01 | comment | added | Ulrich Pennig | @Mikael: Not really an argument. But there are states on $\mathcal{O}_n$ for finite $n$ which yield type $III_{1/n}$-factors. Unfortunately, those are built upon the fact, that $\mathcal{O}_n$ contains a core UHF-algebra and a conditional expectation (via the gauge action) onto it. For $\mathcal{O}_{\infty}$ you only get an AF-algebra. | |
| Aug 31, 2012 at 6:55 | vote | accept | Ulrich Pennig | ||
| Aug 30, 2012 at 19:22 | comment | added | Mikael de la Salle | @Jon: The state constructed in Dima's paper is just the vector state $\langle \cdot \Omega,\Omega\rangle$ restricted to the $C^*$-algebra generated by the operators $s_v + s_v^*$ for $v$ is some real subspace of $H$ (see def 2.3 and remark 2.6 in the paper). I do not believe this is so relevant if one is looking for a state on $\mathcal O_\infty$. | |
| Aug 30, 2012 at 18:41 | comment | added | Jon Bannon | What about these free quasi-free states: msp.berkeley.edu/pjm/1997/177-2/pjm-v177-n2-p07-p.pdf? (I haven't looked carefully, this is an off the cuff comment...) | |
| Aug 30, 2012 at 18:33 | comment | added | Mikael de la Salle | I do not know, but I am not very familiar with $\mathcal O_\infty$ (and with $C^*$-algebras in general). Do you have an argument why such a state should exist? | |
| Aug 30, 2012 at 13:45 | comment | added | Ulrich Pennig | Do you know about any states on $\mathcal{O}_{\infty}$ which would yield a type $III$-von Neumann algebra? | |
| Aug 30, 2012 at 12:06 | comment | added | Ollie | Snap! Nice proof. | |
| Aug 30, 2012 at 11:55 | history | answered | Mikael de la Salle | CC BY-SA 3.0 |