Timeline for Quantization and noncommutative deformations
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 3, 2011 at 8:32 | vote | accept | amine | ||
| Aug 3, 2011 at 8:33 | |||||
| Jul 31, 2011 at 12:57 | vote | accept | amine | ||
| Jul 31, 2011 at 12:57 | |||||
| Jul 29, 2011 at 15:57 | comment | added | Stefan Waldmann | Hi Theo, yeah, things work pretty well to a certain point. Positivity is an entirely <em>algebraic</em> gadget. It is only when you want to talk about competeness etc where analysis really enters. The story can be developed quite far, as lon as you do not ask for a good notion of spectrum. This is where things start to get weird, you need proper Hilbert spaces, self-adjointness etc. for that. But still, one can construct a lot of representations, study their unitary equivalences and so on. | |
| Jul 29, 2011 at 15:31 | comment | added | Theo Johnson-Freyd | @Stefan Waldmann: Oh, great. I've always believed that something like that should be true, but my only introduction to GNS was in Rieffel's C*-algebra class, which had an analytic bent. I am aware that much research has gone into "improving" from formal-power-series to honest Hilbert spaces and the like. As an aside, Graeme Segal once gave a talk at Berkeley in which he said something along the lines of "Never ask physicists what kinds of functions they want." | |
| Jul 29, 2011 at 10:15 | comment | added | Stefan Waldmann | Dear Theo: as a small comment I would just like to add that even in formal DQ there is a perfect notion of GNS construction etc yielding physically the correct results and so on. Of course, you can not expect to get bounded operators on Hilbert spaces but good $^\ast$-representations on pre Hilbert spaces (even over formal power series in $\hbar$ if you want) in the spirit of $O^\ast$-algebras. Your example of diffops is a perfect example for this, the positive functional is the integration over $N$ with respect to some positive density, once you have chosen a good star product for the symbols | |
| Jul 28, 2011 at 23:27 | history | edited | Theo Johnson-Freyd | CC BY-SA 3.0 |
added an estimate
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| Jul 28, 2011 at 17:04 | history | answered | Theo Johnson-Freyd | CC BY-SA 3.0 |