Timeline for Stinespring's dilation without $C^{\ast}$-algebras
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 20, 2015 at 10:01 | comment | added | m07kl | See Theorem 12 in "About the Connes Embedding Conjecture---algebraic approaches" by Ozawa | |
| Jul 14, 2015 at 13:40 | vote | accept | sunspots | ||
| Jul 14, 2015 at 12:34 | comment | added | Nik Weaver | @StefanWaldmann: sure, that makes sense. | |
| Jul 14, 2015 at 6:00 | comment | added | Stefan Waldmann | @Nik Weaver: (complete) positivity is in fact an algebraic notion. For *-algebras over $\mathbb{C}$ this has been investigated a lot by Schmuedgen in his unbounded operator algebra book, for more general *-algebras, please take a look at my answer. | |
| Jul 14, 2015 at 5:58 | answer | added | Stefan Waldmann | timeline score: 8 | |
| Jul 14, 2015 at 1:20 | comment | added | Nik Weaver | How do you define "completely positive" for a topological *-algebra? | |
| Jul 14, 2015 at 0:38 | review | Close votes | |||
| Jul 14, 2015 at 10:47 | |||||
| Jul 14, 2015 at 0:22 | comment | added | Yemon Choi | Finally: why not try running through the proof of Stinepsring's theorem, applied to the star-algebras you are interested in, and seeing how much survives? | |
| Jul 14, 2015 at 0:21 | comment | added | Yemon Choi | If, as I suspect, you are really thinking of topological star-algebra that somehow come from B(H) in some way, e.g. the algebra of operators affiliated to a ${\rm II}_1$-factor, or some kind of pro-Cstar algebra, then you should try to formulate your question in one of these precise settings, and make it clear what statement you actually hope to be true | |
| Jul 14, 2015 at 0:20 | comment | added | Yemon Choi | Banach star-algebras can look very very different from the Cstar case, as @JohannesHahn has alluded. For instance, consider the disc algebra with $f^*(z):=\overline{f(\overline{z})}$. (Palmer volume II is a good source for other examples if you have the patience to dig around in it.) | |
| Jul 13, 2015 at 23:45 | comment | added | Johannes Hahn | I imagine that a sufficiently ugly $\ast$-algebra does not need to have any useful hilbert space representations at all and the theorem could fail for that reason. | |
| Jul 13, 2015 at 22:36 | review | First posts | |||
| Jul 13, 2015 at 23:20 | |||||
| Jul 13, 2015 at 22:36 | history | asked | sunspots | CC BY-SA 3.0 |