Timeline for von neumann algebras and measurable spaces
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 12, 2020 at 3:48 | comment | added | Dmitri Pavlov | “the maps between these spaces have to preserve an extra structure”: such maps are precisely open maps of hyperstonean spaces. So hyperstonean spaces with such morphisms do form a subcategory of topological spaces, albeit not a full subcategory. See Theorem 1.1 in arxiv.org/abs/2005.05284. | |
| Aug 18, 2013 at 11:50 | vote | accept | Issam Ibnouhsein | ||
| Jul 27, 2013 at 16:46 | comment | added | Issam Ibnouhsein | Yes, you are right concerning C*-algebras, though the fact we are restricted is pretty obvious so it didn't really bother me, whereas vNAs got me confused with the subtelties about localizability, integration etc. But now its all clear. Concerning the second point, I thought we could recover the topology from the $\sigma$-algebra through the generator of the $\sigma$-algebra, but your examples of $\mathbb{R}$ and $\mathbb{R}^2$ made me understand I was wrong. Thanks for your answer which helped me on the "psychological" level, and to all others for their (technically) detailed answers! | |
| Jul 27, 2013 at 16:32 | comment | added | Ollie | If you're worried about excluding these pathological measure spaces, you should note the same problem also applies to $C^*$-algebras: we are dualising only compact Hausdorff spaces, not general topological spaces! I'm not sure what you mean by getting the open sets from the generator of the Borel set. The topology of a Borel space can't be recovered from its $\sigma$-algebra - for instance $\mathbb{R}$ and $\mathbb{R}^2$ with their standard topologies are not homeomorphic - however the corresponding measurable spaces are isomorphic. | |
| Jul 27, 2013 at 15:17 | comment | added | Issam Ibnouhsein | And by topology I mean HKB topology, not less rigid versions which clearly can't be reduced to measure theory. PS: I understand Segal's point of pathological cases not being essential, but this view of measure theory might turn out to be too restrictive (who knows how measure theory will develop) so I prefer to consider it in full generality, and explicitely point at the integration part of it when dealing with category equivalences like vNA, hyperstonian spaces etc | |
| Jul 27, 2013 at 15:03 | comment | added | Issam Ibnouhsein | Thanks for your answer Ollie. Your point of HypStone not being a subcategory of Top makes it easier to accept the idea that vNA are not a particular case of C*-algebras, but from your answer I would deduce that a better name for vNAs should be "non-commutative integration theory" and not "non-commutative measure theory" because of the pathological cases I was evoking. One more general question: how come topology is not considered a particular case of measure theory since you can get the primitive notion (open sets) from the generator of the Borel set, hence all other notions (continuity etc)? | |
| Jul 27, 2013 at 6:08 | history | answered | Ollie | CC BY-SA 3.0 |