Timeline for Non-commutative geometry from von Neumann algebras?
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2009 at 1:19 | comment | added | Dave Penneys | agreed. your heuristic is much better. i just wanted to explain why i required diffuseness and separability. | |
| Oct 29, 2009 at 19:18 | comment | added | Dmitri Pavlov | My point was that the general theorem illustrates the statement "Hence the study of von Neumann algebras is sometimes referred to as non-commutative measure theory." better than a particular example of an isomorphism between a von Neumann algebra and an algebra of bounded functions on a measurable space. Using the axiom of choice one can prove that every measurable space is isomorphic to a disjoint union of point and real lines, hence there are more measurable spaces than just [0, 1]. (But not much more since after all we have a complete classification of measurable spaces.) | |
| Oct 29, 2009 at 16:44 | comment | added | Dave Penneys | The isomorphism from an abelian von Neumann algebra to $L^\infty[0,1]$ that I had in mind is not canonical - it's just the spectral theory. For this you need diffuseness and separability. The diffuseness is so we have no atoms, and the separability is so the spectral theorem works nicely. Surely the complex numbers (as a von Neumann algebra) is not isomorphic to $L^\infty[0,1]$! | |
| Oct 29, 2009 at 15:57 | history | answered | Dmitri Pavlov | CC BY-SA 2.5 |