Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Is there any intuitive explanation of the Double Commutant Theorem for Von Neumann Algebras? By intuitive I mean in terms of Quantum Mechanics. For example, duality of states and observables in the case of the Gelfand-Naimark Theorem. http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem

share|improve this question
    
May be worth to add link to theorem or say what is it about .... then you will get my + 1 :) –  Alexander Chervov Jan 6 '13 at 16:33
    
@a.chernov done! –  Koushik Jan 6 '13 at 16:56
    
Thank you! +1:) –  Alexander Chervov Jan 6 '13 at 17:40
    
It seems that what you want (as supplied by Nik Weaver) is an intuitive description rather than an intuitive explanation. I zmean, what is an intuiive explanation of something being WOT-closed? –  Yemon Choi Jan 6 '13 at 18:27
add comment

1 Answer

up vote 9 down vote accepted

Okay, here's an explanation in terms of quantum mechanics. Let ${\cal A}$ be a family of observables, modeled as self-adjoint operators on some Hilbert space, and let ${\cal U}$ be the group of all unitary transformations that leave every observable in ${\cal A}$ invariant. You can consider ${\cal U}$ to be a kind of symmetry group. Mathematically it is the set of unitaries in the first commutant ${\cal A}'$ of ${\cal A}$, and the set of all observables left invariant by ${\cal U}$ is the double commutant of ${\cal A}$. So the double commutant theorem says that the set of all observables left invariant by every transformation that leaves every observable in ${\cal A}$ invariant, is the self-adjoint part of the von Neumann algebra generated by ${\cal A}$.

share|improve this answer
1  
This is a nice description, but I don't really see how it's an explanation... –  Yemon Choi Jan 6 '13 at 18:28
2  
How about "an explanation of the intuitive meaning"? –  Nik Weaver Jan 6 '13 at 20:20
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.