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 GelfandNaimark Theorem. http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem

$\begingroup$ May be worth to add link to theorem or say what is it about .... then you will get my + 1 :) $\endgroup$– Alexander ChervovJan 6, 2013 at 16:33

$\begingroup$ @a.chernov done! $\endgroup$– KoushikJan 6, 2013 at 16:56

$\begingroup$ 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 WOTclosed? $\endgroup$– Yemon ChoiJan 6, 2013 at 18:27

$\begingroup$ @YemonChoi that there is a fair number elements in the topology for a particular purpose? $\endgroup$– user123124May 10, 2017 at 11:38
2 Answers
Okay, here's an explanation in terms of quantum mechanics. Let ${\cal A}$ be a family of observables, modeled as selfadjoint 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 selfadjoint part of the von Neumann algebra generated by ${\cal A}$.

1$\begingroup$ This is a nice description, but I don't really see how it's an explanation... $\endgroup$ Jan 6, 2013 at 18:28

3$\begingroup$ How about "an explanation of the intuitive meaning"? $\endgroup$ Jan 6, 2013 at 20:20
Weak closure is more universal than norm closure. Commutative C* algebra is an algebra of continuous functions on spectrum, which could be any compact space. Von Neumann algebras deal with measure, and measure space is universal  the interval [0,1] with Lebesgue measure (provided measure of a point equals to 0). Check the V.A.Rokhlin theorem: https://en.wikipedia.org/wiki/Standard_probability_space