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
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$\begingroup$ May be worth to add link to theorem or say what is it about .... then you will get my + 1 :) $\endgroup$– Alexander ChervovCommented Jan 6, 2013 at 16:33
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$\begingroup$ @a.chernov done! $\endgroup$– KoushikCommented Jan 6, 2013 at 16:56
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$\begingroup$ 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 WOT-closed? $\endgroup$– Yemon ChoiCommented Jan 6, 2013 at 18:27
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$\begingroup$ @YemonChoi that there is a fair number elements in the topology for a particular purpose? $\endgroup$– user123124Commented May 10, 2017 at 11:38
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2 Answers
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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 ${\cal A}$.
answered Jan 6, 2013 at 16:26
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1$\begingroup$ This is a nice description, but I don't really see how it's an explanation... $\endgroup$ Commented Jan 6, 2013 at 18:28
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3$\begingroup$ How about "an explanation of the intuitive meaning"? $\endgroup$ Commented Jan 6, 2013 at 20:20
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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
