# Intuitive meaning of Double Commutant Theorem

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

• May be worth to add link to theorem or say what is it about .... then you will get my + 1 :) Jan 6, 2013 at 16:33
• @a.chernov done! Jan 6, 2013 at 16:56
• 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? Jan 6, 2013 at 18:27
• @YemonChoi that there is a fair number elements in the topology for a particular purpose? May 10, 2017 at 11:38

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 ${\cal A}$.