Question : What are von Neumann algebras doing in this business?
Answer : In an article called Notes on Some Entanglement Properties of Quantum Field Theory Rev. Mod. Phys. 90(2018) 45003 [arXiv:1803.04993[hep-th]] Edward Witten answered a host of such questions.
The main idea is that when we make the move from quantum mechanics to quantum field theory entanglement is not merely a property of quantum states. Now it is elevated to a property of the operators acting on these states and hence to the algebras of the observables.
Witten's article is about explaining these issues.
Thankfully von Neumann already gave us the answer to some of the mathematical questions.
Witten collects many other essential mathematical details.
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