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Observables are self-adjoint elements of a $C^*$algebra. As such, this structure seems sufficient to describe physics.

A theorem by Gelfand and Naimark says that a $C^*$algebra can always be faithfully represented as bounded operators on a Hilbert space, $B(H)$. Then one can introduces different topologies, and one can see von Neumann algebra as a $C^*$subalgebra of $B(H)$ that is moreover complete in one of those topologies.

In another question of Physics stack exchange https://physics.stackexchange.com/q/2043/2451 someone also talks about the Borel functional calculus, and one also compare von Neumann algebra to "non commutative" measure theory vs "non commutative" for $C^*$algebras.

My question is, is the introduction of von Neumann algebra only a technical thing or has it physical consequences? Or maybe more precisely, why is it important to consider several topologies on our algebra of operators?

ps: I posted the exact same question in physics, but it may be more relevant in math.

Community
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Noix07
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2 Answers2

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There are many $C^*$-algebras, but only few vNa in comparison. $C^*$-algebras are more difficult to understand.

Example: Commutative $C^*$-algebras correspond to locally compact Hausdorff spaces, commutative vNA's to measure spaces, which are all isomorphic to the Lebesgue measure on $[0,1]$ plus counting measure on a discrete set.

Marc Palm
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One consequence of being complete under various topologies, is that one can approximate an element of the von Neumann algebra by converging sequences from a dense subsets. But I guess that the more topologies such that our vN algebra is complete, the more possible dense subsets we have.

Noix07
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