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We usually associate a quantum system with a Hilbert space $\mathcal{H}$ and consider the set of bounded operators on $\mathcal{B}(\mathcal{H})$. Especially the set of unit-trace and positive (semi-definite) operators, $S(\mathcal{H})$, is identified with the set of quantum states of the given quantum system.

Can we say the same with a von Neumann algbera $V$? In particular, can we say that von Neumann algebras with the faithful trace (Type $I_{\text{fin}}$ and Type $II_1$) correspond to a quantum system? Can we pick up unit trace positive elements of $V$ and identify them with quantum states?

If so, what are the examples of quanutum systems that must be described with a von Neumann algebra, not with $\mathcal{B(H)}$ with some Hilbert space $\mathcal{H}$?

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In the axiomatic algebraic formalism, say of the Ostwalder-Schrader axioms of nets of local algebras, it turns that the local algebra must all be isomorphic to the unique hyperfinite factor of type $III_1$. Hence specific theories depend upon how these algebras embed within each other.

For more details, see the paper The Role of Type III Facyors in QFT by Yngvason on the Arxiv and which was presented at the von Neumann centennial conference at Budapest in 2003.

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