# Does a von Neumann algebra correspond to a quantum system?

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 $$\mathcal{H}$$?

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.