Examples of quantum systems modelled with Type II von Neumann algebras What are the examples of quantum systems that should be modelled with a Type $II_1$ or $II_\infty$ von Neumann algebra?
I am pretty much a novice at von Neumann algebra, so I have hard time finding the relevance of von Neumann algebras that are not a full matrix algebra for some Hilbert space. Some literatures say the hyperfinite type $II_1$ von Neumann algebra is the operator space on the Fock space of Fermionic systems, but it's also hard to understand it.
Is the set of bounded operators on $L^2(R)$ (the space of wave functions of a particle in 1D), or some subset of it with sensible constraints, is an exmaple of type $II_1$ von Neumann algebra? I think it is the case since every projector of the form $\int_A |
{x}\rangle\langle{x}|dx$ can be decomposed into two projectors as the definition of type $II$ algebras requires if $A$ can be decomposed into two sets.
 A: Witten has been working with this type of algebras recently, see his last few papers.
In particular, in arXiv:2112.12828 he describes a $\mathrm{II}$ algebra that appears naturally in black hole physics.
He also mentions that such algebras can appear in some matrix models, but gives no other down-to-earth examples, which suggests that this type of algebras does not appear all that often in QM.
A: A type $II_1$ factor $\mathcal{M}$ is a factor of the algebra $\mathcal{B}(\mathcal{H})$ of bounded linear operator on an infinite dimensional Hilbert space $\mathcal{H}$,that admits a unique tracial, faithful, normal state $\tau$. Equivalently a $II_1-factor$ is a finite factor (that is, the identity is finite as a projection) with no minimal projections.
A type $II_{\infty}$ Factor is a von Neumann algebra of $\mathcal{B}(\mathcal{H})$ which is a tensor product of $\mathcal{B}(\mathcal{H})$ and a type $II_1$ factor.
An example of a type $II_1$ factor is the free group factor $\mathbb{F}_n$ where $L(\mathbb{F}_n)$ is a free group on n generators. $L(\mathbb{F}_{\infty})$ is also a type $II$ factor.
There is an intimate bridge between $L(\mathbb{F}_n)$ and large random matrices (including some deterministic matrices). For example, for example the algebras one can recover $L(\mathbb{F}_n)$ from a von Neumann algebra which admits a faithful tracial normal state $\tau$ and generated by n normal elements $a_1, ..., a_n$ which are free according to $\tau$ (in the sense of Voiculescu's freeness). By Voiculescu's theorem, $n$ independent self-adjoint Gaussian matrices converge in distribution to $n$ semicircular elements which are asymptototically free. In the same spirit, one can also used these random matrices to do compression of the $L(\mathbb{F}_n)$.
