The Haag-Kastler approach to quantum field theory (QFT) is one of the oldest approaches to rigorously define what a QFT is, it deals with nets of operator algebras: You start with a spacetime and assign von Neumann algebras (or $C^*$-algebras, but my question is about the von Neumann algebra situation only) to certain subsets of the spacetime subject to certain axioms. I am interested in results about the Murray-von Neumann classification of these algebras, i.e. which kind of factors can occur in the central decomposition (the decomposition of the algebra as a direct integral of factors).

To make this more precise, here is an example: One can define vacuum representations on Minkowski spacetime, for details please see

Haag-Kastler vacuum representation on the nLab.

A net of a vacuum representation is said to satisfy duality, or be a dual net, if one has $\mathcal{M}(\mathcal{O}') = \mathcal{M}'(\mathcal{O})$, put in words: the algebra of the causal complement of a bounded open set $\mathcal{O}$ is the commutant of the algebra associated with $\mathcal{O}$. Then it is a theorem that algebras associated to diamonds can only have factors of type $III_1$ in their central decomposition.

- Is the assumption of duality necessary or is causality enough? Is Haag duality enough? (Haag duality means that the duality condition does not have to hold for all algebras associated to bounded open regions, but for diamonds only).
- What are the necessary assumptions to deduce that algebras associated to diamonds are a factor of type $III_1$, i.e. have trivial center? What are the necessary assumptions to get that these algebras are hyperfinite?
- Are there similar results about the factor decomposition of algebras associated to more general subsets than diamonds of the Minkowski space, like open bounded subsets?
- Are there similar results about more general spacetimes, like globally hyberbolic ones?