Haag-Kastler axioms

Question

In the Haag-Kastler axioms, an algebra of observables $A(O)$ is associated to each open spacetime region $O$ of the Minkowski space. In several treatments, the algebra $A(O)$ is a $C^{*}$ algebra, and in others it is a Von Neumann algebra. Is there a physical intuition or reasoning behind which do we choose depending on situation?

It also depends on the author, whether the chosen spacetime region needs to be only bounded and open, or sometimes aswell casually closed. Is there a physical reasoning as of why to require causally closed sets only?

Finally, in many sources it is stated that this association Is a net. A net is a map from a directed set into a topological space. In order to be a net, the collection of open bounded (if included also causally closed) subsets of Minkowski space should form a directed set. My guess is that it forms a directed set with respect to the subset relation. Reflexivity and transitivity are immediate. How could one verify that for each pair of open bounded casually closed subsets, there exists an upper bound? And the final step to prove that it is a net, is to provide a topological space, where this maps to. Which would be this topological space?

Answer 1

Taking von Neumann algebras rather than $C^*$-algebras has some technical advantages, one of them being that von Neumann algebras contain more projections than $C^*$-algebras. More specifically, if $A$ is a self-adjoint element of a von Neumann algebra, all its spectral projection operators belong to the von Neumann algebra, too. This is of physical relevance because spectral projection operators are needed to compute probabilities of measurement outcomes.

I don't know a physical reason why there should be a difference between considering only causally closed regions or all bounded subsets of Minkowski space. In some situations in which it is more convenient to work with causally closed sets.

Yes, the association from causally closed subsets of Minkowski space to local algebras is a net. The smallest causally closed set $O$ which is an upper bound for a pair $O_1$, $O_2$ of bounded causally closed sets (i.e. $O_1\subset O$ and $O_2\subset O$) is the causal completion of $O_1\cup O_2$. Alternatively, you could consider this association also as functor from the category of subsets of Minkowski space to the category of $C^*$-algebras (or von Neumann algebras).