Local nets of unbounded operator algebras

One approach to a rigorous mathematical foundation to relativistic quantum field theory is to make use of local nets of von Neumann algebras, where, for a given region of spacetime, self-adjoint bounded linear operators in a von Neumann algebra are used as observables (we note that the von Neumann algebra of observables generally depends on the region of spacetime under consideration). The observables are unbounded linear operators on a Hilbert space, so that one would have to consider local nets of unbounded operator algebras in which the closure of the operators is affiliated to a von Neumann algebra (in general, the von Neumann algebra depends on the unbounded operator algebra within the local net under consideration).
The study of unbounded operator algebras started in around the 1960s and is not as well developed as the theory of operator algebras, where only bounded linear operators come into play. There are various types of unbounded operator algebras, two of which are the generalized B*-algebras due to G. R. Allan in 1967, and the GW*-algebras (short for generalized W*-algebras) due to A. Inoue in 1978.
The purpose of this talk is to define local nets of GW*-algebras in a suitable manner, and to give some of their basic properties. For this, one must consider that there are various ways in which to extend the notion of a commutant of a set of bounded linear operators, to those of unbounded linear operators, and then finding the suitable extension of the concept to physical applications. Connections to Wightman theory will also be discussed, which forms a significant component of the talk.