On the net of von Neumann algebras associated with a wedge and wedge-causal manifolds

Borchers, Hans-Jürgen

A wedge in a flat or curved ordered space can be defined with help of two light-rays passing through a point and the double-cones spanned between these light-rays. Only special manifolds have the property that the space-like complement of a wedge is again a wedge as in the flat situation. Such manifolds will be called wedge-causal. Starting from a wedge and its assoviated von Neumann algebra then its properties will be investigated in the flat and the wedge-causal situation. It will be shown, that in the flat situation, all local algebras are of von Neumann type III, and that they are all of the same Connes-von Neumann-type III_1. Here the types can be determined, because the modular group of the wedge-algebra acts local. For the situation of the Minkowski space we will show how to construct from the wedge-algebra the algebra of the double cones. In addition we will show how to construct from a double-cone algebra the algebra of larger double cones and of the wedge. For this we will use either the translations or the modular group of the wedge-algebra and the double cone theorem. All these investigations are dimension independent. Moreover, we will develop new methods determining the von Neumann and the Connes types for the wedge- and double-cone algebras.

Reference:

This paper in PostScript


jh, Last modified: Fri Dec 11 09:53:19 CET 2009