# 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:

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