Endomorphisms on half-sided modular inclusions

Abstract

In algebraic quantum field theory we consider nets of von Neumann algebras indexed over regions of the space time. Wiesbrock [“Conformal quantum field theory and half-sided modular inclusions of von Neumann algebras,” Commun. Math. Phys. 158, 537–543 (1993)] has shown that strongly additive nets of von Neumann algebras on the circle are in correspondence with standard half-sided modular inclusions. We show that a finite index endomorphism on a half-sided modular inclusion extends to a finite index endomorphism on the corresponding net of von Neumann algebras on the circle. Moreover, we present another approach to encoding endomorphisms on nets of von Neumann algebras on the circle into half-sided modular inclusions. There is a natural way to associate a weight to a Möbius covariant endomorphism. The properties of this weight have been studied by Bertozzini et al. [“Covariant sectors with infinite dimension and positivity of the energy,” Commun. Math. Phys. 193, 471–492 (1998)]. In this paper we show the converse, namely, how to associate a Möbius covariant endomorphism to a given weight under certain assumptions, thus obtaining a correspondence between a class of weights on a half-sided modular inclusion and a subclass of the Möbius covariant endomorphisms on the associated net of von Neumann algebras. This allows us to treat Möbius covariant endomorphisms in terms of weights on half-sided modular inclusions. As our aim is to provide a framework for treating endomorphisms on nets of von Neumann algebras in terms of the apparently simpler objects of weights on half-sided modular inclusions, we lastly give some basic results for manipulations with such weights.