Endomorphism Semigroups and Lightlike Translations

TL;DR

This paper analyzes one-parameter semigroups of endomorphisms in von Neumann algebras related to lightlike translations, providing operator-theoretic criteria and a converse to previous results, with applications to local algebra properties.

Contribution

It abstracts and extends Borchers-Wiesbrock results by establishing a von Neumann algebraic converse and criteria for generating semigroups from spatial derivations.

Findings

Criteria for spatial derivations to generate semigroups

A von Neumann algebraic converse to Borchers-Wiesbrock results

Applications to isotony and covariance in local algebras

Abstract

Borchers and Wiesbrock have demonstrated certain results concerning the one-parameter semigroups of endomorphisms of von Neumann algebras that appear as lightlike translations in the theory of algebras of local observables. These results are abstracted and analyzed as essentially operator-theoretic. Criteria are then demonstrated for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms. All this is combined to establish a von Neumann-algebraic converse to the Borchers-Wiesbrock results. This analysis is then applied to questions of isotony and covariance for local algebras.