Endomorphism semigroups and lightlike translations

Borchers and Wiesbrock have studied the one-parameter semigroups of endomorphisms of von Neumann algebras that appear as lightlike translations in the theory of algebras of local observables, showing that they automatically transform under the appropriate modular automorphisms as under velocity transformations. Here, these results are abstracted and analyzed as essentially operator-theoretic. Criteria are then established for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms, and all of this is combined to establish a von Neumann-algebraic converse to the Borchers and Wiesbrock results. This sort of analysis is then applied to questions of isotony and covariance for local algebras, to show that Poincaré covariance together with a domain condition for the translations can imply isotony.This research was partly supported by a fellowship from the Consiglio Nazionale delle Ricerche.