Maps on noncommutative Orlicz spaces

Abstract

A generalization of the Pistone–Sempi argument, demonstrating the utility of noncommutative Orlicz spaces, is presented. In particular, regular quantum statistical systems are described. The question of lifting positive maps defined on von Neumann algebra to maps on corresponding noncommutative Orlicz spaces is discussed. In particular, we describe those Jordan ∗-morphisms on semifinite von Neumann algebras which in a canonical way induce quantum composition operators on noncommutative Orlicz spaces. Consequently, it is proved that the framework of noncommutative Orlicz spaces is well suited for an analysis of a large class of interesting noncommutative dynamical systems.