Maps on noncommutative Orlicz spaces
Louis E. Labuschagne, Wladyslaw A. Majewski
TL;DR
The paper generalizes Pistone–Sempi arguments to noncommutative Orlicz spaces by analyzing how positive maps on semifinite von Neumann algebras lift to actions on noncommutative Orlicz spaces and by characterizing when Jordan $*$-morphisms induce composition operators. It develops a noncommutative Pistone–Sempi framework via the quantum regular variable spaces $L^{quant}_x$ and the weighted Orlicz space $L^{\psi}_x(\tilde{\mathcal{M}})$ with $\psi=\cosh-1$, establishing a quantum analogue of the Pistone–Sempi theorem. A key contribution is the interpolation-based boundedness results showing that unital pure CP maps and a broad class of Jordan morphisms act boundedly on noncommutative Orlicz spaces, coupled with a precise criterion for when a Jordan $*$-morphism induces a bounded composition operator between spaces $L^{\varphi_1}$ and $L^{\varphi_2}$ via $f_J=d(\tau_2\circ J)/d\tau_1$ and the condition $\psi\circ\varphi_2=\varphi_1$. Under $\Delta_2$-type assumptions the criteria become equivalent, providing a robust framework for noncommutative dynamical systems on regular statistical models. These results establish a foundational link between operator-algebraic maps, noncommutative integration theory, and composition-operator dynamics in quantum probability.
Abstract
A generalization of the Pistone-Sempi argument, demonstrating the utility of non-commutative Orlicz spaces, is presented. 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 large class of interesting noncommutative dynamical systems.