Implementability of Liouville evolution, Koopman and Banach-Lamperti theorems in Classical and Quantum dynamics

TL;DR

This paper extends the concept of implementability of evolution semigroups from classical to quantum systems, using Jordan morphisms and isometries on non-commutative L^p spaces, focusing on a quantum analog of the Banach-Lamperti theorem.

Contribution

It introduces a non-commutative framework for the implementability of quantum evolution semigroups, generalizing classical results to quantum dynamics.

Findings

Extension of implementability concept to quantum systems.

Formulation using Jordan morphisms and isometries on non-commutative L^p spaces.

Development of a quantum analog of the Banach-Lamperti theorem.

Abstract

We extend the concept of implementability of semigroups of evolution operators associated with dynamical systems to quantum case. We show that such an extension can be properly formulated in terms of Jordan morphisms and isometries on non-commutative $L^p$ spaces. We focus our attention on a non-commutative analog of the Banach-Lamperti theorem.